Upload
others
View
13
Download
0
Embed Size (px)
Citation preview
MODELLING, IDENTIFICATION AND ACTIVE
CONTROL OF NONLINEAR VIBRATION IN
A FLEXIBLE CANTILEVER BEAM
by
Thanh Lan Vu
May 1998
Department of Mechanical & Materials Engineering
The University of Western Australia
This thesis is presented for the degree of Doctor of Philosophy of
the University of Western Australia
ABSTRACT modelling the dynamics of a flexible cantilever beam has attracted
the attention of researchers in many areas of engineering applications; including robotic
manipulators, satellites, aircraft, etc. Linear theories such as the Rayleigh-Ritz method
and Timoshenko and Euler-Bernoulli beam theory, have for several decades been useful
tools for predicting the behaviour of beams, based on the assumption of small amplitude
vibrations. However, when the beam is subject to a large vibration, many nonlinear
phenomena may occur, such as change of resonance frequency, energy transfer between
higher order modes and lower order modes, modal coupling and frequency modulation.
These nonlinear behaviours preclude an adequate mathematical analysis of the beam
response based on linear models. This fact has led to many theoretical and experimental
investigations into the nonlinear vibration of beams that have been carried out over
more recent years. However, to m y knowledge, none of the nonlinear models for
flexible beams which have been developed so far, are able to adequately describe the
nonlinear behaviour of the beam.
The aim of this work is to develop a nonlinear model for a flexible cantilever beam,
which is able to predict the response of the beam for both the linear and nonlinear cases.
In contrast to other work, the development of the nonlinear model was not only based
on nonlinear theory, but primarily on experimental observation and understanding of the
nonlinear behaviour of the beam. In the process of developing the nonlinear model, a
thorough investigation of the nonlinear vibration of the beam was firstly carried out
from different perspectives to identify various mechanisms in the system. Nonlinear
beam theory was then applied and modified corresponding to the experimental results.
As a result, the developed nonlinear model of the flexible cantilever beam corresponded
very well to the experimental results. In addition, the developed model was expressed
simply in state-space form, which was easily converted to an Auto-Regressive Moving
Average ( A R M A ) model. The A R M A model was then used to predict the response of
the beam on-line using the conventional linear Least Mean Square (LMS) algorithm.
The developed identification scheme is, therefore, conceptually simple. It requires a
small number of weights and is much more efficient than other identification methods
using Finite Impulse Response (FIR) filters, Infinite Impulse Response (IIR) filters and
even nonlinear filters based on Volterra series. It worked well in both the linear and
i
nonlinear case, whereas the other methods failed in the case of nonlinear modal
coupling. This on-line identification scheme would be useful in developing a feed
forward as well as feed-back control scheme for the cancellation of nonlinear vibration
in a flexible cantilever beam. Later in this work, a feed-back controller was developed
and implemented in a dSpace™ Digital Signal Processor (DSP) in order to cancel
nonlinear vibration generated in the flexible beam due to modal coupling. The results
obtained were excellent and represent a significant advance in the field of active
vibration control.
Because of the nature of this research, the work is presented in the following order
covering seven chapters:
• Chapter 1 describes all the nonlinear phenomena including change of resonance
frequency, jump phenomena, energy transfer from higher order modes to lower
order modes, nonlinear modal coupling, frequency modulation, nonlinear stiffness,
etc.; observed during the experiments when the cantilever beam was subject to large
single-modal excitation.
• Chapter 2 examines the behaviour of the beam when at least two or more modes of
the beam are excited.
• Chapter 3 investigates the damping characteristics of the beam for single as well as
multi-frequency excitations. The experimental results show that in the case of
nonlinear modal coupling, the beam exhibits Hysteretic damping in addition to a
combination of Viscous and Quadratic damping in the linear case.
• Chapter 4 describes the process of developing and validating the nonlinear model of
the flexible cantilever beam.
• Chapter 5 presents an on-line identification scheme for the beam based on the
nonlinear model and L M S algorithm. The performance was then evaluated by
comparing the results of the developed identification method with other methods
using IIR and third order Volterra FIR filters.
• Chapter 6 shows h o w the nonlinear vibration, generated in the flexible cantilever
beam due to modal coupling, was cancelled using a feed-back controller
implemented in the DSP.
• The final chapter concludes this research and recommends further development of
this work.
ii
TABLE OF CONTENTS
page Abstract i
Table of contents iii
Acknowledgments vi
1. NONLINEAR BEHAVIOUR OF A FLEXIBLE BEAM -
Single Frequency Excitation 1
1.1 Introduction 1
1.2 Experimental Apparatus 4
1.2.1 Excitation Set-ups 4
1.2.2 Instrumentation 9
1.2.3 Experimental Considerations 12
1.3 Nonlinear behaviour of the Beam 13
1.3.1 Change of the Resonance Frequency 13
1.3.2 Jump Phenomenon 17
1.3.3 Energy Transfer from Higher to Lower Order Modes 20
1.3.4 Nonlinear Modal Coupling 27
1.3.5 Frequency Modulation 29
1.3.6 Nonlinear Stiffness 30
1.4 Conclusions 36
2. NONLINEAR BEHAVIOUR OF A FLEXIBLE BEAM -
Multi-Frequency Excitation 38
2.1 Introduction 38
2.2 Experimental Setup 39
2.3 Linear Response to Multi-Frequency Excitation 4 0
2.3.1 Non-periodic Input Signal 4 0
2.3.2 Periodic Input Signal 5 2
iii
2.4 Nonlinear Response to Multi-Frequency Excitation 54
2.4.1 Non-periodic Input Signal 54
2.4.2 Periodic Input Signal 59
2.5 Conclusions 59
3. DAMPING CHARACTERISTICS OF THE FLEXIBLE CANTILEVER
BEAM 61
3.1 Introduction 61
3.2 Single Frequency Excitation 62
3.2.1 Linear case 62
3.2.2 Nonlinear case 71
3.3 Multi-Frequency Excitation 75
3.3.1 Linear case 75
3.3.2 Nonlinear case 79
3.4 Conclusions 83
4. MODELLING OF NONLINEAR VIBRATION IN A FLEXIBLE
CANTILEVER BEAM 84
4.1 Introduction 84
4.2 Modelling of Nonlinear Vibration in the Beam 92
4.3 State-space Model of the Beam 102
4.4 Verification of the Nonlinear Model 104
4.5 Conclusions 114
5. ON-LINE IDENTIFICATION OF THE FLEXIBLE CANTILEVER
BEAM 116
5.1 Introduction 116
5.2 The Conventional Linear Filters using L M S / R L M S Algorithm 119
5.3 The Conventional Nonlinear Filters 123
5.4 The Developed On-line Identification Scheme for the Flexible Cantilever
Beam 124
iv
5.5 Experimental Setup 127
5.6 Experimental Results 132
5.7 Conclusions 141
6. ACTIVE CONTROL OF NONLINEAR VIBRATION IN THE FLEXIBLE
CANTILEVER BEAM 143
6.1 Introduction 143
6.2 Control Strategy 146
6.3 Experimental Set-up and Results 149
6.4 Conclusions 153
7. SUMMARY AND FUTURE WORK 154
7.1 Summary 154
7.2 Recommendations for Future work 158
REFERENCES 160
APPENDIX A 167
APPENDIX B 169
APPENDIX C 170
v
ACKNOWLEDGMENTS
This thesis would not have been realised without direct or indirect contribution of a
large number of people. First and foremost I would like to thank Associate Professor Jie
Pan for being m y supervisor and his support, guidance and useful comments throughout
m y study.
I would also like to thank Associate Professor James Trevelyan for his support and
guidance during the first year when I joined the department and the Sheep Shearing
Project. Without his support and encouragement, it would be hard for m e to build a
foundation for m y PhD study in a new department at a new university, in a new city and
in a new country. M y thanks also go to all the team members of the Sheep Shearing
Project, especially Ed Tabb, David Elford, Jan Baranski, Daryl Cole, Wai-Chee Yao,
Virginia Shipworth, Ian Hamilton, Professor Brian Stone and again Associate Professor
James Trevelyan for being co-operative, supportive and friendly to m e while conducting
the project. Their support and confidence have encouraged m e to start a new research
direction when the Sheep Shearing project came to an end.
I am also grateful to Professor Brian Stone and Simon Drew for their support and
generosity in lending their equipment. M y research would have been difficult to achieve
without their support.
I am also indebted to the highly skilled electronics and mechanical technicians from the
Mechanical Engineering workshop; in particular, Rob Greenhalgh, Ron D e Pannone,
Dennis Brown, Ian Hamilton, Matt Heme, Derek Goad, Terry Glover, Peter Edmands
and Brian Sambell for their technical skills and help in making m y experimental rigs as
well as their friendships.
I would like to thank the Ex-Head of Department, Mr John Appleyard, the Ex-Acting
Head of Department, Professor Yuri Estrin and the current Head of Department,
Associate Professor Mark Bush for their support during m y PhD.
vi
I would also like to thank m y colleagues and friends Hong Yang, Roshun Paurobally,
Ken Taylor, David Miller, Gorden Fisher, Hui Peng, Nabil Farang, Nicole Kessissoglou,
Dr Ruisen Ming, Dr Chaoying Bao for their support during m y study.
My personal thanks go to Michael Ong for his technical advice in Real-time
programming and for all useful discussions regarding m y experimental rig.
My sincere thanks to my close friend, Finn Haugen in Norway, for his emotional
support and friendship throughout m y PhD study in spite of the distance. His friendship
has given m e the strength to overcome obstacles in m y life. I really appreciate all the
light hearted E-telnet talks as well as open and frank discussions that helped m e to find
the way forward in difficult times.
My most sincere and special thanks to my close friends, Rob Greenhalgh and Simon
Drew for their emotional as well as technical support and their persistent encouragement
over the years. Their friendships, care and love have not only given m e the strength to
overcome difficulties that have arisen during m y study, but also made m y life more
meaningful during m y time at U W A . Their help in proof reading m y thesis is also
gratefully acknowledged.
I would also like to specially thank Tang family for their love and care for the time I
lived in Norway without m y parents. Their sincerity and honesty have touched m y heart
and brought m e to where I a m now. Without these factors I would never have achieved
so much.
My most sincere and heartfelt thanks to my parents and my sisters Huong and Lan who
have given m e a tremendous support, love and care. Last but not least, I would like to
dedicate this thesis to m y parents, m y sister Huong, Tang family, Rob Greenhalgh and
Finn Haugen who have been and will always be special in m y life.
vii
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
Chapter 1
NONLINEAR BEHAVIOUR OF A FLEXIBLE BEAM:
Single Frequency Excitation
1.1 INTRODUCTION
The dynamic response of a non-linear system can exhibit many phenomena that will not
be observed in a linear system. For a linear system, the impulse response and stability
are independent of the magnitude of the system input and the initial conditions.
Conversely, in a non-linear system, the system impulse response and stability are
usually strongly dependent on the magnitude of the input and the initial conditions.
Non-linear systems, generally, have multiple equilibrium conditions, which result in
multiple equilibria for state-space realisations. Some equilibrium states may be stable
whereas others m a y be unstable. Therefore, a non-linear system may have an unstable
forced response although its free (unforced or zero input) response is stable [10]. A
linear system with a periodic input will only exhibit a frequency component at the same
frequency as the input. In contrast, a sinusoidal input to a non-linear system may yield
sub-harmonics, higher harmonics or non-periodic outputs. Jump phenomena may exist
in the sinusoidal frequency response of certain non-linear systems where a discontinuity
in amplitude and phase occurs at different frequencies, depending on whether the
response is measured with increasing or decreasing excitation frequency. For some
multi-degree-of-freedom systems, energy transfer between the modes through non-linear
modal coupling may also be observed.
Earlier works [2-4, 21, 47, 48] have experimentally investigated non-linear vibration in
a flexible cantilever beam and reported many non-linear phenomena such as shifting of
the natural frequency of the resonance modes and energy transfer from higher order
modes to lower order modes. For example, Dugundji and Mukhopadhyay [21]
1
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
investigated the response of a thin cantilever beam where the beam was subjected to
base excitation at a frequency close to the sum of the natural frequencies of the first
bending and first torsional modes, which were approximately in the ratio of 1 to 18.
Their results showed that high frequency excitation could excite low frequency modes
through a non-linear coupling mechanism.
Nayfeh et al. [45-48] experimentally examined the response of axially symmetric
cantilever beams to planar external excitation. They observed a large response from the
first order mode, in addition to the excited higher order mode, when the beam was
excited near the resonance frequencies of the third or any higher order modes.
Moreover, the degree of the coupling between the first order and the higher order modes
increased as the excitation frequency was increased towards the higher resonance
frequencies of the beam.
Anderson et al. [2-4] studied the response of a cantilever beam subjected to a base
excitation. Similar to Nayfeh et al.'s work [45-48], the non-linear coupling between the
excited higher order modes and the first order mode was observed. They believed that
the coupling was due to the energy transfer from higher order modes to lower order
modes. Moreover, they found that the vibration of the first order mode was
accompanied by slow modulation of the amplitude and phase of the high frequency
modes. Anderson et al. [2-4] also experimentally investigated the planar response of a
parametrically excited cantilever beam. They verified that the effective nonlinearity for
the first order mode is of the hardening-type and that the effective nonlinearity for the
second order mode is of the softening-type.
The aim of this chapter is to report on an experimental investigation of the non-linear
behaviour of a flexible cantilever beam. The non-linear phenomena observed included
change of resonance frequency, jump phenomena, energy transfer from higher order
modes to lower order modes, nonlinear modal interactions, frequency modulation, and
nonlinear stiffness and damping characteristics. Although some of these non-linear
phenomena have already been reported in [2-4, 21, 45-48], none of these works showed
2
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
the correspondence between these non-linear phenomena or considered non-linear
stiffness and damping characteristics when developing a nonlinear model for the beam.
This chapter describes all the observed nonlinear phenomena, with both nonlinear
stiffness and hysteric damping characteristics included, in order to establish a
comprehensive understanding of the nonlinear behaviour of the flexible beam.
A thorough investigation into the damping characteristics of the beam for both linear
and nonlinear cases was carried out and is described in Chapter 3. These experimental
observations along with a comprehensive understanding of the nonlinear behaviour of
the flexible beam lay a significant foundation for the development of a nonlinear model
of the beam. To the author's knowledge, none of the nonlinear models for flexible
beams which have been developed so far, are able to adequately describe all the
nonlinear behaviour of the beam. In contrast to other work, the development of the
nonlinear model was not only based on nonlinear theory, but primarily on the
experimental observation and understanding of nonlinear behaviour of the flexible
beam. Nonlinear beam theory was initially applied and then modified corresponding to
the experimental observations. As a result, the developed nonlinear model of the flexible
cantilever beam corresponded very well with the experimental results (refer to Chapter
4).
In addition, the developed model can be simply expressed in state-space form, which is
easily converted into an Auto-Regressive Moving Average ( A R M A ) model. A s will be
shown in Chapter 5, the A R M A model is used to predict the response of the beam on
line using the conventional linear Least Mean Square (LMS) algorithm. The estimated
response based on the developed nonlinear model was excellent for both linear and
nonlinear cases, whereas the results obtained from Finite Impulse Response (FIR) and
nonlinear Volterra filters failed to predict the low frequency vibration induced in the
beam due to nonlinear modal coupling. This on-line identification method based on the
developed nonlinear model, therefore, makes a significant contribution to developing a
feedforward as well as feedback control scheme for the cancellation of nonlinear
vibration in a flexible cantilever beam.
3
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
1.2 EXPERIMENTAL APPARATUS
1.2.1 Excitation Set-ups
Figures 1.1 to 1.6 show various excitation set-ups of a spring steel beam that were used
in the course of the preliminary investigation into the nonlinear dynamics of a slender,
inextensional and flexible cantilever beam (one end of the beam was clamped while the
other end was free), for different excitation methods and beam orientations. The
preliminary experiments were necessary to establish that consistent nonlinear
phenomena would occur independent of the beam position, orientation, and excitation
source. The optimum orientation of the beam offering the lowest sensitivity to the
gravitational effect and mass loading on the beam, was determined.
The following description outlines two excitation arrangements that were used for the
preliminary investigation.
1) Electromagnetic shaker excitation
Figures 1.1 to 1.3 show the beam excited by a Ling Dynamic Systems Model V406
electromagnetic shaker in each of three beam orientations:
the shaker pointing upwards, and the beam pointing horizontally (Figure 1.1),
. the shaker pointing sideways, and the beam pointing horizontally (Figure 1.2),
. the shaker pointing sideways, and the beam pointing vertically up (Figure 1.3).
It was observed that the three different beam orientations had similar nonlinear
effects, despite a slight variation in resonance frequency with a similar percentage
change for all modes. The variation in resonance frequency was assumed to be due
to the gravitational force, the accelerometer mass and the associated cable loading
on the beam. Since the weight of the accelerometer and the cable contributed to the
decrease in resonance frequency, the best setup arrangement was found to be with
the shaker pointing sideways and the beam pointing horizontally (Figure 1.2). With
this arrangement, the measurement was less sensitive to gravity and the mass
4
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
loading on the beam. In this case, only a small difference in resonance frequency
was observed with and without the accelerometer attached at the free end of the
cantilever beam (see Table 1.1). This orientation was then chosen as a basis for the
development of the nonlinear model of the cantilever beam in Chapter 4.
Figure 1.1 The shaker pointing upwards, and the beam pointing horizontally.
Figure 1.2 The shaker pointing sideways, and the beam pointing horizontally.
5
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
Figure 1.3 The shaker pointing sideways, and the beam pointing vertically up.
Beam orientation
Resonance Frequency of 1st order mode Resonance
Frequency of 2 n d order mode
Resonance Frequency of 3 rd order mode
Figure 1.1
with accelerometer
4 Hz
24.6 Hz
69.6 Hz
without accelerometer
4.1Hz
25 Hz
71.6 Hz
Figure 1.2
with accelerometer
4 Hz
24.8 Hz
70.4 Hz
without accelerometer
4.1Hz
25 Hz
71.6 Hz
Figure 1.3
with accelerometer
3.9 H z
24.6 H z
69.8 H z
without accelerometer
4.05 H z
24.8 H z
71.1 Hz
Table 1.2 The resonance frequencies of the beam with / without the Entran
accelerometer for different beam orientations.
6
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
2) Servo motor excitation
The same series of tests were carried out with a Digiplan BLH150 D C motor as an
excitation source for the beam. The D C motor had an adequate bandwidth to excite
the first three resonance modes of the beam. Figures 1.4 to 1.6 show the beam
excited by the D C motor in each of the following positions:
the beam pointing vertically up (Figure 1.4),
the beam pointing horizontally (Figure 1.5),
. the beam pointing vertically down (Figure 1.6).
It was observed that the three different positions of the beam, as shown in Figures 1.4
to 1.6, gave a variation in resonance frequency similar to that obtained with the three
orientations of the electromagnetic shaker (see Table 1.2). Again, the difference in
resonance frequency may be due to the gravitational effect. The vertical and upward
orientation of the beam (Figure 1.4) was finally selected as the optimum measurement
position.
Figure 1.4 The beam pointing vertically up.
7
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
Figure 1.5 The beam pointing horizontally.
Figure 1.6 The beam pointing vertically down.
8
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
Beam
orientation
Resonance Frequency of 1st order mode Resonance
Frequency of 2nd order mode
Resonance Frequency of 3 ^ order mode
Figure 1.4
with accelerometer
3.9 H z
24.6 H z
69.8 H z
without accelerometer
4.05 H z
24.8 H z
71.1 Hz
Figure 1.5
with accelerometer
4 Hz
24.6 Hz
69.6 Hz
without accelerometer
4.1Hz
25 Hz
71.6 Hz
Figure 1.6
with accelerometer
4.1Hz
24.8 H z
70.3 H z
without accelerometer
4.2 H z
25.3 H z
71.7 H z
Table 1.2 The resonance frequencies of the beam with / without the Entran
accelerometer for different beam orientations.
1.2.2 Instrumentation
Figure 1.7 shows a complete experimental set-up of the cantilever with a Ling Dynamic
Systems Model V406 electromagnetic shaker used as the excitation source. The
electromagnetic shaker was driven by a Hewlett Packard arbitrary function generator,
via a Ling Dynamic Systems power amplifier. The system input was measured with an
accelerometer mounted at the clamped end of the beam while the beam response was
picked up with an accelerometer attached to the free end (tip) of the beam. In order to
avoid unnecessary weight loading on the beam, the cable associated with the
accelerometer was glued to the surface of the beam using a thin double-sided adhesive
tape. Since the beam is slender and flexible with a low fundamental resonance
frequency at approximately 4 Hz, the mass and low frequency response of the
accelerometer were essential criteria in selecting the appropriate accelerometer to be
mounted at the tip of the beam. Extended operating range and overange were additional
selection criteria of the accelerometer, since the beam was expected to be subjected to
9
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
large excitation during the investigation of nonlinear behaviour. The critical response of
the beam was identified beforehand by measuring the maximum acceleration of the
beam, and a safety margin was applied in order to ensure that the selected accelerometer
did not exceed its operating range. A n Entran EGAX-250 was chosen on account of its
low mass (0.5 gram), large operating range (250g), and sufficiently broad frequency
response. The frequency range extended from dc to 1 kHz. However, the measured
signal from the accelerometer was highpass filtered at 1 H z and then integrated to
velocity or double integrated to displacement using a conditioning amplifier purposely
built for that particular accelerometer. The conditioning amplifier provided
accelerometer sensitivity scaling, calibration and signal amplification. Since the
frequency range of interest occurred in the 3 H z to 73 H z region, displacement was
chosen as the optimum measurement unit. The beam displacement was physically
measured and cross checked against the electrical output from the accelerometer in
order to validate the results. The analog conditioning amplifier was, in the latter phase
of the research process, replaced by a digital signal conditioning amplifier using a
dSpace™ Digital Signal Processor (DSP) card operating in conjunction with dSpace and
Matlab software. A P C B accelerometer (model no. 309A) was also used for
measurement. It had a mass of 1 gram, a sensitivity of 5mV/g, and a frequency range
from 2 H z to 20 kHz. The experimental results have shown similar nonlinear responses,
except that the response of the beam with the P C B accelerometer had lower resonance
frequencies, due to the larger mass of the P C B accelerometer (see Table 1.3).
Accelerometer type
Resonance frequency of 1st order mode
Resonance frequency of 2nd order mode
Resonance frequency of 3rd order mode
Entran
4 Hz
24.8 Hz
70.4 Hz
PCB
3.8 Hz
23.9 Hz
69.8 Hz
Table 1.3 Comparison of the resonance frequencies of the beam for different
accelerometers.
10
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
Function Generator
Conditioning Amplifier Beam Shaker
Accelerometer
Oscilloscope
Figure 1.7 The experimental set-up of the cantilever beam.
Power Amplifier
The preliminary experiments have shown that the cantilever beam had consistent
nonlinear behaviour independent to its orientation, excitation source and the
accelerometer mass loading on the beam. The nonlinear behaviour was identified by the
following observations:
change of the resonance frequency,
11
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
. jump phenomenon,
energy transfer from higher order modes to lower order modes,
nonlinear modal coupling,
frequency modulation,
. nonlinear stiffness.
The experimental investigation into the nonlinear behaviour of the beam was necessary
for the development of the nonlinear model of the cantilever beam. In order to ensure
that the nonlinear model could be applied to any slender and flexible spring steel
cantilever beam, a similar investigation was carried out for three different sizes of beam
using the excitation arrangement shown in Figure 2. The dimensions of the three beams
were:
. beam 1 : 332 m m (length) x 12.71 m m (width) x 0.45 m m (thickness)
beam 2 : 332 m m (length) x 19.17 m m (width) x 0.75 m m (thickness)
beam 3 : 332 m m (length) x 25.49 m m (width) x 0.6 m m (thickness)
Although the three different beams were slender and flexible, they all have different
resonance modes due to the different ratios between thickness and length. However,
they all exhibited similar nonlinear responses. Beam 3 was then chosen for further
research.
1.2.3 Experimental Considerations
1. Since the magnitude of the frequency response of the P C B accelerometer decreases
to -2.5 dB at 4 H z and -7 dB at 2. 5 Hz, it is necessary to compensate the amplitude
of the first order mode by increasing the gain of the integrator. As a trade off, low
frequency noise and dc-offset are also amplified. In order to reduce the low
frequency noise and eliminate the dc-offset, the integrated signal was high-pass
filtered at 1 Hz.
2. In order to optimise the dynamic range, out of band signals and noise were reduced
by applying a low-pass filter with a cut-off frequency at 200 Hz, since only the first
three modes of the beam were of interest (between 1 H z to 71 Hz).
12
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
3. Since the beam is very thin and flexible, the accelerometer has to be placed exactly
at the tip on the neutral axis of the beam with the accelerometer cable glued along
the neutral axis in order to prevent the beam twisting.
1.3 NONLINEAR BEHAVIOUR OF THE BEAM
1.3.1 Change of the Resonance Frequency
The cantilever beam chosen for this experiment was a thin spring steel beam with a
dimension of 332 m m (length) x 25.49 m m (width) x 0.6 m m (thickness). One end of the
beam was clamped to the top of the Ling Dynamic Systems electromagnetic shaker
while the other end was free, as shown in Figure 1.8.
Function Generator
Power Amplifier
Shaker Beam Accelerometer
•!;: :;r n •:•:• ••:••:•.'••:• ••.••J*l
x :-:.:.:-:-:-:.:.:.:-:-:-:.:.:::::-:.:-:-:.:-:-:-:-:-:-: ri
Conditioning
Amplifier Conditioning Amplifier
Hewlett-Packard Analyser
Figure 1.8 The experimental set-up for the flexible cantilever beam.
Initially, the clamped end of the beam was excited with random noise by the shaker.
T w o accelerometers were used to measure the vibration. One accelerometer was placed
on the top of the clamped end while the other was attached at the tip of the beam. Both
measured signals were fed into a Hewlett-Packard Digital Signal Analyser in order to
obtain the linear frequency response of the cantilever beam.
Figure 1.9 shows the first three resonance peaks of the beam. The resonance frequencies
of the first, second and third order modes were 4 Hz, 24.8 H z and 70.4 Hz, respectively.
13
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
30 4 Hz
24.8 Hz
-40
-50
70.4 Hz
20 40 60 Frequency [Hz]
80 100
Figure 1.9: The linear frequency response of the beam.
The frequency response, using a random noise input, is only valid when the system is
linear, ie. when the system input is coherent with the system output. However, for the
nonlinear case, the system output is no longer coherent with the system input. A
sinusoidal force was then applied to the clamped end of the beam in order to measure
the nonlinear frequency response of the beam.
Figures 1.10, 1.11 and 1.12 show the nonlinear frequency responses of the beam plotted
within the frequency range in the vicinity of the resonance frequency of the first, second
and third order modes, respectively, for different excitation amplitudes. Each curve in
the figures shows the auto spectrum of the displacement measured at the free end of the
beam for one fixed excitation amplitude, when the excitation frequency was swept very
slowly.
14
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
With reference to the scaling of the figures, it should be noted that the accelerometer
used for Figure 1.9 has a different sensitivity from that of Figures 1.10 to 1.12. This, in
conjunction with different conditioning amplifier gain, accounts for the differences in
magnitude scaling factors.
8
6
4
m S 2 *-» 3 Q.
-4
-6
-81 i i 1 1 ' ' 3.4 3.6 3.8 4 4.2 4.4
Excitation frequency [Hz]
Figure 1.10 Frequency response curves in the vicinity of the first order mode for
different excitation amplitudes.
