IMPACT DAMPERS FOR NON-LINEAR VIBRATING SYSTEMS

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Journal of Sound and Vibration(1995) 187(3), 403420

ON IMPACT DAMPER S FOR NON -LINEAR

VIBRATIN G SY STEMS

S. C, A. K. M A. G

Department of Mechanical Engineering,Indian Institute of Technology,Kanpur 208016,India

(Received17 May 1994,and in final form 24 August 1994)

The performance of an impact damper for controlling the vibration of a harmonicallyexcited, hard Duffings oscillator is investigated. Both elastic and inelastic collisions areconsidered. The optimum design is based around a low amplitude, stable solution predictedby the harmonic balance method. The final design is obtained and verified by numericalintegration. Also, the use of an impact damper as an onoff migration controller for a guidedtransition from the resonance to the non-resonance branch of the solution is proposed. Thisscheme obviates the need of continuous, noisy operation of an impact damper.

1995 Academic Press Limited

1. INTRODUCTION

Impact dampers have been successfully used for controlling the high-amplitude vibrationof various systems, such as cutting tools [1], turbine blades [2] and tall flexible structures likechimneys [3]. The vibration of the primary system is controlled by the transfer of momentumto a secondary (loose) mass through repeated impacts. Several research works have beenreported on the design and performance of an impact damper for controlling the vibrationof a harmonically excited linear system. For example, Grubin [4] and Warburton [5]considered a symmetric, two collisions per cycle motion of such a system and derived theoptimum free space for the loose mass to move between collisions. Masri and Caughey [6]

discussed the exact solution and its asymptotic stability for a similar situation. In anotherpaper, Masri [7] obtained all possible types of steady state motions and delineated the zonesin the parameter space where these different types of motion occur. It was concluded thatan impact damper is most effective with two symmetric collisions per cycle. Bapatetal. [8, 9]have considered both single- and multi-unit impact dampers and provided easy-to-use designcharts for the former. Yasuda and Toyoda [10] have provided an experimentally obtained,empirical, equivalent viscous damping coefficient to account for the inelastic collisions. Theuse of bean-bags as the secondary mass, instead of metallic shots, has been proposed byPopplewell and Semercigil [11] for noiseless operation of the impact damper. Masri andIbrahim [12] have considered the performance of an impact damper where the system issubject to a random excitation. Despite several advantages, such as simple constructionalfeatures, theuse of a passive impactdamper is limited dueto itsineffectiveness forbroadbandexcitations. Moreover, the optimal design of a passive impact damper is highly sensitive to

the model of the primary system. To overcome this limitation, an active impact damper withon-line control algorithm has also been proposed [13].

In this paper, the performance and design of an impact damper for controlling the highamplitude vibration of a non-linear oscillator is discussed. Specifically, the primary system

is taken as a harmonically excited, viscously damped, Duffing oscillator. Two different403

0022460X/95/430403+18 $12.00/0 1995 Academic Press Limited


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approaches are considered. First, the secondary mass is used in a passive manner to generatea stable low amplitude solution for the primary system. Towards this end, an initial designobtained by the single term harmonic balance method is improved upon by using thenumerically integrated results. In the second approach, the multi-stability feature of thenon-linear oscillator is utilized. A guided onoff control strategy is followed to use thesecondary mass to change the initial state of the primary oscillator. This change of statebrings the primary oscillator within the basin of attraction of the non-resonance (low

amplitude) branch from the resonance branch. This procedure has been referred to as theguided branch transition. Numerical results are presented to verify the solutions, theirstability and thereby the feasibility of the proposed approaches.

2. THEORETICAL ANALYSIS

2.1.

The model under consideration is shown in Figure 1. For a linear primary system, it hasbeen shown in previous works [7, 9] that an impact damper undergoing elastic collisionsperforms better than the one having inelastic collisions. Accordingly, here also we firstconsider elastic impacts between the primary massm1 and the secondary mass m2. The elasticimpacts are modelled by two linear contact springs, each of stiffnessk2. Later on, we shall

include the impact damping to simulate inelastic collisions having a coefficient of restitution1. The value of is known to control the frequency bandwidth of the effectiveness of

the impact damper [7]. However, the friction force between m1 and m2 is neglected.The equations of motion form1 and m2 are

m1x1 +c1x'1+klx1+k1x31+k2[(yd/2)U(yd/2)+(y+d/2)U(yd/2)]=Fcos t

(1)

and

m2y+k2[(yd/2)U(yd/2)+(y+d/2)U(yd/2)]=m2x1, (2)

where the prime denotes differentiation with respect to timet, and y=x1x2. All othersymbols are explained in Figure 1. In equations (1) and (2), U( ) is the Heaviside stepfunction defined as

U()=01 if0if0.

