MS Thesis_Mustafa Kemal Ozkan

DYNAMIC RESPONSE OF BEAMS WITH PASSIVE TUNED MASS DAMPERS

A Thesis

Submitted to the Faculty

of

Purdue University

by

Mustafa Kemal Ozkan

In Partial Fulfillment of the

Requirements for the Degree

of

Master of Science in Civil Engineering

May 2010

Purdue University

West Lafayette, Indiana

ii

To the memory of my grandfather, Ali Bicer.

To my parents for their endless love, support and encouragement.

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ACKNOWLEDGMENTS

I would here like to express my thanks to the people who have been very

helpful to me during the time it took me to write this thesis.

I would like to express the deepest appreciation to my advisor and mentor,

Professor Ayhan Irfanoglu for giving me the opportunity to work in a very

interesting area and for his support and guidance throughout my graduate

studies at Purdue University.

I also would like to thank the members of my graduate committee,

Professor Mete A. Sozen and Professor Michael E. Kreger, for their time and

suggestions on this thesis.

I wish also to thank Professor Robert J. Connor and Ryan J. Sherman for

kindly sharing data for the high-mast lighting towers.

I would like to express my sincere thanks to my friends and colleagues,

particularly Fabian Consuegra and Bismarck Luna, for providing a very enjoyable

working environment.

I thank to the faculty and staff of the Structural Engineering department,

especially to Molly Stetler, for their kindness and support.

The last, and the most, I want to thank my family for their love, support

and encouragement.

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To my sister Aysenur Ozkan, I appreciate that you are just my sister and

thank you for being there always. I owe so much thanks to my grandmother,

Sabire Bicer, who has always supported me since the start of my life.

I am greatly indebted to my mother, Nurten Ozkan, and my father, Taki

Ozkan, thank you for providing me with the opportunity to be where I am.

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TABLE OF CONTENTS

Page LIST OF TABLES ............................................................................................... viii LIST OF FIGURES ............................................................................................ xxii LIST OF SYMBOLS ......................................................................................... xxvii ABSTRACT ...................................................................................................... xxix CHAPTER 1. INTRODUCTION ............................................................................ 1

1.1. General ....................................................................................................... 1 1.1.1. Sources of Dynamic Excitation ............................................................. 1 1.1.2. Dynamic Loadings ................................................................................ 2 1.1.3. Consequences of Vibration ................................................................... 2 1.1.4. Vibration Control ................................................................................... 3

1.2. Object and Scope ....................................................................................... 4 1.3. Organization ............................................................................................... 5

CHAPTER 2. BACKGROUND AND PREVIOUS RESEARCH ............................. 6 2.1. Human-Structure Dynamic Interaction and Human Induced Vibration ....... 6 2.2. Vibration Criteria ....................................................................................... 12

2.2.1. ISO International Standard ................................................................. 12 2.2.2. Murray’s Criterion ............................................................................... 14 2.2.3. Other Recommendations and Criteria ................................................ 14 2.2.4. Recommended Criteria for Sensitive Laboratory and Healthcare

Facility Floors ...................................................................................... 18 2.3. Vibration Mitigation Techniques ................................................................ 23

2.3.1. Passive Vibration Mitigation Techniques ............................................ 23 2.3.2. Active Vibration Mitigation Techniques ............................................... 29 2.3.3. Semi-Active Vibration Mitigation Techniques ...................................... 31

2.4. Tuned Mass Dampers (TMDs) Overview .................................................. 32 2.4.1. Introduction ......................................................................................... 32 2.4.2. An Introductory Example of a TMD for an Undamped SDOF System ..................................................................................... 37

CHAPTER 3. FREE AND FORCED VIBRATION OF BEAMS WITH ANY NUMBER OF ATTACHED SPRING MASS SYSTEMS SUBJECTED TO DIFFERENT TYPES OF DYNAMIC LOADS ........................................................................ 40

3.1. Introduction ............................................................................................... 40 3.2. Formulation of the Free Vibration Problem for Uniform Beams Carrying Spring-Mass Systems ................................................................. 42

3.2.1. Equations of Motion and Displacement Functions .............................. 42

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Page 3.2.2. Derivation of Eigenfunctions for the Constrained Beam ..................... 43

3.3. Formulation of the Free Vibration Problem for Non-Uniform Beams Carrying Spring-Mass Systems ................................................................. 51

3.3.1. Equations of Motion and Derivation of Eigenfunctions for the Constrained Beam .............................................................................. 51

3.3.2. Coefficient Matrix [Bv] for the v-th Attaching Point .............................. 54 3.3.3. Coefficient Matrix [BL] for the Left End of the Beam............................ 59 3.3.4. Coefficient Matrix [BR] for the Right End of the Beam ......................... 60

3.4. Formulation of the Free Vibration Problem for Uniform Multi-Span Beams Carrying Spring-Mass Systems ................................... 63

3.4.1. Equations of Motion and Displacement Function ................................ 63 3.4.2. Coefficient Matrices and Determination of Natural Frequencies and Mode Shapes ............................................................................... 65

3.5. Forced Vibration of Euler-Bernoulli Beams ............................................... 74 3.5.1. Introduction ......................................................................................... 74 3.5.2. Formulation of Forced Vibration for Beams ........................................ 74

CHAPTER 4. NUMERICAL RESULTS ............................................................... 81 4.1. Introduction ............................................................................................... 81 4.2. Free Vibration Analysis of Single Span Uniform Beam Carrying One, Two and Three Spring-Mass Systems.............................................. 82 4.3. Free Vibration Analysis of Single Span Non-Uniform Beam Carrying

Spring-Mass Systems ............................................................................... 94 4.4. Free Vibration Analysis of Uniform Multi-Span Beam Carrying Spring-Mass Systems ............................................................................. 102

4.4.1. Free Vibration Analysis of Two Span Beam Carrying One Spring-Mass System ................................................................. 102 4.4.2. Free Vibration Analysis of Two Span Beam Carrying Two Spring-Mass Systems ............................................................... 106 4.4.3. Free Vibration Analysis of Three Span Beam Carrying One Spring-Mass Systems ............................................................... 109 4.4.4. Free Vibration Analysis of Three Span Beam Carrying Two Spring-Mass Systems ............................................................... 112

4.5. Forced Vibration Analysis of Single Span Uniform Beam Carrying One, Two and Three Spring-Mass Systems ........................................... 120

4.5.1. Impact Loading ................................................................................. 120 4.5.2. Harmonic Loading ............................................................................ 134 4.5.3. Moving Load ..................................................................................... 148 4.5.4. Moving Pulsating Force .................................................................... 153

4.6. Forced Vibration Analysis of High-Mast Lighting Tower under Wind Load ............................................................................................... 158

4.7. Forced Vibration Analysis of Multi Span Uniform Beams Carrying One and Two Spring-Mass Systems ....................................................... 162

4.7.1. Forced Vibration Analysis of Two Span Beam Carrying One Spring-Mass System ................................................................. 162

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Page 4.7.2. Forced Vibration Analysis of Two Span Beam Carrying Two Spring-Mass Systems ............................................................... 176 4.7.3. Forced Vibration Analysis of Three Span Beam Carrying One Spring-Mass Systems ............................................................... 189 4.7.4. Forced Vibration Analysis of Three Span Beam Carrying Two Spring-Mass Systems ............................................................... 207

CHAPTER 5. SUMMARY AND CONCLUSIONS ............................................. 215 5.1. Summary ................................................................................................ 215 5.2. Conclusion .............................................................................................. 217 5.3. Future Work ............................................................................................ 220

BIBLIOGRAPHY ............................................................................................... 221 APPENDICES

Appendix A. ................................................................................................... 228 Appendix B. ................................................................................................... 234 Appendix C. ................................................................................................... 238 Appendix D. ................................................................................................... 241

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LIST OF TABLES

Table Page 2.1 Recommended values for αi (Murray et al., 1997) ................................... 8 2.2 Recommended acceleration limits for vibration due to rhythmic activities (Allen, 1990) ............................................................................ 17 2.3 Suggested design parameters for rhythmic activities (Allen et al., 1985) .................................................................................. 18 2.4 Minimum recommended natural assembly floor frequencies, Hz (Allen et al., 1985) .................................................................................. 18 2.5 Application and Interpretation of Generic Vibration Criteria (Pan et al.2008) ...................................................................................... 21 4.1 The lowest seven natural frequencies of the uniform beam carrying one spring-mass system (m1/mb=0.01) ................................................... 83 4.2 The lowest seven natural frequencies of the uniform beam carrying one spring-mass system (m1/mb=0.02) ................................................... 83 4.3 The lowest seven natural frequencies of the uniform beam carrying one spring-mass system (m1/mb=0.05) ................................................... 84 4.4 The lowest seven natural frequencies of the uniform beam carrying one spring-mass system (m1/mb=0.1) ..................................................... 84 4.5 The lowest seven natural frequencies of the uniform beam carrying one spring-mass system (m1/mb=0.2) ..................................................... 84 4.6 The lowest seven natural frequencies of the uniform beam carrying two spring-mass systems (m1/mb=m2/mb=0.01) ...................................... 85 4.7 The lowest seven natural frequencies of the uniform beam carrying two spring-mass systems (m1/mb=m2/mb=0.02) ...................................... 85 4.8 The lowest seven natural frequencies of the uniform beam carrying two spring-mass systems (m1/mb=m2/mb=0.05) ...................................... 85 4.9 The lowest seven natural frequencies of the uniform beam carrying two spring-mass systems (m1/mb=m2/mb=0.1) ........................................ 86 4.10 The lowest seven natural frequencies of the uniform beam carrying two spring-mass systems (m1/mb=m2/mb=0.2) ........................................ 86 4.11 The lowest eight natural frequencies of the uniform beam carrying three spring-mass systems (m1/mb=m2/mb=m3/mb=0.01) ....................... 87 4.12 The lowest eight natural frequencies of the uniform beam carrying three spring-mass systems (m1/mb=m2/mb=m3/mb=0.02) ....................... 87 4.13 The lowest eight natural frequencies of the uniform beam carrying three spring-mass systems (m1/mb=m2/mb=m3/mb=0.05) ....................... 88

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Table Page 4.14 The lowest eight natural frequencies of the uniform beam carrying three spring-mass systems (m1/mb=m2/mb=m3/mb=0.1) ......................... 88 4.15 The lowest eight natural frequencies of the uniform beam carrying three spring-mass systems (m1/mb=m2/mb=m3/mb=0.2) ......................... 89 4.16 The lowest six natural frequencies of the bare uniform beam ................ 89 4.17 The lowest six natural frequencies of the non-uniform beam carrying one spring-mass system (m1/mb=0.01) ................................................... 95 4.18 The lowest six natural frequencies of the non-uniform beam carrying one spring-mass system (m1/mb=0.02) ................................................... 95 4.19 The lowest six natural frequencies of the non-uniform beam carrying one spring-mass system (m1/mb=0.05) ................................................... 95 4.20 The lowest six natural frequencies of the non-uniform beam carrying one spring-mass system (m1/mb=0.1) ..................................................... 96 4.21 The lowest six natural frequencies of the non-uniform beam carrying one spring-mass system (m1/mb=0.2) ..................................................... 96 4.22 The lowest five natural frequencies of the bare non-uniform beam ........ 96 4.23 The lowest seven natural frequencies of the high-mast lighting tower

carrying one spring-mass system at the free end ................................. 100 4.24 Comparison of the lowest four natural frequencies of the bare high-mast lighting tower ........................................................................ 100 4.25 The lowest six natural frequencies of the two-span beam carrying one spring-mass system at second span ............................................. 103 4.26 The lowest six natural frequencies of the two-span beam carrying one spring-mass system at first span ................................................... 104 4.27 The lowest six natural frequencies of the two-span beam carrying two spring-mass systems based on case 1 .......................................... 106 4.28 The lowest six natural frequencies of the two-span beam carrying two spring-mass systems based on case 2 .......................................... 107 4.29 The lowest six natural frequencies of the three-span beam carrying one spring-mass system based on case 1 ........................................... 109 4.30 The lowest six natural frequencies of the three-span beam carrying one spring-mass system based on case 2 ........................................... 110 4.31 The lowest six natural frequencies of the three-span beam carrying two spring-mass systems based on case 1 .......................................... 112 4.32 The lowest six natural frequencies of the three-span beam carrying two spring-mass systems based on case 2 .......................................... 113 4.33 The lowest six natural frequencies of the three-span beam carrying two spring-mass systems based on case 3 .......................................... 114 4.34 The lowest six natural frequencies of the three-span beam carrying two spring-mass systems based on case 4 .......................................... 115 4.35 The lowest six natural frequencies of the three-span beam carrying two spring-mass systems based on case 5 .......................................... 116 4.36 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam without any spring mass system-Case 1 ........ 120

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Table Page 4.37 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam without any spring mass system-Case 2 ........ 120 4.38 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under impact loading - Case 3 (m1/mb=0.01) ........................................ 121 4.39 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under impact loading - Case 3 (m1/mb=0.02) ........................................ 122 4.40 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under impact loading - Case 3 (m1/mb=0.05) ........................................ 122 4.41 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under impact loading - Case 3 (m1/mb=0.1) .......................................... 122 4.42 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under impact loading - Case 3 (m1/mb=0.2) .......................................... 123 4.43 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under impact loading - Case 4 (m1/mb=0.01) ........................................ 124 4.44 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under impact loading - Case 4 (m1/mb=0.02) ........................................ 124 4.45 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under impact loading - Case 4 (m1/mb=0.05) ........................................ 124 4.46 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under impact loading - Case 4 (m1/mb=0.1) ......................................... 125 4.47 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under impact loading - Case 4 (m1/mb=0.2) .......................................... 125 4.48 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under impact loading - Case 5 (m1/mb=m2/mb=0.01) ............................ 126 4.49 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under impact loading - Case 5 (m1/mb=m2/mb=0.02) ............................ 126 4.50 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under impact loading - Case 5 (m1/mb=m2/mb=0.05) ............................ 127 4.51 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under impact loading - Case 5 (m1/mb=m2/mb=0.1) .............................. 127

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Table Page 4.52 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under impact loading - Case 5 (m1/mb=m2/mb=0.2) .............................. 127 4.53 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under impact loading - Case 6 (m1/mb=m2/mb=0.01) ............................ 128 4.54 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under impact loading - Case 6 (m1/mb=m2/mb=0.02) ............................ 128 4.55 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under impact loading - Case 6 (m1/mb=m2/mb=0.05) ............................ 129 4.56 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under impact loading - Case 6 (m1/mb=m2/mb=0.1) .............................. 129 4.57 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under impact loading - Case 6 (m1/mb=m2/mb=0.2) .............................. 129 4.58 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under impact loading - Case 7 (m1/mb=m2/mb=m3/mb=0.01) ................ 130 4.59 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under impact loading - Case 7 (m1/mb=m2/mb=m3/mb=0.02) ................ 130 4.60 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under impact loading - Case 7 (m1/mb=m2/mb=m3/mb=0.05) ................ 131 4.61 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under impact loading - Case 7 (m1/mb=m2/mb=m3/mb=0.1) .................. 131 4.62 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under impact loading - Case 7 (m1/mb=m2/mb=m3/mb=0.2) .................. 131 4.63 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under impact loading - Case 8 (m1/mb=m2/mb=m3/mb=0.01) ................ 132 4.64 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under impact loading - Case 8 (m1/mb=m2/mb=m3/mb=0.02) ................ 132 4.65 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under impact loading - Case 8 (m1/mb=m2/mb=m3/mb=0.05) ................ 133

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Table Page 4.66 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under impact loading - Case 8 (m1/mb=m2/mb=m3/mb=0.1) .................. 133 4.67 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under impact loading - Case 8 (m1/mb=m2/mb=m3/mb=0.2) .................. 133 4.68 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam without any spring mass system-Case 1 ........ 134 4.69 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam without any spring mass system-Case 2 ........ 134 4.70 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under

harmonic loading - Case 3 (m1/mb=0.01) .............................................. 135 4.71 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under



harmonic loading - Case 3 (m1/mb=0.1) ................................................ 136 4.74 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under






harmonic loading - Case 4 (m1/mb=0.2) ................................................ 139 4.80 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under

harmonic loading - Case 5 (m1/mb=m2/mb=0.01) .................................. 140

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Table Page 4.81 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under

harmonic loading - Case 5 (m1/mb=m2/mb=0.02) .................................. 140 4.82 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under


harmonic loading - Case 5 (m1/mb=m2/mb=0.1) .................................... 141 4.84 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under






harmonic loading - Case 6 (m1/mb=m2/mb=0.2) .................................... 143 4.90 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under

harmonic loading - Case 7 (m1/mb=m2/mb=m3/mb=0.01) ...................... 144 4.91 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under



harmonic loading - Case 7 (m1/mb=m2/mb=m3/mb=0.1) ........................ 145 4.94 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under

harmonic loading - Case 7 (m1/mb=m2/mb=m3/mb=0.2) ........................ 145

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Table Page 4.95 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under




harmonic loading - Case 8 (m1/mb=m2/mb=m3/mb=0.1) ........................ 147 4.99 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under

harmonic loading - Case 8 (m1/mb=m2/mb=m3/mb=0.2) ........................ 147 4.100 Maximum and RMS responses at x=0.5L for SS beam carrying one spring-mass system under harmonic loading - Case 9 .................. 148 4.101 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam without any spring mass system-Case 1 ........ 148 4.102 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under moving load -Case 2 (m1/mb=0.01) ............................................ 149 4.103 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under moving load–Case 2 (m1/mb=0.02) ............................................. 150 4.104 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under moving load -Case 2 (m1/mb=0.05) ....................................................... 150 4.105 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under moving load -Case 2 (m1/mb=0.1) ......................................................... 150 4.106 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under moving load -Case 2 (m1/mb=0.2) ......................................................... 151 4.107 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under moving load-Case 3 (m1/mb=m2/mb=0.01) ............................................ 151 4.108 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under moving load-Case 3 (m1/mb=m2/mb=0.02) ............................................ 151 4.109 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under moving load-Case 3 (m1/mb=m2/mb=0.05) ............................................ 152

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Table Page 4.110 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under moving load-Case 3 (m1/mb=m2/mb=0.1) .............................................. 152 4.111 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under moving load-Case 3 (m1/mb=m2/mb=0.2) .............................................. 152 4.112 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam without any spring mass system-Case 1 ........ 153 4.113 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under moving pulsating force - Case 2 (m1/mb=0.01) ..................................... 154 4.114 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under moving pulsating force - Case 2 (m1/mb=0.02) ..................................... 154 4.115 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under moving pulsating force - Case 2 (m1/mb=0.05) ..................................... 154 4.116 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under moving pulsating force - Case 2 (m1/mb=0.1) ....................................... 155 4.117 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under moving pulsating force - Case 2 (m1/mb=0.2) ....................................... 155 4.118 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under moving pulsating force - Case 3 (m1/mb=m2/mb=0.01) ......................... 155 4.119 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under moving pulsating force - Case 3 (m1/mb=m2/mb=0.02) ......................... 156 4.120 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under moving pulsating force - Case 3 (m1/mb=m2/mb=0.05) ......................... 156 4.121 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under moving pulsating force - Case 3 (m1/mb=m2/mb=0.1) ........................... 156 4.122 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under moving pulsating force - Case 3 (m1/mb=m2/mb=0.2) ........................... 157 4.123 Maximum and RMS responses at x=0.5L for SS beam carrying one spring-mass system under moving pulsating force - Case 4 ......... 157 4.124 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying one spring-mass system under impact loading - Case 1 ........................................................................ 163

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Table Page 4.125 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying one spring-mass system under impact loading - Case 1 ........................................................................ 163 4.126 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying one spring-mass system under impact loading - Case 2 ........................................................................ 164 4.127 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying one spring-mass system under impact loading - Case 2 ........................................................................ 165 4.128 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying one spring-mass system under impact loading - Case 3 ........................................................................ 166 4.129 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying one spring-mass system under impact loading - Case 3 ........................................................................ 166 4.130 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying one spring-mass system under harmonic loading - Case 1 .................................................................... 167 4.131 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying one spring-mass system under harmonic loading - Case 1 .................................................................... 168 4.132 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying one spring-mass system under harmonic loading - Case 2 .................................................................... 169 4.133 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying one spring-mass system under harmonic loading - Case 2 .................................................................... 169 4.134 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying one spring-mass system under harmonic loading - Case 3 .................................................................... 170 4.135 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying one spring-mass system under harmonic loading - Case 3 .................................................................... 171 4.136 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying one spring-mass system under harmonic loading - Case 4 .................................................................... 171 4.137 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying one spring-mass system under moving load-Case 1 .............................................................................. 172 4.138 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying one spring-mass system under moving load-Case 1 .............................................................................. 173

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Table Page 4.139 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying one spring-mass system under moving pulsating force-Case 1 ............................................................. 174 4.140 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying one spring-mass system under moving pulsating force-Case 1 ............................................................. 174 4.141 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying one spring-mass system under harmonic loading - Case 2 .................................................................... 175 4.142 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying two spring-mass systems under impact loading - Case 1 ........................................................................ 176 4.143 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying two spring-mass systems under impact loading - Case 1 ........................................................................ 177 4.144 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying two spring-mass systems under impact loading - Case 2 ........................................................................ 177 4.145 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying two spring-mass systems under impact loading - Case 2 ........................................................................ 178 4.146 Maximum and RMS responses at both x=0.25L and x=0.75L for two-span uniform beam carrying two spring-mass systems under impact loading - Case 3 ........................................................................ 179 4.147 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying two spring-mass systems under impact loading - Case 4 ........................................................................ 179 4.148 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying two spring-mass systems under impact loading - Case 4 ........................................................................ 180 4.149 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying two spring-mass systems under harmonic loading - Case 1 .................................................................... 181 4.150 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying two spring-mass systems under harmonic loading - Case 1 .................................................................... 181 4.151 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying two spring-mass systems under harmonic loading - Case 2 .................................................................... 182 4.152 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying two spring-mass systems under harmonic loading - Case 2 .................................................................... 182

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Table Page 4.153 Maximum and RMS responses at both x=0.25L and x=0.75L for two-span uniform beam carrying two spring-mass systems under harmonic loading - Case 3 .................................................................... 183 4.154 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying two spring-mass systems under harmonic loading - Case 4 .................................................................... 184 4.155 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying two spring-mass systems under harmonic loading - Case 4 .................................................................... 184 4.156 Maximum and RMS responses at x=0.25L and x=0.75L for two-span uniform beam carrying two spring-mass systems under moving load-Case 1 .............................................................................. 185 4.157 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying two spring-mass systems under moving load-Case 2 .............................................................................. 186 4.158 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying two spring-mass systems under moving load-Case 2 .............................................................................. 186 4.159 Maximum and RMS responses at x=0.25L and x=0.75L for two-span uniform beam carrying two spring-mass systems under moving pulsating force-Case 1 ............................................................. 187 4.160 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying two spring-mass systems under moving pulsating force-Case 2 ............................................................. 188 4.161 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying two spring-mass systems under moving pulsating force-Case 2 ............................................................. 188 4.162 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying one spring-mass system under impact loading - Case 1 ........................................................................ 189 4.163 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying one spring-mass system under impact loading - Case 1 ........................................................................ 190 4.164 Maximum and RMS responses at x=(5/6)L for three-span uniform beam carrying one spring-mass system under impact loading - Case 1 ........................................................................ 190 4.165 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying one spring-mass system under impact loading - Case 2 ........................................................................ 191 4.166 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying one spring-mass system under impact loading - Case 2 ........................................................................ 191

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Table Page 4.167 Maximum and RMS responses at x=(5/6)L for three-span uniform beam carrying one spring-mass system under impact loading - Case 2 ........................................................................ 192 4.168 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying one spring-mass system under impact loading - Case 3 ........................................................................ 193 4.169 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying one spring-mass system under impact loading - Case 3 ........................................................................ 193 4.170 Maximum and RMS responses at x=(5/6)L for three-span uniform beam carrying one spring-mass system under impact loading - Case 3 ........................................................................ 193 4.171 Maximum and RMS responses at x=(1/6)L and x=(5/6)L for three-span uniform beam carrying one spring-mass system under impact loading - Case 4 ........................................................................ 194 4.172 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying one spring-mass system under impact loading - Case 4 ........................................................................ 194 4.173 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying one spring-mass system under harmonic loading - Case 1 .................................................................... 195 4.174 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying one spring-mass system under harmonic loading - Case 1 .................................................................... 196 4.175 Maximum and RMS responses at x=(5/6)L for three-span uniform beam carrying one spring-mass system under harmonic loading - Case 1 .................................................................... 196 4.176 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying one spring-mass system under harmonic loading - Case 2 .................................................................... 197 4.177 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying one spring-mass system under harmonic loading - Case 2 .................................................................... 197 4.178 Maximum and RMS responses at x=(5/6)L for three-span uniform beam carrying one spring-mass system under harmonic loading - Case 2 .................................................................... 198 4.179 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying one spring-mass system under harmonic loading - Case 3 .................................................................... 199 4.180 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying one spring-mass system under harmonic loading - Case 3 .................................................................... 199

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Table Page 4.181 Maximum and RMS responses at x=(5/6)L for three-span uniform beam carrying one spring-mass system under harmonic loading - Case 3 .................................................................... 199 4.182 Maximum and RMS responses at x=(1/6)L and x=(5/6)L for three-span uniform beam carrying one spring-mass system under harmonic loading - Case 4 .................................................................... 200 4.183 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying one spring-mass system under harmonic loading - Case 4 .................................................................... 200 4.184 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying one spring-mass system under moving load-Case 1 .............................................................................. 201 4.185 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying one spring-mass system under moving load-Case 1 .............................................................................. 202 4.186 Maximum and RMS responses at x=(5/6)L for three-span uniform beam carrying one spring-mass system under moving load-Case 1 .............................................................................. 202 4.187 Maximum and RMS responses at x=(1/6)L and x=(5/6)L for three-span uniform beam carrying one spring-mass system under moving load-Case 2 .............................................................................. 203 4.188 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying one spring-mass system under moving load-Case 2 .............................................................................. 203 4.189 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying one spring-mass system under moving pulsating force -Case 1 ............................................................ 204 4.190 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying one spring-mass system under moving pulsating force -Case 1 ............................................................ 205 4.191 Maximum and RMS responses at x=(5/6)L for three-span uniform beam carrying one spring-mass system under moving pulsating force -Case 1 ............................................................ 205 4.192 Maximum and RMS responses at x=(1/6)L and x=(5/6)L for three-span uniform beam carrying one spring-mass system under moving pulsating force -Case 2 ............................................................ 206 4.193 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying one spring-mass system under moving pulsating force -Case 2 ............................................................ 206 4.194 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying two spring-mass systems under impact loading - Case 1 ........................................................................ 208

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Table Page 4.195 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying two spring-mass systems under impact loading - Case 1 ........................................................................ 208 4.196 Maximum and RMS responses at x=(5/6)L for three-span uniform beam carrying two spring-mass systems under impact loading - Case 1 ........................................................................ 208 4.197 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying two spring-mass systems under harmonic loading - Case 1 .................................................................... 209 4.198 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying two spring-mass systems under harmonic loading - Case 1 .................................................................... 210 4.199 Maximum and RMS responses at x=(5/6)L for three-span uniform beam carrying two spring-mass systems under harmonic loading - Case 1 .................................................................... 210 4.200 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying two spring-mass systems under moving load-Case 1 .............................................................................. 211 4.201 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying two spring-mass systems under moving load-Case 1 .............................................................................. 212 4.202 Maximum and RMS responses at x=(5/6)L for three-span uniform beam carrying two spring-mass systems under moving load-Case 1 .............................................................................. 212 4.203 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying two spring-mass systems under moving pulsating force -Case 1 ............................................................ 213 4.204 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying two spring-mass systems under moving pulsating force -Case 1 ............................................................ 214 4.205 Maximum and RMS responses at x=(5/6)L for three-span uniform beam carrying two spring-mass systems under moving pulsating force -Case 1 ............................................................ 214

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LIST OF FIGURES

Figure Page 1.1 Types of Dynamic Loadings (Murray et al.1997) ..................................... 2 2.1 Directions of coordinate systems for vibrations influencing humans

(Naeim, 1991). ...................................................................................... 10 2.2 Average plot of force versus time for heel impact (Naeim, 1991) .......... 11 2.3 Typical floor response to heel impact (Naeim, 1991) ............................ 11 2.4 Recommended peak acceleration for human comfort for vibrations (Allen and Murray, 1993; ISO 2631/2, 1989). ........................................ 13 2.5 Modified Reiher-Meister perceptibility chart (Naeim, 1991) ................... 15 2.6 CSA annoyance criteria chart for floor vibrations (Naeim, 1991)........... 16 2.7 Generic Vibration Criteria of Gordon (Pan et al.2008) ........................... 20 2.8 Perception criteria (Ungar, 2007) .......................................................... 21 2.9 Viscous damper fitted between chevron braces beneath the deck of the London Millennium bridge (Nyawako and Reynolds, 2007) ........ 26 2.10 Free-layer damping and constrained-layer damping systems (Nyawako and Reynolds, 2007) ............................................................ 27 2.11 Friction damper device components and principle of action (Nyawako and Reynolds, 2007) ............................................................ 27 2.12 Illustration of a tuned sloshing damper (Nyawako and Reynolds, 2007) ............................................................ 29 2.13 Operating principles of an active control system (Nyawako and Reynolds, 2007) ............................................................ 30 2.14 Active mass dampers (Nyawako and Reynolds, 2007) ......................... 30 2.15 Uncontrolled and actively controlled velocity response of an office floor (Nyawako and Reynolds, 2007) ...................................... 31 2.16 Semi-active TMD on a vibrating system ................................................ 32 2.17 Undamped and damped vibration absorbers ........................................ 34 2.18 Tuned Mass Dampers beneath the London Millennium Bridge (Nyawako and Reynolds, 2007) ............................................................ 35 2.19 Example of the effect of damping ratio ξ of the vibration absorber on the frequency response of a primary system (Bachmann et al., 1994) ........................................................................ 37 2.20 SDOF-TMD system (Connor, 2003) ...................................................... 37 3.1 A cantilever beam carrying n spring-mass systems (Wu and Chou, 1999) ............................................................................ 42 3.2 A non-uniform cantilever beam carrying n spring-mass systems .......... 51

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Figure Page 3.3 A uniform multi-span beam carrying S spring-mass systems and T pinned supports (Lin and Tsai, 2007) ................................................. 64 3.4 Two-span uniform beam with one intermediate support and one spring-mass system ....................................................................... 73 3.5 Simply-supported beam subjected to step-function force F0 ................. 76 3.6 Simply-supported beam subjected to harmonic force F0sin(Ωt) ............ 77 3.7 Simply-supported beam subjected to moving load ................................ 78 3.8 Simply-supported beam subjected to moving pulsating load ................ 78 3.9 Simply-supported beam subjected to moving pulsating load ................ 79 4.1 Mode Shapes of Uniform SS, CC, CS and CF Beams Carrying One Spring-Mass System ..................................................................... 90 4.2 Mode Shapes of Uniform SS, CC, CS and CF Beams Carrying Two Spring-Mass Systems.................................................................... 91 4.3 Mode Shapes of Uniform SS, CC, CS and CF Beams Carrying Three Spring-Mass Systems ................................................................. 92 4.4 Mode Shapes of Bare SS, CC, CS and CF Uniform Beams ................. 93 4.5 Mode Shapes of Non-Uniform SS, CC, SC and FC Beams Carrying One Spring-Mass System ....................................................... 97 4.6 Mode Shapes of Bare SS, CC, CS and CF Non-Uniform Beams .............................................................................. 98 4.7 Mode Shapes of High-Mast Lighting Tower Carrying One Spring-Mass System on the Top ................................................. 101 4.8 Mode Shapes of Bare High-Mast Lighting Tower ................................ 101 4.9 Two-span beam carrying one spring-mass system attached to second span ..................................................................... 103 4.10 Two-span beam carrying one spring-mass system attached to first span ........................................................................... 104 4.11 Mode shapes of two-span beam carrying one spring-mass system at second span (mtmd=0.01mb) ................................................ 105 4.12 Mode shapes of two-span beam carrying one spring-mass system at first span (mtmd=0.01mb) ...................................................... 105 4.13 Two-span beam carrying two spring-mass systems (Case 1) ............. 106 4.14 Two-span beam carrying two spring-mass systems (Case 2) ............. 107 4.15 Mode shapes of two-span beam carrying two spring-mass systems tuned based on Case 1(m1tmd= m2tmd=0.01mb) ...................... 108 4.16 Mode shapes of two-span beam carrying two spring-mass systems tuned based on Case 2(m1tmd= m2tmd=0.01mb) ...................... 108 4.17 Three-span beam carrying one spring-mass system attached to first span ........................................................................... 109 4.18 Three-span beam carrying one spring-mass system attached to second span ..................................................................... 110 4.19 Mode shapes of three-span beam carrying one spring-mass system at first span (m1tmd=0.01mb) .................................................... 111

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Figure Page 4.20 Mode shapes of three-span beam carrying one spring-mass system at second span (m1tmd=0.01mb) .............................................. 111 4.21 Three-span beam carrying two spring-mass systems attached to first and second span (Case 1) ....................................................... 112 4.22 Three-span beam carrying two spring-mass systems attached to first and second span (Case 2) ....................................................... 113 4.23 Three-span beam carrying two spring-mass systems attached to first and second span (Case 3) ....................................................... 114 4.24 Three-span beam carrying two spring-mass systems attached to first and third span (Case 4) ............................................................ 115 4.25 Three-span beam carrying two spring-mass systems attached to first and third span (Case 5) ............................................................ 116 4.26 Mode shapes of three-span beam carrying two spring-mass systems at first span and second span tuned based on Case 1

(m1tmd=m2tmd=0.01mb) .......................................................................... 117 4.27 Mode shapes of three-span beam carrying two spring-mass systems at first and second span tuned based on Case 2 (m1tmd=m2tmd=0.01mb) .......................................................................... 117 4.28 Mode shapes of three-span beam carrying two spring-mass systems at first span and second span tuned based on Case 3

(m1tmd=m2tmd=0.01mb) .......................................................................... 118 4.29 Mode shapes of three-span beam carrying two spring-mass systems at first and third span tuned based on Case 4 (m1tmd=m2tmd=0.01mb) .......................................................................... 118 4.30 Mode shapes of three-span beam carrying two spring-mass systems at first span and third span tuned based on Case 5 (m1tmd=m2tmd=0.01mb) .......................................................................... 119 4.31 Simply supported beam carrying one spring-mass system subjected to step-function force at x=0.25L ......................................... 121 4.32 Simply supported beam carrying one spring-mass system subjected to step-function force at x=0.5L ........................................... 123 4.33 Simply supported beam carrying one spring-mass system subjected to harmonic force at x=0.25L .............................................. 135 4.34 Simply supported beam carrying one spring-mass system subjected to harmonic force at x=0.5L ................................................ 137 4.35 Simply supported beam carrying one spring-mass system subjected to moving load .................................................................... 149 4.36 Simply supported beam carrying one spring-mass system subjected to moving pulsating force .................................................... 153 4.37 Force and wind velocity profile of HMLT ............................................. 159 4.38 Dynamic responses of bare HMLT under wind load ............................ 160 4.39 Dynamic responses of HMLT carrying one spring-mass system at the shallow end under wind load (m1/mb=0.01) ................... 161

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Figure Page 4.40 Two-span simply supported beam carrying one spring-mass system subjected to step-function force at x=0.25L ............................ 162 4.41 Two-span simply supported beam carrying one spring-mass system subjected to step-function force at x=0.75L ............................ 164 4.42 Two-span simply supported beam carrying one spring-mass system subjected to step-function force at x=0.25L and x=0.75L ........ 165 4.43 Two-span simply supported beam carrying one spring-mass system subjected to harmonic force at x=0.25L .................................. 167 4.44 Two-span simply supported beam carrying one spring-mass system subjected to harmonic force at x=0.75L .................................. 168 4.45 Two-span simply supported beam carrying one spring-mass system subjected to harmonic force at x=0.25L and x=0.75L ............. 170 4.46 Two-span simply supported beam carrying one spring-mass system subjected to moving load ........................................................ 172 4.47 Two-span simply supported beam carrying one spring-mass system subjected to moving pulsating force ........................................ 173 4.48 Two-span simply supported beam carrying two spring-mass systems subjected to step-function force at x=0.75L .......................... 176 4.49 Two-span simply supported beam carrying two spring-mass systems subjected to step-function force at x=0.25L and x=0.75L ...................................................................... 178 4.50 Two-span simply supported beam carrying two spring-mass systems subjected to harmonic force at x=0.75L ................................ 180 4.51 Two-span simply supported beam carrying two spring-mass systems subjected to harmonic force at x=0.25L and x=0.75L ........... 183 4.52 Two-span simply supported beam carrying two spring-mass systems subjected to moving load ...................................................... 185 4.53 Two-span simply supported beam carrying two spring-mass systems subjected to moving pulsating load ....................................... 187 4.54 Three-span simply supported beam carrying one spring-mass system subjected to step-function force at x= (1/6) L .......................... 189 4.55 Three-span simply supported beam carrying one spring-mass system subjected to step-function force at x= (3/6) L .......................... 192 4.56 Three-span simply supported beam carrying one spring-mass system subjected to harmonic force at x= (1/6) L ................................ 195 4.57 Three-span simply supported beam carrying one spring-mass system subjected to harmonic force at x= (3/6) L ................................ 198 4.58 Three-span simply supported beam carrying one spring-mass system subjected to moving load ........................................................ 201 4.59 Three-span simply supported beam carrying one spring-mass system subjected to moving pulsating force ........................................ 204 4.60 Three-span simply supported beam carrying one spring-mass system subjected to step-function force at x= (1/6) L .......................... 207

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Figure Page 4.61 Three-span simply supported beam carrying one spring-mass system subjected to harmonic force at x= (1/6) L ................................ 209 4.62 Three-span simply supported beam carrying one spring-mass system subjected to moving load ........................................................ 211 4.63 Three-span simply supported beam carrying one spring-mass system subjected to moving pulsating force ........................................ 213 Appendix Figure A.1 SDOF-TMD system ............................................................................. 228

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LIST OF SYMBOLS

ω Natural frequency of primary system

c Viscous damping coefficient of primary system

ξ Damping ratio of primary system

k Stiffness of primary system

ωd Natural frequency of tuned mass damper

cd Viscous damping coefficient of tuned mass damper

ξd Damping ratio of tuned mass damper

kd Stiffness of tuned mass damper

Ω Forcing frequency

µ Mass ratio

ξe Equivalent damping ratio

Displacement of SDOF system

Velocity of SDOF system

Acceleration of SDOF system

Displacement of tuned mass damper

Velocity of tuned mass damper

Acceleration of tuned mass damper

Φ Phase angle

E Modulus of elasticity

I Moment of inertia

Beam mass per unit length

mv Point mass of the vth spring-mass system

kv Spring constant of the vth spring-mass system

zv Instantaneous displacement of vth spring-mass system

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Acceleration of vth spring-mass system

yv Displacement of constrained beam at the vth attaching point

Slope of constrained beam at the vth attaching point

Curvature of constrained beam at the vth attaching point

Yv(x) Amplitude of yv

Zv Amplitude of zv

L Represents the left as superscript

R Represents the right as superscript

y(x,t) Instantaneous displacement of the beam

ωv Natural frequency of spring-mass system

[Bv] Coefficient matrix for the vth attaching point

[BL] Coefficient matrix for the left end of the beam

[BR] Coefficient matrix for the right end of the beam

Cvi Integration constants

n Number of spring-mass systems

[B] Overall coefficient matrix

ρ Mass per unit volume

A Cross sectional area

A0 Cross sectional area at x=0

I0 Moment of inertia at x=0

r Radius

α Taper ratio of the beam

t Thickness

ηi(t) Generalized coordinates

Yi(x) ith normal mode shape of a beam

Qi(t) Generalized force corresponding to ηi(t)

w(x,t) Transverse deflection of a beam

f(x,t) External force per unit length

δij Kronecker delta

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ABSTRACT

Ozkan, Mustafa Kemal. M.S.C.E., Purdue University, May 2010. Dynamic Response of Beams with Passive Tuned Mass Dampers. Major Professor: Ayhan Irfanoglu.