It can be seen in Figures 1.10-1.12 that for small excitation amplitudes, the resonance
frequencies (corresponding to the peaks in the response) occurred at 4 Hz, 24.8 H z and
70.4 Hz, which are similar to the first, second and third modes of the beam observed in
the linear frequency response, respectively. However, the resonance frequency then
shifted with increasing excitation amplitude. The resonance frequency of the first order
mode increased with increasing excitation amplitude (see Figure 1.10), whereas the
resonance frequency of the second and third order modes decreased with increasing
excitation amplitude (see Figures 1.11 and 1.12). A s will be shown later in this chapter,
these results are due to the changing beam stiffness for different modes. At the
resonance frequency of the first order mode, the beam has a hardening stiffness
15
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
characteristic, and it changes to a softening characteristic at higher order modes. The
difference in the beam stiffness characteristic between the first order mode and the
higher order modes can be related to nonlinear coupling between the higher order modes
and the first order mode. It will be shown that the nonlinear modal coupling is due to the
energy transfer from higher order modes to the next lower order mode and then to the
first order mode when the excited higher order modes become saturated, but not vice
versa. In other words, only lower order modes can be nonlinearly excited by large
vibrations of the externally excited higher order modes. However, the higher order
modes are not excited by the large excitation amplitude of the first order mode. A s a
result, the first order mode starts to increase its resonance frequency when it becomes
saturated. Further details on nonlinear modal coupling and nonlinear stiffness of the
beam are included in Sections 1.3.4 and 1.3.6, respectively.
23 24 24.5 25 Excitation frequency [Hz]
Figure 1.11 Frequency response curves in the vicinity of the second order mode for
different excitation amplitudes.
16
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
69.5 70 70.5 Excitation frequency [Hz]
Figure 1.12 Frequency response curves in the vicinity of the third order mode for
different excitation amplitudes.
The same series of tests were carried out with the Digiplan BLH150 DC motor as an
excitation source for the beam. In this case, the beam was subject to rotational
acceleration rather than horizontal acceleration, as was the case with the electromagnetic
shaker. Similar results were obtained.
1.3.2 Jump Phenomenon
By sweeping the excitation frequency forward and backward very slowly in the vicinity
of the resonance frequency of one of the higher order modes, such as the second and
third order modes, a jump phenomenon was observed in addition to the change in the
resonance frequency which occurred for a large excitation amplitude.
It can been seen in Figure 1.13 that the magnitude of the tip vibration, at the excited
frequency of the measured output, increased rapidly from point a as the excitation
frequency increased slowly at 0.35 V, and reached its maximum at the resonance
frequency at point b. W h e n the excitation frequency passed the resonance frequency of
17
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
the second order mode, the magnitude then decreased slowly until point c. The
excitation frequency was then decreased slowly, and it was observed that the magnitude
increased slowly back to point b, and kept increasing until point d even when the
excitation frequency passed the resonance frequency. The magnitude then dropped back
to point a again. This is a typical non-linear phenomenon, where there are multiple
values of the magnitude of the response for a given excitation frequency.
Similarly, a jump phenomenon was observed as shown in Figure 1.14 when the
excitation frequency was swept slowly forward and backward in the vicinity of the
resonance frequency of the third order mode (at 0.3 V) .
15
10
m
=- 5 CL +-.
o •o 2 « => 0 w ro d)
-5
-10 23 23.5 24 24.5 25 25.5 26
Excitation frequency [Hz]
Figure 1.13 Jump phenomenon occurred when excitation frequency swept slowly
forward and backward in the vicinity of the second order mode.
18
)K Forward O Backward
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
0.
m T3 3 Q. 3
o 3 (0 CO
cu
-5
-10
-15
I
d
-
a
i
i i i
b
x c
i i i
i
X Forward O Backward
-
*
•
69 69.5 70 70.5 71 Excitation frequency [Hz]
71.5 72
Figure 1.14 Jump phenomenon occurred when the excitation frequency swept
slowly forward and backward in the vicinity of the third order mode.
In contrast to the second and third order modes, the jump phenomenon was not observed
when the excitation frequency was swept slowly forward and backward in the vicinity
of the resonance frequency of the first order mode. However, a jump in magnitude of the
first order mode occurred when the beam was excited at the second or third order modes
with a sufficiently large excitation amplitude. Figure 1.15 shows the magnitude of the
first order mode plotted against the frequency range of the third order mode, when the
excitation frequency was swept slowly from 68.7 H z to 71 H z at 0.35 V, in a similar
way to Figure 1.12. From this figure, it can be seen that there is a large increase in
magnitude of the first order mode when the excitation frequency reached the resonance
frequency of the third order mode (69.6 Hz) at point b. Since the excitation amplitude
was reasonably large, the third order mode became saturated and coupled to the first
order mode. The magnitude of the first order mode, as well as the resonance frequency
of the first order mode, were changing persistently within a range between the line be
and bd until the excitation frequency passed 69.9 Hz. The magnitude then quickly
19
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
dropped to point e. The excitation frequency was then swept slowly backward. The
magnitude of the first order mode initially remained low and then suddenly increased to
point/when the excitation frequency hit the resonance frequency again. Similar to the
case when the excitation frequency was swept forward, the magnitude and frequency of
the first order mode changed persistently within a range between the line^ and fh until
the excitation frequency reached 69.1 Hz. The magnitude of the first order mode then
suddenly disappeared (dropped to approximately -50 dB). It can been seen in the figure
that there were multiple values of magnitudes of the first order mode between the
excitation frequency range of 69.1 Hz to 69.9 Hz. The jump in magnitude was due to
the energy transfer from higher order modes to the first order mode when the excited
higher order mode became saturated. A detailed description of the energy transfer
between modes is described in the following section.
20
10
m 0 73
CD
| -10
W
£-20 cu •a
1 -30 CO
-40
-50
69 69.2 69.4 69.6 69.8 70 70.2 Excitation frequency [Hz]
Figure 1.15 Jump in magnitude of the first order mode occurred when the excitation
frequency swept slowly forward and backward in the vicinity of the third
order mode.
20
8 /-^ *5, ^ Forward O Backward
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
1.3.3 Energy Transfer From Higher To Lower Order Modes
W h e n a sinusoidal signal was fed into the mechanical shaker, the power spectra
measured at the two accelerometers displayed different characteristics, which were
dependent upon the amplitude of the excitation signal. It was observed that the power
spectrum measured at the clamped end had a distinct peak at the excitation frequency
(see Figures 1.16a and 1.17a). This displacement component was the major contributor
(as an inertia force) to the beam vibration measured at the free end. Excitation
harmonics were also visible in the measurement at the clamped end (approximately 36
dB lower than the fundamental). These harmonic components may be attributed to the
nonlinearity in the excitation system (ie. the shaker). However, the measured response at
the clamped end, below the excitation frequency, was at the background noise level (at
least 50 dB below the fundamental).
If the excitation frequency was close to or equal to the resonance frequency of one of
the higher order modes and the excitation amplitude was sufficiently large, lower
frequency peaks were observed at the free end of the beam. Since the lower frequency
displacement levels at the clamped end were small, it was concluded that the lower
frequency peaks were due to the nonlinear coupling between the excited vibration and
the lower order modes of the beam. Figures 1.16b and 1.17b show the power spectra
measured at the tip of the beam corresponding to the power spectra measured at the
clamped end, as shown Figures 1.16a and 1.17a, when the beam was excited with a
frequency of 24 H z (0.35 V ) and 69 H z (0.35 V ) , respectively. It can be seen from the
figures that in addition to the excitation frequency component, peaks at lower
frequencies (corresponding to the resonance frequencies of the first and second order
modes) were observed. The magnitudes and frequencies of these peaks were changing
randomly with time. The change in magnitude can be seen more clearly in Figures 1.18
and 1.19, which captured the time response measured at the tip of the beam when the
beam was excited at 24 H z (0.35 V ) and 69 H z (0.35 V ) , respectively. In order to
observe the frequency shift of the first order mode, the measured displacement was band
filtered simultaneously to select the magnitude of the first order mode separately. The
21
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
frequency of the first order mode was then obtained by measuring the time period tp
(per cycle), between the magnitude peaks in the time history of the first order mode. As
can be seen from Figures 1.20 and 1.21, the frequency of the first order mode was
changing randomly with time when the beam was excited at 24 H z (0.35 V ) and at 69
H z (0.35V), respectively. The change of magnitude and frequency of the peaks in the
response was due to the energy transfer from a higher order mode to the lower order
modes. The details are described as the follows according to the results shown in Table
1.4:
(1) When the beam was excited with a sinusoidal signal at 24.5 Hz (0.2 V), a peak at
24.5 H z only was observed in the auto-spectral density of the displacement of the
beam measured at the free end. A n increase in excitation amplitude to 0.25 V
resulted in the excitation of the first order mode. The magnitude and frequency of
the first order mode varied between the range of 6 m V and 16.54 m V , and 3.28 H z
and 3.635 Hz, respectively. Further increase of the excitation to 0.3V caused a
decrease in resonance frequency of the first order mode to be observed. The
resonance frequency also varied in the range between 2.81 H z and 3.46 Hz, whereas
the magnitude varied in the range between 20.52 m V and 24.52 m V . However, a
further increase in the excitation amplitude (0.35 V ) caused a sudden increase in the
resonance frequency.
(2) W h e n the beam was excited at 69 H z at 0.2 V, only a displacement peak at the
excitation frequency was observed in the measured response. W h e n the excitation
amplitude was increased to 0.25 V, the second order mode started to vibrate at 24.8
Hz. A continued increase in the excitation amplitude caused the magnitude and the
resonance frequency of the second order mode to decrease. This process was
associated with energy transfer to the first order mode. The energy transfer from the
third order mode to the first order mode occurred spontaneously when the excitation
amplitude continued to increase. At the same time the magnitude of the second
order mode dropped considerably. Similar to (1), a further increase in the excitation
22
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
amplitude caused the first order mode to become saturated; as a result, the
resonance frequency of this mode increased.
Excitation
frequency
24.5 Hz
69 Hz
Excitation
amplitude
0.20 V
0.25 V
0.30 V
0.35 V 0.20 V
0.25 V 0.30 V
0.35 V 0.5 V
1st Frequency
—
3.2 - 3.6Hz
2.8-3.5 Hz 3.7-3.8 Hz
—
—
3.3-3.4 Hz 3. 2-3.3 Hz
4Hz
The
mode Amplitude
—
6-16.5mV 20.7 - 24.5 m V 0.8 - 0.9 V
—
—
19-23mV 0.9-1.2 V 10.5 mV
Output
2nd Frequency
—
24.7 - 24.8 Hz 24.7 - 24.8 Hz
24.7 Hz —
mode
Amplitude
—
10.5-15.5 m V
16.5-18.5 m V 5.5- 7.1 raV
—
Table 1.4 The change of the magnitude and resonance frequency of the lower order
modes as a function of the excitation amplitude.
rum [dB]
o
Power Spect
O
o
.(a) :
/ft^Vv4W|yw
>4Hz
i 48 Hz A
t^\%ff» 10
10
20 30 Frequency [Hz]
40 50
20 30 Frequency [Hz]
40 50
23
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
Figure 1.16 (a) Power spectrum measured at the clamped end, (b) The corresponding
power spectrum measured at the tip with an excitation of 24 Hz at 0.35 V.
CO
E 2 o CD CL
CO
1 -100
co 2, E S o cu Q. CO
1 Q.
-50.
•100
3.3 HZ
40 60 Frequency [Hz]
100
69 Hz
65.7 HA 72.3 Hz
40 60 Frequency [Hz]
100
Figure 1.17 (a) Power spectrum measured at the clamped end, (b) The corresponding
power spectrum measured at the tip with an excitation of 69 Hz at 0.35V.
0.2 0.3 0.4 Time [seconds]
0.9 1 1.1 Time [seconds]
Figure 1.18 The measured displacement at the tip with an excitation of 24 Hz at 0.35 V.
24
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
0.2 0.3 0.4 Time [seconds]
0.9 1 1.1 Time [seconds]
Figure 1.19 The measured displacement at the tip with an excitation of 69 H z at 0.35 V.
3.83
3 4 5 Time [seconds]
Figure 1. 20 The frequency of the first order mode, for an excitation of 24 H z at 0.35V,
was changing randomly.
25
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
3.33
3.32
3.31 L
3.3 77
^3.29
| 3.28 £
3.27
3.26
3.25
3.24
-e
Figure 1.21 The frequency of the first order mode for an excitation of 69 H z at 0.35 V
was changing randomly.
In summary, the experiments have shown that the effect of the nonlinear interaction on
the measured auto spectral density, at the free end of the beam, could be described in
terms of energy transfer from a higher order mode to the lower order modes. W h e n the
excitation frequency is close or equal to one of the resonance frequencies of a higher
order mode (in this case the second and third order mode), and the excitation amplitude
is sufficiently large, the beam response at the excitation frequency becomes saturated.
The beam energy at the excitation frequency will then dissipate to the next lower order
mode. The magnitude of the lower order mode is increased up to a point with increasing
excitation amplitude. A further increase in excitation amplitude will then cause the
magnitude of the lower order mode to decrease at the same time the resonance
frequency shifts, and the lower order mode may also become saturated. W h e n saturation
occurs, the energy will then couple to the next lower order mode. In this way, the energy
is dissipated from a higher mode to the next lower mode, and continues to the lowest
order mode as the excitation amplitude increases. A shift in the resonance frequency of
that mode will then occur before the energy is dissipated to the next lower mode. W h e n
26
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
the first order mode eventually becomes saturated, the energy is not dissipated back to
higher order modes. Instead, the resonance frequency of that mode increases. This
energy transfer phenomenon only occurs from a higher order mode to a lower order
mode, but not vice versa, ie. nonlinear coupling exists only between the excitation
vibration and the modes that have a resonance frequency below the excitation
frequency.
1.3.4 Nonlinear Modal Coupling
The first, second, and third mode shapes of the beam for different excitation amplitudes
are shown in Figures 1.22 to 1.24, respectively. The mode shapes were measured using
a digital camera when the beam was excited at its resonance frequencies. As the natural
frequency of the resonance modes shifted with increasing excitation amplitude (as
described in Section 1.3.1), the excitation frequency was, therefore, changed
corresponding to the excitation amplitude, when measuring the mode shapes for
different excitation amplitudes.
x-axis of the beam [cm]
Figure 1.22 The first mode shape of the cantilever beam for different excitation
amplitudes.
27
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
a) without coupling with the 1st order mode.
50
E IE +-•
c CD
E CD O
ro o.
10 15 20 25 30
b) with coupling with the 1st order mode.
35
10 15 20 25 x-axis of the beam [cm]
35
Figure 1.23 The second m o d e shape of the cantilever beam for different excitation
amplitudes.
c CD
E CD O
ro o. in
a) without coupling with the 1st order mode.
1 1 r * lnput:0.4V O lnput:0.6V
10 15 20 25 30
b) with coupling with the 1st order mode. 35
•a 5 -o 3 w co CD
10 15 20 25 x-axis of the beam [cm]
35
Figure 1.24 The third m o d e shape of the cantilever beam for different excitation
amplitudes.
28
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
As can be seen from the figures, the magnitude of the displacement of the beam for all
three mode shapes increased proportionally to the increment in the excitation amplitude.
However, when the beam reached its maximum deformation at the second or third order
resonance, a further increase in the excitation amplitude no longer increased the
deflection of the beam. The beam then started to couple to the first order mode as shown
in Figures 1.23 and 1.24.
In conclusion, the normalised mode shapes of the cantilever beam do not change with
the excitation amplitude. The mode shape functions for a non-linear case are thus the
same as for a linear case. However, when the beam reached its maximum deformation at
the higher order modes, modal interactions occurred due to energy transfer from the
higher order mode to the lower order mode(s), when the higher order mode became
saturated. The deflection of the beam was observed to be a linear summation of the
deflections of the excitation mode and the coupled modes.
Based on the experimental observations, the lateral deflection of the beam, W(x,t), can
OO
be assumed to be expressed as W(x,t) = £ 0 ; (x)f; (t), where 0;(x) is the i mode shape i= l
function, and f;(t) is the time-variant function of the mode /'. Hence, the mode shape
functions ®;(x) are derived using the linear theory of the cantilever beam. Only the time-
variant functions fj(t) and the resonance frequencies need to be determined as a function
of the excitation amplitude and frequency. This assumption has formed a basis for the
development of the nonlinear model of the cantilever beam.
1.3.5 Frequency Modulation
As described in Sections 1.3.3 and 1.3.4, the beam energy dissipates from higher order
modes to lower order modes, but not vice versa. W h e n the beam reaches its maximum
deformation at the higher order mode, the beam energy at the excitation frequency will
then dissipate to the next lower order mode, right down to the first order mode. This can
be seen in Figures 1.16 and 1.17. In addition to the peak at the first order mode,
29
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
sidebands were observed either side of the higher order modes as a result of frequency
modulation between the first order and higher order modes. As will be shown in Chapter
4, the frequency modulation is a nonlinear phenomenon of the beam. However, the
magnitude of the sidebands were insignificant compared to other frequency
components, and the modulation was impossible to identify in the measured time series
(see Figures 1.18 and 1.19).
1.3.6 Nonlinear Stiffness
In this section, the stiffness of the beam, which is defined in this work as the
relationship between the acceleration applied to the clamped end of the beam and the
displacement measured at the tip of the beam, is examined.
Figure 1.25 shows the displacement of the beam as a function of the excitation
amplitude (acceleration applied to the clamped end of the beam) when the beam was
excited at 4 H z ( the resonance frequency of the first order mode). From this figure, it
can be seen that there is a piecewise linear relationship between the excitation amplitude
and the displacement at the tip. It can also be seen from the figure that the slope of the
displacement is steepest from point a to b. W h e n the excitation amplitude reached 0.04
V at point b, the displacement increased less for the same increment of excitation
amplitude. The slope of the displacement then started to decrease slightly (from point b
to point c), and then flattened when the excitation amplitude passed 0.07 V. This
indicates that a hardening-type stiffness is present at the first order mode. This result
explains the experimental results in Section 1.3.1, where the resonance frequency of the
first order mode increased with increasing excitation amplitude. The increase in
resonance frequency is a typical nonlinear phenomenon of a hardening-type system.
30
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
, , F o^ Q. *-< CD JC * J
-*—' CO TJ (l> 3 in CO CD E -4-*
c CD E CD O CO Q.
w Q
10
9
8
/
6
5
4
3
2
1 "a
0 0.02 0.04 0.06 0.08 0.1 Acceleration measured at the clamped end [Volts]
Figure 1.25 Displacement measured at the tip of the beam as a function of the excitation
amplitude when the beam was excited at 4 Hz.
Figures 1.26a and 1.26b show the displacements of the magnitude of the second and
first order mode as a function of the excitation amplitude, respectively, when the beam
was excited at 24.8 H z (the resonance frequency of the second order mode). Similar to
the experimental results in Section 1.3.4, the displacement of the beam increased
proportionally to the increment in the excitation amplitude (from point a and point b as
shown in Figure 1.26a). W h e n the beam reached its maximum deformation (at point c),
a further increase in excitation amplitude then caused the beam to couple with the first
order mode through internal energy transfer between the modes (refer to Section 1.3.3).
In other words, when the excitation amplitude reached the coupling threshold amplitude
of 0.06 V, a further increase in excitation amplitude did not increase the displacement of
the beam at 24.8 H z (as shown in Figure 1.26a). Instead, the beam started to couple with
the first order mode (see Figure 1.26b). The magnitude of the first order mode was
changing randomly in between lines ce and cf. The total displacement at the tip was a
summation of the displacements of the first and second order modes as shown in Figure
1.27. From the figure, it can be seen that the slope of the displacement between point b
31
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
and c became steeper when the excitation amplitude reached the coupling threshold
value. This is a typical softening characteristic of nonlinear stiffness. This result also
corresponds to the experimental results in Section 1.3.1, where the resonance frequency
of the second order mode decreased with increasing excitation amplitude. It appears that
the mechanism of the softening characteristic is the nonlinear coupling from a higher
order mode to the first order mode. The large increase in the displacement of the first
order mode, due to the nonlinear modal coupling, caused the increase in the slope
between point b and c of the total displacement.
Similar to the second order mode, the third order mode also has a softening stiffness
characteristic as shown in Figure 1.28. From this figure, it can be seen that the slope of
the displacement at the tip between point b and c (when the beam started to couple with
the first order mode) was steeper than the slope of displacement shown in Figure 1.27.
This indicates that the coupling between the third order mode and the first order mode
was stronger than that between the second order mode and the first order mode.
E
Q.
CD .C +-. +-.
CO
xs 9>
0.02 0.04 0.06 0.08 Acceleration measured at the clamped end [Volts]
0.02 0.04 0.06 0.08 Acceleration measured at the clamped end [Volts]
0.1
0.1
Figure 1.26 Displacement measured at the tip of the beam as a function of the excitation
amplitude when the beam was excited at 24.8 Hz.
32
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
0.02 0.04 0.06 0.08 Acceleration measured at the clamped end [Volts]
0.1
Figure 1.27 Total displacement measured at the tip of the beam as a function of the
excitation amplitude when the beam was excited at 24.8 Hz.
5
4.5
I 4 Q.
© 3.5 .£
ro 3
meas
ured
1 1-5
Displace
o
i
a ^ ^ ^
7 l
1
b /
i
c
•
d e
J •&
f
_
_
-
-
0.02 0.04 0.06 0.08 Acceleration measured at the clamped end [Volts]
0.1
Figure 1.28 Total displacement measured at the tip of the beam as a function of the
excitation amplitude when the beam was excited at 70.3 Hz.
33
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
In addition to the softening stiffness characteristic, a hysteretic stiffness characteristic
was also observed when decreasing the excitation amplitude. Figures 1.29a and 1.29b
show the displacement of the magnitudes of the second and first order mode as a
function of the excitation amplitude, respectively, with excitation of the second order
mode and decreasing excitation amplitude. A s can be seen from the figures, the
magnitudes of the first and second order modes remained large even when the excitation
amplitude was decreased to 0.06 V (which was the threshold of nonlinear coupling - see
Figure 1.26). The coupling did not disappear until the excitation amplitude reduced to
0.02 V. The beam then vibrated at the second order mode only, indicating that the beam
has a decoupling threshold smaller than the coupling threshold. For comparison, the
total displacement (the summation of deflections of the first and second modes)
measured at the tip with increasing amplitude was plotted against that with decreasing
excitation amplitude, as shown in Figure 1.30. It can be seen from the figure that the
magnitude of the displacement with decreasing excitation amplitude did not follow the
same path as with increasing excitation amplitude. This is a typical characteristic of
hysteresis. The hysteretic characteristic was observed in parallel with the softening
stiffness characteristic, ie only with large excitation of the second or third order mode,
but not with the first order mode.
In conclusion, the experimental results have shown that the beam has a different
nonlinear characteristic of stiffness when it is excited at different resonance frequencies.
At the first order mode of the beam, the stiffness has a hardening characteristic.
However, the stiffness changes from a hardening to a softening characteristic at higher
order modes. The change in the stiffness characteristic corresponds to the change in the
resonance frequencies of the beam. In addition to the softening stiffness characteristic,
hysteretic characteristics were also observed with large excitation of the higher order
modes.
34
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
E
g. CD
ra T3
0.02 0.04 0.06 0.08 Acceleration measured at the clamped end [Volts]
0.02 0.04 0.06 0.08 Acceleration measured at the clamped end [Volts]
0.1
0.1
Figure 1.29 Displacement at the tip of the beam as a function of the excitation amplitude
(decreasing from 0.09V to 0V) when the beam was excited at 24.8 Hz.
4.5
4
^ Forward O Backward
0.02 0.04 0.06 0.08 Acceleration measured at the clamped end [Volts]
0.1
Figure 1.30 Total displacement measured at the tip of the beam as a function of the
excitation amplitude when the beam was excited at 24.8 Hz.
35
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
1.4 CONCLUSIONS
A number of nonlinear phenomena have been observed in the experiments when the
beam was subjected to large vibration deformations. These include change in the
resonance frequency, jump phenomenon, modal coupling, modulation, nonlinear
stiffness and hysteric damping characteristics. Many of these nonlinear phenomena
verified the results of other work. For instance, Anderson et al [2-4] observed modal
interactions and change in resonance frequencies in a vertically mounted cantilever
beam when the beam was subject to harmonic vertical base motion. Nayfeh and M o o k
[48-49] found energy transfer from higher order modes to lower order modes in flexible
structures. However, the inter-relationship between these nonlinear phenomena was not
discussed. Therefore, the significance of the current work is to provide an understanding
of this relationship.
The degree of nonlinearity is dependent upon the stiffness of the beam. The
experimental results have shown that there is a nonlinear relationship between the
acceleration applied at the clamped end and the displacement measured at the tip of the
beam, and the nonlinearity is different for each mode. At the first order mode, the beam
has a hardening stiffness characteristic. However, the stiffness characteristic changes
from hardening to softening when the beam is subjected to large amplitude vibration at
higher order modes. The change in the beam stiffness corresponds to the change in
resonance frequency of the modes with increasing excitation amplitude. The resonance
frequency of the first order mode was observed to increase with increasing excitation
amplitude, whereas the resonance frequencies of the second and third order modes
decreased with increasing excitation amplitude. Increasing and decreasing the resonance
frequency can be qualitatively related to the increasing and decreasing of the kinetic
energy of the free vibrating mode f — mcor2A 2 J, where cor is the resonance frequency
and A is the amplitude of the vibration. Increasing the input amplitude causes an
increase of the kinetic energy. For the first mode, it is not possible to transfer energy to
other higher order modes. The beam can only increase the resonance frequency to store
more kinetic energy if the modal amplitude cannot be further increased. O n the other
36
Chapter 1. Nonlinear behaviour of a flexible beam: Single Frequency Excitation
hand, the higher order modes use the nonlinear coupling to transfer energy when the
modal amplitude can no longer be increased. It appears that the higher order modes even
reduce the kinetic energy by reducing the resonance frequency to maintain the energy
transfer to the lower order modes.
In addition to the change in the resonance frequency of the beam, nonlinear modal
interactions were observed. W h e n the beam was excited at one of the higher order
modes and reached maximum deformation, the beam then started to couple to the first
order mode. This was due to the energy cascading from the higher order modes to the
lower order modes, right down to the first order mode. During the energy transfer, the
magnitudes of the resonance peaks were changing continuously corresponding to their
frequency shift. As a result of this, jump phenomena occurred (multiple values of
magnitudes obtained for a given excitation frequency). In addition, hysteresis (multiple
values of magnitudes obtained for a given excitation amplitude) was also observed.
W h e n the magnitude of the first order mode was sufficiently large, the frequency of the
first order mode was modulated with higher frequency components and subsequently
created sidebands. Hence, the beam can be described as a multi-degree-of-freedom
system with each mode of the beam corresponding to one degree-of-freedom. The
nonlinearity of the beam is different for each degree-of-freedom (mode). W h e n the
beam is subject to a large excitation at the first order mode, the beam will inherit a
hardening stiffness characteristic. Conversely, the beam has a softening stiffness
characteristic at higher order modes due to the energy transfer between modes. The
nonlinearity of the beam at higher order modes is dominated by the nonlinear coupling
between modes as a result of energy loss from higher order modes to lower order
modes.
Furthermore, in this work, it has been shown that the normalised mode shapes did not
change with excitation amplitude and, therefore, could be derived by using the linear
boundary theory of the cantilever beam (see Section 1.3.4). Only the resonance
frequency and time-variant function need to be determined as functions of excitation
amplitude. This assumption and other experimental observations provide the useful
foundation of the development for a nonlinear model of the flexible cantilever beam.
37
Chapter 2. Nonlinear behaviour of a flexible beam: Multi-Frequency Excitation
Chapter 2
NONLINEAR BEHAVIOUR OF A FLEXIBLE BEAM:
Multi-Frequency Excitation
2.1 INTRODUCTION
In the previous chapter, a sinusoidal signal was used to excite one vibration mode of the
beam at a time, in order to distinguish the different nonlinear characteristics contributed
by each mode. Using a single sinusoidal signal is more advantageous than random
noise, since the energy transfer phenomenon from higher order modes to lower order
modes is more easily identified. However, in practice, a number of beam modes may be
excited simultaneously. Consequently, it is useful to examine the behaviour of the beam
when two or more modes of the beam are excited. Furthermore, the experimental results
described in Chapter 1 have shown that in the nonlinear case, the beam can have a
multi-frequency response for a single frequency excitation. To actively control the
nonlinear frequency components induced in the beam, due to nonlinear modal coupling,
another control input at a frequency different from the primary excitation frequency may
be introduced. Therefore, it is necessary to study the response of the beam structure
under multi-frequency excitation.