Figure 1. A model of an elastic impact damper.


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Substituting the non-dimensional parameters Z1=x1/x0, Z2=y/x0, 0t=T,

0=(k1x20 /m1), r1=k2/m120 , rm=m2/m1, d0=d/2x0, h=c1/m10, =/0,2l=kl/(m120 ) andx0=(F/k1)1/3 in equations (1) and (2) yields

Z1+hZ 1+2lZ1+Z31+r1(Z2)=cosT, Z2+(r1/rm )(Z2)=Z1, (3, 4)

where(Z2)=(Z2d0)U(Z2d0)+(Z2+d0)U(Z2d0) andthe dotdenotes differentiationwith respect toT.

After substituting into equations (3) and (4) solutions of the form

Z1=a cos (T+1), Z2=bcos (T+2) (5)

one can expand (Z2) in the Fourier series, retaining only the first harmonic. Thereafter,equating, respectively, the coefficients of sin (T+1), sin (T+2), cos (T+1) andcos (T+2) from both sides one obtains thefollowing set of non-linear algebraic equations:

a2la2+34a3+r1B(b) cos (21)=cos1, (6a)

har1B(b) sin (21)=sin1, (6b)

b2+(r1/rm )B(b)=a2 cos (12), sin (12)=0, (6c, d)

where

B(b)=b[1(2/){(d0/b)1(d0/b)2+sin1 (d0/b)}]

0ifd0/b 1ifd0/b 1, (6e)

Equation (6d) implies12=n,n=0, 1, 2, . . .. If12=0, 2, 4, . . . , then equations(6a), (6b) and (6c) yield

a2la2+34a3+r1B(b)=cos1, ha=sin1 (7a, b)

b2+(r1/rm )B(b)=a2. (7c)

Similarly, with12=, 3, 5, . . . , one obtains

a2la2+34a3r1B(b)=cos2, ha=sin2, b2+(r1/rm )B(b)=a2,

(8ac)

which, witha=a*, can be rewritten as

a*2la*2+34a*3+r1B(b)=cos2, ha*=sin2,

b2+(r1/rm )B(b)=a*2. (9ac)

Thus, one can see that the solution set (a,b,1) obtained from equations (7) is the same asthe solution set (a*,b,2) obtained from equations (9). Therefore, one can use onlyequations (7) to obtain all the solutions.

In all real life situations, the value ofr1 is very high, implying

b=d0+, where 01. (10)

Using equation (10) in equation (6e) yields

B(b)=(2/)2d0+o() ford0b, (11)where is obtained from equation (7c) as

=rm(ad0)2/22d0r1. (12)


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Using relations (11) and (12) in equations (7a) and (7b), one finally obtains

a2la2+34a3rm2(ad0)=cos1, ha=sin1. (13a, b)

By substituting=tan (1/2) in equations (13a) and (13b) and eliminating the variable a ,the following equation in is obtained:

6(+1)+5+(3+1)4+(+2)3+(31)2++(1)=0. (14)

Here =(2/h)(2l2rm2), =6/h33 and =rm2d0. After solving thepolynomial equation (14) for , one can find the values of the amplitudes (a) from equation(13b). Thus, all the symmetric, two impacts per cycle motions are obtained.

Before proceeding on to the stability analysis of these solutions (presented in the nextsection), it may be worthwhile to note that equations (13a) and (13b) represent the singleterm harmonic balance solution of the following differential equation:

x+hx+2lx+x3=cosTrmxrm2d0cos (T+1). (15)

Equation (15) implies that the addition of an impact damper to the primary oscillator isequivalentto the addition of acceleration feedback and force-modification mechanisms. Thisconfirms the use of the term acceleration damper [4, 5] as a synonym for an impact

damper.

2.2.

