Vibration suppression of flexible structures using internal resonance

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VIBRATION SUPPRESSION OF RESONANCE

Vol. 18, (2/3), 135-143, 1991 Printed in the USA Copyright (c) 1991 Pergamon Press plc

FLEXIBLE STRUCTURES

USING

INTERNAL

M. Farid Golnaraghi Department of Mechanical Engineering University of Waterloo, Waterloo, ON, Canada N2L 3G1 (Received 21 June 1990; accepted for print 19 November 1990)

Introduction

Due to the recent developments in the space program, flexible structures have become the focus of numerous research studies [1-3]. Low inherent damping, small natural frequencies, and extreme light weights are among common characteristics of these systems which make them vulnerable to any external/internal disturbances such as slewing manoeuvres, impacts, etc. Thus, a great deal of research is currently in progress on designing active and passive vibration control mechanisms to deal with this problem [4-6]. The conventional passive dampers (e.g. coatings and elastomers) have limited capabilities in dissipating the oscillation energy. The success of active controllers is also restricted by physical constraints (e.g. light structural mass and limited actuator force). Hence, large amplitude vibrations control is not an easy task. It is valuable to propose a rather unconventional active/passive vibration suppression technique for cantilever beam based on simple principles in non-linear vibrations due to the limitation of capabilities in dissipating the oscillatory energy for the conventional passive dampers and the restriction of physical constraints for the use of the active controllers. The proposed controller is a sliding mass-spring-dashpot mechanism placed at the free end of the beam, introducing Coriolis, inertia, and centripetal nonlinearities into the system. The main objective of the study in this paper is to explore the nonlinear terms in the system which may be used to eliminate the transient vibration of the beam when the natural frequency of the slider is twice the fundamental beam frequency (internal resonance). Furthermore the amplitude, R 1, of the zeroth order solution for ¢~ (related to the original beam variable 0 by a linear transformation) can be adjusted by using the damping term in the equation for R 2, the amplitude of zeroth order solution of p (related to the original slider variable, r). Numerical results show that this technique may be used to increase the effective damping ratio of the structure from 0.5% to - 6%. The controller is particularly successful in reducing large amplitudes of oscillation to levels which may be handled by conventional methods. 135

136

M.F. GOLNARAGHI

Description of the Mathematical Model We will simplify the model of a cantilever beam to demonstrate the effectiveness of the technique in suppressing the oscillations of the fundamental mode. The model consists of a massless rigid bar of length £, with mass M 1 concentrated at the centre and a torsional spring K0 representing the beam stiffness (Figure 1). A dashpot C o represents structural damping. The controller, a slider mechanism with mass M 2, is placed at the free end of the bar. The radial stiffness is I% and damping is C r. Note that if active control is used, mass M 2 will be that of the actuator, position feedback gain will provide the stiffness 1% and damping will be the velocity feedback gain. The system is therefore a simple sliding-pendulum with two degrees of freedom represented by 0 and r. F

¥

Kr C

r

× C8 K8 Figure 1. A simplified model of the system. The nonlinear equations of motion of the mechanism are obtained using Lagrangian dynamics. They are then normalized to yield + 2~xo~O + ~o~0 + m ( r 2 + 2r)O + 2m(1 + r):O = O ,

(l-a)

(l-b)

f + 2 ~ 2 0 2 !: + ¢o~r - (1 + r)O 2 = O,

where, C, 2M 2 to

m = M1 4

'

'

(2-a)

(2-b)

VIBRATION SUPPRESSION IN FLEXIBLE STRUCTURES

2

13,

2

(2-c)

(D1 =

CO0

(Dr = 1 , (D2 = ~ , COO (DO

(2-d)

md where (-) represents derivatives with respect to nondimensional time ~', so that r = (Dot. thus (Do and (Dr are the natural frequencies of the bar and mass, respectively, (DI and (D2 are the lormalized natural frequencies, ~l and ~2 are the system damping ratios, and m is the lormalized mass. ¢¢e now proceed to scale equations (1). For the scaling process, we use a small dimensionless ~arameter e in (1), e < < 1, which represents the order of nonlinearities and coupling. This ;tep is important for the theoretical analysis which is based on the two variable expansion )erturbation method. We choose e so that we can perturb off the linear equations for small e. ro begin, we posit a change of variables such that 0 = e ~ , r = ep (3)

in order to perturb off the undamped linear equations, we choose to scale the damping ratios ~1 md ~2 as ~1 = e ~ l ,

~2 = ¢ ~ 2 "

(4)

leith these assumptions, the equations of motion take the following form: (5 -a)

2

+ 2e~2(D215 + ¢~2P - ¢(1 + e p ) + 2 = 0 .