Although there are numerous works on single frequency or random noise excitation, to
the author's knowledge, there is no published work on the response of the cantilever
beam to multi-frequency excitation. Because the linear response of the flexible
cantilever beam to multi-frequency excitation has not been investigated, this chapter
will start with the study of linear response of the beam, followed by examination of
nonlinear response of the beam to multi-frequency excitation. A series of tests, where
the beam was excited with multiple frequency excitations, ie. combinations of two or
38
Chapter 2. Nonlinear behaviour of a flexible beam: Multi-Frequency Excitation
three modes excited in parallel, were carried out to identify interactions between the
modes of the beam and the effect of internal beam properties on the interactions. Both
periodic and non-periodic signals were used for the experiments.
As will be shown in this chapter, the response of the beam to a periodic signal is the
same as to a non-periodic excitation. However, there was energy transfer from lower
order modes to higher order modes in the linear case, which is contradictory to the
traditional understanding of the linear response of a single degree of freedom system.
This is also in contrast to the nonlinear case where the energy transfers from higher
order modes to lower order modes.
Furthermore, this study has led to two useful concepts for the development of an active
control algorithm for nonlinear vibration cancellation in the flexible beam. The first is
using the low frequency vibration induced in the beam due to nonlinear modal coupling
to cancel the induced nonlinear vibration. The second is to increase the stiffness of the
first resonance mode of the beam by exciting the beam at the higher order modes with a
small excitation amplitude.
2.2 EXPERIMENTAL SETUP Figure 2.1 shows the experimental setup of the flexible cantilever beam for multi-
frequency excitation. In this experiment, the shaker was driven by a multi-frequency
excitation input via the power amplifier. The multi-frequency excitation input was
obtained by using a summing amplifier in order to combine two or more sinusoidal
frequency components, each of which was generated using a function generator.
In order to distinguish between the magnitudes of different modes in the time domain,
the measured signals were passed through three different bandpass filters as shown in
Figure 2.1. The outputs of the bandpass filters at 4 Hz, 24 H z and 70 H z were the
magnitude of the first, second and third order modes, respectively.
39
Chapter 2. Nonlinear behaviour of a flexible beam: Multi-Frequency Excitation
Function Generator
Measured signal
Function Generator
Function Generator
Summing Amplifier
Power Amplifier
Shaker
1 .».•.•••. ••.*.•.'••.•
Beam Accelerometer
m ' ' • ' • ' ' • ' ' . ' • ' . ' • ' . ' . ' ' • ' • ' ' • ' . ' • ' • ' • ' • ' • ' • ' • ' • ' . '
Conditioning Amplifier
Conditioning Amplifier
Bandpass Filter (at 4 Hz)
Bandpass Filter (at 24 Hz)
-^ Magnitude of 1st order mode
Bandpass Filter (at 70 Hz)
-• Magnitude of 2 order mode
II
-^ Magnitude of 3rd order mode
Hewlett-Packard Analyser
Figure 2.1 The experimental setup for the flexible beam for multi-frequency excitation.
2.3 LINEAR RESPONSE TO MULTI-FREQUENCY EXCITATION
2.3.1 Non-periodic Input Signal
In this experiment, the beam was excited with a non-periodic signal which was
comprised of two or more frequency components close to or at the resonance
frequencies. For instance, signal combinations of 4 H z and 24.5 Hz, 4 H z and 70.3 Hz,
24.5 H z and 70.3 Hz, and 4 Hz, 24.5 H z and 70.3 H z were used to excite the first and
second order modes, first and third order modes, second and third order modes, and
40
Chapter 2. Nonlinear behaviour of a flexible beam: Multi-Frequency Excitation
first, second and third order modes, respectively. Because the ratios between the
combined frequency components were not integers, the summation of those frequency
components was non-periodic.
In order to ensure that the beam response was linear, the excitation amplitude (input
voltage to the shaker) was well below the nonlinear threshold level (at which the beam
response became nonlinear, ie. frequency shift or modal coupling occurred). Table 2.1
shows the threshold value of the excitation amplitude at different excitation frequencies.
Excitation frequency
4 Hz
24 Hz
24.5 Hz
70 Hz
70.3 Hz
Nonlinear threshold level
0.45 V
0.35 V
0.4 V
0.35 V
0.4 V
Table 2.1 The threshold of nonlinear excitation amplitude for different excitation
frequencies.
In the experiment of multi-modal excitation, it was observed that the magnitude of a
lower order mode decreased with increasing excitation of a higher order mode, whereas
the magnitude of a higher order mode increased with increasing excitation of a lower
order mode. For instance, Figures 2.2 and 2.3 show the magnitude of the first order
mode measured at the tip of the beam as a function of the excitation amplitude of 24.5
H z and 70.3 Hz, respectively, when the beam was excited with combinations of 4 H z
and 24.5 H z and at 4 H z and 70.3 Hz. In this experiment, the excitation amplitudes at
24.5 H z and 70.3 H z were increased from 0.05 V to 0.15 V while the excitation
amplitude of 4 H z remained fixed at 0.05 V. As can be seen from the figures, the
41
Chapter 2. Nonlinear behaviour of a flexible beam: Multi-Frequency Excitation
magnitude of the first order mode decreased with increasing excitation amplitude of the
second and third order modes.
In contrast to Figures 2.2 and 2.3, Figures 2.5 and 2.6 show the magnitude of the second
and third order modes, respectively, as a function of the excitation amplitude at 4 Hz.
The results were plotted when the excitation amplitude at 4 H z was increased from
0.05V to 0.15 V, while the excitation amplitudes at 24.5 H z and 70.3 H z were fixed at
0.05 V. A s can be seen from the figures, the magnitude of the second and third order
modes increased with increasing excitation amplitude of the first order mode.
Similar results were observed in the case of excitation at combined frequencies of 24.5
H z and 70.3 Hz. A s can be seen in Figure 2.4, the magnitude of the second order mode
decreased with increasing excitation amplitude at 70.3 Hz, whereas the magnitude of the
third order mode increased with increasing excitation amplitude at 24.5 H z (see Figure
2.7).
0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 Excitation amplitude of 24.5 Hz [Volts]
Figure 2.2 Magnitude of the first order mode as a function of excitation amplitude at
24.5 Hz, while the excitation amplitude at 4 H z was constant at 0.05 V.
42
Chapter 2. Nonlinear behaviour of a flexible beam: Multi-Frequency Excitation
0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 Excitation amplitude of 70.3 Hz [Volts]
Figure 2.3 Magnitude of the first order mode as a function of excitation amplitude at
70.3 Hz, while the excitation amplitude at 4 Hz was constant at 0.05 V.
0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 Excitation amplitude of 70.3 Hz [Volts]
Figure 2.4 Magnitude of the second order mode as a function of excitation amplitude at
70.3 Hz, while the excitation amplitude at 24.5 Hz was constant at 0.1 V.
43
Chapter 2. Nonlinear behaviour of a flexible beam: Multi-Frequency Excitation
0.25
•»• 0.245 +-»
o £. 0.24 <D T3 | 0.235 i_
<u •2 0.23 o TJ § 0.225 o <D o 0.22 .c +-•
'S 0.215 a> •o
i 0.21c c O) | 0.205
0.2
i i i i i i i i i
_
-
_ _
^0*JQ>*~*'^
——-0""""
- - y **" -
**-*" _
• • i i • • • i i
0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 Excitation amplitude of 4 Hz [Volts]
Figure 2.5 Magnitude of the second order mode as a function of excitation amplitude at
4 Hz.
.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 Excitation amplitude of 4 Hz [Volts]
Figure 2.6 Magnitude of the third order m o d e as a function of excitation amplitude at
4Hz.
44
Chapter 2. Nonlinear behaviour of a flexible beam: Multi-Frequency Excitation
</> •*—«
2 Cl)
n E 0 " o "2 xr *-< CD
>•-O a) T3 3
c ra CO
2
0.21
0.208
0.206
0.204
0?0?
0.2
0.198
0.196
0.194
0.192
0.19(«— i i i i i i i i i 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14
Excitation amplitude of 24.5 Hz [Volts]
Figure 2.7 Magnitude of the third order mode as a function of excitation amplitude at
24.5 Hz.
These experimental observations have indicated an energy transfer from lower order
modes to higher order modes, which is in contrast to the nonlinear case where the
energy transfer was from higher order modes to lower order modes.
In order to ensure that this energy transfer occurred internally in the beam structure (as
measured at the tip) rather than in the excitation mechanism (as measured at the
clamped end), the beam was initially excited individually at 4 H z (0.05 V ) and 24.5 H z
(0.1 V ) . Displacement was measured at both the clamped end and the tip of the beam.
Figures 2.8a, 2.9a, 2.10a and 2.11a show the displacement measured in the time and
frequency domain at the clamped end for single frequency excitation at 4 H z and 24.5
Hz, respectively.
45
Chapter 2. Nonlinear behaviour of a flexible beam: Multi-Frequency Excitation
The beam was then simultaneously excited at the second order mode in parallel with
excitation of the first order mode. As can be seen from Figures 2.8b and 2.10b, the
magnitude of the first and second order modes measured at the clamped end remained
the same as that for single frequency excitation (as shown in Figures 2.8a and 2.10a).
The same results were also obtained from the power spectra measured at the clamped
end for multi-frequency excitation at 4 H z and 24.5 Hz, as shown in Figures 2.9b and
2.11b.
However, the magnitude of the first order mode measured at the beam tip, due to the
multi-frequency excitation, reduced significantly compared to the case of single-modal
excitation of first order mode (Figure 2.12). In contrast, the magnitude of the second
order mode increased very slightly for the case of multi-modal excitation (see Figure
2.13). It appears that in this linear excitation case, the energy from the first order mode
transfers to the second order mode through internal damping. This is in contrast to the
results described in section 1.3.3, where the energy was observed to transfer from higher
order modes to lower order modes under nonlinear excitation.
0.02
Time [seconds]
Time [seconds]
Figure 2.8 The magnitude of the first order mode measured at the clamped end for an
excitation at: (a) 4 H z (0.05V) only, (b) 4 H z (0.05V) and 24.5 H z (0.1V).
46
Chapter 2. Nonlinear behaviour of a flexible beam: Multi-Frequency Excitation
oo 0
E 2
I -50
§ °- -100
(
-(a)
r
4 Hz '
' "vyy% ) 10
~i
1ty4h{Jc 20
1
\X.«JLJA /I
^SA/v-30
i
A «IW^JM>
40
-
J[u**MK t 50
m 0 2, E 2 §_ -50 CO
i °- -100
10
Frequency [Hz]
-(b)
N 4 Hz '
_J ._ i
rtfrt
24.5 Hz
t '"
V'SvVrW •*f*wt 20 30 Frequency [Hz]
40 50
Figure 2.9 The power spectra measured at the clamped end when the beam was excited
at: (a) 4 Hz (0.05 V) only, (b) 4 Hz (0.05 V) and 24.5 Hz (0.1 V).
£ 0.02
0 0.05 0.1 0.15 0.2 0.25 Time [seconds]
0.3 0.35 0.4
£ 0.02
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Time [seconds]
Figure 2.10 The magnitude of the second order mode measured at the clamped end for
excitations at: (a) 24.5Hz (0.1V) only,( b) 4Hz (0.05V) and 24.5Hz (0.1V).
47
Chapter 2. Nonlinear behaviour of a flexible beam: Multi-Frequency Excitation
20 30 Frequency [Hz]
20 30 Frequency [Hz]
50
50
Figure 2.11 The power spectra measured at the clamped end when the beam was excited
at: (a) 24.5 Hz (0.1 V) only, (b) 4 Hz (0.05 V) and 24.5 Hz (0.1 V).
0.2
[volts]
o
<
| 0 CO o
i-o-1 (0
b-n? (
(a) A
)
A' r\
V , V 0.5
mA f
1
\ I I J I / "
1.5 2
0.2 W
9 0.1
£ 0 a> o f-0.1 b-0.2
Time [seconds]
*A /I 0
AA i
0.5
AA
/v\ 1
1
AAA-N\i\
i
1.5 2 Time [seconds]
Figure 2.12 The magnitude of the first order mode measured at the tip when the beam
was excited at: (a) 4 Hz (0.05V) only, (b) 4 Hz (0.05V) and 24.5 Hz (0.1V).
48
Chapter 2. Nonlinear behaviour of a flexible beam: Multi-Frequency Excitation
0 0.05 0.1 0.15 0.2 0.25 Time [seconds]
0.4
0.05 0.1 0.15 0.2 0.25 Time [seconds]
0.4
Figure 2.13 The magnitude of the second order mode measured at the tip when the beam
was excited at: (a) 24.5 H z (0.1V) only, (b) 4 H z (0.05V) and 24.5 H z
(0.1V).
Similar results were observed when the beam was excited simultaneously at the first and
third order modes, and at the second and third order modes. The magnitude of the first
and second order modes were reduced significantly as soon as the third order mode was
excited (Figures 2.14 and 2.15, respectively).
The filtered signals shown in Figures 2.8b, 2.12b and 2.15b were distorted due to
leakage at the stop-band of the bandpass filters.
49
Chapter 2. Nonlinear behaviour of a flexible beam: Multi-Frequency Excitation
i, 0.1 *-»
| 0 co o £ -0.1 CO b-0.2
,(a) A
" \ /
0 V , v
0.5
7? ""'/
T / \ j
1
L W ' f\ f\
V i vf V
1.5 2 Time [seconds]
Time [seconds]
Figure 2.14 The magnitude of the first order mode measured at the tip when the beam
was excited at: (a) 4 Hz (0.05V) only, (b) 4 Hz (0.05V) and 70.3 Hz (0.1V).
Q -0.2.
0.05 0.1 0.15 0.2 0.25 Time [seconds]
0.05 0.15 0.2 0.25 Time [seconds]
0.4
Figure 2.15 Magnitude of the second order mode measured at the tip for an excitation at:
(a) 24.5 Hz (0.1V) only, (b) 24.5 Hz (0.1V) and 70.3 Hz (0.1V).
50
Chapter 2. Nonlinear behaviour of a flexible beam: Multi-Frequency Excitation
For comparison, the experimental results described above are presented in Table 2.2.
From the table, it can be seen that the displacements measured at the tip for a single
sinusoidal signal of 4 Hz, 24.5 H z and 70.3 H z were 0.198 V, 0.205 V and 0.18 V,
respectively.
When the beam was excited by a nonperiodic signal combination of 4 Hz and 24.5 Hz,
the magnitude of the first order mode dropped from 0.198 V to 0.165 V. This decrease
in the magnitude of the first order mode caused a slight increase in magnitude of the
second order mode from 0.205 V to 0.21 V. Given that both the modes were excited
with equal kinetic energy, the higher frequency, second order mode, had a smaller
displacement than the first order mode.
A similar decrease in the magnitude of the first order mode (from 0.198 V to 0.15 V)
was also observed when the beam was excited at the first and third order modes.
Because the frequency of the third order mode of the beam was much higher than the
frequency of the first order mode, the amount of increase in the magnitude of the third
order mode (0.009 V ) was insignificant compared to the amount of decrease in the
magnitude of the first order mode (0.048 V).
When the beam was excited at the second and third order modes, the reduction in
magnitude of the second order mode caused a greater increase in magnitude of the third
order mode, compared to the proportional decrease in the magnitude of the first order
mode (ie. in the case of excitation of the first and third order modes).
In summary, the experimental results have shown that the energy transfers from lower
order modes to higher order modes in the linear case. This seems to be contradictory to
the traditional understanding of the linear response of the single degree of freedom
system. At this stage, the mechanism involved in this energy transfer phenomenon is
still unclear. However, the experimental evidence with multi-frequency excitation will
allow open discussion and encourage other comments and opinions.
51
Chapter 2. Nonlinear behaviour of a flexible beam: Multi-Frequency Excitation
Single
frequency
excitation
Multi-
frequency
excitation
Excitation Frequency
&
Amplitude
4 Hz (0.05V)
24.5 Hz (0. IV)
70.3 Hz (0. IV)
4 Hz (0.05V) + 24.5 Hz (0.1 V)
4 Hz (0.05V) + 70.3 Hz (0.1V)
24.5 Hz (0.1 V) + 70.3 Hz (0.1 V)
4 Hz (0.05V) + 24.5 Hz (0.1 V)
+ 70.3 Hz (0.1 V)
Magnitude of
1st order
mode
0.198 V
0.165 V
0.15 V
0.11 V
Magnitude
of 2nd order
mode
0.205 V
0.21V
0.165 V
0.21V
Magnitude
of 3r" order
mode
0.18V
0.189 V
0.199 V
0.2 V
Table 2.2 The displacement measured at the tip for different single- and multi-modal
excitations.
2.3.2 Periodic Input Signal
Similar experiments to those described in section 2.2.1 were repeated with periodic
excitation signals used for multi-frequency excitation of the beam. In other words, the
signal had to be a summation of two or more frequency components whose ratio was an
integer. For instance, the excitation signals used in this experiment were summations of
4 Hz and 24 Hz, 4 Hz and 72 Hz, and 24 Hz and 72 Hz.
Figures 2.16, 2.17 and 2.18 show the periodic excitation signals used in these
experiments.
As in the case of a nonperiodic signal, the magnitude of the first order mode started to
decrease as soon as the second or the third order mode was excited, although the
excitation amplitude of the first order mode still remained the same. This was due to
energy transfer from the lower order modes to higher order modes when the beam was
subject to small excitation. In other words, for linear excitation the stiffness of the first
order modes was increased if higher order modes were excited in parallel.
52
Chapter 2. Nonlinear behaviour of a flexible beam: Multi-Frequency Excitation
0.2 0.3 Time [seconds]
Figure 2.16 A combined excitation signal of 4 H z and 24 Hz.
CO
c CO
E a> o ro Q. co
b
0.25
0.2
0.1511
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25 r
llll
H 0.05 0.1 0.15 0.2 0.25
Time [seconds]
I
0.3 0.35 0.4
Figure 2.17 A combined excitation signal of 4 H z and 72 Hz.
53
Chapter 2. Nonlinear behaviour of a flexible beam: Multi-Frequency Excitation
0.1 Time [seconds]
Figure 2.18 A combined excitation signal of 24 H z and 72 Hz.
Because the responses were very similar to those shown in Section 2.3.1 (non-periodic
signals), they have not been presented here.
2.4 NONLINEAR RESPONSE TO MULTI-FREQUENCY EXCITATION
2.4.1 Non-periodic Signal
In Chapter 1, it was observed that excited higher order modes (such as the second and
third order modes), coupled to the first order mode when the excited mode became
saturated, but not vice versa. In this experiment, higher order modes were initially
excited with a sufficiently large amplitude to allow the beam to couple to the first order
mode. The beam was then excited at a small amplitude at the first order mode (4 H z at
0.07 V ) together with nonlinear excitation of the higher order mode.
54
Chapter 2. Nonlinear behaviour of a flexible beam: Multi-Frequency Excitation
Figure 2.19a shows the tip magnitude of the first order mode, due to coupling from the
second order mode, for a nonlinear single sinusoidal excitation of the second order
mode (24 Hz at 0.35 V). As soon as the beam was excited with a multi-frequency of 4
Hz and 24 Hz, a beating effect between the vibration induced in the beam due to the
nonlinear coupling and the vibration due to the excitation of 4 Hz, was observed (see
Figure 2.19b). The induced nonlinear vibration varied between 3.76 Hz to 3.82 Hz,
resulting in a beat frequency of approximately 0.2 Hz.
The induced vibration due to the nonlinear modal coupling could be decreased or
increased by exciting the beam with a multi-frequency of 3.8 Hz (0.1 V) and 24 Hz
(0.35 V ) as well as changing the phase of the 3.8 Hz excitation. A decrease in the
magnitude of the first order mode was observed when the phase of 3.8 Hz was between
195 and 220 degrees (see Figure 2.20), whereas an increase in the magnitude occurred
with the other phase values of 3.8 Hz (see Figure 2.21). Because the frequency and
magnitude of the induced nonlinear component varied rapidly with time, it was difficult
to achieve a complete reduction of the induced vibration without feedback.
co *-»
o > CO E CO
o ro a. co
b
1
0.5
0
-0.5
-1 IL
\
3 4 5 6 Time [seconds]
3 4 5 Time [seconds]
Figure 2.19 Magnitude of the first order mode with an excitation at: (a) 24 Hz (0.35 V)
only, (b) 4 Hz (0.07 V) and 24 Hz (0.35 V).
55
Chapter 2. Nonlinear behaviour of a flexible beam: Multi-Frequency Excitation
co • * - .
o > c CO
E CO
o ro a. co b
1
0.5
0
-0.5
-1
CO
o > c CO
E CO
o ra Q. co b
1
0.5
0
-0.5
-1
L LL 2 3 4 5 6 Time [seconds]
Time [seconds]
u 0 I
11 _V v V V V V u
0 1
I
i W 2
• i
1 J V u U
3
i V V V u
4
i
i i/vyy 5
1 V V V \i 1
6
..J.
Mi v v V v v. 7 8
Figure 2.20 Magnitude of the first order mode with an excitation at: (a) 24 H z (0.35V)
only, (b) 3.8 Hz (0.1 V) with 196 degrees phase shift and 24 Hz (0.35 V).
£ 0.5.
co E CO
o 2 -0.5 Q. CO b -1
0 -n i l l
i in
u 11 3 4 5 Time [seconds]
6
2 3 4 5 Time [seconds]
Figure 2.21 Magnitude of the first order mode with an excitation at: (a) 24 Hz (0.35V)
only, (b) 3.8 Hz (0.1 V) with 86 degrees phase shift and 24Hz (0.35 V).
56
Chapter 2. Nonlinear behaviour of a flexible beam: Multi-Frequency Excitation
A similar beating effect was observed when the beam was excited with a multi-
frequency of 3.8 H z (0.1 V) and 69 Hz (0.35 V) ( as shown in Figure 2.22). Because the
induced vibration due to the coupling with 69 Hz had a frequency range between 3.2 Hz
and 3.3 H z (see Figure 1.21), a combined excitation frequency of 3.3 Hz (0.1V) and 69
Hz (0.35 V ) was selected in order to obtain the attenuation of the magnitude of the first
order mode due to the nonlinear modal coupling. In contrast to the case of combined
excitation of the first and second order modes, the attenuation of the first order mode
was observed only when the excitation frequency of 3.3 Hz had a phase shift between
80 and 100 degrees (see Figures 2.23 and 2.24). This indicates that the vibration induced
in the beam due to coupling with the second order mode has different frequency and
phase than the one due to coupling with the third order mode.
lllllllllllllll
3 4 5 Time [seconds]
3 4 5 Time [seconds]
Figure 2.22 Magnitude of the first order mode with an excitation at: (a) 69 Hz (0.35 V)
only, (b) 3.8 Hz (0.1 V ) and 69 Hz (0.35V).
57
Chapter 2. Nonlinear behaviour of a flexible beam: Multi-Frequency Excitation
3 4 5 Time [seconds]
Figure 2.23 Magnitude of the first order mode with an excitation at: (a) 69 H z (0.35 V)
only, (b) 3.3 Hz (0.1 V) with 86 degrees phase shift and 69 Hz (0.35 V).
3 4 5 Time [seconds]
2 3 4 5 Time [seconds]
Figure 2.24 Magnitude of the first order mode with an excitation at: (a) 69 Hz (0.35V)
only, (b) 3.3 Hz (0.1 V) with 196 degrees phase shift and 69 Hz (0.35 V).
58
Chapter 2. Nonlinear behaviour of a flexible beam: Multi-Frequency Excitation
2.4.2 Periodic Input Signal
In the case of nonlinear periodic multi-frequency excitation, a signal combination of 4
H z and 24 H z was used for excitation of the first and second order modes. Similarly, a
signal combination of 4 H z and 68 H z was used to excite the first and third order modes.
The results were similar to the results obtained from the case where nonperiodic signals
were used for excitation. In other words, the response of the beam became nonperiodic
as a result of nonlinear coupling, although the excitation signal was periodic.
2.5 CONCLUSIONS
Both nonperiodic and periodic signals were used for linear and nonlinear multi-modal
excitation. The experimental results have shown that the type of signal (whether the
signal is periodic or nonperiodic) has no effect on the response of the beam. However,
in the case of linear multi-modal excitation, it was observed that the magnitude of first
order mode decreased, ie. the stiffness of the first order mode increased, as soon as
higher order modes became excited. This was due to energy transfer from lower order
modes to higher order modes for linear excitation. This is in contrast to the case of
nonlinear excitation where the energy transferred from higher order modes to lower
order modes. W h e n the excited higher order mode became saturated, the beam started to
couple to the first order mode. The displacement of the first order mode resulting from
coupling of the second order mode exhibited different frequency and phase than the
displacement resulting from coupling with the third order mode. This displacement was
reduced or amplified in the case of multi-frequency excitation depending on the phase
of the excitation frequency of the first order mode. In order to obtain the maximum
cancellation of the induced vibration due to the nonlinear coupling, the induced
nonlinear vibration signal needs to be fed back, since the frequency of this induced
vibration varies randomly with time (as shown in Section 1.3.3).
In conclusion, the observations of the behaviour of the cantilever beam with multi-
frequency excitation, have led to two useful concepts for the development of an active
control algorithm for nonlinear vibration cancellation in the flexible beam. They are:
59
Chapter 2. Nonlinear behaviour of a flexible beam: Multi-Frequency Excitation
1. Using the low frequency vibration, resulting from nonlinear coupling from
higher order modes, to cancel the low frequency vibration.
2. Increasing the stiffness of the first order mode by exciting the beam at higher
order modes with small excitation amplitude. As a consequence, the beam
becomes less flexible and the amplitude decreases.
60
Chapter 3. Damping characteristics of the flexible cantilever beam
Chapter 3
DAMPING CHARACTERISTICS OF THE FLEXIBLE
CANTILEVER BEAM
3.1 INTRODUCTION
The interest in and knowledge of damping have increased rapidly in recent years, since
damping exists in all vibrating systems. The effect of damping is to remove energy from
the system. The loss of energy from an oscillatory system results in a decay of the
amplitude of free vibration.
In the case where the damping of the system is proportional to the velocity of vibration,
the damping force can be modelled as Equivalent Viscous Damping. If the damping of
the system is proportional to the square of the velocity, the damping force can be
modelled as Quadratic Damping. In some systems, the damping force opposing the
motion has a constant magnitude, this damping force is then referred as the Coulomb
Friction Force. In a solid, some of the energy loss is attributed to the imperfect elasticity
or internal friction of the material. The damping force may be considered to be
proportional to the amplitude and independent of the frequency. This kind of damping
can be referred as Structural Damping.
While Viscous damping is a linear damping, Quadratic, Coulomb and Structural
damping are classified as nonlinear damping. Although Coulomb damping is a nonlinear
damping, the amplitude of a system with Coulomb damping decreases linearly with
time; whereas a system with Viscous damping has an exponential decay. However, the
decay rate for systems with Coulomb or Viscous damping are not affected by the
magnitude of the vibration. In contrast, in systems with Quadratic damping, the
61
Chapter 3. Damping characteristics of the flexible cantilever beam
amplitude decays algebraically rather than exponentially with time; and the decay rate is
proportional to the magnitude of the vibration [45].
Damping of a real system is a complex phenomenon involving several kinds of damping
mechanisms. Predicting the magnitude of the damping force could, therefore, be
difficult. In order to be able to obtain a good estimate of damping force, it is usually
necessary to rely on experiments.
The aim of this chapter is to examine the damping force of the cantilever beam for a
variety of excitation conditions. Because damping determines the dynamic behaviour of
the beam, it is necessary to investigate experimentally the damping of the beam, in
particular in the presence of nonlinear modal interactions.
As described in previous chapters, the cantilever beam can be treated as a multi-degree-
of-freedom system, where each mode of the beam represents a degree-of-freedom. In
this work, the decay rate of the first three resonance modes of the beam was examined,
when the beam was subject to single frequency excitation as well as multi-frequency
excitation, for both linear and nonlinear cases. As will be shown in this chapter, the
beam exhibits a combination of Viscous and Quadratic damping even in the linear case.
This investigation is a fundamental part of the development of a nonlinear model of the
cantilever beam; although many interesting nonlinear phenomena, such as change of
resonance frequency, modal coupling, nonlinear stiffness and energy transfer from
higher order modes to lower order modes, have already been examined.