To analyze the stability of the solutions obtained in the previous section, one considers

Z1=a(T) cos (T+1(T)), Z2=b(T) cos (T+2(T)) (16a, b)

witha(T),b(T),1(T) and2(T) as slowly varying functions of time. Substituting equations(16a) and (16b) in equations (3) and (4) and carrying out the harmonic balance afterneglecting small terms such as a(T), b(T), 1(T), 2(T), ha, hb , etc., one obtains the followingfour autonomous differential equations:

a=(1/2){ha+r1B(b) sin (21)+sin1}, (17a)

1=(1/2a){cos1r1B(b) cos (21)a2

l3

4a3

+a2

}, (17b)

b=(1/2){sin2+hacos (12)+(a2l+34a3) sin (12)}, (17c)

2=(1/2b){b2r1B(b)(1+1/rm )+hasin (12)

(a2l34a3) cos (21)+cos2}. (17d)

While deriving equations (17a)(17d), the Fourier series of the function (Z2), withZ2=b(T) cos (T+2(T)), has to be worked out. However, the Fourier coefficients of(Z2)cannot be obtained in a closed form. Hence, the following approximation was made. Sincer1is very large, the contact time during an impact is very small. Consequently,(Z2) is zerofor most of the time during the period 2/and assumes a non-zero value only for a verysmall interval of time during which the slowly varying functions b(T) and 2(T) can

effectively be considered as constants. This approximation makes the closed form evaluationof the Fourier coefficients possible. The stability of the solutions can thus be judged fromthe signs of the real part of the eigenvalues of the Jacobian (see Appendix A) (evaluated at(a,b,1,2)) of the right side of equations (17). If the real parts of all the eigenvalues arenegative, the solution is stable and otherwise it is unstable.


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3. DESIGN OF AN ELASTIC IMPACT ABSORBER

By design of the impact absorber, we mean the determination of the values of the

parametersrm andd0 for which the response of the primary system, i.e., the value of theamplitude (a), is rendered small. Before going into the details of this design, the followingpoints should be mentioned.

(i) The variation of the amplitude of the primary system with the forcing frequency, inthe absence of the impact absorber, has the general trend shown in Figure 2 (for not toohigh values of h and l). The frequency corresponding to the maximum value of theamplitude is defined as the peak frequencyp .

(ii) The relative amount of non-linear contribution in the restoring force of the primarysystem can be controlled by the parameterl. A higher value oflimplies a lower relativenon-linearity andvice versa.

(iii) To integrate numerically the equations of motion, the fourth order RungeKuttaMerson algorithm of NAG with adaptive step-size control was used. We have randomlyselected a large number of initial conditions within a ball of radius 2 around the origin of

the phase space. All these initial conditions resulted in motions where collisions occurred.There may exist certain special combinations of initial conditions (depending on the valueof other parameters) where no collision occurs. In this work, no further attention has beengiven to such non-colliding motions.

(iv) The steady state of the motion has been ascertained by considering the Poincaresection (stroboscopic map with 2/as the strobe period) of the phase trajectories. Whenthe iterates of the Poincare map reach a fixed point, the corresponding motion obviouslyhas a period 2/. Similarly, the existence ofn fixed points in the iterates implies a motionhaving a period 2n/, which is used to locate the subharmonic solutions.

From the approximate solution given by equation (14), it is easy to see that ifrmand d0are chosen to satisfy the relation

rm2d0=1 (18)

then one of the values for the amplitude (a) turns out to be zero. Of course under such asituation, i.e., when equation (18) holds good, the solution of equations (7) does not givethe amplitude as exactly zero, but does reveal a very small amplitude of the primary system.By following the stability analysis presented in section 2.2, this small amplitude solution is

found to be stable. The existence of such a stable small amplitude solution is also confirmedby obtaining it through direct numerical integration. It was also observed that the otherimpacting solutions, different from this small amplitude solution, obtained from equations(14) and (7) match very closely.

Figure 2. A typical frequency response curve of a sinusoidally driven Duffing oscillator. , Stable branch; , unstable branch; , one-third subharmonic. l=0,h=0215.


