A time-domain non-linear viscoelastic damper model

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Smart Mater. Struct. 5 (1996) 517–528. Printed in the UK

A time-domain non-linear viscoelasticdamper model ∗

Farhan Gandhi † and Inderjit Chopra ‡Center for Rotorcraft Education and Research, Department of AerospaceEngineering, University of Maryland, College Park, MD 20742, USA

Received 6 May 1996, accepted for publication 22 August 1996

Abstract. A viscoelastic solid model, comprising of a combination of linear andnon-linear springs and dashpots, is developed to represent an elastomeric damper.A non-linear constitutive differential equation is derived to characterize the damperbehaviour in the time domain. A system identification method is presented todetermine the spring–dashpot parameters (coefficients of the constitutive equation)from experimental data. The model is able to predict the non-linearamplitude-dependent behaviour of elastomeric dampers under single as well asdual-frequency excitations. A ‘two-level implicit–implicit’ scheme is developed forthe integration of the non-linear damper model into a structural dynamic analysis.With increase in amplitude of excitation, softening behaviour of the lead springresults in lesser motion in the Kelvin chain, lower damping levels, and slower decayof initial perturbations. The baseline model is augmented with additional series andparallel non-linear springs to represent the reduction in damping (G ′′) at very smallamplitudes, and the occurrence of limit cycle oscillations.

1. Introduction

Elastomeric materials are becoming increasingly popular for structural vibration attenuation applications due tothe tremendous advantages they offer over mechanicaldampers. For example, elastomeric dampers arerapidly replacing conventional hydraulic dampers toaugment the lag mode damping of helicopter rotors.Under dynamic conditions, elastomers exhibit viscoelasticbehaviour, dissipating energy through hysteresis. Besidesdisplaying the frequency and temperature-dependence ofcharacteristics of viscoelastic materials, these materialsare nonlinear with respect to the amplitude of motion.Consequently, it has been difficult to develop analyticalmodels that can adequately characterize the behaviourof these elastomeric dampers. The requirement for anonlinear viscoelastic damper model that can be easily andconveniently integrated into a structural dynamic analysisposes an even greater challenge.

For viscoelastic materials, it is well known that theconstitutive relations depend not only on the instantaneousvalues of stress/strain, but also on the stress/strain timehistories. These constitutive relations can be expressedeither in an integral form, or as a differential equation [1].The disadvantage of the differential equation representationis the possible presence of several higher derivatives of

∗ Modified version of paper presented at the SPIE North AmericanConference on Smart Structures and Materials (Orlando, Florida, 1994).† Currently Assistant Professor of Aerospace Engineering, ThePennsylvania State University.‡ Professor and Director.

stress and strain. The integral representation circumventshigher stress/strain derivatives and is useful for creep orrelaxation studies of viscoelastic materials. However, in thestudy of structural dynamics, we encounter a set of second-order differential equations, and incorporating an integralrepresentation of viscoelastic material behaviour leads tointegro-differential equations, the solutions of which arecumbersome and computationally expensive.

Over the last couple of decades, the theory ofviscoelasticity has developed to a mature level [2, 3], withapproaches such as the ‘complex modulus’ and ‘generalizedor fractional derivative’ approach having gained wideacceptance for dynamic analyses. The complex modulusapproach, a frequency domain approach, is most suitablefor steady-state harmonic response of linear viscoelasticmaterials. For non-linear viscoelastic materials, thecomplex modulus componentsG′ and G′′ can showconsiderable variation with the amplitude of excitation.Additionally, the variation ofG′ and G′′ becomes moreinvolved under multi-frequency excitation conditions [4]making this method quite unattractive. The fractionalderivative approach [5, 6], being a time-domain approach,is suitable for transient analyses as well, but is applicableonly to linear viscoelastic materials. A few other variedattempts have been made to calculate the dynamics ofviscoelastic structures [7–9]. The Golla–Hughes–McTavishmethod [9] proposes a mini-oscillator model, and usesa transfer function approach, replacing higher derivativeswith internal dissipation coordinates. However, all of thesemethods are developed for linear viscoelastic materialsonly.

0964-1726/96/050517+12$19.50 c© 1996 IOP Publishing Ltd 517

F Gandhi and I Chopra

Most of the theories proposed to model the behaviour ofnon-linear viscoelastic materials are based on the hereditaryintegral representation. The modified superpositionprinciple by Findley and Lai [10], the Bernstein–Kearsley–Zapas theory [11], and the Schapery thermodynamictheory [12], all describe the non-linear stress/strainconstitutive equation in integral form. More recently,Glockner and Szyszkowski [13, 14] presented semi-empirical constitutive models, in integral form, to predictthe creep, strain softening and relaxation behaviour of non-linear viscoelastic materials. These non-linear viscoelastictheories, being in the integral form, cannot be convenientlyincorporated into a structural dynamic analysis.