3.2 SINGLE FREQUENCY EXCITATION
3.2.1 Linear case
In this experiment, the beam was initially excited with a single sinusoidal signal whose
frequency was close to or at the resonance frequency of the beam. A step change1
(reduction) in excitation amplitude was then applied to the shaker. The displacement
1 Implemented by quickly turning the power amplifier gain to minimum.
62
Chapter 3. Damping characteristics of the flexible cantilever beam
measured at the tip of the beam was recorded using a D S P in order to observe the
decayed amplitude of the measured displacement over a period of time. Since the mass
of the accelerometer, the asscociated cable, the conditioning amplifier and the shaker
could have some influence on the decay slope of the measured displacement, they were,
therefore, included as part of the system. However, when the beam was excited at one of
its resonance frequencies, the response of the beam was very lightly damped compared
to the responses of the accelerometer and the shaker. The displacement measured at the
clamped end (the displacement of the shaker) decayed almost immediately as the power
amplifier was turned down to minimum. Therefore, the decay slope of the displacement
measured at tip was mainly influenced by the damping characteristics of the beam.
In order to examine how the excitation frequency and amplitude influence the
displacement decay slope of the beam, a variety of excitation frequencies and
amplitudes were used.
Firstly, the beam was excited at 4 Hz at different excitation amplitudes. Figure 3.1
shows the decay time histories of the displacement measured at the tip of the beam when
the beam was excited at 0.1 V and 0.3 V. For comparison, the magnitude for the
excitation amplitude of 0.1 V was multiplied by 2.4 and plotted versus the magnitude
for the excitation amplitude of 0.3 V, as shown in Figure 3.1. As can be seen from the
figure, the decay slope was steeper in the case of an excitation amplitude of 0.3 V
compared to an excitation amplitude of 0.1 V. In effect, the rate of decay in magnitude
of the measured displacement increased with increasing excitation amplitude. The
increase in excitation amplitude resulted in increased magnitude of displacement
measured at the tip. In other words, the decay rate was proportional to the magnitude of
the measured displacement. This indicates a Quadratic damping characteristic. The
equation of motion for a system with Quadratic damping is:
x + e|x|x + co02x = 0 (3.1)
The solution of Eq.(3.1) is:
63
Chapter 3. Damping characteristics of the flexible cantilever beam
X(t) = torn x cos((°ot + <P) (3.2)
3rc
where x 0 is the initial condition, (p is the phase angle and £ is the Quadratic damping
factor.
It can be seen from Eq.(3.2) that the larger the initial magnitude is, the faster it decays.
In order to illustrate the Quadratic damping characteristic, Figure 3.2 shows the decay
slopes of Quadratic damping with e = 0.036, for the initial magnitudes of 0.52 and 0.7.
1 r
0.8 -
Time [seconds]
Figure 3.1 Decayed magnitude of the first mode for excitations of 0.1 V and 0.3V at
4Hz(magnitude of excitation at 0.1 V scaled by 2.4 for overlay comparison).
64
Chapter 3. Damping characteristics of the flexible cantilever beam
ni 1 1 1 1 1 1 1 1
0 2 4 6 8 10 12 14 16 Time [s]
Figure 3.2 Decay slopes of Quadratic damping for different initial magnitudes
(e=0.036).
Figure 3.3 shows the decayed magnitude of the first order mode of the beam with an
excitation of 4 H z at 0.2V, against two different decay slopes of Quadratic damping
(where £=0.036 and £=0.066). As can be seen from the figure, the decay slope with
£=0.036 only matched the first part of the decayed magnitude of the first order mode. In
contrast, the decay slope with a larger damping factor (£=0.066) matched the end part of
the decayed magnitude, but it had a steeper slope at the beginning. This indicates that
the first order mode does not only exhibit Quadratic damping, but a combination of
Quadratic and Viscous damping.
The equation of motion for a system with Viscous and Quadratic damping is:
x + 2<;G)0x + £|x|x + co02x = 0 (3.3)
65
Chapter 3. Damping characteristics of the flexible cantilever beam
According to [45], the solution of Eq.(3.3) is:
-SOW
x(t) = x0e I | 4 £ C Q o x o /i 0-<;co0t
•cos(co0t + 9 ) , (3.4)
371 . ( 1 _ e - W )
where £ is the Viscous damping factor.
0.6
-0.6
£=0.036
£=0.066
6 8 10 12 14 Time [s]
16
Figure 3.3 Decayed magnitude of the first mode for excitation of 4 H z at 0.2 V plotted
against the decay slope of Quadratic damping.
Figure 3.4 shows the decayed magnitude of the first order mode against the decay slope
of a combination of Viscous and Quadratic damping where <; = 0.0048 and £ = 0.066.
The values of <; and £ were estimated by trial and error. This was not a unique
combination of c, and £; there were many other possible combinations of c; and £ which
also gave a similar decay slope.
Similarly, Figure 3.5 shows the decayed magnitude of the first order mode of the beam
with an excitation of 3.7 H z at 0.2 V against the decay slope of a combination of
Viscous and Quadratic damping where <; = 0.0048 and £ = 0.07.
66
Chapter 3. Damping characteristics of the flexible cantilever beam
6 8 10 Time [s]
Figure 3.4 Decayed magnitude of the first mode for excitation of 4 H z at 0.2 V plotted
against the decay slope of combined Viscous and Quadratic damping.
4 6 Time [s]
Figure 3.5 Decayed magnitude of the first mode for excitation of 3.7 H z at 0.2 V plotted
against the decay slope of combined Viscous and Quadratic damping.
67
Chapter 3. Damping characteristics of the flexible cantilever beam
Like the first order mode, the decay rate of the second order mode is proportional to the
magnitude of the displacement, as shown in Figure 3.6. Hence, the second order mode
has also a combination of Viscous and Quadratic damping.
Figure 3.7 plots the decayed magnitude of the second order mode with excitation of 24.5
H z at 0.2 V, against the decay slope of a combination of Viscous and Quadratic
damping where q = 0.005 and £ = 0.002.
0 0.2 0.4 0.6 0.8 1 Time [seconds]
Figure 3.6 Decayed magnitude of the second order mode for excitation of 0.1V at 24.5
H z versus an excitation of 0.2V at 24.5 H z (magnitude for excitation of
0.1V multiplied by 2 for overlay comparison).
68
Chapter 3. Damping characteristics of the flexible cantilever beam
(0
9.
c (1) E a> 3 Q. OT O
0.5
0.4
0.3
0.2
0.1
U
-0.1
-0.2
-0.3
-0.4
-0.5
0 0.5 1 1.5 2
Time [seconds]
Figure 3.7 Decayed magnitude of the second order mode with excitation of 24.5 H z at
0.2 V against the decay slope of combined Viscous and Quadratic damping.
As in the case of the first and second order modes, the third order mode also exhibited a
combination of Viscous and Quadratic damping (see Figure 3.9) where c, = 0.0056 and E
= 2.67X10"4. The decay rate is also proportional to the magnitude of the vibration as
shown in Figure 3.8, where the decayed magnitude of the third order mode for excitation
of 70.3 H z at 0.1 V is plotted versus an excitation of 70.3 H z at 0.2V.
For a better overview of the decay rate of different modes of the beam, Table 3.1 shows
the Viscous and Quadratic damping factors for different excitation frequencies.
Excitation Frequency
4 Hz
24.5 Hz
70.3 Hz
C 4xl03
5xl0"3
5.68xl0"3
£
66xl0~3
20x10"3
26.7X104
69
Chapter 3. Damping characteristics of the flexible cantilever beam
0.25 -ii: — 0.1V ---0.2V
-02 llljlili! -0.25 I lill'1 .Hi1
If 0 0.2 0.4 0.6
Time [seconds] 0.8 1
Figure 3.8 Decayed magnitude of the third order m o d e for excitation of 0.1V versus
0.2V at 70.3 Hz.
0 0.2 0.4 0.6
Time [seconds]
0.25
0 2 r
0.15 [I
2 0.1 1
r 0.05 I
1 o 1 S -0.05 I a. Q -0.1 |
-0.15 I
-0.2
-0.25
i- 1 1 i
£=2.67x10^
£=0.00568
_
^1+4J
\\\ 1 1 1 n M1 luVr^-^ II nII ii i nII nII & i K A iii II 1 1 1 1 • i •**-*.
II' II II
• • • •
0.8
Figure 3.9 Decayed magnitude of the third order m o d e for excitation of 70.3 H z at 0.2V
plotted against the decay slope of combined Viscous and Quadratic damping.
70
Chapter 3. Damping characteristics of the flexible cantilever beam
In summary, the first three modes of the beam exhibited a combination of Viscous and
Quadratic damping in the linear case. The decay rate is different for different modes -
the higher order modes have faster decay rate than the lower order modes. In other
words, the decay rate is proportional to the magnitude and frequency of the vibration.
3.2.2 Nonlinear case
In this experiment, the beam was excited at the second or third order modes at such a
large amplitude that the beam started to couple with the first order mode. The
displacement at the tip of the beam was measured and filtered by the bandpass filters
with centre frequencies coinciding with the resonance frequency of each mode as shown
in Figure 2.1, to allow each mode to be identified separately from the measured signal.
Since the bandpass filters have a damping coefficient of unity (1.0) (which is larger than
the damping coefficients of the beam), the decay slopes of the filtered signals were,
therefore, mainly influenced by the damping characteristics of the beam. The decay time
history of the first order mode and the excited mode were captured simultaneously using
the D S P immediately after removal of excitation.
Figures 3.10 and 3.11 show the decay in magnitudes of the first and second order
modes, respectively, for an excitation of 24 H z at 0.35 V. As can be seen from the
figures, the magnitude of the second order mode started to decay in a similar way as in
the linear case (as soon as the power amplifier was turned down at t = 0.4 seconds),
while the magnitude of the first order mode continued varying due to the nonlinear
coupling between the modes. The decay rate of the magnitude of the second order mode
was not affected by the presence of vibration of the first order mode. W h e n the
magnitude of the second order mode reduced to approximately 0.21 V (at t = 1.3
seconds), the magnitude of the first order mode then started to decay at a similar rate to
the case of linear single-modal excitation shown in Figure 3.5. This phenomenon
corresponds to the hysteretic stiffness characteristic described in the section 1.3.6. The
displacement of the second order mode was observed to increase proportionally to the
increment in excitation amplitude, until the excitation amplitude reached the coupling
threshold value. The beam then started to couple with the first order mode. W h e n the
71
Chapter 3. Damping characteristics of the flexible cantilever beam
excitation amplitude was gradually decreased, the coupling did not disappear until the
excitation amplitude reached the decoupling threshold value (which was lower than the
coupling threshold value). Once the decoupling started, the beam response became
linear again. For a better overview, Figures 3.12a and 3.12b show the measured
displacement in the frequency domain captured at the times when the beam started to
couple and decouple with the first order mode, respectively.
A similar result was obtained when the beam was excited at 69 Hz at 0.35 V. Both the
first and third order modes decayed in the same way as in the linear case, as soon as the
magnitude of the third order mode decayed to the decoupling threshold value at t = 1.3
seconds (see Figures 3.13 and 3.14).
In summary, in addition to Viscous and Quadratic damping the beam also exhibited
hysteretic damping characteristics in the nonlinear case.
1 n 1 1 1 1
OT *^ o > 4_.
C CD
E CD CJ to Q_
OT Q
0.6
04
0.2
0
-0?
-0.4
-0.6
-0.8
-1
Time [seconds]
Figure 3.10 Decayed magnitude of the first order mode for excitation of 24 H z at 0.35V.
72
Chapter 3. Damping characteristics of the flexible cantilever beam
1.5 2 Time [seconds]
Figure 3.11 Decay in magnitude of the second order mode for excitation of 24 Hz at
0.35 V (with coupling to the first order mode).
~ 1 OT +-. O > CD ^ 0.5 4-»
c D> (0 2
n
-(a) Coupling"with the 1st order
-
^ . _ _ . i . . . i
node
-
i i .—..
10 20 30 Frequency [Hz]
10 20 30 Frequency [Hz]
40 50
Figure 3.12 The measured displacement captured at the time the beam started to:
(a) couple and (b) decouple with the first order mode.
73
Chapter 3. Damping characteristics of the flexible cantilever beam
0 4 6 Time [seconds]
8
Figure 3.13 Decayed magnitude of the first order mode for an excitation of 69 H z at
0.35 V.
Time [seconds]
Figure 3.14 Decay in magnitude of the third order mode for an excitation of 69 H z at
0.35 V (with coupling to first order mode).
74
Chapter 3. Damping characteristics of the flexible cantilever beam
3.3 MULTI-FREQUENCY EXCITATION
3.3.1 Linear case
In this experiment, frequency component combinations of 4 H z (0.05 V ) and 24.5 H z
(0.1 V ) , 4 H z (0.05 V ) and 70.3 H z (0.1 V ) , and 24.5 H z (0.1 V ) and 70.3 H z (0.1 V )
were used in order to excite the beam at the first and second order modes, the first and
third order modes, and the second and third order modes, respectively.
Figures 3.15 and 3.16 show the decay in magnitude of the first order mode for excitation
of first and second order modes, and first and third modes, respectively. As can be seen
from the figures, the magnitudes of the first order mode decayed in a similar way as in
the linear case of single-frequency excitation. However, a slight increase in magnitude
of the first order mode was observed immediately after the magnitude of the higher
order modes had dropped. This was due to the fact that the stiffness of the first order
mode increased due the presence of the vibration of the higher order mode (as described
in Chapter 2). In other words, the magnitude of the first order mode would increase (or
the stiffness would decrease) as soon as the vibration of the second or third mode
became insignificant.
Similarly, Figures 3.17 and 3.18 show the decay in magnitude of the second order mode
for combined excitation of the first and second order modes, and the second and third
order modes, respectively. As can be seen from the figures, the decay rate of the
magnitude of the second order mode was not affected by vibration of the first or third
order modes - the magnitude decayed in the same way as in the linear case of single-
frequency excitation. However, in the case of excitation at 4 H z and 24.5 Hz, the decay
in magnitude of the second order mode was superimposed with the decay magnitude of
the first order mode due to leakage at the stop-band of the 25 H z bandpass filter. The
superimposed signal became more prominent as the magnitude of the second order
mode diminished.
75
Chapter 3. Damping characteristics of the flexible cantilever beam
Figure 3.15 Decay in magnitude of the first order mode for combined excitation at 4 Hz
(0.05 V ) and 24.5 Hz (0.1 V).
Figure 3.16 Decay in magnitude of the first order mode for combined excitation at 4 Hz
(0.05 V ) and 70.3 Hz (0.1 V).
76
Chapter 3. Damping characteristics of the flexible cantilever beam
0.25
0.4 0.6
Time [seconds] 0.8
Figure 3.17 Decay in magnitude of the second order mode for a combined excitation at
4 Hz (0.05 V) and 24.5 Hz (0.1 V).
0.25
0.2 0.4 0.6
Time [seconds]
Figure 3.18 Decay in magnitude of the second order mode for combined excitation at
24.5 Hz (0.1 V) and 70.3 Hz (0.1 V).
77
Chapter 3. Damping characteristics of the flexible cantilever beam
Similarly to the second order mode, the magnitude of the third order mode (see Figures
3.19 and 3.20) also decayed in the same way as in the linear case of single frequency
excitation previously shown in Figure 3.8. Similar to the results shown in Figure 3.17,
the results in Figure 3.20 show the decay in magnitude of the third order mode
superimposed with the decay magnitude of the second order mode due to leakage in the
stop-band of the 70 H z bandpass filter.
0.2
0.15
0.1
CO +-»
Q +m*
c CD
E CD O <o a. <n D
0.05
0
-0.05
-0.1
-0.15
0 0.5 1 1.5 2 Time [seconds]
Figure 3.19 Decay in magnitude of the third order mode for combined excitation at 4 Hz
(0.05 V ) and 70.3 Hz (0.1 V).
78
Chapter 3. Damping characteristics of the flexible cantilever beam
0.2
0.15
0.1
9 0.05
I o CD
li S--0.05 b
-0.1
-0.15
-0.2 0 0.2 0.4 0.6 0.8 1
Time [seconds]
Figure 3.20 Decay in magnitude of the third order mode for a combined excitation at
24.5 H z (0.1 V ) and 70.3 H z (0.1 V).
3.3.2 Nonlinear case
The beam was initially excited at the second order mode (24 Hz), third order mode
(69Hz), or a combination of the second and third order modes with sufficiently large
amplitude that the beam started to couple with the first order mode. The beam was then
excited at the first order mode in parallel with excitation of the second or third order
mode or a combination of the second and third modes.
Similar to the results obtained in the nonlinear case of single frequency excitation, the
magnitude of the first order mode continued to be excited when the magnitudes of the
higher order modes (second and third order modes) were still above the coupling
threshold values. As soon the magnitudes of the higher order modes decayed to the
decoupling threshold values, the magnitude of the first order mode started to decay in
the same way as in the linear case of single frequency excitation (see Figures 3.21 to
3.23).
79
Chapter 3. Damping characteristics of the flexible cantilever beam
Figure 3.21 Decay in magnitude of the first order mode for combined excitation of 4 Hz
at 0.05 V and 24 H z at 0.35 V.
Figure 3.22 Decay in magnitude of the first order mode for combined excitation of 4 Hz
at 0.05 V and 69 H z at 0.35 V.
80
Chapter 3. Damping characteristics of the flexible cantilever beam
Figure 3.23 Decayed magnitude of the first order mode for combined excitation at 24 Hz
(0.25 V ) and 69 H z (0.35 V).
Figures 3.24 and 3.25 show the decay in magnitude of the second order mode for a
combined excitation at 4 H z (0.05V) and 24 H z (0.35 V) , and a combination of 24 Hz
(0.25 v) and 69 H z (0.35 V) , respectively. As can be seen in the figures, the decay
envelopes were similar to the linear case.
Figure 3.26 shows the decay curve for the third order mode using multi-frequency
excitation. Again, the magnitude of the third order mode decayed in a similar way as in
the linear case.
81
Chapter 3. Damping characteristics of the flexible cantilever beam
OT ••—*
O .>. +_.
c cu
E CU CJ J5 Q. OT Q
1
0.8
0.6
0.4
U.2
0
-0.2
-0.4
-0.6
-0.8
-1 0 0.5 1 1.5 2 2.5 3
Time [seconds]
Figure 3.24 Decay in magnitude of the second mode for combined excitation at 24 H z
(0.35V) and 4 H z (0.05V).
i 1 1 ' • 1~~ —\
1
0.8
0.6
o
r °-2
1 o cu o OT
i=> -0.4 -0.6 -0.8
-1
0 0.5 1 1.5 2 2.5 3 Time [seconds]
Figure 3.25 Decay in magnitude of the second mode for combined excitation at 24 H z
(0.25V) and 69 H z (0.35V).
82
Chapter 3. Damping characteristics of the flexible cantilever beam
0.4
0.3
W
£ 0.1
I o CU
u •5.-0.1 CO
b -0.2 -0.3
-0.4
0 0.5 1 1.5 Time [seconds]
Figure 3.26 Decay in magnitude of the third order mode for combined excitation at 4 H z
(0.05V) and 69 H z (0.35V).
3.4 CONCLUSIONS
The damping of the first three modes of the beam has been investigated for single as
well as multi-frequency excitation in both the linear and nonlinear cases. The
experimental results have shown that all of the modes decayed independently of each
other and at different rates - the higher order modes have a faster decay rate than the
lower order modes. In addition, for the same excitation frequency, the larger the
displacement amplitude, then the faster the magnitude decayed. Because the decay rate
of each mode was proportional to both the amplitude and frequency of the vibration, the
damping of each mode of the beam can be modelled as a combination of Viscous and
Quadratic damping in the linear case. In the nonlinear case, where nonlinear modal
coupling occurred, the beam also exhibited Hysteretic damping characteristics above the
coupling/decoupling threshold in addition to Viscous and Quadratic damping. However,
as soon as the magnitude of the displacement decayed to the decoupling threshold level,
the beam then exhibited only Viscous and Quadratic damping.
83
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
Chapter 4
MODELLING OF NONLINEAR VIBRATION IN
A FLEXIBLE CANTILEVER BEAM
4.1 INTRODUCTION
The problem of modelling the dynamics of flexible beams has attracted the attention of
researchers in many areas of engineering applications. Examples are robotic
manipulators, satellites and aircraft stmctural dynamics. Many linear theories such as
Timoshenko beam theory, Euler-Bernoulli beam theory and the Rayleigh-Ritz method,
have for several decades been useful tools for predicting the behaviour of beams,
assuming that the beam undergoes small amplitude vibrations. W h e n the amplitude of
the vibration becomes large, the beam no longer behaves like a linear system. For a
single sinusoidal excitation, the response of the beam may comprise lower order
harmonics, higher order harmonics, or other induced frequency components. Jump
phenomena may exist in the sinusoidal frequency response of the beam such that
discontinuities in amplitude and phase shift occur at different frequencies, depending on
whether the response is measured with increasing or decreasing frequency. Furthermore,
many other nonlinear phenomena such as change in resonance frequency, modal
interaction, frequency modulation between higher order modes and the first order mode,
energy transfer from higher order modes to lower order modes, would occur when the
beam is subject to a parametric excitation (where the excitation frequency is close to one
of the resonance frequencies of the beam). These nonlinear behaviours preclude an
adequate mathematical analysis of the beam response based on linear models.
Many theoretical and experimental investigations into nonlinear vibration of beams have
been carried out over the years. Under assumptions contradictory to linear theory,
84
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
Eringen [23] has developed a solution for free flexural vibration of elastic bars having
simply supported ends by means of a perturbation method. The solution of this problem
adequately described those motions in which the changes in axial tension as well as in
deflection, were large. In his solution, the rotatory inertia was taken into account while
the shear effect were excluded. The equations of motion for bars with simply supported
boundary condition were later corrected by Woodall and solved by Ray and Bert [54] by
means of three different approximate solutions: (i) Assumed Space Mode, (ii) Assumed
Time Mode, and (iii) Ritz-Galerklin method. They also experimentally verified all the
solutions by comparing the theoretical resonance frequency and stiffness of the first
order mode for each solution with the experimental results. Rather than using simply
supported boundary conditions, Evensen [25] investigated the nonlinear vibration of
beams with various boundary conditions. Like Eringen [23], he applied a perturbation
method in order to derive approximate amplitude-frequency relations for nonlinear
vibration of uniform beams with clamped-clamped and clamped-supported boundary
conditions. The theoretical results have shown that the clamped-clamped beam
exhibited a smaller change in resonance frequency with increasing excitation amplitude
than the simply supported beam. Also, for higher order modes of vibration, the
amplitude-frequency curves for the clamped-clamped or clamped-supported beams
tended to approach that of the simply supported beam.
Because a great variety of nonlinearities may enter the equations of motion, different
authors often deal with different types of nonlinearities depending on the nature of the
problem and the objectives of the analysis. For bars with simply supported boundary
conditions, the nonlinearity may arise due to axial stretching of the beam during the
vibration. Woinowsky-Krieger [76] has studied the effect of the axial stretching on the
vibrations of hinged bars and found that the resonance frequency of vibration increased
with increasing amplitude. In addition to the axial stretching, McDonald and Raleigh
[41] investigated the nonlinear mode shapes of a uniform hinged beam and also found
that the resonance frequency of each mode was dependent on the amplitude of vibration.
85
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
In contrast to the work of Woinowsky-Krieger [76] and McDonald and Raleigh [41],
where the nonlinearity considered was only the effect of the axial stretching, Atluri [5]
included the effects of large curvature, longitudinal inertia and rotary inertia while
ignoring the effects of axial stretching and transverse shear deformation. Atluri has also
classified the nonlinearity in beam vibration due to moderately large curvatures and
longitudinal elastic forces, generated by longitudinally immovable supports, as elastic
nonlinearities; and the longitudinal and rotary inertia forces as inertia nonlinearities. In
contrast to the case where only nonlinearity due to axial stretching is considered,
Atluri's results have shown the nonlinearities due to large curvature, longitudinal inertia
and rotary inertia were softening-type rather than hardening-type, as predicted by
McDonald and Raleigh [41] and Woinowsky-Krieger [76].
In the case of large deflection of free-free beams, Hu and Kirmser [31] have derived a
nonlinear partial differential equation to describe the natural vibrations of the free-free
beam based on the nonlinear relationship between the bending moment and the
curvature of the beam. Similar to Atluri's work, where the nonlinear partial differential
equation was reduced to a nonlinear ordinary differential equation using a Galerkin
method, H u and Kirmser used Duffing and Ritz-Kantorovich methods to develop the
nonlinear ordinary differential equation of the motion of the free-free beam. The
ordinary differential equation was then solved using perturbation and shooting methods.
Their solutions showed that the mode shape functions for nonlinear vibrations differed
from the linear case. The mode shape functions changed slightly for different
frequencies depending on the amplitude. However, the higher order modes of vibration
were not presented in the solutions for the nonlinear vibration of the beam. Wagner [72]
included axial inertia and nonlinear curvature in the analysis of large amplitude free
vibrations of a straight elastic beam having free-free or clamped-free end conditions,
with negligible shear and rotary inertia effects. A combination of Hamilton's principle
and Buhnov's method was used to obtain a uni-modal nonlinear differential equation. It
is noted that the nonlinear terms in the equation are hardening-type, none is of
softening-type. This contradicts to the experimental results described in Chapter 1,
86
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
where the beam was observed to have a hardening characteristic at the first order mode,
and softening characteristic at higher order modes.
As the concept of normal modes of motion is well developed and widely used to
estimate the motion of a continuous beam, the study of the effect of large vibration on
the mode shape and resonance frequency has attracted the interest of many researchers
[8, 11, 47]. They measured mode shapes at large amplitude vibration and found that the
mode shapes were different to the calculated mode shapes based upon linear theory.
Since the mode shapes were amplitude dependent, different approaches were developed
to derive the nonlinear mode shapes and resonance frequencies of beams. A common
approach to such a problem is to assume the beam motion to be harmonic of unknown
frequency, then apply the harmonic balance method in order to obtain the nonlinear
mode shapes and resonance frequencies suitable for the boundary conditions. This
approach was used by Lewandowski [37], Bennouna and White [11] and Benamar and
Bennouna [8] while investigating the effects of large amplitude vibration on the mode
shapes and the resonance frequencies of beams.
Unlike Benamar and Bennouna [8] and Bennouna and White [11], Nayfeh et al [47]
used a Galerkin procedure to convert partial differential equations into ordinary
differential equations, and used perturbation and invariant manifold techniques in order
to obtain the nonlinear mode shapes. Their alternative to determine the nonlinear mode
shapes and resonance frequencies was to apply a multiple scale method directly to the
governing partial differential equations and boundary conditions. Another common
method called Rayleigh-Ritz, has been used by Bhat [13] to obtain approximate values
for the resonance frequencies and mode shapes. His study presents some insight into the
nature of the resonance frequencies obtained and their dependence on the assumed mode
shape functions.
In contrast to Benamar and Bennouna [8] and Nayfeh et al [47], Bennett [10] believed
that the nonlinear motion of the beam could be described by only nonlinear time-variant
differential equations, while assuming that the mode shapes were amplitude
87
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
independent. Based on the assumption that the mode shapes for the nonlinear case were
the same as for the linear case, he developed a multi-degree-of-freedom beam model
where the partial differential equations were reduced to time-variant ordinary
differential equations using the Galerkin method and the mode shape functions satisfied
the linear boundary conditions. In his motion equations for the beam, he included the
nonlinear effects due to the axial stretching at the clamped end and neglected the shear
deformations and longitudinal inertia. Both his experimental and theoretical results
showed the softening characteristics of the first, second and third order modes, and
nonlinear coupling between the modes of a clamped-clamped beam.
In addition, recent experimental and theoretical studies [2, 3, 4, 47, 48] have shown the
nonlinear coupling between high order modes and low order modes, due to the energy
transfer from the excited high order mode to the lower order modes. In parallel with the
modal coupling, frequency modulation between the higher order modes and the first
order mode was also observed. Therefore, multi-mode approaches have significant
advantages over single-mode methods as presented in [12, 23, 25, 31, 37, 41, 51, 54,
74], since they allow for interactions between the modes and different types of
nonlinearities to enter the mode equations.