(5-b)

the Two-Variable Exp~,nsion-Perturbation Method L/sing the two-variable expansion perturbation method [7-10], we replace the independent /ariable r by two new variables, E and rl , such that = T,11

= e~

;

(6)

~¢here E is just r and 11 is a slow time variable. The idea is to permit the dependent variables and ¢~ to depend explicitly on fwo time scales, E and 11. For example, periodic steady-state

138

M.F. GOLNARAGHI

behaviour will occur in ~, while approach to steady-state will occur in ~ . Using the chain rule, we can rewrite the time derivation of p (~, rl ) and ¢, (~, rl ) as d

_

d'~

c3 + e --~--o aT I

¢9~

d2

- --02 + 2 e ~ 0 2

a~ 2

d't "2

+ O ( e 2)

(7)

0~tgTi

We also expand p and ~ as

P = Po +epl +0( e2),4' = %

+e4h +O(e 2).

(8)

Upon substitution of (7) and (8) into (5) and collecting terms of same powers of E to O(E2), we obtain the following equations. For order e0: 2 ¢~0CC + ¢~0 : 0 , POCC + 0 2 P o

(9)

: 0 ;

For order e : 4h~ + ~I = - (2qbo~. + 2mPo~4~o~ + 2~'i~o~ + 2 m P o ~ o ~ )

Pl~

+ tO~Pl : t~2~ - 2 ( 2 0 2 P 0 ~

,

(lO-a)

(10-b)

- 2Po~. "

Here the subscripts represent the derivative with respect to ~ and 1] . The solution of (9) is in the form tb0 = R t ( ~ ) c o s ( ~ + qt~(~)), Po = R2(TI)c°s(t°2~ + q~201)) (11)

We now substitute (11) into (10), simplify the trigonometric terms and suppress the secular terms (ie. coefficients of sin ~, cos ~, sin ta 2 ~, and cos ta 2 ~). Furthermore, we will distinguish between internal resonance to 2 = 2 and the nonresonant case (ie. to 2 away from 2).

The Nonresonant Case In this case, to 2 is away from 2. In order to avoid the secular terms the following conditions must hold Rl. = - ~lRl , R2n = - ~2 c°2 R2 • (12) Thus, R l = A 1 eap(-¢lrl)

, ~

= A 2 e-xP(-¢2%rl)

,

(13)

or

=A l

exp(-e C ~)cos(~

+ %) ÷0 (14)

p

=A 2exp(-e

~2 ~ ) c o s ( t ° 2 ~ + qt2) + 0

VIBRATION SUPPRESSION IN FLEXIBLE STRUCTURES

13'

The solutions (14) are basically those of the linear uncouol~ systems, and to order e the two modes do not affect one another. As time 7 --..o, ¢ and-p will approach zero.

The Internal Resonance C a ~ This is the case when to2 is close to 2. We introduce a detuning parameter o to show the nearness of the two frequencies. Hence, to2 = 2

+ eo.

(15)

In this case, the solvability conditions which are used to eliminate the secular terms are: m

Rl,~ = - (1 R1 "~" RIRz sin 0 ,

(16-a)

1 2 R,21q = - ~2¢D2R2 + "~ R 1 ~

(16-b)

(

11/ ,

',2)

(16-c)

= 2 ~1 - ~ 2 "

(17)

in the absence of damping (ie. (l = (2 = 0), the exact solution of (16) can be obtained [10]. ~owever, due to the length of these expressions the solution is not shown here. It is sufficient o show that for the undamped system, equations (16) could be combined, upon dividing (16-a) ~y (16-b), eliminating time and integrating so that R~ + 4 m P ~