Passive tuned mass damper (TMD) is a stand-alone vibrating system

attached to a primary structure and designed to reduce vibration of the structure

at selected frequency. This study focuses on the application of single or multiple

TMDs on Euler-Bernoulli beams and examines their effectiveness based on free

and forced vibration characteristics of the beams, i.e., the primary structures.

There is a gap in the existing literature in terms of free and forced vibration

analysis of beams carrying any number of concentrated elements. There are

methods developed for the free vibration analysis but they are not practical due

to the complex mathematical expressions. Numerical assembly method (Wu and

Chou, 1999) is used to determine free vibration characteristics of beams in order

to get over the drawbacks of other approaches in the literature and forced

vibration response is obtained based on modal analysis approach and

orthogonality condition.

The free-vibration formulations for uniform, non-uniform single-span and

multi-span continuous beams carrying any number of elastically mounted

masses are derived for various boundary conditions. Numerical solutions for

dynamic responses of these beams subjected to impulsive, harmonic, moving

and moving pulsating loads are presented. A numerical eigenvalue solution is

used to obtain the modal properties of the entire beam at its fundamental and

lower normal modes. The modal analysis approach allows calculating

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displacement, velocity, acceleration and jerk responses at any point on the

beam. The resultant dynamic responses of beams with and without TMDs are

compared with each other in order to observe the performance of TMDs.

Numerical examples are given to confirm the validity and efficiency of the

proposed method. Natural frequencies and mode shapes of several structures

studied in literature are calculated and compared with those in existing literature

to verify the accuracy of the developed algorithm. The illustrative forcing

functions are considered as human-induced dynamic loads for uniform single and

multiple span beams. The results demonstrate that passive TMDs are efficient in

reducing the dynamic responses of beams subjected to harmonic excitations.

However, passive TMDs do not show the same level of performance under non-

harmonic loads. Additionally, wind load analysis is performed for a sample high

mast lighting tower (HMLT) represented as a cantilever non-uniform beam in this

study and the efficiency of attached TMD is analyzed. Experimental wind velocity

data is used to generate the wind induced dynamic load on the HMLT. Results

indicate that properly tuned passive TMDs may be an option to reduce dynamic

response in wind-excited HMLTs.

1

CHAPTER 1. INTRODUCTION

1.1. General

Annoying motions and audible resonant behavior are two common

concerns in structures under service-level dynamic loads. In more serious cases,

excessive vibrations or fatigue which may cause structural failure may exist.

These unwanted vibrations should be eliminated or at least reduced below

corresponding threshold levels to avoid serious structural problems or simply to

meet system performance requirements. It may be easier to modify the design of

a yet to be built structure to mitigate against possible unwanted vibrations

compared to modifying an actual, existing structure. The latter case may be

called for if the existing structure has insufficient design or, is subjected to

change in functionality, or due to changes in environmental conditions.

1.1.1. Sources of Dynamic Excitation

In practical engineering design one of the most important requirements is

to define the sources of dynamic excitation and to analyze their magnitude and

significance by comparing them with the static loads. It is usually much easier to

deal with static loads rather than dynamic loads. Some of the structures,

especially the flexible and lightly damped ones, may exhibit large amplifications

against dynamic loads. The use of a structure, such as a laboratory housing

sensitive equipments or hospitals with sensitive operating rooms such as those

for neurosurgery or microsurgery, is another issue to consider in vibration

analysis. Therefore, the relation between the sources of dynamic excitation, the

structural form and the purpose of the structure should be considered at the

design stage.

2

1.1.2. Dynamic Loadings

Dynamic loads can be categorized as harmonic, periodic, transient and

impulsive as illustrated in Figure 1.1. Rotating machinery can be identified as

harmonic or sinusoidal loads. Rhythmic human activities such as dancing and

aerobics and impactive machinery cause periodic loads. Transient loads consist

of movement of people including walking and running. Single jumps and heel-

drop impacts can be given examples of impulsive loads.

Figure 1.1 Types of Dynamic Loadings (Murray et al.1997)

1.1.3. Consequences of Vibration

Vibration of structures is undesirable for a number of reasons. For

example, overstressing and collapse of structures or simply cracking or other

damages requiring repair can be given as consequences of earthquake-induced

vibrations. To give another example, damage to safety-related equipment is

another problem occurring in nuclear plants during earthquakes. Excessive

structural vibrations in hospitals and other medical facilities can interfere with the

performance of medical procedures, impair the operation of sensitive equipment

and have adverse effects on patient comfort. Adverse human response is also a

3

phenomenon in structures such as health clubs, gymnasiums, stadiums, dance

floors and even office buildings due to the human activities.

Human beings are highly sensitive to vibration. Therefore, adverse human

response should be seriously thought at the design stage although vibrations that

are disturbing for occupants of buildings usually cause small stresses. However,

if the structure is subjected to large number of cycles of loads above certain

thresholds, fatigue fracture problem may occur as another phenomenon. Fatigue

fracture usually occurs in welded steel structures where tiny cracks, which are

initially difficult to see, grow in size under the repetitions of stress until they are

large enough to be seen or cause rupture. For example, high-mast lighting

towers are subjected to wind load which, over time, may cause millions of cycles

of significant stress and could result in structural failure.

1.1.4. Vibration Control

The first step to design the structure which is sensitive to vibrations is to

identify the dynamic loads in terms of frequency and amplitude or measured

variation in time. Analyzing the response of the structure to obtain dynamic

deflections, stresses, frequencies and accelerations come next. Finally, it is

essential to check the calculated or measured performance by using specified

criteria to guarantee that there are no adverse consequences of vibration.

It is important to think ahead during the early stage of conceptual design

and make necessary design adjustments in order to minimize the vibration

susceptibility. Some structures, such as dancing floors, are affected over a

confined frequency range. It may be feasible to increase the structural depth in

dance floors in order to improve stiffness of the structure so as to keep the

frequency of the structure above predominant dancing frequency.

4

Active control over the natural frequency of buildings may be provided by

increasing the stiffness or reducing the mass but this method is usually difficult or

uneconomic to obtain the optimum value. It may be more efficient to design and

use special vibration-absorbing devices, called passive tuned mass dampers

(TMDs) or tuned vibration absorbers (TVAs), as part of the structure to reduce

effects of dynamic loads. Some of the construction techniques, such as welded

steelwork, may be more delicate to vibration because of their lack of inherent

damping capacity. Therefore, it may sometimes be more effective to choose

materials with high damping or to install artificial damping devices.

1.2. Object and Scope

The main motivation of this study is to investigate the effects of tuned

vibration absorbers (TVAs), also called tuned mass dampers (TMDs), in terms of

controlling the dynamic deflections, stresses, frequencies and accelerations of

structural elements or structures which are subjected to different types of

dynamic loadings.

The scope of the study includes (a) analytical and numerical procedures to

solve the free and forced vibration of uniform and non-uniform single and multi-

span beams, which are subjected to different types of dynamic loadings, with

attached spring-mass systems, in particular, passive TMDs (b) some applications

of these analytical procedures in real structural elements.

The free vibration analysis of structural elements, such as beams, is

commonly found in existing literature. A critical aspect of the presented study is

to analyze the forced vibration of these structural elements with attached spring-

mass systems and to check that whether these spring-mass systems, which can

also be called as TVA or TMD, work towards reducing the consequences of

adverse vibration.

5

1.3. Organization

Chapter 2 reviews and describes the existing research on practical

approaches for vibration reduction together with detailed explanation of vibration

excitation sources which are human-induced vibration, machinery induced

vibration and wind-induced vibration.

Chapter 3 discusses the analytical approaches to determine natural

frequencies, mode shapes and responses of beams with attached spring mass

systems under different types of dynamic loads. Applications of passive TVAs to

uniform and non-uniform beams are presented. The effect of single TVA versus

multiple TVAs is discussed.

Chapter 4 includes the numerical results for free and forced vibration of

beams carrying single or multiple spring mass systems. Illustrative numerical

examples are presented for step-function forces (i.e., impact loading), harmonic

forces, moving loads and moving pulsating forces.

Summary and conclusions are presented in Chapter 5.

6

CHAPTER 2. BACKGROUND AND PREVIOUS RESEARCH

2.1. Human-Structure Dynamic Interaction and Human Induced Vibration

Human-induced forces, such as walking, running, jumping, dancing and

other similar activities, cause unwanted vibrations in civil engineering structures,

such as floors, footbridges, grandstands, stairs. These kinds of rhythmical human

activities can generate significant resonant, transient, steady-state or impulsive

responses (Nyawako and Reynolds, 2007). According to Nyawako and Reynolds

(2007), pacing frequency for walking can be considered in the range of 1.5-3.0

Hz and it is above 3 Hz for activities such as running or jogging. However, this

range can decrase to the range of 1.25-1.5 Hz for offices based on Smith’s

(1998) and Murray’s (1998) studies.

Human-structure interaction concept is a considerably important issue for

slender structures which are subjected to human-induced forces (R.Sachse et al.

2003). Today’s new construction techniques provide us to design light, slender,

long-spanned structures, however these opportunities have increased the

susceptibility level of structures to detrimental vibrations (Firth, 2002; Naeim,

1991; Tuan and Saul, 1985; Setareh et al. 2006a). When these lighter and longer

floor systems together with their less damping are considered with the rhythmical

activities of the occupants, there is a significant increase in attention to the

vibration level during the design stage to reduce floor vibration problems because

of the greater possibility of increased vibration annoyance in occupants (Naeim,

1991). Moreover, human-induced vibrations may cause serviceability and safety

problems, in terms of annoying level of vibrations for occupants and fatigue

behavior of structures, respectively (Smith, 1998; Bachmann, 1992). The major

reasons of annoying vibrations in civil engineering structures can be explained

with three main factors which are increased human activities, such as aerobics or

7

audience participation, reduced natural frequency due to longer floor systems

and reduced damping and mass provided by new construction techniques (Allen,

1990). Pedestrian structures, office buildings, footbridges, gymnasiums and sport

halls, shopping malls, airport terminals, dance halls and concert halls can be

given as an example for civil engineering structures which tend to be susceptible

to human-induced vibration (Bachmann, 1992; Kerr and Bishop, 2001; Pavic et

al. 2002a; Hanagan et al. 2003; Ebrahimpour and Sack, 2005).

Another case that designers need to pay attention is to consider the

usability of civil engineering structures for different types of occupation for both

economic and serviceability reasons. For instance, building owners or occupants

might need to convert an office floor into a gymnasium or a dance facility in the

future (Webster and Levy, 1992).

It is also necessary to know how human induced forces can be identified

in order to figure out the response of the structure under human induced

vibration. Human induced force-time histories have been obtained for the last

three decades and these experimental results have been analyzed by Fourier

series. Because of this approximation for many experimental results, most of the

researchers came up with the common idea that human induced forces are

perfectly periodic (Sachse et al., 2003). On the other hand, there are some

opposite assumptions questioning that human induced forces are perfectly

periodic because of the fact that these forces are inherently narrow-band

(Eriksson, 1994). Sometimes, auto-spectral density functions are used to

represent human-induced forces in the frequency domain (Tuan and Saul, 1985;

Mouring and Ellingwood, 1994; Eriksson, 1994). Moreover, according to Murray

et al. (1997), dynamic forces which cause floor vibration problems are generally

repeated forces, such as machinery or human induced forces, and they are

usually sinusoidal or nearly sinusoidal. Thereby, such repeated forces can be

defined as sum of sinusoidal forces. Their forcing frequencies can be considered

as multiples of the fundamental frequency of the force repetition such as step

frequency for human activities. Murray et al. (1997) defined time-dependent

8

harmonic force component matching the fundamental frequency of the floor by

using Fourier series as given in Equation 2.1.

F cos 2 Eq. 2.1

where P (person’s weight) can be taken as 0.7 kN (157 pounds) and

recommended values for are given in Table 2.1.

Table 2.1 Recommended values for α (Murray et al., 1997)

Common Forcing Frequencies (f) and Dynamic Coefficients* (αi)

Harmonic

i

Person Walking Aerobic Class Group Dancing

f, Hz (αi) f, Hz (αi) f, Hz (αi)

1 1.6-2.2 0.5 2.2-2.8 1.5 1.8-2.8 0.5

2 3.2-4.4 0.2 4.4-5.6 0.6 3.6-5.6 0.1

3 4.8-6.6 0.1 6.6-8.4 0.1 - -

4 6.4-8.8 0.05 - - - -

*Dynamic Coefficients = Peak Sinusoidal force/weight of person(s).

Previous researches show that the number of people has an effect on

dynamic loads. Activities of a group of people generate more dynamic loads than

individuals. However, it does not mean that dynamic loads are increasing with

the number of people linearly (Sachse et al., 2003). In addition, synchronization

of people is also an important issue that influences dynamic loads and it can be

classified as deliberate and unintentional. Coordinated human activities such as

in aerobic classes can be given as an example for deliberate synchronization and

these types of human activities has been studied for a long time (Sachse et al.,

2003). However, unintentional synchronization is a recent subject that

researchers have considered to be important. This type of synchronization can

cause serviceability problems due to considerably strong vibration level and

safety problems due to panic (Dallard et al. 2000). There have been many cases

9

observed and reported in civil engineering structures due to unintentional

synchronization (Bachmann, 1992; Fujino et al., 1993; Dallard et al., 2000,

Curtis, 2001, New Civil Engineer, 2001).

Excessive structural vibration is also very important problem for healthcare

facility floors. It might negatively affect the performance of medical procedures,

make highly vibration sensitive equipments unusable and cause negative effects

on patient comfort. These problems pose newer vibration criteria that force

structural engineers to accomplish more strict requirements to provide

serviceability for highly vibration sensitive high-tech equipments than before. Pan

et al. (2008) studied a long-spanned biotechnology laboratory floor which is

supported by reinforced concrete beams and evaluated the performance of the

laboratory instruments while the floor is subjected to human induced vibration,

specifically walking. Footfall forces are applied to finite element model of beams

and floors, and then time history analysis results are compared with the

appropriate vibration criteria. According to results, the floor performed well

enough to satisfy the required vibration criteria. Yazdanniyaz et al. (2004) studied

footfall induced vibrations and discussed different vibration criteria and prediction

methods including AISC method, BBN method and analytical modal analysis.

This study also compares these prediction methods with measured floor vibration

levels for both composite and concrete laboratory floor. According to

Yazdanniyaz et al. (2004), designers should avoid predicting a vibration level

greater than it is or applying more strict vibration criteria than it is needed by

considering possible increase for building cost.

Vibration perceptibility is another subject of human-structure dynamic

interaction and needs to be considered to understand what kind of factors

influence the level of perception and sensitivity of people. This issue provides

some improvement during design stage by considering sensitivity of people and

preventing some possible increase in structural vibration. According to Naeim

(1991), vibration perceptibility mostly depends on position of human body,

characteristics of excitation sources and floor systems, exposure time, level of

10

expectancy and type of activity that people perform. As it is shown in Figure 2.1

human-body coordinate system has three different axes which are x, y and z

showing back to chest direction, right side to left side direction and foot to head

direction, respectively. According to International Standard ISO-2631/1-1985(E)

and International Standard ISO-2631/2-1989(E), humans’ sensitivity to

acceleration is experienced significantly when the frequency range is between 4

to 8 Hz for vibration along the z-axis and 0 to 2 Hz for vibration along the x or y

axes. Human activities being influenced by the vibration along the x or y axes,

such as sleeping, make these two axes as important as the z axis which is more

important than others for the design of office buildings.

Figure 2.1 Directions of coordinate systems for vibrations influencing humans

(Naeim, 1991).

11

Standard heel drop impact test is a practical way to model impulsive

forces caused by walking and to approximate to the real. In this test, a person

raises his heel about 2.5 inches and drops his weight through his heels to the

floor. Before initiating the impulse, the person, who weighs 170 pounds, supports

his weight on his toes. The result of the heel drop impact and a typical floor

response to heel impact are in Figure 2.2 and Figure 2.3 (Naeim, 1991).

Figure 2.2 Average plot of force versus time for heel impact (Naeim, 1991)

Figure 2.3 Typical floor response to heel impact (Naeim, 1991)

12

2.2. Vibration Criteria

2.2.1. ISO International Standard

Naeim (1991) indicates the classification of human response to vibrations

by referring International Standard Organization (ISO-2631/1). ISO-2631/1

categorizes human response with respect to the limitations which increase step

by step by considering the perception levels of humans. These categories can be

defined as the limit for reduced comfort, the limit that affects the working

conditions negatively and the limit that cause health or safety problems. These

categories were conceived depending on the studies related with transportation

industries which provide higher level of tolerance than the acceptable level for

buildings. The magnitudes of these limits are slightly exceeding the minimum

levels of human tolerance and they are determined with respect to minimum

adverse comment level of occupants.

Murray et al. (1997) also states that the perception level of people and

how they react to vibration considerably depend on what they are doing.

According to Murray et al. (1997), the vibration level that disturbs people in

offices and residences is 0.5 percent of the acceleration of gravity, g. On the

other hand, this limit might be approximately 10 times higher than that (5 percent

of g) when people are taking part in an activity. In addition, duration of vibration

and distance from the vibration source are other factors that affect the sensitivity

of the occupants. If people are very close to vibration excitation source, the limit

of annoyance level will be about 1.5 percent of g. The frequency range for these

limits indicated in the above is between 4Hz and 8 Hz. Higher vibration

acceleration levels are accepted for the outside of this frequency range. ISO-

2631/2 provides the recommended peak acceleration limits for different types of

structures and occupancies in terms of “root mean square” (RMS) acceleration

as it is shown in Figure 2.4. The limitations can vary depending on duration and

frequency of vibration by the range between 0.8 and 1.5 times the suggested

values.

13

According to ISO, vibrations can be classified as transient and continuous

(steady-state) in terms of human response depending on the type of excitation

and its duration. If a structural system is subjected to a continuous harmonic

force, the resulting motion has constant frequency and constant maximum

amplitude. This type of vibration is defined as continuous vibration. On the other

hand, transient vibration is defined as an instantaneous increase to a peak

followed by a damped decay in a short time. For example, floors subjected to

operating machines have continuous vibration because machines usually run

continuously for a long period of time. On the contrary, the floor vibration for the

residential and office buildings can be categorized in transient vibration because

of the intermittent movement of a small number of occupants. There are also

some cases that walking-induced excitation is rhythmic with an approximately

constant frequency and this type of vibration can be defined as mostly steady-

state.

Figure 2.4 Recommended peak acceleration for human comfort for vibrations

(Allen and Murray, 1993; ISO 2631/2, 1989).

14

2.2.2. Murray’s Criterion

According to Naeim (1991), the primary aim of Murray’s categorization of

human response is to be more design oriented. Therefore, this factor makes his

categorization more useful than others. There are four categories of human

response defined with respect to perception and discomfort levels of occupants.

For the first category, vibration is not perceived by the occupants and for the

second one, vibration is perceived but does not annoy. These two categories are

acceptable for design. For the third category, vibration just disturbs but for the

fourth one, it also makes occupants ill.

Murray’s method evaluates the potential floor vibration problems by

providing a step by step procedure. Field measurements and human response

studies for approximately 100 floor systems conceived the basis of this method.

Murray’s criteria is a very common and recommended for especially residential

and office buildings (Naeim, 1991).

2.2.3. Other Recommendations and Criteria

Ellingwood et al. (1986) recommended a vibration criterion for commercial

floor systems such as shopping malls. According to this criterion, if the maximum

deflection for anywhere on the floor, which is subjected to 450 lb force, does not

exceed 0.2 inches, the criterion is satisfied. In addition, Canadian Standards

Association and Murray also recommend that the natural frequency of

commercial floor systems should be greater than 8 Hz in order to reduce the

possibility of resonance due to walking (Naeim, 1991).

Another criterion called Wiss-Parmelee (1974) rating factor was based on

a laboratory study which was conceived to investigate human perception to

transient floor vibrations. The perception levels of 40 people were observed by

using platform motions corresponding to walking induced floor vibration. This

criterion was adopted by United States Department of Housing and Urban

Development to verify whether the floor system is in acceptable limits or not.

15

Reiher and Meister (1949) also studied human perceptibility to steady

state vibration in early 1930s. The forcing frequency range and the displacement

amplitude range of this study were 3 to 100 Hz and 0.0004 inches to 0.4 inches,

respectively. In early 1960s, Lenzen (1966) stated that if Reiher and Meister’s

amplitude scale was extended with a factor of 10, the resulting scale would be

applicable to correlate human perceptibility with natural frequency and

displacement amplitude for lightly damped floor systems. The new scale, called

Modified Reiher-Meister scale, shown in Figure 2.5 is usually used with an

additional method, like Murray’s acceptability criterion, in the design environment

to satisfy critical situations. The scale is being criticized negatively due to lack of

explicit consideration of damping which is considered to be quite important

(Naeim, 1991).

Figure 2.5 Modified Reiher-Meister perceptibility chart (Naeim, 1991)

Another annoyance criterion that is related with Allen and Rainer’s studies

about floor vibrations was adopted by Canadian Standards Association (CSA) for

residential, office and school rooms. This criterion has peak acceleration limit for

16

each floor system in terms of its natural frequency and damping ratio. As it is

shown in figure 2.6, there are three limit curves and their corresponding damping

ratios for walking vibration and a base curve for continuous vibration. Any floor

system below the corresponding limit curve is considered to be satisfactory

(Naeim, 1991).

Tolaymat (1988) also conceived his acceptability criteria by using the

results of 96 composite floor systems studied by Murray. Instead of studying just

a single heel drop impact, Tolaymat used heel drop impacts following one after

another to simulate walking induced vibration more sensitive. Therefore, his new

criterion is providing better results in terms of correlation between test results and

reported human perceptibility levels (Naeim, 1991).

Figure 2.6 CSA annoyance criteria chart for floor vibrations (Naeim, 1991)

17

According to Naeim (1991), the reason of human annoyance on the floors

subjected to rhythmic human activities is usually resonance or near resonance

situation which generates significant dynamic amplification. There should be

enough difference, for the most applicable design, between natural frequency of

the floor and the dominant frequencies caused by human activities to be sure

that there will not be any resonance. Because of that reason, to design a

multipurpose floor system, being used for both rhythmic activities such as

aerobics class and office space, is a difficult task to consider. Allen (1990)

presented a design guide which is also adopted as a serviceability criterion by

National Building Code of Canada. In addition, this design guideline is

considered to be the most comprehensive one to design multipurpose facilities.

Allen’s recommendations include the tables of the maximum acceptable

acceleration limits for floor vibration (Table 2.2), dominant range of forcing

frequency (Table 2.3) and minimum recommended natural assembly floor

frequencies (Table 2.4).

Table 2.2 Recommended acceleration limits for vibration due to rhythmic activities (Allen, 1990)

Occupancies affected by the vibration Acceleration Limit, percent gravity

Office and residential 0.4 to 0.7

Dining, Dancing, Weight-lifting 1.5 to 2.5

Aerobics, rhythmic activities only 4 to 7

Mixed use occupancies housing aerobics 2

18

Table 2.3 Suggested design parameters for rhythmic activities (Allen et al., 1985)

Activity Forcing

frequency ff, Hz

Weight of participants*

wp, psf

Dynamic load factor**

α

Dynamic load α wp, psf

Dancing 1.5-3.0 12.5 (27 ft2/couple) 0.5 6.25

Lively concert or sport event

1.5-3.0 31.3 (5 ft2/person) 0.25 7.83

Aerobics 1st Harmonic 2nd Harmonic 3rd Harmonic

2-2.75 4.5.80 6-8.25

4.2 (42 ft2/person)*** 4.2 (42 ft2/person)*** 4.2 (42 ft2/person)***

1.5 0.6 0.1

6.30 2.52 0.42

*Density of participants is for commonly encountered conditions. For special events the density of participants can be greater. **Values of α are based on commonly encountered events involving a minimum of about 20 participants. Values of α should be increased for well-coordinated events (e.g. jump dances) or for fewer than 20 participants. ***Suggested revision to the 1985 supplement of CSA code.

Table 2.4 Minimum recommended natural assembly floor frequencies, Hz (Allen et al., 1985)

Type of floor construction Dance floors*,Gymnasia**

Stadia, arenas**

Composite (steel-concrete) 9 6

Solid Concrete 7 5

Wood 12 8

*Limiting peak acceleration 0.02 g

**Limiting peak acceleration 0.05 g

2.2.4. Recommended Criteria for Sensitive Laboratory and Healthcare

Facility Floors

Because of the highly vibration sensitive high-tech equipments operated in

healthcare facilities or biotechnology laboratories, vibration criteria for these

19

critical areas are much more restrictive than before. This phenomenon was

carried out after the first vibration problem experienced in advanced technology

facilities by Intel in Livermore and Aloha, in late 1970s. Therefore, more

restrictive criteria were adopted by industry with the rapid development of

science and technology and these strict requirements posed new challenges on

structural engineers (Pan et al.2008). Total system approach in a single

analytical model, including both the support system and the equipment, is an

appropriate way to get more accurate response if a particular piece of equipment

was accommodated in a specific space (Medearis, 1995). The vibration criteria

provided by the manufacturer are then compared with the response results of the

equipment. However, sometimes the equipment hasn’t been selected yet or the

supporting structure might be required for more flexible usage. Therefore,

structural engineers want to use more comprehensive criteria which cover the

requirements of all equipments in just one particular category (Pan et al.2008).

Spectrum based generic criterion is one of the criteria that can be

classified into two groups in terms of expression. One of them is discrete

frequencies and the other is frequency bands. Design Guide 11 (AISC 1997) can

be given as an example of discrete frequencies. One-third octave bandwidth

spectra is one of the frequency bands that is commonly adopted by industries.

The one third octave band criteria were first presented in 1983 and then Gordon

and Ungar developed and republished them (Ungar et al. 1990; Gordon 1991).

Institute of Environmental Sciences (IES) also accepted these criteria and

published them in 1993 and 1998 (IES 1998).

As it is shown in Figure 2.7, there is a set of root mean square (RMS)

velocity spectra defined as vibration criteria curves VC-A – VC-E and table 2.5

shows the explanation of each curve of generic vibration criteria (IES 1998).

These criteria were posed by reviewing many particular equipment criteria which

were provided by equipment manufacturers (Pan et al.2008).

20

Figure 2.7 Generic Vibration Criteria of Gordon (Pan et al.2008)

The criteria for healthcare facilities are based on American National

Standard Institute (ANSI) Standard S3.29-1983 titled as ‘Guide to the Evaluation

of Human Exposure to Vibration in Buildings’ which shows base-response curve

values related with human perception. According to ANSI, the base response

curve values correspond to the approximate threshold of perception for the most

sensitive humans and approximately one half of the mean threshold of

perception. This standard has similarities with International Standard ISO 2631-

2:1989(E), ‘Evaluation of Human Exposure to Whole-Body Vibration-Part 2:

Continuous and Shock-Induced Vibration in Buildings’ (Ungar, 2007). The

frequency variations of two base-response-curve RMS accelerations which are in

units of micro-g are shown in Figure 2.8.

21

Table 2.5 Application and Interpretation of Generic Vibration Criteria (Pan et al.2008)

Criterion curve

RMS amplitude* (µm)

Detail size (µm)

Description of use

Office (ISO)

400 N/A** Perceptible vibration. Appropriate for offices and non-sensitive areas.

VC-A 50 8 Adequate in most instances for optical microscopes to 400x, microbalances, optical balances, proximity and projection aligners, etc.

VC-B 25 3 Appropriate standard for optical microscopes to 1000x, inspection and lithography equipment (including steppers) to 3 µm line widths.

VC-C 12.5 1 A good standard for most lithography and inspection equipment (including electron microscopes) to 1 µm detail size.

VC-D 6 0.3

Suitable in most instances for the most demanding equipment, including electron microscopes (TEMs and SEMs) and e-beam systems, operating to the limits of their capability.

VC-E 3 0.1

A difficult criterion to achieve in most instances. Assumed to be adequate for the most demanding of sensitive systems including long-path, laser-based small target systems and other systems requiring extraordinary dynamic stability.

Figure 2.8 Perception criteria (Ungar, 2007)

22

ANSI S3.29-1983 also recommends that the vibrations of operating room

floors should not be more than between 0.7 and 1.0 times the base-response

curve values. However, ISO which has many similarities with ANSI standard

suggests that vibrations of operating rooms should not exceed the base

response values. One-fourth of the limiting values for ordinary operating rooms

have been used for sensitive operating rooms such as neurosurgery and

microsurgery.

For the floor vibration in rooms where the patients, who are very

susceptible to vibration and should not be disturbed, are located, vibrations

should also be within the limits of criterion as indicated in the above. Greater

vibrations may be acceptable in rooms for less sensitive patients. For the footfall-

induced floor vibrations, the RMS footfall-induced vibrations and the peak footfall-

induced vibrations should not exceed 4000 μin/sec and 5600 μin/sec

respectively. The corresponding limiting values for sensitive operating rooms are

1000 μin/sec and 1400 μin/sec respectively (Ungar, 2007).

Floor vibration above the operating rooms can affect sensitive instruments

which are supported from the operating room ceiling. Vibration sensitivity of the

equipment and the transmission characteristics of the instruments which are

used in order to mount the equipment are the main factors to be considered in

order to satisfy adopted criteria. There are not any specific criteria for vibrations

taking place on areas above operating rooms but it is suggested to meet

requirements of criteria for sensitive operating rooms. If these instruments do not

contact with the ceiling directly, there is no need to use such criteria (Ungar,

2007).

For MRI Systems, there are different criteria for different suppliers. Ungar

(2007), summarized some of the MRI supplier’s criteria and corresponding MRI

systems’ criteria shown in his study.

23

2.3. Vibration Mitigation Techniques

Advancements in civil engineering, especially the development of high

strength light-weight materials together with highly developed computer aided

design, have provided to design more light-weight and longer spanned floor

structures. Therefore, decreasing structural mass and damping cause some

serviceability problems in structures. There are some conventional methods that

can be used for mitigation. For example, changing mass and stiffness might be a

solution in order to overcome excessive floor vibration problems. However, there

might not be enough space for new structural elements in order to increase the

stiffness, and also additional mass might create overstress in structural

members. Moreover, it is not usually possible to use these additional

nonstructural elements because of architectural requirements (Setareh et al.

2006).

Vibration mitigation techniques are used in order to improve the

performance of the structures which are susceptible to excessive vibration.

These improvements cover reduction of annoyance level of occupants,

prevention of safety problems or providing convenient spaces for highly vibration

sensitive equipments. Vibration mitigation techniques can be applied in order to

improve vibration serviceability problems of existing structures or considered

during the design stage of new structures tending to be susceptible to vibrations

(Nyawako and Reynolds, 2007). According to Nyawako and Reynolds, vibration

mitigation techniques can generally be classified into three main categories as

passive, active and semi-active.

2.3.1. Passive Vibration Mitigation Techniques

Passive vibration mitigation techniques decrease energy dissipation demands on

primary structures. Instead of dissipating all energy by itself, most of the energy

is absorbed by using these techniques (Nyawako and Reynolds, 2007).

Additional damping devices absorb most of the input energy. These systems

24

improve the performance of primary structures by providing additional flexibility

and energy absorption capability through reduction in energy levels transmitted

to primary structures (Yang, 2001). The most important reason that these

devices differ from others is that there is no change in their dynamic

characteristics and also there is no need for external energy to operate them

(Housner et al., 1997; Tentor, 2001; Setareh, 2002). This factor is one of the

limitations of these devices because of their lack of control the characteristics of

external loading, for example when the excitation frequency changes. Viscous

dampers, friction dampers, viscoelastic treatments, tuned mass

dampers/vibration absorbers, pendulum tuned mass dampers, tuned liquid

dampers, yield plates, unbonded braces, constrained layer damping, impact

dampers and some other simple methods such as the use of wall partitions,

props, pendulum tuned mass dampers and additional concrete, damping posts

may be given as an example of passive vibration mitigation techniques (Nyawako

and Reynolds, 2007).

Viscous dampers are one of the passive mitigation techniques that are

generally known as fluid dampers or viscous fluid dampers in structural

engineering environment. These devices dissipate energy with dampers mounted

between two vibrating parts. Hence, increased damping improves performance of

the structure by creating extension and compression forces due to relative

motions. There isn’t any significant contribution to strength or stiffness of primary

structures. However, they achieve to improve their vibration performance

(Jarukovski et al., 1986; Gibson and Austin 1993; Uetani et al., 2003; Weber et

al., 2006). Fluid dampers have been mostly used for mitigation of wind and

seismic induced vibration. The first utilization of fluid dampers for seismic

induced vibration was in 1993 and the most comprehensive implementation of

fluid dampers is the one in five buildings of the San Bernardino County Medical

Center with 233 dampers (Taylor and Constantinou, 2005). Taylor and

Constantinou (2005) studied the applications of fluid dampers which are used as

seismic energy dissipation systems for buildings and bridges. The experimental

25

study includes shake table testing of one-story and three-story building models

and also bridge models. Experimental results have shown that there is a

significant improvement in the energy dissipation capability of structures with fluid

dampers and a substantial reduction in story drifts and story shears. There are

also some other implementations to mitigate human-induced vibrations in civil

engineering structures. The use of 37 viscous dampers in order to suppress the

problematic low-frequency modes of the London Millennium footbridge is one of

the good example that improve the dynamic performance of the structure

(Nyawako and Reynolds, 2007). One of the dampers used in London Millennium

footbridge is shown in Figure 2.9.

Another passive vibration mitigation technique is viscoelastic treatments.

