active control of nonlinear forced vibration in a flexibl

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Active control of nonlinear forced vibration in aflexible beam using piezoelectric materialFeng-Ming Lia, Guo Yaob & Yimin Zhangba College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, Chinab School of Mechanical Engineering and Automation, Northeastern University, Shenyang110819, ChinaAccepted author version posted online: 18 Nov 2014.

To cite this article: Feng-Ming Li, Guo Yao & Yimin Zhang (2014): Active control of nonlinear forced vibration in a flexiblebeam using piezoelectric material, Mechanics of Advanced Materials and Structures, DOI: 10.1080/15376494.2014.981613

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ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT 1

Active control of nonlinear forced vibration in a flexible beam using piezoelectric material

Feng-Ming Li 1

, Guo Yao 2, Yimin Zhang

2

1. College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, China

2. School of Mechanical Engineering and Automation, Northeastern University, Shenyang

110819, China

Abstract: Geometric nonlinearities of the beam and piezoelectric patch are considered. Velocity

feedback control algorithm is implemented applying piezoelectric materials. The equation of

motion of the system is established using Hamiltons principle. The effects of control gains on

primary resonance properties of the beam are studied. It is observed that with the amplitude of

external excitation increasing, the amplitude of resonance curve increases. The velocity feedback

control can improve unstable resonance of the beam. When the control gain is increased to a

certain value, the unstable regions in the resonance and amplitude-frequency curves disappear.

Key words: Beam; piezoelectric material; nonlinear vibration; active control; multiple-scales

method.

Corresponding author. Tel.: +86 10 67392704.

Email address: [email protected] (F.-M. Li). Dr., Professor of Beijing University of Technology, Beijing, China.

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1 Introduction

The nonlinear forced vibrations of the mechanical structures are commonly seen in many

mechanical applications and the nonlinear dynamic characteristics of these structural systems are

focused by numerous researchers. Maccari [1] studied the time-delayed feedback control of the

primary resonance of a cantilevered beam. By applying the asymptotic perturbation method, the

amplitude-frequency relation of the system was obtained and the effects of the time delay and

feedback control gains on the resonance of the beam were studied. El-Bassiouny [2] studied the

resonances and control of a cantilevered beam by using the multiple-scales method and found

that the quantic velocity feedback control can stabilize the unstable amplitude. Yao and Li [3]

studied the bifurcation and chaotic motions of the two-dimensional composite laminated plates in

subsonic flow. They obtained the critical instability velocities of the plate by computing a

generalized double integral. Li and Liu [4] studied the nonlinear vibration and active control of a

simply supported beam subjected to axial harmonic excitation. The effects of the velocity

feedback control gains on the stability of the beam were discussed.

Based on the Euler-Bernoulli beam theory, Rafiee et al. [5] studied the nonlinear free and

forced vibration of a nanotube by using the multiple-scales method. The effects of the spring

constant of the elastic foundation and the aspect ratio of the nanotube on the nonlinear oscillation

properties of the system were discussed. Shooshtari and Rafiee [6] investigated the primary and

secondary resonances of a symmetric functionally graded beam by using the multiple-scales

method. The amplitude-frequency relation of the system was obtained and the effects of the

system parameters on the resonance properties of the beam were studied. Emam and Nayfeh [7]

studied the nonlinear vibration of a clamped beam subjected to uniform axial load and transverse

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harmonic excitation. The bifurcation and chaos of the buckled beam were analysed.

The nonlinear vibration and its control of the plates and shells were also extensively studied.

Abe et al. [8] investigated the primary resonance of a composite rectangular plate by using the

multiple-scales method. Zhang [9] analysed the chaotic motion of a composite plate subjected to

parametric excitation. Tang and Chen [10] researched the nonlinear vibration of a plate

considering the internal resonance. Singha and Daripa [11] studied the nonlinear forced vibration

of a plate under transverse harmonic excitation and in-plane periodical load. Amabili [12-14]

conducted a series of relevant investigations in the nonlinear vibration of the cylindrical shells.