With this background, the objective of the presentstudy is to develop a non-linear spring–dashpot viscoelasticmodel for an elastomeric damper, a differential equationrepresentation for the model, and a solution scheme forstructural dynamic equations with the inclusion of thisnon-linear viscoelastic damper. It should be mentionedthat Szyszkowski, Dost and Glockner [15] presented anon-linear spring–dashpot viscoelastic fluid model and itsconstitutive differential equation, to represent the primaryand secondary creep stages of ice. However, no techniquewas presented to obtain the model parameters, whichwere adjusted to match experimental data. Also, nomethodology was presented for the integration of the non-linear viscoelastic constitutive equation into a structuraldynamic analysis.

2. Elastomeric damper model and constitutiveequation

Figure 1 shows a simple representation of a standardviscoelastic solid. When all of the elements (springsS1, S2, and dashpot D2) in this model are linear, weget the standard 3-parameter linear viscoelastic solid. Itis hypothesized that it should be possible to representa non-linear viscoelastic material by a combination oflinear and non-linear elements. When the model infigure 1 is subjected to a static load, the spring S1, alone,deforms instantaneously, at timet = 0. The Kelvinchain (composed of spring S2 and dashpot D2) wouldsubsequently creep. Since the static stress/strain curve ofan elastomeric material is non-linear, a phenomenologicalapproach would require that the spring S1, be non-linear.If S2 and D2 could be assumed linear, a concise non-linear differential equation could be obtained for the model,and considerable simplification would be possible in thesolution process. The validity of such an assumption isto be tested by comparison with experimental data for theelastomeric damper sought to be modelled.

When a viscoelastic damper model consisting of non-linear spring S1 in series with a Kelvin chain (comprisingof linear springK2 and linear dashpotC2) is subjected toa force D, the spring S1 and the Kelvin chain undergodeformations x1 and x2 respectively. The non-linearforce/displacement relation of the spring S1 is assumed tobe

x1 = f (D) = sign(D)[c1|D| + c2|D|2 + c3|D|3 + c4|D|4

].

(1)

Figure 1. Spring–dashpot model for a viscoelastic solid.

The above form is based on the assumption that theinstantaneous response of the non-linear viscoelasticmaterial under positive and negative forces is the same.The behaviour of the Kelvin chain is given by

K2x2 + C2x2 = D. (2)

Using (1), (2), andx = x1 +x2, the constitutive differentialequation relating to the total damper displacement,x, to thedamper force,D, can be obtained as

K2x + C2x = K2f (D) + C2df

dDD + D. (3)

For a viscous damper, the damping force can be explicitlyobtained asD = G(x), and this expression for dampingforce can be directly introduced into the appropriateoscillator (or structural dynamics) governing differentialequation. For a viscoelastic damper, such as the oneunder consideration, since an explicit expression for damperforce is not available for introduction into the oscillatorequation, the damper constitutive equation, (3), needs tobe solved simultaneously with the oscillator equation, withthe damper force,D, appearing as an independent state.Alternatively, an internal displacement (such asx1 of x2)can be used as an independent state in place of the damperforce. For example, ifx1 is the independent variable, (1),(2) and (3) can be written as

D = h(x1) = sign

(x1)

[c1|x1| + c2|x2

‖ + c3|x1|3 + c4|x1|4]

(1a)

K2x2 + C2x2 = K2(x − x1

) + C2(x − x1

) = D (2a)

K2(x − x1

) + C2(x − x1

) = h(x1

). (3a)

The viscoelastic constitutive differential equation, (3a), isthen solved simultaneously with the oscillator (or structuraldynamic) equation, with the damper force in the oscillatorequation expressed by (1a).

For representation of viscoelastic behaviour over abroader frequency range, additional Kelvin chains wouldhave to be included in the viscoelastic solid model. In thiscase additional states (higher derivatives of damper force,or additional internal dissipation coordinates,x2, x3, . . .)would be present in the damper constitutive differentialequations. For example, when a second Kelvin chain(comprising of linear springK3 and linear dashpotC3) is

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Non-linear viscoelastic damper model

introduced, the damper non-linear constitutive differentialequation can be represented as

a1x + a2x + a3x = a1f (D) + a2df

dDD

+ a3

(df

dDD + d2f

dD2D2

)+ a4D + a5D (4)

where a1 = K2K3, a2 = K2C3 + K3C2, a3 = C2C3,a4 = K2 + K3 and a5 = C2 + C3. The derivationof equations (3) and (4) and method for obtaining theconstitutive differential equation for a model with non-linear spring S1 and an arbitrary number of Kelvin chainsis outlined in Appendix A.

Thus, viscoelastic solids can be modelled using oneor several Kelvin chains (depending on the frequencybandwidth required); with either damper force andderivatives, or internal dissipation coordinates used asindependent states. In the present paper, a non-linearviscoelastic damper model with a single Kelvin chain isdeveloped and its behavioural characteristics are examined.Damper force, D, it used as the independent state.However, the methodology described can be extended toa viscoelastic damper model with several Kelvin chains,or with internal dissipation coordinates as the independentstates.