Damping is provided by strain energy that is transmitted through viscoelastic

materials. The use of viscoelastic materials can be classified in three different

ways which are free-layer damping treatment (FLD), constrained-layer damping

treatment (CLD) and tuned viscoelastic dampers (Rao,2001; Renninger, 2005;

Joyal,2006). For FLD, energy is dissipated with extension and compression

forces occurring in viscoelastic/damping material through flexural stress of the

primary structure. The degree of damping depends on the thickness and weight

of the material (Rao, 2001; Renninger, 2005). FLD is applied by attaching it to

the surface of the structure with a strong bonding implement. On the other hand,

CLD systems dissipate energy through shear deformation of the material which

arises from shear strains in the damping layer during vibration. FLD and CLD are

both shown in Figure 2.10. Tuned viscoelastic dampers are much like dynamic

absorbers. Utilization of viscoleastic material makes the difference among them

(Nyawako and Reynolds, 2007).

26

Figure 2.9 Viscous damper fitted between chevron braces beneath the deck of

the London Millennium bridge (Nyawako and Reynolds, 2007)

Friction dampers are another common passive vibration control systems

which dissipate energy through friction forces. These forces are generated with

moving parts by sliding over each other. The energy dissipated by a friction

damper reduces the energy demand on the structure and damps the structural

response. The friction damper system includes the friction unit and a structural

system in order to integrate the friction unit with the structure. The structural

system can be either steel braces bolted to corner regions of the open bay space

in the frame or an infill wall with gaps around the edges to prevent stiffness

interaction of the wall with the frame members. Friction dampers are used as

sacrificial or non-sacrificial elements. Their utilization as sacrificial elements is a

very common attitude in civil engineering environment. In earthquake

engineering applications, some of the structural members might be sacrificed in

order to prevent the collapse of entire structure. These structural members

absorb and dissipate the transmitted energy through plastic deformation in

specially detailed regions (Nyawako and Reynolds, 2007). Location of the friction

27

damper and stiffness of the braces which are used in order to install dampers are

the main factors that affect the design parameters of the damper. Figure 2.11

shows an example of friction damper device and the principle of action.

Figure 2.10 Free-layer damping and constrained-layer damping systems

(Nyawako and Reynolds, 2007)

Figure 2.11 Friction damper device components and principle of action


Most applications of friction dampers have focused on absorption and

dissipation of large amount of energies such as wind or earthquake induced

vibrations. Therefore, these structural members are designed as sacrificial

members to yield or fail during dissipation of such energies. This action provides

28

primary structure to limit their response within elastic range. Hence, friction

damper systems might not be appropriate for mitigation of human induced

vibrations. However, they may be utilized to suppress human-induced vibration

when extreme human activities occurs (Nyawako and Reynolds, 2007).

Passive vibration absorber, also known as tuned mass damper (TMD), is

a device which is mounted in structures in order to reduce the dynamic response

of structures and it comprises a mass, a spring and a damper (Hunt, 1979;

Nishimura et al., 1998; CSA Engineering, 2005; Weber et al., 2006). TMDs have

been very successful implementations to mitigate vibration problems in civil

engineering structures. The frequency of the TMD is tuned to a resonant

frequency of a particular mode of structure and energy is dissipated through

inertia force of TMD which is attached to the primary structure. This principle is

first discovered and applied by Frahm (1911) who also conceived the frequency

splitter (Nyawako and Reynolds, 2007). More study and research about TMDs

will be discussed in section 2.4.

Tuned liquid dampers (TLD) can also be classified in passive control

devices. These devices are basically a different type of TMDs. Liquid is used

instead of mass and it is likewise tuned to a critical frequency of the structure

which is required to be controlled (So and Semercigil, 2004). Tuned sloshing

dampers (TSDs) and tuned liquid column dampers (TLCDs) are the different

types of TLDs. They introduce additional damping to primary structures and

improve the dynamic performance of the structure (Kareem et al., 2005). Figure

2.12 shows an illustration of tuned sloshing damper. TLDs are considerably

successful for mitigation of human, wind and earthquake induced vibration in civil

engineering structures. Moreover, the maintenance is easy and inexpensive

together with their rare failure. The major disadvantage is the use of liquid more

than needed because of the fact that not all liquid is utilized in order to suppress

structural motion (Nyawako and Reynolds, 2007).

29

Figure 2.12 Illustration of a tuned sloshing damper


2.3.2. Active Vibration Mitigation Techniques

Passive vibration mitigation techniques are preferred because of their

simplicity, reliability and also their ability to remain functional without any external

source. On the other hand, these systems cannot maintain their functionality

when environmental factors, such as excitation source, change. Therefore, active

control systems, which are more efficient and more capable of adopting such

changes, were posed in order to overcome the limitations of passive vibration

control systems. These systems are capable of responding to changes almost

instantaneously in order to provide continuous vibration reduction (Hanagan,

1994; Housner et al., 1996; Symans and Constantinou, 1999; Preumont, 2002;

Petkovski, 2004).

Active control systems consist of sensors, a controller and actuators. The

sensors are utilized in order to identify the particular characteristics of the

vibration and measure the structural responses, and then the information is sent

to the controller which computes the control forces based on the given

information by using composed algorithm. After that controller processes the

information and sends the counteracting signal to the actuators which generate

the required control forces. This loop goes on in order to suppress unwanted

30

vibration and it is shown in Figure 2.13 (Soong and Spencer, 2002; Preumont,

2002).

Active mass dampers (AMDs) were initially conceived in order to improve

vibration control performance of TMDs by adding an active control system

generating the required control forces. As indicated in the above, instead of

damping and spring devices of TMDs, AMDs comprise actuators as shown in

Figure 2.14 (Morison and Karnopp, 1973; Nishimura et al., 1992; Nishimura et

al., 1998; Lametrie, 2001). Based on previous research studies, AMDs are quite

successful of reducing structural responses under wind-induced excitations and

also providing some improvement for structures subjected to seismic induced

vibrations (Cao and Li, 2004; Kareem et al., 2005).

Figure 2.13 Operating principles of an active control system


Figure 2.14 Active mass dampers (Nyawako and Reynolds, 2007)

31

There are successful examples of AMD implementations for an office floor

and a chemistry laboratory in order to reduce structural response due to human

induced vibration. Almost 70% vibration reduction was provided for the controlled

office floor with respect to uncontrolled case. Moreover, the vibration reduction

level was greater than 75% for the controlled chemistry floor (Hanagan and

Murray, 1998; Hanagan et al., 2003). Figure 2.15 shows the velocity responses

of uncontrolled and actively controlled office floor to point out such

improvements.

Figure 2.15 Uncontrolled and actively controlled velocity response of an office

floor (Nyawako and Reynolds, 2007)

2.3.3. Semi-Active Vibration Mitigation Techniques

Semi-active vibration control systems comprise passive and active control

systems. Passive vibration system provides the reliability and active control

system can be used due to their capability for varying excitation cases or

structural dynamic characteristics. These systems can also be called as

controllable passive dampers because of their lack of ability to generate arbitrary

forces which is the most important factor distinguishing them from active control

systems. Moreover, the device has adjustable damping and stiffness so that it

can be tuned to the excitation frequency (Housner et al., 1997; Preumont, 2002;

Weber, 2002).

Semi-active TMDs are capable of changing their dynamical parameters

through given algorithm. Therefore, semi-active TMDs can likewise maintain their

32

functionality by adapting to variations in structural characteristics and provide

vibration control over a wide range of frequencies. These devices are preferred

because of their simple hardware requirements and low operational costs.

Actively controlled dampers are used instead of passive fixed dampers in order

to mitigate human induced floor vibrations and therefore, they are able to

overcome the variations in human loading (Setareh,2002; Jiang et al., 2006).

Figure 2.16 shows equivalent single degree of freedom system and TMD

with semi active variable damping. Here, f(t) is the representation of the actively

controlled friction damper force. According to Jiang et al. (2006) studies, the

utilization of semi-active TMD improved the performance and provided 98% and

87% reductions in velocity responses for two floors subjected to walking-induced

vibration by simulating heel-drop test. These studies also showed that semi-

active TMDs performed well for the resonant steady state responses of each two

floor subjected to pedestrian induced vibration.

Figure 2.16 Semi-active TMD on a vibrating system

2.4. Tuned Mass Dampers (TMDs) Overview

2.4.1. Introduction

Tuned mass dampers which are mentioned shortly in section 2.3.1 will be

discussed in detail here. A tuned mass damper is a vibratory device which is

33

attached to the larger primary structure and it consists of a mass, a spring and a

damper. The frequency of the damper is tuned to a particular structural frequency

so that TMD generates inertia forces which resist the forces applied to the

primary structure. Energy is dissipated through generated inertia forces (Connor,

2003).

The first application of TMD was applied by Frahm in 1909 in order to

reduce the rolling motion of ships and also ship hull vibration. The vibration

control devices that Frahm used for his applications were small vibrating springs

attached to the highly-vibrated points of the hull of the ship and didn’t have any

inherent damping. These devices were effective only when absorber’s natural

frequency was very close to the excitation frequency. Ormondroyd and Den

Hartog (1928) ,later on, introduced optimal tuning and damping parameters in

addition to Frahm’s absorber and response at resonance, which was very large

in Frahm’s absorber when excitation frequency approximates any of the two

natural frequency of structure-absorber system, could also be reduced

significantly with utilization of damping (Connor,2003; Setareh et al. 2006). The

most important characteristic that distinguishes damped and undamped vibration

absorbers from each other is their capability to absorb energy. Undamped

vibration absorbers doesn’t absorb any energy, on the other hand, damped

vibration absorbers are designed to absorb energy through increased damping

level of the primary structure (Nyawako and Reynolds, 2007). The damped and

undamped vibration absorbers are shown in Figure 2.17.

34

Figure 2.17 Undamped and damped vibration absorbers

TMDs have been used for mitigation of wind, earthquake, human and

machinery induced vibrations. There have been several successful

implementations of TMDs since Frahm applied the first TMD concept to reduce

the ship hull vibration. Different types of applications of TMDs exist in the

literature for floor vibration control. Lenzen (1966) implemented small TMDs

made of steel and supported by springs hanging from the floor beams. The total

mass of all TMDs was about 2% of the floor mass and damping was provided by

dashpots. According to Lenzen’s studies, significant and satisfactory vibration

reduction was achieved for the floors susceptible to excessive vibration. Allen

and Swallow (1975) used TMDs in order to reduce floor vibration by using steel

boxes containing concrete blocks as mass and attached to each point at the

corners of the floor through springs. Matsumoto et al. (1978) and Allen and

Pernica (1984) also used TMDs in order to reduce excessive vibrations of

structures subjected to walking induced vibration and Setareh and Hanson

(1992) designed TMDs made of steel boxes in order to control floor vibrations

due to dancing on the floor of an auditorium.

35

In addition to previous applications of TMDs, 8 TMDs were used in

London Millennium Bridge in order to prevent excessive vertical vibrations due to

pedestrian induced excitations and provide additional damping. One of the

dampers is shown in Figure 2.18.

Figure 2.18 Tuned Mass Dampers beneath the London Millennium Bridge


The floor vibration due to dancing in the ballroom of Park Building in New

York was suppressed by 60% and also a successful implementation which

reduced the floor vibration by 30-60% due to various human activities was noted

for an auditorium in Osaka International Convention Center (Nyawako and

Reynolds, 2007).

TMDs are also considered to suppress wind and earthquake induced

vibrations as well as human induced vibrations. Widespread implementations can

be found for high-rise buildings, airports and long-spanned bridges. TMDs have

been successfully implemented in order to reduce wind induced vibrations for 7-

36

star Hotel Burj-al-Arab Dubai, Washington National Airport Control Tower,

Petronas Towers and Taipei 101 (Nyawako and Reynolds, 2007).

The effectiveness of a single TMD is restricted to control only a particular

mode. TMDs are generally designed to control fundamental frequency of

structures. However, there might be significant vibration in both the fundamental

and higher modes of structures. Therefore, a single TMD tuned to the

fundamental frequency of a structure is not able to suppress the vibration of

higher modes. It has been shown that the use of multiple TMDs which have

same total mass with a single TMD can be more effective and robust design than

a conventional single TMD. Moreover, multiple TMDs might be more convenient

for uncertainties in the parameter of the system (Nyawako and Reynolds, 2007).

Spring stiffness, mass and damping ratio are the fundamental properties

that define TMD devices. Den Hartog (1947) developed a formula of optimum

damper parameters f and ξd which reduce the steady state response of primary

mass subjected to harmonic excitation. These formulas are used in order to

determine initial properties of TMD for a given mass ratio. After damping ratio is

determined, spring stiffness and damping coefficient can be easily found.

However, if excitation frequency remains constant, damping is not necessarily

defined. Figure 2.19 shows the effect of damping ratio of the vibration absorber.

The optimum parameters are given as;

11 Eq. 2.2

Eq. 2.3

where µ is the mass ratio.

37

Figure 2.19 Example of the effect of damping ratio ξ of the vibration absorber on

the frequency response of a primary system (Bachmann et al., 1994)

2.4.2. An Introductory Example of a TMD for an Undamped SDOF System

As it is shown in Figure 2.20, the structure and TMD are represented as

two-mass system and a SDOF system is used in order to characterize the whole

structure. TMD parameters are given with subscript d by following equations

(Connor, 2003).

Figure 2.20 SDOF-TMD system (Connor, 2003)

38

Eq. 2.4

2 Eq. 2.5

Eq. 2.6

2 Eq. 2.7

Eq. 2.8

Governing equations of motions can be written as;

For primary mass,

1 2 Eq. 2.9

For tuned mass,

2 Eq. 2.10

The optimal natural frequency of tuned mass damper can be assumed as

ω in this design procedure which means that TMD is tuned to fundamental

frequency of the primary structure and therefore, stiffness of the damper and the

primary mass can be correlated by following equation;

Eq. 2.11

Forcing function and the corresponding response of TMD and primary

structure can be given as;

Eq. 2.12

Eq. 2.13

Eq. 2.14

where is the displacement amplitude and is the phase shift.

The worst case here is the equal frequencies of forcing Ω and the

structure ω which is the resonant condition. The responses for this case are as

follows;

1

1 2 12

Eq. 2.15

39

12

Eq. 2.16

And the response without damper is;

12

Eq. 2.17

Eq.2.15 can be expressed in terms of equivalent damping ratio in order to

compare the cases with and without damper.

12

Eq. 2.18

21

2 12

Eq. 2.19

As it is shown in equation 2.19, any increase in mass ratio increases the

total damping. Moreover, decreasing the damping ratio of the damper increases

the damping as well. However, decreasing the damping ratio also increase the

relative motion of the damper according to equation 2.16. Therefore, it is

important to consider this motion during the design stage and select TMD

parameters based on available circumstances. The detailed procedures to obtain

the responses of TMD and primary structure are shown in Appendix A.

40

CHAPTER 3. FREE AND FORCED VIBRATION OF BEAMS WITH ANY NUMBER OF ATTACHED SPRING MASS SYSTEMS SUBJECTED TO DIFFERENT TYPES OF DYNAMIC LOADS

3.1. Introduction

This chapter describes the free and the forced transverse vibrations of

uniform and non-uniform beams carrying single or multiple spring-mass systems.

Tuned vibration absorbers (TVAs) are represented as spring-mass systems in

this study. The beam and attached TVAs are considered as a single system

together in order to show how TVAs contribute to the mitigation of vibration in

terms of controlling the responses of the primary structure by comparing the

cases with and without TVAs. The equations of motion of a beam are derived

according to Euler-Bernoulli beam theory. Hence, rotary inertia and shear

deflection are neglected. Also plane sections are assumed to remain plane and

normal to the longitudinal axis. In this chapter, the free vibration analysis of

uniform, non-uniform single-span and multi-span continuous beams with single or

multiple attached spring mass systems is performed first and then the forced

vibration of these beams is discussed with numerical examples.

The free vibration characteristics of a uniform beam can be easily

obtained but the problem gets more difficult when the beam carries concentrated

elements such as elastically mounted point masses which refer TVAs in this

study. Although there are various techniques presented to solve the eigenvalue

problem for beams carrying any number of concentrated masses, these

techniques are not practical to apply due to complex mathematical expressions

and excessive computing time. Most of the existing research consists of no more

than two concentrated masses because of these reasons. However, Wu and

Chou (1999) found the natural frequencies and mode shapes of a uniform single

41

span beam carrying any number of elastically attached point masses through a

numerical assembly method. This study adopts the same method to obtain the

exact solutions for the free vibration characteristics of any type of beams with

attached spring-mass systems mentioned here.

The numerical assembly method is used in order to obtain the eigenvalue

equation 0 for a uniform Euler-Bernoulli beam carrying multiple spring-

mass systems. The eigenvalues and the corresponding eigenvectors are

obtained by MATHEMATICA via symbolic computation. A coefficient matrix is

composed for the left and right side of the points to which spring-mass system is

attached. Moreover, two more coefficient matrices are generated for each end of

the beam and the conventional assembly technique for finite element method is

used in order to obtain overall coefficient matrix, . On the other hand, the

integration constants and the mode displacement of spring-masses compose

vector. Any value of ω, angular frequency, which makes determinant of the

coefficient matrix equal to zero indicates one of the natural frequency of the

beam together with all the attached spring-mass systems. Moreover, when the

obtained natural frequency is introduced into the coefficient matrix, the vector

that satisfies the eigenvalue equation represents the corresponding mode shape

as well (Wu and Chou, 1999). The same procedure is also used for the free

vibration of non-uniform and multi-span uniform beams carrying spring-mass

systems. After that, forced vibration response of beam subjected to different

types of dynamic loads is obtained and the resultant responses are compared for

the cases with and without spring-mass systems.

42

3.2. Formulation of the Free Vibration Problem for Uniform Beams Carrying

Spring-Mass Systems

3.2.1. Equations of Motion and Displacement Functions

Based on Wu and Chou (1999)’s study, figure 3.1 shows the sketch of a

uniform cantilever beam carrying n spring-mass systems. The whole beam with

length L is divided into n+1 segments. v enclosed in circle and v in parentheses

represent the attaching point and the corresponding segment, respectively.

Moreover, the left and the right ends of the beam are shown by the letters L and

R, respectively.

The equation of motion for a bare uniform Euler-Bernoulli beam and the v-

th spring-mass are given by

, ,0

Eq. 3.1

0 Eq. 3.2

Figure 3.1 A cantilever beam carrying n spring-mass systems (Wu and Chou, 1999)

43

The continuity of the deformations at the attaching point requires that

For the deflection , , Eq. 3.3

For the slope , , Eq. 3.4

For the curvature , , Eq. 3.5

The force equilibrium at the attaching point requires that

, , Eq. 3.6

The boundary conditions for the fixed and free ends of the beam are given

by

0, 0 0, 0 Eq. 3.7

, 0 , 0 Eq. 3.8

3.2.2. Derivation of Eigenfunctions for the Constrained Beam

The free vibration of the beam and v-th spring-mass take the form

, Eq. 3.9

Eq. 3.10

The insertion of equation 3.9 into equation 3.1 leads to

0 Eq. 3.11

0 Eq. 3.12

where

Eq. 3.13

and the substitution of equation 3.9 and 3.10 into equation 3.2 gives

0 Eq. 3.14

1 0 Eq. 3.15

where

and Eq. 3.16

44

If equations 3.9 and 3.10 are substituted into equations 3.3-6, the

compatibility equations and force equilibrium at the attaching point are given by

Eq. 3.17

Eq. 3.18

Eq. 3.19

0 Eq. 3.20

where

and Eq. 3.21

After introducing equation 3.9 into equations 3.7 and 3.8, the boundary

conditions of the cantilever beam become

0 0 0 0 Eq. 3.22

0 0 Eq. 3.23

The solution of the differential equation 3.12 is given by (Murphy, 1960)

and takes the form as follows

sin cos sinh cosh Eq. 3.24

For the vth segment, the solution can be written as follows


where

Eq. 3.26

The derivatives of can be also written as following equations

1 1 Eq. 3.27

1 1 1 1 Eq. 3.28

1 1 1 1 Eq. 3.29

45

Therefore, the following equations can be written from equations 3.24-29

1cos sin cosh sinh

Eq. 3.30

1sin cos sinh cosh

Eq. 3.31

1cos sin cosh sinh

Eq. 3.32

As it is shown in Figure 3.1, the left end of the beam is corresponding with

the first segment of the beam. If boundary conditions (Equation 3.22) of the left

end of the beam are introduced into equations 3.25 and 3.30, one obtains

0 0 Eq. 3.33

0 0 0 0 Eq. 3.34

The last two expressions can be written in matrix form as follows

0 Eq. 3.35

where

1 2 3 4

0 1 0 1

1 0 1 0

1

2

Eq. 3.36

Eq. 3.37

The following equations are obtained by introducing equations 3.3-6 into

equations 3.24-29. The segments on the left and the right side of the v-th

attaching point located at x=xv are represented by v and v+1, respectively,

because of the segments that they belong. Therefore, the related coefficients are

shown by Cvj and Cv+1,j (j=1~4), respectively.

sin cos sinh cosh , sin

, cos , sinh , cosh 0

Eq. 3.38

46

cos sin cosh sinh , cos

, sin , cosh , sinh 0

Eq. 3.39

sin cos sinh cosh , sin

, cos , sinh , cosh 0

Eq. 3.40

cos sin cosh sinh , cos

, sin , cosh , sinh 0

Eq. 3.41

Moreover, the substitution of equation 3.25 into equation 3.15 gives

sin cos sinh cosh 1 0

Eq. 3.42

where

Eq. 3.43

If equations 3.38-42 are written in matrix form, one obtains

0 Eq. 3.44

where and are shown on the next page

4 3 4 2 4 1 4 4 1 4 2 4 3 4 4 4 5

sin cos sinh cosh sin cos sinh cosh 0

cos sin cosh sinh cos sin cosh sinh 0


cos sin cosh sinh cos sin cosh sinh

sin cos sinh cosh 0 0 0 0 1

5 2

5 1

5

5 1

5 2

Eq. 3.45

, , , , Eq. 3.46

Eq. 3.47

47

48

The right end of the beam belongs to the (n+1)th segment as it is shown in

Figure 3.1. If boundary conditions (Equation 3.23) of the right end of the beam

are introduced into equations 3.31 and 3.32, one obtains

, sin , cos , sinh , cosh 0

Eq. 3.48

, cos , sin , cosh , sinh 0

Eq. 3.49


0 Eq. 3.50

where

4 1 4 2 4 3 4 4

sin cos sinh cosh

cos sin cosh sinh

1

Eq. 3.51

, , , ,

Eq. 3.52

where

5 4 Eq. 3.53

Here, p represents the total number of equations. There are five equations

for any attaching point for a spring-mass system, including three compatibility

equations, one force equilibrium equation and one governing equation for the

sprung mass. Moreover, there are two more equations for each boundary of the

beam. Therefore, there are 5n+4 equations in all for the whole beam to obtain

integration constants Cvi and modal displacements Zv where v=1~n and i=1~4. In

other words, based on equation 3.25 and the governing equation of spring-mass,

there are four unknown integration constants for each beam segment and there

is one additional unknown Zv. If there is n spring-mass system, it means that

there is n+1 segment in the whole beam. Therefore, the total number of

unknowns for the beam carrying n spring-mass systems is equal to

4(n+1)+n=5n+4 (n unknown for Zv and 4(n+1) unknown for integration constants).

49

Hence, if all of the unknowns (Cvi and Zv) indicated in equations 3.37, 3.47

and 3.52 can be written as column vector and the matrices , and

are assembled by using the conventional assembly technique for direct stiffness

matrix method. Then, the following equation can be written for the entire system

in order to obtain required natural frequencies and mode shapes

0 Eq. 3.54

The non-trivial solution of the equation 3.54 is

|B| 0 Eq. 3.55

The natural frequencies of the entire vibrating system (i=1, 2,…) are

obtained by solving the equation 3.55. The integration constants are also

obtained by substituting each value of the natural frequencies into equation 3.54.

Once the integration constants are obtained, they are introduced into the

equation 3.24 and the corresponding mode shapes for each segment can be

defined. The coefficient matrix for a cantilever beam carrying one spring

mass system is given on the next page.

and are shown for beams which have various boundary

conditions in Appendix B. Moreover, the entire coefficient matrices ( ) for these

beams carrying more than one spring-mass system are also indicated in

Appendix B.

0 1 0 1 0 0 0 0 0

1 0 1 0 0 0 0 0 0






0 0 0 0 sin cos sinh cosh 0

0 0 0 0 cos sin cosh sinh 0

1

2

3

4

5

6

7

8

9

Eq. 3.56

50

51

3.3. Formulation of the Free Vibration Problem for Non-Uniform Beams Carrying

Spring-Mass Systems

3.3.1. Equations of Motion and Derivation of Eigenfunctions for the

Constrained Beam

The free vibration problem of uniform beams carrying various

concentrated elements has been studied by many researchers. On the other

hand, the literature regarding the free vibration characteristics of non-uniform

beams carrying any number of spring-mass systems is very rare. However, Chen

and Wu (2002) applied numerical assembly method for non-uniform beams

carrying spring-mass systems. Based on their studies, the same procedure that

is used in order to obtain the free vibration characteristics of uniform beams is

applied to non-uniform beams in this study. As it is shown in Figure 3.2, a non-

uniform beam carrying n spring-mass systems is divided into (n+1) segments.

The total length is L; v on the top represents the attaching point and the

corresponding segment. The left and the right ends of the beam are shown by

the letters L and R, respectively.

Figure 3.2 A non-uniform cantilever beam carrying n spring-mass systems

52

The equation of motion for a bare non-uniform beam is given by

, ,0 Eq. 3.57

where and are the cross-sectional area at position x and moment of

inertia of . and are given as follows

2 Eq. 3.58

4

Eq. 3.59

where

2 Eq. 3.60

2 Eq. 3.61

00

0 11

Eq. 3.62

where

0 Eq. 3.63

When equation 3.61 and 3.62 are introduced into equation 3.59, can

be written as

44 Eq. 3.64

Because of the fact that t3 is much smaller than r(x), the term, r(x) t3, can

be neglected. Hence, the moment of inertia is given by

Eq. 3.65

and the substitution of equation 3.62 into equations 3.58 and 3.65 gives

2 0 11

11

Eq. 3.66

0 11

11

Eq. 3.67

Similar to equation 3.9, the free vibration of the beam takes the form

, Eq. 3.68

53

If equations 3.66-68 are substituted into equation 3.57, one obtains

0

Eq. 3.69

1 3 1 6 1

3 1 1 0

Eq. 3.70

The coefficient ξ is given by

11

Eq. 3.71

then the derivatives of can be also written as following equations

Eq. 3.74

Eq. 3.75

Insertion of equations 3.71-75 into equation 3.70 gives

6 6 0 Eq. 3.76

6 6 0 Eq. 3.77

where

and Eq. 3.78



1 1Eq. 3.72

1 1 1 1

Eq. 3.73

54

12 1 1 2 1 3 1 4 1 Eq. 3.79

where Ci (i=1~4) are the integration constants, J1 and Y1 are the first order of

Bessel function of first and second kinds and I1 and K1 are the first order modified

Bessel function of first and second kinds.

For an arbitrary point, equation 3.79 can be written as follows for the v-th

segment

Eq. 3.80

where

11

Eq. 3.81

and the derivatives of equation 3.80 with respect to ξv

Eq. 3.82

Eq. 3.83

Eq. 3.84

3.3.2. Coefficient Matrix [Bv] for the v-th Attaching Point

Similar to equations 3.17-19, the compatibility equations at the attaching

point for non-uniform beam are given by

Eq. 3.85

Eq. 3.86

Eq. 3.87

55

The force equilibrium at the attaching point is as follows

Eq. 3.88

Eq. 3.89

If equations 3.71-74 are introduced into equation 3.89, one obtains

3 1 1

3 1 1

Eq. 3.90

3 1 1 3 1

1

Eq. 3.91

The equation of motion for the v-th spring-mass system is given by

0 Eq. 3.92

Similar to equation 3.10, free vibration of the v-th spring-mass system is

Eq. 3.93

The substitution of equations 3.68 and 3.93 into equation 3.92, one

obtains

0 Eq. 3.94

0 Eq. 3.95

1 0 Eq. 3.96

Where

Eq. 3.97

Insertion of equation 3.78 into equation 3.97 gives

12 1

Eq. 3.98

where

56

Eq. 3.99

12 1

where

2 2 11

Eq. 3.100

12

Eq. 3.101

After introducing equations 3.78, 3.93, 3.96 and 3.99, the interactive force

Fs, in equation 3.91, between the beam and the attached spring-mass system is

given by

1

12 1

1

12 1

Eq. 3.102

If equations 3.80-84 are inserted into equations 3.85-87, 3.91 and 3.96

,

, , , 0

Eq. 3.103

,

, , , 0

Eq. 3.104

,

, , , 0

Eq. 3.105

6

57

8

6 , , ,

, , , ,

, 0

Eq. 3.106

1 0

Eq. 3.107

where

12 1

1

12 1

11

Eq. 3.108

Equations 3.103-108 consist of integration coefficients represented by Cvi

and Cv+1,i (i=1~4) for the left and right side of the v-th attaching point,

respectively. The left side of the attaching point belongs to the segment (v) and

the right side belongs to the segment (v+1).

Similar to equation 3.44, equations 3.103-107 can be written in matrix

form as follows

0 Eq. 3.109

where and are shown on the next page.

4 3 4 2 4 1 4 4 1 4 2 4 3 4 4 4 5

0

0

0

∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ 0

0 0 0 0 1

5 2

5 1

5

5 1

5 2

Eq. 3.110

, , , , Eq. 3.111

Eq. 3.112

58

59

where

∆ 6 8 Eq. 3.113

∆ 6 8 Eq. 3.114

∆ 6 8 Eq. 3.115

∆ 6 8 Eq. 3.116

∆ 6 Eq. 3.117

∆ 6 Eq. 3.118

∆ 6 Eq. 3.119

∆ 6 Eq. 3.120

Eq. 3.121

3.3.3. Coefficient Matrix [BL] for the Left End of the Beam

The left end of the cantilever beam belongs to the first segment. The

boundary conditions for the cantilever beam with left end free are given as

follows

0 1

Therefore

1 0 Eq. 3.122

0 Eq. 3.123

substitution of equation 3.67, 3.73 and 3.74 into equation 3.123

3 1 1 0 Eq. 3.124

If equations 3.83 and 3.84 are inserted into equations 3.122 and 3.124,

respectively

0 Eq. 3.125

60

6

0

Eq. 3.126


0 Eq. 3.127

where

1 2 3 4

1

2

Eq. 3.128

Eq. 3.129

where

6 Eq. 3.130

6 Eq. 3.131

6 Eq. 3.132

6 Eq. 3.133

3.3.4. Coefficient Matrix [BR] for the Right End of the Beam

The right end of the cantilever beam belongs to the (n+1)th segment of the

beam carrying n spring-spring mass systems. The boundary conditions for the

cantilever beam with right end clamped are given as follows

therefore

For the deflection 0 Eq. 3.134

For the slope 0 Eq. 3.135

The substitution of equations 3.80 and 3.82 into equations equation 3.134

and 3.135, one obtains

, √ , √ , √ , √ 0 Eq. 3.136

, √ , √ , √ , √ 0 Eq. 3.137

61


0 Eq. 3.138

where

4 1 4 2 4 3 4 4

√ √ √ √

√ √ √ √

1

Eq. 3.139

, , , , Eq. 3.140

where p is defined as indicated in equation 3.53.

Similar to the definitions for uniform beam, here p represents the total

number of equations. Two boundary conditions for each boundary together with

compatibility equations, force equilibrium and governing equation of motion for

the sprung mass for each attaching point compose 5n+4 equations for the entire

beam in order to obtain integration constants and modal displacements.

As it is indicated for uniform beams, overall coefficient matrix [B] of the

entire beam is composed by using direct stiffness matrix method. Similar to the

equation 3.54 and 3.55, the following equations can be used in order to obtain

the natural frequencies and corresponding mode shapes for the entire system

0 Eq. 3.141


|B| 0 Eq. 3.142

Once one obtains the overall coefficient matrix, the solution of equation

3.142 gives the natural frequencies of the entire system and the insertion of that

solution into equation 3.141, one obtains the corresponding mode shapes for

each natural frequency for each segment. The coefficient matrix B for a

cantilever beam carrying one spring mass system is given on the next page.

Coefficient matrices for various boundary conditions are shown in

Appendix B. Moreover, the entire coefficient matrices ( B ) for these beams

carrying more than one spring-mass system are also indicated in Appendix B for

non-uniform beams.

0 0 0 0 0

0 0 0 0 0

0

0

0

∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ 0

0 0 0 0 1

0 0 0 0 √ √ √ √ 0

0 0 0 0 √ √ √ √ 0

1

2

3

4

5

6

7

8

9

Eq. 3.143

62

63

3.4. Formulation of the Free Vibration Problem for Uniform Multi-Span Beams

Carrying Spring-Mass Systems

3.4.1. Equations of Motion and Displacement Function

As it is indicated in the sections 3.2 and 3.3, the exact solutions of the free

vibration characteristics of uniform and non-uniform single span beams carrying

any number of spring-mass systems can be obtained through numerical

assembly method determined by Wu and Chou (1999) for uniform beams and

Chen and Wu (2001) for non-uniform beams. Based on these studies and the

numerical assembly method, Lin and Tsai (2007) studied the free vibration

characteristics of the multi-span uniform beam carrying any number of spring-

mass systems. This study also adopts the same method in order to obtain the

free vibration characteristics of constrained multi-span uniform beams.

As it is shown in Figure 3.3, the total length of the beam is L and the multi-

span uniform beam is supported by T pinned supports and carrying S spring-

mass systems. Each point that T pinned supports or S spring-mass systems

located is called station and each station is represented by 1~ .

On the other hand, each spring-mass system and pin support is located at

1~ and 1~ , respectively.

The equation of motion for a uniform Euler-Bernoulli beam and the pth

spring-mass system are given by

, ,0 Eq. 3.144

0 Eq. 3.145

where zp represents the displacement of the pth spring-mass relative to its static

equilibrium position and yp is the transverse deflection of the beam at the pth

attaching point.

64

The free vibration of the beam and the pth spring-mass system takes the

form

, Eq. 3.146

Eq. 3.147

where Y(x) and Zp are the amplitudes and ω is the natural frequency of the entire

vibrating system.

Figure 3.3 A uniform multi-span beam carrying S spring-mass systems and T pinned supports (Lin and Tsai, 2007)

The insertion of equation 3.146 into equation 3.144 leads to

0 Eq. 3.148

0 Eq. 3.149

65

where

Eq. 3.150




3.4.2. Coefficient Matrices and Determination of Natural Frequencies and

Mode Shapes

For any of the station point, the solution can be written as follows

, sin , cos , sinh

, cosh Eq. 3.152

where

Eq. 3.153

Eq. 3.154

The derivatives of can be written as following equations

, cos , sin , cosh , sinh

Eq. 3.155

, sin , cos , sinh , cosh

Eq. 3.156


Eq. 3.157

The left end of the beam belongs to station 1’ and is pin supported as it is

shown in Figure 3.3. The boundary conditions for the left end of the beam are

given by

0 0

therefore

Deflection 0 0 Eq. 3.158

66

Bending 0 0 Eq. 3.159

The substitution of equations 3.152 and 3.156 into equations 3.158 and

3.159 gives

0 , , 0 Eq. 3.160

0 , , 0 0 , , 0 Eq. 3.161

The last two expressions can be also represented as

0 Eq. 3.162

where

1 2 3 4

0 1 0 1

0 1 0 1

1

2

Eq. 3.163

, , , , Eq. 3.164

The compatibility and force equilibrium equations for the (p)th spring-mass

system requires that

Eq. 3.165

Eq. 3.166

Eq. 3.167

0 Eq. 3.168

where

and Eq. 3.169

If equations 3.146 and 3.147 are substituted into equation 3.145, one can

obtain

0 Eq. 3.170

1 0 Eq. 3.171

where

and Eq. 3.172

where ωp is the natural frequency of the (p)th spring-mass system.

67

The insertion of equations 3.152-157 into equations 3.165-168 gives



Eq. 3.173



Eq. 3.174



Eq. 3.175



Eq. 3.176

and the substitution of equation 3.152 into equation 3.171 gives

, sin , cos sinh cosh

1 0

Eq. 3.177

The last expressions can be written in matrix form

0 Eq. 3.178

where

4 3 4 2 4 1 4 4 1 4 2 4 3 4 4 4 5






4 1

4

4 1

4 2

4 3

, , , , , , , , Eq. 3.180

Eq. 3.179

68

69

Similar to the (p)th spring-mass system, compatibility equations for (r)th

intermediate support are given by

0 Eq. 3.181

Eq. 3.182

Eq. 3.183

Similarly, if equations 3.152-157 are substituted into equations 3.181-182,

one obtains


Eq. 3.184


Eq. 3.185



Eq. 3.186



Eq. 3.187

If equations 3.184-187 are written in matrix form, one obtains

0 Eq. 3.188

where and are shown on the next page.

