Active vibration control of a dynamical system subjected ... · Sayed and Kamel [14] investigated...

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 4 (2017) pp. 434-442

© Research India Publications. http://www.ripublication.com

434

Active vibration control of a dynamical system subjected to simultaneous

excitation forces

Y. S. Hamed 1, 2 , M. R. Alharthi 2 and H. K. AlKhathami 3

1Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering,

Menoufia University, Menouf 32952, Egypt.

2Department of Mathematics and Statistics, Faculty of Science, Taif University, Taif ,

El-Haweiah, P.O. Box 888, Zip Code 21974, Kingdom of Saudi Arabia.

3Department of Mathematics, Faculty of Science and Home Economic, Bisha University,

61922, Kingdom of Saudi Arabia.

Abstract

In this paper, we investigated the effects of an active vibration

control on a nonlinear two-degree-of-freedom system

described by a nonlinear differential equations subjected to

mixed excitation forces. The method of multiple scale

perturbation technique is applied to determine the

approximate solutions of the coupled nonlinear differential

equations up to the second order approximation. The

frequency response equations and phase plane technique at the

worst resonance cases are used to study the stability of the

vibrating system. Numerical simulations show the steady state

response amplitude versus the detuning parameter and the

effects of the parameters system and controller.

Keywords: two-degree-of-freedom system, Vibration,

Stability, multi excitation forces.

INTRODUCTION

Active control system is characterized essentially in terms of

certain amount of external power or energy is required. It

follows that active control contains a broad range of

technologies. The most important areas in the active control

design are the modeling of control system and control law

design. A fully active control strategy involves using force

actuators requiring external energy. The actuation force is

used to improve the absorption level, and it is traditionally

applied between the absorber's mass and the primary system

parallel to the elastic (resilient) element that supports the

absorber mass. Active vibration control for the suppression of

helicopter rotor blade flapping vibrations has been

investigated and studied [1-3]. Eissa and Sayed [4-6], studied

the effects of different active controllers on simple and spring

pendulum at the primary resonance via negative velocity

feedback or its square or cubic. Yabuno et al. [7] proposed a

non-linear active cancellation method to stabilize the principal

parametric resonance in a magnetically levitated body

subjected to an unsymmetrical restoring force. Eissa et al.

[8, 9], studied mathematically the vibrations of a cantilever

beam or the aircraft wing and investigated the saturation

phenomena that suppresses these vibrations at one of the

extracted resonance cases. Also, Eissa et al. [10, 11], have

studied both passive and active controllers of the vibrating

systems of an aircraft wing. Jun et al. [12], introduced the

non-linear saturation-based control strategy for the

suppression of the self-excited vibration of a van der Pol

oscillator. It is demonstrated that the saturation-based control

method is effective in reducing the vibration response of the

self-excited plant when the absorber’s frequency is exactly

tuned to one-half the natural frequency of the plant. An active

non-linear vibration absorber for suppressing the high-

amplitude vibration of the non-linear plant when subjected to

primary external excitation is studied [13]. The absorber is

based on the saturation phenomenon associated with the

dynamical systems with quadratic non-linearities and 2:1

internal resonance. Sayed and Kamel [14] investigated the

effect of linear absorber on the vibrating system and the

saturation control of a linear absorber to reduce vibrations due

to rotor blade flapping motion. The stability of the obtained

numerical solution is investigated using both phase plane

methods and frequency response equations. Variation of some

parameters leads to the bending of the frequency response

curves and hence to the jump phenomenon occurrence. Sayed

and Kamel [15] applied active control for suppressing the

vibration of the non-linear plant when subjected to external

and parametric excitations in the presence of 1:2 and 1:3

internal resonance. The method of multiple scale perturbation

technique is applied to determine four first-order non-linear

ordinary differential equations that govern the modulation of

the amplitudes and phases in the presence of internal

resonance of the two systems with quadratic and cubic order

of control. Hamed and Amer [16] they used different types of

control to suppress the vibrations of a flexible composite

beam system. Hamed et al. [17] investigated the nonlinear

vibrations, energy transfer and stability of the MEMS

gyroscope system under multi-parametric excitations. Also,



435

they obtained the frequency response equations using the

averaging method.

MATHEMATICAL MODEL The equations of motion for the two degrees-of freedom

system are modified by dimensionless form [10]:

2 3

1 1 1 1 1 1 1 1 12 cosx x x x f t

2 1 1 3 2 3cos cos sinf x t f t t

2

1 1 2( )x x (1)

2 2

2 2 2 2 2 2 2 2 12x x x x x (2)

With initial condition

1 2 1 2x (0)=0.5,x (0)=0.5,x (0)=0.5,x (0)=0.0 .