3. System identification

Having obtained the constitutive differential equation, (3),of the non-linear viscoelastic damper model symbolically, asystematic method is developed to evaluate the coefficientsof this equation. These coefficients can be related directlyto the parameters in the spring–dashpot model, and canbe calculated from available experimental data. Mostexperimental data for elastomeric dampers are presentedin terms of the variation of complex modulus componentsG′ (stiffness) andG′′ (damping) with dynamic amplitude[4, 16]. Thus, the first step in the process for theidentification of system parameters is to obtain analyticalexpressions for the complex modulus componentsG′ andG′′, in terms of the unknown parameters. To evaluatethe complex modulus components, the system needs to besubjected to a harmonic excitation.

It is well known that the response of a linear viscoelasticsystem to a harmonic excitation forceD = D0 sin�t ,would be of the form,x = xc cos�t +xs sin�t . Since non-linear system response would contain higher harmonics, itis assumed to be

x =∑

n

{xnc cosn�t + xns sinn�t

}. (5)

The expressions for damper forceD = D0 sin�t ,assumed damper displacement (5), and their derivatives,are introduced into the non-linear constitutive equation (3).Since we are only interested in the coefficientsx1c andx1s,(hereafter referred to asxc and xs), (3) is multiplied bycos�t and sin�t respectively, and integrated fromt = 0

to 2π/� to yield

K2xc + (C2�

)xs = (

C2�)c1D0 + 8

3π

(C2�

)c2D

20

+3

4

(C2�

)c3D

30 + 32

15π

(C2�

)c4D

40 (6a)

K2xs − (C2�

)xc = (

K2c1 + 1)D0 + 8

3πK2c2D

20

+3

4K2c3D

30 + 32

15πK2c4D

40. (6b)

In (1), since the definition off (D) differs over thedomain (depending on the sign ofD), the integrationperiod [0, 2π/�] was sub-divided into two parts, [0, π/�]and [π/�, 2π/�]. In the first part, sin�t and D =D0 sin�t ≥ 0, while in the second part,D ≤ 0. Ifappropriate care is not taken in this regard, the quadraticand quartic terms in (6a) and (6b) will drop out. Solving(6) simultaneously forxc andxs

xc = −(C2�)D0

K22 + (C2�)2

(7a)

xs = K2D0

K22 + (C2�)2

+ c1D0 + 8

3πc2D

20

+3

4c3D

30 + 32

15πc2D

40. (7b)

With these expressions forxc and xs, G′ and G′′ can bedetermined as

G′ = D0

∣∣xs

∣∣/(x2

c + x2s

)(8a)

G′′ = D0

∣∣xc

∣∣/(x2

c + x2s

). (8b)

Note thatxc andxs, and thereforeG′ andG′′, are functionsof the system parametersc1, c2, c3, c4, K2 andC2 whichwe seek to determine, as well as the amplitudeD0, ofthe excitation force. At different amplitudes of excitation,experimental valuesG′

exp and G′′exp are available in [16].

We seek to minimize the function

F =N∑

i=1

[(G′

i − G′iexp

)2+

(G′′

i − G′′iexp

)2](9)

where N denotes the number of experimental datapoints considered (at different amplitudes of excitation).Minimization of F implies differentiating (9) with respectto each of the coefficientsc1, c2, c3, c4, K2 andC2, andequating to zero.

dF

dcj=

N∑i=1

[2(G′

i − G′iexp

)dG′i

dcj+ 2

(G′′

i − G′′iexp

)dG′′j

dcj

]= 0 (j = 1, 4)

dF

dK2=

N∑i=1

[2(G′

i − G′iexp

) dG′i

dK2+ 2

(G′′

i − G′′iexp

)dG′′i

dK2

]= 0

dF

dC2=

N∑i=1

[2(G′

i − G′iexp

)dG′i

dC2+ 2

(G′′

i − G′′iexp

)dG′′i

dC2

]= 0. (10)

Expressions for derivativesdG′i

dcj, dG′′

i

dcj, dG′

dK2, dG′′

i

dK2, dG′

i

dC2, and

dG′′i

dC2, are given in appendix B.

519

F Gandhi and I Chopra

Table 1. Coefficients of the damper non-linear differentialequation.

c1 = 1.2525996 × 10−6 rad(N m)−1

c2 = 9.6533565 × 10−9 rad(N m)−2

c3 = −2.9611342 × 10−12 rad(N m)−3

c4 = 2.9745904 × 10−16 rad(N m)−4

K2 = 22216.08 N m rad−1

C2 = 7221.62 N m sec rad−1

Figure 2. Variation of G ′ and G ′′ with dynamic amplitude.

An unconstrained optimization problem may sometimeslead to negative values of parameters such asK2 and C2.Since this would have no physical meaning, additionalconstraint such asK2 > 0, C2 > 0, c1 > 0, could beused as required. The minimization ofF with inequalityconstraints could then be solved using slack variables andLagrange multipliers [17].

