Experimental Investigations On Free Vibration Of Beams

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Research and Reviews on Experimental and Applied Mechanics

Volume 2 Issue 2

Experimental Investigations On Free Vibration Of Beams

Kemparaju H.R1, Prasanta Kumar Samal

2

1Associate Engineer, HCL Technology, Bengaluru, India.

2Assistant Professor, Department of Mechanical Engineering, NIE Mysuru, India.

E-mail: [email protected], [email protected]

ABSTRACT

In this article experimental investigation of the natural frequency, damping ratio and damping

constant of beams in different material i.e. steel, aluminium, copper with different boundary

conditions like Clamped-free, Clamped-clamped, clamped-simply support, Simply support-

simply support, and free-free(by using sponge, and by using rubber band) has been

investigated. Natural frequencies obtained using accelerometer, NI-DAQ (9234), and NI-

DAQ chassis (9137),and LABVIEW and MATLAB. Then the main objective of this paper is

to provide experimental data that can be used for checking the accuracy and reliability of

different theories and approaches like analytical, finite element method by using MATLAB

and ANSYS. The effects of different geometrical parameters including density are discussed

in above mentioned all boundary conditions in details with up to first 3 natural frequencies.

This study may provide valuable information for researchers and engineers in design

applications.

Keywords: Natural frequency, damping ratio, damping constant, and LABVIEW.

INTRODUCTION During last three decades the subject

Vibration Analysis has undergone

significant development. There has been

an urge and urgency of designing present

day modern and complex structures and

systems in their proper perspective, a task

which could not be conceived even three

decades back, this has been possible due to

the advent of the electronic digital

computer.

Daniel Ambrosini [1] . In this paper author

is carried out the experimental modal

analysis of the non-symmetrical thin

walled beam by using aluminium material.

The experiment is carried out in two

boundary conditions i.e. fixed-free and

fixed-fixed. Here accelerometer Kyowa

AS-GB and PCM�DAS16D=16 of 16 bit

data acquisition system is used to a modal

testing of beam. Then the experimental

results are compared with the analytical

method of vlasovs theory of thin walled

beam and numerical method by using

finite element software SAP2000N and

compared results. Here the author is shows

that the results are good agreement

between the all three methods. Mehmet

Avcar [2]. The free vibration analysis of

the square cross sectioned aluminium

beam is investigated by the analytical and

numerically under four boundary

conditions i.e. C-C, C-F, C-SS, SS-SS.

The Euler Bernoulli beam theory and

Newton Raphson methods are used to

analytical method. And finite element

method based software ANSYS is used to

find out numerical results for Free

Vibration Analysis of Beams. Then he

obtained a natural frequency and he

discussed first three modes of the

including boundary conditions, geometric

characteristics i.e. length, cross sectional

etc. Mr. P.Kumar, Dr. S.Bhaduri [3]. In

this paper the natural frequencies of two

different cantilever beams made of

Aluminum and Iron are measured

experimentally with and without the

presence of end masses. The finite element



Volume 2 Issue 2

analysis is done using the developed codes

in MATLAB and also by ANSYS

software. In experimental setup consists of

FFT analyser with computer and its

accessories. For further analysis of the

effect of different masses on the natural

frequency of cantilever beam, free end of

cantilever is loaded with different masses

along with accelerometer and the result of

this experiment is observed on the

computer screen for the aluminium and

iron beams. Then the author is concluded

that numerical and experimental results are

much closed together. Yasha- vantha

kumar G.A [4]. The numerical study of the

free vibration analysis of the smart

composite beam by using the ANSYS is

explained in this paper. Here the author

composite beam made up of glass epoxy

and PZT patches are added in the surface

of the beam. Then the vibration analysis is

carried out under the clamped-free

condition of the beam. Kotambkar [5]. He

studied the mass loading effect of the

accelerometer on the natural frequency of

the beam under free-free boundary

condition. The analytical calculation is

taken b Euler-Bernoulli equation for

uniform cross section beam. For

experimental analysis accelerometer (B

and K make; weight 27(gram), FFT

analyser (DI22000) is used to obtained the

results. In most of the case mass of the

accelerometer is ignored,however when

lighter structures are investigated this

effect must be considered. Nikil T,

Chandrahas T,[6]. Here the author is

design and develop a test rig for

determining the vibration characteristics of

the beam with different boundary

conditions like C-F, C-C, and SS-SS. In

the test rig he provided a adjustable

eccentric weights are provided to vary the

force with approximately same frequency.

