MODELLING, IDENTIFICATION AND ACTIVE CONTROL OF … · and Timoshenko and Euler-Bernoulli beam theory, have for several decades been useful ... but primarily on experimental observation

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


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


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


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


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


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


4.1Hz

25 Hz

71.6 Hz

Figure 1.3

with accelerometer

3.9 H z

24.6 H z

69.8 H z


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


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


Figure 1.5 The beam pointing horizontally.

Figure 1.6 The beam pointing vertically down.

8


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


4.05 H z

24.8 H z

71.1 Hz

Figure 1.5

with accelerometer

4 Hz

24.6 Hz

69.6 Hz


4.1Hz

25 Hz

71.6 Hz

Figure 1.6

with accelerometer

4.1Hz

24.8 H z

70.3 H z


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


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


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


. 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


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


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


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


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


16


69.5 70 70.5 Excitation frequency [Hz]

Figure 1.12 Frequency response curves in the vicinity of the third order mode for


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


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


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


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


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


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


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


40 50

23


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


100

69 Hz

65.7 HA 72.3 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



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


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


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


x-axis of the beam [cm]

Figure 1.22 The first mode shape of the cantilever beam for different excitation

amplitudes.

27


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


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


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


, , 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


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]


0.1

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



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

_

_

-

-


0.1



33


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


E

g. CD

ra T3



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


0.1



35


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


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


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


Function Generator

Measured signal

Function Generator

Function Generator

Summing Amplifier

Power Amplifier

Shaker

1 .».•.•••. ••.*.•.'••.•

Beam Accelerometer

m ' ' • ' • ' ' • ' ' . ' • ' . ' • ' . ' . ' ' • ' • ' ' • ' . ' • ' • ' • ' • ' • ' • ' • ' • ' . '

Conditioning Amplifier


Bandpass Filter (at 4 Hz)


-^ Magnitude of 1st order mode


-• 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


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


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



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.


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


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


</> •*—«

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


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


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




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


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


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


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


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


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


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


53


0.1 Time [seconds]


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


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]


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


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


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


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



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



Figure 2.23 Magnitude of the first order mode with an excitation at: (a) 69 H z (0.35 V)




Figure 2.24 Magnitude of the first order mode with an excitation at: (a) 69 Hz (0.35V)


58


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


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


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


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


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


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


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


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


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


(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


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


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


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



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 .—..



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



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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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) = Ôi(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


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


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


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=-î^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


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 uî 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


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


(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


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


(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


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


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


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


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


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


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


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



Figure 4.3 The response of the model in the vicinity of the first order mode for



Figure 4.4 The response of the model in the vicinity of the second order mode for


107


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


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


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^yiî^yiAih,

28 Hz

J^W^(^M^M\L 10 20 30


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



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.4JiiUîîtoi*.Ûîii<Aj<i»hLiJ \tk

100

iMk*t&k\AlkM 20 40 60


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


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


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


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


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


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


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


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


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


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


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


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


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


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


x 5 k and x 6 k are the measured displacement and velocity signals of the third


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


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


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


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


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


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


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


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


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


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


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


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Time [seconds]


IIR filter for the linear case.

0.15 0.2 0.25 Time [seconds]

0.35 0.4


Volterra FIR filter for the linear case.

136


. 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


NMI method for the nonlinear case.

137


Time [seconds]


IIR filter for the nonlinear case.

Time [seconds]


Volterra FIR filter for the nonlinear case.

138


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


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


(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


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


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


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


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


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


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


• _ 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


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


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


-20

-120

with control without contra


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.


0.05 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



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


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


(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 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


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


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

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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.

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