The parametersc1, c2, c3, c4, K2 and C2 obtainedthrough the optimization process completely determine thenon-linear viscoelastic damper system represented by (3)and (1), and are given in table 1. The experimental data[16] used to determinec1, c2, c3, c4, K2 and C2, werefor an elastomeric lag damper for helicopter rotor blades.Thus ‘x’ and ‘D’ in the damper governing equation (3),represent the rotor blade lag angle and the lag-dampermoment, respectively. For correlation with experimentaldata, the system is subjected to harmonic excitations ofincreasing amplitudes and a fast Fourier transform analysisof the response is carried out in each case to determinethe values ofG′ and G′′. As can be seen in figure 2,the analytical model compares well with the experimentaldata [16] it sought to reproduce. Improved correlationat low amplitudes can be obtained by using non-uniformweighting functions in (9). Figure 3 shows the non-linearforce/displacement curve for the spring S1, represented by(1).

For large frequency bandwidth applications, the methoddescribed in this section for identification of systemparameters can be easily extended to a non-linearviscoelastic model with additional Kelvin chains. Forexample, when a model with two Kelvin chains is subjectedto a harmonic forceD = D0 sin�t , equations (7) forxc and

Figure 3. Non-linear force/displacement curve for springS1.

xs become

xc = −(C2�)D0

K22 + (C2�)2

+ −(C3�)D0

K23 + (C3�)2

(11a)

xs = K2D0

K22 + (C2�)2

+ K3D0

K23 + (C3�)2

+ c1D0 + 8

3πc2D

20

+3

4c3D

30 + 32

15πc4D

40. (11b)

The above expressions can be used to obtainG′ and G′′,equation (8), followed by the minimization of

F =N�∑j=1

N∑i=1

[(G′

i,j −G′i,jexp

)2+

(G′′

i,j −G′′i,jexp

)2](11c)

at N� different frequencies, and atN different amplitudesat each of the frequencies.

4. Dual-frequency excitation

The variation with dynamic amplitude of complexmodulus componentsG′ and G′′ in figure 2, wereobtained using single-frequency excitation. In practice,the elastomeric damper may experience multi-frequencyexcitation conditions. For example, an elastomeric lagdamper on the rotor blade of a helicopter in forward flightcould experience excitations at the rotational frequency aswell as the lag natural frequency. A previous study [4] hasshown that the behaviour of the elastomeric damper underdual-frequency excitation differs from the single-frequencyexcitation behaviour. In this section, the behaviour of thenon-linear viscoelastic damper model is examined underdual-frequency excitation conditions. The excitation is ofthe form

x = x1 sin�1t + x2 sin�2t. (12)

�1 is taken as 315 rpm, a typical value of rotational speedfor a helicopter rotor, and the second frequency�2 is takenas �2 = 0.6�1, since this is a typical value of the lagnatural frequency of a soft-in-plane hingeless rotor blade.Figures 4(a) and 4(b) show the variation ofG′ and G′′

520

Non-linear viscoelastic damper model

(a) (b)

Figure 4. (a) Variation of G ′ under dual-frequency excitation conditions. (b) Variation of G ′′ under dual-frequency excitationconditions.

respectively, with amplitudex1, for increasing values ofx2 = 0◦, 0.25◦, 0.50◦, 0.75◦ and 1◦. For the present non-linear viscoelastic damper model, the change in behaviourof G′ andG′′ with increase inx2 shows similar qualitativetrends to previously reported experimental results for anelastomeric damper under dual-frequency excitation [4].

5. Numerical solution scheme for oscillator withnon-linear viscoelastic damper

Figure 5 shows a schematic of a dynamic oscillator withthe inclusion of the non-linear viscoelastic damper. Thissimple model can be used to represent several real lifesystems. For example, it could model the lag dynamics ofa helicopter rotor blade, with the mass,M, representing theblade lag inertia, and the springK representing the bladelag stiffness. In this section a numerical solution scheme isdeveloped for the dynamic oscillator shown in figure 5. Thesystem dynamics are described by the following equations.

Mx + Kx + D = P (13a)

Kx + Cx = Kf (D) + Cdf

dDD + D. (13b)

Equations (13a) and (13b) need to be solved together.Standard differential equation solvers employing Gear orRunge–Kutta methods are not very efficient for the dampermodel under consideration, as explained later in thissection.

To avoid numerical instability, an implicit solutionscheme such as the trapezoidal rule could be chosen. Thetrapezoidal rule for second-order differential equations canbe written as follows.

xik+1 = xk + xk + xi−1

k+1

21t (14a)

xik+1 = xk + xk + xi

k+1

21t (14b)

xik+1 = F

(xi

k+1, xik+1, Pk+1

). (14c)

Figure 5. Dynamic oscillator with non-linear viscoelasticdamper.

The iterations are started using the Euler approximationxi

k+1 = xk + xk1t . The right hand side of (14c) is obtainedusing the system governing differential equation, (13a).

xik+1 = Pk+1 − Kxi

k+1 − Dik+1

M. (14d)

Note that on the right hand side of (14d) the damper force,Di

k+1, is unknown and needs to be evaluated by solvingthe damper equation, (13b). To avoid numerical instabilityproblems, equation (13b) is solved using an implicit scheme(trapezoidal rule) as described below.