Li and Yao [15] studied the 1/3 subharmonic resonance of a composite cylindrical shell subjected

to subsonic airflow. The necessary condition for the 1/3 subharmonic resonance of the shell was

obtained.

From the literatures mentioned above, it can be seen that the unstable nonlinear resonances

of the flexible structures are common in the mechanical engineering. So it is necessary to

investigate the characteristics of the nonlinear resonance and to search for appropriate control

algorithm to stabilize the nonlinear vibration system. In the present study, the equation of motion

of the beam bonded with piezoelectric patch is established using the Hamiltons principle. The

nonlinear forced vibration of the beam is studied by applying the multiple-scales method. The

velocity feedback control algorithm is adopted to stabilize the unstable resonance and the applied

external control voltages on the piezoelectric actuators are presented to show the feasibility of

the active control strategy.

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2 Structural equation of motion

The simply supported beam with rectangular cross section as shown in Fig. 1 is considered.

The piezoelectric patch is well bonded on the surface of the beam. Fig. 1 shows the Cartesian

coordinates, in which the x-axis is established at the neutral surface of the base beam and along

the axial direction. The harmonic force F(t) = F0cos(t) is applied transversely at the position x

= xF of the base beam. In the study, the effects of the axial deformation on the transverse

displacement of the beam are neglected.

The material of the base beam is homogenous and isotropic. The normal stress and normal

strain in the beam are expressed as

xx E , 2

2

2

)(2

1

x

w

x

wzx

, (1)

where w is the transverse deflection of the beam, and x, x and E are the normal stress, normal

strain and elastic modulus, respectively.

The piezoelectric material is transversely isotropic and the polarization direction is in the

z-axis. The constitutive equation of the piezoelectric material is expressed as

z

p

x

p

x Eec 3111 , zp

xz EeD 3331 , (2)

where px and p

x are the normal stress and normal strain of the piezoelectric material, c11, e31

and 33 are the elastic constant, piezoelectric constant and dielectric constant of the piezoelectric

material, Dz is the electric displacement in the z direction, and Ez = V0(t)/hp is the electric

intensity, in which V0 is the external applied voltage on the piezoelectric layer and hp is the

thickness of the piezoelectric layer. Since the piezoelectric material is well bonded on the surface

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of the base beam and the influences of the axial deformation on the transverse displacement of

the beam are neglected, it is assumed that the normal strain px of the piezoelectric material is

the same as that of the base beam, i.e. the deformations between the piezoelectric material and

the base beam are consistent.

The kinetic and potential energies of the total structure can be obtained as [4]

L

xt

wmT

0

2d)(2

1, (3a)

pp V

zzV

xpx

Vxx VEDVVU d

2

1d

2

1d

2

1

LLL

xx

w

x

wax

x

wVax

x

wa

0

2

2

2

30 2

2

020

2

2

2

1 d)(dd)(

206

0

45

0

204 d)(d)( Vax

x

wax

x

wVa

LL

, (3b)

where L is the length of the base beam, m = S +pSp is the mass per unit length of the base beam

and piezoelectric material, and p are the mass densities of the base beam and piezoelectric

material, S and Sp are the section areas of the base beam and piezoelectric patch, and the

coefficients can be expressed as

24

)364( 22113

1

hhhhbhcEbha

ppp ,

2

)2(312

hhbea

p ,

4

)(113

hhhbca

pp ,

2

314

bea ,

8

11

5

pbhcEbha

,

ph

bLa

2

336

, (4)

where h is the thickness of the base beam and b is the width of the base beam and piezoelectric

patch.

The work done by the external applied force F is written as

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FxxFFwW . (5)

Hamiltons principle is written as

0dd)(2

1

2

1

tWtUT Ft

t

t

t , (6)

where denotes the first variation, and t1 and t2 are the integration time limits.