(18)

= E = const. ,

vhere E is the integration constant proportional to the initial energy of the system. Using the ransformation

R,+

P4: E

(1-;t),

(19)

qayfeh and Mook [10] show that when ¢i = (z = O, one can reduce (16) to one equation. ~'hus we get

4 (dX) 2 = j.2(I_X)_4m(C Em

d:l

oE(1-X).)2

E3 =F 2(j.) - G2(j.)

m ,

(20)

140

M.F. GOLNARAGHI

where F2(k) ~- G2(~.) for realistic motion. Note that C is an integration constant obtained in the intermediate steps which are not shown here. Rewriting (19) in terms of the equilibrium values of ~. we get 4 Em

(d~.) 2

~

= (X3-~.)(~.-X2)(k-~.t)

,

(21)

where ~1, ~-2, and ~'3 are the roots of the right hand side polynomial in (20), and depend on the system parameters. Furthermore, Nayfeh and Mook [10] show that ~.~, k2, and ~'3 satisfy three possibilities: 1) ~'t < ~'2 < k3, 2) ~'1 = ~'2 < ~'3, and 3) ~-i < ~-2 = ~3, where all cases possess a closed form solution. In the first case, the closed form solution to (21) can be expressed using Jacobi elliptic function sn as - RI2 - ~.3_(k3_~.2) E

sn 2

[t:(t_to);rl ]

(22)

where K -

¢m2 ~ E(~'3-~'l)4m

(23)

In this case the motion is bounded, and R~ and R 2 will have a stable periodic solution as shown in Figure 2-a. The energy in the system is exchanged continuously between R~ and R 2. This suggests that the response of the system to O(e), equations (11), be quasi-periodic (Figure 2-b). As shown, the response of 0 for an initial disturbance is aperiodic. In the second case, the two modes approach a steady state value of R I = 0 and R 2 = (E/4m) ~:2 as time goes to infinity. In this case the oscillatory energy of O to O(e) is expected to be absorbed by r. Finally for the last case, Nayfeh and Mook show that R1 = E~--~2 '

p~ = ~ _ ~ m ( l _ ~ . 2 ) '

(24)

In this case, the motion of (11) is expected to be periodic. Cases 2 and 3, however, are not stable and any small disturbance leads to a motion defined by case 1 [10]. Thus, we expect to see periodic motion and exchange of energy between the two modes R l and R 2 which implies O and r to be aperiodic.

The Internal Resonance Controller (Damoed Case) Once damping is added to the system, the motion will dissipate. However, energy continues to be exchanged between R 1 and R 2. In this case the system contains a stable equilibrium value ( R~ = R 2 = 0) for arbitrary phase. An interesting result which forms the basis of this study is that the rate at which R l dissipates can be adjusted using the damping term in the equation for R 2 (ie. (2)- As Figure 3 shows, A faster dissipation of R 1 implies a faster reduction in the amplitude of response of 0.

VIBRATION SUPPRESSION IN FLEXIBLE STRUCTURES

a)

1.5

b) 2'

U3

=,

14

R2 R1 --'

1.0

I'

.

0.5 o

0

-2

0

50

0

100

2'0

Slow Time ( 11 )

Figure 2.

,~

6'o

s'0

lot

Nondtmen$|onal Tim-, •

a) Modal amplitudes R l and R2, when (1 = (2 = 0, e = -1.5, m = 0.3, and Rl(0) = I and R2(0 ) = 0. b) Quasi-periodic motion o f the system in the absence o f damping (l = (2 = 0, (D2 1.85, m = 0.3, 0(0) ---- 1 and r(0) = 0. =

1.5 m

e¢ a

x7 =m

c

£L

E

<

0.5

"0 0

0

I

I

0

50

100

I

150

200

Slow Time ( 11 ) :igure 3.

Effect o f damping (2 on modal amplitude R 1. a) uncoupled linear case, (1 = 0.01, b) (1 = 0.01, (2 = 0, c) (i = 0.01, (2 = 0.01, and d) (i = 0.01, (2 = 0.1.