He used the accelerometer and NI 9234 to

acquire the vibration data. Then he

compared the results of transverse

vibration of aluminium and mild steel

beam and he also did the effect of the

geometric characteristics (length, c/s area)

on the frequency of the beam. M.N.

Hamdan [7]. Here the author showed that

the mass effect on the natural frequency of

the beam with different end conditions by

Galerkins methods. And compared the

results with the Euler-Bernoulli beam

equation results. H. Navi [8]. Study of this

paper shows that the effect of the crack

depth on the natural frequency of the beam

i.e. frequency decrease with the increase

crack depth. Then he compared the

frequency of the cantilever beam on with

crack without crack. And he also had done

the numerical results by using the

MATLAB.

In this paper, the effects of different

geometrical parameters including density

are discussed in above mentioned all

boundary conditions in details with up to

first 3 natural frequencies.

EXPERIMENTAL DETAILS

Material Testing and Selection

Aluminium, Steel, and Copper are selected

as component material.

Description of specimens

For BEAMS,

Table1: Dimensions of BEAMS

Dimensions in mm No of steel No of copper No of Aluminium Total

350*20*3 1 1 1 3

550*30*3 1 1 1 3

550*40*3 1 1 1 3

Tensile Testing

Tensile test were carried out using

Universal Testing Machine at NIE

Mysuru. Then calculating the young’s



Volume 2 Issue 2

modulus (E).after tensile testing young’s

modulus of the material is get as given in

below table

Fig. 1: Material tested in UTM

Fig. 2: Tested specimen

Table 2: Properties of Materials

sl.no Material Young’s modulus in N/m2 Density in kg/m3

1 Steel (ASTM-A36) 2X1011 7850

2 Copper 1.2X1011

8940

3 Aluminium 0.7X1011 2720

Experimental setup

Experimental modal analysis is a

technique used to study the dynamic

characteristics of a mechanical structure.



Volume 2 Issue 2

This technique can be used to describe a

structure in terms of its natural

characteristics, namely natural frequency,

damping ratio, and mode shapes. Two

widely used methods for performing

experimental modal analysis are impact

test and vibration shaker test.

In our experiments the tri axial

accelerometer (356A15) mounted at a test

specimens of Beams with suitable

position. The accelerometer connected to a

DAQ-9234 and the impact hammer is used

to give a initial disturbance. The impact

hammer also connected to a DAQ-9234.

And the DAQ-9234 is mounting on the

cDAQ-9178 chassis, the power supply is

given to the chassis. Then draw a block

diagrammed of the data acquisition

process on the LABVIEW block

diagrammed window. Here to obtain a

frequency response to FFT analyser is

used to convert time domain signal to

frequency signal. We can also save the

data into measurement files by using write

to measurement files tool. And later we

can plot it. The below figures shows the

experimental setups of beams with

different end conditions and corresponding

experimental results of natural frequency.

The below Figure3 shows the experimental

setup of clamped-free and clamped-

clamped beam and Figure4 shows the

experimental setup of free-free vibration in

two types c) mounting on a sponge, d) by

using on a rubber band. And Figure6

shows the simply supported at both ends

and clamped-simply supported.

Fig. 3: Photo Copy of Beam with Different Boundary Conditions a) C-F conditions, b) C-C

Fig. 4: Photo Copy of Beam with Free-Free Boundary Conditions c) by Using Sponge, d) by

Using Rubber Band



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Fig.5: Photo Copy of Beam with Different Boundary Conditions c) SS-SS, d) C-SS

Analytical and FEM methods

For beams, we shall assume that the

deflection is small. In addition, we shall

neglect the rotation of the beam elements

during oscillation. We shall consider the

deflections only due to the bending

moment [14] page (497-500).

1

Moreover, since we are neglecting the

rotational motion of the element, the total

moment about the y-axis must be zero.

2

Further, from the elementary beam theory,

we get (with the sign convention used).

3

Where I is the second moment of beam

cross-section about its neutral axis. Using

(2) in (1) in conjunction with (3), the final

equation of motion we obtain is

4

Assuming a normal mode vibration in the

form x (z, t) = X(z) cos t, we can rewrite

the foregoing equation as

5

√

6

The general solution of (5) can be

rewritten as

X = cosh z+ sinh z+ cos z+

sin z 7

The values of can be obtained when the

boundary conditions of the beam are

prescribed. Once the is known, the

natural frequencies can be computed from

(6). Let us consider the following common

types of boundary conditions:

Both ends Simply-Supported

In this situation, the deflections and the

bending moments at the support cross-

sections must be equal to zero. Thus, at

such ends,

x = 0, M = 0, for all t.