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With the design frequency selected asp , the parametersrmand d0are chosen to satisfythe relation

rmd0=1/2p . (19)

The frequency responses obtained from equation (7) for a fixed value ofrmwithd0given byequation (19) are shown in Figures 3(a)3(c) for various value ofl. With the impact damperin operation, only the stable solutions are shown. It is seen that, irrespective of the value

of l, a stable small amplitude solution is always present around the peak frequencyp .Apart from this low amplitude solution, a high amplitude stable solution is also present.However, there is always a region aroundp in which only the low amplitude solution exists.We define the frequency range, aroundpwhere the impact damper introduces a single low

amplitude solution, in place of the resonance branch of the primary system, as thesuppression band (s ). These suppression bands, indicated in Figures 3(a)3(c), widen withincreasingl(i.e., decreasing relative non-linearity). However, within the suppression band,the percentage reduction in the amplitude is relatively insensitive to the degree ofnon-linearity. A decrease inlreduces the value ofp . Consequently, for a given value ofrm , the optimum value ofd0 obviously increases (see equation (19)) as can also be noted fromFigures 3(a)3(c). It is known from previous works [9] on linear systems, that near theresonance frequency, the loose mass undergoes two symmetric collisions per cycle. Toexamine what happens in the presence of a non-linear restoring force, the phase diagramsof the primary system were also studied. A representative diagram is shown in Figure 4,which clearly reveals the existence of two symmetric impacts/cycle.

From now onwards, the primary system will be considered to have only a cubic restoring

force: i.e., l=0. The frequency responses of the system obtained from equation (7) fordifferent combinations ofrmandd0satisfying equation (19) are shown in Figures 5(a)5(c).The results obtained by the direct numerical integration are also shown in the same figures.The small amplitude motion is vindicated by the direct numerical integration (revealing two

Figure 3. The frequency response of a primary system with and without the impact damper. , Stable branch(harmonic balance) with damper; , stable branch without damper; , unstable branch without damper.h=0215,r1=2000,rm=012. (a) l=15,d0=069; (b) l=1,d0=12; (c) l=0,d0=2.


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Figure4. A phaseplotof thelowamplitudemotion ofthe primary system. rm=012, d0=2,=2, h=0215,l=0,r1=2000. Initial conditions (1, 05, 0).

symmetric impacts per cycle). However, the high amplitude solution, obtained by integratingthe equation of motion (although numerically matched by those obtained by the harmonicbalance method), revealed more than two impacts per cycle. The frequency regions havingdifferent number of impacts per cycle were also seen to be separated by narrow chaotic

regimes.The number of impacts per cycle is not an important issue in the method of harmonic

balance. For a single-term approximation to yield meaningful results, only the energycontent in the fundamental frequency is of primary importance. Of course, the stabilityanalysis carried out in this work is approximate, and therefore is not able to capture all thedelicate instability regions. However, both the amplitude and the width of the suppression

Figure 5. Thefrequency response of the primary system withand withoutthe impact damper., Stable branch(harmonic balance); , numerical integration; , stable branch without damper; , unstable branchwithout damper;, one-third subharmonic without damper; , one-third subharmonic with damper. h=0215,l=0,r1=2000. (a)rm=012, d0=2; (b)rm=025, d0=1; (c)rm=04,d0=0625.


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band are estimated accurately by the harmonic balance method, which goes a long way toproviding an initial design. Both the harmonic balance and numerically integrated resultsshowed that the motion of the primary and the loose mass are almost in anti-phase forpand almost in phase forp .

One can define the set of values of the pair (rm ,d0) satisfying equation (19) as theoptimum set. In the following discussion we vary (rm ,d0) only in that set. A comparisonof Figure 5(a) with Figure 5(b) reveals that the suppression band widens with increasing

values of rm . Thus, a high value of rm is to be preferred for a wider suppressionband. Moreover, the higher the value of rm , the lower is the value of the oscillationamplitude in the suppression band. However, too high a value of rm makes the systemmore prone to subharmonic oscillations (within the suppression band) as indicated in

Figure 5(c).To determine the optimum values ofrmandd0, we compute the response of the primary

system at the design frequency for three different values ofrmwith varying values ofd0. Themaximum displacement of the primary mass is obtained both from equations (7) andnumerical integration. These results are shown in Figure 6. The numerically integratedresults areseen to yield a slightly highervalue of theoptimumd0 as compared to that obtainedfrom the harmonic balance method. However, the motion of the primary mass atpis seento be rather insensitive to the value ofrmif the optimum value ofd0is used. Thus, dependingon the required suppression bandwidth and available space, a value ofrmcan be chosen andthe first estimate ofd0should be obtained from equation (19). Around these values ofrmandd0, the exact optimum values can then be obtained from the numerically integrated results.The insensitivity of the primary response near the optimum design ensures satisfactory

performance of the impact damper, even if the actual parameters are a little away from thetheoretical optimum values. These observations remain valid for different values of theparameterh , as can be seen from Figure 7.