Dj

k+1,i = Dk + Dk + Dj−1k+1,i

21t (15a)

Dj

k+1,i = Kxik+1 + Cxi

k+1 − Kf (Dj

k+1,i ) − Dj

k+1,i

Cdf

dD

∣∣∣∣D

j

k+1,i

. (15b)

Equations (15a) and (15b) are solved iteratively forDk+1,i .The iterations can be started using Euler approximation,D1

k+1,i = Dk + Dk1t , or the approximationDj

k+1,i =

521

F Gandhi and I Chopra

Dk + Dk+Dk+1,i−1

2 1t . The solution scheme can be thoughtof as ‘two-level’: i iterations of equations (14a), (14b) and(14d) for xk+1, xk+1, and j iterations of equations (15a)and (15b) forDi

k+1, within each of thei iterations. Sinceimplicit method is used in both the levels, this solutionscheme can be referred to as ‘two-level implicit–implicit’.

In equation (15b), separate definitions need to be usedfor f (D) and df/dD depending on whetherD > 0 orD < 0. Thus each timeDj

k+1,i is evaluated using (15a), itis compared withDk. If the two are of opposite sign, thenthe crossover time1t1 (< 1t), at which D = 0, needsto be determined. Subsequent iterations would continueusing the other definitions off (D) and df/dD. 1t1 canbe initially estimated by settingDj

k+1,i = 0 in (15a).

1t1 = − 2Dk

Dk + Dj−1k+1,i

. (16)

Note that this is only an initial approximation for1t1 asD

j−1k+1,i in (16), evaluated using (15b), depended onxi

k+1

and xik+1 which were calculated at1t . To calculate1t1

accurately, as well asx andx t 1t1, the two-level implicit–implicit iterative scheme is used, with a better estimate1tm+1

1 being obtained by settingDk+1tm1to zero at each

iteration.

1tm+11 = − 2Dk

Dk + Dk+1tm1

. (17)

These iterations are continued till1t1 is obtained such thatDk+1t1 is zero. The procedure described above performsbetter than standard differential equation solvers whichneed infinitesimally small time steps and many iterationsin each time step near the crossover, to obtain a solution.This becomes particularly advantageous if the steady statesolution is required, and numerical integration needs to becarried out several cycles to ensure that the transient hasdecayed.

The solution procedure described in this section canbe easily extended to a viscoelastic damper model withadditional Kelvin chains. For example, with two Kelvinchains, the damper behaviour is described by (4).Di

k+1,required in (14d), is obtained by iteration of the followingset of equations.

Dj

k+1,i = Dk + Dk + Dj−1k+1,i

21t (18a)

Dj

k+1,i = Dk + Dk + Dj

k+1,i

2(18b)

Dj

k+1,i =[a1x

ik+1 + a2x

ik+1 + a3x

i−1k+1 − a1f

(D

j

k+1,i

)−a2

df

dD

∣∣∣∣D

j

k+1,i

Dj

k+1,i − a3d2f

dD2

∣∣∣∣D

j

k+1,i

(D

j

k+1,i

)2

−a4Dj

k+1,i − a5Dj

k+1,i

](a3

df

dD

∣∣∣∣D

j

k+1.i

)−1

. (18c)

It should be noted that in this case the non-linear springshould not have a quadratic term. This implies that in (1),c2 = 0. If this condition is not satisfied d2f/dD2 will bediscontinuous atD = 0. However, additional terms inD5,D6, etc., can be introduced in (1) to accurately representthe behaviour of the non-linear spring S1.

Figure 6. Forced response of the non-linear viscoelasticdamper and the linearized model, over one cycle.

Figure 7. Damper force over one cycle for linearized andnon-linear dampers.

6. Dynamic response of oscillator withelastomeric damper

In this section, the influence of the non-linear viscoelasticdamper model on the behaviour of the dynamic oscillator(figure 5) is examined. To understand the effects ofthe non-linear viscoelastic damper on steady-state andtransient response of the oscillator, the behaviour of theindividual components of the damper (the non-linear springS1 and the Kelvin chain) are examined. Additionally,to get a better understanding of the influence of thenon-linear characterization of the viscoelastic damperon the oscillator response, comparisons are carried outwith the response of an oscillator with a linearizedperturbation damper model. This model, obtained bylinearizing the damper governing equation, (3), about zeroequilibrium displacement, is tantamount to approximatingthe force/displacement behaviour of the non-linear spring,(1), by the linear relationx1 = c1D. For numerical resultspresented in this section,M was taken to be 2034 kg m2, atypical value for the lag inertia of a helicopter rotor blade.

522

Non-linear viscoelastic damper model

(a) (b)

(c) (d)

Figure 8. Transient response of oscillator under initial lag displacement of (a) 0.005 rad; (b) 0.010 rad; (c) 0.015 rad; (d)0.020 rad.