The increasing interest in the development of a light-weight robot arm for high speed
operation has promoted work on the nonlinear vibration of a flexible cantilever beam
carrying a lumped mass at the tip. For example, Zavodney and Nayfeh [79] investigated
theoretically and experimentally the nonlinear response of a slender cantilever beam
carrying a lumped mass for a parametric vertical base excitation. The Euler-Bernoulli
theory was used to derive the governing nonlinear partial differential equation of the
beam for an arbitrary position of the lumped mass. The equation contained nonlinear
terms due to large geometric curvature and axial inertia up to the third order, but
excluded the shear deformations and rotary inertia. The governing partial differential
equation was then discretized to a second order ordinary differential equation using the
Galerkin method. The ordinary differential equation was then solved using the
88
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
perturbation method. Both theoretical and experimental results showed chaotic
behaviour of the beam when it was subject to a parametric excitation.
Hamdan and Shabaneh [80] investigated the nonlinear period (which is inversely
proportional to the resonance frequency) of the first four modes of a slender inextensible
cantilever beam, with a rotational flexible root and carrying a lumped mass at an
intermediate position along its span, when it is subject to a large excitation amplitude.
By taking into account axial inertia and nonlinear curvature, two different approaches
were used to formulate the equation of motion. In the first approach, Hamilton's
principle, which does not account for the inextensibility condition, was used to obtain
the governing partial differential equation. The partial differential equation was then
reduced to a nonlinear uni-modal Duffing-type equation by using a single-mode
approximation in conjunction with the Rayleigh-Ritz method. In the second approach,
an assumed single-mode Lagrange method, taking into account the inextensibility
condition, was used to directly form the fifth order nonlinear uni-modal equation. Their
theoretical results showed that the base stiffness, the magnitude and the position of the
attached mass had similar effects on the period of a nonlinear case as in a linear case
(when the amplitude of motion was small), but their effects became more pronounced in
the nonlinear case when the amplitude of motion was relatively large.
Recently, Finite Elemept methods have been widely used for analysis of nonlinear
vibration of beams. For example, Sarma and Varadan [60] have introduced the Ritz-type
Finite Element approach to the problem of nonlinear vibrations of simply supported
beams. The nonlinear equations of beam vibration were based on Lagrange's principle
and solved by using two different methods. The first method, called the direct iteration
technique, was used to compute a numerically exact fundamental mode shape and the
corresponding frequency. The second one, a Rayleigh quotient type of formulation,
evaluated the frequency of vibration of the fundamental mode. Their theoretical results
showed that the mode shape for the case of hinged-hinged end conditions remained the
same, whereas the mode shapes for the clamped-hinged and clamped-clamped end
conditions increased with increasing amplitude.
89
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
Similar to Sarma and Varadan [60], Heyliger and Reddy [81] developed a finite element
model for the static and dynamic response of rectangular beams using a higher order
shear deformation theory. They claimed that for static or low frequency deformation,
the higher order theory yielded more accurate and consistent results than the
Timoshenko theory. However, the higher order theory provided a poor approximation to
the true shear stress distribution for higher order modes.
The aim of this work is to develop a nonlinear model for a flexible cantilever beam with
a response which corresponds to the nonlinear behaviours observed during the
experiments described in Chapter 1. None of the nonlinear models for flexible beams
which have been developed so far, are able to adequately describe the nonlinear
behaviour of the flexible cantilever beam; such as change in resonance frequency
(increase in resonance frequency of the first order mode, and decrease in resonance
frequency of higher order modes), a continuous energy transfer between higher order
modes and lower order modes, modal interactions, frequency modulations between
higher order modes and the first order mode, nonlinear stiffness and hysteretic damping.
For example, Rao et al [50], To [67], Bruch and Mitchell [15], Nayfeh et al [47, 48]
among others only investigated nonlinear mode shape and resonance frequency of a
cantilever beam. Their models did not contain coupling terms between the modes or
predict the changing stiffness characteristic (from hardening to softening). Berdichevsky
and Kim [12] developed a nonlinear one-degree-of-freedom beam model for a cantilever
beam. The model is valid so long as the beam is excited at the first order mode only.
W h e n the beam is excited at higher order modes at reasonably large amplitude, the beam
would couple to the first order mode as observed during the experimental phase of this
work. In this case, the one-degree-of-freedom model becomes inadequate. In references
[2, 18], the equations of motion contained all the coupling terms between the modes.
However, in the case of the cantilever beam, the coupling between the second and third
order modes and all the quadratic terms can be neglected since they were observed in
the experiments to be insignificant. Like many other works [12, 15, 47, 50, 67], they did
not consider hysteretic damping which can contribute significantly to the nonlinear
90
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
behaviour of the beam. As described in Chapter 1, when the magnitude of vibration of
the higher order modes reached the coupling threshold level, the beam started to couple
to the first order mode. Conversely, the decoupling did not occur until the magnitude of
the vibration of the excited higher order mode decreased to the decoupling threshold
value (which was smaller than the coupling threshold value). This phenomenon was due
to the property of hysteretic damping existing in the beam. Furthermore, neither Viscous
or Quadratic damping were taken into account in the nonlinear motion equations of
other work.
In this work, the lateral deflection of the cantilever beam with base force excitation,
W(x,t), is obtained by superimposing the responses of the first three individual modes of
the beam since the responses of the fourth and higher order modes are insignificant. In
other words, the response of the cantilever beam is assumed to be expressed as
3
W(x,t) = ^Oi(x)f i(t), where Oi(x) is the im mode shape function, and fi(t) is the
i= l
time-variant function of mode / resulting from the applied force. As described in
section 1.3.1, the resonance frequencies change with increasing excitation amplitude.
Therefore, the excitation frequency was changed corresponding to the excitation
amplitude in order to measure the resonance response of the cantilever beam. It was
observed that the normalised mode shapes for the nonlinear case were the same as for
the linear case. The mode shape functions can, therefore, be derived based on the linear
theory. Hence, only the resonance frequencies and the time-variant motion functions of
the beam, which are amplitude-dependent, need to be determined as functions of the
excitation amplitude and frequency. In contrast to other work, the development of the
nonlinear model was not only based on nonlinear theory, but primarily on experimental
observation and understanding of the nonlinear behaviour of the beam. In the process of
developing the nonlinear model, nonlinear beam theory was firstly applied and then
modified corresponding to experimental results observed in Chapters 1 to 3. It will be
shown in this chapter that the developed nonlinear model of the flexible cantilever beam
corresponded very well with the experimental results.
91
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
4.2 MODELLING OF NONLINEAR VIBRATION IN THE BEAM
In this section, the nonlinear response of the cantilever beam will be determined based
on the experimental results described in Chapter 1, when a sinusoidal inertia force F(t)
is applied perpendicular to the beam axis (as shown in Figure 1.2).
Due to the nature of the experimental set-up, the following assumptions can be made:
1) There is no twisting deformation.
2) The thickness of the beam is very small compared to it's length. Hence, the
rotary inertia and shear effects are ignored.
3) The beam has a uniform density.
4) Since the beam was pointing horizontally (see Figure 1.2), the gravity effect
on the transverse vibration is negligible.
The nonlinear model of the cantilever beam is determined in the following sequence:
(D Excitation in the vicinity of the first resonance
From the experiments it was observed that the resonance frequency of the first order
mode increased with increasing excitation amplitude. This indicates that the beam
has a hardening stiffness characteristic at the first order mode. Furthermore, it was
also observed (see Chapter 3) that the beam had combined viscous and quadratic
damping characteristics for the first order mode.
Assume that the lateral deflection of the flexible cantilever beam with a force
excitation F(t) = A cos fit (here the excitation frequency fi is in the vicinity of the
resonance frequency of the first order mode ooO at the clamped end can be expressed
as
W(x,t) = 01(x)f1(t). (4.1)
92
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
Hence, the kinetic energy for the beam with base motion is
1 L 1 L
T = - JpS W(x,t)2 dx = - JpS(0,(x))2 f ,2 dx, (4.2)
where p is mass density, S is cross sectional area, and L is the length of the beam.
Let M , = JpSO,(x)2dx and obtain
1 T = -M1f1
2 l
(4.3)
The potential energy is
1 L
U = -jMd6, (4.4)
where M is the bending moment.
Brazier [13] and Hu and Kirmser [24] have shown that for large deformation, the
relationship between the bending moment and the curvature of the beam can be
approximated as
( *2
M = EI dzW
V dx 2 <"C1
' d 2 W ^ 2
v d x2 ,
+ c, 'd2w^r
Vdx2, (4.5)
where E is Young's modulus, I is the area inertia moment of the beam, R is the
curvature of the beam, and ci and c2 are nonlinear parameters.
The potential energy can be expressed as
93
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
U = - E I 2
=IEI 2
L-'d2W^ ^ d 2 W V
Jhr dx+cJh?- dx+cJ L/^2
o v d x y
L »
dx^
d'W dx2
dx
L » L »
f12J{01(x)}
2dx + c1f13J{01(x)}
3dx + c2f14J{01(x)}
4dx
(4.6)
where 0,(x) is the second derivative of the mode shape function of the first order
mode; and the mode shape function 3>,(x) is determined based on the clamped-free
boundary condition in the linear case (see Appendix A).
L » L » L »
Let K „ = ElJ{0,(x)}2dx, K21 =£10,1(0,(x)}3dx and K3, =EIcJ{0,(x)}
4dx
The potential energy can be rewritten as
U = iK„f,2+lK2,f,3-f-iK3,f,4. (4.7)
As shown in Chapter 3, the first three resonance modes of the beam exhibited both
Viscous and Quadratic damping, the Lagrange's equation for the beam with
excitation of the first resonance then becomes
f, +d,f, + e, f, K, F(t)
^<^4^2+2^3=-^i^Wdx' 2M, M, M, { (4.8)
where co = I — — is the resonance frequency of the first order mode, di is the
VMi
viscous damping factor, ei is the quadratic damping factor.
94
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
3 K K Let u., = —, a, =2—2- and F(t) = A cos fit. The equation (4.8) can then be
2 M, M, written as
f,+d,f, + £, fi fi2Acosfit r 2r r2 r 3 " ACOSl^l f
f,+co,2f, + u.,f,2 +a,f,3 = JO,(x)d> M l 0
= b, A cos fit, (4.9)
fi2L where b, = [o,(x)dx , and (jLi and ai are positive since the beam has a
M i o
hardening stiffness characteristic at first order mode. It will be shown in the
following that \i\ and cti had to be positive to correspond to the increase in the
resonance frequency of the first order mode.
For convenience, the change of the resonance frequency as a function of u^i and (Xi
was illustrated using the motion equation of the beam for free vibration as follows:
f,+d, *•, + £, f, f, + co,2f, + p.,f,2 +a,f,3 = 0. (4.10)
Using the harmonic balance method [46] and assuming that the steady-state solution
ofEq.(4.10)is
f, = A, coscot + A2cos2cot + A3cos3cot, (4.11)
then
f, =-A,©sin©t-2A2cosin2©t-3A3©sin3Q)t, (4.12)
f, = -A,co2 costot-4A2co2 cos2cot-9A3co
2 cos3cot, (4.13)
and
f, f, = A,2co2|sincot|sinfit + 2A1A2a)2|sincot|sin2cot + 3A1A3C0
2|sincotjsin3cot
+ 2 A, A2co2 |sin 2cot| sin cot + 4A2
2co2 |sin 2cot| sin 2cot + 6 A 2 A3co2 Jsin 2cot| sin 3cot
+ 3A, A3co2 |sin 3cot| sin cot + 6A 2 A3co
2 |sin 3cot| sin 2cot + 9 A32co2 |sin 3cot| sin 3cot.
(4.14)
95
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
Let y/= cot and define
signl(\|/) = 1 for \p = [0, TC]
-1 for \j/ =< TC, 2TC >'
sign2(\|/) = 1 for\j/ = [0,7c/2]and[7i,37c/2]
-1 for \|/ =< n 12, n> and < 3n / 2,2n>'
sign3(\[0 = 1 for\|/ = [0,7i/3], [27t/3,7c]and[47t/3,57c/3]
-1 for \j/ =< 7C / 3, 2TC / 3 >, < it, 47C / 3 > and < 5TC / 3, 2TC> '
.2 1 1 and using sin x sin y = cos(x + y) - cos(x - y) and sin x = — - —cos 2x, Eq.(4.14)
can be rewritten as
|f, f, = signl(v)
(
'A,2co2 A,2co2
cos2cot + A, A2co2 cos cot - A, A2co
2 cos3cot
V )
+ signl(v) 3A,A3co
z „ 3A,A3coz
— - — - — cos 2cot •—-— cos 4cot
+ sign2(\|/)(A, A2co2 coscot - A, A2co
2 cos3a* + 2A22co2 - 2A2
2co2 cos4cot)
+ sign2(\|0(3A2A3co2 coscot + 3A2A3co
2 cos5cot)
„, /3A,A3co2 „ t 3A,A3co
+ sign3(\|/)—hr— •cos2cot-cos4cot
J
+ sign3(\|/) 3A, A,co2 coscot - 3A, A3co
2 cos5cot + L2"3'
9A32co2 9A3
2co2
— cos 9cot
(4.15)
Substituting Eqs.(4.11) into (4.13) and (4.15) into Eq.(4.10) and equating the
coefficients of coscot on each of Eq.(4.10), we obtain
96
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
(co,2 -co2)A, +e,co2(2A,A2 + 6A2A3) + u.,(A,A2 + A2A3)+-a,(A,3 + A,2A3) = 0
co2 =co,24 ' A,
e,C02(2A,A2 + 6 A 2 A 3 ) + u.,(A,A2 + A 2 A 3 ) + -a,(A,3 + A, 2A 3 )
(4.16a)
c IK K
For \|/ = (—,— \3 2
co2 =co,2 -\ 1
1 A, e,co22A,A2 +p,(A,A2 + A 2 A 3 ) + -cc,(A,
3 + A, 2A 3) (4.16b)
ForV = (ftf):
co2 =co,2-f 1
1 A,L e,co2(-6A2A3)-i-ji,(A,A2 + A 2 A 3 ) + -a,(A,
3 + A, 2A 3) (4.16c)
For \j/ = 2K 4K\
LT'T/:
co2=co2 1
H,(A,A2 + A2A3) + -a,(A,3 + A,2A3)
A, L 4 (4.16d)
For \|/ = 47t 3K
T'T co2 =co,2-t
1 e1co
2(6A2A3) + iI(A1A2 + A 2 A 3 ) + -a,(A13 + A,2A3)
A, L
(4.16e)
3K 5K
For \j/ = (—,— Y \2 3
co2 = CO,2 -I 1
1 A, e,co2(-2A,A2) + jx,(A,A2 + A 2 A 3 ) + -a,(A,
3 + A,2A3)
(4.16f)
97
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
For y =
2 i t CO = C 0 , -T A,
e,co2(-2A,A2 -6A2A3) + u.,(A,A2 + A2A3) + -a,(A,3 + A,2A3)
(4.16g)
Any cubic terms containing A2 2 or A32, such as Ai A22, A1A32, A 22A3 and A 2 A 3
2
were ignored in Eqs.(4.16a-g), since the magnitude Ai was observed in the
experimental results to be much greater than A 2 and A3. It can be seen from the
equations that the term —a,(A,3 + A, 2 A 3 ) is always larger than the term
p,,(A,A 2 + A 2 A 3 ) if jp,,|<|a,|. A s a result, when |p,,|»|a,j, the resonance
frequency is increases with increasing magnitude of vibration if both p.i and cci are
positive and decreases if they are negative. If j|X, | < |cc, |, only 0C1 has to be positive to
correspond to the increase in the resonance frequency of the first order mode. W h e n
|0,i, cci and the magnitude of A 2 and A3 are zero (ie. the linear response), co becomes
equal to co,. Otherwise, the resonance frequency varies in different time ranges,
depending on the magnitude of Ai, A 2 and A3. This result also explains the change of
the resonance frequency of the first order mode due to the nonlinear modal coupling,
as shown in Figures 1.20 and 1.21.
(2) Excitation at the second resonance
(a) Without modal coupling
In contrast to the first order mode, the resonance frequency of the second order
mode decreases with increasing excitation amplitude. In other words, the stiffness
characteristic of the second order mode is softening-type rather than hardening-
type. Without modal coupling, the motion equation of the flexible cantilever beam
with excitation of the second order mode can be expressed as in Eq.(4.9). The
resonance frequency is also derived from Eqs.(4.16a-g). However, |i2 and cx2 in
this case are negative due to the softening-type stiffness characteristic.
98
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
(b) With modal coupling
As described in section 1.3.3, when the excited second order mode reached it's
maximum deformation, the beam started to couple to the first order mode due to
energy transfer between the modes (see section 1.3.4). Hence, the displacement of
the beam with a large excitation of the second order mode, can be assumed as
W(x,t) = 0,(x)f,(t) + 02(x)f2(t). (4.17)
Let M y = J p S O ^ x ) 4>j(x)dx and obtain the kinetic energy
L i=l j=l
(4.18)
Since the coupling between the modes is due to the nonlinear stiffness
characteristic of the beam, M u = 0 for i j as the result of an orthogonal
relationship [48]. Eq.(4.17) can be reduced to
1 2 2
T = -£Mufi ^ i=l
(4.19)
The potential energy is
L 2
U= -EI 2 o i*i
L 2
id 0 i=l
J{Xfi^(x)}2dx+cJ^fi0i(x)}3dx+c2J{£fi^(x)}4dx V 2
0 1=1
(4.20)
The Lagrange's equation of motion associated with mode i is
f: +d, f. + e, f, f'i + co,2^ + J i, f,2 + o, fs
3 +C U j f j +C2>ijfj2 +C3>ijf/ +C4>ijfifj
+ C54jf,2fJ+C(M,f1fJ
1=b1AcosQt
(4.21)
when i = 1, j = 2 (and vice versa),
99
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
3c,EIr_ t - 2c2EILr x 4 j
^ ="iM~J°i(x) dx'ai =ivr"l^(x ) dx>
^1V1ii 0 1V1ii 0
EI C, jj = J Oi(x)Oj(x)dx = 0 (because of the orthogonality property),
u o
C^ = ^rJ°^ x)°jW 2 d x' c3.ii = ^ ^ i ( x ) O j ( X )3 d x ,
Z M i i 0 Mii 0
c<« =^J 5 J*.W , *.(*)*.c i l =^J<»,(x)3a.i(x)dx,
1V1ii 0 z,ivlii o
C6.U =% Ejk(x) 2O j(x)2dx, b, = -£-J*,(x)dx.
M u M u 0
The experimental results have shown that the autospectrum of the beam response
mainly contained peaks corresponding to the first and second order modes, and
peaks on the side-bands of the second order mode; whereas the frequency peaks
(2CO1+CO2), (2CO1-CO2), (20^+coi) and (2a)2-C0i) corresponding to the coupling terms
f 1 f2 and fif2 were insignificant. They can be, therefore, ignored in the motion
equations. As a resulf, the motion equations of the flexible cantilever with a large
excitation of the second order mode (Eq.(4.21)) can be simplified as follows:
f,+d,f, + £, fi f, + co,2f, +|i1f1
2 +a,f,3 + C2i,2f22 + C3il2f2
3 + C4>12f, f2
= bjAcosfit,
(4.22a)
f2+d2f2 + e2 f>co22f2+p2f2
2+a2f23 + C2i2,f,
2 + C3i2,f,3 + C4>2,f,f2
= b2Acosfit.
(4.22b)
100
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
Again, p., > Oandoe, > 0 (since the first order mode has hardening-type stiffness
characteristic), whereas (i2 < 0 a n d a 2 < 0 (because the second order mode has a
softening-type stiffness characteristic).
As shown in Chapter 1, the beam coupled with the first order mode when the
excited second order mode became saturated, due to the energy transfer from the
saturated second order mode to the first order mode. This energy transfer can be
described by the coupling terms C2,i2f22 and C3, i2f2
3 in Eq.(4.22a).
In addition to the nonlinear modal coupling, peaks on the side bands of the second
order mode, due to the modulation of the first and second order modes, was also
observed. This can be described by the coupling term C4,2ifif2.
Because the energy only transferred from the second order mode to the first order
mode, but not vice versa, the coupling terms C2,2ifi and C3)2ifi in Eq.(4.22b) can
be ignored. The Equations (4.22a-b) can be simplified as
f, +d,f, + e, f,
f 2 + d 2 f 2 + e 2
f, + co,2f, + p, f,2 + a, f,3 + C212f22 + C312f2
3 + C4>12f, f2
= b,Acosfit,
(4.23a)
f2+co22f2 +u,2f2
2 + a2f23 + C421f,f2 = b2Acosfit. (4.23b)
(3) Excitation at the third resonance
Similar to the second order mode, the resonance frequency of the third order mode
decreased with increasing excitation amplitude. Also, the beam started to couple
with the first order mode when the displacement of the third order mode reached
it's maximum deformation due to the energy transfer from the third order mode
101
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
through the second order mode to the first order mode. The motion equations of
the beam with a large excitation of the third order mode become
f, +d,f,+e, f.
f 2 + d 2 f 2 + e 2
f', + co,2f, +u,,f,2 +a,f,3 + C2>12f22 + C312f2
3 + C4,,2f,f2
+ C5,13f32 + C6,13f33 + C7,13flf3 = b, A COS fit,
(4.24a)
4 + co22f2 + m f 2
2 +a2f23+C4t 2,f,f2 + C523f3
2 + C6j23f33
= b2Acosfit,
(4.24b)
f3 + d 3 f3+e3 f3+co32f3 + u.3f3
2 + cx3f33 + C73,f,f3 = b3Acosfit, (4.24c)
where m,ai, u.2 and a2are as in case (2), p 3 < 0 and a 3 < 0 (since the third order
mode has a softening-type stiffness characteristic).
Similar to the case of excitation of the second order mode, the coupling terms
f22f3, fif3
2, f22f3, f2f3
2 and fif2f3 corresponding to the peaks (2CO2+CO3), (2C02-C03),
(2CO3+C0i), (2CO3-C0i), (2CO2-CO3), (2CO3+CO2), (2CO3-CO2), (CO1+CO2+CO3), (CO1+CO2-CO3),
(CO1-CO2+CO3) and (CO1-CO2-CO3), respectively, were ignored in Eqs. (4.24a-b). The
coupling terms C2,i2f22, C3,i2f2
3, C5,i3f32 C6,i3f3
3, C5,23f32 and C6,23f3
3 correspond to
the energy transfer from the excited third order mode to the first order mode via
the second order mode.
4.3 STATE-SPACE MODEL OF THE BEAM
The displacement of the flexible cantilever beam with a force excitation
F(t) = A cos fit at the clamped end is
W(x, t) = O, (x)f, (t) + 02 (x)f2 (t) + 03 (x)f3 (t), (4.25a)
where
102
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
fi+difi + Ei f. f. + cOj^+jx.f^+a^.' + C^f^ + C^f.' + C^fjf,
+ C513f32 + C613f3
3 + C713f,f3 = b,Acosfit,
f 2 + d 2 f 2 + e 2
f 3 + d 3 f 3 + e 3
(4.25b)
f2+co22f2 +u.2f2
2 + a2f23 + C42,f,f2 + C523f3
2 + C623f33
= b2Acosfit,
(4.25c)
f3+co32f3 + U-3f3
2 + a 3 f 33 + C73,f,f3 = b3Acosfit, (4.25d)
and Oi(x), 02(x) and <&3(x) are determined by the linear boundary conditions.
Let x,=f,, x2=fi, x3=f2, x4=f2, x5=f3 x6=f3, a, =0,(x), a2=02(x),
a3 = 0 3 (x) and y = W(x, t), Eqs.(4.25a-d) can be rewritten as
Xi = x 2 , (4.26a)
X2 — ~CO, X, — d,X2 — £,|X2|X2 — U.,X, -CX,X, C2,2X3 -3,12X3 *-4,12XlX3
- C5,,3X52 " C6,13
X52 ~ C7.13X1X5 + b,ACOSfit,
X3 = X 4 ,
(4.26b)
(4.26c)
X4 — —co2 x3 — d2x4 — e2|x4|x4 —p.2x3 — oc2x3 C42,x,x3 C52 3x5 C6 2 3x5
+ b2Acosfit,
(4.26d)
x5 = x6, (4.26e)
x6 =-co32x5-d3x6-e3|x6|x6-u.3x5
2 -oc3x63 -C7 3 1x,x3 + b3Acosfit, (4.26f)
y = a,x,+a3x3+a5x5. (4.26g)
Eqs.(4.26a-g) represents a state space model of the flexible cantilever beam with a
force excitation at the clamped end. In the model, only the first three resonance
modes were taken into account while higher order modes were negligible. It will be
shown in the next section that the developed model is able to adequately describe the
103
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
nonlinear behaviour of the beam. In addition, this model can easily be converted to
an Auto-Regressive Moving Average ( A R M A ) model which is later used to predict
the response of the beam on-line for both linear and nonlinear cases.
Because the aim of this work is to obtain a nonlinear model for system identification
of the beam, an accurate model structure is more important than accurate parameter
values. In order to evaluate the model, the simulation result will be compared with
the experimental results in the next section.
4.4 VERIFICATION OF THE NONLINEAR MODEL
In order to verify the nonlinear model of the flexible cantilever beam, the observed
nonlinear behaviour of the beam such as change of resonance frequency, jump
phenomenon, energy transfer from higher order modes to lower order modes and
hysteretic characteristic, will be examined in a simulation of the model.
Figure 4.1 shows the simulation set-up of the cantilever beam where the state-space
model of the cantilever beam (Eqs.(4.26a-g)) was implemented in C + + and down
loaded into the DSP. The signal from a function generator was fed into the D S P via the
A/D converter and used as the input signal of the model. The response of the model was
fed into an oscilloscope or an analyser via the D/A converter. Since the model is a sixth-
order system, selection of step size and simulation method were critical for simulation
stability. In this simulation, a step size of 2.0x10"4 was selected, and a third/four-order
Runga-Kutta method was used for integration.
Function Generator
|
A/D converter
— • Beam Model — •
DSP
D/A converter
1—•
— •
Oscilloscope
Analyser
Figure 4.1 The simulation set-up of the cantilever beam.
104
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
In the process of verifying the nonlinear model, the parameters in the model were
chosen by trial and error following the same sequence as in the process of developing
the nonlinear model. In other words, di, ei, |Xi and cci were initially determined in such a
way to correspond to the response of the first order mode, while setting the other
parameters equal to zero. For instance, (ii and cti were chosen to be positive since the
resonance frequency of the first order mode increased with increasing excitation
amplitude. It was also observed that the first order mode had a combination of Viscous
and Quadratic damping with a slow decay rate; di and £1 should, therefore, be less than
1 and greater than 0, respectively.
Once the combination of di, £i, u.i and cti, giving a response of the first order mode
similar to the experimental results, were selected, d2, £2, p.2 and a 2 were then
determined, while setting the other parameters equal to zero. In contrast to the first order
mode, (X2 and a 2 were chosen to be negative, because the resonance frequency of the
second order mode decreased with increasing excitation amplitude. It was also found in
Chapter 3 that the second order mode had a slightly higher viscous damping factor and
smaller quadratic damping factor than the first order mode. Hence, di < d2, and £2 < £i.
The next step was to find the values of d3, £3, p,3 and 0C3. Again, they were determined in
a similar way as in the case of the second order mode. However, d2 < d3, and £3 < £2.
Finally, the coupling parameters C2,i2, C3,i2, Cs,i3, C6>23, Cs>23 and C6,23 were determined.
From the experiment, it was observed that coupling only occurred when the magnitude
of the excited second or third order modes reached the coupling threshold value V c ^ or
Vc3a, respectively. Hence, C2,i2 and C3,i2; and C5>i3, C6,23, C 5 3 and C6,23 were set equal
to zero when I f2l < Vc2a, and I f3l < Vc3a, respectively. Otherwise, C2>i2= Kci2a and
C3,i2= Kd2b when I f2l > Vc2a; and C5,i3= Kci3a, C6,i3= K^t,, Cs>23= Kc23a and C6,23= Kg23b
when I f3l > Vc3a. Once the coupling occurred, the decoupling did not happen until the
magnitude of the second or third order mode decreased to the decoupling threshold
value Vc2b or Vc3b, respectively, due to the hysteretic damping characteristic. For a
105
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
better overview, Figure 4.2 shows the value of the coupling parameter C2)i2 as a function
of the magnitude of the second order mode. The other coupling parameters had similar
characteristics to C2,i2. The coupling values K d ^ , Kd2b, Kd3a, Kd3b, Kc23a and Kc23b
were chosen to be reasonably large so that the first order mode became excited when the
magnitudes of x2 and X3 were larger than the coupling threshold values Vc2 a and Vc3a,
respectively. The details of the parameters are shown in Appendix C.