4 3 4 2 4 1 4 4 1 4 2 4 3 4 4

sin cos sinh cosh 0 0 0 0

0 0 0 0 sin cos sinh cosh


sin cos sinh cosh sin cos sinh cosh

4 1

4

4 1

4 2

Eq. 3.189

, , , , , , , , Eq. 3.190

70

71

The right end of the beam belongs to station N’ and is pin supported as

shown in Figure 3.3. The boundary conditions for the right end of the beam are

given by

1

Therefore

Deflection 1 0 Eq. 3.191

Bending 1 0 Eq. 3.192

The substitution of equations 3.152 and 3.156 into equations 3.190 and

3.191 gives

1 , sin , cos , sinh , cosh 0

Eq. 3.193

1 , sin , cos , sinh , cosh

Eq. 3.194


0 Eq. 3.195

where

4 1 4 2 4 3 4 4

sin cos sinh cosh

sin cos sinh cosh

1

Eq. 3.196

, , , , Eq. 3.197

where and q represents the total number of intermediate supports and total

number of equations, respectively. As indicated before, shows the total

number of stations and consists of total spring-mass system and pinned

supports. There are four equations to write for intermediate supports, five

equations for spring-mass systems and two equations for each boundary.

Therefore the total number of equations is given by

4 2 5 4 Eq. 3.198

The integration constant matrices for intermediate supports, spring-mass

systems and each boundaries ( , , , ) compose the overall

coefficient matrix for the entire beam through numerical assembly method.

72

0 Eq. 3.199


|B| 0 Eq. 3.200

The free vibration characteristics of the entire beam can be obtained by

solving equations 3.199 and 3.200. The non-trivial solution of equation 3.200

gives the natural frequencies of the entire beam and the substitution of the

solution into equation 3.199 gives the corresponding mode shapes for the whole

beam. The associated coefficient matrix for a multi-span uniform pinned

supported beam carrying one spring-mass system and consisting of one

intermediate support is given on the next page.

Figure 3.4 Two-span uniform beam with one intermediate support and one spring-mass system

0 1 0 1 0 0 0 0 0 0 0 0 0

0 1 0 1 0 0 0 0 0 0 0 0 0

sin cos sinh cosh 0 0 0 0 0 0 0 0 0

0 0 0 0 sin cos sinh cosh 0 0 0 0 0

cos sin cosh sinh cos sin cosh sinh 0 0 0 0 0

sin cos sinh cosh sin cos sinh cosh 0 0 0 0 0

0 0 0 0 sin cos sinh cosh sin cos sinh cosh 0

0 0 0 0 cos sin cosh sinh cos sin cosh sinh 0

0 0 0 0 sin cos sinh cosh sin cos sinh cosh 0

0 0 0 0 cos sin cosh sinh cos sin cosh sinh

0 0 0 0 sin cos sinh cosh 0 0 0 0 1

0 0 0 0 0 0 0 0 sin cos sinh cosh 0

0 0 0 0 0 0 0 0 sin cos sinh cosh 0

Eq. 3.201

73

74

3.5. Forced Vibration of Euler-Bernoulli Beams

3.5.1. Introduction

The forced vibration of the beams which are subdivided into segments in

the previous sections is analyzed for each segment separately by using

corresponding mode shape and then forced vibration response for the entire

beam is obtained by combining each response for each segment through

MATHEMATICA via symbolic computation. Rao (2007) discussed solution of the

forced vibration of beams through modal analysis approach and orthogonality

condition. This study adopts the same method in order to obtain the exact

solutions for the forced vibration of any types of beams with attached spring-

mass systems indicated in the previous sections.

3.5.2. Formulation of Forced Vibration for Beams

The equation of motion for Euler-Bernoulli beam subjected to distributed

transverse force takes the form

, ,, Eq. 3.202

The solution of equation 3.202 can be defined as linear combination of normal

modes of the beam as follows

, Eq. 3.203

where are the mode shapes that are obtained by using the methods

indicated in the previous sections for different types of beams carrying attached

spring-mass systems and are the modal participation coefficients.

0 Eq. 3.204

75

If equation 3.203 is inserted into equation 3.202, one obtains

, Eq. 3.205

From equations 3.203 and 3.204

, Eq. 3.206

Based on the orthogonality condition, for a beam with total length L

Eq. 3.207

where is the Kronecker delta that is

0 and 1 Eq. 3.208

If equation 3.206 is multiplied with integrated from 0 to L

,

Eq. 3.209

According to equation 3.208, the left side of equation 3.209 is only valid

when i=j,

Eq. 3.210

where

, Eq. 3.211

The solution of equation 3.210 is given by

cos sin1

sin Eq. 3.212

therefore the solution of equation 3.202 is given by substituting equation 3.212

into equation 3.203

, ∑ cos sin sin

Eq. 3.213

76

It is important to note that the first two terms inside the brackets in

equation 3.213 indicate the free vibration (homogeneous solution), and the third

term indicates the forced vibration of the beam. and can be computed

through initial conditions of the beam. In this study, the initial conditions are

assumed to be zero, hence

, 0 0 and , 0 0 Eq. 3.214

cos 0 sin 0 0 Eq. 3.215

sin 0 cos 0 0 Eq. 3.216

from equations 3.215 and 3.216, and are equal to zero. Therefore the

equation 3.213 can be simplified as

,1

sin Eq. 3.217

3.5.2.1. Impact Loading (Step-Function Force) and Harmonic Loading

Figure 3.5 Simply-supported beam subjected to step-function force F0

77

Figure 3.6 Simply-supported beam subjected to harmonic force F0sin(Ωt)

The step-function force acting on the simply supported beam can be

represented as

, Eq. 3.218

and the beam subjected to harmonic force can be written as

, sin Eq. 3.219

therefore the generalized force corresponding to the ith mode can be determined

by using equation 3.211 as below

For step-function force Eq. 3.220

For harmonic force sin Eq. 3.221

and the generalized coordinate in the ith mode is represented as below assuming

that initial conditions of the beam are zero

1

sin Eq. 3.222

Hence, the response of the beam subjected to step-function force can be

expressed by

,1

sin Eq. 3.223

78

3.5.2.2. Moving and Moving Pulsating Load

Figure 3.7 Simply-supported beam subjected to moving load

Figure 3.8 Simply-supported beam subjected to moving pulsating load

As it is shown in figures 3.7 and 3.8 the concentrated moving and moving

pulsating loads move with a constant speed v0. f(x) can be assumed as uniformly

distributed load applied over an elemental length 2∆x and centered at x=d as it is

shown in figure 3.9. f(x) can be represented as below

79

Figure 3.9 Simply-supported beam subjected to moving pulsating load

0 0 ∆

2∆∆ ∆

0 ∆

Eq. 3.224

f(x) can be represented as Fourier series and defined over the interval 0 to L by

expanding f(x) for all values of x in terms of only sine terms as shown below

sin / Eq. 3.225

where fn is defined by

2sin / Eq. 3.226

therefore fn can be obtained by substituting equation 3.224 into equation 3.226

as below

22∆

sin /∆

∆ Eq. 3.227

sin /∆

∆

2 sin sin ∆

∆ Eq. 3.228

80

If ∆x converges to 0 in equation 3.226

lim∆

sin ∆

∆1 Eq. 3.229

therefore for constant moving load fn can be defined as

2sin Eq. 3.230

and for moving pulsating force

2sin sin Eq. 3.231

then the Fourier series expansion of f(x) can be written as below for constant

moving load

2sin sin / Eq. 3.232

and for moving pulsating load

,2

sin sin sin / Eq. 3.233

After d is defined as v0t, equation 3.232 and equation 3.233 are written as

,2

/ Eq. 3.234

,2

sin sin sin / Eq. 3.235

81

CHAPTER 4. NUMERICAL RESULTS

4.1. Introduction

The free and forced vibration analyses of different types of beams carrying

multiple spring-mass systems are performed by using an algorithm coded in

MATHEMATICA. Algorithms are developed for each type of the beams based on

the methodologies described in Chapter 3. The results for the beams with

multiple spring-mass systems are compared with the bare beam. Considering

normal modes with cumulative effective modal mass adding up to 90% of the

total mass is assumed to be sufficient for the forced vibration response analysis

of the structure. The effective modal mass is computed after normalizing the

eigenvectors based on following equations

Г Eq. 4.1

where Гn is the participation factor, is the corresponding mass-normalized

mode shape (normalized eigenvector) and is mass per unit length.

The effective modal mass is as follows

, Eq. 4.2

where normalized eigenvectors are required to be

1 Eq. 4.3

Therefore, effective modal mass can be simplified as

, Eq. 4.4

82

4.2. Free Vibration Analysis of Single Span Uniform Beam Carrying One, Two

and Three Spring-Mass Systems

The boundary conditions of uniform beams studied in this section are

single span pinned-pinned (SS), clamped-clamped (CC), clamped-pinned (CS)

and clamped-free (CF) at their two ends. The spring-mass systems are attached

at their mid point for SS, CC and CS and its free end for the CF. The beam

evaluated here has a total length of 16.5 meters and a constant cross-sectional

area of 0.375 m2. The physical properties of the uniform beam studied in this

section are as follows

Young’s modulus E=30x109 N/m2

Moment of inertia of the cross-sectional area I=9x10-3 m4

Mass density of beam material ρ=2400 kg/m3

Mass per unit length m= ρA=900 kg/m

The total mass of the beam mb=mL=14850 kg

The free vibration characteristics of uniform beams with one, two and

three spring-mass systems are compared with each other and with bare uniform

beam as well. If n spring-mass system is attached to the uniform beam, the first n

mode is under the influence of n spring-mass system and the (n+1)th mode is

actually the first mode of the uniform beam. Therefore, each additional spring-

mass system’s natural frequency is tuned to the natural frequency of (n+1)th

mode of the uniform beam carrying one spring-mass system less than it is

attached. For example, the natural frequency of the spring-mass system attached

to a bare uniform beam is tuned to first natural frequency of the bare uniform

beam and if one more spring-mass system is attached to that beam, the

additional spring-mass system is tuned to the second natural frequency of the

beam carrying one spring-mass system. This method is performed in this study in

order to reduce the dynamic response of the beam subjected to forced vibration

83

by finding the adequate number of spring-mass systems to be attached to the

uniform beam.

For the cases of uniform beam carrying one, two and three spring-mass

systems and for a bare uniform beam, the natural frequencies and mode shapes

are given as follows

Table 4.1 The lowest seven natural frequencies of the uniform beam carrying one spring-mass system (m1/mb=0.01)

Boundary

Conditions ω1 ω 2 ω 3 ω 4 ω 5 ω 6 ω 7 ω tmd1

SS 18.50 21.31 79.42 178.73 317.70 496.41 714.82 19.86

CC 41.56 48.70 124.08 243.32 402.09 600.68 838.5 44.986

CS 28.85 33.33 100.54 209.77 358.65 547.30 775.63 31.01

CF 6.40 7.82 44.35 124.13 243.24 402.51 602.50 7.08


Boundary


SS 17.97 21.94 79.42 178.75 317.70 496.42 714.82 19.86

CC 40.21 50.31 124.08 243.41 402.09 600.71 839.5 44.986

CS 28.00 34.33 100.56 209.81 358.66 547.31 775.63 31.01

CF 6.14 8.15 44.38 124.14 243.24 401.03 602.50 7.08

84


Boundary


SS 16.96 23.23 79.42 178.82 317.70 496.44 714.82 19.86

CC 37.68 53.62 124.08 243.66 402.09 600.53 839.5 44.986

CS 26.38 36.38 100.61 209.93 358.67 547.36 775.64 31.01

CF 5.66 8.82 44.45 124.17 243.35 402.90 602.50 7.08


Boundary


SS 15.90 24.76 79.42 178.93 317.70 496.48 714.82 19.86

CC 35.04 57.53 124.08 244.09 402.09 600.99 839.5 44.986

CS 24.69 38.80 100.70 210.13 358.69 547.43 775.64 31.01

CF 5.17 9.63 44.56 124.21 243.28 402.12 604.00 7.08


Boundary


SS 14.53 27.05 79.42 179.15 317.70 496.56 714.82 19.86

CC 31.68 63.35 124.08 244.96 402.09 601.33 839.5 44.986

CS 22.51 42.37 100.88 210.53 358.73 547.58 775.66 31.01

CF 4.57 10.83 44.80 124.29 243.32 402.28 602.50 7.08

85

Table 4.6 The lowest seven natural frequencies of the uniform beam carrying two spring-mass systems (m1/mb=m2/mb=0.01)

Boundary

Conditions ω1 ω 2 ω 3 ω 4 ω 5 ω 6 ω 7 ω tmd1 ω tmd2

SS 18.16 20.47 22.60 79.42 178.75 317.70 496.42 19.86 21.31

CC 40.69 46.55 52.01 124.08 243.42 402.09 600.72 44.986 48.70

CS 28.30 31.99 35.38 100.56 209.82 358.66 547.31 31.01 33.326

CF 6.23 7.39 8.50 44.38 124.14 243.24 402.09 7.08 7.82


Boundary


SS 17.49 20.72 23.86 79.42 178.80 317.70 496.43 19.86 21.94

CC 39.00 47.17 55.27 124.08 243.62 402.09 600.80 44.986 50.31

CS 27.23 32.38 37.38 100.60 209.91 358.67 547.35 31.01 34.328

CF 5.91 7.51 9.18 44.44 124.16 243.25 402.10 7.08 8.15


Boundary


SS 16.22 21.19 26.60 79.42 178.97 317.70 496.48 19.86 23.23

CC 35.82 48.36 62.35 124.08 244.28 402.09 601.06 44.986 53.62

CS 25.20 33.13 41.70 100.73 210.21 358.70 547.46 31.01 36.38

CF 5.31 7.74 10.68 44.63 124.23 243.29 402.12 7.08 8.82

86


Boundary


SS 14.90 21.70 30.08 79.42 179.28 317.70 496.61 19.86 24.76

CC 32.53 49.62 71.32 124.08 245.54 402.09 601.54 44.986 57.53

CS 23.09 33.93 47.11 101.00 210.77 358.75 547.67 31.01 38.80

CF 4.71 7.98 12.62 45.00 124.35 243.35 402.15 7.08 9.63


Boundary


SS 13.22 22.38 35.73 79.42 180.00 317.70 496.86 19.86 27.05

CC 28.45 51.26 85.62 124.08 248.60 402.09 602.70 44.986 63.35

CS 20.42 34.98 55.63 101.67 212.09 358.88 548.16 31.01 42.37

CF 3.996 8.27 15.73 45.95 124.66 243.50 402.24 7.08 10.83

Table 4.11 The lowest eight natural frequencies of the uniform beam carrying three spring-mass systems

(m1/mb=m2/mb=m3/mb=0.01)

Boundary

Conditions ω1 ω 2 ω 3 ω 4 ω 5 ω 6 ω 7 ω 8 ω tmd1 ω tmd2 ω tmd3

SS 17.91 20.36 21.85 23.81 79.64 178.11 317.98 496.97 19.86 21.31 22.59

CC 40.04 46.28 50.09 55.15 124.08 244.70 402.50 602.29 44.986 48.701 52.013

CS 27.90 31.82 34.19 37.31 100.75 209.83 358.30 547.16 31.01 33.326 35.376

CF 6.11 7.33 8.10 9.16 44.47 124.10 243.61 402.09 7.08 7.818 8.488

Table 4.12 The lowest eight natural frequencies of the uniform beam carrying three spring-mass systems (m1/mb=m2/mb=m3/mb=0.02)

Boundary


SS 17.13 20.56 22.73 25.74 79.71 179.12 317.98 496.29 19.86 21.94 23.86

CC 38.09 46.77 52.35 60.19 124.08 244.12 402.50 600.67 44.986 50.308 55.269

CS 26.66 32.13 35.59 40.36 100.70 210.91 358.50 547.16 31.01 34.328 37.382

CF 5.73 7.43 8.56 10.23 44.53 125.10 243.24 402.10 7.08 8.147 9.175

87

Table 4.13 The lowest eight natural frequencies of the uniform beam carrying three spring-mass systems

(m1/mb=m2/mb=m3/mb=0.05)

Boundary


SS 15.67 20.94 24.57 30.13 79.87 179.81 318.21 495.62 19.86 23.23 26.60

CC 34.42 47.69 57.06 71.58 124.08 245.42 402.50 603.10 44.986 53.621 62.351

CS 24.32 32.72 38.49 47.22 100.96 209.74 359.54 547.07 31.01 36.38 41.71

CF 5.05 7.60 9.53 12.77 45.00 126.00 246.06 402.12 7.08 8.820 10.678


Boundary


SS 14.15 21.32 26.78 36.07 79.17 180.14 318.00 498.20 19.86 24.76 30.08

CC 30.67 48.63 62.69 86.95 124.08 253.04 402.50 600.45 44.986 57.533 71.321

CS 21.89 33.32 41.96 56.22 101.42 212.03 359.55 549.99 31.01 38.80 47.11

CF 4.36 7.77 10.70 16.15 45.79 126.00 246.07 402.25 7.08 9.625 12.620

88


Boundary


SS 12.27 21.80 30.14 46.40 79.62 181.75 318.00 498.01 19.86 27.05 35.73

CC 26.12 49.75 71.15 111.50 124.08 252.20 402.81 602.66 44.986 63.351 85.621

CS 18.89 34.06 47.12 70.36 103.66 214.78 361.40 546.51 31.01 42.37 55.63

CF 3.60 7.96 12.45 21.53 45.77 124.10 243.50 403.04 7.08 10.828 15.735

Table 4.16 The lowest six natural frequencies of the bare uniform beam

Boundary

Conditionsω1 ω 2 ω 3 ω 4 ω 5 ω 6

SS 19.86 79.42 178.70 317.70 496.40 714.82

CC 45.01 124.08 243.24 402.09 600.93 839.5

CS 31.02 100.52 209.73 358.65 547.28 775.63

CF 7.07 44.33 124.13 243.24 402.17 602.00

89

Figure 4.1 Mode Shapes of Uniform SS, CC, CS and CF Beams Carrying One Spring-Mass System

0 5 10 15

-0.010

-0.005

0.000

0.005

0.010

0 5 10 15

-0.010

-0.005

0.000

0.005

0.010

0 5 10 15

-0.010

-0.005

0.000

0.005

0.010

0 5 10 15

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

Mod

al D

ispl

acem

ent

Length (m)

a) Mode shapes of simply supported beam (m1/mb=0.01)

Mod

al D

ispl

acem

ent

Length (m)

b) Mode shapes of clamped-clamped beam (m1/mb=0.01)

Mod

al D

ispl

acem

ent

Length (m)

Mod

al D

ispl

acem

ent

Length (m)

1st mode

2nd mode

3rd mode

4th mode

5th mode

6th mode

7th mode

1st mode

2nd mode

3rd mode

4th mode

5th mode

6th mode

7th mode

c) Mode shapes of clamped-pinned beam (m1/mb=0.01) d) Mode shapes of clamped-free beam (m1/mb=0.01)

90

Figure 4.2 Mode Shapes of Uniform SS, CC, CS and CF Beams Carrying Two Spring-Mass Systems

0 5 10 15

-0.010

-0.005

0.000

0.005

0.010

0 5 10 15

-0.010

-0.005

0.000

0.005

0.010

0 5 10 15

-0.010

-0.005

0.000

0.005

0.010

0 5 10 15

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

Mod

al D

ispl

acem

ent

Length (m)

a) Mode shapes of simply supported beam (m1/mb= m2/mb =0.01)

Mod

al D

ispl

acem

ent

Length (m)

b) Mode shapes of clamped-clamped beam (m1/mb= m2/mb =0.01)

Mod

al D

ispl

acem

ent

Length (m)

Mod

al D

ispl

acem

ent

Length (m)

1st mode

2nd mode

3rd mode

4th mode

5th mode

6th mode

7th mode

2nd mode

3rd mode

4th mode

5th mode

6th mode

7th mode

c) Mode shapes of clamped-pinned beam (m1/mb= m2/mb =0.01) d) Mode shapes of clamped-free beam (m1/mb= m2/mb =0.01)

1st mode

91

Figure 4.3 Mode Shapes of Uniform SS, CC, CS and CF Beams Carrying Three Spring-Mass Systems

0 5 10 15

-0.010

-0.005

0.000

0.005

0.010

0 5 10 15

-0.010

-0.005

0.000

0.005

0.010

0 5 10 15

-0.010

-0.005

0.000

0.005

0.010

0 5 10 15

-0.015

-0.010

-0.005

0.000

0.005

0.010

Mod

al D

ispl

acem

ent

Length (m)

a) Mode shapes of simply supported beam (m1/mb=m2/mb=m3/mb=0.01)

Mod

al D

ispl

acem

ent

Length (m)

b) Mode shapes of clamped-clamped beam (m1/mb=m2/mb=m3/mb=0.01)

Mod

al D

ispl

acem

ent

Length (m)

Mod

al D

ispl

acem

ent

Length (m)

1st mode

2nd mode

3rd mode

4th mode

5th mode

6th mode

7th mode

2nd mode

3rd mode

4th mode

5th mode

6th mode

7th mode

c) Mode shapes of clamped-pinned beam (m1/mb=m2/mb=m3/mb=0.01) d) Mode shapes of clamped-free beam (m1/mb=m2/mb=m3/mb=0.01)

8th mode

8th mode

1st mode

92

Figure 4.4 Mode Shapes of Bare SS, CC, CS and CF Uniform Beams

0 5 10 15

-0.010

-0.005

0.000

0.005

0.010

0 5 10 15

-0.010

-0.005

0.000

0.005

0.010

0.015

0 5 10 15

-0.010

-0.005

0.000

0.005

0.010

0 5 10 15

-0.010

-0.005

0.000

0.005

0.010

0.015

Mod

al D

ispl

acem

ent

Length (m)

a) Mode shapes of bare simply supported uniform beam

Mod

al D

ispl

acem

ent

Length (m)

b) Mode shapes of bare clamped-clamped uniform beam

Mod

al D

ispl

acem

ent

Length (m)

Mod

al D

ispl

acem

ent

Length (m)

1st mode

2nd mode

3rd mode

4th mode

5th mode

6th mode

2nd mode

3rd mode

4th mode

5th mode

6th mode

c) Mode shapes of bare clamped-pinned uniform beam d) Mode shapes of bare clamped-free uniform beam

1st mode

93

94

4.3. Free Vibration Analysis of Single Span Non-Uniform Beam Carrying

Spring-Mass Systems

The non-uniform beams studied in this section are single span pinned-

pinned (SS), clamped-clamped (CC), pinned-clamped (SC) and free-clamped

(FC) at their two ends. The spring-mass systems are attached at their mid point

for SS, CC and SC boundaries and its free end for the FC boundary. Two

different types of structures are used for this section.

First non-uniform beam has a length of 40 in and the cross-sectional area

at the shallow end (A0) is 1.5 in2. The taper ratio of the beam α=rav(L)/rav(0) is

assumed to be 2. The physical properties of the non-uniform beam with SS, CC,

SC and FC types of boundaries are as follows

Young’s modulus E=2.9x107 psi

Moment of inertia of the cross-sectional area at the shallow end Io=0.28125 in4

Mass density of beam material ρ=0.283 lb/in3

The total mass of the beam mb=ρ A0 L ((α+1)/2)=25.47 lb

Similar to uniform beams, non-uniform beams with one spring-mass

system is compared with bare non-uniform beam as well. The same method in

order to tune the spring-mass system for uniform beams is also performed for

non-uniform beams.

The lowest natural frequencies and corresponding mode shapes of non-

uniform beams carrying one spring-mass system and a bare non-uniform beam

are given as follows for four types of boundary conditions. The mode shapes are

normalized by making the maximum value of each mode equal to unity.

95

Table 4.17 The lowest six natural frequencies of the non-uniform beam carrying one spring-mass system (m1/mb=0.01)

Boundary

Conditions ω1 ω 2 ω 3 ω 4 ω 5 ω 6 ω tmd1

SS 36.36 41.85 158.64 356.07 632.03 986.75 39.00

CC 82.81 96.70 246.50 483.06 798.16 1192.15 89.50

SC 63.93 73.45 209.23 429.84 730.50 1111.23 68.56

FC 19.70 27.70 100.61 259.07 495.65 810.96 20.95


Boundary


SS 35.31 43.08 158.64 356.11 632.04 986.76 39.00

CC 80.18 99.80 246.54 483.19 798.19 1192.19 89.50

SC 62.10 75.56 209.33 429.89 730.54 1111.24 68.56

FC 18.84 28.88 100.85 259.16 495.69 810.99 20.95


Boundary


SS 33.34 45.62 158.64 356.22 632.06 986.79 39.00

CC 75.22 106.20 246.67 483.59 798.28 1192.30 89.50

SC 58.64 79.87 209.62 430.05 730.66 1111.27 68.56

FC 17.09 31.55 101.59 259.44 495.84 811.08 20.95

96


Boundary


SS 31.26 48.61 158.66 356.41 632.08 986.84 39.00

CC 70.05 113.72 246.89 484.26 798.45 1192.48 89.50

SC 54.98 84.89 210.11 430.32 730.87 1111.32 68.56

FC 15.28 34.78 102.83 259.91 496.09 811.23 20.95


Boundary


SS 28.58 53.08 158.68 356.80 632.13 986.94 39.00

CC 63.45 124.84 247.35 485.61 798.77 1192.85 89.50

SC 50.25 92.24 211.12 430.86 731.28 1111.41 68.56

FC 13.09 39.40 105.33 260.86 496.58 811.53 20.95

Table 4.22 The lowest five natural frequencies of the bare non-uniform beam

Boundary

Conditions ω1 ω 2 ω 3 ω 4 ω 5

SS 39.02 158.63 356.03 632.03 986.74

CC 89.51 246.46 482.93 798.13 1192.12

SC 68.55 209.14 429.79 730.46 1111.22

FC 20.95 100.36 258.97 495.60 810.93

Figure 4.5 Mode Shapes of Non-Uniform SS, CC, SC and FC Beams Carrying One Spring-Mass System

0 10 20 30 40

-1.0

-0.5

0.0

0.5

1.0

0 10 20 30 40-1.0

-0.5

0.0

0.5

1.0

0 10 20 30 40-1.0

-0.5

0.0

0.5

1.0

0 10 20 30 40-1.0

-0.5

0.0

0.5

1.0

Mod

al D

ispl

acem

ent

Length (in)

a) Mode shapes of simply supported beam (m1/mb=0.01)

Mod

al D

ispl

acem

ent

Length (in)

b) Mode shapes of clamped-clamped beam (m1/mb=0.01)

Mod

al D

ispl

acem

ent

Length (in)

Mod

al D

ispl

acem

ent

Length (in)

1st mode

2nd mode

3rd mode

4th mode

5th mode

6th mode

1st mode

2nd mode

3rd mode

4th mode

5th mode

6th mode

c) Mode shapes of pinned-clamped beam (m1/mb=0.01) d) Mode shapes of free-clamped beam (m1/mb=0.01)

97

Figure 4.6 Mode Shapes of Bare SS, CC, CS and CF Non-Uniform Beams

0 10 20 30 40-1.0

-0.5

0.0

0.5

1.0

0 10 20 30 40

-0.5

0.0

0.5

1.0

0 10 20 30 40-1.0

-0.5

0.0

0.5

1.0

0 10 20 30 40

-1.0

-0.5

0.0

0.5

1.0

Mod

al D

ispl

acem

ent

Length (in)

a) Mode shapes of bare simply supported non-uniform beam

Mod

al D

ispl

acem

ent

Length (in)

b) Mode shapes of bare clamped-clamped non-uniform beam

Mod

al D

ispl

acem

ent

Length (in)

Mod

al D

ispl

acem

ent

Length (in)

1st mode

2nd mode

3rd mode

4th mode

5th mode

2nd mode

3rd mode

4th mode

5th mode

c) Mode shapes of bare pinned-clamped non-uniform beam d) Mode shapes of bare clamped-free non-uniform beam

1st mode

98

99

In addition, the sample high-mast lighting tower is studied for evaluation of

spring-mass system effect by modeling the real high-mast lighting tower as FC

type of non-uniform beam. The dimensions and physical properties of the

structure were taken from Sherman’s master thesis (Sherman, 2009) and the

resultant natural frequencies of the present method are compared with that study

in order to verify the developed algorithm.

The outer diameter of the shallow end OD1=6 in

The outer diameter of the clamped end OD2=24.75 in

Thickness t=0.18 in

Taper ratio α=4.222

Length L=140 ft

Mass density of the material ρ=0.283 lb/in3

The total mass of the structure mb=ρ A0 L ((α+1)/2)=4100 lb

A 687 pound luminary is supported by structure.

The high-mast lighting tower is a 140-foot tall structure with a 687 pound

luminary at the shallow end. 16-sided cross sectional area is considered as

cylindrical section. The lowest seven natural frequencies of high-mast lighting

tower with one spring-mass system on the top of the structure and the

comparison of the lowest four natural frequencies of bare high-mast lighting

tower in this study with ABAQUS and SAP2000 results which are given by

Sherman (2009) are shown in Tables 4.23 and 4.24, respectively. Figures 4.5

and 4.6 show the corresponding mode shapes which are scaled to unity.

100

Table 4.23 The lowest seven natural frequencies of the high-mast lighting tower carrying one spring-mass system at the free end

Boundary

Condition m1/mb ω1 ω 2 ω 3 ω 4 ω 5 ω 6 ω 7 ω tmd1

FC 0.01 1.47 1.79 7.06 18.80 37.19 62.23 93.94 1.62

FC 0.02 1.42 1.86 7.06 18.80 37.19 62.23 93.94 1.62

FC 0.05 1.31 2.01 7.07 18.80 37.19 62.23 93.94 1.62

FC 0.1 1.20 2.18 7.08 18.80 37.19 62.23 93.94 1.62

FC 0.2 1.06 2.45 7.10 18.81 37.19 62.23 93.94 1.62

FC 0 1.62 7.06 18.80 37.19 62.23 93.94 132.31 0

Table 4.24 Comparison of the lowest four natural frequencies of the bare high-mast lighting tower

Boundary

Condition Methods ω1 ω 2 ω 3 ω 4

FC This study 1.62 7.05 18.80 37.19

FC ABAQUS (Sherman,

2009) 1.63 7.15 19.02 37.47

FC SAP2000 (Sherman,

2009) 1.58 7.01 18.60 39.01

101

Figure 4.7 Mode Shapes of High-Mast Lighting Tower Carrying One Spring-Mass System on the Top

Figure 4.8 Mode Shapes of Bare High-Mast Lighting Tower

0 500 1000 1500-1.0

-0.5

0.0

0.5

1.0

0 500 1000 1500-1.0

-0.5

0.0

0.5

1.0

1st mode

2nd mode

3rd mode

4th mode

5th mode

6th mode

Mod

al D

isp

lace

men

t

Length (in)

1st mode

2nd mode

3rd mode

4th mode

5th mode

Mod

al D

isp

lace

men

t

Length (in)

102

4.4. Free Vibration Analysis of Uniform Multi-Span Beam Carrying

Spring-Mass Systems

The dimensions and physical properties of the uniform beam indicated in

section 4.2 are also same for multi-span uniform beams in this section. Each

span has a total length of 16.5 m, a constant cross-sectional area of 0.375 m2

and each multi-span uniform beam is pinned at its two ends. The physical

properties are as follows

Young’s modulus E=30x109 N/m2

Moment of inertia of the cross-sectional area I=9x10-3 m4

Mass density of beam material ρ=2400 kg/m3

Mass per unit length m= ρA=900 kg/m

The total mass of each span mb=mL=14850 kg

Similar to single span uniform beams, multi-span uniform beams with one

and two spring-mass systems and one and two intermediate pinned supports are

also compared with each other and with bare multi-span uniform beam as well.

The same method for tuning the spring-mass system for single span uniform

beams is also performed for multi-span uniform beams.

4.4.1. Free Vibration Analysis of Two Span Beam Carrying

One Spring-Mass System

The beam studied in this section is two-span uniform beam pinned at its

two ends and at ξ1’=x1’/L=0.5 and carries one intermediate spring-mass system at

ξ1*=0.75 as shown in Figure 4.9.

103

Case 1: Spring-mass system is attached to second span

Figure 4.9 Two-span beam carrying one spring-mass system attached to second span

Table 4.25 The lowest six natural frequencies of the two-span beam carrying one spring-mass system at second span

m1/mb ω1 ω 2 ω 3 ω 4 ω 5 ω 6 ω tmd

x1=0.75L

0.01 18.44 21.22 31.25 79.42 100.53 178.70 19.86

0.02 17.86 21.74 31.48 79.42 100.53 178.83 19.86

0.05 16.73 22.68 32.18 79.42 100.56 178.87 19.86

0.1 15.53 23.52 33.37 79.42 100.59 178.78 19.86

0 19.86 31.02 79.42 100.52 178.70 210.11 0.00

104

Case 2: Spring-mass system is attached to first span

Figure 4.10 Two-span beam carrying one spring-mass system attached to first span

Table 4.26 The lowest six natural frequencies of the two-span beam carrying one spring-mass system at first span

m1/mb ω1 ω 2 ω 3 ω 4 ω 5 ω 6 ω tmd

x1=0.25L

0.01 18.44 21.22 31.25 79.42 100.53 178.73 19.86

0.02 17.86 21.74 31.48 79.42 100.53 178.75 19.86

0.05 16.73 22.68 32.18 79.42 100.56 178.82 19.86

0.1 15.53 23.52 33.37 79.42 100.59 178.93 19.86

0 19.86 31.02 79.42 100.52 178.70 210.11 0.00

105

Figure 4.11 Mode shapes of two-span beam carrying one spring-mass system at second span (mtmd=0.01mb)

Figure 4.12 Mode shapes of two-span beam carrying one spring-mass system at first span (mtmd=0.01mb)

0 5 10 15 20 25 30

-0.005

0.000

0.005

0 5 10 15 20 25 30

-0.005

0.000

0.005

1st mode

4th mode

3rd mode

2nd mode

5th mode

6th mode

Length (m)

Mod

al D

isp

lace

men

t

1st mode

4th mode

3rd mode

2nd mode

5th mode

6th mode

Length (m)

Mod

al D

isp

lace

men

t

106

4.4.2. Free Vibration Analysis of Two Span Beam Carrying

Two Spring-Mass Systems

Case 1: Both of the spring-mass systems are tuned to constant frequency which

is same with the first natural frequency of two-span bare uniform beam.

Figure 4.13 Two-span beam carrying two spring-mass systems (Case 1)

Table 4.27 The lowest six natural frequencies of the two-span beam carrying two spring-mass systems based on case 1

m1/mb=

m2/mb ω1 ω 2 ω 3 ω 4 ω 5 ω 6

ω tmd1

x1=0.25L

ω tmd2

x2=0.75L

0.01 17.97 19.57 21.94 31.46 79.42 100.53 19.85 19.85

0.02 17.24 19.31 22.85 31.87 79.42 100.55 19.85 19.85

0.05 15.90 18.62 24.76 33.02 79.42 100.59 19.85 19.85

0.1 14.53 17.69 27.05 34.70 79.42 100.66 19.85 19.85

0 19.86 31.02 79.42 100.52 178.70 210.11 0.00 0.00

107

Case 2: The first spring-mass system is tuned to the first natural frequency of

two-span bare uniform beam and the second one is tuned to the second natural

frequency of the two-span uniform beam carrying one spring-mass system.

Figure 4.14 Two-span beam carrying two spring-mass systems (Case 2)

Table 4.28 The lowest six natural frequencies of the two-span beam carrying two spring-mass systems based on case 2

m1/mb=

m2/mb ω1 ω 2 ω 3 ω 4 ω 5 ω 6

ω tmd1

x1=0.25L

ω tmd2

x2=0.75L

0.01 18.14 20.15 22.50 31.51 79.42 100.54 19.85 21.22

0.02 17.46 20.07 23.66 32.02 79.42 100.55 19.85 21.74

0.05 16.17 19.59 26.00 33.54 79.42 100.60 19.85 22.68

0.1 14.82 18.71 28.66 35.91 79.42 100.69 19.85 23.52

0 19.86 31.02 79.42 100.52 178.70 210.11 0.00 0.00

108

Figure 4.15 Mode shapes of two-span beam carrying two spring-mass systems tuned based on Case 1(m1tmd= m2tmd=0.01mb)

Figure 4.16 Mode shapes of two-span beam carrying two spring-mass systems tuned based on Case 2(m1tmd= m2tmd=0.01mb)

0 5 10 15 20 25 30

-0.005

0.000

0.005

0 5 10 15 20 25 30

-0.005

0.000

0.005

1st mode

4th mode

3rd mode

2nd mode

5th mode

6th mode

Length (m)

Mod

al D

isp

lace

men

t

1st mode

4th mode

3rd mode

2nd mode

5th mode

6th mode

Length (m)

Mod

al D

isp

lace

men

t

109

4.4.3. Free Vibration Analysis of Three Span Beam Carrying

One Spring-Mass Systems

Case 1: Spring-mass system is attached to first span

Figure 4.17 Three-span beam carrying one spring-mass system attached to first span

Table 4.29 The lowest six natural frequencies of the three-span beam carrying one spring-mass system based on case 1

m1/mb=

m2/mb ω1 ω 2 ω 3 ω 4 ω 5 ω 6

ω tmd1

x1=1/6 L

0.01 18.30 21.00 26.03 37.24 79.42 90.52 19.86

0.02 17.62 21.30 26.58 37.33 79.42 90.52 19.86

0.05 16.28 21.66 28.03 37.63 79.42 90.53 19.86

0.1 14.87 21.86 29.83 38.26 79.42 90.55 19.86

0 19.86 25.45 37.16 79.42 90.54 110.30 0.00

110

Case 2: Spring-mass system is attached to second span

Figure 4.18 Three-span beam carrying one spring-mass system attached to second span

Table 4.30 The lowest six natural frequencies of the three-span beam carrying one spring-mass system based on case 2

m1/mb=

m2/mb ω1 ω 2 ω 3 ω 4 ω 5 ω 6

ω tmd1

x1=1/6 L

0.01 18.42 21.20 25.45 37.48 79.42 90.53 19.86

0.02 17.83 21.72 25.45 37.81 79.42 90.54 19.86

0.05 16.66 22.63 25.45 38.77 79.42 90.58 19.86

0.1 15.40 23.47 25.45 40.34 79.42 90.65 19.86

0 19.86 25.45 37.16 79.42 90.54 110.30 0.00

111

Figure 4.19 Mode shapes of three-span beam carrying one spring-mass system at first span (m1tmd=0.01mb)

Figure 4.20 Mode shapes of three-span beam carrying one spring-mass system at second span (m1tmd=0.01mb)

0 10 20 30 40 50

-0.005

0.000

0.005

0 10 20 30 40 50-0.010

-0.005

0.000

0.005

0.010

1st mode

4th mode

3rd mode

2nd mode

5th mode

6th mode

Length (m)

Mod

al D

isp

lace

men

t

1st mode

4th mode

3rd mode

2nd mode

5th mode

6th mode

Length (m)

Mod

al D

isp

lace

men

t

112

4.4.4. Free Vibration Analysis of Three Span Beam Carrying


Case 1: Spring-mass systems are attached to first and second span. Both of the

spring-mass systems are tuned to constant frequency which is same with the first

natural frequency of three-span bare uniform beam.