The oscillator is subjected to a harmonic excitationforce, P = P0ej�t , and the steady state response of boththe non-linear as well as the linearized model are examinedover one cycle of the forcing function. The forcing

Figure 9. Spring–dashpot representation of the augmentedelastomeric damper model.

frequency, �, is taken as 315 rpm, and the amplitudeof vibratory force P0 = 15000 N m. Figure 6 showsthe displacementx1 of the spring S1, the displacementx2 of the Kelvin chain, as well as the total displacement,x = x1 + x1, for both models. For the linearized modelwe can see that the amplitude ofx2 is about 3.5 timeslarger thanx1, since the spring S1 is very stiff at zerodisplacement. However, for the non-linear damper, thespring S1 softens rapidly, allowing much larger deformationx1, without generating large damper force,D. This resultsin smaller deformations in the Kelvin chain, as compared tothe linearized model. In figure 6, it can be clearly seen thatfor the non-linear response,x1 is about 1.5 times largerthan x2. Since motion of the dissipating element,C2, inthe Kelvin chain has decreased, the energy dissipated bythe non-linear system is less than that dissipated by thelinearized system. Figure 7 shows the damper force,D,at two different amplitudes of vibratory force,P0 = 15000and 30000 N m, for both the linearized as well as the non-linear damper models. Since the spring S1 softens, thedamper force generated by the non-linear damper is lessthan the force in the linearized model for the same loadP0.It can also be seen from figure 7 that when amplitude of the

523

F Gandhi and I Chopra

vibratory force,P0, doubles in value, the non-linear damperforce does not increase proportionately. It should be notedthat if the linearized model was developed based on thesecant stiffness of the non-linear spring S1 (rather than thetangent stiffness about zero displacement condition), thelinear estimates of damper force in figure 7 (as well as thedisplacementsx1 and x2 in figure 6) would be closer tothe non-linear estimates. However, this would require ana priori knowledge of the amplitude of the oscillations (orthe damping force).

Next, the transient response of the oscillator isexamined. Increasing values of initial displacement,x =0.005, 0.010, 0.015 and 0.020 radian, are applied, and thenon-linear and linearized response for each case is shownin figure 8(a)–(d), respectively. For the present dampermodel, it can be seen that as initial displacement increases,the transient decays much slower. This can be explainedby recalling that for larger displacements, motion of thesoftening spring S1 increases, damper forces generated arelow, and motion in the dissipating element in the Kelvinchain decreases, reducing damping.

7. Augmented elastomeric damper model

Recent investigations have revealed that the imaginarypart, G′′, of the complex modulus can be considerablydegraded at very low dynamic amplitudes [19, 20]. Thisbehaviour has not been represented in the model developedand discussed in the previous sections. It has also beenobserved that perturbation of helicopter rotor blades withelastomeric dampers can result in a transient that settlesto constant amplitude stable limit cycle oscillations (jitterphenomenon). The elastomeric damper representation infigure 1 is augmented with additional non-linear springs S3and S4 (figure 9) to model these behavioural characteristics.

An understanding of the non-linear damping mecha-nism in the model shown in figure 1 is a useful guide indetermining the behaviour of the augmenting springs S3 andS4. The high stiffness of S1 at low displacement amplituderesults in considerable damping (highG′′) as the majorityof the damper motion occurs in the Kelvin chain (compris-ing of linear springsK2 and dashpotC2). As displacementamplitude increases, significant motion is obtained in thespring S1 due to its softening behaviour, causing a decreasein G′′ (damping) associated with the reduced motion in theKelvin chain.

It is hypothesized that a reduction inG′′ at smalldisplacement amplitudes could be obtained by introducing aseries spring, S3, that is very soft in the small displacementregime (so that the majority of the damper motion occursin this spring rather than the dashpot in the Kelvin chain)and rapidly hardens thereafter, rendering it ineffective inthe large displacement regime. The force/displacementbehaviour of spring S3 is shown in figure 10(a) and isdescribed by

D = (signx3

)c5

(ec6|x3| − 1

)(19a)

or

x3 = (signD)1

c6ln

( |D|c5

+ 1

)(19b)

where D is the force andx3 is the displacement in S3.While augmenting the model with such a springalone, itexpectedly reducesG′′ at low amplitudes of excitation, itresults in a decrease inG′, as well.

To alleviate the reduction inG′ at small amplitudes,spring S4 is introducedin parallel with the seriesconfiguration consisting of the Kelvin chain and springsS1 and S3, as shown in figure 9. The non-linear behaviourof spring S4 is expressed as

DS4 = g(x) = (signx)c7(1 − e−c8|x|) (20)

and is shown in figure 10(b). In the above equation,DS4

is the force in the spring S4, andx is the total damperdisplacement. The high stiffness at small displacementamplitudes prevents the decrease ofG′, whereas the verylow stiffness at higher amplitudes renders the branchvirtually ineffective at larger displacements.