Where 1x and 2x are displacement. ,n nx x (n=1,2) are

the first and second derivatives, 1 and 2

are linear

damping coefficients, 1 is non-linear parameter, is a

small perturbation (0<𝜀<<1), jf (j=1,2,3) are the excitation

amplitudes, 1 2, are the natural frequencies, , j

(j=1,2,3) are excitation frequencies and 1 and 2 are

active control coefficients.

2.1. Perturbation analysis

The MSPT method [18-19] is applied to obtain a first-order

approximation for the system, which is a powerful tool in

determining periodic solutions of small amplitude. We seek an

approximate solution of Eqs. (1)- (2) in the form 2

1 10 0 1 11 0 1; , , ( )x t x T T x T T O (4)

2

2 20 0 1 21 0 1x t;ε =x T ,T +εx T ,T +O(ε ) (5)

The derivatives will be in the form

0 1

220 0 12

d=D +εD +...

dt

d=D +2εD D +...

dt

(6)

For the first-order approximation, we introduce two time

scales, where n

nT t and the derivatives n nD T ,

(n= 0, 1).

Substituting (4)–(6) into (1)–(2) and equating the coefficients

of equal powers of leads to

0( )O

2 2

0 1 10( ) 0D x (7a)

2 2

0 2 20( ) 0D x (7b)

1( )O

2 2

0 1 11 0 1 10 1 1 0 10( ) 2 2D x D D x D x

3

1 10 1 2 10 1cos cosx f t f x t

2

3 2 3 1 10 0 20cos sin ( )f t t x D x (8a)

2 2

0 2 21 0 1 20 2 2 0 202 2D x D D x D x

2

2 20 0 10x D x (8b)

The general solutions of (7) can be written in the form

10 0 1 1 0 0 1 1 0( )exp ( )expx A T i T A T i T (9a)

20 0 1 2 0 0 1 2 0( )exp ( )expx B T i T B T i T (9b)

Where 0 0 0, ,A B A and 0B are complex function in 1T .

Substituting Eqs. (9a)- (9b) into Eqs. (8a)- (8b), we obtain:

2 2

0 1 11 1 1 0 1 1 0[ 2 ( )D x i D A A

2 2

1 0 1 0 0 2 1 00 03 2 exp]A BA B A i T

2

0 01 1 1 001 1[ ( )2 3A A Ai D A2 3

1 0 2 1 0 10 0 10 0] [ )2 exp exp(3B i T A i TB A

31

0 1 0 0)exp( 3 ] exp( )2

fA i T i T

2 010 0 1 1exp( e) (xp

2)

2

f Afi T iT

2 0 2 00 1 1 0 1 1( ) ( )exp exp

2 2

f A f AiT iT

2 0 30 1 1 0 3 2( ) ( )exp exp

2 4

f A ifiT iT

3 30 3 2 0 3 2( ) ( )exp exp

4 4

if ifiT iT

2 230 3 2 1 0 0 2 0 1 2exp exp 2

4( ) ( )

ifiT B A iT

2 2

1 2 0 1 2 1 0

2

0 0 2

2

0 (exp )2 BiT AB A

2 2

0 1 2 1 0 0 2 0 1 2exp 2 e 2( ) )x (piT B A iT

(10a)

2 2

0 2 21 2 1 0 2 2 0 2 0( )2 expD x i D B B i T

0 02 1 2 2 2 02 e( ) xpi D iB TB

2

1 2 0 0 0 2 1 1 2

2

0 0exp 2( ) BB A iT i Ai

2

0 2 1 1 2 0 00 2 1( ) ( )exp 2 exp 2iT i B iTA2

01 2 0 0 2 1 1 2 0 00exp 2 2( )B Bi A iT i B A

1 0 1 2 0 00 1 0exp 2 expi T i B iB TA

(10b)



436

After eliminating the secular terms in Eqs. (10a)- (10b), then

the particular solutions will be in the form:

11 1 0 1 1 01exp expx A i T A i T

3310 1 0 02

1

1 0exp(3 exp( 3 ][ ) )8

A i T A i T

10 02 2

1

[exp( exp( )

) ](2

)f

i T i T

20 0 1 12 2

1 1 1

[ exp2( )

( )( )

fA iT

20 0 1 1 2 2

1 1 1

exp ]2(

( ))( )

fA iT

0 0 1 1 0 0 1 1[ exp e) x ]( ( )pA iT A iT

30 3 22 2

1 3 2

( )( )