The first order expansion of the transverse displacement of the simply supported beam is

considered. So the displacement w can be expressed as

L

xtWtxw

sin)(),( , (7)

where W(t) is the displacement amplitude, i.e. the generalized coordinates.

Substituting Eqs. (3), (5) and (7) into Eq. (6), and performing the variation operation in

terms of W, the following nonlinear equation of motion of the whole structure can be obtained:

tFWaWaWVaWaVat

W cos

d

d1

3

11

2

10098072

2

, (8)

where 2

27

4

mL

aa

,

4

4

18

2

mL

aa

,

2

2

49

2

mL

aa

,

4

3

310

4

mL

aa

,

4

4

511

3

mL

aa

, and

L

x

mL

FF F

sin

2 01 .

One can observe that the second and fourth terms in Eq. (8) are the electromechanical

coupling terms which relate the external applied voltage to the structural deformation, and

indicate that the external applied voltage will affect the nonlinear dynamic properties of the

structural system. Furthermore, it can be seen that the structural vibration equation is cubic

nonlinear.

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3 Controller design

The piezoelectric material can be activated by appropriate external control voltage to obtain

active damping and improve structural vibration stability. To achieve this aim, a velocity

feedback control strategy can be applied. The velocity at position x0 of the beam is measured by

a velocity sensor and fed back to the controller. The controller calculates control voltage with

velocity feedback control method, and exerts the voltage to the piezoelectric actuator patch to

produce control force. The control voltage exerted to the piezoelectric actuator is proportional to

the velocity at the position x0, and can be expressed as

t

W

L

xK

t

txwKV

d

dsin

d

),(d 000

, (9)

where K is the feedback control gain for the piezoelectric actuator.

Substituting Eq. (9) into Eq. (8), and combining with Eq. (7), the equation of motion is

changed to the following cubic nonlinear equation with active damping

tFWaWat

WWaWa

t

Wc

t

Wp cos

d

d

d

d

d

d1

3

11

2

101282

2

, (10)

where L

xKacp

07 sin

and

L

xKaa 0912 sin

.

The second term in Eq. (10) is the active damping force due to the velocity feedback, and

the coefficient cp is the active damping coefficient which is related to the feedback control gain K.

We can see from Eq. (10) that by applying the piezoelectric material and velocity feedback

control, the active damping can be produced to influence the structural dynamic response. By

actively adjusting the feedback control gain K, the structural dynamic properties will be changed.

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4 Active control of nonlinear vibration

The active control for the nonlinear vibration of the beam will be investigated by solving

the steady response of the equation of motion, i.e. Eq. (10). By defining 82

0 a , Eq. (10) can

be changed to the following form:

tFWaWat

WWaW

t

Wc

t

Wp cos

d

d

d

d

d

d1

3

11

2

1012

2

02

2

. (11)

Defining the following non-dimensional variables:

h

WW ,

1

00

, tt 1 ,

1

, (12)

where 4

2

1mL

EI is the first natural frequency of the base beam, in which

12

3bhI is the

moment of inertia for the cross section of the base beam. Substituting the above non-dimensional

variables into Eq. (11) results in the following dimensionless equation of motion:

tFWWt

WWW

t

Wc

t

W p cos

d

d

d

d

d

d 3220

1

2

2

, (13)

where , and are the dimensionless coefficients and expressed as: 1

12

ha

, 2

1

10

ha

and 2

1

2

11

ha

, and 2

1

1h

FF is the dimensionless force amplitude. For the sake of the

convenience of writing, the symbol over the non-dimensional variables will be omitted in

what follows.