The resulting variation ofG′ and G′′ with dynamicamplitude is shown in figure 11. Comparison of figures 2and 11 reveals that the additional springs in the augmentedmodel effectively decreaseG′′ at small amplitudes, whilehaving virtually no effect on the variation ofG′ or G′′

with increasing dynamic amplitude. The instantaneousforce/displacement behaviour of the augmented model(figure 12) represents the softening behaviour of elastomersand is very similar to the instantaneous force/displacementcharacteristics of the original model, represented by thebehaviour of the single non-linear spring S1, (figure 3).

For analysis purposes, the two non-linear springs S1and S3 can be treated as a single non-linear spring. Ifxnon = x1+x3 is the total displacement in the two non-linearsprings S1 and S3, andD is the force, from equations (1)and (19b)

xnon = f (D) = (signD)(c1|D| + c2|D|2 + c3|D|3

+c4|D|4)

+ (signD)1

c6ln

( |D|c5

+ 1

). (21)

The relation betweenD and the total damper displacementx is then given by (3), with the new definition off (D),(21), being used. It should be noted, however, that the totalelastomeric damper force is the sum of the forcesD (in thebranch containing the Kelvin chain) andDS4 (in the parallelspring S4), as shown in figure 9.

8. Limit cycle oscillations

To examine whether the augmented elastomeric dampermodel can predict limit cycle oscillations, the damper non-linear differential equation is solved simultaneously with astructural dynamic equation representative of a helicopterrotor blade lead–lag motion. The blade lead–lag equationcan be written as

x + ν2ζ x + DS4 + D = Mζ (22)

whereD + DS4 is the total elastomeric damper force (non-dimensional),x is the lag displacement, andνζ is therotating lag frequency. Introducing (20) into (22) yields

x + ν2ζ x + g(x) + D = Mζ . (23)

524

Non-linear viscoelastic damper model

Figure 10. Force/displacement behaviour of non-linear spring (a) S3; (b) S4 of the augmented model.

Figure 11. Variation of complex moduli G ′ and G ′′ of theaugmented damper model with amplitude.

Figure 12. Instantaneous force/displacement behaviour ofaugmented damper model (figure 9).

Equation (23) is solved simultaneously with the non-lineardifferential equation, (3), for the force,D, in the Kelvin

chain branch of the damper. To evaluate the transientresponse of the system, the aerodynamic lead–lag moment,Mζ , is set to zero, and the system is successively givenvarying initial perturbationsx0 = 0.0005 radian, and0.003 radian. In both cases, within a short time the transientsettles to stable self-sustained limit cycle oscillations, asshown in figures 13(a) and 14(a). Figures 13(b) and14(b) show the transient responses represented in thephase-plane (velocity/displacement plane). Limit cycles areknown to be characterized by ‘orbits’ in the phase plane,implying a periodic notion. In figures 13 and 14, differentinitial perturbations, expectedly, yield stable limit cycleoscillations, of the same amplitude, as it is known that theamplitude is dependent only on the characteristics of thenon-linear system and is independent of the magnitude ofthe initial perturbation.

9. Non-linear hysteresis cycles

The force/displacement hysteresis cycles of the augmentedelastomeric damper model can be calculated by theapplication of a single frequency sinusoidal excitationx = x0ei�t to (3) and (20); with (21) being usedfor f (D) in (3). For a linear viscoelastic material,the hysteresis cycles are ellipses, and can be describedmathematically by standard equations. However, for thenon-linear elastomeric damper, the nature of the hysteresiscycles changes with both dynamic amplitude and staticpreload. Figure 15 shows hysteresis cycles of differentamplitudes at zero static preload, for the augmented dampermodel. These cycles show some qualitative similarity toexperimentally measured hysteresis cycles [19], softeningat higher dynamic amplitudes.

10. Summary and concluding remarks

A spring-dashpot model is developed to represent non-linear viscoelastic behaviour of an elastomeric damper.A single non-linear softening spring in series with linearKelvin chains was adequate to characterize the non-linear

525

F Gandhi and I Chopra

Figure 13. Transient responses for an initial displacement, x0 = 0.0005 radian.

Figure 14. Transient responses for an initial displacement, x0 = 0.003 radian.

Figure 15. Non-linear hysteresis cycles at differentdynamic load levels of 1000, 1500 and 2000 N m (zerostatic load).

amplitude dependence of the complex-moduliG′ and G′′

of an elastomeric damper. The non-linear constitutivedifferential equation of the damper model was obtained,and system parameters were identified by minimizingan error sum, at different dynamic amplitudes, between

experimental values and analytically obtained expressionsof complex moduli. The system parameters thus obtained,can reproduce experimental data satisfactorily.

When subjected to dual frequency excitations, thevariations in complex moduli (G′ andG′′) with amplitudeof excitation predicted by the present non-linear dampermodel, were qualitatively very similar to previouslyreported experimental results.

A ‘two-level implicit–implicit’ scheme is developed tosolve structural dynamic equations with the inclusion of thenon-linear viscoelastic damper model. The scheme is verysuitable for the present damper model, whose piecewisedefinition over the domain requires the crossover point(time at whichD = 0) to be determined.