Figure 4.21 Three-span beam carrying two spring-mass systems attached to first and second span (Case 1)

Table 4.31 The lowest six natural frequencies of the three-span beam carrying two spring-mass systems based on case 1

m1/mb=

m2/mb ω1 ω 2 ω 3 ω 4 ω 5 ω 6

ω tmd1

x1=1/6 L

ω tmd2

x2=3/6 L

0.01 17.88 19.45 21.75 26.04 37.56 79.42 19.86 19.86

0.02 17.09 19.09 22.42 26.61 37.96 79.42 19.86 19.86

0.05 15.61 18.21 23.45 28.26 39.11 79.42 19.86 19.86

0.1 14.11 17.09 24.20 30.70 40.91 79.42 19.86 19.86

0 19.86 25.45 37.16 79.42 90.54 110.30 0.00 0.00

113

Case 2: Spring-mass systems are attached to first and second span. The first

spring-mass system is tuned to the first natural frequency of the three-span bare

uniform beam and the second spring-mass system is tuned to the second natural

frequency of the three-span beam carrying one spring-mass system which is

attached to first span of the beam.



m1/mb=

m2/mb ω1 ω 2 ω 3 ω 4 ω 5 ω 6

ω tmd1

x1=1/6 L

ω tmd2

x2=3/6 L

0.01 18.02 19.93 22.23 26.04 37.62 79.42 19.86 21.00

0.02 17.25 19.66 23.02 26.63 38.10 79.42 19.86 21.30

0.05 15.78 18.84 24.12 28.36 39.52 79.42 19.86 21.66

0.1 14.26 17.69 24.78 30.94 41.71 79.42 19.86 21.86

0 19.86 25.45 37.16 79.42 90.54 110.30 0.00 0.00

114

Case 3: Spring-mass systems are attached to first and second span. The first




attached to second span of the beam.



m1/mb=

m2/mb ω1 ω 2 ω 3 ω 4 ω 5 ω 6

ω tmd1

x1=1/6 L

ω tmd2

x2=3/6 L

0.01 18.04 20.00 22.34 26.04 37.63 79.42 19.86 21.20

0.02 17.28 19.80 23.24 26.63 38.14 79.42 19.86 21.72

0.05 15.83 19.12 24.53 28.43 39.77 79.42 19.86 22.63

0.1 14.33 18.07 25.25 31.16 42.47 79.42 19.86 23.47

0 19.86 25.45 37.16 79.42 90.54 110.30 0.00 0.00

115

Case 4: Spring-mass systems are attached to first and third span. Both of the



Figure 4.24 Three-span beam carrying two spring-mass systems attached to first and third span (Case 4)


m1/mb=

m2/mb ω1 ω 2 ω 3 ω 4 ω 5 ω 6

ω tmd1

x1=1/6 L

ω tmd2

x2=5/6 L

0.01 17.93 19.06 21.88 26.50 37.32 79.43 19.86 19.86

0.02 17.18 18.45 22.72 27.36 37.49 79.44 19.86 19.86

0.05 15.77 17.13 24.36 29.43 38.03 79.46 19.86 19.86

0.1 14.34 15.66 26.07 32.10 39.02 79.41 19.86 19.86

0 19.86 25.45 37.16 79.42 90.54 110.30 0.00 0.00

116

Case 5: Spring-mass systems are attached to first and third span. The first




attached to first span of the beam.

Figure 4.25 Three-span beam carrying two spring-mass systems attached to first and third span (Case 5)


m1/mb=

m2/mb ω1 ω 2 ω 3 ω 4 ω 5 ω 6

ω tmd1

x1=1/6 L

ω tmd2

x2=5/6 L

0.01 18.08 19.51 22.28 26.64 37.34 79.31 19.86 21.00

0.02 17.35 18.95 23.21 27.66 37.53 79.39 19.86 21.30

0.05 15.96 17.61 24.94 30.03 38.17 79.32 19.86 21.66

0.1 14.51 16.07 26.66 32.97 39.37 79.41 19.86 21.86

0 19.86 25.45 37.16 79.42 90.54 110.30 0.00 0.00

117

Figure 4.26 Mode shapes of three-span beam carrying two spring-mass systems at first span and second span tuned based on Case 1 (m1tmd=m2tmd=0.01mb)

Figure 4.27 Mode shapes of three-span beam carrying two spring-mass systems at first and second span tuned based on Case 2 (m1tmd=m2tmd=0.01mb)

0 10 20 30 40 50

-0.005

0.000

0.005

0 10 20 30 40 50

-0.005

0.000

0.005

1st mode

4th mode

3rd mode

2nd mode

5th mode

6th mode

Length (m)

Mod

al D

isp

lace

men

t

1st mode

4th mode

3rd mode

2nd mode

5th mode

6th mode

Length (m)

Mod

al D

isp

lace

men

t

118

Figure 4.28 Mode shapes of three-span beam carrying two spring-mass systems at first span and second span tuned based on Case 3 (m1tmd=m2tmd=0.01mb)

Figure 4.29 Mode shapes of three-span beam carrying two spring-mass systems at first and third span tuned based on Case 4 (m1tmd=m2tmd=0.01mb)

0 10 20 30 40 50

-0.005

0.000

0.005

0 10 20 30 40 50

-0.010

-0.005

0.000

0.005

1st mode

4th mode

3rd mode

2nd mode

5th mode

6th mode

Length (m)

Mod

al D

isp

lace

men

t

1st mode

4th mode

3rd mode

2nd mode

5th mode

6th mode

Length (m)

Mod

al D

isp

lace

men

t

119

Figure 4.30 Mode shapes of three-span beam carrying two spring-mass systems at first span and third span tuned based on Case 5 (m1tmd=m2tmd=0.01mb)

0 10 20 30 40 50

-0.005

0.000

0.005

1st mode

4th mode

3rd mode

2nd mode

5th mode

6th mode

Length (m)

Mod

al D

isp

lace

men

t

120

4.5. Forced Vibration Analysis of Single Span Uniform Beam Carrying One, Two

and Three Spring-Mass Systems

F0 is assumed to be 350 N for harmonic loading and 700 N for impact

loading, moving load and moving pulsating loads.

4.5.1. Impact Loading

Case 1: Step-function force is applied to x=0.5L for SS, CC and CS bare uniform

beams and x=L for CF uniform beam.

Table 4.36 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam without any spring mass system - Case 1

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 4.8x10-4 5.4x10-3 0.28 65 2.97x10-4 3.39x10-3 0.115 35.19

CC 1.2x10-4 3.1x10-3 0.3 82 7.36x10-5 1.89x10-3 0.126 43.23

CS 2.1x10-4 3.6x10-3 0.27 75 1.29x10-4 2.26x10-3 0.108 34.24

CF 3.8x10-3 1.6x10-2 0.48 130 2.39x10-3 9.50x10-3 0.164 52.67

Case 2: Step-function force is applied to x=0.25L for SS, CC and CS bare

uniform beams and x=0.5L for CF uniform beam.


Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 3.3x10-4 3.6x10-3 0.19 45 2.06x10-4 2.39x10-3 0.0816 24.88

CC 6.3x10-5 1.8x10-3 0.23 78 3.78x10-5 1.05x10-3 0.0884 34.03

CS 9.8x10-5 2.2x10-3 0.20 53 5.77x10-5 1.08x10-3 0.0798 21.41

CF 2.5x10-3 1.2x10-2 0.45 120 1.54x10-3 6.72x10-3 0.171 62.01

121

Case 3: Step-function force is applied to x=0.25L and one spring-mass system is

attached to x=0.5L for SS, CC and CS beams. For CF beam, the step-function

force is applied to x=0.5L and one spring-mass system is attached to x=L.

Figure 4.31 Simply supported beam carrying one spring-mass system subjected to step-function force at x=0.25L

Table 4.38 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under impact loading -

Case 3 (m1/mb=0.01)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 3.3x10-4 3.8x10-3 0.19 45 1.89x10-4 1.76x10-3 0.075 24.87

CC 6.1x10-5 1.7x10-3 0.18 45 3.42x10-5 7.56x10-4 0.076 22.33

CS 9.8x10-5 2.0x10-3 0.19 36 5.27x10-5 8.10x10-4 0.074 15.92

CF 2.5x10-3 1.2x10-2 0.34 45 1.37x10-3 5.16x10-3 0.154 28.50

122


Case 3 (m1/mb=0.02)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 3.3x10-4 3.8x10-3 0.19 45 1.87x10-4 1.73x10-3 0.075 24.87

CC 6.1x10-5 1.8x10-3 0.20 47 3.41x10-5 7.73x10-4 0.082 23.47

CS 9.8x10-5 2.1x10-3 0.19 36 5.25x10-5 8.16x10-4 0.074 15.86

CF 2.5x10-3 1.2x10-2 0.32 38 1.38x10-3 4.88x10-3 0.138 23.32


Case 3 (m1/mb=0.05)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 3.3x10-4 3.8x10-3 0.19 45 1.87x10-4 1.68x10-3 0.075 24.88

CC 6.3x10-5 1.8x10-3 0.19 48 3.42x10-5 7.62x10-4 0.082 23.47

CS 9.8x10-5 2.1x10-3 0.19 38 5.29x10-5 8.07x10-4 0.075 15.95

CF 2.5x10-3 1.2x10-2 0.32 38 1.38x10-3 4.92x10-3 0.138 23.37


Case 3 (m1/mb=0.1)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 3.3x10-4 3.6x10-3 0.19 45 1.89x10-4 1.66x10-3 0.075 24.90

CC 6.3x10-5 1.8x10-3 0.195 48 3.43x10-5 7.56x10-4 0.083 23.57

CS 9.8x10-5 2.1x10-3 0.19 38 5.32x10-5 8.08x10-4 0.077 16.56

CF 2.5x10-3 1.2x10-2 0.32 38 1.37x10-3 4.83x10-3 0.138 23.34

123


Case 3 (m1/mb=0.2)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 3.3x10-4 3.5x10-3 0.19 45 1.89x10-4 1.62x10-3 0.075 24.90

CC 4.8x10-5 1.6x10-3 0.19 46 3.45x10-5 7.32x10-4 0.078 22.58

CS 9.0x10-5 1.9x10-3 0.19 40 5.33x10-5 7.85x10-4 0.078 16.55

CF 2.5x10-3 1.2x10-2 0.32 38 1.43x10-3 4.65x10-3 0.139 23.37

Case 4: Step-function force is applied to x=0.5L and one spring-mass system is

attached to x=0.5L for SS, CC and CS beams. For CF beam, the step-function

force is applied to x=L and one spring-mass system is attached to x=L.

Figure 4.32 Simply supported beam carrying one spring-mass system subjected to step-function force at x=0.5L

124


Case 4 (m1/mb=0.01)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 4.8x10-4 5.4x10-3 0.27 63 2.74x10-4 2.49x10-3 0.106 35.18

CC 1.2x10-4 3.0x10-3 0.30 85 6.72x10-5 1.33x10-3 0.113 45.56

CS 2.1x10-4 3.6x10-3 0.27 70 1.18x10-4 1.60x10-3 0.096 34.47

CF 3.8x10-3 1.6x10-2 0.47 90 2.15x10-3 7.08x10-3 0.166 38.79


Case 4 (m1/mb=0.02)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 4.8x10-4 5.4x10-3 0.27 63 2.70x10-4 2.45x10-3 0.106 35.19

CC 1.2x10-4 3.1x10-3 0.31 86 6.75x10-5 1.35x10-3 0.118 46.18

CS 2.1x10-4 3.6x10-3 0.26 65 1.17x10-4 1.61x10-3 0.095 33.61

CF 3.6x10-3 1.5x10-2 0.40 73 2.17x10-3 6.60x10-3 0.141 32.54


Case 4 (m1/mb=0.05)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 4.8x10-4 5.0x10-3 0.27 63 2.71x10-4 2.37x10-3 0.106 35.21

CC 1.2x10-4 3.0x10-3 0.31 88 6.77x10-5 1.33x10-3 0.118 46.20

CS 2.1x10-4 3.6x10-3 0.27 65 1.18x10-4 1.59x10-3 0.095 33.75

CF 3.8x10-3 1.5x10-2 0.42 80 2.17x10-3 6.66x10-3 0.145 34.96

125


Case 4 (m1/mb=0.1)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 4.8x10-4 5.0x10-3 0.27 63 2.74x10-4 2.34x10-3 0.106 35.23

CC 1.2x10-4 3.0x10-3 0.30 88 6.80x10-5 1.31x10-3 0.118 46.43

CS 2.1x10-4 3.5x10-3 0.27 68 1.19x10-4 1.58x10-3 0.098 34.11

CF 3.8x10-3 1.4x10-2 0.45 75 2.16x10-3 6.45x10-3 0.142 32.61


Case 4 (m1/mb=0.2)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 4.8x10-4 5.0x10-3 0.27 63 2.74x10-4 2.27x10-3 0.107 35.43

CC 9.0x10-5 2.6x10-3 0.30 85 6.79x10-5 1.26x10-3 0.115 46.13

CS 2.1x10-4 3.6x10-3 0.26 70 1.19x10-4 1.52x10-3 0.097 33.78

CF 3.8x10-3 1.3x10-2 0.40 75 2.25x10-3 6.07x10-3 0.144 32.80

126

Case 5: Step-function force is applied to x=0.25L and two spring-mass systems

are attached to x=0.5L for SS, CC and CS beams. For CF beam, the step-

function force is applied to x=0.5L and two spring-mass systems are attached to

x=L.

Table 4.48 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under impact loading -

Case 5 (m1/mb=m2/mb=0.01)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 3.3x10-4 3.6x10-3 0.19 45 1.84x10-4 1.55x10-3 0.073 24.87

CC 6.0x10-5 1.7x10-3 0.19 48 3.37x10-5 7.24x10-4 0.081 23.26

CS 9.5x10-5 2.1x10-3 0.18 38 5.23x10-5 7.75x10-4 0.077 16.60

CF 2.4x10-3 1.2x10-2 0.32 38 1.36x10-3 4.53x10-3 0.136 23.32



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 3.3x10-4 3.6x10-3 0.19 45 1.87x10-4 1.58x10-3 0.073 24.88

CC 6.1x10-5 1.7x10-3 0.19 47 3.38x10-5 7.20x10-4 0.081 23.39

CS 9.7x10-5 2.0x10-3 0.18 38 5.30x10-5 7.85x10-4 0.078 16.62

CF 2.4x10-3 1.2x10-2 0.32 38 1.39x10-3 4.70x10-3 0.137 23.32

127



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 3.3x10-4 3.6x10-3 0.19 45 1.89x10-4 1.58x10-3 0.073 24.90

CC 6.2x10-5 1.7x10-3 0.18 47 3.42x10-5 7.11x10-4 0.081 23.57

CS 9.0x10-5 2.1x10-3 0.18 38 5.30x10-5 7.75x10-4 0.078 16.65

CF 2.0x10-3 1.1x10-2 0.32 38 1.35x10-3 4.62x10-3 0.138 23.35



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 3.1x10-4 3.4x10-3 0.19 45 1.92x10-4 1.54x10-3 0.073 24.93

CC 6.2x10-5 1.7x10-3 0.19 49 3.45x10-5 6.92x10-4 0.081 23.63

CS 9.5x10-5 1.9x10-3 0.20 40 5.33x10-5 7.61x10-4 0.078 16.70

CF 2.3x10-3 1.1x10-2 0.32 38 1.43x10-3 4.43x10-3 0.138 23.39



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 3.2x10-4 3.2x10-3 0.19 45 1.95x10-4 1.48x10-3 0.074 25.00

CC 6.3x10-5 1.5x10-3 0.19 49 3.51x10-5 6.54x10-4 0.079 23.45

CS 9.8x10-5 1.9x10-3 0.19 40 5.40x10-5 7.26x10-4 0.076 16.30

CF 2.3x10-3 9.0x10-3 0.32 38 1.38x10-3 4.14x10-3 0.139 23.49

128

Case 6: Step-function force is applied to x=0.5L and two spring-mass systems


function force is applied to x=L and two spring-mass systems are attached to

x=L.



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 4.8x10-4 5.4x10-3 0.27 63 2.67x10-4 2.19x10-3 0.103 35.18

CC 1.2x10-4 3.0x10-3 0.30 88 6.68x10-5 1.26x10-3 0.115 46.28

CS 2.0x10-4 3.6x10-3 0.26 70 1.17x10-4 1.48x10-3 0.096 34.71

CF 3.6x10-3 1.5x10-2 0.40 73 2.15x10-3 6.08x10-3 0.139 32.13



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 4.8x10-4 5.0x10-3 0.27 63 2.70x10-4 2.24x10-3 0.104 35.20

CC 1.2x10-4 3.0x10-3 0.31 85 6.71x10-5 1.25x10-3 0.115 46.29

CS 2.1x10-4 3.6x10-3 0.26 70 1.18x10-4 1.50x10-3 0.096 34.74

CF 3.8x10-3 1.5x10-2 0.40 74 2.19x10-3 6.34x10-3 0.140 32.48

129



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 4.7x10-4 5.0x10-3 0.27 63 2.73x10-4 2.23x10-3 0.104 35.24

CC 1.2x10-4 3.0x10-3 0.30 88 6.78x10-5 1.23x10-3 0.115 46.47

CS 2.0x10-4 3.4x10-3 0.26 70 1.18x10-4 1.48x10-3 0.096 34.85

CF 3.1x10-3 1.4x10-2 0.42 75 2.14x10-3 6.22x10-3 0.141 32.78



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 4.4x10-4 4.4x10-3 0.27 63 2.77x10-4 2.18x10-3 0.104 35.33

CC 1.2x10-4 2.6x10-3 0.30 90 6.85x10-5 1.19x10-3 0.116 46.97

CS 2.1x10-4 3.2x10-3 0.26 68 1.19x10-4 1.43x10-3 0.096 35.02

CF 3.5x10-3 1.3x10-2 0.40 75 2.26x10-3 5.83x10-3 0.142 32.87



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 4.6x10-4 4.4x10-3 0.27 63 2.83x10-4 2.07x10-3 0.105 35.54

CC 1.2x10-4 2.5x10-3 0.30 90 6.95x10-5 1.11x10-3 0.115 47.62

CS 2.0x10-4 2.8x10-3 0.25 70 1.20x10-4 1.34x10-3 0.095 35.19

CF 3.4x10-3 1.3x10-2 0.34 40 2.19x10-3 5.29x10-3 0.127 18.85

130

Case 7: Step-function force is applied to x=0.25L and three spring-mass systems


function force is applied to x=0.5L and three spring-mass systems are attached

to x=L.

Table 4.58 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under impact loading -

Case 7 (m1/mb=m2/mb=m3/mb=0.01)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 3.2x10-4 3.5x10-3 0.19 45 1.84x10-4 1.52x10-3 0.073 24.92

CC 6.8x10-5 1.8x10-3 0.19 50 3.81x10-5 7.99x10-4 0.086 24.60

CS 1.0x10-4 2.0x10-3 0.20 40 5.56x10-5 8.16x10-4 0.078 16.61

CF 2.4x10-3 1.2x10-2 0.32 38 1.38x10-3 4.62x10-3 0.137 23.37



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 3.1x10-4 3.5x10-3 0.19 45 1.87x10-4 1.52x10-3 0.073 24.87

CC 6.6x10-5 1.8x10-3 0.20 49 3.79x10-5 7.88x10-4 0.087 24.58

CS 1.0x10-4 2.0x10-3 0.20 40 5.60x10-5 8.08x10-4 0.078 16.65

CF 2.3x10-3 1.1x10-2 0.30 38 1.40x10-3 4.62x10-3 0.137 23.29

131



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 3.3x10-4 3.6x10-3 0.18 45 1.89x10-4 1.52x10-3 0.073 24.78

CC 7.0x10-5 1.8x10-3 0.20 50 3.89x10-5 7.66x10-4 0.082 24.34

CS 1.0x10-4 2.0x10-3 0.19 39 5.70x10-5 8.04x10-4 0.078 16.58

CF 2.3x10-3 1.0x10-2 0.31 40 1.39x10-3 4.41x10-3 0.138 23.69



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 3.3x10-4 3.2x10-3 0.18 45 1.92x10-4 1.47x10-3 0.073 25.21

CC 7.0x10-5 1.8x10-3 0.19 50 3.97x10-5 7.36x10-4 0.082 24.20

CS 1.0x10-4 1.9x10-3 0.19 40 5.84x10-5 7.89x10-4 0.078 16.94

CF 2.4x10-3 9.0x10-3 0.32 40 1.41x10-3 4.18x10-3 0.138 23.78



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 3.4x10-4 2.7x10-3 0.18 45 2.01x10-4 1.40x10-3 0.073 25.24

CC 7.0x10-5 1.6x10-3 0.20 52 4.12x10-5 6.87x10-4 0.082 24.94

CS 1.1x10-4 1.9x10-3 0.19 40 6.04x10-5 7.64x10-4 0.080 16.83

CF 2.5x10-3 1.0x10-2 0.32 40 1.53x10-3 3.96x10-3 0.141 23.62

132

Case 8: Step-function force is applied to x=0.5L and three spring-mass systems


function force is applied to x=L and three spring-mass systems are attached to

x=L.



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 4.6x10-4 5.0x10-3 0.20 40 2.66x10-4 2.15x10-3 0.082 20.15

CC 1.3x10-4 3.2x10-3 0.26 58 7.51x10-5 1.40x10-3 0.097 27.35

CS 2.1x10-4 3.5x10-3 0.25 55 1.24x10-4 1.58x10-3 0.087 26.28

CF 3.5x10-3 1.5x10-2 0.40 73 2.17x10-3 6.25x10-3 0.139 32.78



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 4.4x10-4 5.0x10-3 0.22 40 2.71x10-4 2.15x10-3 0.085 20.42

CC 1.3x10-4 3.0x10-3 0.25 55 7.47x10-5 1.37x10-3 0.096 27.32

CS 2.2x10-4 3.4x10-3 0.20 40 1.25x10-4 1.56x10-3 0.079 16.36

CF 3.6x10-3 1.4x10-2 0.38 73 2.19x10-3 6.22x10-3 0.137 33.50

133



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 4.6x10-4 4.5x10-3 0.22 40 2.74x10-4 2.16x10-3 0.085 20.45

CC 1.3x10-4 3.0x10-3 0.25 60 7.67x10-5 1.35x10-3 0.100 29.37

CS 2.2x10-4 3.3x10-3 0.24 56 1.27x10-4 1.54x10-3 0.086 26.24

CF 3.5x10-3 1.3x10-2 0.40 75 2.21x10-3 5.91x10-3 0.138 32.78



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 4.6x10-4 4.0x10-3 0.22 40 2.78x10-4 2.08x10-3 0.086 20.81

CC 1.3x10-4 2.8x10-3 0.26 60 7.83x10-5 1.29x10-3 0.101 29.88

CS 2.2x10-4 3.4x10-3 0.22 40 1.30x10-4 1.49x10-3 0.079 16.94

CF 3.8x10-3 1.2x10-2 0.34 45 2.24x10-3 5.50x10-3 0.134 22.56



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 4.8x10-4 4x10-3 0.22 42 2.91x10-4 1.96x10-3 0.087 21.19

CC 1.3x10-4 2.6x10-3 0.25 70 8.15x10-5 1.18x10-3 0.104 33.66

CS 2.2x10-4 3.0x10-3 0.23 60 1.34x10-4 1.38x10-3 0.087 27.20

CF 3.7x10-3 1.1x10-2 0.40 80 2.28x10-3 4.88x10-3 0.158 37.45

134

4.5.2. Harmonic Loading

Case 1: Harmonic force is applied to x=0.5L for SS, CC and CS bare uniform

beams and x=L for CF uniform beam.


Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 2.2x10-2 0.43 8.5 190 9.30x10-3 0.185 3.66 74.82 6.3π

CC 3.1x10-3 0.14 6.25 300 1.74x10-3 0.078 3.53 160.36 14.4π

CS 3.8x10-3 0.12 3.8 145 1.86x10-3 0.058 1.81 58.99 10π

CF 4.2x10-2 0.30 2.4 80 2.13x10-2 0.152 1.09 31.19 2.3π

Case 2: Harmonic force is applied to x=0.25L for SS, CC and CS bare uniform

beams and x=0.5L for CF uniform beam.


Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 1.5x10-2 0.30 6.0 135 6.58x10-3 0.131 2.58 52.91 6.3π

CC 5.7x10-3 0.26 11.5 550 3.19x10-3 0.144 6.50 294.28 14.4π

CS 1.8x10-3 0.05 1.75 75 8.36x10-4 0.026 0.815 27.61 10π

CF 2.9x10-2 0.20 1.6 75 1.45x10-2 0.104 0.747 37.87 2.3π

135

Case 3: Harmonic force is applied to x=0.25L and one spring-mass system is

attached to x=0.5L for SS, CC and CS beams. For CF beam, the harmonic force

is applied to x=0.5L and one spring-mass system is attached to x=L.

Figure 4.33 Simply supported beam carrying one spring-mass system subjected to harmonic force at x=0.25L

Table 4.70 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under harmonic loading -

Case 3 (m1/mb=0.01)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 6.0x10-4 1.2x10-2 0.28 26 3.00x10-4 5.97x10-3 0.124 12.70 6.3π

CC 1.1x10-4 5.0x10-3 0.28 32 5.06x10-5 2.32x10-3 0.113 12.43 14.4π

CS 1.8x10-4 6.0x10-3 0.24 23 8.43x10-5 2.69x10-3 0.094 8.58 10π

CF 3.8x10-3 2.8x10-2 0.34 24 1.73x10-3 1.30x10-3 0.123 14.29 2.3π

136


Case 3 (m1/mb=0.02)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 4.2x10-4 8.5x10-3 0.21 24 2.13x10-4 4.24x10-3 0.092 12.60 6.3π

CC 6.8x10-5 3.4x10-3 0.20 30 3.53x10-5 1.64x10-3 0.086 12.53 14.4π

CS 1.3x10-4 4.0x10-3 0.20 22 5.71x10-5 1.84x10-3 0.070 8.32 10π

CF 2.4x10-3 2.0x10-2 0.26 20 1.16x10-3 8.75x10-3 0.095 11.70 2.3π


Case 3 (m1/mb=0.05)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 2.6x10-4 5.5x10-3 0.17 24 1.33x10-4 2.69x10-3 0.066 12.54 6.3π

CC 4.5x10-5 2.2x10-3 0.15 28 2.22x10-5 1.06x10-3 0.065 12.25 14.4π

CS 7.2x10-5 2.6x10-3 0.14 20 3.53x10-5 1.16x10-3 0.053 8.24 10π

CF 1.4x10-3 1.2x10-2 0.20 20 7.01x10-4 5.54x10-3 0.081 11.71 2.3π


Case 3 (m1/mb=0.1)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 1.8x10-4 4.0x10-3 0.14 24 9.23x10-5 1.92x10-3 0.055 12.52 6.3π

CC 3.0x10-5 1.6x10-3 0.14 26 1.54x10-5 7.67x10-4 0.056 12.21 14.4π

CS 5.0x10-5 2.0x10-3 0.13 20 2.43x10-5 8.31x10-4 0.048 8.50 10π

CF 1.0x10-3 9.0x10-3 0.18 20 4.82x10-4 4.07x10-3 0.077 11.70 2.3π

137


Case 3 (m1/mb=0.2)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 1.3x10-4 2.8x10-3 0.12 23 6.41x10-5 1.40x10-3 0.048 12.51 6.3π

CC 1.4x10-5 1.2x10-3 0.12 26 1.07x10-5 5.68x10-4 0.049 11.65 14.4π

CS 3.6x10-5 1.5x10-3 0.12 20 1.66x10-5 6.06x10-4 0.044 8.49 10π

CF 7.0x10-4 6.5x10-3 0.18 20 3.30x10-4 3.12x10-3 0.075 11.71 2.3π

Case 4: Harmonic force is applied to x=0.5L and one spring-mass system is


is applied to x=L and one spring-mass system is attached to x=L.

Figure 4.34 Simply supported beam carrying one spring-mass system subjected to harmonic force at x=0.5L

138


Case 4 (m1/mb=0.01)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 8.5x10-4 1.7x10-2 0.40 36 4.26x10-4 8.44x10-3 0.175 17.97 6.3π

CC 2.0x10-4 9.0x10-3 0.45 58 9.29x10-5 4.25x10-3 0.202 24.72 14.4π

CS 4.0x10-4 1.3x10-2 0.45 45 1.87x10-4 5.97x10-3 0.196 18.40 10π

CF 5.5x10-3 4.0x10-2 0.45 45 2.53x10-3 1.88x10-2 0.162 19.46 2.3π


Case 4 (m1/mb=0.02)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 6.0x10-4 1.2x10-2 0.30 34 3.00x10-4 5.99x10-3 0.130 17.82 6.3π

CC 1.3x10-4 6.0x10-3 0.36 55 6.47x10-5 2.98x10-3 0.149 24.23 14.4π

CS 2.8x10-4 9.0x10-3 0.34 40 1.27x10-4 4.07x10-3 0.139 17.45 10π

CF 3.6x10-3 2.8x10-2 0.34 37 1.68x10-3 1.26x10-2 0.118 16.31 2.3π


Case 4 (m1/mb=0.05)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 3.8x10-4 7.8x10-3 0.24 34 1.87x10-4 3.79x10-3 0.093 17.74 6.3π

CC 8.0x10-5 4.0x10-3 0.26 50 4.05x10-5 1.89x10-3 0.106 23.76 14.4π

CS 1.6x10-4 5.0x10-3 0.24 34 7.83x10-5 2.54x10-3 0.095 17.21 10π

CF 2.0x10-3 1.6x10-2 0.26 40 1.02x10-3 7.80x10-3 0.094 17.51 2.3π

139


Case 4 (m1/mb=0.1)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 2.8x10-4 5.5x10-3 0.20 33 1.30x10-4 2.70x10-3 0.077 17.72 6.3π

CC 5.5x10-5 2.8x10-3 0.22 48 2.81x10-5 1.35x10-3 0.088 23.72 14.4π

CS 1.1x10-4 3.6x10-3 0.20 36 5.39x10-5 1.80x10-3 0.077 17.30 10π

CF 1.4x10-3 1.2x10-2 0.23 38 6.91x10-4 5.54x10-3 0.083 16.32 2.3π


Case 4 (m1/mb=0.2)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 1.8x10-4 4.0x10-3 0.17 33 9.00x10-5 1.95x10-3 0.067 17.72 6.3π

CC 2.6x10-5 2.0x10-3 0.19 47 1.91x10-5 9.65x10-4 0.075 23.48 14.4π

CS 7.5x10-5 2.6x10-3 0.17 34 3.68x10-5 1.27x10-3 0.065 17.09 10π

CF 9.5x10-4 8.5x10-3 0.24 38 4.65x10-4 4.00x10-3 0.079 16.42 2.3π

140

Case 5: Harmonic force is applied to x=0.25L and two spring-mass systems are


is applied to x=0.5L and two spring-mass systems are attached to x=L.

Table 4.80 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under harmonic loading -


Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 5.0x10-4 1.0x10-2 0.26 25 2.22x10-4 4.26x10-3 0.089 12.58 6.3π

CC 9.5x10-5 4.5x10-3 0.25 30 3.81x10-5 1.69x10-3 0.085 12.36 14.4π

CS 1.8x10-4 5.5x10-3 0.22 22 7.32x10-5 2.28x10-3 0.081 8.76 10π

CF 3.5x10-3 2.6x10-2 0.30 20 1.55x10-3 1.11x10-2 0.105 11.70 2.3π



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 3.4x10-4 6.5x10-3 0.18 24 1.56x10-4 2.96x10-3 0.067 12.53 6.3π

CC 5.5x10-5 2.8x10-3 0.17 28 2.62x10-5 1.16x10-3 0.065 12.19 14.4π

CS 1.2x10-4 3.6x10-3 0.17 20 4.50x10-5 1.39x10-3 0.058 8.59 10π

CF 2.0x10-3 1.7x10-2 0.22 20 9.32x10-4 6.74x10-3 0.084 11.69 2.3π

141



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 2.2x10-4 4.5x10-3 0.15 24 9.77x10-5 1.85x10-3 0.051 12.51 6.3π

CC 3.6x10-5 1.8x10-3 0.14 27 1.60x10-5 7.18x10-4 0.052 12.15 14.4π

CS 6.5x10-5 2.3x10-3 0.13 20 2.64x10-5 8.26x10-4 0.047 8.54 10π

CF 1.1x10-3 1.0x10-2 0.20 20 5.28x10-4 3.94x10-3 0.075 11.70 2.3π



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 1.5x10-4 3.0x10-3 0.12 23 6.69x10-5 1.27x10-3 0.044 12.52 6.3π

CC 2.6x10-5 1.3x10-3 0.11 16 1.09x10-5 5.06x10-4 0.045 9.19 14.4π

CS 4.0x10-5 1.7x10-3 0.12 20 1.78x10-5 5.76x10-4 0.043 8.55 10π

CF 8.0x10-4 7.0x10-3 0.17 20 3.35x10-4 2.73x10-3 0.072 11.72 2.3π



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 1.0x10-4 2.2x10-3 0.11 22 4.51x10-5 8.86x10-4 0.041 12.55 6.3π

CC 1.7x10-5 9.8x10-4 0.09 15 7.22x10-6 3.65x10-4 0.041 8.85 14.4π

CS 2.8x10-5 1.3x10-3 0.10 20 1.17x10-5 4.14x10-4 0.041 8.34 10π

CF 5.0x10-4 5.5x10-3 0.17 20 2.18x10-4 2.06x10-3 0.072 11.77 2.3π

142

Case 6: Harmonic force is applied to x=0.5L and two spring-mass systems are


is applied to x=L and two spring-mass systems are attached to x=L.



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 6.3x10-4 1.4x10-2 0.36 36 3.15x10-4 6.03x10-3 0.126 17.79 6.3π

CC 1.7x10-4 7.5x10-3 0.40 55 7.03x10-5 3.10x10-3 0.148 24.19 14.4π

CS 4.0x10-4 1.2x10-3 0.40 43 1.63x10-4 5.05x10-3 0.164 18.16 10π

CF 5.0x10-3 3.8x10-2 0.40 37 2.27x10-3 1.62x10-2 0.135 16.10 2.3π



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 4.5x10-4 9.5x10-3 0.26 34 2.21x10-4 4.19x10-3 0.095 17.73 6.3π

CC 1.0x10-4 5.0x10-3 0.28 48 4.85x10-5 2.12x10-3 0.109 23.77 14.4π

CS 2.6x10-4 8.0x10-3 0.32 37 9.98x10-5 3.06x10-3 0.106 17.74 10π

CF 2.9x10-3 2.2x10-2 0.29 37 1.37x10-3 9.75x10-3 0.098 16.26 2.3π

143



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 3.0x10-4 6.0x10-3 0.20 34 1.39x10-4 2.61x10-3 0.071 17.71 6.3π

CC 7.0x10-5 3.2x10-3 0.24 46 2.97x10-5 1.29x10-3 0.080 23.65 14.4π

CS 1.4x10-4 4.5x10-3 0.22 36 5.85x10-5 1.78x10-3 0.072 17.62 10π

CF 1.7x10-3 1.4x10-2 0.27 38 7.76x10-4 5.49x10-3 0.080 16.40 2.3π



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 2.0x10-4 4.5x10-3 0.17 33 9.49x10-5 1.79x10-3 0.062 17.74 6.3π

CC 4.5x10-5 2.2x10-3 0.15 20 2.01x10-5 8.74x10-4 0.059 10.25 14.4π

CS 9.5x10-5 3.1x10-3 0.17 36 3.94x10-5 1.19x10-3 0.060 17.67 10π

CF 1.2x10-3 8.5x10-3 0.22 38 4.91x10-4 3.54x10-3 0.075 16.44 2.3π



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 1.5x10-4 3.1x10-3 0.15 33 6.39x10-5 1.23x10-3 0.058 17.83 6.3π

CC 3.0x10-5 1.5x10-3 0.13 18 1.33x10-5 5.87x10-4 0.052 9.93 14.4π

CS 6.0x10-5 2.2x10-3 0.15 35 2.59x10-5 7.88x10-4 0.053 17.74 10π

CF 7.0x10-4 6.0x10-3 0.19 20 3.16x10-4 2.37x10-3 0.066 9.45 2.3π

144

Case 7: Harmonic force is applied to x=0.25L and three spring-mass systems are


is applied to x=0.5L and three spring-mass systems are attached to x=L.