The method of multiple-scales [16, 17] will be used to solve the nonlinear equation of

motion. So the following substitutions are made:

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ccp

1

, , , , fF , (14)

where is a small dimensionless perturbation parameter. Then Eq. (13) can be changed to

tfWWt

WWW

t

Wc

t

W cos

d

d

d

d

d

d 32202

2

. (15)

Suppose the solution of Eq. (15) can be expressed as the following form:

),,,(),,,(),,,(),( 21022

21012100 TTTWTTTWTTTWtW , (16)

where Tn = nt (n = 0, 1, 2, ), and the first two time scales are taken. T0 = t is the fast varying

time scale and T1 = t is the slowly varying time scale. So the time derivatives are written as

follows:

10

10d

dDD

TTt ,

10

2

0

10

2

2

0

2

2

2

22d

dDDD

TTTt . (17)

In this paper, the nonlinear primary resonance is studied. So we set

220 , (18)

where is the tuning parameter describing the approaching degree between and .

Substitution of Eqs. (16)-(18) into Eq. (15), after equating coefficients of same power of ,

results in the following differential equations:

002

0

2

0 WWD , (19)

)cos(2 03

0

2

0000000101

2

1

2

0 TfWWWDWWcDWDDWWD . (20)

The complex solution of Eq. (19) can be obtained as

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)iexp()()iexp()( 01010 TTaTTaW , (21)

where a is the complex conjugate of a.

Substituting Eq. (21) into Eq. (20) yields the following equation:

)i2exp()i()iexp()2

13ii2( 0

22

0

2

1

2

1

2

0 TaaTfaaaacaWWD

ccaaTa )i3exp( 03 , (22)

where a is the derivative of a with respect to T1, and cc indicates the complex conjugate part at

the right hand side of Eq. (22). Eliminating the secular terms in Eq. (22) leads to

02

13ii2 2 faaaaca . (23)

In order to solve function a, one writes a in exponential function form as follows:

)iexp(2

1Aa , (24)

where A and are all real function of T1. Substituting Eq. (24) into Eq. (23) and dividing the real

parts from the imaginary parts yield

sin2

1

2

1fcAA

, (25)

cos228

3 2

A

fA . (26)

By letting 0 A and substituting

22

0 in Eqs. (25) and (26), the stationary

solution can be obtained as

0])[()(2

3

16

9 2222222

0

22

0426

fAcAA

. (27)

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The above formulation is called as the resonant curve equation which relates the displacement

amplitude A with the force amplitude f and frequency . From Eq. (27) one can see that the

feedback control gain K (which is related to the damping coefficient c) has effects on the

structural dynamic response.

When the active control is implemented, the external applied voltage amplitude can be

obtained by combining Eqs. (9), (21) and (24) as

L

xKAhV 010 sin

. (28)

Next, the influences of the different parameters on the structural primary resonant curve

characteristics will be analyzed.

5 Example and discussions

A simply supported beam with geometric nonlinearity bonded with piezoelectric layer

subjected to external force F is shown in Fig. 1. The parameters for the base beam and

piezoelectric material are shown in Table 1. The velocity sensor is located at the middle position

of the beam, i.e. x0 = L/2. The external force F is assumed to be also applied at the middle

position of the beam, i.e. xF = L/2.

5.1 Validity of the present methodology

In this section, the validity of the present study is verified by comparing the amplitude-frequency

curve obtained from Eq. (27) with that by numerical simulation. In numerical calculation, the

time-domain responses of the beam under certain excitation frequencies are computed using the

4th order Runge-Kutta method. The response amplitudes corresponding to different excitation

frequencies are extracted from the time-domain responses. Fig. 2 shows the comparison of the

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amplitude-frequency response at the center of the beam obtained by this method with that by the

numerical calculation. In Fig. 2 the solid curve represents the stable response of the beam, the

dashed line denotes the unstable response, and the dots represent the vibration amplitude of the

beam obtained from the numerical simulation. From Fig. 2 it can be seen that the

amplitude-frequency curve obtained by the present method is in agreement with that from the

numerical simulation, which verifies the validity of the present methodology.