Another important feature of the impact-damper for a non-linear primary system is thesuppression of the one-third subharmonic response as discussed below. Figure 2 reveals thatthe primary system without the damper can have one-third subharmonic oscillation near thedesign frequency. In Figures 5(a) and 5(b) it is clearly shown that the one-third subharmonicmotion can even be completely suppressed by a properly designed impact damper. With

Figure 6. A comparative study of analytical and numerical optimum design values, h=0215, =2.rm=025: , analytical,, numerical. rm=018: , analytical,, numerical. rm=012: , analytical,, numerical.


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Figure 7. A comparative study of analytical and numerical optimum design for various values of the parameterh.h=0107,=p (=28):rm=018, , analytical, , numerical;rm=012, , analytical, , numerical.

h=015, =p (=237):rm=018, , analytical, numerical;rm=012, , analytical, , numerical.

higher values ofrm , the one-third subharmonic motion may appear but with considerablylower amplitude. Such control of the one-third subharmonic response is prominentlydisplayed with lower values ofh as indicated in Figure 8.

Figure 8. The frequency responseof the primary system withand without the impact damper. h=0107, r1=2000,rm=012. . Stable solution without damper; , unstable solution without damper; , analytical stablesolution with damper;, numerical low amplitude solution with damper;, numerical high amplitude solutionwith damper; , one-third subharmonic solution without damper; , one-third subharmonic solution withdamper.


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Figure 9. A model of an inelastic impact damper.

4. INELASTIC IMPACT DAMPER

A mathematical model of an impact damper undergoing inelastic collisions is shown inFigure 9, where the impact is modelled by a linear spring with stiffnessk2and a dashpot withviscous damping coefficient c2. The non-dimensionalized equations of motion can be writtenas

Z1+hZ 1+Z3

1+1(Z2,Z 2)=cosT, Z2+2(Z2,Z 2)=Z1, (20, 21)with

1(Z2,Z 2)=r1(Z2)+2cr1rmM(Z2)Z 2, (22)

2(Z2,Z 2)=(r1/rm )(Z2)+2cr1/rmM(Z2)Z 2, (23)

M(Z2)=U(Z2d0)+U(Z2d0). (24)

As shown in Appendix B, the damping ratioc is obtained in terms of the coefficient of

restitution () as

c=ln ()/(1+rm )[2+(ln ())2]. (25)

To study the effects of the coefficient of restitution on the performance of the impactdamper, equations (20) and (21) were numerically integrated to obtain the primary response.

The frequency response of the primary system is plotted in Figure 10 for=075. One canobserve from this figure that the suppression band with1 is wider than that with=1.However, the response amplitude with inelastic collisions is marginally higher than that with

Figure 10. The frequency response of an inelastic impact damper. =075, rm=025,d0=1,h=0215. ,Stable branch without damper; , unstable branch without damper; , stable branch with elastic damper[analytical];, numerical with elastic damper, , one-third subharmonic without damper; +, numerical withinelastic damper.


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Figure 11. Numerical optimum design values of inelastic damper. , rm=012; + ,rm=018; ,rm=025; all for=075. ,rm=012; ,rm=018 all for =09.

elastic collisions. In general, the higher the impact damping (i.e., the lower the value of)the wider is the suppression band, butthe larger is the response amplitude (in the suppressionband). A similar behaviour was also seen for a linear primary system [7]. Thus, a high valueof, signifying almost elastic collisions, is preferable for optimum performance in the caseof a synchronous operation.

The variations of the amplitude of the primary system withd0for various values ofrmandare shown in Figure 11. From this figure it may be concluded that the higher the impactdamping, the lower is the required value ofd0for optimum performance. Moreover, as theimpact damping increases, the primary response becomes more insensitive to the variationind0 (for a near optimal design).