Table 4.90 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under harmonic loading -


Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 4.2x10-4 9.0x10-3 0.20 23 1.99x10-4 3.75x10-3 0.079 12.57 6.3π

CC 8.5x10-5 4.0x10-3 0.23 30 3.78x10-5 1.63x10-3 0.082 12.96 14.4π

CS 1.8x10-4 6.0x10-3 0.24 22 7.31x10-5 2.26x10-3 0.080 8.75 10π

CF 3.9x10-3 2.8x10-2 0.32 20 1.42x10-3 1.01x10-2 0.099 11.71 2.3π



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 3.2x10-4 6.0x10-3 0.17 24 1.38x10-4 2.56x10-3 0.059 12.50 6.3π

CC 6.0x10-5 2.8x10-3 0.18 28 2.56x10-5 1.10x10-3 0.064 12.75 14.4π

CS 1.1x10-4 3.8x10-3 0.17 22 4.39x10-5 1.34x10-3 0.056 8.59 10π

CF 2.1x10-3 1.6x10-2 0.24 20 9.48x10-4 6.74x10-3 0.084 11.67 2.3π

145



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 1.8x10-4 3.8x10-3 0.13 23 8.43x10-5 1.53x10-3 0.045 12.44 6.3π

CC 3.4x10-5 1.7x10-3 0.14 28 1.54x10-5 6.54x10-4 0.049 12.51 14.4π

CS 6.0x10-5 2.0x10-3 0.13 20 2.51x10-5 7.59x10-4 0.045 8.49 10π

CF 1.2x10-3 9.5x10-3 0.19 20 4.56x10-4 3.34x10-3 0.073 11.86 2.3π



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 1.2x10-4 2.8x10-3 0.11 23 5.73x10-5 1.03x10-3 0.041 12.65 6.3π

CC 2.3x10-5 1.2x10-3 0.12 26 1.03x10-5 4.45x10-4 0.045 12.41 14.4π

CS 4.0x10-5 1.6x10-3 0.11 20 1.65x10-5 5.14x10-4 0.042 8.66 10π

CF 7.2x10-4 7.0x10-3 0.17 20 3.03x10-4 2.39x10-3 0.072 11.91 2.3π



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 8.0x10-5 1.7x10-3 0.10 23 3.76x10-5 6.90x10-4 0.039 12.66 6.3π

CC 1.5x10-5 9.0x10-4 0.10 28 6.62x10-6 3.16x10-4 0.044 12.77 14.4π

CS 2.6x10-5 1.1x10-3 0.10 20 1.08x10-5 3.78x10-4 0.042 8.61 10π

CF 4.5x10-4 4.5x10-3 0.17 20 1.84x10-4 1.74x10-3 0.072 11.83 2.3π

146

Case 8: Harmonic force is applied to x=0.5L and three spring-mass systems are


is applied to x=L and three spring-mass systems are attached to x=L.



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 6.0x10-4 1.2x10-2 0.28 22 2.83x10-4 5.31x10-3 0.108 10.41 6.3π

CC 1.5x10-4 7.2x10-3 0.36 38 7.01x10-5 3.01x10-3 0.137 15.05 14.4π

CS 4.0x10-4 1.3x10-2 0.45 38 1.62x10-3 5.00x10-3 0.160 14.09 10π

CF 5.5x10-3 4.0x10-2 0.40 36 2.08x10-3 1.47x10-2 0.125 16.41 2.3π



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 4.5x10-4 9.0x10-3 0.22 22 1.96x10-4 3.63x10-3 0.079 10.35 6.3π

CC 1.1x10-4 4.6x10-3 0.26 35 4.77x10-5 2.00x10-3 0.096 14.41 14.4π

CS 2.4x10-4 8.5x10-3 0.30 24 9.76x10-5 2.95x10-3 0.097 8.77 10π

CF 3.0x10-3 2.2x10-2 0.30 40 1.37x10-3 9.57x10-3 0.096 16.76 2.3π

147



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 2.6x10-4 5.2x10-3 0.17 21 1.20x10-4 2.17x10-3 0.057 10.31 6.3π

CC 6.0x10-5 2.8x10-3 0.21 34 2.88x10-5 1.17x10-3 0.068 15.14 14.4π

CS 1.3x10-4 3.8x10-3 0.14 30 5.57x10-5 1.62x10-3 0.063 13.32 10π

CF 1.7x10-3 1.2x10-2 0.24 36 6.77x10-4 4.61x10-3 0.077 16.40 2.3π



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 1.7x10-4 3.8x10-3 0.14 21 8.16x10-5 1.45x10-3 0.050 10.48 6.3π

CC 4.0x10-5 2.0x10-3 0.17 34 1.92x10-5 7.66x10-4 0.058 15.32 14.4π

CS 8.0x10-5 2.8x10-3 0.14 20 3.66x10-5 1.04x10-3 0.049 8.68 10π

CF 1.0x10-3 8.0x10-3 0.18 24 4.43x10-4 3.02x10-3 0.070 11.31 2.3π



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 1.2x10-4 2.2x10-3 0.13 22 5.35x10-5 9.49x10-4 0.047 10.67 6.3π

CC 2.6x10-5 1.3x10-3 0.15 36 1.24x10-5 4.92x10-4 0.055 17.18 14.4π

CS 5.0x10-5 1.9x10-3 0.13 30 2.36x10-5 6.51x10-4 0.047 13.74 10π

CF 6.0x10-4 5.5x10-3 0.23 40 2.62x10-4 1.98x10-3 0.080 18.74 2.3π

148

Case 9: Harmonic force is applied to x=0.5L and one spring-mass systems is

attached to x=0.5L for SS beam. Forcing frequency is different from the resonant

frequency of the beam.

Table 4.100 Maximum and RMS responses at x=0.5L for SS beam carrying one spring-mass system under harmonic loading - Case 9

m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 1.5x10-4 4.0x10-3 0.19 34 7.33x10-5 1.94x10-3 0.074 17.80 10π

0.02 1.4x10-4 3.8x10-3 0.18 33 6.88x10-5 1.83x10-3 0.072 17.81 10π

0.05 1.2x10-4 3.2x10-3 0.17 34 5.85x10-5 1.59x10-3 0.068 17.82 10π

0.10 9.5x10-5 2.8x10-3 0.16 33 4.82x10-5 1.35x10-3 0.065 17.78 10π

0.20 7.2x10-5 2.2x10-3 0.15 34 3.66x10-5 1.09x10-3 0.062 17.95 10π

No TVA 1.5x10-4 4.0x10-3 0.20 34 6.35x10-5 1.82x10-3 0.062 16.59 10π

4.5.3. Moving Load

Case 1: The velocity of moving load is 1.333 m/sec and applied to SS, CC, CS

and CF bare uniform beams.


Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 2.4x10-4 1.2x10-4 1.2x10-3 0.024 1.65x10-4 6.15x10-5 8.53x10-4 0.0169

CC 5.0x10-5 2.6x10-5 6.0x10-4 0.033 3.54x10-5 1.32x10-5 4.16x10-4 0.0190

CS 1.0x10-4 5.0x10-5 7.5x10-4 0.028 6.78x10-5 2.52x10-5 5.44x10-4 0.0170

CF 1.8x10-3 8.8x10-4 3.8x10-3 0.053 1.19x10-3 4.49x10-4 2.28x10-3 0.0239

149

Case 2: The velocity of moving load is 1.333 m/sec and one spring-mass system

is attached to x=0.5L for SS, CC and CS beams and x=L for CF beam.

Figure 4.35 Simply supported beam carrying one spring-mass system subjected to moving load

Table 4.102 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under moving load -Case 2

(m1/mb=0.01)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 2.4x10-4 1.2x10-4 1.2x10-3 0.024 1.65x10-4 5.34x10-5 5.99x10-4 0.0120

CC 5.0x10-5 2.6x10-5 6x10-4 0.030 3.54x10-5 1.15x10-5 2.94x10-4 0.0136

CS 1.0x10-4 5.0x10-5 7.8x10-4 0.028 6.78x10-5 2.19x10-5 3.84x10-4 0.0121

CF 1.75x10-3 8.5x10-4 3.6x10-3 0.05 1.19x10-3 3.88x10-4 1.63x10-3 0.0211

150

Table 4.103 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under moving load–Case 2

(m1/mb=0.02)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 2.4x10-4 1.2x10-4 1.2x10-3 0.024 1.65x10-4 5.36x10-5 6.01x10-4 0.0121

CC 5.0x10-5 2.6x10-5 6x10-4 0.030 3.54x10-5 1.15x10-5 2.93x10-4 0.0136

CS 1.0x10-4 4.8x10-5 7.8x10-4 0.027 6.78x10-5 2.20x10-5 3.83x10-4 0.0121

CF 1.8x10-3 8.5x10-4 3.7x10-3 0.05 1.19x10-3 3.89x10-4 1.61x10-3 0.0211


(m1/mb=0.05)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 2.4x10-4 1.2x10-4 1.2x10-3 0.024 1.65x10-4 5.36x10-5 5.95x10-4 0.0121

CC 5.0x10-5 2.6x10-5 6x10-4 0.030 3.54x10-5 1.15x10-5 2.90x10-4 0.0137

CS 1.0x10-4 4.8x10-5 7.8x10-4 0.027 6.78x10-5 2.20x10-5 3.79x10-4 0.0122

CF 1.8x10-3 8.5x10-4 3.6x10-3 0.05 1.19x10-3 3.93x10-4 1.60x10-3 0.0213


(m1/mb=0.1)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 2.4x10-4 1.2x10-4 1.2x10-3 0.024 1.65x10-4 5.39x10-5 5.88x10-4 0.0122

CC 5.0x10-5 2.6x10-5 6x10-4 0.030 3.54x10-5 1.16x10-5 2.85x10-4 0.0138

CS 1.0x10-4 4.8x10-5 7.5x10-4 0.027 6.78x10-5 2.21x10-5 3.73x10-4 0.0122

CF 1.75x10-3 8.0x10-4 3.6x10-3 0.05 1.19x10-3 3.94x10-4 1.57x10-3 0.0214

151


(m1/mb=0.2)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 2.4x10-4 1.2x10-4 1.2x10-3 0.023 1.65x10-4 5.42x10-5 5.73x10-4 0.0124

CC 5.0x10-5 2.6x10-5 5.5x10-4 0.030 3.54x10-5 1.17x10-5 2.75x10-4 0.0139

CS 1.0x10-4 4.8x10-5 7.5x10-4 0.027 6.78x10-5 2.23x10-5 3.63x10-4 0.0123

CF 1.8x10-3 8.5x10-4 3.4x10-3 0.05 1.19x10-3 3.99x10-4 1.50x10-3 0.0215

Case 3: The velocity of moving load is 1.333 m/sec and two spring-mass

systems are attached to x=0.5L for SS, CC and CS beams and x=L for CF beam.

Table 4.107 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under moving load-Case 3

(m1/mb=m2/mb=0.01)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 2.4x10-4 1.2x10-4 1.2x10-3 0.024 1.65x10-4 5.27x10-5 5.58x10-4 0.0109

CC 5.0x10-5 2.5x10-5 5.9x10-4 0.030 3.54x10-5 1.13x10-5 2.72x10-4 0.0124

CS 1.0x10-4 4.7x10-5 7.5x10-4 0.026 6.78x10-5 2.16x10-5 3.56x10-4 0.0111

CF 1.75x10-3 8.5x10-4 3.6x10-3 0.05 1.19x10-3 3.85x10-4 1.53x10-3 0.0206


(m1/mb=m2/mb=0.02)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 2.4x10-4 1.2x10-4 1.2x10-3 0.024 1.65x10-4 5.30x10-5 5.56x10-4 0.0109

CC 5.0x10-5 2.5x10-5 5.9x10-4 0.030 3.54x10-5 1.14x10-5 2.70x10-4 0.0123

CS 1.0x10-4 4.7x10-5 7.5x10-4 0.026 6.78x10-5 2.17x10-5 3.54x10-4 0.0110

CF 1.75x10-3 8.5x10-4 3.6x10-3 0.05 1.19x10-3 3.87x10-4 1.50x10-3 0.0205

152


(m1/mb=m2/mb=0.05)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 2.4x10-4 1.2x10-4 1.2x10-3 0.024 1.66x10-4 5.38x10-5 5.51x10-4 0.0109

CC 5.0x10-5 2.5x10-5 5.5x10-4 0.030 3.54x10-5 1.15x10-5 2.65x10-4 0.0123

CS 1.0x10-4 4.5x10-5 7.5x10-4 0.026 6.78x10-5 2.20x10-5 3.49x10-4 0.0109

CF 1.75x10-3 8.5x10-4 3.5x10-3 0.05 1.19x10-3 3.92x10-4 1.47x10-3 0.0206


(m1/mb=m2/mb=0.1)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 2.4x10-4 1.1x10-4 1.2x10-3 0.024 1.67x10-4 5.47x10-5 5.40x10-4 0.0110

CC 5.0x10-5 2.5x10-5 5.3x10-4 0.029 3.54x10-5 1.17x10-5 2.56x10-4 0.0122

CS 1.0x10-4 4.7x10-5 7.2x10-4 0.026 6.78x10-5 2.23x10-5 3.38x10-4 0.0108

CF 1.75x10-3 8.5x10-4 3.4x10-3 0.05 1.19x10-3 4.00x10-4 1.40x10-3 0.0207


(m1/mb=m2/mb=0.2)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

SS 2.4x10-4 1.2x10-4 1.1x10-3 0.024 1.69x10-4 5.65x10-5 5.20x10-4 0.0114

CC 5.0x10-5 2.5x10-5 5.0x10-4 0.028 3.54x10-5 1.20x10-5 2.39x10-4 0.0121

CS 1.0x10-4 4.5x10-5 6.8x10-4 0.024 6.78x10-5 2.27x10-5 3.18x10-4 0.0106

CF 1.75x10-3 7.5x10-4 3.1x10-3 0.05 1.19x10-3 4.11x10-4 1.28x10-3 0.0206

153

4.5.4. Moving Pulsating Force

Case 1: The velocity of moving pulsating force is 1.333 m/sec and applied to SS,

CC, CS and CF bare uniform beams.


Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 1.8x10-2 0.36 7.0 140 9.18x10-3 0.183 3.63 72.25 6.3π

CC 7.8x10-3 0.35 15.5 700 3.52x10-3 0.158 7.09 318.67 14.2π

CS 1.6x10-2 0.50 16.0 460 6.35x10-3 0.196 6.10 189.03 9.8π

CF 8.0x10-2 0.57 4.0 29 4.12x10-2 0.292 2.08 14.73 2.2π


is attached to x=0.5L for SS, CC and CS beams and x=L for CF beam.

Figure 4.36 Simply supported beam carrying one spring-mass system subjected to moving pulsating force

154

Table 4.113 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and

at x=L for CF beam carrying one spring-mass system under moving pulsating

force - Case 2 (m1/mb=0.01)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 5.8x10-4 1.2x10-2 0.22 4.5 2.28x10-4 4.47x10-3 0.088 1.74 6.3π

CC 5.6x10-5 2.4x10-3 0.11 5.0 2.55x10-5 1.13x10-3 0.050 2.21 14.2π

CS 1.5x10-4 4.5x10-3 0.14 4.3 6.87x10-5 2.09x10-3 0.064 1.95 9.8π

CF 1.0x10-2 6.7x10-2 0.45 3.0 4.60x10-3 3.04x10-2 0.200 1.33 2.2π

Table 4.114 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under moving pulsating


Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 3.0x10-4 5.8x10-3 0.115 2.2 1.13x10-4 2.21x10-3 0.044 0.87 6.3π

CC 2.9x10-5 1.3x10-3 0.058 2.5 1.24x10-5 5.48x10-4 0.024 1.08 14.2π

CS 7.5x10-5 2.3x10-3 0.07 2.2 3.35x10-5 1.02x10-3 0.031 0.95 9.8π

CF 3.6x10-3 2.3x10-2 0.155 1.05 1.66x10-3 1.09x10-2 0.072 0.486 2.2π



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 1.2x10-4 2.4x10-3 0.046 0.9 4.44x10-4 8.63x10-3 0.172 0.35 6.3π

CC 1.1x10-5 5.0x10-4 0.022 1.0 4.93x10-6 2.16x10-4 0.0096 0.435 14.2π

CS 3.0x10-5 9.0x10-4 0.028 0.9 1.32x10-5 3.99x10-4 0.012 0.383 9.8π

CF 1.4x10-3 9.0x10-3 0.060 0.42 5.95x10-4 3.87x10-3 0.026 0.182 2.2π

155



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 5.8x10-5 1.2x10-3 0.022 0.45 2.24x10-5 4.29x10-4 0.0086 0.182 6.3π

CC 5.7x10-6 2.5x10-4 0.011 0.50 2.48x10-6 1.08x10-4 0.0048 0.224 14.2π

CS 1. 5x10-5 4.5x10-4 0.014 0.45 6.64x10-6 1.98x10-4 0.0061 0.196 9.8π

CF 6.0x10-4 4.0x10-3 0.028 0.21 2.89x10-4 1.85x10-3 0.0127 0.095 2.2π



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 2.9x10-5 5.5x10-4 0.011 0.24 1.13x10-5 2.11x10-4 0.0043 0.0972 6.3π

CC 2.8x10-6 1.2x10-4 0.005 0.26 1.23x10-6 5.32x10-5 0.0024 0.116 14.2π

CS 7.5x10-6 2.2x10-4 0.007 0.23 3.34x10-6 9.81x10-5 0.0031 0.104 9.8π

CF 3.0x10-4 1.9x10-3 0.014 0.12 1.42x10-4 8.89x10-4 0.0063 0.053 2.2π

Case 3: The velocity of moving load is 1.333 m/sec and two spring-mass

systems are attached to x=0.5L for SS, CC and CS beams and x=L for CF beam.

Table 4.118 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and

at x=L for CF beam carrying two spring-mass systems under moving pulsating

force - Case 3 (m1/mb=m2/mb=0.01)

Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 5.0x10-4 1.0x10-2 0.20 4.0 2.03x10-4 3.99x10-3 0.079 1.56 6.3π

CC 4.6x10-5 2.1x10-3 0.09 4.0 2.13x10-5 9.38x10-4 0.042 1.84 14.2π

CS 1.3x10-4 3.8x10-3 0.12 3.6 5.56x10-5 1.69x10-3 0.051 1.57 9.8π

CF 6.0x10-3 4.0x10-2 0.27 1.85 2.72x10-3 1.79x10-2 0.119 0.79 2.2π

156

Table 4.119 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under moving pulsating


Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 2.8x10-4 5.5x10-3 0.11 2.2 1.03x10-4 2.01x10-3 0.040 0.792 6.3π

CC 2.4x10-5 1.1x10-3 0.05 2.1 1.08x10-5 4.76x10-4 0.021 0.940 14.2π

CS 6.5x10-5 2.0x10-3 0.06 1.9 2.85x10-5 8.65x10-4 0.026 0.809 9.8π

CF 2.8x10-3 1.9x10-2 0.13 0.88 1.19x10-3 7.79x10-3 0.052 0.35 2.2π



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 1.1x10-4 2.1x10-3 0.04 0.8 4.09x10-5 8.01x10-4 0.016 0.321 6.3π

CC 1.0x10-5 4.5x10-4 0.02 0.9 4.44x10-6 1.96x10-4 0.0087 0.39 14.2π

CS 2.6x10-5 8.0x10-4 0.024 0.73 1.16x10-5 3.51x10-4 0.011 0.333 9.8π

CF 1.1x10-3 7.5x10-3 0.050 0.36 4.60x10-4 3.02x10-3 0.020 0.141 2.2π



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 5.3x10-5 1.1x10-3 0.022 0.45 2.08x10-5 4.05x10-4 0.0081 0.166 6.3π

CC 5.0x10-6 2.2x10-4 0.01 0.47 2.28x10-6 1.01x10-4 0.0047 0.201 14.2π

CS 1.4x10-5 4.0x10-4 0.013 0.40 5.97x10-6 1.80x10-4 0.056 0.175 9.8π

CF 5.5x10-4 3.4x10-3 0.024 0.18 2.27x10-4 1.48x10-3 0.0101 0.073 2.2π

157



Boundary

Conditions

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

SS 2.8x10-5 5.5x10-4 0.011 0.23 1.06x10-5 2.06x10-4 0.0042 0.088 6.3π

CC 2.6x10-6 1.2x10-4 0.005 0.24 1.16x10-6 5.15x10-5 0.0024 0.105 14.2π

CS 7.2x10-6 2.0x10-4 0.007 0.21 2.99x10-6 8.99x10-5 0.0028 0.091 9.8π

CF 2.8x10-4 1.8x10-3 0.012 0.11 1.16x10-4 7.53x10-4 0.0053 0.041 2.2π


is attached to x=0.5L for SS beam. Forcing frequency is different from the

resonant frequency of the beam. Spring-mass system is tuned to the excitation

frequency.

Table 4.123 Maximum and RMS responses at x=0.5L for SS beam carrying one spring-mass system under moving pulsating force - Case 4

m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 2.0x10-4 6.5x10-3 0.20 6.5 1.02x10-4 3.24x10-3 0.103 3.28 10π

0.02 8.0x10-5 2.5x10-3 0.08 2.6 3.40x10-5 1.09x10-3 0.035 1.12 10π

0.05 3.0x10-5 9.5x10-4 0.03 0.95 1.22x10-5 3.95x10-4 0.013 0.425 10π

0.10 1.5x10-5 4.8x10-4 0.015 0.50 5.75x10-6 1.88x10-4 0.006 0.215 10π

0.20 7.5x10-6 2.2x10-4 0.007 0.24 2.77x10-6 8.93x10-5 0.003 0.112 10π

No TVA 1.6x10-4 5.0x10-3 0.16 4.75 8.22x10-5 2.58x10-3 0.081 2.55 10π

158

4.6. Forced Vibration Analysis of High-Mast Lighting Tower under Wind Load

The high-mast lighting (HMLT) tower indicated in free vibration of non-

uniform beams (Section 4.3) is subjected to wind load by using wind velocity data

given by Prof.Robert J. Connor. The wind profile is obtained by using the

empirical power-law profile which is particularly used because of its simplicity

such as in the Canadian code NBC 1990 (Dyrbye and Hansen, 1997). The wind

velocity at any height of the structure can be obtained by using the empirical

equation given as below

Eq. 4.5

where zref is a reference height and the parameter α is considered as 0.16 which

is used for a terrain category including farmland with boundary hedges,

occasional small farm structures, houses or trees (Dyrbye and Hansen, 1997).

The given wind velocity data is assumed to be at 30 ft which is usually used for

reference height.

The dynamic response of a structure subjected to wind loading cannot be

represented in a definite form. Wind can be expressed in terms of its mean

velocity and turbulence component and HMLT can be considered as line-like

structure. The wind load per unit length for line like structures is obtained by

,12

, , Eq. 4.6

where U(z)+u(z,t) is the wind velocity which is the sum of mean wind velocity

U(z) and the longitudinal turbulence component u(z,t). The given wind velocity

data in this study is assumed to be U(z)+u(z,t) because the mean wind velocity is

generally much larger than the turbulence component(Dyrbye and Hansen,

1996). ξdef(z,t) is the structural velocity which is used in order to find the relative

wind velocity with respect to structure. The structural velocity is ignored in this

study and U(z)+u(z,t) is defined as V(z,t). d(z) is the width of the structure

perpendicular to the wind direction and C(z) is the shape factor which is 0.6 for

circular sections. Therefore equation 4.6 can be written as

159

,12

, Eq. 4.7

The HMLT is subdivided into six segments as it is shown in figure 4.37

and wind velocity variation with respect to time at 14 ft, 42 ft, 70 ft, 98 ft and 126

ft are found respectively based on equation 4.5. Wind load functions with respect

to time are obtained for each point with MATLAB curve fitting tool by using

Fourier series after the corresponding wind load variations at each point with

respect to time are obtained by using equation 4.7. Each forcing function is

applied at corresponding heights and displacement, velocity, acceleration and

jerk responses at any point of the HMLT are found by using the developed

algorithm in MATHEMATICA. The resultant responses for bare HMLT and HMLT

with one spring-mass system on the top of the structure are given in figures 4.38

and 39.

Figure 4.37 Force and wind velocity profile of HMLT

Figure 4.38 Dynamic responses of bare HMLT under wind load

0 20 40 60 80 100 120 140

0

5

10

15

0 20 40 60 80 100 120 140

-10

-5

0

5

10

0 20 40 60 80 100 120 140

-20

-10

0

10

20

0 20 40 60 80 100 120 140

-60

-40

-20

0

20

40

60

Time (Second)

Dis

plac

emen

t (in

)

Vel

ocity

(in

/s)

Time (Second)

Acc

eler

atio

n (in

/s2 )

Time (Second)

Jerk

(in

/s3 )

Time (Second)

160

Figure 4.39 Dynamic responses of HMLT carrying one spring-mass system at the shallow end under wind load

(m1/mb=0.01)

0 20 40 60 80 100 120 140

0

5

10

15

0 20 40 60 80 100 120 140

-10

-5

0

5

10

0 20 40 60 80 100 120 140

-20

-10

0

10

20

0 20 40 60 80 100 120 140

-60

-40

-20

0

20

40

60

Time (Second)

Dis

plac

emen

t (in

)

Vel

ocity

(in

/s)

Time (Second)

Acc

eler

atio

n (in

/s2 )

Time (Second)

Jerk

(in

/s3 )

Time (Second)

161

162

4.7. Forced Vibration Analysis of Multi Span Uniform Beams Carrying One and


F0 is assumed to be 350 N for harmonic loading and 700 N for impact

loading, moving load and moving pulsating loads.

4.7.1. Forced Vibration Analysis of Two Span Beam Carrying

One Spring-Mass System

4.7.1.1. Impact Loading

Case 1: Spring-mass system is attached to second span (x=0.75L) and the step-

function force is applied to x=0.25L.

Figure 4.40 Two-span simply supported beam carrying one spring-mass system subjected to step-function force at x=0.25L

163

Table 4.124 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying one spring-mass system under impact loading - Case 1

m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 4.4x10-4 9.0x10-3 0.26 7.5 2.03x10-4 5.06x10-3 0.156 4.83

0.02 4.2x10-4 8.5x10-3 0.24 7.0 1.98x10-4 4.87x10-3 0.151 4.72

0.05 4.0x10-4 7.8x10-3 0.22 6.75 1.86x10-4 4.39x10-3 0.138 4.41

0.10 4.1x10-4 7.8x10-3 0.22 6.5 1.75x10-4 3.82x10-3 0.121 3.97

No TVA 4.6x10-4 1.0x10-2 0.32 13 2.18x10-4 5.43x10-3 0.167 6.24


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 3.3x10-4 4x10-3 0.09 2.4 1.85x10-4 1.64x10-3 0.042 1.16

0.02 3.4x10-4 4x10-3 0.095 2.4 1.85x10-4 1.68x10-3 0.042 1.15

0.05 3.4x10-4 3.8x10-3 0.092 2.4 1.87x10-4 1.77x10-3 0.043 1.15

0.10 3.4x10-4 3.8x10-3 0.095 2.4 1.90x10-4 1.93x10-3 0.047 1.19

No TVA 3.6x10-4 4.0x10-3 0.10 2.5 2.05x10-4 2.13x10-3 0.050 1.29

164


function force is applied to x=0.75L.

Figure 4.41 Two-span simply supported beam carrying one spring-mass system subjected to step-function force at x=0.75L


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 3.3x10-4 3.9x10-3 0.095 2.4 1.85x10-4 1.63x10-3 0.042 1.21

0.02 3.4x10-4 3.8x10-3 0.095 2.4 1.86x10-4 1.66x10-3 0.043 1.24

0.05 3.35x10-4 3.8x10-3 0.090 2.5 1.86x10-4 1.68x10-3 0.045 1.34

0.10 3.4x10-4 3.7x10-3 0.090 2.6 1.88x10-4 1.71x10-3 0.047 1.49

No TVA 3.2x10-4 3.7x10-3 0.090 2.4 1.83x10-4 1.91x10-3 0.046 1.24

165


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 5.0x10-4 1.0x10-2 0.28 8.2 2.29x10-4 5.75x10-3 0.175 5.42

0.02 5.3x10-4 1.1x10-2 0.31 9.0 2.47x10-4 6.21x10-3 0.189 5.90

0.05 6.7x10-4 1.4x10-2 0.40 11.5 2.98x10-4 7.57x10-3 0.234 7.45

0.10 8.2x10-4 1.8x10-2 0.50 15.5 3.69x10-4 9.68x10-3 0.311 10.28

No TVA 4.7x10-4 1.0x10-2 0.33 14.5 2.21x10-4 5.40x10-3 0.169 6.98


function force is applied to both x=0.25L and x=0.75L.

Figure 4.42 Two-span simply supported beam carrying one spring-mass system subjected to step-function force at x=0.25L and x=0.75L

166


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 8.5x10-4 9x10-3 0.27 8.5 3.44x10-4 6.09x10-3 0.190 5.94

0.02 8.8x10-4 9x10-3 0.275 8.5 3.41x10-4 5.94x10-3 0.187 5.88

0.05 8.5x10-4 8.8x10-3 0.27 8.5 3.33x10-4 5.54x10-3 0.177 5.70

0.10 8.0x10-4 8.5x10-3 0.27 8.5 3.24x10-4 5.00x10-3 0.164 5.42

No TVA 9.0x10-4 9.0x10-3 0.34 15 3.54x10-4 6.25x10-3 0.200 7.80


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 7.0x10-4 1.2x10-2 0.34 10 3.63x10-4 6.70x10-3 0.207 6.43

0.02 7.8x10-4 1.3x10-2 0.37 10.5 3.76x10-4 7.14x10-3 0.220 6.88

0.05 9.3x10-4 1.6x10-2 0.45 13.5 4.17x10-4 8.42x10-3 0.262 8.33

0.10 1.1x10-3 2x10-2 0.60 17.5 4.77x10-4 1.04x10-2 0.334 11.00

No TVA 9.0x10-4 9.0x10-3 0.34 15 3.54x10-4 6.25x10-3 0.200 7.80

167

4.7.1.2. Harmonic Loading

Case 1: Spring-mass system is attached to second span (x=0.75L) and the

harmonic force is applied to x=0.25L.

Figure 4.43 Two-span simply supported beam carrying one spring-mass system subjected to harmonic force at x=0.25L

Table 4.130 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying one spring-mass system under harmonic loading - Case 1

m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 7.0x10-4 1.7x10-2 0.40 10 3.24x10-4 7.07x10-3 0.171 4.58 6.3π

0.02 6.0x10-4 1.4x10-2 0.32 8.5 2.82x10-4 6.28x10-3 0.156 4.31 6.3π

0.05 5.5x10-4 1.3x10-2 0.30 8.0 2.54x10-4 5.64x10-3 0.141 3.94 6.3π

0.10 5.0x10-4 1.3x10-2 0.30 8.0 2.46x10-4 5.39x10-3 0.131 3.59 6.3π

No TVA 1.1x10-2 0.22 4.25 85 4.60x10-3 9.16x10-2 1.813 36.30 6.3π

168


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 6.0x10-4 1.2x10-2 0.24 5.5 2.61x10-4 5.36x10-3 0.112 2.39 6.3π

0.02 4.0x10-4 8.5x10-3 0.19 4.25 2.13x10-4 4.47x10-3 0.095 2.10 6.3π

0.05 4.2x10-4 9.0x10-2 0.20 4.6 1.86x10-4 4.00x10-3 0.088 1.97 6.3π

0.10 4.0x10-4 8.0x10-2 0.19 4.5 1.80x10-4 3.95x10-3 0.088 2.01 6.3π

No TVA 1.1x10-2 0.22 4.25 85 4.66x10-3 9.27x10-2 1.832 36.44 6.3π



Figure 4.44 Two-span simply supported beam carrying one spring-mass system subjected to harmonic force at x=0.75L

169


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 4.5x10-4 9.0x10-3 0.20 4.2 2.14x10-4 4.24x10-3 0.086 1.86 6.3π

0.02 3.2x10-4 6.5x10-3 0.15 3.4 1.49x10-4 2.95x10-3 0.062 1.45 6.3π

0.05 2.0x10-4 4.5x10-3 0.10 2.6 9.39x10-5 1.90x10-3 0.045 1.21 6.3π

0.10 1.5x10-4 3.4x10-3 0.08 2.2 6.66x10-5 1.42x10-3 0.038 1.18 6.3π

No TVA 1.1x10-2 0.22 4.25 85 4.66x10-3 9.27x10-2 1.833 36.47 6.3π


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 1.2x10-3 2.4x10-2 0.55 13 5.07x10-4 1.05x10-2 0.232 5.72 6.3π

0.02 1.1x10-3 2.4x10-2 0.55 13 4.81x10-4 1.01x10-2 0.230 5.88 6.3π

0.05 1.1x10-3 2.4x10-2 0.55 14 4.76x10-4 1.03x10-2 0.248 6.79 6.3π

0.10 1.1x10-3 2.4x10-2 0.60 17 4.82x10-4 1.10x10-2 0.288 8.53 6.3π

No TVA 1.1x10-2 0.22 4.25 85 4.60x10-3 9.17x10-2 1.814 36.32 6.3π

170


harmonic force is applied to both x=0.25L and x=0.75L.

Figure 4.45 Two-span simply supported beam carrying one spring-mass system subjected to harmonic force at x=0.25L and x=0.75L


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 5.8x10-4 1.4x10-2 0.36 9.8 2.68x10-4 6.60x10-3 0.179 5.23 6.3π

0.02 5.8x10-4 1.4x10-2 0.36 9.8 2.64x10-4 6.48x10-3 0.176 5.14 6.3π

0.05 5.8x10-4 1.4x10-2 0.36 9.8 2.59x10-4 6.24x10-3 0.168 4.92 6.3π

0.10 5.8x10-4 1.4x10-2 0.34 9.8 2.57x10-4 6.04x10-3 0.158 4.63 6.3π

No TVA 4.7x10-4 0.0125 0.35 12 2.43x10-4 6.29x10-3 0.179 5.84 6.3π

171

Table 4.135 Maximum and RMS responses at x=0.25L for two-span uniform

beam carrying one spring-mass system under harmonic loading - Case 3

m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 1.4x10-3 3.0x10-2 0.70 16.5 5.88x10-4 1.24x10-2 0.279 6.92 6.3π

0.02 1.4x10-3 3.0x10-2 0.70 17 5.87x10-4 1.25x10-2 0.284 7.16 6.3π

0.05 1.4x10-3 3.1x10-2 0.72 18 5.99x10-4 1.29x10-2 0.305 8.05 6.3π

0.10 1.4x10-3 3.1x10-2 0.78 20 6.11x10-4 1.36x10-2 0.340 9.61 6.3π

No TVA 4.7x10-4 0.0125 0.35 12 2.43x10-4 6.29x10-3 0.179 5.84 6.3π


harmonic force is applied to x=0.25L. Forcing frequency is different from the

resonant frequency of the beam. Spring-mass system is tuned to the excitation

frequency.


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 1.0x10-3 3.2x10-2 1.0 30 4.70x10-4 1.42x10-2 0.434 13.32 10π

0.02 9.0x10-4 2.6x10-2 0.8 24 3.87x10-4 1.15x10-2 0.348 10.57 10π

0.05 7.0x10-4 2.0x10-2 0.6 20 3.26x10-4 9.54x10-3 0.282 8.51 10π

0.10 6.0x10-4 1.8x10-2 0.55 17 3.00x10-4 8.69x10-3 0.255 7.63 10π

No TVA 9.0x10-3 0.28 8.5 270 4.20x10-3 1.31x10-2 4.105 128.05 10π

172

4.7.1.3. Moving Load

Case 1: The velocity of moving load is 1.333 m/sec and the spring-mass system

is attached to second span at x=0.75L.