5.2 Nonlinear dynamic properties of the uncontrolled beam

In this section, the nonlinear dynamic properties of the uncontrolled beam under harmonic

external excitation are investigated. From Eq. (27), one can obtain the relation between the

amplitude of the resonance displacement and the amplitude of the external excitation. For =

1.2, it implies that the frequency of the external excitation is 1.2 times of the fundamental natural

frequency of the beam with piezoelectric actuator. Fig. 3 shows the variations of the

displacement amplitude A with the force amplitude f of the uncontrolled dynamic system. It can

be seen in Fig. 3 that the displacement amplitude is a multi-valued function of the external force

amplitude. This may lead to instability of the system. For example, with the force amplitude f

increasing from 0, the amplitude of the displacement A increases continuously from 0. When f

increases to 1.75, the displacement amplitude rapidly changes from 0.15 to 0.32 and then

increases gradually. Similarly, when the non-dimensional amplitude of the external excitation f

decreases from 6, the displacement amplitude A of the beam decreases continuously first and

jumps suddenly from 0.27 to 0 when f decreases to 0. This is called the jump phenomenon. The

discontinuous changes of the displacement amplitude with respect to the force amplitude may

make damage to the structural system.

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Fig. 4 shows the amplitude-force curves of the beam for different non-dimensional external

excitation frequencies. It can be seen in Fig. 4 that when =1.1, the amplitude-force curve is a

single-valued curve and the displacement amplitude of the beam increases with the external

excitation amplitude increasing. When = 1.15 and = 1.2, the amplitude-force curves are

multi-valued ones. For = 1.2, the interval of the external excitation amplitude for the

multi-valued range of the steady state response is larger than that for = 1.15. Fig. 5 shows the

amplitude-frequency curves of the beam under different force amplitudes. In Fig. 5, the solid

lines represent the stable displacement amplitudes and the dashed lines represent the unstable

amplitudes of the displacement. It can be seen in Fig. 5 that with the frequency of the external

excitation increasing, the vibration amplitude increases first and then decreases. The beam

exhibits the hardening-type nonlinearity and with the amplitude of the external excitation

increasing, the vibration amplitude increases also and the frequency interval for the unstable

region is prolonged.

5.3 Active vibration control

The effects of the velocity feedback control on the resonance curve and the

amplitude-frequency relations of the beam are investigated. Fig. 6 shows the variations of the

displacement amplitude A with the force amplitude f under different feedback control gains. In

the simulation, the frequency of the external excitation is chosen as = 1.2. It can be seen in Fig.

6 that with the velocity feedback control gain increasing, the interval for the unstable resonance

amplitude of the beam is reduced. When K is increased to 1000, the displacement amplitude is a

single-valued function of the force amplitude. In such situation, the jump phenomenon of the

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displacement amplitude with respect to the amplitude of the external force will disappear.

Fig. 7 shows the external applied control voltage amplitudes with respect to the force

amplitude f for different feedback control gains when = 1.2. It can be seen that with the

feedback control gain increasing, the amplitude of the external applied voltage increases and the

maximal amplitude of the control voltage is about 450V.

Fig. 8 shows the amplitude-frequency curve of the beam with different feedback control

gains when F0 = 2N. It can be seen in Fig. 8 that with the feedback control gain increasing, the

amplitude of the resonance is decreased and the unstable region of the displacement amplitude is

reduced. When K is increased to 1000, the unstable region of the vibration amplitude disappears.

Fig. 9 represents the relations between the amplitudes of the applied external voltages and

the frequency of the external force under different control gains. It can be seen in Fig. 9 that the

peaks of the voltage amplitudes are about 420V for different control gains. But for K = 1000, the

unstable region of the voltage amplitude will disappear. So the effects of the velocity feedback

control gain on the stability of the applied external voltage can not be ignored.

Fig. 10 shows the displacement-time histories at the center of the beam with and without

velocity feedback control. It can be seen in Fig. 10 that for the uncontrolled beam, the maximum

non-dimensional amplitude of the transverse vibration is about 0.5. When the feedback control

gain K is chosen to be 1000, the maximum amplitude of the vibration is fallen to about 0.25,

which implies that the velocity feedback control strategy can stabilize the nonlinear primary

resonance of the beam.