[exp4( )

ifiT

30 3 2 2 2

1 3 2

exp ]4(

( )( ) )

ifiT

0 3 2 0 3 2[exp exp( ) ( ) ]iT iT

221 20 0 0 1 22 2

1 1 2

[ exp 2( 2 )

( )( )

B A iT

2

1 20 1 2 2 2

1

2

1

0 0

2

exp 2 ]( 2

() )

)(

iB A T

2

0

2

0 0 1 2 0 0 0 1 2[ exp 2 exp 2( ]) ( )B AA iT B iT

(11a)

21 1 2 0 1 2 0exp expB i B ix T T

21 20 0 0 2 12 2

2 2 1

[ exp 2( 2 )

( )( )

iB A iT

1 20 2 1 2

2 2 1

2

0 0 2exp 2 ]( )

(2 )( )

iiTB A

2

0

2

0 0 2 1 0 10 0 2( ) ( )[ exp 2 exp 2 ]B iT A iTA B

1 20 0 1 0 0 1 02 2 0 0

2 1

0

2[ exp exp ]

( )B

iB Bi B iAA T T

(11b)

Where 1 1 1, ,A B A and 1B are complex function in 1T . From

the above-derived solution many resonance cases can be

deduced. The reported resonance cases at this approximation

order are:

(a) Primary resonance: 1

(b) Sub-harmonic resonance: 1 12

(c) Internal resonance: 1 2 1 23,

(d) Combined resonance: 2 3 1

Any combination of the above resonance cases is considered

as simultaneous resonance.

Periodic solutions:

The stability from the first-order approximation solution is

studied at, the worst resonance case, the primary, sub-

harmonic, internal and combined resonance case

1 1 1 2 3 1 1 2, ,( 2 3, ) .

Introducing the detuning parameters 1 2 3, , and 4

according to

1 1 1 1 1 2, ,

2 3 1 3 2 1 43, (12)

Substituting Eq. (12) into Eqs. (8a)-(8b) and eliminating the

secular terms leads to solvability conditions for the first and

second-order expansions as:

2

0 0

2

1 1 0 1 1 0 1 0 1 0 0 22 3 2i D A A A B AA B

11 0 1 1 0exp( exp

2) ( )

fi T i T

320 1 2 1 030exp ( ) exp ( ) 0

2 4

iffA i T i T

(13a)

2 1 0 2 2 0 2 02 e )xp(i D B B i T

1 2 0 2 4 0

2

0 exp ( ) 0A i TBi (13b)

To analyze the solution of (13), it is convenient to express

1( )A T and 1( )B T in the form

0 1 1 0 1 1

1 1( )exp(( ( )), ( )exp( ( ))

2 2A a T i T B b T i T

(14)

Where a and b are the steady-state amplitudes, and

are the phases of the motions.

Inserting (14) into (13) and equating the real and imaginary

parts, the following equations are obtained as the following:

31 21 1 1 2 3

1 1 1

sin sin cos2 4 4

ff f aa a

(15a) 3 2 2

1 2 1 11

1 1 1

3cos

8 4 2

a ab fa

322 3

1 1

cos4 4

ff asin (15b)

2

1 22 2 4

2

cos8

abb b (16a)



437

2

1 24

2

sin8

abb (16b)

Where 1 1 1 2 2 1 3 3 1, 2 ,T T T

and 4 4 13 T .

Thus, the zero approximation periodic solution can be written

in the form

1 1cos( )x a t (17a)

1 42 cos( )

3 3

tx b (17b)

Where 1, ,a b and 4 are obtained by the solutions of (15)–

(16).

Stability of the fixed points :

The fixed point of our dynamical system of equations (15)–

(16) is obtained when 0a b and 0n where

(n=1, …,4) as the following:

31 21 1 1 2 3

1 1 1

sin sin cos2 4 4

ff f aa

(18a) 3 2 2

1 2 1 11

1 1 1

3cos

8 4 2

a ab fa

322 3

1 1

cos sin4 4

ff a (18b)

2

1 22 2 4

2

cos8

abb (19a)

2

4 1 24

2

( )sin

3 8

abb (19b)

The frequency response equations for the practical case

( 0, 0a b ) are given by:

2 28 6 2 2 21 1 2

1 12 211 1

9 3 1

64 4 16

fa a

4 2 6 2 4 2

1 2 1 4 1 2 1 2 1 2 14 2 4 2 4 2

1 2 1 2 1 2

124 4

3 34 4 2

4 31 2 1 4 1 2 2 1 44 2 2 3 2

1 2 1 1 2

8 1 32

3 4 9

f fa a

2 2 4 23 2 1 4 12 3 2 2

1 1 2 1

1 64 1

16 9 4

f f

6 2 3 4 10 2 2 222 1 2 2 1 2 1 2

3 2 3 2 6 4

1 2 1 2 1 2

32 32 512

9 9a

10 2 2 10 2 2 2

2 1 2 4 2 1 2 46 4 6 4

1 2 1 2

1024 512

9 912 2 4 8 2 3 8 2 2 2

2 1 2 2 1 4 2 1 46 4 6 4 6 4

1 2 1 2 1 2

256 1024 512

81 278 2 3 8 2 4 8 2 4

2 1 4 2 1 2 1 46 4 6 4 6 4

1 2 1 2 1 2

1024 256 2560

81 81 81 (20a)

2 2 22 2

22 2 1

4 2 2

2

24 2

92 9 0

64

a b

(20b)

Stability of linear solution :

To study the stability of the linear solution, one investigates

the solution of the linearized form of Eqs. (13a)- (13b) as

1 1 0 1 1 0 1 02 ex )p(i D A A i T

1 21 1 0 0 1 2 0( )exp exp ( )

2 2

f fi T A i T

33

1 0exp ( ) 04

ifi T (21a)

2 1 0 2 2 0 2 02 e )xp(i D B B i T

(21b)

We express 0 0,A B in the form

0 1 1 1 1 0 2 2 2 1

1 1( )exp( ), ( )exp( )2 2

A p iq iv T B p iq iv T

(22)

Where,

41 2,

3v v . Separating real and

imaginary parts into expression, we obtain the autonomous

equation of the modulation of the amplitudes and phases of

the response

321 1 1 1

1 1

1 1

1

04 4

ffp vp q

q (23a)

2 11 1 1 1 1 1

1 1 1

04 2

qf f

q v pp

(23b)

2 22 2 2 2 0p vp q (24a)

2 22 2 2 2 0q vq p (24b)

The stability of a particular equilibrium solution is ascertained

by investigating the eigenvalues of the Jacobin matrix of the

right-hand sides of Eqs. (23)- (24).

The eigenvalues of the above system of equations are given

by the equation 4 3 2

1 2 3 4 0r r r r (25)



438

Where 1 2 3, ,r r r and 4r are constants. If the real part of the

eigenvalues is negative, then the corresponding periodic

solution is stable; otherwise become unstable. According to

the Routh–Huriwitz criterion, the necessary and sufficient

conditions for all the roots of (25) to have been negative real

parts are satisfied as: 2

1 1 2 3 3 1 2 3 1 4 40, 0, 0, 0r r r r r r r r r r r

(26)

RESULTS AND DISCUSSIONS

In this section, the oscillations of the two-degree-of-freedom

non-linear system under multi excitation force are studied.

Equations (1) and (2) are numerically integrated using a fourth

order Runge–Kutta algorithm to verify the analytic predictions

and to analyze the periodic and chaotic motions of the system.

Fig. 1 illustrates the response and the phase-plane for the

main system without controller at principal parametric

resonance case where 1 1 1, 2 and

2 3 1. It is observed from this Figure that the

steady state amplitude of the composite beam is about 900%

of the maximum excitation force amplitude 1f the

oscillation response starts with increasing amplitude and

becomes stable and the phase plane shows limit cycle.

Figure 1. System behavior without controller at primary, sub-

harmonic and combined resonance case

( 1 1 1 2 3 1, ,2 )

Effects of the magnetorheological (MR) controller on

system :

Fig. 2 illustrates the results at simultaneous primary , internal,

sub-harmonic and combined resonance case when the

controller is effective, i.e., when

1 1 1 2 3 12, , , and 2 13 .

It can be seen for the main system that the steady state

amplitude is 20%, but the steady state amplitude of the

controller is about 2500% of excitation amplitude 1f . This

means that the effectiveness of the absorber aE (aE = the

steady state amplitude of the main system without

controller/the steady state amplitude of main system with

controller) is about 45.

Figure 2. The response of the system and controller at

primary, sub-harmonic, combined and internal resonance case

( 1 1 1 2 1 13 2, , , 32 ),

1 2 10.00625, 0.000625, 0.02,

1 2 1 2 35, 0.4, 0.05, 0.004, 0.003,f f f

1 2 1 2 33, 1, 3, 6, 4, 1

Response curves of system

In this section, the stability zone and effects of the different

parameters are discussed using frequency response equations.