5. USE OF IMPACT DAMPER FOR GUIDED BRANCH TRANSITION

The concept of a guided branch transition, or migration control as it is called, is recent

in the literature [14, 15]. Non-linear systems generally have multiple, steady state, periodicor strange (chaotic) attractors for fixed parameter values and all such attractors coexist withtheir respective basins of attraction. The role of a migration controller is to lead the systemto a pre-assigned attractor from any initial condition. If the initial condition falls within the

basin of an undesirable attractor, then the migration controller operates and guides thesystem to the basin of attraction of the desired attractor. Once this is achieved, the controlleris released. The whole concept can be recast mathematically as follows:

A forced non-linear oscillator can be described by an autonomous first-order vectordifferential equation as

Z =f(Z,p()), = ,

whereZ Rn and S1,f: RnS1Rn, andp :S1Rl withln. Here the dot representsdifferentiation with respect to timeTandis a constant. With control variablesZcand the

control forcefc (Zc ), the system equations take the formZ =f(Z,p())+fc (Zc ), Z c=f*c (,Z,Zc ), = , (27)

whereZc Rm,fc : RmRn andf*c : RnRmS1Rm. We represent the basin of attraction ofthe undesirable attractorAudof equations (26) by the set BAud and that of the attractor


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Figure 12. The migration control scheme.

Acof equations (27) byBAc . Let the basin of attraction of the goal attractorAdof equations(26) be represented byBAd. Thus,BAudR

nS1,BAdRnS1 andBAcR

n+mS1. We definethe flow represented by equations (26) and (27) byTwc and

Tc, respectively. Let the initial

conditions be represented by a pointPr1(Z0,0) BAudsuch that limTTwc (Pr1(Z0,0))Aud.

However, if the control force is added, Pr1 gets extended to P1(Z0,0,Zc0) BAc andlimT

Tc(P1(Z0,0,Zc0))AcBAc . If the control is released when the motion reaches a

point P2 [AcBAd], then P2 is restricted to Pr2(Z1,1) [ BAd] when

limTTwc ((Pr2(Z1,1))Ad [BAd]. The whole control action is schematically shown in

Figure 12.In sections 2 and 4, we have seen that an impact damper can reduce the response of

a harmonically driven, Duffing oscillator, especially in a frequency band correspondingto the high amplitude resonance branch. Using the multiplicity features of the steadystate motion of Duffing oscillator, here we use an elastic impact damper as a migrationcontroller. It will be seen how, after starting from the basin of attraction of the resonancesolution, the system goes to the non-resonance branch when an impact damper is usedas a controller. It is generally seen that, near the high frequency part of the resonance

branch, the probability of finding an initial condition for the non-resonance motion ishigher near the trivial (zero) initial conditions. Around these trivial initial conditions, awide zone exists which forms the subset of the basin of attraction of the non-resonancemotion [16]. It has already been seen in sections 2 and 3 that an impact damper brings the

primary system to a near-trivial motion. Thus, the points on the steady state attractor ofthe system (including the impact damper) might be completely embedded within thebasin of attraction of the non-resonance motion of the primary system (without the impactdamper). At the steady state, the primary motion might choose the initial point suitablefor the non-resonance oscillation when the impact damper can be switched off. However,as the system is non-autonomous, the phase of the excitation at the time of withdrawalof the control is very important. The phase should correspond to the desired goal. Thus,using the migration control one can avoid the continuous noisy operation of an impactdamper.

A numerical example showing the feasibility of the above scheme is as follows: say

Z=Z1Z1, Z1=Z 1 and f= Z1f1(Z1,Z1,), =,


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Figure 13. The oscillator signature of the primary system with migration control. =195,h=0215,rm=012,d=2,r1=2000.

wheref1(Z1,Z1,)=Z31hZ1+cos. If an impact damper is used as a controller

Zc=Z2Z2, Z2=Z 2, fc (Zc )= 0fc 1(Z2)and

f*c (,Z,Zc )= z2(1+1/rm )fc1(Z2)f1(Z1,Z1,),where fc 1(Z2)=r1{(Z2d0)U(Z2d0)+(Z2+d0)U(Z2d0)}. Let us take r1=2000,h=0215, d0=2 and rm=012. We consider the operation of the impact damper as amigration controller at three different values of the excitation frequency.