142

M.F. GOLNARAGHI

Numerical integration of the differential equations of motion, with and without the controller show that the slider mechanism could significantly reduce the large amplitude oscillations of the beam. For the structure without controller, the response is lightly damped with (~ = 0.5% (Figure 4-a). Using the slider mechanism, when damping ratios (t = 0.5% and (2 = 50%, the effective damping of 0 could be increased to - 6%, as shown in Figure (4-b). This value was obtained using the logarithmic decrement method. Thus the controller is useful in reducing large amplitudes of oscillations to more tolerable levels which could be eliminated using other passive and/or active systems.

a)

b)

,4 m

4

2 E ,< iD

0

g

o

¢D

E

:2

£ -4 2oo Nondtmenstonal Time •

Figure 4.

!

o

i

ioo

2q

Nonatmenslonel T~rne

Response of the system: a) without the controller: (~ = 0.5%, 0(0) = 1; b) with the controller: (l = 0.5%, (2 = 50%, t~2 = 1.85, m = 0.3, 0(0) = 1 and r(0) = 0.

Conclusions A sliding mechanism is proposed for controlling the oscillations of the fundamental mode of a cantilever beam. The controller, by introducing Coriolis and centripetal kinematic nonlinearities into the system, has been shown to be effective at ~2 ~" 2 internal resonance where the nonlinear coupling between the slider and beam is maximized, allowing the oscillational energy to be transferred from the beam to the slider. The damping in the slider is used to quench the oscillations of the beam. Although the nonlinear behaviour of systems similar to our proposed model have been studied in the literature (e.g. [10]), we believe using internal resonance to control structures is a new approach and has not been addressed. It is hoped that our results prompt future work in this area.

VIBRATION SUPPRESSION IN FLEXIBLE STRUCTURES

14~

Acknowledgement Fhe author wishes to thank Professors Raj Dubey, Glenn Heppler, Jan Huissoon, Frank Moon ~nd Richard Rand for their discussions. The author also likes to acknowledge the helpful ;uggestion of the referees which enhanced the quality of this paper. This work was supported t,y the Natural Science and Engineering Research Council of Canada. References

[1] [2] C3] [4] [5] [6]

[7] [8] [9] [10]

G.A. Foelsche, J.H. Griffin and J. Bielak, "Transient Response of Joint-Dominated Space Structures: A New Linearization Technique," AIAA Journal, Vol. 26, No. 10, 1988. P.M. Bainum, "A Review of Modelling Techniques for the Open and Closed-Loop Dynamics of Large Space Structures," Springer Series in Computational Mechanics, S.N. Atluri, A.K. Amos (Eds.) Large Space Structures: Dynamics and Control, 1988. M.F. Golnaraghi, F.C. Moon and R.H. Rand, "Resonance in a High-Speed FlexibleArm Robot," The Int. J. of Dynamics and Stability of Systems, Vol. 4, Nos. 3 & 4, pp. 169-188, 1989. A.Y. Chert and F.C. Moon, "Experimental Vibration Suppression in a Space Truss Using Self-equilibrated Collocated Feedback Control," under review, AIAA Journal. A.H. Von Flotow and B. Schafer, "Wave-Absorbing Controllers for a Flexible Beam," J. Guidance Control and Dynamics, Vol. 9, No. 6, pp. 673-680, 1986. A.A. Fen'i, "Damping and Vibration of Beams with Various Types of Frictional Support Conditions," Proceedings of the AIAA/ASME/ASCE/AHS/ASC 30th Structures, Structural Dynamics and Materials Conference, Mobile, AL, April 30, 1989. R.H. Rand and D. Armbruster, Perturbation Methods, BifurCation Theory_and Comouter Algebra Perturbation Methods, Applied Mathematical Sciences, 65, Springer-Verlag, New York, 1987. R.H. Rand, Computer Algebra in Applied Mathematics: An Introduction to MACSYMA, Research Notes in Mathematics, 94, Pitman Publishing, Boston, 1984. J. Kevorkian and J.D. Cole, Perturbation Methods in Applied Mathematics, SpringerVerlag, New York, 1981. A.H. Nayfeh and D.T. Mook, Nonlinear Oscillations, John Wiley and Sons, New York, 1979.