Also, the stability of the numerical solution is studied using

the phase-plane method. The steady state response of the

given system at various parameters near the simultaneous

primary, sub-harmonic, combined and internal resonance case

is investigated and studied. The frequency response equation

given by (20a) is solved numerically. Figure 3(a) shows the

effects of the detuning parameter on the steady state

amplitude of the main system a, for the practical case, where



439

( 0a and 0b ). Figs. 3(b, c) show that the steady state

amplitude of the main system are monotonic decreasing

function in the linear damping coefficient 1 and the natural

frequencies 1 and 2 . The steady state amplitude of the

main system are monotonic increasing function in the

damping coefficient 2 , the non-linear parameter 1 and

the excitation amplitudes ( 1,2,3)jf j as shown in Figs.

3(d, e, f, g, h). For positive and negative values of the non-

linear parameters 1 ,the frequency response curves are bent

to left or right leading to the jump phenomena occurrence and

multivalued amplitudes as shown in Fig .3(i). For increasing

values of the non-linear parameters 2 ,the steady state

amplitude of the main system is decreased and the curves are

shifted to the left as shown in Fig .3(j).

Fig. 3a. Effects of the detuning parameter

Figure 3b. Effects of the damping coefficient 1

Figure 3c. Effects of the natural frequencies 1 and 2

Figure 3d. Effects of the damping coefficient 2

Figure 3e. Effects of the nonlinear parameter 1

Figure 3f. Effects of the excitation force amplitude 1f



440

Figure 3g. Effects of the excitation force amplitude 2f

Figure 3h. Effects of the excitation force amplitude 3f

Figure 3i. Effects of the nonlinear parameter 1

Figure 3g. Effects of the nonlinear parameter 2

Figure 3. Effect of system parameters on the frequency

response curves of the main system at the practical case

( 0a and 0b )

Response curves of the magnetorheological (MR)

controller :

In this section, the numerical results are plotted in figure 4,

which represent the variation of the amplitudes b with the

detuning parameter 4

for the practical case, where ( 0a

and 0b ). The frequency response equation given by (20b)

is solved numerically.

In fig. 4(a), the response amplitude has two continuous curves

which is show the effects of the detuning parameter 2 on the

steady state amplitude of the absorber b for the practical

case, where ( 0a and 0b ). The steady state amplitude

of the controller is a monotonic increasing function in the

damping coefficient 2 as shown in Fig 3.4(b). Fig. 3.4(c)

show that the steady state amplitude of the controller is a

monotonic decreasing function in the non-linear parameters

1 .

Figure 4a. Effects of the detuning parameter 4

Figure 4b. Effects of the damping coefficient 2



441

Figure 4c. Effects of the nonlinear parameter 2

Figure 4. Effect of system parameters on the frequency

response curves of the absorber at (1 0a and

2 0a )

CONCLUSIONS

The nonlinear dynamics and vibration suppression of a

nonlinear electromechanical system under harmonic and

parametric excitations has been investigated. The considered

model consists of an electrical part coupled to mechanical one

and modeled by a coupled nonlinear ordinary differential

equations. The analytical up to second order approximate

solutions are sought applying the method of multiple scales

method. We used the time-series and method of averaging to

analyze the response and stability of the solutions at the worst

resonance cases. The results of perturbation solution have

been verified through numerical simulations, where different

effects of the system parameters have been reported.

Comparison between analytical and numerical solutions is

obtained. From this study the following may be included:

For uncontrolled system the steady state amplitude at

simultaneous resonance case

1 1 1 2 3 1, ,2 is about 900% of

the maximum excitation force amplitude1f , which is one

of the worst resonance cases and it should be avoided in

design.

The effectiveness of the controller ( = the steady

state amplitude of the main system without controller

/the steady state amplitude of the main system with

controller ) is about 45.

The steady state amplitude of the main system are

monotonic decreasing function in the linear damping

coefficient 1

and the natural frequencies 1

and 2

.

The steady state amplitude of the main system are

monotonic increasing function in the damping

coefficient 2 , the non-linear parameter 1 and the

excitation amplitudes ( 1,2,3)jf j .

For positive and negative values of the non-

linear parameters 1 ,the frequency response

curves are bent to left or right leading to the

jump phenomena occurrence and multivalued

amplitudes.

For increasing values of the non-linear

parameters 2 ,the steady state amplitude of

the main system is decreased and the curves are

shifted to the left.

The steady state amplitude of the controller is a

monotonic increasing function in the damping

coefficient 2 .

The steady state amplitude of the controller is a

monotonic decreasing function in the non-linear

parameters 1 .

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