(i) =195. With the above-mentioned functional form and prescribed parameter values,we numerically integrate equations (26) starting from an initial condition. Let the initialcondition be within the basin of the resonance motion. After the motion has reached thesecond state, we turn the controller (i.e., the impact damper) on and again numericallyintegrateequations (27) up to thesteady state.When thecontrol is switched off at some point,thesubsequent motionleads to thenon-resonance branch. We repeatthe procedure to ensurethat even if the control is released at any arbitrary forcing phase, the motion still migratesto the non-resonance branch. One example showing the primary response (Z1) plottedagainst the non-dimensional time (T) is shown in Figure 13. The instants of switching on

and off the impact damper are indicated in this figure.(ii) =17. The steady state attractor of the system with the controller (equations (27))

passes through the basin of one-periodic non-resonance motion and three-periodicmotion of the system, described by equations (26), as shown in Figure 14. Therefore,

depending on the time of release of the control, the system might choose initial conditions


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Figure 14. The steady state attractor of the primary. =17,rm=012, d0=2,h=0215. , The points whichthe motion without the impact damper goes to one-third subharmonic oscillation; , the points from which themotion without the impact damper goes to the non-resonance motion.

(definitely on the above-mentioned attractor) leading to either of the two states. Thus,one may need several onoffs to reach the non-resonance one-periodic motion, as shownin Figure 15.

(iii) =185. The response is plotted against time in Figure 16 for the case in which the

control is released before the steady state is reached. The motion is seen to reach thethree-periodic solution. Several switchings of the controller ultimately lead the system to thenon-resonance one-periodic motion.

For a number of parameter values tested, it was observed that several onoff switchingsspanning, in total, not more than 350 cycles of operation could lead the response to thenon-resonance branch. For a safe application of an impact damper as a migrationcontroller,

Figure 15. The oscillatory signature of the primary with migration control. =17; other parameters as inFigure 13.


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Figure 16. The oscillatory signature of the primary with migration control. =185; other parameters as in

Figure 13.

one needs to design the impact damper for its optimum performance. Thereafter, a hit andtry control algorithm depicted in Figure 17 can be followed for migrating to the desiredmotion. An experimental investigation of the proposed scheme is in progress.

Figure 17. An algorithm of migration control.


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6. CONCLUSIONS

The performance of an elastic impact damper attached to a harmonically excited Duffing

oscillator has been presented. The existence and stability of a low amplitude motion nearthe high frequency, large amplitude resonance branch of the primary system has beenestablished by using the harmonic balance method. The optimum design predicted by theharmonic balance solution is refined by numerical integration. For a given mass ratio (rm ),the optimum free space for the loose mass (d0) is obtained. The primary response is seen tobe rather insensitive to the values of bothrm and d0 near this optimal design. With theoptimum design, two symmetric collisions per cycle are always seen to occur. Thesuppression band of the impact damper narrows down with increasing non-linearity in therestoring force, but the percentage reduction of amplitude at the design frequency is seento be insensitive to the degree of non-linearity. An impact damper is very effective forreducing (or even eliminating) the one-third subharmonic response. An inelastic damperwidens the suppression band at the cost of higher primary response. An onoff migrationcontrol strategy, using an impact damper, for a guided transition from one solution branch

to theother, is proposed.The feasibility of the proposed schemeis verified through numericalexamples. This scheme, applicable to other non-linear primary systems having multi-stablesolutions, obviates the need for continuous, noisy operation of an impact damper.

ACKNOWLEDGMENT

The authors gratefully acknowledge the suggestions of the reviewers (unknown) on anearlier version of the paper.

REFERENCES

1. M. M. S 1972Machinery120, 152161. Impact dampers for controlling vibration in machinetools.

2. A. L. P1937Engineering 143, 305307. Vibration in steam turbine buckets and damping byimpacts.

3. H. H. R 1967 Wind Effects on Buildings and Structures, Proceedings of Internal ResearchSeminar, Ottawa. Hanging chain impact dampersa simple method of damping tall, flexiblestructures.