This implies X=0, and

at the

simply supported ends. Therefore, the

boundary conditions can be written as

X=0,

Substituting these conditions in (1), we

get, for the nontrivial solutions, sin L = 0

or

(i=1, 2, 3…….) 8



Volume 2 Issue 2

First three roots of this equation are

Hence, using (1) in (2), we find the natural

frequencies come out as

√

One end fixed and the other end free

(cantilever)

When one end of the beam is clamped,

both x and

should be zero for all t. This

yields, at the clamped end,

X=0,

For the free end of the beam, the bending

moment and the shear force are zero which

Yield

=0 for all t.

Therefore, the boundary conditions can be

written as,

X=0,

Using these conditions in (2) and

considering the nontrivial solutions, the

frequency equation we get has the form

Cos L cosh L = -1 9

First three roots of this equation are

For i > 3, the roots of 4 can be

approximated as

(

) 10

Both ends fixed

Both ends fixed proceeding in a manner

similar to that in previous conditions, the

frequency equation we obtain is

Cos L cosh L = 1 11

The first three roots and the asymptotic

solutions of (5) are

(

) i > 3 12

One end fixed and other end simply-

supported Here, the frequency equation is given by

tan L = tan L 13

The first three roots and the asymptotic

solutions of (7) are

(

) i > 3 14

Both ends free

The first three roots for Free-Free

boundary conditions.

(

) i =1, 2, 3 15

FEM method to use calculates natural

frequencies of beams.

The stiffness matrices of beam.



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[ ]

[

]

16

The mass matrices of beam.

[ ]

[

]

17

The nodal displacement vector,

q(t) = [ ]T

18

The equilibrium equations of motion of

entire beam is given by,

[M]= ̈ [ ] 19

To solve this above equation by applying a

boundary conditions of the beam we get

the solution.

ANSYS results

The below figures shows the ANSYS

results of aluminium beam (550*40*3 in

mm) in different boundary conditions.

Fig.6: Cantilever Beam Analysis Results

Fig. 7: Analysis of C-C Boundary Condition Results



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Fig. 8: Analysis of C-SS Boundary Condition Results

Fig. 9: Analysis of F-F Boundary Condition Results

Fig.10: Analysis of SS-SS Boundary Condition Results

Experimental Results The below figures shows the experimental

results of Time domain signals and

Frequency domain signals of aluminium

beam (550*40*3 in mm) with different

boundary conditions. Here due to the space

constrained not shown the other specimen

results. We discussed detailed in

aluminium beam (550*40*3 mm) similar

steps followed for others.



Volume 2 Issue 2

Fig. 11: Cantilever Beam Experimental Results a) Time Domain Signal, b) Frequency

Domain Signal

Fig.12: C-C Beam Experimental Results c) Time Domain Signal, d) Frequency Domain

Signal

Results and Discussion

Free vibration analysis of beams

Below table shows the free vibration

analysis results of beams by varying

boundary conditions and geometry

conditions with analytical, FEM results by

using MATLAB, ANSYS and

experimental results are given in below

table’s.

Aluminium beam results B=20mm, L=350mm, t=3mm.

Table 3: Frequency Results of Aluminium C-F Beam in Hz Modes Analytical MATLAB ANSYS Experimental

1 20.3 20.13 20 18

2 129.3 112.87 125 110

3 356 311.35 350.18 345

Table 4: Frequency Results of Aluminium C-C Beam in Hz Modes Analytical MATLAB ANSYS Experimental

1 129.3 112.87 127.8 103

2 356 311.35 350.18 345

3 698.2 660.38 689.71 612



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Steel beam results B=20mm, L=350mm, t=3mm.

Table 5: Frequency Results of Steel C-F Beam in Hz Modes Analytical MATLAB ANSYS Experimental

1 19.6 19.5 19.73 17

2 125.1 109.23 123.63 110

3 344.3 301.2 346 317

Table 6: Frequency Results of Steel C-C Beam in Hz Modes Analytical MATLAB ANSYS Experimental

1 125.1 109.23 123.63 110

2 344.3 301.2 346 317

3 675 638.74 679.7 558

Copper beam results B=20mm, L=350mm, t=3mm.