C2,i2
/is
-Vc2a -Vc2b
Kcl2a
Vc2 b Vc 2 a
-•-Kcl2a
Figure 4.2 The coupling term C2ji2 as a function of f2.
(1) Change of resonance frequencies
While having fixed input amplitude, the input frequency was swept slowly in the
vicinity of the resonance frequency of the first, second and third order modes, and the
response of the model was plotted.
Figures 4.3 to 4.5 show the response of the beam for different input amplitudes. From
the figures, it can be seen that the resonance frequency of the first order mode
increased whereas the resonance frequencies of the second and third order modes
decreased with increasing input amplitude. These results correspond to the
experimental results shown in Figures 1.10-1.12.
106
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
3.8 4 4.2 Excitation frequency [Hz]
Figure 4.3 The response of the model in the vicinity of the first order mode for
different excitation amplitudes.
23.5 24 24.5 Excitation frequency [Hz]
Figure 4.4 The response of the model in the vicinity of the second order mode for
different excitation amplitudes.
107
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
T r — 1 1 1 1 1 r
J I I I I I I I
69.2 69.4 69.6 69.8 70 70.2 70.4 70.6 70.8 Excitation frequency [Hz]
Figure 4.5 The response of the model in the vicinity of the third order mode for different
excitation amplitudes.
(2) Jump phenomenon
Figure 4.6 shows the response of the beam model measured when sweeping the input
frequency forward and backward slowly in the vicinity of the second order mode.
Similar to the experimental results, two steady-state magnitudes of the response of
the model were observed for a given input frequency.
Similar results were also observed when sweeping the input frequency forward and
backward slowly in the vicinity of the third order mode.
u
-5-m
CL
o (D
-10
-1K
108
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
1 1 1
XL Forward O Backward
23.5 24 24.5 25 Excitation frequency [Hz]
Figure 4.6 Jump phenomenon occurred when input frequency swept slowly forward
and backward in the vicinity of the second order mode.
(3) Energy transfer from higher order modes to lower order modes
Figures 4.7a-b show the autospectra of the response of the model with the excitation
frequency of 24 H z for input amplitudes of 0.5V and 0.8V, respectively. As can be
seen from the figures, the response of the beam is linear for the input amplitude of
0.5V. Only a peak at 24 H z was observed in the autospectmm of the response. W h e n
the input amplitude increased to 0.8V, a peak at the resonance frequency of the first
order mode and sidebands peaks of 24 H z were observed, in addition to the peak at
24Hz, in the autospectmm of the response. This corresponds to the energy transfer
phenomenon observed during the experiment where the excited second order mode
became saturated (see Figure 1.16b).
IU
5
m •o » ' +^ 3 Q. * 3 O CD
0
-5
-10
-•IK
109
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
m 0
E i-20 Q. CO
1-40 o Q.
eg 0
E
1-20 Q. CO
1^0 o Q.
(a) Without coupling 24 Hz
\tkykjMkuwAfi ^JrA4i^Y(A^hnrm^m 10 20 30
Frequency [Hz] 40 50
(b) With coupling
4Hz
1
24 Hz
Wi
20 Hz
wy^yi^i^yiAih,
28 Hz
J^W^(^M^M\L 10 20 30
Frequency [Hz] 40 50
Figure 4.7 Autospectra of the response of the model for excitation at 24 Hz: (a) Input
at 0.5 V, and (b) Input at 0.8 V.
Correspondingly, Figure 4.8 shows the response of the model in time domain for the
input amplitude of 0.8 V. Similar to the experimental results, the magnitude of the
peaks changed randomly with time in the simulation results.
Similar results were obtained for excitation frequency at 70 Hz. As can be seen from
Figure 4.9, peaks at the resonance frequency of the first and second modes were
observed in addition to the peak at 70 Hz in the autospectmm of the response. Again,
the magnitudes of these peaks changed randomly with time (see Figure 4.10).
110
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
0.2 0.3 0.4 Time [seconds]
0.8 0.9 1 Time [seconds]
1.2
Figure 4.8 The time response of the model for excitation at 24 Hz with coupling.
S Oh
1-4 8-20t CL OT
(a) Without coupling
i yLiLMJiiuiuji JLLL ULLIAID L L U J U U A 20 40 60
Frequency [Hz] 80
2. 0
|-10
8-20 CL OT
I o Q.
-30
4 Hz (b) With coupling 70 Hz
66 H; 74 Hz
-4Q Ul.4JiiU^i^itoi*.^U^iii<Aj<i»hLiJ \tk
100
iMk*t&k\AlkM 20 40 60
Frequency [Hz] 80 100
Figure 4.9 Autospectra of the response of the model for excitation at 70 Hz: (a) Input
at 0.5 V, and (b) Input at 0.8 V.
Ill
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
0.1 0.2 0.3 0.4 0.5
Time [seconds] 0.8
Figure 4.10 The time response of the model for excitation at 70 H z with coupling.
(4) Stiffness
Figure 4.11 shows the relationship between the response of the model and the input
amplitude for an input frequency of 4 Hz. As in the experimental results, the
simulation results also showed that the first order mode had a hardening-type
stiffness characteristic.
Figures 4.12a and 4.12b show the magnitude of the first and second order modes,
respectively, as a function of increasing input amplitude at 24 Hz. Similar to the
experimental results (Figure 1.24), the magnitude of the second order mode increased
proportionally to the increment of the input amplitude until point b (ie. the coupling
threshold). A further increase in input amplitude caused a very slight increase in the
magnitude of the second order mode, but instead a large increase in the magnitude of
the first order mode. The magnitudes of the first and second order modes were
changing randomly between the lines ce and cf, de and df, respectively. These results
corresponded to the experimental results shown in Section 1.3.6.
112
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
0.2 0.4 0.6 0.8 The input amplitude [Volts]
Figure 4.11 The output of the model as a function of the input amplitude at 4 Hz.
0.2 0.4 0.6 0.8 The input amplitude [Volts]
Figure 4.12 The output of the model as a function of the input amplitude at 24 H z
(where the input amplitude increased from 0V to 0.9V).
113
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
Similarly, Figures 4.13a - 4.13b show the magnitude of the first and second order
modes as a function of the input amplitude, respectively, at 24 H z when the input
amplitude decreased from 0.9V to 0V. The results shown in Figures 4.13a and 4.13b
were different to Figures 4.12a and 4.12b due to the hysteretic damping
characteristic.
o > "35 T3
o E
I.O
1
0.5
1 1 1 —
(a) Magnitude of the 2nd order mode
b n ft-—8 %T fo—• —' v \J \J
s 1 1 1
— © —
1
o—
e
— © f
0.2 0.4 0.6
The input amplitude [Volts]
0.2 0.4 0.6
The input amplitude [Volts]
0.8
Figure 4.13 The output of the model as a function of the input amplitude at 24 H z
(where the input amplitude decreased from 0.9V to 0V).
4.5 CONCLUSIONS
Unlike the other nonlinear models which have been developed so far [10, 14, 32, 33, 36
and 47], the nonlinear model described in Eqs.(4.26a-g) is able to demonstrate the
nonlinear behaviour of the flexible cantilever beam observed when the beam is subject
114
Chapter 4. Modelling of nonlinear vibration in a flexible cantilever beam
to a parametric harmonic excitation at the clamped end. Although the simulation results
and the experimental results had different magnitude scales, they both had the same
nonlinear behaviour patterns. This nonlinear model can easily be converted to an Auto-
Regressive Moving Average ( A R M A ) model (see Chapter 5), which can be used to
predict the response of the beam on-line, using the conventional linear Least Mean
Square (LMS) algorithm. It will be shown in Chapter 5 that the on-line estimated
response of the beam based on the developed model is much more accurate than other
conventional filters, which are commonly used for system identification. The
identification method based on the developed nonlinear model works well in both linear
and nonlinear cases; whereas the other methods, such as Finite Impulse Response (FIR)
and nonlinear Volterra FIR filters, failed in the case of nonlinear modal coupling.
115
Chapter 5. On-line identification of the flexible cantilever beam
Chapter 5
ON-LINE IDENTIFICATION OF THE FLEXIBLE
CANTILEVER BEAM
5.1 INTRODUCTION
System identification is to determine a mathematical model describing the response of a
physical system based on the relationships between the system input and output. It
involves two steps: modelling and parameter estimation. In the process of modelling, a
model structure of the physical system is determined based on physical laws and prior
knowledge of system response. Once the model has been validated, the parameters of
the model can then be estimated on or off-line, dependent on the properties of the
estimated parameters. If the estimated parameters vary very slowly with time, they can
be regarded as time-invariant and need to be estimated only once. O n the other hand,
on-line identification is required for time-variant parameters as they need to be updated
frequently.
There are many system identification methods used to estimate the parameter on-line.
They are, to name a few, Least Mean Square (LMS) and Recursive Least Mean Square
( R L M S ) algorithms, Kalman Filter and Self-Tuning methods, etc. A m o n g them, L M S
and R L M S have been proven to be one of the most popular algorithms because of their
simplicity. They are widely used to identify an unknown system, in other words, to
derive the parameters of the model of the unknown system in such a way that the output
of the model is close to the actual response of the system as possible. This idea has also
been applied to applications such as Active Control of Noise & Vibration ( A C N V ) in
order to calculate a control output signal, which is actually an output of an Finite
116
Chapter 5. On-line identification of the flexible cantilever beam
Impulse Response (FIR) filter or an Infinite Impulse Response (IIR) filter. This control
signal, in turn, drives another cancelling source such as loudspeaker or an actuator, in
order to attenuate noise or vibration generated in the system from a noise source.
Originally, the L M S algorithm was developed by Widrow-Hoff in 1959 and applied to a
pattern recognition scheme. Later in 1975, several applications of A C N V using L M S
algorithm were presented [7, 22, 35, 39, 55, 63, 70, 71]. Redman-While et al [55], V o n
Flowtow et al [71] and Elliot et al [22] developed a feed-forward control scheme based
on L M S algorithm in order to cancel the bending motion of infinite or serm-infinite thin
beams. Unlike the application of system identification, the error signal in A C N V
applications, which is used to update the parameters (weights/coefficients) of adaptive
filters, is first filtered by the same transfer function as the cancelling source/secondary
path between the filter and the error sensor, such as the loudspeaker, actuator,
microphone or accelerometer, before being used to update the parameters. This modified
L M S algorithm was referred as "Filtered-X" algorithm [57, 66].
In order to increase the convergence rate and performance, IIR filters are used instead of
FIR filters. The coefficients of the IIR filters are updated using R L M S method. Similar
to L M S , the filtered measured error signal is used to update the coefficients of an IIR
filter, rather than the measured error signal in applications of A C N V . The modified
algorithm was named "Filtered-U". The Filtered-U has been widely used and proven to
give better convergence rate than the Filtered-X. However, the Filtered-U does not
guarantee stability since the poles of the IIR filter can possibly move outside the unit
circle. In order to avoid the stability problem, Vipperman et al [70] filtered the error
signal, with the poles of the plant estimated off-line using an IIR filter, before updating
the coefficients of an FIR filter. The output of the FIR filter was then used to drive a
cancelling actuator in order to cancel vibration in a simply supported beam. The
experimental results showed that the modified algorithm did not achieve as much
vibration reduction as the conventional Filtered-X algorithm. However, it did ensure
stability even when some of the poles of the estimated plant moved outside the unit
circle.
117
Chapter 5. On-line identification of the flexible cantilever beam
Conventional adaptive L M S / R L M S filters use a fixed convergence factor which is a
very critical parameter. For each iteration, the coefficients of the adaptive filter are
increased or decreased by the convergence factor that multiplies the error and the input
signal. The choice of the convergence factor reflects a trade off between the steady-state
misadjustment and the convergence speed which is inversely proportional to the
convergence time. A small convergence factor gives smaller misadjustment, but also
larger convergence time, which is defined as the time the coefficients take to converge
to the optimum value. Recently, a number of publications [20,24,26,28, 33, 36,40,44,
77] have proposed alternative adaptive convergence factor methods to be employed in
L M S algorithm in order to find an optimum convergence factor and consequently
increase performance, such as convergence speed, steady-state misadjustment and
tracking capability.
Although linear filters have been very useful in a large variety of applications, there are
several applications in which they do not give good performance, especially in nonlinear
systems. For these applications, nonlinear filters are required. A very c o m m o n system
model that has been employed with relatively good success in nonlinear filtering
applications is the Volterra Series model [30]. For instance, Baik et al [6] have
presented an Adaptive Lattice Bilinear Filter where the output of the filter can be
expressed in terms of a second-order Volterra series expansion of the input and the past
samples of the output. A s the Adaptive Lattice Bilinear Filter was then transformed into
equivalent multi-channel linear filters, the coefficients of the nonlinear filter were able
to be updated using linear L M S algorithm. Similarly, Tan and Jiang [66] developed a
Filtered-X second-order Volterra adaptive algorithm for a multi-channel application of
noise cancellation. In their simulations, the Filtered-X second-order Volterra algorithms
were proven to give better performance than the conventional linear Filtered-X
algorithm.
Despite the extensive application of LMS/RLMS within the area of ACNV, to my
knowledge these algorithms have not been used to estimate the nonlinear response of
the flexible cantilever beam. This chapter describes h o w the state-space nonlinear model
118
Chapter 5. On-line identification of the flexible cantilever beam
of the flexible cantilever beam, developed in the previous chapter, is converted to an
auto-regressive moving average model ( A R M A model) which is later used to predict the
response of the beam on-line using L M S algorithm. The developed identification
scheme based on the A R M A model is named as the Nonlinear Modal Identification
(NMI) method, since the response of the beam is predicted as a summation of the
responses of all the excited modes. It is conceptually simple and requires only a small
number of estimated parameters (the number of the coefficients of the filter), but still
has the ability to achieve high performance in terms of accuracy, convergence speed and
computational time. In order to evaluate the performance of the N M I method, the
experimental results of the N M I method were compared with those obtained from other
methods using IIR filter and nonlinear filters based on Volterra series. It will be shown
in this chapter that the N M I method is the most efficient identification method. It
worked well in the nonlinear as well as linear cases, whereas the other methods failed
in the case of nonlinear modal interaction. In addition, it has been proven
experimentally that this new scheme has faster convergence speed, requires less
computational time and is more accurate than the other methods in the application of
identification of nonlinear vibration in a flexible cantilever beam.
5.2 THE CONVENTIONAL LINEAR FILTERS USING LMS/RLMS
ALGORITHM
The L M S algorithm is the most popular algorithm in adaptive signal processing. The
goal of the adaptive algorithm is to minimise the system mean square error, which is
defined as the ensemble average of the squared value of the difference between the
actual system response and the estimated response.
c;k=E{ek2} = E{(dk-yk)
2}. (5.1)
The estimated response yk is the output of an Finite Impulse Response (FIR) filter
which is a linear combination of the input samples.
119
Chapter 5. On-line identification of the flexible cantilever beam
yk = wo xk + w, xk_, +• • -+wn xk_n. (5.2)
The input signal is sampled, providing a discrete filter input value xk. This value is
propagated through the filter processing stages with each new sample taken. Thus, at
any time k, the value on the delay chain can be represented as a vector X k defined as
Xk=[xkxk_,--xk_JT, (5.3)
where n is the number of stages in the filter. Each time a new sample enters the
transversal filter, the previous n samples are shifted one position, and the values at each
stage are multiplied by a coefficient assigned to that stage. The results are summed to
produce a filtered output. Representing the coefficients at time k as a vector W k
Wk=[w1w2-wJ, (5.4)
the output of the filter is
yk = WkXk. (5.5)
Substituting Eq.(5.5) into Eq.(5.1), the mean square error can be expressed as
tlk = E{ek2} = E{dk
2} + WkTE{XkXk
T}Wk -2E{dkXkT}Wk. (5.6)
Defining the cross correlation between the system response and the input vector, P,
P = E{dkXkT}, (5.7)
and the input auto-correlation matrix, R,
120
Chapter 5. On-line identification of the flexible cantilever beam
R = E { X k X kT } , (5.8)
the mean square error can be rewritten as
£k =E{dk2} + Wk
TRWk -2PWk. (5.9)
The aim of the adaptive LMS algorithm is to derive an optimum set of weight
coefficients, W , in such a way that the value of the mean square error is minimum. The
optimum weight coefficient vector can be found by differentiating the mean square error
with respect to the coefficient vector and setting the resulting gradient equal to zero
-^ = 2RWk-2P = 0. (5.10) d W k
The optimum weight coefficient vector is then
W*=R"1P. (5.11)
This is the discrete form of the solution to the Weiner-Hopf integral equation. It is
usually impractical to solve Eq.(5.11) to obtain the optimum weight coefficient vector,
W * , owing to the required averaging and matrix inversion. Rather, the optimum weight
coefficient vector is found by some numerical search routine. Normally, a simple
steepest descent algorithm is used since the error surface is hyper-paraboloid.
The algorithm begins with arbitrary initial weights, W0, then descends down the sides
of the error surface, eventually arriving at the bottom of the surface (the location of the
optimum weight coefficients). The new coefficient vector, W,, is found by adding to
the initial one, W 0 , with an increment proportional to the negative of the gradient slope.
Another new value , W 2 , is then derived in the same way by adding to W , with the
gradient slope measured at iteration k = 1. This procedure is repeated until the optimum
value, W * , is reached.
121
Chapter 5. On-line identification of the flexible cantilever beam
The repetitive or iterative gradient search procedure described above can be represented
algebraically as
Wk+1=Wk+p(-Vk), (5.12)
where k is the step or iteration number. Thus Wk is the "present" derived coefficient
vector while W k + , is the "new" (or "updated") vector. The gradient at iteration k is
designated by V k . p is an update rate constant or convergence factor which governs
the rate of convergence and stability.
The gradient, Vk, is obtained by differentiating the mean square error with respect to
W k
_ c ^ = a e k2 _ = J9ek_ =
k awk awk kawk
k k
Substituting Eq.(5.12) into Eq.(2.11) yields
Wk+1=Wk+2uekXk. (5.14)
The selection of p is very critical for the LMS algorithm. A small p will ensure small
misadjustment (error) in steady-state, but the algorithm will converge slowly and may
not track the non-stationary behaviour of the operating environment very well. O n the
other hand, a large p will in general provide faster convergence and better tracking
capability at the cost of higher misadjustment.
For some applications where Infinite Impulse Response (IIR) filters are applied for
system identification, the R L M S algorithm is then used to estimate the parameters of the
filter. For IIR filters, the relationship between the output and input is expressed as a
linear combination of the input samples and the previous output samples,
122
Chapter 5. On-line identification of the flexible cantilever beam
yk = a, yk_i+-+aB yk_mb0 xk +b, xk_,+-+bn xk_n. (5.15)
Equation (5.15) can be rewritten in a vector form as
yk=WkUk, (5.16)
where W k = [ a , a 2 - a m , b 0 b 1 - b j and U k =[yk_1--yk_mxkxk_1--xk_JT.
The weight vector, Wk, is updated as follows:
Wk+1=Wk+2pekUk. (5.17)
5.3 THE CONVENTIONAL NONLINEAR FILTERS
Although linear filters have been very useful in a large variety of applications, there are
several applications in which they can not perform well at all, especially for nonlinear
systems. In these applications, nonlinear filters are more efficient. There are a number of
representations for nonlinear systems that are suitable for system identification
purposes. A very well-known representation is the Volterra series [9] which has been
widely employed in many nonlinear filtering applications.
The output of such a nonlinear filter using Volterra series expansion of the input signal
can be expressed as:
a n n yk = ZhiXk_i + Z Z hijXk-i
xk-j + Z Z Z hi)j,mxk_ixk_jxk_m+-". (5.18) i=0 i=0 j=o i=0 j=o m=0
The output yk can be rewritten in a vector form as follows:
y k = W k X k , (5.19)
123
Chapter 5. On-line identification of the flexible cantilever beam
where W k =[h0.-hn|h0>0..-hn)n|h0,0)0-hn>n>nh-], (5.20)
and X k =[xk-xk_Jxk2,xkxk_1,.--xk_n
2|xk3,--xk_n
3|--]T. (5.21)
Although the output of the filter is nonlinear to the input, the parameters of the filter are
linear. Hence, the weight vector in Eq. (5.20), according to Hsia [9], can be estimated
using the linear L M S algorithm as described in Eq.(5.14).
5.4 THE DEVELOPED ON-LINE IDENTIFICATION SCHEME FOR
THE FLEXIBLE CANTILEVER BEAM
Using the Euler integration method:
+. _ Xk+1 Xk
At
the state-space model of the cantilever beam developed in Chapter 4 (Eqs. 4.26a-g) can
be converted into an Auto-Regressive Moving Average model ( A R M A model) as
follows:
x,,k+, =x1>k+x2kAt, (5.22a)
x2k+i = x2,k + (-0)I x1(k — djX2k-e1 x2>k|x2)k-pjXj^ —ajXlk -C212x3k 2
(5.22b) — C3(12x3)k — C41 2xl kx3 ) k — C 5 1 3 x 5 k — C 6 1 3 x 5 k — 7>i3XlkX5k + DjUjjAt,
x3,k+i=x3,k+x4)kAt, (5.22c)
X4,k+1 =X4,k+(-°)22x3,k-d2X4,k-e2Kk|X4,k-^2X3,k2 -« 2X3,1/
"~ ^4,21Xl,kX3,k ~~ ^5,23X5,k ~" ^6,23X5,k + D2Uk)At,
x5,k+. =x5,k+x6,kAt, (5-22e)
124
Chapter 5. On-line identification of the flexible cantilever beam
X6,k+1 = X6,k +(—®3 X5,k — ^3X6,k _e3X6,kX6,k —H"3X5,k — a3X6,k (5.22f) -C7,31Xl,kX3,k+ b3Uk)At,
yk = aixi,k + a3
x3,k + a5
x5,k • (5-22g)
Substituting Eqs.(5.22a-f) into Eq.(5.22g) yields
yk = wi,oxi,k + wuxi,k-i + w u x i , k-i 2 +w1>3x1>k_,
3 +w2>1x2>k_, +w2>2|x2;k_1|x2)k_1
+ w 3 0 x 3 k + w3,x 3 k_, + w 3 2 x 3 k _ ,2 + w 3 3 x 3 k _ ,
3 +w 4 r l x 4 k _! +w4j 2|x4k_, X4,k-1
+ w 5 0 x 5 k + w 5 1 x 5 k _ , +w 5 2x 5 ) k _,2 + w 5 3 x 5 k _ 1
3 + w 6 , x 6 k _ , H-w^lx^^lx^., + W13,lX1)k_iX3k_j + w,5,xlk_,x5k_1 +bu k_j,
(5.22h)
where y k is the estimated response of the beam at time step t = k,
Xj k and x 2 k are the measured displacement and velocity signals of the first
order m o d e sampled at time step t = k, respectively,
x 3 k and x 4 k are the measured displacement and velocity signals of the second
order m o d e sampled at time step t = k, respectively,
x 5 k and x 6 k are the measured displacement and velocity signals of the third
order m o d e sampled at time step t = k, respectively,
u k is the sampled input, and
Wl,0 > Wl,l > Wl,2 > Wl,3 > W2,l ' W2,2 ' W3,0 »W3,l > W3,2 ' W3,3 ' W4,l ' W4.2 ' W5,0 ' W5,l > W5,2 ' W5,3'
W e 1' w 6 2' wi31»Wis i and are e s y s t e m parameters to be estimated.
In order to obtain the sampled velocity and displacement for each mode, the measured
velocity at the tip of the beam is filtered by three band-pass filters simultaneously. Each
filter has a bandwidth corresponding to the resonance frequencies of the first, second
and third order m o d e of the beam as shown in Figure 5.1. The outputs of the band-pass
filters are the measured velocity signals of each m o d e of the beam. These velocity
signals are then integrated to displacements.
125
Chapter 5. On-line identification of the flexible cantilever beam
A s the coupling terms w 1 3 jX, k_,x3 k_, and w 1 5 jX, k_jX5 k_, represent sidebands of the
second and third order modes, respectively, where the sideband frequencies are close to
the resonance frequencies of the second and third order modes, they pass through the
band-pass filters unattenuated. The terms w,31xlk_1x3k_1 and w ^ x ^ . ^ ^ , can
therefore be combined with the terms w31x3k_! and w51x5k_1, respectively. The
Eq.(5.22) can be reduced and rewritten in a vector form as
yk=wkxk\ (5.23)
where W k = [wlio,...,Wi)3,w2jl,w22,w3>0,...,w33,w41,w42,w5;0,...,w53,w61,w6>2,b],
(5.24)
and X k = [x1jc,x1)k_1,...,x1(k_1 »x2k_j,|x2k_1|x2k_1,x3k,...,|x4)k_1|x4k_],x5k,...,uk_1j.
(5.25)
Again, as in the case of the nonlinear Volterra filter, the weight vector, W k , is updated
using the conventional L M S algorithm as described in Eq.(5.14). Figure 5.1 shows the
developed on-line identification scheme using L M S algorithm. With this scheme, the
response of the beam is predicted as a summation of the responses of all the excited
modes which are identified simultaneously. The developed identification scheme can,
therefore, be named as the Nonlinear Modal Identification (NMI) method.
Input
PLANT Output
Bandpass
filter
^
Bandpass
filter
1 Bandpass
filter
— T Z Z | W I
LMS
6
Figure 5.1 The developed on-line identification scheme using L M S algorithm.
126
Chapter 5. On-line identification of the flexible cantilever beam
5.5 EXPERIMENTAL SET-UP
In this experiment, the D C motor was used to excite a thin spring steel beam with a
dimensions 332 m m (length) x 25.49 m m (width) x 0.6 m m (thickness). The beam was
clamped and attached to the shaft of the motor as shown in Figure 1.4. The motor
included an optical encoder with a quadrature digital output to detect the position of the
shaft. The encoder signal was fed to the D S P and used to derive the control signal of the
motor which is proportional to the difference between the reference generated from a
Hewlett-Packard function generator and the position of the shaft. The control signal was
then fed to the D C motor via the D/A converter and servo amplifier. The servo amplifier
has potentiometer adjustments for current limit, input signal gain, tachometer signal
gain, damping and time constant. Using these adjustments, the response of the motor
was set up to be critically damped.
The vibration at the tip of the beam was measured using the Entran accelerometer. The
measured vibration signal was then connected to the signal conditioning system, which
included inbuilt integrators, and fourth order high-pass and low-pass filters, in order to
integrate the measured vibration signal to velocity/displacement and select the useful
frequency range (between 2 H z to 200 Hz). In the experiment for testing the N M I
method, velocity was selected as the output of the conditioning system and fed to the
D S P via the A/D converter. This filtered velocity signal was then simultaneously passed
through three band-pass filters in order to collect the measured velocity for the first,
second and third order modes of the beam separately. The velocity of each mode was
then integrated to displacement using a different integrator gain for each mode. Both
band-pass filters and integrators were implemented in the DSP. The reference signal,
and the velocity and displacement signals were then used to update the weight vector in
such a way that the output of the filter was as close to the measured displacement as
possible (see Figure 5.3).
127
Chapter 5. On-line identification of the flexible cantilever beam
In order to compensate for the frequency response of the D C motor including the servo
amplifier, the reference signal was filtered by the estimated transfer-function of the
closed-loop transfer-function of the D C motor as follows:
Hc(*) = 7.9xl0"V - 20.7xl0-3.s +18
44.1xl0~8s3 + 118.16xlO_V + 9 1 . 7 2 X 1 0 - 2 J + 72.91 (5.26)
Figure 5.2 shows the measured closed-loop frequency response of the D C motor plotted
versus the estimated transfer-function.