4. C. G 1956Journal of Applied Mechanics 23, 373378. On the theory of acceleration damper.5. G. B. W1957Journal of Applied Mechanics 24, 322324. On the theory of acceleration

damper.6. S. F. M and T. K. C 1966 Transactions of the American Society of Mechanical

Engineers,Journal of Applied Mechanics E33, 586592. On the stability of the impact damper.7. S. F. M1970Journal of the Acoustical Society of America 47, 229237. General motion of

impact dampers.8. C. N. B, S. Sand N. P1983 53rd Shock and Vibration Bulletin 4, 112.

Experimental investigation of controlling vibrations using multi-unit impact damper.9. C. N. Band S. S1985Journal of Sound and Vibration 99, 8594. Single unit impact

damper in free and forced vibration.10. K. Y and M. T 1978 Bulletin of the Japan Society of Mechanical Engineers 21,

424430. The damping effect of an impact damper.11. N. P and S. E. S 1989 Journal of Sound and Vibration 133, 193223.

Performance of the bean bag impact damper for a sinusoidal external force.

12. S.F.M

andA.M.I

1973 Earthquake Engineering and Structural Dynamics1

, 337346.Stochastic excitation of simple system with impact damper.13. S. F. M, R. K. M, T. J. Dand T. K. C 1989 Transactions of the

American Society of Mechanical Engineers, Journal of Applied Mechanics 56, 656666. Activeparameter control of nonlinear vibrating structures.

14. E. A. J1991Physica D 50, 341366. On the control of complex dynamic systems.


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15. L. K, A. Sand L. O. C1993International Journal of Bifurcation and Chaos 3,479483. Transitions in dynamical regimes by driving: a unified method of control andsynchronization of chaos.

16. T. F and E. H. D1987 International Journal of Non -linear Mechanics 22, 267274.Numerical simulations of jump phenomena in stable duffing system.

APPENDIX A

The elements of the JacobianJof the right side of equations (17) are as follows:

J=

J11J21J31J41

J12J22J32J42

J13J23J33J43

J14J24J34J44

,

where

J11=h2

, J12=r1B(b)

2 cos(21)

12

cos1, J13=r12

sin(21)B(b)b

,

J14=r1B(b)

2 cos(21), J21=

cos1r1B(b)cos(21)2a2

+34

a

,

J22= 1

2a{sin1r1B(b)sin(21)}, J23=

r1cos(21)2a

B(b)b

,

J24=r1B(b)

2a sin(21), J31=

h2

cos(12)9a28+2l2sin(12),

J32=ha2

sin(12)3a38+a2l2cos(12), J33=0,J34=

1

2{cos2+hasin(12)(a2l+34a

3)cos(12)},

J41= 1

2b{hsin(12)(2l+94a

2)cos(21)},

J42= 1

2b{hacos(12)(a2l+34a

3)sin(21)},

J43= 1

2b2b21rm+1r1B(b)bf5,

J44=ha2b cos(12)

a2l2b+

3a3

8bsin(21)+

12bsin2,

f5=b2r1B(b)1+ 1rmhasin(12)(a2l+34a3)cos(21)+cos2.


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Figure B1. A model of an impact pair.

APPENDIX B

The mathematical model of an impact pair is shown in Figure B1. The contact dynamicsof the two massesm2 and m1 are represented by the differential equations

m2X2+c2(X 2X 1)+k2(X2X1)=0, (B1)

m1X1+c2(X 1X 2)+k2(X1X2)=0, (B2)

where the dot denotes differentiation with respect to timet .SubstitutingYc=X2X1 and rm=m2/m1 in equations (B1) and (B2) yields

Yc+2nY c+2nYc=0, (B3)

where =c21+rm/2m2k2 and n=(k2/m2)(1+rm). At the beginning of the contactphase one can write that att=0,Yc=0,Y c=V, and at the end of the contact phase, whenYc is again zero, one obtainsY c=Ven/d whered=n12. Thus, according to thedefinition of the coefficient of restitution (), =exp[/12], which givesc=c2/2k2m2=ln()/[2+ln()2]1/2(1+rm)1/2. This model of an impact pair is seen to besatisfactory only for high values of the coefficient of restitution. The model seems to fail forlow values of implying a high value ofc. A high value of damping enhances the time ofcontact and thus violates the condition of zero contact time during impact.

IMPACT DAMPERS FOR NON-LINEAR VIBRATING SYSTEMS

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