Figure 4.46 Two-span simply supported beam carrying one spring-mass system subjected to moving load

Table 4.137 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying one spring-mass system under moving load-Case 1

m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 1.0x10-4 2.6x10-5 4.0x10-4 0.013 7.51x10-5 1.26x10-5 2.8x10-4 0.0087

0.02 1.0x10-4 2.6x10-5 4.0x10-4 0.013 7.51x10-5 1.24x10-5 2.7x10-4 0.0085

0.05 1.0x10-4 2.6x10-5 4.0x10-4 0.013 7.51x10-5 1.20x10-5 2.5x10-4 0.0081

0.10 1.0x10-4 2.6x10-5 4.0x10-4 0.013 7.51x10-5 1.14x10-5 2.3x10-4 0.0074

No TVA 1.1x10-4 2.6x10-5 4.5x10-4 0.017 7.76x10-5 1.31x10-5 2.9x10-4 0.0094

173

Table 4.138 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying one spring-mass system under moving load-Case 1

m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 1.1x10-4 3.0x10-5 5.0x10-4 0.015 7.62x10-5 1.34x10-5 3.1x10-4 0.0095

0.02 1.1x10-4 3.3x10-5 5.5x10-4 0.016 7.73x10-5 1.40x10-5 3.2x10-4 0.010

0.05 1.1x10-4 3.8x10-5 6.8x10-4 0.020 8.07x10-5 1.56x10-5 3.7x10-4 0.012

0.10 1.2x10-4 4.5x10-5 8.5x10-4 0.025 8.63x10-5 1.80x10-5 4.6x10-4 0.015

No TVA 1.1x10-4 2.6x10-5 4.5x10-4 0.017 7.76x10-5 1.31x10-5 2.9x10-4 0.0094

4.7.1.4. Moving Pulsating Force

Case 1: The velocity of moving pulsating load is 1.333 m/sec and the spring-

mass system is attached to second span at x=0.75L.

Figure 4.47 Two-span simply supported beam carrying one spring-mass system subjected to moving pulsating force

174

Table 4.139 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying one spring-mass system under moving pulsating force-Case 1

m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 2.4x10-4 4.6x10-3 0.092 1.9 1.21x10-4 2.39x10-3 0.047 0.94 6.3π

0.02 2.4x10-4 4.6x10-3 0.092 1.9 1.21x10-4 2.39x10-3 0.047 0.94 6.3π

0.05 2.4x10-4 4.6x10-3 0.092 1.85 1.21x10-4 2.39x10-3 0.047 0.94 6.3π

0.10 2.4x10-4 4.6x10-3 0.092 1.85 1.21x10-4 2.39x10-3 0.047 0.94 6.3π

No TVA 4.0x10-3 8.0x10-2 1.6 33 2.15x10-4 4.25x10-2 0.842 16.73 6.3π

Table 4.140 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying one spring-mass system under moving pulsating force-Case 1

m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 4.5x10-4 9.0x10-3 0.180 3.6 2.28x10-4 4.51x10-3 0.089 1.77 6.3π

0.02 4.5x10-4 9.0x10-3 0.180 3.6 2.28x10-4 4.52x10-3 0.089 1.77 6.3π

0.05 4.5x10-4 9.0x10-3 0.175 3.4 2.30x10-4 4.55x10-3 0.090 1.78 6.3π

0.10 4.5x10-4 9.0x10-3 0.175 3.5 2.34x10-4 4.63x10-3 0.092 1.81 6.3π

No TVA 4.0x10-3 8.0x10-2 1.6 33 2.15x10-4 4.25x10-2 0.842 16.73 6.3π

175


mass system is attached to second span at x=0.75L. Forcing frequency is

different from the resonant frequency of the beam. Spring-mass system is tuned

to the excitation frequency.


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 2.4x10-4 7.5x10-3 0.23 7.5 1.11x10-4 3.46x10-3 0.108 3.40 10π

0.02 2.2x10-4 6.7x10-3 0.21 6.75 1.08x10-4 3.40x10-3 0.107 3.34 10π

0.05 2.0x10-4 6.5x10-3 0.20 6.5 1.08x10-4 3.38x10-3 0.106 3.32 10π

0.10 2.0x10-4 6.5x10-3 0.20 6.2 1.07x10-4 3.37x10-3 0.106 3.32 10π

No TVA 5.7x10-3 0.18 5.5 175 2.69x10-3 8.42x10-2 2.637 82.47 10π

176

4.7.2. Forced Vibration Analysis of Two Span Beam Carrying





natural frequency of two-span bare uniform beam and the step-function force is

applied to x=0.75L.

Figure 4.48 Two-span simply supported beam carrying two spring-mass systems subjected to step-function force at x=0.75L

Table 4.142 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying two spring-mass systems under impact loading - Case 1

m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 4.7x10-4 9.5x10-3 0.27 8.0 2.23x10-4 5.54x10-3 0.170 5.30

0.02 5.3x10-4 1.1x10-2 0.30 8.8 2.36x10-4 5.83x10-3 0.179 5.66

0.05 6.5x10-4 1.4x10-2 0.38 11 2.71x10-4 6.66x10-3 0.209 6.79

0.10 8.0x10-4 1.7x10-2 0.48 15 3.21x10-4 7.97x10-3 0.260 8.80

No TVA 4.7x10-4 1.0x10-2 0.33 14.5 2.21x10-4 5.40x10-3 0.169 6.98

177


m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 3.4x10-4 3.8x10-3 0.09 2.4 1.84x10-4 1.60x10-3 0.042 1.18

0.02 3.4x10-4 3.8x10-3 0.09 2.4 1.84x10-4 1.57x10-3 0.041 1.18

0.05 3.4x10-4 3.8x10-3 0.09 2.45 1.84x10-4 1.52x10-3 0.040 1.20

0.10 3.4x10-4 3.6x10-3 0.09 2.6 1.84x10-4 1.45x10-3 0.040 1.23

No TVA 3.2x10-4 3.7x10-3 0.090 2.4 1.83x10-4 1.91x10-3 0.046 1.24



frequency of the two-span uniform beam carrying one spring-mass system. The

step-function force is applied to x=0.75L.


m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 5.0x10-4 1.0x10-2 0.29 8.2 2.24x10-4 5.63x10-3 0.173 5.41

0.02 5.5x10-4 1.1x10-2 0.33 9.3 2.40x10-4 6.07x10-3 0.188 5.98

0.05 7.0x10-4 1.5x10-2 0.42 13 2.87x10-4 7.52x10-3 0.243 8.09

0.10 8.7x10-4 2.0x10-2 0.60 20 3.61x10-4 1.01x10-2 0.352 12.55

No TVA 4.7x10-4 1.0x10-2 0.33 14.5 2.21x10-4 5.40x10-3 0.169 6.98

178


m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 3.3x10-4 3.8x10-3 0.09 2.4 1.83x10-4 1.51x10-3 0.040 1.15

0.02 3.4x10-4 3.6x10-3 0.09 2.3 1.83x10-4 1.49x10-3 0.040 1.17

0.05 3.3x10-4 3.6x10-3 0.09 2.5 1.83x10-4 1.45x10-3 0.040 1.22

0.10 3.4x10-4 3.6x10-3 0.09 2.6 1.83x10-4 1.42x10-3 0.040 1.35

No TVA 3.2x10-4 3.7x10-3 0.090 2.4 1.83x10-4 1.91x10-3 0.046 1.24


is same with the first natural frequency of two-span bare uniform beam and the

step-function force is applied to both x=0.25L and x=0.75L.

Figure 4.49 Two-span simply supported beam carrying two spring-mass systems subjected to step-function force at x=0.25L and x=0.75L

179

Table 4.146 Maximum and RMS responses at both x=0.25L and x=0.75L for two-span uniform beam carrying two spring-mass systems under impact loading -

Case 3

m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 7.0x10-4 1.5x10-2 0.33 10 3.57x10-4 6.51x10-3 0.203 6.36

0.02 7.2x10-4 1.5x10-2 0.35 10.5 3.66x10-4 6.77x10-3 0.213 6.74

0.05 8.0x10-4 1.5x10-2 0.42 13 3.92x10-4 7.58x10-3 0.243 7.91

0.10 9.8x10-4 1.8x10-2 0.55 17.5 4.32x10-4 8.86x10-3 0.294 9.98

No TVA 9.0x10-4 9.0x10-3 0.34 15 3.54x10-4 6.25x10-3 0.200 7.80




step-function force is applied to both x=0.25L and x=0.75L.


beam carrying two spring-mass systems under impact loading - Case 4

m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 6.7x10-4 1.1x10-2 0.33 10 3.58x10-4 6.58x10-3 0.206 6.46

0.02 7.5x10-4 1.3x10-2 0.37 11 3.70x10-4 6.97x10-3 0.220 7.01

0.05 8.7x10-4 1.6x10-2 0.48 15 4.05x10-4 8.32x10-3 0.273 9.07

0.10 1.0x10-3 2.0x10-2 0.65 22 4.65x10-4 1.08x10-2 0.378 13.43

No TVA 9.0x10-4 9.0x10-3 0.34 15 3.54x10-4 6.25x10-3 0.200 7.80

180


m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 6.8x10-4 1.1x10-2 0.32 9.8 3.56x10-4 6.46x10-3 0.201 6.33

0.02 7.3x10-4 1.3x10-2 0.35 10.5 3.64x10-4 6.63x10-3 0.209 6.63

0.05 8.5x10-4 1.5x10-2 0.42 13 3.87x10-4 7.14x10-3 0.228 7.49

0.10 1.0x10-3 1.8x10-2 0.53 17.5 4.26x10-4 8.15x10-3 0.265 9.00

No TVA 9.0x10-4 9.0x10-3 0.34 15 3.54x10-4 6.25x10-3 0.200 7.80




natural frequency of two-span bare uniform beam and the harmonic force is

applied to x=0.75L.

Figure 4.50 Two-span simply supported beam carrying two spring-mass systems subjected to harmonic force at x=0.75L

181

Table 4.149 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying two spring-mass systems under harmonic loading - Case 1

m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 1.1x10-3 2.2x10-2 0.50 12.5 4.99x10-4 1.03x10-2 0.229 5.62 6.3π

0.02 1.0x10-3 2.2x10-2 0.50 12 4.59x10-4 9.63x10-3 0.220 5.63 6.3π

0.05 1.0x10-3 2.2x10-2 0.50 14 4.53x10-4 9.72x10-3 0.231 6.23 6.3π

0.10 1.0x10-3 2.4x10-2 0.60 16 4.58x10-4 1.01x10-2 0.255 7.39 6.3π

No TVA 1.1x10-2 0.22 4.25 85 4.60x10-3 9.17x10-2 1.814 36.32 6.3π


m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 4.0x10-4 8.0x10-3 0.18 4 1.60x10-4 3.25x10-3 0.069 1.58 6.3π

0.02 3.0x10-4 6.0x10-3 0.13 3.2 1.13x10-4 2.35x10-3 0.053 1.30 6.3π

0.05 1.9x10-4 4.0x10-3 0.10 2.5 7.36x10-5 1.62x10-3 0.040 1.11 6.3π

0.10 1.4x10-4 3.2x10-3 0.078 2.2 5.33x10-5 1.23x10-3 0.034 1.02 6.3π

No TVA 1.1x10-2 0.22 4.25 85 4.66x10-3 9.27x10-2 1.833 36.47 6.3π

182






m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 1.0x10-3 2.2x10-2 0.50 12 3.86x10-4 8.28x10-3 0.196 5.18 6.3π

0.02 1.1x10-3 2.6x10-2 0.55 14 5.13x10-4 1.08x10-2 0.245 6.17 6.3π

0.05 3.5x10-3 7.0x10-2 1.50 33 1.73x10-3 3.43x10-2 0.693 14.57 6.3π

0.10 1.6x10-3 3.4x10-2 0.80 22 6.69x10-4 1.44x10-2 0.351 10.23 6.3π

No TVA 1.1x10-2 0.22 4.25 85 4.60x10-3 9.17x10-2 1.814 36.32 6.3π


m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 8.0x10-4 1.7x10-2 0.34 7.25 3.12x10-4 6.22x10-3 0.125 2.60 6.3π

0.02 1.1x10-3 2.2x10-2 0.45 9.8 5.25x10-4 1.05x10-2 0.210 4.24 6.3π

0.05 1.8x10-3 3.5x10-2 0.70 14 9.35x10-4 1.84x10-2 0.363 7.20 6.3π

0.10 4.0x10-4 8.0x10-3 0.17 4 1.81x10-4 3.52x10-3 0.072 1.60 6.3π

No TVA 1.1x10-2 0.22 4.25 85 4.66x10-3 9.27x10-2 1.833 36.47 6.3π

183


is same with the first natural frequency of two-span bare uniform beam and the


Figure 4.51 Two-span simply supported beam carrying two spring-mass systems subjected to harmonic force at x=0.25L and x=0.75L

Table 4.153 Maximum and RMS responses at both x=0.25L and x=0.75L for two-span uniform beam carrying two spring-mass systems under harmonic loading -

Case 3

m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 1.1x10-3 2.4x10-2 0.55 14 4.62x10-4 1.00x10-2 0.238 6.25 6.3π

0.02 1.0x10-3 2.3x10-2 0.55 14 4.51x10-4 9.91x10-3 0.239 6.40 6.3π

0.05 1.0x10-3 2.4x10-2 0.60 15 4.55x10-4 1.02x10-2 0.253 7.07 6.3π

0.10 1.0x10-3 2.5x10-2 0.65 18 4.62x10-4 1.06x10-2 0.278 8.25 6.3π

No TVA 4.7x10-4 0.0125 0.35 12 2.43x10-4 6.29x10-3 0.179 5.84 6.3π

184






m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 9.0x10-4 2.1x10-2 0.50 13 3.35x10-4 7.91x10-3 0.206 5.83 6.3π

0.02 1.2x10-3 2.5x10-2 0.60 15 5.02x10-4 1.10x10-2 0.261 6.86 6.3π

0.05 3.4x10-3 7.0x10-2 1.5 34 1.72x10-3 3.43x10-2 0.698 14.90 6.3π

0.10 1.6x10-3 3.4x10-2 0.90 24 6.66x10-4 1.46x10-2 0.365 10.83 6.3π

No TVA 4.7x10-4 0.0125 0.35 12 2.43x10-4 6.29x10-3 0.179 5.84 6.3π


m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 1.5x10-3 3.2x10-2 0.70 17 6.39x10-4 1.33x10-2 0.293 7.09 6.3π

0.02 2.0x10-3 4.0x10-2 0.90 20 8.80x10-4 1.79x10-2 0.380 8.64 6.3π

0.05 1.5x10-3 3.4x10-2 0.75 19 6.74x10-4 1.39x10-2 0.308 7.67 6.3π

0.10 9.0x10-4 2.2x10-2 0.60 17 3.68x10-4 8.75x10-3 0.238 7.28 6.3π

No TVA 4.7x10-4 0.0125 0.35 12 2.43x10-4 6.29x10-3 0.179 5.84 6.3π

185


Case 1: The velocity of moving load is 1.333 m/sec and the spring-mass systems

are attached to first and second span at x=0.25L and x=0.75L, respectively. Both

of the spring-mass systems are tuned to constant frequency which is same with

the first natural frequency of two-span bare uniform beam.

Figure 4.52 Two-span simply supported beam carrying two spring-mass systems subjected to moving load

Table 4.156 Maximum and RMS responses at x=0.25L and x=0.75L for two-span

uniform beam carrying two spring-mass systems under moving load-Case 1

m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 1.0x10-4 3.0x10-5 4.8x10-4 0.015 7.62x10-5 1.32x10-5 3.0x10-4 0.0093

0.02 1.1x10-4 3.1x10-5 5.3x10-4 0.016 7.73x10-5 1.35x10-5 3.1x10-4 0.0097

0.05 1.1x10-4 3.6x10-5 6.3x10-4 0.019 8.07x10-5 1.44x10-5 3.4x10-4 0.0109

0.10 1.2x10-4 4.0x10-5 7.8x10-4 0.023 8.63x10-5 1.59x10-5 3.9x10-4 0.0130

No TVA 1.1x10-4 2.6x10-5 4.5x10-4 0.017 7.76x10-5 1.31x10-5 2.9x10-4 0.0094

186


are attached to first and second span at x=0.25L and x=0.75L, respectively. The

first spring-mass system is tuned to the first natural frequency of two-span bare

uniform beam and the second one is tuned to the second natural frequency of

the two-span uniform beam carrying one spring-mass system.


beam carrying two spring-mass systems under moving load-Case 2

m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 1.0x10-4 3.0x10-5 4.8x10-4 0.015 7.62x10-5 1.31x10-5 3.0x10-4 0.0092

0.02 1.1x10-4 3.1x10-5 5.5x10-4 0.016 7.73x10-5 1.34x10-5 3.0x10-4 0.0095

0.05 1.1x10-4 3.6x10-5 6.3x10-4 0.019 8.07x10-5 1.41x10-5 3.2x10-4 0.0102

0.10 1.2x10-4 4.0x10-5 7.9x10-4 0.023 8.62x10-5 1.55x10-5 3.5x10-4 0.0117

No TVA 1.1x10-4 2.6x10-5 4.5x10-4 0.017 7.76x10-5 1.31x10-5 2.9x10-4 0.0094



m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 1.0x10-4 3.0x10-5 5.0x10-4 0.015 7.64x10-5 1.32x10-5 3.0x10-4 0.0094

0.02 1.1x10-4 3.1x10-5 4.9x10-4 0.016 7.78x10-5 1.37x10-5 3.2x10-4 0.0100

0.05 1.1x10-4 3.6x10-5 6.7x10-4 0.020 8.24x10-5 1.50x10-5 3.7x10-4 0.0122

0.10 1.3x10-4 4.3x10-5 8.8x10-4 0.028 9.09x10-5 1.75x10-5 4.7x10-4 0.0168

No TVA 1.1x10-4 2.6x10-5 4.5x10-4 0.017 7.76x10-5 1.31x10-5 2.9x10-4 0.0094

187



are attached to first and second span at x=0.25L and x=0.75L, respectively. Both

of the spring-mass systems are tuned to constant frequency which is same with

the first natural frequency of two-span bare uniform beam.

Figure 4.53 Two-span simply supported beam carrying two spring-mass systems subjected to moving pulsating load

Table 4.159 Maximum and RMS responses at x=0.25L and x=0.75L for two-span uniform beam carrying two spring-mass systems under moving pulsating force-

Case 1

m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 3.6x10-4 7.0x10-3 0.14 2.8 1.85x10-4 3.66x10-3 0.072 1.43 6.3π

0.02 3.6x10-4 7.0x10-3 0.14 2.7 1.80x10-4 3.57x10-3 0.071 1.40 6.3π

0.05 3.5x10-4 6.7x10-3 0.14 2.7 1.82x10-4 3.61x10-3 0.071 1.41 6.3π

0.10 3.5x10-4 7.0x10-3 0.14 2.8 1.86x10-4 3.68x10-3 0.073 1.44 6.3π

No TVA 4.0x10-3 8.0x10-2 1.6 33 2.15x10-4 4.25x10-2 0.842 16.73 6.3π

188


are attached to first and second span at x=0.25L and x=0.75L, respectively. The

first spring-mass system is tuned to the first natural frequency of two-span bare

uniform beam and the second one is tuned to the second natural frequency of

the two-span uniform beam carrying one spring-mass system.


beam carrying two spring-mass systems under moving pulsating force-Case 2

m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 5.8x10-4 1.2x10-2 0.22 4.5 2.73x10-4 5.41x10-3 0.107 2.13 6.3π

0.02 7.5x10-4 1.5x10-2 0.30 6.0 3.76x10-4 7.47x10-3 0.148 2.95 6.3π

0.05 8.5x10-4 1.6x10-2 0.32 6.25 4.28x10-4 8.39x10-3 0.165 3.23 6.3π

0.10 2.7x10-4 5.3x10-3 0.105 2.1 1.38x10-4 2.73x10-3 0.054 1.07 6.3π

No TVA 4.0x10-3 8.0x10-2 1.6 33 2.15x10-4 4.25x10-2 0.842 16.73 6.3π


beam carrying two spring-mass systems under moving pulsating force-Case 2

m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 1.8x10-4 3.6x10-3 0.07 1.4 9.11x10-5 1.80x10-3 0.036 0.708 6.3π

0.02 3.0x10-4 6.0x10-3 0.12 2.4 1.39x10-4 2.79x10-3 0.056 1.11 6.3π

0.05 1.9x10-3 3.6x10-2 0.70 14 8.30x10-4 1.63x10-2 0.321 6.31 6.3π

0.10 4.7x10-4 9.5x10-3 0.19 3.7 2.48x10-4 4.90x10-3 0.097 1.92 6.3π

No TVA 4.0x10-3 8.0x10-2 1.6 33 2.15x10-4 4.25x10-2 0.842 16.73 6.3π

189

4.7.3. Forced Vibration Analysis of Three Span Beam Carrying

One Spring-Mass Systems


Case 1: Spring-mass system is attached to first span at x= (1/6) L and the step-

function force is applied to x=(1/6)L.

Figure 4.54 Three-span simply supported beam carrying one spring-mass system subjected to step-function force at x= (1/6) L

Table 4.162 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying one spring-mass system under impact loading - Case 1

m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 3.6x10-4 4.5x10-3 0.12 3.4 2.01x10-4 1.82x10-3 0.051 1.56

0.02 3.7x10-4 4.5x10-3 0.12 3.5 2.02x10-4 1.85x10-3 0.052 1.61

0.05 3.7x10-4 4.2x10-3 0.12 3.6 2.02x10-4 1.82x10-3 0.052 1.68

0.10 3.5x10-4 4.1x10-3 0.12 3.6 2.00x10-4 1.76x10-3 0.052 1.78

No TVA 3.7x10-4 4.5x10-3 0.12 3.4 2.03x10-4 1.93x10-3 0.052 1.53

190


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 8.5x10-4 1.6x10-2 0.53 18 4.05x10-4 8.29x10-3 0.298 10.89

0.02 9.5x10-4 1.8x10-2 0.60 20 4.32x10-4 8.97x10-3 0.319 11.63

0.05 1.1x10-3 2.2x10-2 0.70 24 4.86x10-4 1.06x10-2 0.375 13.73

0.10 1.3x10-3 2.7x10-2 0.90 31 5.66x10-4 1.32x10-2 0.476 17.80

No TVA 7.5x10-4 1.5x10-2 0.47 16.5 3.81x10-4 7.70x10-3 0.278 10.23


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 5.7x10-2 1.0 34 1200 3.23x10-2 0.576 21.16 782.3

0.02 6.1x10-2 1.05 36 1300 3.46x10-2 0.616 22.61 836.9

0.05 7.3x10-2 1.3 45 1600 4.05x10-2 0.729 26.91 1002.4

0.10 9.0x10-2 1.65 60 2200 5.08x10-2 0.946 35.59 1349.4

No TVA 5.2x10-2 0.90 31 1100 3.00x10-2 0.539 19.80 732.0

191

Case 2: Spring-mass system is attached to first span at x= (1/6) L and the step-



m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 9.5x10-4 1.8x10-2 0.67 25 5.21x10-4 1.25x10-2 0.466 17.34

0.02 9.5x10-4 1.9x10-2 0.67 24 5.00x10-4 1.21x10-2 0.450 16.80

0.05 8.5x10-4 1.8x10-2 0.60 22 4.32x10-4 1.06x10-2 0.398 14.96

0.10 7.2x10-4 1.5x10-2 0.50 18.5 3.36x10-4 8.46x10-3 0.320 12.22

No TVA 9.8x10-4 2.0x10-2 0.70 25 5.43x10-4 1.30x10-2 0.482 17.88


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 4.0x10-4 6.0x10-3 0.20 7 2.27x10-4 3.39x10-3 0.123 4.57

0.02 3.9x10-4 5.9x10-3 0.20 6.5 2.20x10-4 3.23x10-3 0.117 4.35

0.05 3.5x10-4 5.3x10-3 0.17 6 2.00x10-4 2.72x10-3 0.100 3.68

0.10 3.1x10-3 4.4x10-3 0.15 4.7 1.72x10-4 2.06x10-3 0.074 2.74

No TVA 4.2x10-4 6.5x10-3 0.21 7.3 2.37x10-4 3.63x10-3 0.130 4.79

192


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 9.5x10-4 1.9x10-2 0.68 24 5.15x10-4 1.24x10-2 0.459 17.08

0.02 9.3x10-4 1.9x10-2 0.67 24 4.89x10-4 1.18x10-2 0.437 16.27

0.05 8.0x10-4 1.7x10-2 0.60 21 4.07x10-4 9.88x10-3 0.367 13.76

0.10 6.7x10-4 1.5x10-2 0.50 17.5 3.03x10-4 7.36x10-3 0.272 10.26

No TVA 9.8x10-4 2.0x10-2 0.70 25 5.43x10-4 1.30x10-2 0.482 17.88

Case 3: Spring-mass system is attached to mid span at x= (3/6) L and the step-



193


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 3.5x10-4 4.0x10-3 0.12 3.2 1.96x10-4 1.74x10-3 0.047 1.42

0.02 3.5x10-4 4.1x10-3 0.11 3.2 1.93x10-4 1.72x10-3 0.046 1.35

0.05 3.5x10-4 4.0x10-3 0.11 3.0 1.88x10-4 1.68x10-3 0.044 1.23

0.10 3.3x10-4 3.9x10-3 0.10 2.7 1.85x10-4 1.71x10-3 0.043 1.15

No TVA 3.7x10-4 4.5x10-3 0.12 3.4 2.03x10-4 1.93x10-3 0.052 1.53


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 7.0x10-4 1.3x10-2 0.43 15 3.48x10-4 6.98x10-3 0.254 9.42

0.02 7.0x10-4 1.2x10-2 0.40 14.5 3.24x10-4 6.44x10-3 0.236 8.78

0.05 5.8x10-4 1.1x10-2 0.35 13 2.77x10-4 5.32x10-3 0.196 7.43

0.10 5.2x10-4 9.8x10-3 0.30 10 2.34x10-4 4.23x10-3 0.155 6.00

No TVA 7.5x10-4 1.5x10-2 0.47 16.5 3.81x10-4 7.70x10-3 0.278 10.23


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 4.9x10-2 0.8 30 1050 2.81x10-2 0.497 18.38 684.8

0.02 4.6x10-2 0.8 28 1000 2.65x10-2 0.464 17.25 647.8

0.05 4.2x10-2 0.7 25 900 2.33x10-2 0.394 14.93 573.0

0.10 3.7x10-2 0.65 22 800 2.03x10-2 0.323 12.52 496.7

No TVA 5.2x10-2 0.90 31 1100 3.00x10-2 0.539 19.80 732.0

194

Case 4: Spring-mass system is attached to mid span at x= (3/6) L and the step-


Table 4.171 Maximum and RMS responses at x=(1/6)L and x=(5/6)L for three-span uniform beam carrying one spring-mass system under impact loading -

Case 4

m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 1.1x10-3 2.1x10-2 0.75 27 5.59x10-4 1.37x10-2 0.510 19.11

0.02 1.1x10-3 2.3x10-2 0.80 29 5.77x10-4 1.44x10-2 0.540 20.41

0.05 1.3x10-3 2.7x10-2 0.97 36 6.30x10-4 1.65x10-2 0.633 24.52

0.10 1.5x10-3 3.2x10-2 1.2 45 6.95x10-4 1.95x10-2 0.777 31.30

No TVA 9.8x10-4 2.0x10-2 0.70 25 5.43x10-4 1.30x10-2 0.482 17.88


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 4.0x10-4 6.0x10-3 0.20 7.0 2.24x10-4 3.33x10-3 0.122 4.56

0.02 3.8x10-4 5.8x10-3 0.19 6.5 2.16x10-4 3.18x10-3 0.117 4.40

0.05 3.6x10-4 5.2x10-3 0.17 6.0 2.00x10-4 2.84x10-3 0.106 4.10

0.10 3.3x10-4 4.5x10-3 0.15 5.5 1.82x10-4 2.45x10-3 0.094 3.78

No TVA 4.2x10-4 6.5x10-3 0.21 7.3 2.37x10-4 3.63x10-3 0.130 4.79

195


Case 1: Spring-mass system is attached to first span at x= (1/6) L and the

harmonic load is applied to x=(1/6)L.

Figure 4.56 Three-span simply supported beam carrying one spring-mass system subjected to harmonic force at x= (1/6) L

Table 4.173 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying one spring-mass system under harmonic loading - Case 1

m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 3.6x10-4 7.8x10-3 0.18 4.3 1.56x10-4 3.18x10-3 0.069 1.68 6.3π

0.02 2.7x10-4 6.0x10-3 0.15 3.6 1.17x10-4 2.43x10-3 0.057 1.50 6.3π

0.05 1.8x10-4 4.0x10-3 0.11 3.0 7.86x10-5 1.72x10-3 0.045 1.36 6.3π

0.10 1.4x10-4 3.1x10-3 0.09 2.7 5.65x10-5 1.31x10-3 0.039 1.31 6.3π

No TVA 7.2x10-3 0.15 2.9 56 3.13x10-3 0.0623 1.23 24.49 6.3π

196


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 1.4x10-3 3.0x10-2 0.78 20 5.80x10-4 1.24x10-2 0.301 8.79 6.3π

0.02 1.3x10-3 3.0x10-2 0.78 20 5.69x10-4 1.23x10-2 0.309 9.22 6.3π

0.05 1.3x10-3 3.1x10-2 0.80 23.5 5.66x10-4 1.28x10-2 0.336 10.53 6.3π

0.10 1.4x10-3 3.4x10-2 0.90 28 5.84x10-4 1.38x10-2 0.391 13.04 6.3π

No TVA 7.3x10-3 0.15 3.0 65 3.03x10-3 0.0605 1.21 24.79 6.3π


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 5.0x10-2 1.30 36 1100 2.29x10-2 0.570 16.92 572.0 6.3π

0.02 5.5x10-2 1.35 40 1200 2.40x10-2 0.603 18.01 611.1 6.3π

0.05 6.0x10-2 1.55 45 1400 2.66x10-2 0.684 20.96 723.5 6.3π

0.10 7.0x10-2 1.90 58 1800 3.13x10-2 0.835 26.70 952.3 6.3π

No TVA 4.5x10-2 1.2 34 1000 2.00x10-2 0.511 15.52 531.7 6.3π

197

Case 2: Spring-mass system is attached to first span at x= (1/6) L and the



m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 1.0x10-3 2.6x10-2 0.70 22 4.57x10-4 1.16x10-2 0.358 12.45 6.3π

0.02 1.0x10-3 2.2x10-2 0.70 21 4.32x10-4 1.11x10-2 0.344 12.02 6.3π

0.05 9.0x10-4 2.4x10-2 0.65 20 3.90x10-4 9.93x10-3 0.306 10.69 6.3π

0.10 8.0x10-4 2.0x10-2 0.50 17 3.41x10-4 8.39x10-3 0.251 8.71 6.3π

No TVA 7.3x10-3 0.15 3.1 70 3.03x10-3 0.061 1.236 26.76 6.3π


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 5.5x10-4 1.3x10-2 0.31 8.2 2.35x10-4 5.13x10-3 0.127 3.70 6.3π

0.02 5.0x10-4 1.2x10-2 0.29 7.8 2.15x10-4 4.74x10-3 0.118 3.50 6.3π

0.05 4.3x10-4 1.0x10-2 0.24 6.75 1.91x10-4 4.19x10-3 0.103 3.00 6.3π

0.10 4.0x10-4 9.0x10-3 0.21 6.0 1.75x10-4 3.75x10-3 0.088 2.40 6.3π

No TVA 7.2x10-3 0.15 2.9 60 3.12x10-3 0.062 1.230 24.64 6.3π

198


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 7.8x10-4 2.1x10-2 0.65 20 3.39x10-4 1.00x10-2 0.338 12.12 6.3π

0.02 7.0x10-4 1.9x10-2 0.60 19 3.21x10-4 9.56x10-3 0.321 11.54 6.3π

0.05 6.5x10-4 1.8x10-2 0.55 17.5 2.96x10-4 8.46x10-3 0.276 9.81 6.3π

0.10 6.5x10-4 1.7x10-2 0.50 15 2.76x10-4 7.22x10-3 0.218 7.46 6.3π

No TVA 7.3x10-3 0.15 3.1 70 3.03x10-3 0.061 1.236 26.76 6.3π

Case 3: Spring-mass system is attached to mid span at x= (3/6) L and the



199


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 4.8x10-4 1.1x10-2 0.23 5.5 2.06x10-4 4.29x10-3 0.092 2.08 6.3π

0.02 4.2x10-4 9.0x10-3 0.21 5.0 1.80x10-4 3.80x10-3 0.083 1.91 6.3π

0.05 3.4x10-4 7.5x10-3 0.17 4.1 1.64x10-4 3.51x10-3 0.078 1.79 6.3π

0.10 3.8x10-4 8.0x10-3 0.19 4.2 1.59x10-4 3.44x10-3 0.077 1.76 6.3π

No TVA 7.2x10-3 0.15 2.9 56 3.13x10-3 0.0623 1.23 24.49 6.3π


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 8.0x10-4 2.0x10-2 0.50 15 3.59x10-4 8.22x10-3 0.222 7.11 6.3π

0.02 7.8x10-4 1.8x10-2 0.50 13.8 3.29x10-4 7.55x10-3 0.205 6.59 6.3π

0.05 6.8x10-4 1.6x10-2 0.42 12 2.94x10-4 6.64x10-3 0.176 5.58 6.3π

0.10 6.0x10-4 1.5x10-2 0.37 11 2.71x10-4 5.97x10-3 0.150 4.61 6.3π

No TVA 7.3x10-3 0.15 3.0 65 3.03x10-3 0.0605 1.21 24.79 6.3π


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 4.5x10-2 1.1 33 950 2.00x10-2 0.495 14.65 498.1 6.3π

0.02 4.5x10-2 1.1 32 900 1.90x10-2 0.469 13.80 469.7 6.3π

0.05 3.8x10-2 0.98 27 800 1.74x10-2 0.418 12.08 412.1 6.3π

0.10 3.6x10-2 0.9 25 750 1.59x10-2 0.371 10.41 354.2 6.3π

No TVA 4.5x10-2 1.2 34 1000 2.00x10-2 0.511 15.52 531.7 6.3π

200

Case 4: Spring-mass system is attached to mid span at x=(3/6)L and the


Table 4.182 Maximum and RMS responses at x=(1/6)L and x=(5/6)L for three-span uniform beam carrying one spring-mass system under harmonic loading -

Case 4

m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 1.5x10-3 3.6x10-2 0.90 27 6.52x10-4 1.51x10-2 0.421 13.99 6.3π

0.02 1.5x10-3 3.6x10-2 0.95 27 6.42x10-4 1.51x10-2 0.434 14.73 6.3π

0.05 1.5x10-3 3.6x10-2 1.0 31 6.50x10-4 1.59x10-2 0.483 17.16 6.3π

0.10 1.5x10-3 3.7x10-2 1.1 36 6.57x10-4 1.70x10-2 0.557 21.07 6.3π

No TVA 7.3x10-3 0.15 3.1 70 3.03x10-3 0.061 1.236 26.76 6.3π


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 3.7x10-4 8.2x10-3 0.22 6.25 1.57x10-4 3.60x10-3 0.101 3.34 6.3π

0.02 2.8x10-4 6.7x10-3 0.18 5.8 1.17x10-4 2.88x10-3 0.089 3.12 6.3π

0.05 2.0x10-4 5.0x10-3 0.15 4.75 7.98x10-5 2.18x10-3 0.075 2.80 6.3π

0.10 1.4x10-4 3.6x10-3 0.11 4.0 5.80x10-5 1.70x10-3 0.063 2.50 6.3π

No TVA 7.2x10-3 0.15 2.9 60 3.12x10-3 0.062 1.230 24.64 6.3π

201



is attached to first span at x=(1/6)L.

Figure 4.58 Three-span simply supported beam carrying one spring-mass system subjected to moving load

Table 4.184 Maximum and RMS responses at x=(1/6)L for three-span uniform

beam carrying one spring-mass system under moving load-Case 1

m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 4.2x10-3 7.0x10-4 1.2x10-2 0.40 2.84x10-3 3.20x10-4 0.0068 0.251

0.02 4.0x10-3 6.5x10-4 1.1x10-2 0.40 2.75x10-3 3.07x10-4 0.0065 0.238

0.05 3.6x10-3 6.0x10-4 1.0x10-2 0.35 2.45x10-3 2.68x10-4 0.0054 0.199

0.10 3.0x10-3 5.0x10-4 8.0x10-3 0.30 2.09x10-3 2.21x10-4 0.0041 0.148

No TVA 4.3x10-3 7.1x10-4 1.2x10-2 0.41 2.93x10-3 3.33x10-4 0.0072 0.264

202

Table 4.185 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying one spring-mass system under moving load-Case 1

m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 1.7x10-4 3.0x10-5 5.0x10-4 0.018 1.17x10-4 1.34x10-5 3.0x10-4 0.0109

0.02 1.7x10-4 3.0x10-5 5.3x10-4 0.018 1.17x10-4 1.34x10-5 2.9x10-4 0.0108

0.05 1.5x10-4 3.0x10-5 5.5x10-4 0.018 1.11x10-4 1.29x10-5 2.9x10-4 0.0105

0.10 1.5x10-4 2.8x10-5 5.5x10-4 0.020 1.05x10-4 1.25x10-5 2.8x10-4 0.0106

No TVA 1.8x10-4 2.9x10-5 5.0x10-4 0.018 1.18x10-4 1.35x10-5 3.0x10-4 0.0109


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 4.6x10-3 8.0x10-4 0.017 0.45 3.16x10-3 3.60x10-4 0.0078 0.285

0.02 5.0x10-3 8.2x10-4 0.015 0.50 3.38x10-3 3.85x10-4 0.0084 0.307

0.05 5.8x10-3 1.0x10-3 0.018 0.60 3.96x10-3 4.53x10-4 0.0101 0.373

0.10 7.1x10-3 1.2x10-3 0.022 0.80 4.91x10-3 5.68x10-4 0.0132 0.499

No TVA 4.3x10-3 7.1x10-4 1.2x10-2 0.41 2.93x10-3 3.33x10-4 0.0072 0.264

203


is attached to mid span at x=(3/6)L.