10
0. CQ
"§ -10
1-20
-30
10
10 20 30 40 Frequency [Hz]
50
20 30 40 50 Frequency [Hz]
, 1 1 1 1
-\
^^^
1 ' — The measured — The estimated -
"
60 70
Figure 5.2 The measured and estimated closed-loop frequency responses of the D C
motor.
128
Chapter 5. On-line identification of the flexible cantilever beam
Similarly, the output of the filter is filtered by the transfer-function of the accelerometer,
including the conditioning amplifier, before subtraction from the measured
displacement. However, since the response of the accelerometer and conditioning
amplifier are linear, the output of the filter can be easily compensated by increasing the
number of weights of the filter. In this way, the accelerometer and conditioning
amplifier are included into the filter (model).
Although the nonlinear model of the cantilever beam, used in system identification, was
originally developed for the case of horizontal excitation, the model was still applicable
to this case since the beam had small rotational movements.
Figure 5.3 shows a functional block diagram of the experimental set-up for on-line
identification of the flexible cantilever beam using the N M I method.
For comparison, an IIR filter and a second order Volterra IIR filter were implemented to
predict the response of the experimental cantilever beam. In contrast to the N M I
method, the measured vibration in the IIR filter and Volterra filter schemes was
integrated twice to displacement and used to update the weights of the filters as shown
in Figures 5.4 and 5.5, respectively.
129
Chapter 5. On-line identification of the flexible cantilever beam
Velocity Conditioning
Amplifier
Function
Generator
Beam Accelerometer
•. t:-?inrT|'.''.**.•:• :"!"P'F!;i;!'i:i i!i;i!i;i;'imv*i*i:i>«>i*i;i>i>i;i i:i!i;':»:':':'!i:i:i:i;i:i;i:':i:i:i:i:i:i:i:i;i;iii:i;i;i:i;i:i;i:i;i:ii;i:i:i:iii;i;i;i;<
Motor
Encoder
" ' • •
Power
Amplifier
A/D
(Channel 1)
(Channel 2)
(Channel 3)
(Channel 4)
Incremental
Encoder
Interface
J.
D/A
(Channel 1)
(Channel 2)
(Channel 3)
(Channel 4)
Transfer-
function of the motor
Proportional
controller
Band-pass Filter (4Hz)
Velocity of 1st mode
* Integrator _r^
Band-pass Filter
(24Hz)
Velocity of 2nd mode
Integrator
Integrator
Band-pass
Filter
(70Hz)
Velocity of3rd mode
Integrator
w
LMS «
Transfer-function of
accelerometer &
conditioning
hC
JQSE.
Figure 5.3 Functional block diagram of the experimental set-up for on-line identification
of the flexible cantilever beam using the NMI method.
130
Chapter 5. On-line identification of the flexible cantilever beam
Displacement Conditioning Amplifier
Function Generator
n Beam Accelerometer
rV!M:!;!"!l!l!M'{?!?!i|!|S|!|^V?i*,*,Ti;'!l!lIl!l!l?l!l!l?l!l!l?|!|!|!l!l!| ijijjjjjjjijijijijjjiijjijijijijijfo^
Motor
Encoder
Power Amplifier
A/D
(Channel 1)
(Channel 2)
(Channel 3)
(Channel 4)
£ Transfer-function of the motor
-T>i-
••ir
Incremental
Encoder
Interface
Proportional
controller
D/A
(Channel 1)
(Channel 2)
(Channel 3)
(Channel 4)
IIR filter Transfer-function of
! accelerometer
w> RLMS
conditioning ...ampHfier.
> w,
ngh feL
W
* w
-C
Transfer-function of
accelerometer
& conditioning
amplifier
*0
LMS Third order Volterra FIR filter |
DSP
Figure 5.4 Experimental set-up for on-line identification using IIR and Volterra Filters.
131
Chapter 5. On-line identification of the flexible cantilever beam
5.6 EXPERIMENTAL RESULTS
In order to evaluate the performance of the N M I method, the following performance
criteria were examined:
(i) Convergence speed,
(ii) Accuracy,
(iii) Computational time.
The results of the NMI method were compared with those obtained from other methods
using IIR and third order Volterra FIR filters for both linear and nonlinear cases.
(i) Convergence speed
As the convergence factor and sampling frequency could influence the performance
of the system identification, all three identification methods had the same sampling
frequency at 1 kHz, but different convergence factors. The convergence factor for
each method was selected as large as possible.
Figures 5.5, 5.6 and 5.7 show the measured response of the beam plotted versus the
estimated response, using the N M I method, the IIR and Volterra FIR filters,
respectively, for an sinusoidal excitation. As can be seen from the figures, the
estimated response of the N M I method converged fastest to the measured response
compared to the two other methods. Next was the Volterra FIR filter; the IIR filter
had the slowest convergence speed.
132
Chapter 5. On-line identification of the flexible cantilever beam
0.6
0.4.
rtn *-• 9 *-. c a> E CD o (0 a. CO
b
0.2
0
-0.2
-0.4
-0.6
— Measured Estimated
0.5 1 1.5 2 Time [seconds]
Figure 5.5 The measured and estimated response of the beam using the N M I method.
0.6
0.4
To 0.2
9
| 0 CD O ro
f-0.2
-0.4
-0.6
Measured Estimated
0.5 1 1.5 2 Time [seconds]
2.5
Figure 5.6 The measured and estimated response of the beam using the IIR filter.
133
Chapter 5. On-line identification of the flexible cantilever beam
i 1 1 •
— Measured I ll Estimated.
L I nil' ll ft ^ M h rt H 'i
"' V ^ l( 1/ 11/1 I
P|| If! J i 1 1 —
1 1.5 2 2.5 3 Time [seconds]
Figure 5.7 The measured and estimated response of the beam using the Volterra FIR filter.
(ii) Accuracy
Figures 5.8, 5.9 and 5.10 show the linear response of the beam plotted versus the
estimated response, as well as the error, using the N M I method, the IIR and Volterra
FIR filters, respectively, when the estimates converged to the measured response of
the beam. From the figures, it can be seen that these three methods performed equally
well in the linear case. For comparison, the errors for all three methods were plotted
together in Figure 5.11.
When the response of the beam became distorted and was no longer linear, the
performance of the IIR filter deteriorated significantly whereas the N M I method
continued to perform as well as in the linear case. Although the Volterra FIR filter
did not perform as well as the N M I method, it was significantly better than the IIR
filter (see Figures 5.12, 5.13 and 5.14). Again, the errors for these three methods
were plotted together and shown in Figure 5.15. A s can be seen from the figure, the
134
Chapter 5. On-line identification of the flexible cantilever beam
N M I method had least error compared to two other methods; the next best was the
Volterra filter while the IIR filter had the worst performance.
When the beam was excited at the third order mode and started to couple with the
first order mode, both IIR and Volterra filters could only estimate the high frequency
component due to the excitation of the third order mode, but failed to estimate the
low frequency component due to nonlinear coupling between the modes. In contrast,
the N M I method still performed well in the case of nonlinear coupling between the
modes. It could estimate the low frequency component as well as the high frequency
component (see Figures 5.16, 5.17 and 5.18).
. Measured Estimated Error
0.15 0.2 0.25 Time [seconds]
0.4
Figure 5.8 The measured and estimated response of the beam and the error using the
N M I method for the linear case.
135
Chapter 5. On-line identification of the flexible cantilever beam
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Time [seconds]
Figure 5.9 The measured and estimated response of the beam and the error using the
IIR filter for the linear case.
0.15 0.2 0.25 Time [seconds]
0.35 0.4
Figure 5.10 The measured and estimated response of the beam and the error using the
Volterra FIR filter for the linear case.
136
Chapter 5. On-line identification of the flexible cantilever beam
. Developed scheme Volterra filter IIR filter
-0.06
-0.08
-0.1 0.05 0.1 0.15 0.2 0.25 Time [seconds]
0.3 0.35 0.4
Figure 5.11 The respective errors for the N M I method, IIR filter and Volterra FIR filter
in the linear case.
0.5
Measured Estimated Error
0.15 0.2 0.25 Time [seconds]
0.4
Figure 5.12 The measured and estimated response of the beam and the error using the
NMI method for the nonlinear case.
137
Chapter 5. On-line identification of the flexible cantilever beam
Time [seconds]
Figure 5.13 The measured and estimated response of the beam and the error using the
IIR filter for the nonlinear case.
Time [seconds]
Figure 5.14 The measured and estimated response of the beam and the error using the
Volterra FIR filter for the nonlinear case.
138
Chapter 5. On-line identification of the flexible cantilever beam
0.25|-
0.2
0.15
. Developed scheme Volterra filter IIR filter
-0.15|-
-0.2
-0.25
0.05 0.1 0.15 0.2 0.25 Time [seconds]
0.3 0.35 0.4
Figure 5.15 The respective errors for the N M I method, IIR filter and Volterra FIR filter
in the nonlinear case.
0.3 Measured Estimated
0.05 0.1 0.15 0.2 Time [seconds]
0.25
Figure 5.16 The measured and estimated response of the b e a m using the N M I method in
the case of nonlinear modal coupling.
139
Chapter 5. On-line identification of the flexible cantilever beam
0.05 0.1 0.15 0.2 0.25 0.3 Time [seconds]
Figure 5.17 The measured and estimated response of the beam using the IIR filter in the
case of nonlinear modal coupling.
0.05 0.1 0.15 0.2 0.25 0.3 Time [seconds]
Figure 5.18 The measured and estimated response of the beam using the Volterra FIR
filter in the case of nonlinear modal coupling.
140
Chapter 5. On-line identification of the flexible cantilever beam
(iii) Computational time
Method
NMI
IIR filter
Volterra filter
Computational time
288 ps
257 ps
626 ps
No. of weights1
16
54
134
Table 5.1 Comparison of the computational time and number of weights used for
different identification methods.
As can be seen from Table 5.1, the IIR filter had 54 weights and required the least
computational time. The N M I method used only 16 weights and had a slightly longer
computational time than the IIR filter, whereas the Volterra FIR filter had the
computational time and number of weights approximately 2.5 times longer than IIR
filter. However, the computational time of the N M I method could be reduced
significantly if the digital bandpass filters and integrators currently implemented in
the D S P (see Figure 5.3) were replaced with hardware filters and integrators.
5.7 CONCLUSIONS It has been demonstrated experimentally that the N M I method is much better than the
IIR and the third order Volterra FIR filters. It works well in the nonlinear case as well
as the linear case, whereas both the IIR filter and Volterra FIR filter failed to estimate
the low frequency component in the case of nonlinear modal coupling.
1 The number of weights was selected to achieve best error performance and meet the DSP capacity requirement.
141
Chapter 5. On-line identification of the flexible cantilever beam
It is a fact that an IIR filter is a linear system, in which case it would not be excepted to
work well when applied to a nonlinear system. This was verified by the experimental
results, as the IIR filter only worked in the linear case when the output was coherent
with the input. Because the beam has nonlinear coupling of higher order modes to the
first order mode, a model which is able to predict this nonlinear modal coupling, is
required. Although the Volterra FIR filter was based on the nonlinear Volterra series
which is suitable for system identification of some nonlinear systems, the output of the
filter was not able to model the nonlinear modal coupling of the beam. The experimental
results have shown that the N M I method was very accurate, only because it incorporated
a valid nonlinear model of the beam.
Although development of the NMI method was based on the nonlinear model of the
beam, this method used the conventional L M S algorithm to estimate the parameters.
The method is, therefore, conceptually simple. In addition, the method has faster
convergence speed, is more accurate and requires less computational time compared to
the other identification methods. This new identification scheme would be useful for
developing a feed-forward as well as feed-back control scheme for cancelling vibration
in the flexible cantilever beam.
142
Chapter 6. Active control of nonlinear vibration in the flexible cantilever beam
Chapter 6
ACTIVE CONTROL OF NONLINEAR VIBRATION IN
THE FLEXIBLE CANTILEVER BEAM
6.1 INTRODUCTION
Vibration in a robot arm has always been of great concern to robotics researchers and
fabricators in design optimisation. Most industrial robots are designed to carry heavy
loads at the same time as having fast and accurate motion responses. In order to achieve
accurate motion response without suffering from structural vibration due to high speed
operation, the robot arm has to be stiff and rigid. Consequently, heavy arms and larger
actuators are needed and thus higher energy consumption occurs.
Instead, many researchers and robotics fabricators have attempted to design a flexible
robot with a lightweight arm to reduce the power consumption and the cost of
production and operation. However, when a lightweight arm carrying a heavy load
moves quickly, the inertia force in the flexible structure can excite many of the
resonance modes, resulting in a large vibration at the end of the motion. As a result, a
more sophisticated control scheme for the flexible manipulator is required in order to
achieve the same performance as a rigid manipulator.
For instance, Cannon and Schmitz [17] attempted to develop a feedback control scheme
for position control of the end-point of a flexible single robot arm using a linear pinned-
free beam model. The error between the measured position of the end-point of the
flexible arm was detected by an optical sensor and the output of the estimator was fed
back in order to control the end-point position. To improve the performance of the
143
Chapter 6. Active control of nonlinear vibration in the flexible cantilever beam
control system, a feedback loop using the hub angle sensor was also employed in
addition to the end-point position feedback loop.
Similar to Cannon and Schmitz [17], Sakawa et al [59] designed a feedback
compensator in order to attenuate vibration in a flexible arm using a linear Euler-
Bernoulli model, in which rotary inertia and shear deformation effects and Coulomb
friction and backlash of the gears were neglected. Instead of using the optical encoder,
they used a strain gauge to measure the vibration at the end-point of the flexible arm. In
addition, they used a rotary encoder and a tachometer to detect the position and speed of
the motor, respectively. These three sensing signals were fed into a micro-computer via
an A/D converter in order to derive the control output used to drive a D C motor in such
a way that the vibration in the arm is stabilised.
Later, Rovner and Cannon [56] improved the feedback control scheme by using a PD-
controller instead of the P-controllers used in [17]. In contrast to Cannon and Schmitz's
work, they estimated the transfer-function between the torque and the end-point of the
flexible manipulator off-line using R L M S algorithm. Once the estimated parameters
converged, the estimated model was then used to derive the control output of the P D
controller using an adaptive Linear Quadratic Gaussian algorithm. Their experimental
results showed that the performance of their adaptive PD-controller was better than the
fixed gain P-controllers.
Similar to Rovner and Cannon [56], Kotnik et al [34] also identified the poles and zeros
of the transfer-function from the motor input current to the end-point position of a
flexible manipulator arm off-line. The model was then used to derive a control gain for a
P-controller where the acceleration of the end-point was used as feedback control signal.
In addition to the acceleration feedback loop, they also employed a shaft position
feedback loop in order to control the rigid body motion of the manipulator arm. Both
measured acceleration and position signals were filtered by a Butterworth filter before
being fed into the controller.
144
Chapter 6. Active control of nonlinear vibration in the flexible cantilever beam
In contrast to other work [17, 34, 56, 59], Yurkovich and Pacheco [78] developed a
controller tuning method for a flexible manipulator arm carrying an unknown and
varying payload. Similar to Rovner and Cannon's work [56], the manipulator arm was
represented in a A R M A model. The parameters of the model were estimated both on
line and off-line using R L M S algorithm. To control the end-point position, they
employed a cascaded control scheme, where both the end-point acceleration and shaft
motor angle signals were fed back to PID-controller. In order to compensate for the
payload at the end-point, the gains of the PID-controller were automatically tuned
corresponding the payload.
Although considerable work on active control of flexible manipulators have been
carried out over the past decades, none has dealt with nonlinear vibration generated in
the flexible manipulators. Most of them used linear models which are not suitable for
cases when significant nonlinearities exist in the system.
In Chapters 1-3, it has been shown experimentally that the tested cantilever beam
exhibited numerous nonlinear phenomena that are commonly observed in many flexible
structures. These include shift of the resonance frequency, jump phenomena, energy
transfer from higher order modes to lower order modes, modal coupling, frequency
modulation, hysteric damping and nonlinear stiffness. A m o n g these nonlinear
characteristics, modal coupling is the issue of greatest concern for flexible structures.
In contrast to other work, this research aims to cancel the nonlinear low frequency
vibration, generated in the tested flexible cantilever beam, due to nonlinear modal
coupling, while the high frequency vibration, due to the reference signal, remains
unattenuated. The work documented in Chapter 2 has led to a useful concept: cancelling
the low frequency vibration generated in the flexible cantilever beam, due to nonlinear
coupling between the modes, by feeding the low frequency vibration back to the system.
This chapter describes the design and implementation of a feedback control scheme, in
a dSpace™ Digital Signal Processor, for cancelling the nonlinear vibration in the beam.
It has been demonstrated experimentally that the feedback controller was capable of
145
Chapter 6. Acrive control of nonlinear vibration in the flexible cantilever beam
cancelling the low frequency vibration generated in the flexible cantilever beam due to
nonlinear interaction between the modes of the beam. The results obtained were
excellent and represent a significant advancement in the field of active nonlinear
vibration control.
6.2 CONTROL STRATEGY
Since the aim of this work was to deal only with the nonlinear vibration generated in the
beam, the beam needed to be excited close to one of the resonance frequencies of a
higher order mode (such as the second or third order modes) in such a way that the beam
was coupled with the first order mode.
For an excitation input of A cos cot (where co is close to the resonance frequency of the
second order mode) the displacement measured at the tip of the beam can be
approximated as
y(t) = B, coscot + B2 cos2c0jt + B3 cos3co,t
+ B 4 cos(cot + (p) + B 5 cos 2cot + B 6 cos3cot (6.1)
+ B 7 cos(oo - co, )t + B 8 cos(a> + co, )t
where oo, is the resonance frequency of the first order mode,
q> is the phase difference between the input and output at the driven frequency.
As shown in Eq.(6.1), the response of the beam contained two parts: (a) a linear
response to the excitation frequency and (b) a nonlinear response which comprised the
harmonic components with different frequencies to the excitation frequency. These
harmonic components are defined as the remmants of the system.
The sinusoidal describing function, which is defined as the relationship between the
input and the fundamental component of the output at the excitation frequency [82], is:
146
Chapter 6. Active control of nonlinear vibration in the flexible cantilever beam
HN(co) = B 4 cos(cot + (p)
A cos cot (6.2)
This describing function HN(co) includes the transfer function of the shaker, power
amplifier, accelerometer, conditioning amplifier and the beam at co.
Hence, the remmants of the system are:
TX , , B, coseo,t + B, cos2co,t + B, cos3co,t + B,cos2cot+ •••+B8cos(co + co,)t HR(co) = — - ! - 5 — .(6.3)
A cos cot
From the experimental results, it was observed that the magnitudes B5 and B6 are much
smaller than Bi and B4, they could be ignored. Hence, Eq.(6.3) can be rewritten as
B, cosco,t + B 2 cos2co,t + B 3 cos3(D,t + B 7 cos(co - oo, )t + B 8 cos(co + co, )t HR(co) = A cos cot
(6.4)
The open loop "frequency response" between the input and output can be expressed as
H0(co) = HN(co) + HR(co). (6.5)
Figure 6.1 shows a feedback control scheme for cancelling nonlinear vibration of the
flexible cantilever beam.
Reference signal
Figure 6.1 A block diagram for a closed-loop transfer-function of the flexible beam
with feedback controller.
147
Chapter 6. Active control of nonlinear vibration in the flexible cantilever beam
The purpose of the feedback controller is to cancel the nonlinear low frequency
vibration while the high frequency vibration due to the reference signal remained
unattenuated. In other word, the feedback controller was designed in such a way that the
relationship between the input and output of the closed-loop system at co became:
Hclosed(CD) = — g o _ = B4cos(a)t + (p2) closed v i _ H 0 H c A cos cot
Hence, the relationship between the input and output of the controller for the excitation
frequency co2 is:
Hc((D) = JW Ho((0) H L ( G»
B, cosco,t + B 2 cos2co,t + B 3 cos3oo,t + B 7 cos(co - co, )t + B 8 cos(co + co, )t 1
B, cosco,t+-+B8cos(co + oo,)t HL(co)
(6.6)
As can be seen in Eq.(6.6), the feedback controller comprised two parts. The first part is
a set of bandpass filters in parallel with central frequencies at coi, 2coi, 3coi, (co-coO and
(co+coi). The purpose of these filters is to select the vibration frequency components
generated in the beam, which are to be cancelled. However, because of the nature of the
nonlinearity, only the first harmonic of the first order mode, ©i, need to be cancelled.
Once the first harmonic is cancelled, the other higher harmonics 2coi, 3coi, and
sidebands (co-coO, (co+C0i) will be consequently cancelled. The set of bandpass filters
can be reduced to one single bandpass or lowpass filter with cut-off frequency at coi.
The output of the filter has an opposite phase to the cancelled frequency components.
The second part of the controller is the inverse relationship between the reference signal
and the output of the beam at oo.
148
Chapter 6. Active control of nonlinear vibration in the flexible cantilever beam
6.3 EXPERIMENTAL SET-UP AND RESULTS
Figure 6.2 shows a schematic diagram of the experimental set-up for cancellation of
nonlinear vibration in the flexible cantilever beam with feedback controller. The beam
was initially excited at 70.3 H z with an excitation amplitude of 0.7V. The displacement
at the tip of the beam was measured using the P C B accelerometer. Both the reference
signal and the measured signal were passed through a conditioning amplifier and fed to
the D S P via a A/D converter and used for system identification of the system for that
particular excitation frequency and amplitude. The system identification could be done
off-line providing that the reference signal and the gains of the power amplifier and
condition amplifier were fixed; otherwise, it had to be done on-line. In this experiment,
the linear frequency response of the shaker, power amplifier, accelerometer and
conditioning amplifier was estimated off-line whereas the frequency response of the
beam was estimated on-line.
In the process of cancellation, the measured signal was filtered by a digital lowpass filter
in order to select only the response of the low frequency component. The filtered signal
was then passed through a compensator. Both the lowpass filter and the compensator
were implemented in the DSP. The compensator was used to compensate for the
frequency response of the beam, the accelerometer (including the conditioning
amplifier), the power amplifier and the shaker. The control output was a summation of
the output of the compensator and the reference signal. The control output was then fed
to the power amplifier via a D/A converter in order to drive the shaker to cancel the low
frequency vibration generated in the beam.
The controller, including the lowpass filter and compensator, was expressed in state-
space form as follows:
xi =-125.7x,-3947.8x2+u
X2 =x,
X3 =x4
149
Chapter 6. Active control of nonlinear vibration in the flexible cantilever beam
• _ 3947.84K 1 b X4 — X2 X3 X4
a a a y = 3947.84Kx2-2bx4
where x,, x2, x3 and x4 are the states of the controller; u is the input of the controller;
y is the output of the controller; and a, b and K are the parameters which were selected
to obtain the optimum cancellation. In other words, to minimise the error between the
response of the beam to the controller output and the low frequency vibration induced in
the beam due to the nonlinear modal coupling. It has been shown in Chapter 1 that Bi
and C0i were dependent on the excitation frequency, and they were also changing
randomly with time. Parameters a, b and K were, therefore, varied dependent on the
excitation frequency. While a and b determined the phase of the controller output, K
provided the gain of the controller. The higher the value that K had, the more sensitive
the system became. In order to ensure stability and experimental safety, the values of a,
b and K were allowed to vary between the minimum and maximum range. This
minimum/maximum range was determined off-line using the estimated response of the
beam, as described in Chapter 5.
Power Amplifier
i k
Shaker
* k i
Beam A i Accele
r
Conditioning Amplifier
Function Generator
'
romeier
-* A/D -• Lowpass Filter
— * • Compensator
Reference
J* D/A
D S P board
Figure 6.2 A functional block diagram of the experimental set-up for the cancellation of
nonlinear vibration in the flexible cantilever beam using feedback controller.
150
Chapter 6. Active control of nonlinear vibration in the flexible cantilever beam
Figure 6.3 shows the response measured at the tip of the beam with and without
feedback control, when the beam was excited at 70.3 Hz. It can be seen that the low
frequency vibration was almost attenuated while the high frequency vibration, due to the
reference signal, remained constant. Similarly, the auto spectra of the responses with
and without control were plotted as shown in Figure 6.4. The figure shows a significant
reduction of the vibration of the first order mode (approximately 50 dB) as well as
removing all the sub-harmonic and coupling frequency components.
The control scheme was also used to cancel the low frequency vibration generated when
the beam was excited at its second order mode (23.5 Hz). Figures 6.5 and 6.6 show the
response measured at the tip of the beam, with and without feedback control, in the time
and frequency domains, respectively.
0 0.1 0.2 0.3 0.4 0.5 0.6 Time [seconds]
1 1 r
(b) with control
j i 1 1
0 0.05 0.1 0.15 0.2 Time [seconds]
Figure 6.3 Time response measured at the tip of the beam: (a) without control, (b) with
control.
CO
% 0.5
g--0.5 5
1 o a>
151
Chapter 6. Active control of nonlinear vibration in the flexible cantilever beam
-20
-120
with control without contra
20 40 60 Frequency [Hz]
80 100
Figure 6.4 The auto-spectra measured at the tip of the beam with and without control for
an excitation frequency of 70.3 Hz.
0.3 0.4 0.5 Time [seconds]
0.05 0.1 Time [seconds]
0.15
Figure 6.5 Time response measured at the tip of the beam: (a) without control, (b) with
control for an excitation frequency of 24 Hz .
152
Chapter 6. Active control of nonlinear vibration in the flexible cantilever beam
10 20 30 Frequency [Hz]
Figure 6.6 The auto-spectra measured at the tip of the beam with and without control for
an excitation frequency of 24 Hz.
In summary, the results show that the feedback controller was capable of cancelling the
nonlinear low frequency vibration induced in the beam when the beam was excited with
an excitation frequency close to or at the resonance frequency of one of the higher order
modes. However, it was observed during the experiment that the response of the beam
without control was nonlinear and very sensitive to a large step change in excitation
input. It was, therefore, necessary to ensure that the power amplifier to the shaker is
turned off when changing the excitation input from 24 H z to 69 Hz, or vice versa, even
when the system was in the control mode.
6.4 CONCLUSIONS
It has been demonstrated experimentally that the on-line feedback controller was
capable of cancelling the low frequency vibration induced in the beam, due to nonlinear
interaction between the modes of the beam. This control scheme is relatively simple and
has considerable potential for cancelling nonlinear vibration in large structures such as
aircraft, ships, etc. in order to reduce stress and fatigue in these structures.
153
Chapter 7. Summary and conclusions
Chapter 7
SUMMARY AND FUTURE WORK
7.1 SUMMARY
Chapter 1 showed various experimental set-ups of the spring steel beam that were used
in the course of the preliminary investigation into the nonlinear dynamics of a slender
and flexible cantilever beam, for different excitation methods and beam orientations.
The preUminary experimental results have shown that the flexible cantilever beam had
consistent nonlinear behaviour independent of its orientation, excitation source and the
accelerometer mass loading on the beam. The nonlinear phenomena observed during the
experiment were change of resonance frequency, jump phenomenon, energy transfer
from higher order modes to lower order modes, nonlinear stiffness, hysteretic damping,
modal coupling and frequency modulation.
The degree of nonlinearity is dependent upon the nonlinear relationship between the
acceleration applied at the clamped end and the displacement measured at the tip of the
beam, ie. the stiffness of the beam. The experimental results have shown that the beam
stiffness is different for each mode. At the first order mode, the beam has a hardening
stiffness characteristic. However, the stiffness characteristic changes from hardening to
softening when the beam is subjected to large bending at higher order modes. The
change in the beam stiffness corresponds to the change in resonance frequencies of the
modes with increasing excitation amplitude. The resonance frequency of the first order
mode was observed to increase with increasing excitation amplitude, whereas the
resonance frequencies of the second and third order modes decreased with increasing
excitation amplitude. Increasing and decreasing the resonance frequency can be
quantitatively related to the increasing and decreasing the kinetic energy of the free
154
Chapter 7. Summary and conclusions
vibrating mode — mcor2A2 ]. Increasing the input amplitude causes an increase of the
kinetic energy. For the first mode, it is not possible to transfer energy to other higher
order modes. The beam can only increase the resonance frequency to store more kinetic
energy if the modal amplitude cannot be further increased. O n the other hand, the higher
order modes use the nonlinear coupling to transfer energy when the modal amplitude
can no longer be increased. It appears that the higher order modes even reduce the
kinetic energy, by reducing the resonance frequency, to maintain energy transfer to the
lower order modes.