Table 7: Frequency Results of Copper C-F Beam in Hz Modes Analytical MATLAB ANSYS Experimental

1 14.5 14 14.5 13

2 92.4 80.7 90.6 84

3 254.5 222.5 253.7 244

Table 8: Frequency Results of Copper C-C Beam in Hz Modes Analytical MATLAB ANSYS Experimental

1 92.4 80.68 90.6 84

2 254.5 222.5 253.7 244

3 499 472 498.8 468

Aluminium beam B=20mm, L=550mm, t=3mm.


1 8.22 8.72 8.1 7

2 52 45.85 50.9 44

3 144 126 142.68 130


1 52.34 45.85 50.9 44

2 144 126 142 130

3 282.7 267 280 265

Steel beam B=20mm, L=550mm, t=3mm.


1 7.95 8.72 7.7 7

2 50.6 44 48 44

3 139.4 121.7 134.8 128


1 50.6 44 48 44

2 139.4 121.9 134.8 128

3 273.5 258.6 264.6 255

Copper B=20mm, L=550mm, t=3mm.


1 5.8 5.9 5.8 6



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2 37.4 32.6 36.7 36

3 103 90 102.6 107


1 37.4 32.7 36.7 36

2 103 90 102.6 107

3 201 191 201 211

Aluminium B=40mm, L=550mm, t=3mm.


1 8.2 8.7 7.9 8

2 52 45.8 49 45

3 144 126 138 133


1 52 45 49.5 45

2 144 126 138 133

3 282.7 267.4 271.9 281

Steel B=40mm, L=550mm, t=3mm.


1 7.9 8.7 7.8 7

2 50.6 44 48.7 44

3 139.4 121.9 136 126


1 50.6 44 49.7 42

2 139 121.9 136.8 120

3 273.5 258.6 268.3 235

Copper B=40mm, L=550mm, t=3mm.


1 5.9 5.9 5.7 6

2 37 32.7 35 41

3 103 90 99 119


1 37 32.7 36 36

2 103 90 99.8 108

3 201 191 195 210

Comparison of beam results

Comparison of free vibration results of

beam with by varying Density, Length,

Cross section area, and different boundary

conditions as given below,

1. Above all table results shows that there

are good agreement between the all four

analytical, FEM methods by using

MATLAB, ANSYS, and experimental

results.

2. Comparison of Density v/s Frequency of

the BEAMS.

The ANSYS and experimental results are

taken to compare the results.

Beam dimension B=20mm, L=350mm,

t=3mm, C-F, BC’S.



Volume 2 Issue 2

Aluminium (2700 kg/m3)

Steel (6585 kg/m3)

Copper 8933 kg/m3)

Table 21: Density v/s Frequency of the Cantilever Beam in Hz Mode

Number

ANSYS

(AL) Exp (AL)

ANSYS

(STL) Exp (STL) ANSYS (COP)

Exp (COP)

1 20 18 19.7 17 14.5 13

2 125 110 123.6 110 90.6 84

3 350 345 346 317 253.7 244

From above Table21 we can observe that

density increases natural frequency of the

system decreases.

Comparison of Length v/s Frequency of

the Cantilever beams. Beam, B=20mm,

t=3mm,

L1 = 350mm; and L2 = 550mm;

Table 22: Length v/s Frequency of the Cantilever Beam in Hz Mode umber AL(L1) AL(L2) STL(L1) STL(L2) COP(L1) COP(L2)

1 18 7 17 7 13 6

2 110 44 110 44 84 36

3 345 130 317 128 244 107


length increases natural frequency of the

system decreases.

4. Comparison of c/s Area v/s Frequency

of the beam. Beam L=550mm, t=3mm,

B1 = 40mm; and B2 = 20mm.

Table 23: C/s area v/s Frequency of the Cantilever Beam in Hz Mode umber AL (B1) AL (B2) STL (B1) STL (B2) COP (B1) COP (B2)

1 7 8 7 7 6 6

2 44 45 44 44 36 41

3 130 133 128 126 107 119


c/s area increases natural frequency of the

system decreases slightly. But in steel it is

same for first 2 frequency and 3 frequency

is reverse to the others

5. Free-Free vibration analysis of beam.

Dimension of Aluminium beam

L=550mm, B=40mm, t=3mm.

Free vibration analysis of beam under

Free-Free boundary conditions by using

rubber band and mounting on a sponge.

Then the results are compared with the

analytical and ANSYS results are shown

in below Table24. Both experimental

results are near to the analytical and

ANSYS results.