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6 Conclusions

The nonlinear primary resonance properties and active control of a flexible beam under

transverse harmonic excitation are studied. The equation of motion of the beam with geometric

nonlinearity is established using the Hamiltons principle. The method of multiple-scales is

applied to analyze the primary resonance properties of the beam, and velocity feedback control is

used to stabilize the nonlinear vibration of the beam. From the theoretical analysis and numerical

simulations, the following conclusions can be drawn:

(1) For the uncontrolled system, the jump phenomenon appears. When the excitation frequency

> 1, the vibration amplitude can be multi-valued function of the external excitation

amplitude and the frequency interval for the unstable region is prolonged with the excitation

frequency increasing.

(2) The beam exhibits the hardening-type of nonlinearity. With the amplitude of the external

excitation increasing, the amplitude of the steady state response increases and the

hardening-type of nonlinearity is strengthened.

(3) The velocity feedback control can stabilize the unstable primary resonance. With the

feedback control gain increasing, the amplitude of the resonance curve and the peak values of

the amplitude-frequency curves are reduced.

(4) When the feedback control gain is increased to a certain value, the unstable region of the

vibration amplitude will disappear, and the jump phenomenon is also avoided. The applied

external voltages are in reasonable range, which shows the feasibility of the active control.

Acknowledgements

This research is supported by the National Basic Research Program of China (No.

2011CB711100) and the National Natural Science Foundation of China (No. 11172084 and

10672017).

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[4] Li F.M., Liu C.C. Parametric vibration stability and active control of nonlinear beams.

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primary-resonance excitation. Nonlinear Dynamics, 2005, 35 (1): 1-17.

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110-126.

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The list of the figures

x

z

O

F(t) Piezoelectric material

Base beam

Fig. 1 The schematic diagram of a beam with piezoelectric patch.

xF

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

w/h

Fig. 2 Comparison of the amplitude-frequency curve of the beam calculated by the present method with that by

numerical simulation (F0 = 5N, K = 1103).

Present result

Numerical simulation

0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

f

A

Fig. 3 The variation of the displacement amplitude A with the force amplitude f when = 1.2.

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

= 1.1

= 1.15

= 1.2

f

A

Fig. 4 Amplitude-force curves of the beam for different external excitation frequencies.

A

F0 = 1N

F0 = 2N

F0 = 3N

Fig. 5 Amplitude-frequency curves of the beam for different force amplitudes.

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0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

f

A

K = 0

K = 300

K = 600

K = 1000

Fig. 6 The variations of the displacement amplitude A with the force amplitude f under different feedback

control gains when = 1.2.

0 1 2 3 4 50

100

200

300

400

500

V0(V

)

f

K = 300

K = 600

K = 1000

Fig. 7 The amplitudes of the applied external voltages with respect to the force amplitude f for different

feedback control gains when = 1.2. Dow

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A

K = 300

K = 600

K = 1000

Fig. 8 Amplitude-frequency curves of the beam with different feedback control gains when F0 = 2N.

V0(V

) K = 300

K = 600

K = 1000

Fig. 9 Variations of the amplitudes of the applied external voltages with the frequency of the external force for

different feedback control gains when F0 = 2N.

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0 50 100 150 200 250 300

-0.4

-0.2

0

0.2

0.4

0.6

K = 0 K = 1000

t

w/h

Fig. 10 Displacement-time histories at the center of the beam under different feedback control gains.

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Table 1 The parameters for the base beam and piezoelectric material.

Elastic constant Mass density Piezoelectric constant Length Width Thickness

E or c11/(GPa) /(kgms2) e31/(Cm

2) L/(m) b/(m) h/(m)

Base beam(Al) 71 2710 0.5 0.01 0.005

Piezoelectric layer(PZT-5H) 126 7500 6.50 0.5 0.01 0.001

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active control of nonlinear forced vibration in a flexibl

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