In addition to the change in the resonance frequency of the beam, nonlinear modal
interactions were observed. W h e n the beam was excited at one of the higher order
modes and reached m a x i m u m deformation, the beam then started to couple to the first
order mode. This was due to the energy cascading from the higher order modes to the
lower order modes, right down to the first order mode. During the energy transfer, the
magnitudes of the resonance peaks were changing continuously, corresponding to their
frequency shift. As a result, jump phenomena were observed (multiple values of
magnitude obtained for a given excitation frequency). In addition, hysteresis (multiple
values of magnitude obtained for a given excitation amplitude) was also observed.
W h e n the magnitude of the first order mode was sufficiently large, the frequency of the
first order mode was modulated with higher frequency components and subsequently
created sidebands. These nonlinear observations were useful for the development of the
nonlinear model of the cantilever beam. In order to ensure that the nonlinear model
could be applied to any slender and flexible spring steel cantilever beam, a similar
investigation was carried out for three different sizes of beam. Although the three
different beams were slender and flexible, they all had different resonance modes due to
different ratios between their thickness and their length. However, they all exhibited
similar nonlinear responses.
Chapter 2 examined the behaviour of the beam when two or more modes of the beam
were excited. Both nonperiodic and periodic signals were used for linear and nonlinear
multi-modal excitation. The experimental results have shown that the type of signal
155
Chapter 7. Summary and conclusions
(whether the signal is periodic or nonperiodic) has no effect on the response of the
beam. However, for the case of linear multi-modal excitation, it was observed that the
magnitude of first order mode decreased, ie. the stiffness of the first order mode
increased, with the excitation of higher order modes. This was due to energy transfer
from lower order modes to higher order modes with linear excitation. In contrast, in the
case of nonlinear excitation the energy is transferred from higher order modes to lower
order modes. Observations of the behaviour of the cantilever beam, when it is subject to
multi-modal excitation, have identified two useful concepts for the development of an
active control algorithm for nonlinear vibration cancellation in the flexible beam. These
are: (i) using the low frequency vibration, resulting from nonlinear coupling from higher
order modes, to cancel the low frequency vibration; (ii) increasing the stiffness of the
first order mode by exciting the beam at higher order modes with small excitation
amplitude.
Chapter 3 investigated the damping of the first three modes of the beam for single, as
well as multi-frequency excitation, in both linear and nonlinear cases. The experimental
results have shown that all of the modes decayed independently of each other and at
different rates - the higher order modes had a faster decay rate than the lower order
modes. In addition, the larger the amplitude of the displacement was, the faster the
magnitude decayed. Because the decay rate of each mode was proportional to both the
amplitude and frequency of the vibration, the damping of each mode of the beam can be
modelled as a combination of Viscous and Quadratic damping in the linear case. In the
nonlinear case, when nonlinear modal coupling occurred, the magnitude of the first
order mode (due to coupling from the higher order modes) did not decay until the
magnitude of the excited higher order modes decayed to the decoupling threshold level.
W h e n the magnitude of the excited higher order modes reached the decoupling
threshold value, the magnitude of the first order mode then proceeded to decay in a
similar way as in the case of linear excitation. Hence, the beam also exhibited Hysteretic
damping characteristics above the coupling/decoupling threshold in addition to Viscous
and Quadratic damping. However, as soon as the magnitude of the displacement
decayed to the decoupling threshold level, the beam then exhibited only Viscous and
Quadratic damping.
156
Chapter 7. Summary and conclusions
Chapter 4 described the development process of the nonlinear model for the flexible
cantilever beam. In contrast to other work, the development of the nonlinear model was
not only based on nonlinear theory, but primarily on experimental observation and
understanding of the nonlinear behaviour of the beam. In the process of developing the
nonlinear model, nonlinear theory of the beam was applied and then modified
corresponding to the experimental results. In the process of validating the model, the
nonlinear model, developed in state-space form, was implemented in C + + and down
loaded into the DSP. A third/fourth order Runga-Kutta method with a step size of
2.0x10"4 was used for integration. It has been demonstrated experimentally that the
output of the model exhibited all the nonlinear behaviour of the flexible cantilever
beam. Although the simulation results and the experimental results had different
magnitude scales, they both had the same nonlinear behaviour patterns. This nonlinear
model can easily be converted to an Auto-Regressive Moving Average ( A R M A ) model
which can be used to predict the response of the beam on-line, using the conventional
linear Least Mean Square (LMS) algorithm.
In chapter 5, the nonlinear state-space model of the beam was converted to an ARMA
model using the Euler integration method. The parameters of the A R M A model were
then easily estimated using the conventional linear L M S algorithm. With this new
identification scheme, each mode of the beam was identified separately and then added
together to obtain an estimated displacement of the beam. The method is conceptually
simple and requires only a small number of estimated parameters, but it is still able to
achieve high performance in both linear and nonlinear cases. Although the nonlinear
model of the cantilever beam, used for identification, was originally developed for the
case of horizontal excitation, the model was still applicable to the case of rotational
excitation. It has been demonstrated experimentally that the developed identification
method is much more advanced than the IIR and the third order Volterra FIR filters. It
works well in the nonlinear case as well as the linear one, whereas both the IIR filter and
Volterra FIR filter failed to estimate the low frequency component in the case of
nonlinear modal coupling, hi addition, the developed identification method has faster
157
Chapter 7. Summary and conclusions
convergence speed, is more accurate and requires less computational time compared to
the other identification methods.
Chapter 6 describes the design and implementation of a feedback control scheme, in a
dSpace™ Digital Signal Processor, for cancelling the nonlinear vibration in the beam. It
has been demonstrated experimentally that the feedback controller was capable of
cancelling the low frequency vibration generated in the flexible cantilever beam due to
nonlinear interaction between the modes of the beam. This control scheme is simple and
has considerable potential for cancelling nonlinear vibration in large structures such as
aircraft, ships, etc. in order to reduce stress and fatigue failure in these structures.
7.2 RECOMMENDATIONS FOR FUTURE WORK
A number of recommendations for future work can be made as a result of this research.
In this research, only the clamped-free end condition was investigated. Relevant areas of
investigation would include the nonlinear behaviour of the beam for different boundary
conditions, including clamped-clamped, free-free, clamped-simply supported ends.
Since the nonlinear model of the cantilever beam used for identification, which was
originally developed for the case of horizontal excitation, worked well in the case of
rotational excitation, it is recommended that the developed identification scheme be
applied to beams with different boundary conditions other than clamped-free ends.
In Chapter 2, the experimental results have shown that the energy transfers from lower
order modes to higher order modes in the linear case. This is contradictory to the
traditional understanding of the linear response of the single degree of freedom system.
A further investigation on multi-frequency excitation of single as well as multi-degree
of freedom systems is suggested.
Chapter 3 examined the damping characteristics of the beam in air. As the air in the
room can influence the damping characteristics, it is recommended to investigate the
damping characteristics of the beam when the beam is isolated in a vacuum. In this way,
158
Chapter 7. Summary and conclusions
the effect of the structural damping and the air damping on the system can be separated
from each other.
The nonlinear model of the beam described in Chapter 4 was developed for system
identification. The parameters of the model were estimated on-line in order to predict
the response of the beam in both the linear and nonlinear cases. However, it would be
useful if the parameters of the model could be exactly derived using the beam constants
and mode shapes.
It has been demonstrated experimentally that the feedback controller was capable of
cancelling low frequency vibration, generated in the flexible cantilever beam due to
nonlinear interaction between the modes of the beam. However, in some cases, where
the reference signal needs to be attenuated, a feedforward controller is preferable,
especially for high frequency vibration. A further recommendation for future work
includes the implementation of a feedforward controller using the developed
identification (NMI) method to derive the controller output. The output of the controller
then drives a piezo-ceramic actuator as a cancelling source. Because the beam is
flexible, the mass of the piezo-ceramics actuator would lower the resonance frequency
of the beam which, in turn, requires a more powerful piezo-ceramic actuator. In practice,
a thin piezo-ceramic actuator may not provide enough power to cancel the low
frequency vibration. In order to overcome this problem, a high voltage piezo driver is
recommended to excite the piezo-ceramic actuator.
Finally, this work lays the foundation for future development of a combined
feedforward and feedback controller. Such a hybrid scheme would combine the
advantages of both feedforward and feedback schemes to produce a comprehensive
controller. In this proposed system, the feedforward controller would handle the
cancellation of high frequency vibration using the piezo-ceramic actuator, while the
feedback controller would drive the shaker (also used as a reference source) to cancel
low frequency vibration.
159
REFERENCES
1. Al-Bedoor, B.O. and Khulief, Y.A., "Vibrational motion of an elastic beam with
prismatic and revolute joints", Journal of Sound and Vibration, 190(2), 1996, 195-
206.
2. Anderson, T.J., Balachandran, B., and Nayfeh, A.H., "Nonlinear resonances in a
flexible cantilver beam", Journal of Vibration and Acoustics, 16,1994,480-484.
3. Anderson, T.J., Nayfeh, A.H. and Balachandran, B., "Coupling between High-
Frequency Modes and a Low-Frequency Mode: Theory and Experiment, Nonlinear
Dynamics, 11,1996,17-36.
4. Anderson, T.J., Nayfeh, A.H. and Balachandran, B., "Experimental Verification of
the Importance of the Nonlinear Curvature in the Response of a Cantilever Beam",
Journal of Vibration and Acoustics, Vol. 118, Janurary 1996,21-27.
5. Atluri, S., "Nonlinear vibrations of a hinged beam including nonlinear inertia
effects", Jounral of Applied Mechanics, 40,1973,121-126.
6. Baik, H.K. and Mathews, V.J., "Adaptive Lattice Bilinear Filters", IEEE
Transactions on Signal Processing, Vol. 41, No. 6, June 1993,2033-2046.
7. Balachandran, B. and Nayfeh, A.H., "Identification of Nonlinear Interactions in
Structures", Journal of Guidance, Control and Dynamics, Vol. 17, No. 2, March-
April 1994, 257-262.
8. Benamar, R. and Bennouna, M.M.K., "The effects of large vibration amplitudes on
the mode shapes and natural frequencies of thin elastic structures - Part I: Simply
supported and clamped-clamped beams", Journal of Sound and Vibration, 149(2),
1991, 179-195.
9. Bennett, J. A., "A multiple degree of freedom approach to nonlinear beam
vibrations", American Institute of Aeronautics and Astronautics Journal, 8, 1970,
734-739.
10. Bennett, J.A. and Eisley, J.G., "Stability of large amplitude forced motion of a
simply supported beam", International Journal of Nonlinear Mechanics, 5, 1970,
645-657.
160
11. Bennouna, M.M. and White, R.G, "The effect of large vibration amplitudes on the
fundamental mode shape of a clamped-clamped uniform beam", Journal of Sound
and Vibration, 96,1984, 309-331.
12. Berdichevsky, V.L. and Kim, W.W., "Dynamical potential for non-linear vibrations
of cantilevered beams", Journal of Sound and Vibration, 179(1), 1995,151-164.
13. Bhat, R.B., "Effects of normal mode contents in assumed deflection shapes in
Rayleigh-Ritz method", Journal of Sound and Vibration, 189(3), 1996,407-419.
14. Birman, V., "On the Effects of Nonlinear Elastic Foundation on Free Vibration of
Beams", ASME Journal of Applied Mechanics, 53,1986,471-473.
15. Bruch, J.C. and Mitchell, T.P., "Vibrations of a mass-loaded clamped-free
Timoshenko beam", Journal of Sound and Vibration, 114(2), 1987, 341-345.
16. Burton, T. D. and Hamdan, M . N., "On the calculation of non-linear normal modes
in continuous systems", Journal of Sound and Vibration, 197,1996,117-130.
17. Cannon, R. H, Jr. and Schmitz, E., "Initial Experiments on the End-Point Control of
a Flexible One-Link Robot", The International Journal of Robotics Research, 3(3),
1984, 62-75.
18. Crespo Da Silva, M . R. M. and Zaretzky, C. L., "Nonlinear modal coupling in
planar and non-planar responses of inextensional beams", International Journal of
Nonlinear Mechanics, 25, 1990, 227-339.
19. Cusumano, J.P. and Moon, F.C., "Chaotic non-planar vibrations of the thin elastica.
Part 1: Experimental observation of planar instability", Journal of Sound and
Vibration, 179, 1995,185-208.
20.Diniz, P.S.R. and Biscainho, L.W.P., "Optimal Variable Step Size for the
LMS/Newton Algorithm with Application to Subband Adaptive Filtering", IEEE
Transactions on Signal Processing, Vol. 40, No. 11, November 1992,2825-2829.
21. Dugundji, J. and Mukhopadhyay, "Lateral bending-torsion vibrations of a thin beam
under parametric excitation", Journal of Applied Mechanics, Vol. 40, 1973, 693-
698.
22. Elliot, S.J., Stothers, LM. and Billet, L., "Adaptive feedforward control of flexural
wave propagating in a beam", Proceedings of the Institute of Acoustics, 12(1), 1990,
613-622.
161
23. Eringen, A.C., "On the non-linear vibration of elastic bars", Quarterly of Applied
Mathematics, Vol.9, No.4,1952, 361-369.
24. Evans, J.B. and Liu, B., "Variable Step Size methods for the L M S adaptive
algorithm, Proceedings of IEEE Int. Symp. Circuits Systems, May 1987,422-425.
25. Evensen, D. A., "Nonlinear Vibrations of Beams with Various Boundary
Conditions", AIAAA Journal, 6,1968,371-372.
26. Farhang-Boroujeny, B., "Fast LMS/Newton Algorithms Based on Autoregressive
Modeling and Their Application to Acoustic Echo Cancellation", IEEE
Transactions on Signal Processing, Vol. 45, No. 8, August 1997,1987-2000.
27. Haddow, A. G., Barr, A. D. S. and Mook, D. T., "Theoretical and Experimental
Study of Modal Interaction in A Two-Degree-Of-Freedom Structure", Journal of
Sound and Vibration, 97,1984,451-473.
28. Harris, R.W., Chabries, D.M. and Bishop, F.A., "A Variable Step (VS) Adaptive
Filter Algorithm", IEEE Transactions on Acoustics, Speech and Signal Processing,
Vol. ASSP-34, No.2, April 1986,309-316.
29. Hoa, S. V, "Vibration of a rotating beam with tip mass", Journal of Sound and
Vibration , 1979,67, 369-381.
30. Hsia, T.C., "System Identification", Lexington Books, D.C. Health and Company,
Lexington, 1977.
31. Hu, K.K, and Kirmser, P.G., "On the Nonlinear Vibrations of Free-Free Beams",
Journal pf Applied Mechanics, Transactions of the A S M E , June 1971,461-466.
32. Kane, T. R., Ryan, R. R., and Banerjee, A. K., "Dynamics of a Cantilever Beam
Attached to a Moving Base", AIAA Journal of Guidance, Control and Dynamics,
Vol. 10, No. 2, 1987,139-151.
33. Kami, S. and Zeng, G., "A N e w Convergence Factor for Adaptive Filters", IEEE
Transactions on Circuits and Systems, Vol. 36, No. 7, July 1989,1011-1012.
34. Kotnik, P.T., Yurkovich, S. and Ozguner, U., "Acceleration Feedback for Control of
a Flexible Manipulator Arm", Journal of Robotic Systems, 5(3), 1988, 181-196.
35.Kwang, H.K. and Li, Q.P., "New Nonlinear Adaptive FIR Digital Filter for
Broadband Noise Cancellation", IEEE Transactions on Circuits and Systems- II:
Analog and Digital Signal Processing, Vol.41, No. 5, May 1994, 355-360.
162
36.Kwong, R. and Johnston, E., "A Variable Step Size L M S Algorithm", IEEE
Transactions on Signal Processing, Vol. 40, No. 7, July 1992,1633-1642.
37. Lewandowshi, R., "Application of the Ritz method to the analysis of non-linear
free vibrations of beams", Journal of Sound and Vibration, 114(1), 1987,91-101.
38.Lim, T.W., Bosse, A. and Fisher, S., "Adaptive Filters for Real-Time System
Identification and Control", Journal of Guidance, Control and Dynamics, Vol. 20,
No. 1, January-February 1997,61-66.
39. Marcos, S. and Macchi, O., "Tracking Capability of the Least Mean Square
Algorithm: Application: Application to an Asynchronous Echo Canceller", IEEE
Transactions on Acoustics, Speech and Signal Processing, Vol. ASSP-35, No.ll,
November 1987,1570-1578.
40. Mathews, V.J. and Xie, Z., "A stochastic Gradient Adaptive Filter with Gradient
Adaptive Step Size", IEEE Transactions on Signal Processing, Vol. 40, No. 6, June
1993,2075-2087.
41. McDonald, P. H., Jr. and Raleigh, R.C., "Nonlinear Dynamic Coupling in a Beam
Vibration", Journal of Applied Mechanics, 22, T R A N S A S M E , Vol. 77, 1955, 573-
578.
42. McLachlan, N.W., "Ordinary non-linear differential equations in engineering and
physical sciences", second edition, Oxford University Press, London, 1956
43. Meirovitch, L. amd Baruh, H., " Nonlinear Natural Control of an Experimental
Beam", Journal of Guidance, Control, and Dynamics, Vol. 7, No.4,437-442.
44. Mikhael, W.B., W u , F. H., Kazovsky, L.G., Kang, G.S. and Fransen L., "Adaptive
Filters with Individual Adaptation of Parameters", IEEE Transactions on Circuits
and Systems, Vol. CAS-33, No. 7, July 1986, 677-685.
45. Nayfeh, A.H and Mook, D.T., Nonlinear Oscillations, Wiley, Newyork, 1979.
46. Nayfeh, A.H. and Pai, P.H., "Nonlinear non-planar parametric response of an
inextensional beam", International Journal of Nonlinear Mechanics, 24, 1989, 139-
158.
47. Nayfeh, A.H., Chin, C. and Nayfeh, S.A., "Nonlinear Normal Modes of a Cantilever
beam", Journal of Vibration and Acoustics, No\. 117, October 1995,447-481.
163
48. Nayfeh, S.A. and Nayfeh, A.H., "Energy Transfer From High to Low-Frequency
Modes in a Flexible Structure via Modulation", Journal of Vibration and Acoustics,
Vol. 116, April 1994,203-297.
49. Pai, P.H. and Nayfeh, A.H., "Nonlinear non-planar oscillations of a cantilever beam
under lateral base exciations", International Journal of Nonlinear Mechanics, 25,
1990,227-339.
50. Rao, B. N. and Rao, G. V., " Large amplitude vibrations of clamped-free and free-
free uniform beams", Journal of Sound and Vibration, 134,1989,353-358.
51. Rao, B. N., Shastry, B.P. and Rao, G. V., " Large deflections of a cantilever beam
subjected to a tip concentrated rotational load", Aeronautical Journal,
August/September 1986,262-266.
52. Rao, G. V., Raju, K. K. and Raju, I. S., "Nonlinear vibrations of beams considering
shear deformation and rotary intertia", Journal of American Institute of Aeronautics
and Astronautics, 14,1976,685-687.
53. Rao, J.S. and Carnegie, W . "Nonlinear vibrations of rotating cantilever beams",
Aeronautical Journal of the Royal Aeronautical Society, 1970,161-165.
54. Ray, J.D. and Bert, C.W., "Nonlinear Vibrations of a Beam With Pinned Ends",
Journal of Engineering for Industry, Transactions of the A S M E , November 1969,
997-1004.
55. Redman-While, W., Nelson, P.A. and Curtis, A.R.D., "Experiments on active
control of flexural wave power", Journal of Sound and Vibration, 112, 1987, 181-
187.
56. Rovner, D.M. and Cannon, R. H, Jr., "Experiments Toward On-Line Identification
and Control of a Very Flxible One-Link Manipulator", The International Journal of
Robotics Research, 6(4), 1987,3-19.
57. Rupp, M., "Saving Complexity of Modified Filtered-X L M S and Delayed Update
L M S Algorithms", IEEE Transactions on Circuits and Systems- II: Analog and
Digital Signal Processing, Vol.44, No. 1, January 1997, 57-60.
58. Saito, H, Sato, K. amd Yutani, T., "Nonlinear forced vibrations of a beam carrying
concentrated mass under gravity", Journal of Sound and Vibration, 46, 1976, 515-
525.
164
59. Sakawa, Y., Matsuno, F. and Fukushima, S., "Modeling and Feedback Control, of a
Flexible Arm", Journal of Robotic Systems, 2(4), 1985,453-472.
60. Sarma, B.S. and Varadan, T.K., "Lagrange-type formulation for finite element
analysis of non-linear beam vibrations", Journal of Sound and Vibration, 86(1),
1983,61-70.
61. Shaw, S.W and Pierre, C , "Normal modes of vibration for non-linear continuous
systems", Journal of Sound and Vibration, 169,1994, 319-348.
62. Shaw, W., "Chaotic dynamics of a slender beam rotating about its longitudinal
axis", Journal of Sound and Vibration, 124,1988,329-343.
63. Snyder, S.D. and Hansen, C.H., "The influence of transducer transfer functions and
acoustic time delays on the implementation of the L M S algorithm in Active Noise
Control systems", Journal of Sound and Vibration, 140(3), 1990,1-16.
64. Takahashi, K., "A method of stability analysis of non-linear vibration of beams",
Journal of Sound and Vibration, 67,1979,43-54.
65. Takahashi, K., "Non-linear free vibration of inextensible beams", Journal of Sound
and Vibration, 64,1979, 31-34.
66. Tan, L. and Jiang, J., "Filtered-X second-order Volterra adaptive algorithms",
Electronics Letters, Vol. 33, No. 8,10th April 1997, 671-672.
67. To, C. W.S., "Vibration of a cantilever beam with base excitation and tip mass",
Journal of Sound and Vibration, 83,1982,445-460.
68. Tseng, W . Y. and Dugundji, J., "Nonlinear vibrations of a buckled beam under
harmonic excitation", Journal of Applied Mechanics, 38,1971,467-476.
69. Tseng, W.Y. and Dugundji, J., "Nonlinear vibrations of a beam under harmonic
excitation", Journal of Applied Mechanics, 37, 1970,292-297.
70. Vipperman, J.S., Burdisso, R A . and Fuller, C.R., "Active control of broadband
structural vibration using the L M S adaptive algorithm", Journal of Sound and
Vibration, 166(2), 1993,283-299.
71. Von Flowtow, A.H. and Schafer, B., "Wave-absorbing controller for a flexible
beam", Journal of Guidance, Control and Dynamics, 9,1986, 673-680.
72. Wagner, H., "Large-amplitude free vibration of a beam", Journal of Applied
Mechanics, 32, 1965, 887.
165
73. Wang, CM., Lam, K.Y., He,X.Q. and Chucheepsakul, "Large deflections of an end
supported beam subjected to a point load", International Journal of Nonlinear
Mechanics, Vol. 32, No.l, 1997, 63-72.
74. Wang, P. K. C. and Wei, J. D., "Vibrations in a moving flexible robot arm", Journal
of Sound and Vibration, 116,1987,149-160.
75. Widrow, B. and Steams, S.D.,"Adaptive Signal Processing", Englewood Cliffs,
NJ:Prentice-Hall, 1985.
76. Woinowsky- Krieger, S., "The effect of an axial force on the vibration of hinged
bars", Journal of Applied Mechanics, 17,1950, 35-36.
77. Yassa, F.F., "Optimality in the Choice of the Convergence Factor for Gradient-
Based Adaptive Algorithms", IEEE Transactions on Acoustics, Speech and Signal
Processing, Vol. ASSP-35, No.l, January 1987,48-59.
78. Yurkovich, S. and Pacheco, F.E., "On Controller Tuning for a Flexible-Link
Manipulator with Varying Payload", Journal of Robotic Systems, 6(3), 1989, 233-
254.
79. Zavodney, L.D. and Nayfeh, A.H., "The non-linear response of a slender beam
carrying a lumped mass to a principal parametric excitation: theory and experiment",
International Journal of Nonlinear Mechanics, Vol. 24, No.2,1989,105-125.
80. Hu, K.K, and Kirmser, P.G., "On the Nonlinear Vibrations of Free-Free Beams",
Journal pf Applied Mechanics, Transactions of the ASME, June 1971,461-466.
81. Kane, T. R., Ryan, R. R., and Banerjee, A. K., "Dynamics of a Cantilever Beam
Attached to a Moving Base", AIAA Journal of Guidance, Control and Dynamics,
Vol. 10, No. 2,1987,139-151.
82. Graham, D., "Analysis of nonlinear control systems", John Wiley & Sons, Inc.,
USA, 1961.
166
Appendix A Normal modes of uniform beams
APPENDIX A
NORMAL MODES OF UNIFORM BEAMS
The linear free flexural vibrations of a uniform beam are governed by Euler's
differential equation:
EId*W(x,t) + p Sd2W(x,t)
dx4 K dt2
where W(x,t) is the lateral deflection of the beam, E is Young's modulus, I is the area
inertia moment of the beam, p is the mass density and S is the cross sectional area.
Let the partial differential equation in Eq.(A.l) be satisfied by the functions of the form
W(x,t) = 0(x)f(t), (A.2)
where O(x) is a function of the space variable x alone and f(t) is a function of time, t,
alone.
Substituting Eq.(A.2) into Eq.(A.l) and separating the variables, we obtain
^ ^ ) - P 4 ( D ( X ) = 0 , (A3) dx 4
where B4 = P , and co corresponds to the resonance frequency when boundary H EI
conditions are applied.
167
Appendix A Normal modes of uniform beams
The general solution of Eq.(A.3) is
O(x) = C, sin px + C2 cospx + C3 sinh Px + C4 cosh px. (A.4)
The boundary conditions for a cantilever beam with a length L are
(1) W(0,t) = 0,
(2)W(0,t) = 0,
(3) W"(L,f) = 0,
(4) W"(L,f) = 0.
Substituting these boundary conditions into the general solution from Eq.(A.4), we
obtain the frequency equation
cospLcoshpL = -l, (A5)
and the mode shape function
cp(x) = A(cos px - cosh px) + (sin px - sinh px), (A.6)
sin pL + sinh pL cos pL + coshpL where A = — — — = . OT . r OT
cos pL + cosh pL sin pL - sinh PL
168
Appendix B Bandpass Filters
APPENDIX B
BANDPASS FILTERS
The transfer functions of the bandpass filters implemented in the D S P as shown in
Figure 2.1 are:
H,(s) = 2.8xl06s3
s6 + 150.8s5 + 94.7 x 102s4 + 31.7 x 1 0 V + 59.8 x 104s2 + 60.2 x 106s + 25.2 x 107
H2(s) = 3.8xl09s3
s6 +92.4 x 102s5 +35.6 x 104s4 + 73 x 1 0 V + 84.2 x 108s2 + 51.9 x 1010s +13.3 x 1012
2.8xl0ns3
HB(S) S6 + 26.4 x 1 0 V + 29 x 1 0 V +17 x 108s3 + 56.1 x 10,0s2 + 98.8 x 1012s + 72.4 x 1014
The centre frequencies of H,(s), H2(s) and H3(s) are 4 Hz, 24.5 H z and 70 Hz,
respectively.
169
APPENDIX C
VALUES OF THE PARAMETERS USED IN THE SIMULATION
The values of the parameters which were used in the simulation described in Section 4.4
are as follows:
InEq.(4.26b):
co, = 4, d, = 0.15, s, = 0.0002, p, = 10, a, =20.1, C2>12 = 1000, C3>12 = 55000,
C4,12 = 500, C5>13 = 1000, C W 3 = 55000, C6>13 = 500, b, = 10.
InEq.(4.26d):
co2 = 24.5, d2 = 1,82 = 0.0003, p2 = -500, a2 = -710, C4>21 = 500, C5>23 = 100,
C6>23= 1500, b2 = 200.
InEq.(4.26f):
co3 = 70.3, d3 = 7, e3 = 0.000002, p3 = -500, a3 = -1550, C7>,3 = 1500, b3 = 280.
InEq.(4.26g):
a, = 2, a3 = -2, a3 = 2.
170
UNIVERSITY OF W.A.
LIBRARY