Table 24: Frequency of Free-Free BC’S beam results in Hz Mode Number Analytical ANSYS by using Rubber by using Sponge

1 51.9 49.8 48 48

2 144 137.4 138 137

3 282.5 269.6 258 262

6. Both the ends Simply Supported BEAM


L=550mm,

B=40mm, t=3mm.

Free vibration analysis of beam under both

the ends simply supported boundary

conditions. Then the results are compared

with the analytical and ANSYS results are

shown in below Table25.Experimental


ANSYS results.



Volume 2 Issue 2

Table 25: Frequency of both the ends simply supported beam results in Hz Mode Number Analytical ANSYS Experimental

1 23 22.8 23

2 92 91.2 117

3 207.5 205.3 196

7. Clamped and Simply Supported BEAM


L=550mm, B=40mm, t=3mm.

Free vibration analysis of beam under

Free-Free boundary conditions by using

rubber band and mounting on a sponge.

Then the results are compared with the

analytical and ANSYS results are shown

in below Table26. Both experimental


ANSYS results.

Table 26: Frequency of Clamped-Simply supported beam results in Hz Mode Number Analytical ANSYS Experimental

1 35.9 34.5 32

2 116.8 111.8 107

3 242.98 233.5 233

8. Experimental results of aluminium

beam L=550mm, B=40mm, t=3mm, in all

Boundary conditions as shown in below

table

From below Table27 we can observe that

natural frequency is higher at free-free

boundary conditions and next clamped-

clamped boundary conditions next

clamped-simply support then simply

support-simply supported and clamped-

free conditions results respectively.

Table 27: Frequency of Aluminium Beam (L=550mm,B=40mm, t=3mm)in all Boundary

Conditions in Hz Mode umber C-F C-C C-SS SS-SS F-F by Rubber F-F by sponge

1 8 45 32 23 48 48

2 45 133 107 117 138 137

3 133 281 233 196 258 262

Experimental results of Damping factor (

), and Damping constant(C) for Beams

Logarithmic decrement method is one of

the most basic method used to measure the

damping factor of an under-damped

system.

Fig.13: Logarithmic Decrement Method of Calculating Damping Factor

In such a system, the vibration amplitude

decays with time and the natural log of

amplitudes of any two successive peak is

called the logarithmic decrement or log



Volume 2 Issue 2

decrement. Logarithmic decrement ( ) can

be calculated by the equation

20

Where d is the log decrement ( ), is the

maximum amplitude of first cycle and

X is the maximum amplitude of nth cycle.

Damping ratio, ( ) can be calculated using

log decrement as follows

√

21

Then the critical damping constant (Cc)

can be obtained

Cc = 2mωn in

22

Thus the damping constant(C) is given by,

C = Cc in

23

The below Table28 shows the dimension

of specimens.

Table 28: Dimension wise Beam Naming Specimen Name Dimensions (L*B*h) in mm

1 350*20*3

2 550*20*3

3 550*40*3

9. Experimental results of Damping factor

( ), and Damping constant(C) for beams

the below Table29 shows the Damping

factor ( ), Critical damping factor (Cc),

damping constant(C) for aluminium, steel,

and copper with C-F and C-C. In that table

we can observe that the values of damping

factor ( ) and critical damping factor (Cc)

in

, and damping constant (C) in

clamped-clamped boundary conditions is

more than the clamped-free boundary

conditions.

Table 29: Experimental results of beam for ( ),Cc, C for CC and CF boundary conditions

Specimen for CC Cc

for

CC

C

for

CC for CF

Cc

for

CF C

for CF

AL 1 0.01997 11.68 0.2332 0.00705 2.0412 0.0144

AL 2 0.034 7.841 0.2666 0.0148 1.2474 0.18495

AL 3 0.01979 16.038 0.317 0.0102 2.8512 0.0291

STL 1 0.0107 36.267 0.38806 0.00294 5.605 0.0165

STL 2 0.264 22.801 0.6024 0.0147 3.63 0.05321

STL 3 0.0244 43.52 1.0618 0.01929 7.25 0.1399

COP 1 0.01554 31.51 0.4896 0.01324 4.877 0.0646

COP 2 0.0405 21.23 0.8591 0.02354 3.5376 0.0833

COP 3 0.0413 42.45 0.1399 0.0413 7.25 0.087

Error Causes of Results

It can see from the above table the

experimental results are not exactly equal

to the numerical results due to some of the

error occurred during the experimental

process. And some of assumption taken in

analytical method to simplify the

calculations. The causes of error are listed

in below

1. Signal Leakage error.

2. In calculation vibration damping is

neglected, but in actual practice damping

occurred.