Table 4.187 Maximum and RMS responses at x=(1/6)L and x=(5/6)L for three-

span uniform beam carrying one spring-mass system under moving load-Case 2

m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 4.0x10-3 6.7x10-4 1.2x10-2 0.40 2.77x10-3 3.11x10-4 0.0067 0.245

0.02 3.8x10-3 6.5x10-4 1.1x10-2 0.38 2.63x10-3 2.94x10-4 0.0062 0.229

0.05 3.4x10-3 6.0x10-4 1.0x10-2 0.35 2.36x10-3 2.60x10-4 0.0053 0.199

0.10 3.0x10-3 5.5x10-4 9.0x10-3 0.30 2.10x10-3 2.28x10-4 0.0044 0.167

No TVA 4.3x10-3 7.1x10-4 1.2x10-2 0.41 2.93x10-3 3.33x10-4 0.0072 0.264


m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 1.6x10-4 2.7x10-5 4.5x10-4 0.017 1.11x10-4 1.26x10-5 2.7x10-4 0.0101

0.02 1.5x10-4 2.5x10-5 4.5x10-4 0.015 1.05x10-4 1.18x10-5 2.5x10-4 0.0094

0.05 1.4x10-4 2.3x10-5 4.0x10-4 0.014 9.34x10-5 1.03x10-5 2.1x10-4 0.0080

0.10 1.2x10-4 2.0x10-5 3.4x10-4 0.012 8.20x10-5 8.80x10-6 1.7x10-4 0.0065

No TVA 1.8x10-4 2.9x10-5 5.0x10-4 0.018 1.18x10-4 1.35x10-5 3.0x10-4 0.0109

204



mass system is attached to first span at x=(1/6)L.

Figure 4.59 Three-span simply supported beam carrying one spring-mass system subjected to moving pulsating force


beam carrying one spring-mass system under moving pulsating force -Case 1

m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 8.8x10-3 0.17 3.4 67.5 4.21x10-3 8.34x10-2 1.65 32.67 6.3π

0.02 8.5x10-3 0.17 3.4 65 4.12x10-3 8.16x10-2 1.61 31.97 6.3π

0.05 8.0x10-3 0.15 3.0 60 3.81x10-3 7.54x10-2 1.49 29.57 6.3π

0.10 7.0x10-3 0.14 2.8 55 3.45x10-3 6.84x10-2 1.35 26.80 6.3π

No TVA 0.013 0.25 5.0 100 6.28x10-3 0.124 2.46 48.76 6.3π

205



m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 4.0x10-4 8.0x10-3 0.16 3.2 1.99x10-4 3.95x10-3 0.078 1.55 6.3π

0.02 4.0x10-4 8.0x10-3 0.16 3.2 1.98x10-4 3.92x10-3 0.078 1.54 6.3π

0.05 4.0x10-4 7.5x10-3 0.15 3.0 1.92x10-4 3.80x10-3 0.075 1.49 6.3π

0.10 3.8x10-4 7.5x10-3 0.15 3.0 1.85x10-4 3.66x10-3 0.072 1.43 6.3π

No TVA 3.1x10-4 6.0x10-3 0.12 2.4 1.50x10-4 2.98x10-3 0.059 1.17 6.3π



m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 7.5x10-3 0.15 3.0 60 3.72x10-3 7.36x10-2 1.457 28.85 6.3π

0.02 8.0x10-3 0.16 3.1 63 3.92x10-3 7.75x10-2 1.534 30.38 6.3π

0.05 9.0x10-3 0.18 3.6 70 4.38x10-3 8.67x10-2 1.717 33.98 6.3π

0.10 1.1x10-2 0.20 4.0 70 5.14x10-3 1.02x10-1 2.016 39.91 6.3π

No TVA 0.016 0.32 6.25 125 7.82x10-3 0.155 3.06 60.65 6.3π

206


mass system is attached to mid span at x=(3/6)L.

Table 4.192 Maximum and RMS responses at x=(1/6)L and x=(5/6)L for three-

span uniform beam carrying one spring-mass system under moving pulsating

force -Case 2

m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 8.8x10-3 0.17 3.5 67.5 4.24x10-3 8.40x10-2 1.663 32.92 6.3π

0.02 8.2x10-3 0.17 3.3 65 4.10x10-3 8.12x10-2 1.607 31.81 6.3π

0.05 8.0x10-3 0.15 3.0 60 3.82x10-3 7.56x10-2 1.496 29.62 6.3π

0.10 7.5x10-3 0.145 2.9 58 3.55x10-3 7.04x10-2 1.393 27.58 6.3π

No TVA 0.013 0.25 5.0 100 6.28x10-3 0.124 2.46 48.76 6.3π



m1/mb wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 3.2x10-4 6.2x10-3 0.125 2.5 1.56x10-4 3.09x10-3 0.061 1.209 6.3π

0.02 3.0x10-4 6.0x10-3 0.120 2.4 1.50x10-4 2.96x10-3 0.059 1.162 6.3π

0.05 2.8x10-4 5.5x10-3 0.110 2.2 1.38x10-4 2.72x10-3 0.054 1.067 6.3π

0.10 2.6x10-4 5.0x10-3 0.100 2.0 1.26x10-4 2.50x10-3 0.049 0.979 6.3π

No TVA 3.1x10-4 6.0x10-3 0.12 2.4 1.50x10-4 2.98x10-3 0.059 1.17 6.3π

207

4.7.4. Forced Vibration Analysis of Three Span Beam Carrying



Case 1: Spring-mass systems are attached to first and second span at x=(1/6) L

and x=(3/6) L and the step-function force is applied to x=(1/6)L. Both of the




208

Table 4.194 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying two spring-mass systems under impact loading - Case 1

m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 3.5x10-4 4.0x10-3 0.11 3.3 1.97x10-4 1.76x10-3 0.048 1.47

0.02 3.5x10-4 4.0x10-3 0.11 3.2 1.95x10-4 1.76x10-3 0.048 1.47

0.05 3.4x10-4 4.0x10-3 0.11 3.2 1.92x10-4 1.72x10-3 0.048 1.47

0.10 3.3x10-4 3.8x10-3 0.10 3.0 1.88x10-4 1.62x10-3 0.046 1.50

No TVA 3.7x10-4 4.5x10-3 0.12 3.4 2.03x10-4 1.93x10-3 0.052 1.53


m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 8.0x10-4 1.5x10-2 0.50 17.0 3.77x10-4 7.68x10-3 0.276 10.16

0.02 8.5x10-4 1.6x10-2 0.52 17.5 3.79x10-4 7.82x10-3 0.277 10.19

0.05 9.5x10-4 1.9x10-2 0.60 20.0 4.03x10-4 8.71x10-3 0.301 11.06

0.10 1.05x10-3 2.4x10-2 0.75 26.0 4.53x10-4 1.05x10-2 0.364 13.44

No TVA 7.5x10-4 1.5x10-2 0.47 16.5 3.81x10-4 7.70x10-3 0.278 10.23


m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 5.4x10-2 0.90 32 1150 3.04x10-2 0.537 19.87 739.9

0.02 5.6x10-2 1.00 32 1200 3.11x10-2 0.541 20.05 752.2

0.05 6.4x10-2 1.10 40 1400 3.46x10-2 0.593 22.17 845.3

0.10 7.5x10-2 1.40 50 1900 4.20x10-2 0.725 27.56 1076.1

No TVA 5.2x10-2 0.90 31 1100 3.00x10-2 0.539 19.80 732.0

209


Case 1: Spring-mass systems are attached to first and second span at x=(1/6) L

and x=(3/6) L and the harmonic load is applied to x=(1/6)L. Both of the spring-

mass systems are tuned to constant frequency which is same with the first



Table 4.197 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying two spring-mass systems under harmonic loading - Case 1

m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 3.0x10-4 6.5x10-3 0.16 3.8 1.20x10-4 2.55x10-3 0.059 1.51 6.3π

0.02 2.3x10-4 5.0x10-3 0.13 3.2 9.45x10-5 2.09x10-3 0.051 1.39 6.3π

0.05 1.6x10-4 3.8x10-3 0.10 2.7 6.78x10-5 1.61x10-3 0.043 1.27 6.3π

0.10 1.2x10-4 2.8x10-3 0.08 2.3 4.95x10-5 1.25x10-3 0.037 1.19 6.3π

No TVA 7.2x10-3 0.15 2.9 56 3.13x10-3 0.0623 1.23 24.49 6.3π

210


m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 1.2x10-3 2.4x10-2 0.65 19 5.37x10-4 1.15x10-2 0.28 8.19 6.3π

0.02 1.2x10-3 2.6x10-2 0.70 19 5.27x10-4 1.14x10-2 0.28 8.20 6.3π

0.05 1.1x10-3 2.8x10-2 0.7 20 5.19x10-4 1.15x10-2 0.29 8.75 6.3π

0.10 1.15x10-3 2.9x10-2 0.78 23 5.22x10-4 1.20x10-2 0.32 10.24 6.3π

No TVA 7.3x10-3 0.15 3.0 65 3.03x10-3 0.0605 1.21 24.79 6.3π


m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 5.0x10-2 1.25 35 1000 2.17x10-2 0.538 15.91 539.04 6.3π

0.02 5.0x10-2 1.23 34 1100 2.20x10-2 0.545 16.10 546.75 6.3π

0.05 5.0x10-2 1.30 40 1200 2.35x10-2 0.586 17.54 605.68 6.3π

0.10 6.0x10-2 1.60 48 1500 2.66x10-2 0.682 21.11 753.16 6.3π

No TVA 4.5x10-2 1.2 34 1000 2.00x10-2 0.511 15.52 531.7 6.3π

211



are attached to first and second span at x=(1/6) L and x=(3/6) L. Both of the



Figure 4.62 Three-span simply supported beam carrying one spring-mass system subjected to moving load



m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 4.0x10-3 6.5x10-4 1.1x10-2 0.40 2.70x10-3 3.02x10-4 0.0064 0.236

0.02 3.6x10-3 6.0x10-4 1.0x10-2 0.36 2.52x10-3 2.77x10-4 0.0057 0.213

0.05 3.2x10-3 5.5x10-4 9.0x10-3 0.32 2.21x10-3 2.39x10-4 0.0046 0.174

0.10 2.8x10-3 4.8x10-4 8.0x10-3 0.28 1.96x10-3 2.14x10-4 0.0039 0.143

No TVA 4.3x10-3 7.1x10-4 1.2x10-2 0.41 2.93x10-3 3.33x10-4 0.0072 0.264

212

Table 4.201 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying two spring-mass systems under moving load-Case 1

m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 1.6x10-4 2.8x10-5 4.8x10-4 0.017 1.11x10-4 1.26x10-5 2.8x10-4 0.0101

0.02 1.5x10-4 2.6x10-5 4.8x10-4 0.016 1.05x10-4 1.19x10-5 2.6x10-4 0.0095

0.05 1.4x10-4 2.6x10-5 4.8x10-4 0.016 9.61x10-5 1.09x10-5 2.3x10-4 0.0086

0.10 1.3x10-4 2.4x10-5 4.8x10-4 0.016 9.00x10-5 1.03x10-5 2.2x10-4 0.0083

No TVA 1.8x10-4 2.9x10-5 5.0x10-4 0.018 1.18x10-4 1.35x10-5 3.0x10-4 0.0109

Table 4.202 Maximum and RMS responses at x=(5/6)L for three-span uniform beam carrying two spring-mass systems under moving load-Case 1

m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

0.01 4.4x10-3 7.4x10-4 1.3x10-2 0.45 3.00x10-3 3.39x10-4 0.0073 0.268

0.02 4.5x10-3 7.5x10-4 1.4x10-2 0.45 3.07x10-3 3.46x10-4 0.0074 0.272

0.05 5.0x10-3 8.5x10-4 1.5x10-2 0.53 3.42x10-3 3.84x10-4 0.0082 0.306

0.10 6.0x10-3 1.0x10-3 2.0x10-2 0.70 4.14x10-3 4.64x10-4 0.010 0.385

No TVA 4.3x10-3 7.1x10-4 1.2x10-2 0.41 2.93x10-3 3.33x10-4 0.0072 0.264

213



mass systems are attached to first and second span at x=(1/6) L and x=(3/6) L.

Both of the spring-mass systems are tuned to constant frequency which is same

with the first natural frequency of three-span bare uniform beam.

Figure 4.63 Three-span simply supported beam carrying one spring-mass system subjected to moving pulsating force


beam carrying two spring-mass systems under moving pulsating force -Case 1

m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 8.5 x10-3 0.17 3.4 65 4.17x10-3 8.26x10-2 1.64 32.37 6.3π

0.02 8.0 x10-3 0.16 3.2 63 3.98x10-3 7.88x10-2 1.56 30.87 6.3π

0.05 7.5x10-3 0.15 3.0 60 3.66x10-3 7.24x10-2 1.43 28.38 6.3π

0.10 7.0 x10-3 0.14 2.7 55 3.41x10-3 6.75x10-2 1.34 26.46 6.3π

No TVA 0.013 0.25 5.0 100 6.28x10-3 0.124 2.46 48.76 6.3π

214



m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 3.6x10-4 7.2x10-3 0.14 2.8 1.78x10-4 3.53x10-3 0.070 1.38 6.3π

0.02 3.5x10-4 7.0x10-3 0.14 2.8 1.72x10-4 3.40x10-3 0.067 1.33 6.3π

0.05 3.3x10-4 6.5x10-3 0.13 2.5 1.62x10-4 3.20x10-3 0.063 1.25 6.3π

0.10 3.2x10-4 6.2x10-3 0.12 2.4 1.54x10-4 3.05x10-3 0.060 1.20 6.3π

No TVA 3.1x10-4 6.0x10-3 0.12 2.4 1.50x10-4 2.98x10-3 0.059 1.17 6.3π



m1/mb=

m2/mb

wmax

(m)

vmax

(m/s)

a max

(m/s2)

j max

(m/s3)

wrms

(m)

vrms

(m/s)

a rms

(m/s2)

j rms

(m/s3)

Ω

(rad/s)

0.01 8.0x10-3 0.16 3.2 65 4.04x10-3 7.99x10-2 1.581 31.30 6.3π

0.02 8.0x10-3 0.16 3.2 65 4.07x10-3 8.06x10-2 1.595 31.57 6.3π

0.05 8.5x10-3 0.18 3.4 67.5 4.31x10-3 8.53x10-2 1.688 33.43 6.3π

0.10 1.0x10-2 0.19 3.8 75 4.83x10-3 9.57x10-2 1.894 37.50 6.3π

No TVA 0.016 0.32 6.25 125 7.82x10-3 0.155 3.06 60.65 6.3π

215

CHAPTER 5. SUMMARY AND CONCLUSIONS

5.1. Summary

The main objective of this study is to investigate the effects of spring-mass

systems (Tuned Vibration Absorbers, or TVAs) attached to Euler-Bernoulli

beams in order to control the response due to excessive vibrations. Effectiveness

of tuned vibration absorbers has been studied and their performance evaluated

through comparisons on an extensive combination of loading dynamic types,

span configurations, and TVA distributions. The proposed method includes the

exact solutions of natural frequencies and mode shapes of uniform and non-

uniform beams carrying any number of passive tuned mass dampers (TMD) and

forced vibration of these beams based on their free vibration data. A

mathematical formulation has been presented for free and forced vibration of

beams. An algorithm has been developed through MATHEMATICA and

numerical results have been obtained for various forcing systems and boundary

conditions.

Several numerical examples have been provided in order to evaluate the

performance of TMDs under free and forced vibration of uniform and non-uniform

beams carrying single or multiple spring-mass systems with various boundary

conditions. Free vibration characteristics of beams carrying elastically attached

point masses are obtained through numerical assembly method (Wu and Chou,

1999). Overall coefficient matrix is generated by combining the coefficient

matrices of each boundaries of the beam and each attaching points for a spring-

mass system through conventional assembly technique for finite element

method. Numerical assembly method is used in order to derive the eigenvalue

equation and then the developed algorithm is used for the solution of the

216

eigenvalues and the corresponding mode shapes. The accuracy of the

developed algorithm in this study is evaluated by comparing its numerical results

with existing literature.

The first part of the present study deals with the determination of natural

frequencies and corresponding mode shapes of single-span uniform beams,

single-span non-uniform beams and multi-span uniform beams carrying any

number of spring-mass systems and the second part calculates the forced

vibration responses of uniform beams under the excitation of step-function forces

(Impact Loading), harmonic forces, moving loads and moving pulsating forces.

First and second part also includes the free and forced vibration of a high mast

lighting tower (HMLT) which is subjected to wind induced dynamic load and

represented as non-uniform cantilever beam. The beams are considered as

continuous structural elements and both free and forced vibration solutions are

analyzed using thin beam (Euler-Bernoulli) theory. For single-span beams, four

boundary conditions are studied including simply supported-simply supported,

clamped-clamped, clamped-simply supported and clamped-free boundaries. On

the other hand, each intermediate support is assumed as simply supported for

multi-span beams. For the force vibration response of the structural elements,

90% of modal mass contribution is considered to be sufficient. Forced vibration of

the entire beam is obtained by using normal mode approach and linear

combination of the normal modes. Displacement, velocity, acceleration and jerk

responses of the entire beam with TMDs are calculated and the resultant

responses are compared with the beam without TMDs.

The illustrative numerical examples presented in this study are based on

human induced loads for single-span and multi-span uniform beams. Moreover,

single-span non-uniform beams are subjected to wind-induced loads to evaluate

the passive vibration control of HMLTs. Harmonic forces are considered as

repeated forces caused by human activities such as walking or dancing and

217

represented as time–dependent sinusoidal forcing. Harmonic moving loads are

also represented as concentrated loads with sinusoidally varying amplitude and

moving with a constant speed v0 which is the average human walking speed,

80m/min. Moreover, footstep impulse vibration has defined as step-function force

and non-harmonic moving load is considered to simulate pedestrian walking load

with a constant speed. The step-function and harmonic forces are represented by

using Dirac delta function and moving load and moving pulsating force are

expressed by using Fourier series. To evaluate the performance of TMDs, the

frequency component of the exciting forces is selected to match with the natural

frequency of the beams in order to simulate and approximate the condition for

resonance which could be the worst scenario causing significant vibration

amplification.

In addition, the performance of TMD under wind induced vibration has been

investigated through an analysis performed for a HMLT structure which is

assumed to be a non-uniform cantilever beam to carry out the proposed method

in this study. Wind induced dynamic loads are estimated using available wind

velocity data obtained at a certain height of the HMLT. The wind velocity profile

has been generated with empirical power-law method based on five selected

points throughout the height of HMLT and the corresponding forcing functions

with respect to time are defined by Fourier series using obtained wind profiles for

each selected points. A TMD is attached to top of the structure and dynamic

response at that point has been compared with the results obtained from bare

HMLT under wind-induced vibration.

5.2. Conclusion

This study presents the free and forced vibration of beams carrying any

number of spring-mass systems and the resultant responses were compared

with the bare uniform and non-uniform beams. Based on the observations from

218

numerical results for single span uniform, single span non-uniform and multi-

span uniform beams, the following conclusions are made;

1. It is observed that TMDs are very effective when they are properly

tuned to the excitation frequency. The effectiveness level of TMD

increases if the excitation frequency converges to the any normal

mode frequency causing the condition for resonance.

2. When TMD is tuned to the exact excitation frequency, it can be

concluded that single TMD application is more effective than multiple

TMDs of the same total mass ratio based on the peak resultant

responses or RMS values obtained from the main structure for

harmonic excitations. On the other hand, it is difficult to estimate the

exact excitation frequency in practice. Therefore it may be more

effective to implement multiple TMDs within a small frequency range in

order to overcome randomly varying excitations such as wind induced

or human induced vibrations.

3. TMD loses its effectiveness when the structure is subjected to non-

harmonic excitations such as step-function forces or constant moving

loads.

4. If the natural frequency of the TMD diverges further from the excitation

frequency or fundamental frequency of the structure, the performance

of TMD significantly decreases.

5. Single or multiple TMD application is more effective and robust when

the attached mass is increased without changing natural frequency of

properly tuned TMD under harmonic excitations.

219

6. Based on the results obtained from wind induced vibration analysis of

HMLT structure, it can be confirmed that a passive TMD can be

effective in reducing the dynamic response of HMLTs when it is

properly tuned to the fundamental frequency of the HMLT it is installed

in.

7. It is also observed that a TMD attached to top of the HMLT structure

having 1% mass of the total mass of the structure decreases the wind

induced dynamic response by about 50%.

8. The results for HMLT structure also show that the dynamic response of

the main structure does not change with the increase of the mass of

TMD under wind induced vibration. The relative motion of TMD is also

in practical limit and able to be accommodated in the actual structure.

9. For multi-span beams, when TMDs are used for each span, one of the

normal mode frequencies being dominated by any of the TMDs may

converge to the fundamental frequency of the structure and this may

cause undesirable responses. Therefore, this case requires a careful

consideration on selecting the final dynamic characteristics of the

TMDs for multi-span beams.

10. Although the first four lowest natural frequencies and corresponding

mode shapes of the structures are found without having any problem

by solving the eigenvalue equation using the developed algorithm in

MATHEMATICA, some numerical difficulties are encountered in finding

the roots of determinant expression of coefficient matrix in the

eigenvalue equation. The precise starting values for finding the roots of

determinant expression increase the computational time in case of

more than two TMDs particularly.

220

5.3. Future Work

No damping characteristics of the structure or the TMDs have been

included in the proposed method of this study. Based on the numerous examples

given in this study, the use of single or multiple TMDs is significantly effective in

reducing the dynamic response of the main structure under harmonic excitations

and wind induced vibration. However, the study may be extended by considering

the effects of damping characteristics of the structure and TMD in order to

generalize the results of this study.

Experimental study is needed to extend this research and to obtain more

information in order to understand the performance of TMD in actuality.

Moreover, experimental results are necessary to study the level of reliability of

the analytical and numerical method proposed in this study.

This study may also be extended for the free and forced vibration of

uniform rectangular plates with attached TMDs located at arbitrary points and the

performance of TMDs can be also evaluated for floor vibration control of

structures subjected to human induced vibration particularly.

In this study, it is also indicated that the use of single TMD has a

disadvantage when the excitation frequency is not known exactly. However,

multiple TMDs within a small frequency range may perform better for randomly

varying excitations. As a result of this, it is recommended to investigate the

optimum parameters of multiple TMDs having different dynamic characteristics in

order to improve the effectiveness of vibration control.

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221

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APPENDICES

228

Appendix A.

A.1. Detailed formulation for the responses of SDOF-TMD system

Figure A.1 SDOF-TMD system

Parameters of primary structure and TMD;

Eq. A.1

2 Eq. A.2

Eq. A.3

2 Eq. A.4

The mass ratio, µ, is defined as;

Eq. A.5

The governing equations of motion for the SDOF-TMD system are as

follows;

0 Eq. A.6

0 Eq. A.7

From equations A.6 and A.7 one obtains;

Eq. A.8

Dividing equations A.6 and A.7 by md the governing equations of motions

are as follows;

229

Primary mass,

1 2 Eq. A.9

Tuned mass,

2 Eq. A.10

The optimal approximation for the damper is assumed as,

Eq. A.11

The stiffness relation between damper and structure is defined as,

Eq. A.12

And the periodic excitation can be shown as follows,

sin Eq. A.13

The responses of the structure and the damper is given by,

sin Eq. A.14

sin Eq. A.15

The critical scenario is the equality of Ω and ω which is resonant condition

and the solutions for this case are as follows,

sin cos Eq. A.16

sin cos Eq. A.17

cos sin Eq. A.18

cos sin Eq. A.19

Eq. A.20

Eq. A.21

sin Eq. A.22

sin Eq. A.23

When equations A.16 and A.17 are introduced into equations A.9 and

A.10, one obtains,

1 sin cos 2 cos sin

sin cos sin sin cos

Eq. A.24

230

sin cos sin cos sin

cos cos sin 0

Eq. A.25

If equations A.24 and A.25 are simplified, one obtains,

1 2

Eq. A.26

1 2 Eq. A.27

0 Eq. A.28

0 Eq. A.29

where,

Eq. A.30

tan Eq. A.31

tan Eq. A.32

2 Eq. A.33

2 Eq. A.34

If equation A.30 is substituted into equations A.26, A.27, A28 and A29,

one obtains,

2 Eq. A.35

2 0 Eq. A.36

2 0 Eq. A.37

2 0 Eq. A.38

The simplified forms of equations A.35, A.36, A.37 and A.38,

2 Eq. A.39

2 0 Eq. A.40

2 0 Eq. A.41

2 0 Eq. A.42

A, B, C and D are as follows after solving equations A.39, A.40, A.41 and

A.42;

231

4

8 4 16 Eq. A.43

2 4

8 4 16 Eq. A.44

4

8 4 16 Eq. A.45

2

8 4 16 Eq. A.46

And the responses are given by;

2

8 16 1 4 Eq. A.47

The simplified form of equation A.47 is given by,

1

1 2 12

Eq. A.48

8 16 1 4 Eq. A.49

12

Eq. A.50

The response for no damper is as follows;

0 Eq. A.51

2sin Eq. A.52

sin cos sin Eq. A.53

cos sin Eq. A.54

232

When equation A.53 is introduced into equation A.52, one obtains,

sin cos 2 cos sin sin

cos sin

Eq. A.55

The simplified form of equation A.55 is given by,

2 0 Eq. A.56

2 Eq. A.57

where , and A and B are as follows after solving equations A.56 and

A.57,

0 Eq. A.58

12

1

2

2

Eq. A.59

And the response for no damper is given by;

2

12

Eq. A.60

Equation A.60 can be expressed in terms of equivalent damping ratio in

order to compare the cases with and without damper.

12

Eq. A.61

where,

21

2 12

Eq. A.62

If equation A.54 substituted in equation A.53, one obtains,

sin cos cos sin Eq. A.63

tan Eq. A.64

tan Eq. A.65

2 Eq. A.66

233

Example

Assume that 0 and 0.1, the relation between and is as

follows

21

2 12

0.1 Eq. A.67

where

12

Eq. A.68

Inserting equation Eq.A.68 into equation Eq.A.67 and assuming that

0 as indicated above gives,

21 0.1 Eq. A.69

Since is greater than 1, equation A.69 can be written as,

20.1 Eq. A.70

If is assumed to be 10, then equation A.70 gives an estimate for ,

2 0.110

0.02 Eq. A.71

and from equation A.68 and A.12,

12

0.12

0.05 Eq. A.72

0.02 Eq. A.73

Therefore, 2% of the primary mass provides an effective damping ratio of

10% as it is shown in the above. On the other hand, the large relative motion of

the damper mass should be considered during design stage in order to control

this motion in a real structure.

234

Appendix B.

B.1. Determination of and for Different Boundary Conditions for

Uniform Beams

a) Clamped-Clamped beam

1 2 3 4

0 1 0 1

1 0 1 0

1

2

Eq.B.1

4 1 4 2 4 3 4 4

sin cos sinh cosh

cos sin cosh sinh

1

Eq.B.2

b) Simply supported-Simply Supported beam

1 2 3 4

0 1 0 1

0 1 0 1

1

2

Eq.B.3

4 1 4 2 4 3 4 4

sin cos sinh cosh

sin cos sinh cosh

1

Eq.B.4

c) Clamped-Simply Supported beam

1 2 3 4

0 1 0 1

1 0 1 0

1

2

Eq.B.5

4 1 4 2 4 3 4 4

sin cos sinh cosh

sin cos sinh cosh

1

Eq.B.6

235

d) Clamped-Free End with Attached Mass (M) beam

0 0 0 0 Eq.B.7

0 Eq.B.8

1 2 3 4

0 1 0 1

1 0 1 0

1

2

Eq.B.9

4 1 4 2 4 3 4 4

sin cos sinh cosh 1

Eq.B.10

where

cos sin Eq.B.11

sin cos Eq.B.12

cosh sinh Eq.B.13

sinh cosh Eq.B.14

B.2. Coefficient Matrix for Uniform Simply Supported Beams Carrying Multiple Spring-Mass Systems

Case 1-Uniform Simply Supported Beam Carrying Two Spring-Mass Systems

1

0 1 0 1 0 0 0 0 0 0 0 0 0 0

0 1 0 1 0 0 0 0 0 0 0 0 0 0

sin 1 cos 1 sinh 1 cosh 1 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0

cos 1 sin 1 cosh 1 sinh 1 cos 1 sin 1 cosh 1 sinh 1 0 0 0 0 0 0

sin 1 cos 1 sinh 1 cosh 1 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0

cos 1 sin 1 cosh 1 sinh 1 cos 1 sin 1 cosh 1 sinh 1 0 0 0 0 1 0

sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 12 1 0

0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 sin 2 cos 2 sinh 2 cosh 2 0 0

0 0 0 0 cos 2 sin 2 cosh 2 sinh 2 cos 2 sin 2 cosh 2 sinh 2 0 0

0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 sin 2 cos 2 sinh 2 cosh 2 0 0

0 0 0 0 cos 2 sin 2 cosh 2 sinh 2 cos 2 sin 2 cosh 2 sinh 2 0 2

0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 0 0 0 0 0 22 1

0 0 0 0 0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0 0

0 0 0 0 0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0 0

236

Case 2-Uniform Simply Supported Beam Carrying Three Spring-Mass Systems

1

0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sin 1 cos 1 sinh 1 cosh 1 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0 0 0

cos 1 sin 1 cosh 1 sinh 1 cos 1 sin 1 cosh 1 sinh 1 0 0 0 0 0 0 0 0 0 0 0

sin 1 cos 1 sinh 1 cosh 1 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0 0 0

cos 1 sin 1 cosh 1 sinh 1 cos 1 sin 1 cosh 1 sinh 1 0 0 0 0 0 0 0 0 1 0 0

sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0 0 0 0 12 1 0 0

0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 sin 2 cos 2 sinh 2 cosh 2 0 0 0 0 0 0 0

0 0 0 0 cos 2 sin 2 cosh 2 sinh 2 cos 2 sin 2 cosh 2 sinh 2 0 0 0 0 0 0 0

0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 sin 2 cos 2 sinh 2 cosh 2 0 0 0 0 0 0 0

0 0 0 0 cos 2 sin 2 cosh 2 sinh 2 cos 2 sin 2 cosh 2 sinh 2 0 0 0 0 0 2 0

0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 0 0 0 0 0 0 0 0 0 22 1 0

0 0 0 0 0 0 0 0 sin 3 cos 3 sinh 3 cosh 3 sin 3 cos 3 sinh 2 cosh 2 0 0 0

0 0 0 0 0 0 0 0 cos 3 sin 3 cosh 3 sinh 3 cos 3 sin 3 cosh 2 sinh 2 0 0 0

0 0 0 0 0 0 0 0 sin 3 cos 3 sinh 3 cosh 3 sin 3 cos 3 sinh 2 cosh 2 0 0 0

0 0 0 0 0 0 0 0 cos 3 sin 3 cosh 3 sinh 3 cos 3 sin 3 cosh 2 sinh 2 0 0 3

0 0 0 0 0 0 0 0 sin 3 cos 3 sinh 3 cosh 3 0 0 0 0 0 0 32 1

0 0 0 0 0 0 0 0 0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0 0 0

237

238

Appendix C.

C.1. Determination of and for Different Boundary Conditions for Non-

Uniform Beams

a) Clamped-Clamped beam

1 2 3 4

1

2

Eq.C.1

4 1 4 2 4 3 4 4

√ √ √ √

√ √ √ √

1

Eq.C.2

b) Simply supported-Simply Supported beam

1 2 3 4

1

2

Eq.C.3

4 1 4 2 4 3 4 4

√ √ √ √

√ √ √ √

1

Eq.C.4

c) Simply Supported-Clamped beam

1 2 3 4

1

2

Eq.C.5

4 1 4 2 4 3 4 4

√ √ √ √

√ √ √ √

1

Eq.C.6

239

d) Free End with Attached Mass (M)-Clamped beam

For free end

0 1

therefore

Shear,

1 0 Eq.C.7

Bending,

Eq.C.8

0 Eq.C.9

6

Eq.C.10

1 2 3 4

1

2

Eq.C.11

4 1 4 2 4 3 4 4

√ √ √ √

√ √ √ √

1

Eq.C.12

where

68

1 Eq.C.13

68

1 Eq.C.14

68

1 Eq.C.15

68

1 Eq.C.16

C.2. Coefficient Matrix for Non-Uniform Free-Clamped Beams Carrying Multiple Spring-Mass Systems

Case 1-Non-Uniform Free-Clamped Beam Carrying Two Spring-Mass Systems

1

3 3 3 3 0 0 0 0 0 0 0 0 0 0

1 2 3 4 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0

2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 0 0 0 0 0 0

3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 0 0 0 0 0 0

∆11 ∆21 ∆31 ∆41 ∆51 ∆61 ∆71 ∆81 0 0 0 0 0 0

1

12 1 1 1

12 1 1 1

12 1 1 1

12 1 1 0 0 0 0 0 0 0 0 1

2 1 0

0 0 0 0 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 0 0

0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 0

0 0 0 0 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 0 0

0 0 0 0 ∆12 ∆22 ∆32 ∆42 ∆52 ∆62 ∆72 ∆82 0 0

0 0 0 0 2

12 1 2 2

12 1 2 2

12 1 2 2

12 1 2 0 0 0 0 0 2

2 1

0 0 0 0 0 0 0 0 1 √ 1 √ 1 √ 1 √ 0 0

0 0 0 0 0 0 0 0 2 √ 2 √ 2 √ 2 √ 0 0

240

Appendix D.

D.1. Coefficient Matrix for Multi-Span Uniform Beams Carrying Multiple Spring-Mass Systems

Case 1-Two-Span Uniform Beam Carrying Two Spring-Mass Systems

1

0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sin 1 cos 1 sinh 1 cosh 1 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0 0

cos 1 sin 1 cosh 1 sinh 1 cos 1 sin 1 cosh 1 sinh 1 0 0 0 0 0 0 0 0 0 0

sin 1 cos 1 sinh 1 cosh 1 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0 0

cos 1 sin 1 cosh 1 sinh 1 cos 1 sin 1 cosh 1 sinh 1 0 0 0 0 0 0 0 0 1 0

sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0 0 0 0 12 1 0

0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0

0 0 0 0 cos 1 sin 1 cosh 1 sinh 1 cos 1 sin 1 cosh 1 sinh 1 0 0 0 0 0 0

0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 sin 2 cos 2 sinh 2 cosh 2 0 0

0 0 0 0 0 0 0 0 cos 2 sin 2 cosh 2 sinh 2 cos 2 sin 2 cosh 2 sinh 2 0 0

0 0 0 0 0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 sin 2 cos 2 sinh 2 cosh 2 0 0

0 0 0 0 0 0 0 0 cos 2 sin 2 cosh 2 sinh 2 cos 2 sin 2 cosh 2 sinh 2 0 2

0 0 0 0 0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 0 0 0 0 0 22 1

0 0 0 0 0 0 0 0 0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0 0

241

Case 2-Three-Span Uniform Beam Carrying One Spring-Mass System attached to First Span

1

0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

sin 1 cos 1 sinh 1 cosh 1 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0

cos 1 sin 1 cosh 1 sinh 1 cos 1 sin 1 cosh 1 sinh 1 0 0 0 0 0 0 0 0 0



sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0 0 0 0 12 1

0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0

0 0 0 0 cos 1 sin 1 cosh 1 sinh 1 cos 1 sin 1 cosh 1 sinh 1 0 0 0 0 0

0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0

0 0 0 0 0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 0

0 0 0 0 0 0 0 0 cos 2 sin 2 cosh 2 sinh 2 cos 2 sin 2 cosh 2 sinh 2 0

0 0 0 0 0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 sin 2 cos 2 sinh 2 cosh 2 0

0 0 0 0 0 0 0 0 0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0

0 0 0 0 0 0 0 0 0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0

242

Case 3-Three-Span Uniform Beam Carrying One Spring-Mass System attached to Mid Span

1

0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0







0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 12 1

0 0 0 0 0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 0

0 0 0 0 0 0 0 0 cos 2 sin 2 cosh 2 sinh 2 cos 2 sin 2 cosh 2 sinh 2 0

0 0 0 0 0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 sin 2 cos 2 sinh 2 cosh 2 0

0 0 0 0 0 0 0 0 0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0

0 0 0 0 0 0 0 0 0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0

243

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MS Thesis_Mustafa Kemal Ozkan