3. Weight of the accelerometer.

4. Electrical noise.

5. Geometrical imperfection.

6. Environmental effect. Etc...



Volume 2 Issue 2

Leakage error

1. Difference between the time period of

captured signal and original signal is

called leakage error.

2. To avoid this leakage error to match the

time period or multiply with the integer

after capturing the signal.

3. Windowing techniques are used to

reduce the leakage error by multiplying the

weights.

Ex:- middle of signal multiply with higher

value and end of the signal multiply with

smaller value then joining this type of two

signal we get smooth joining curve so

leakage is less.

Strategies for Choosing Windows

Fig.14: Types windows available in LABVIEW

Each window has its own characteristics,

and different windows are used for

different applications. To choose a spectral

window, you must guess the signal

frequency content. If the signal contains

strong interfering frequency components

distant from the frequency of interest,

choose a window with a high side lobe

roll-off rate. If there are strong interfering

signals near the frequency of interest,

choose a window with a low maximum

side lobe level.

Selecting a window function is not a

simple task. In fact, there is no universal

approach for doing so. However, Table

below can help you in your initial choice.

Always compare the performance of

different window functions to find the best

one for the application.

Table 30: Initial Window Choice Based on Signal Content Window Signal Content

Hanning 1. Sine wave or combination of sine wave.

2. Narrow band random signal (vibration data)

3. Unknown data.

Flat top Sine wave (amplitude accuracy is important).

Uniform Broadband random (white noise).

uniform, Hanning Closely spaced sine waves.

Force Excitation signals (hammer blow).

Exponential Response signal.



Volume 2 Issue 2

Electrical noise.

1. Random electron motion (micro volts) it

creates heat on surface during measure.

2. Local magnetic fields arcing (in mille

volts).

3. Earth loop faults.

Geometrical imperfection

1. Chemical combination of material is not

linear.

2. Dimension of specimen is not accurate.

Environmental effect.

1. Temperature variation.

2. Noise in wind flow.

3. Dust particle etc.

CONCLUSIONS

The following are the important

conclusions of this study:

4.1. Free vibration analysis of BEAMS

1. Experimental results are closer to the

numerical results.

2. C-F Beam 2nd frequency is first

frequency of the C-C beam modal

analysis.

3. Density increases frequency decreases.

4. For Free-Free BCs placed on Sponge

and hanging by rubber band both are gives

same results.

5. All boundary conditions of beam BCs is

experimentally satisfied.

6. C/s area increases frequency is slightly

decreases.

7. We can observe that natural frequency

is higher at free-free boundary conditions

and next clamped-clamped boundary

conditions next clamped-simply support

then simply support-simply supported and

clamped-free conditions results

respectively.

8. We can observe that the values of

damping factor ( ) and critical damping

factor (Cc) in

, and damping constant

(C) in clamped-clamped boundary

conditions is more than the clamped-free

boundary conditions.

ACKNOWLEDGEMENTS This work was performed using facilities

at NIE, Mysuru, funded by the NIE,

TEQUIP-2. The authors are grateful to the

NIE, Mysuru for their support and

encouragement.

REFERENCES

1. Daniel Ambrosini, ”Experimental

validation of free vibrations from non-

symmetrical

Thin walled beams”. Engineering

Structures 32(2010)1324-1332

2. Mehmet Avcar, ”Free Vibration

Analysis of Beams Considering

Different Geometric Characteristics

and Boundary Conditions ”. DOI:

10:5923= j:mechanics:20140403:03

3. Mr. P.Kumar, Dr. S.Bhaduri, Dr. A.

Kumar,”Vibration Analysis of

Cantilever

Beam: An Experimental Study”. ISSN

: 2321-9653

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Sathish Kumarb,”Free vibration

analysis of

Smart composite beam”. Materials

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5. Prof. M. S. Kotambkar,”Effect of

mass attachment on natural frequency

of free-free beam: analytical,

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investigation”.

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ISSN22498974

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Procedia Engineering

144(2016)312320

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Volume 2 Issue 2

8. H. Nahvi, M. Jabbari,”Crack detection

in beams using experimental modal

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Documents

Experimental Investigations On Free Vibration Of Beams