Introduction of the baseline non-linear viscoelasticdamper model in a dynamic oscillator revealed that asamplitude of oscillatory force increased, the non-linearspring rapidly softened, allowing large motions of the leadspring without generating large overall damper forces, ascompared to a linearized model. Since the motion ofthe Kelvin chain was smaller in case of the non-lineardamper, this resulted in a lower dissipation of energyand lower damping. For the linearized damper, motionx2 in the Kelvin chain was 3.5 times larger than motion

526

Non-linear viscoelastic damper model

x1 in the spring. For the non-linear damper,x1 in thesoftening spring was 1.5 times greater thanx2 in the Kelvinchain. When the system is subjected to initial perturbations,the transient decay is slower for the non-linear damper,confirming that the non-linear viscoelastic damper modelwith softening spring and linear Kelvin chain has dampingreduced for larger motion.

Augmenting the baseline spring-dashpot model withadditional non-linear series and parallel springs, it waspossible to simulate degradation ofG′′ at low dynamicamplitudes, and the occurrence of limit cycle oscillationswhen subject to perturbations.

Acknowledgments

This research work is supported by the Army ResearchOffice under the Center for Rotorcraft Education andResearch, Grant No. DAAH04-93-G-0001; TechnicalMonitors Dr Tom Doligagalski and Dr Robert Singleton.

Appendix A.

Consider a viscoelastic damper model with non-linearspring S1 in series with a number of Kelvin chains.D

is the force acting on the damper (and therefore actingon S1 and on each of the Kelvin chains), andx is thetotal displacement of the damper. The force displacementrelation of the non-linear spring isx1 = f (D). Ifx2, x3, . . . are the displacements of the Kelvin chains, then,K2x2 +C2x2 = D, K3x3 +C3x3 = D, . . .. Taking Laplacetransform, (

K2 + sC2)X2(s) = D(s)(

K3 + sC3)X3(s) = D(s).

The total motion in all the linear Kelvin chains is given as

X2(s) + X3(s) + . . . = D(s)

[1

(K2 + sC2)

+ 1

(K3 + sC3)+ . . .

]= X(s) − X1(s).

For example, for one Kelvin chain:

D(s)

[1

(K2 + sC2)

]= X(s) − X1(s) or D(s) = [

X(s)

−X1(s)](

K2 + sC2).

Taking inverse transform,

D = K2(x − x1

) + C2(x − x1

)or K2x + C2x

= K2x1 + C2x1 + D.

Substituting

x1 = f (D), x1 = df

dDD

gives:

K2x + C2x = K2f (D) + C2df

dDD + D.

Similarly, for two Kelvin chains,

D(s)

[1

(K2 + sC2)+ 1

(K3 + sC3)

]= X(s) − X1(s)

or

D(s)

[(K2 + K3) + s(C2 + C3)

K2K3 + s(K2C2 + K3C2) + s2C2C3

]= X(s) − X1(s)

or[K2K3 + s

(K2C3 + K3C2

) + s2C2C3

]X(s)

= D(s)[(

K2 + K + 3) + s

(C2 + C3

)]+[

K2K3 + s(K2C3 + K3C2

) + s2C2C3]X1(s).

Taking inverse Laplace transform,

a1x + a2x + a3x = a4D + a5D + a1x1 + a2x1 + a3x1

where a1 = K2K3, a2 = K2C3 + K3C2, a3 = C2C3,a4 = K2 + K3 anda5 = C2 + C3. Substitutingx1 = f (D),x1 = (df/dD)D and x1 = (df/dD)D + (d2f/dD2)D2

gives

a1x + a2x + a3x = a1f (D) + a2df

dDD

+a3

(df

dDD + d2f

dD2D2

)+ a4D + a5D.

Appendix B.

dG′i

dc1= D2

0i

(x2ci

− x2si)

(x2ci

+ x2si)2

dG′′i

dc1= −D2

0i

2xcixsi

(x2ci

+ x2si)2

dG′i

dc2= 8

3πD3

0i

(x2ci

− x2si)

x2ci

+ x2si)2

dG′′i

dc2= − 8

3πD3

0i

2xcixsi

(x2ci

+ x2si)2

dG′i

dc2= 3

4D4

0i

(x2ci

− x2si)

(x2ci

+ xssi)2

dG′′j

dc2= −3

4D4

0i

2xcixsi

(x2ci

+ x2si)2

dG′i

dc4= 32

15πD5

0i

(x2ci − x2si)

(x2ci

+ x2si)2

dG′′i

dc4= − 32

15πD2

0i

2xcixsi

(x2ci

+ x2si)2

dG′idK

= D0i

{(x2

ci− x2

si)

dxsidK

− 2xcixsi

dxcidK

}(x2

ci+ xsi

)2

dG′′i

dK= D0i

{(x2

ci− x2

si)

dxcidK

− 2xcixsi

dxsidK

}(x2

ci+ x2

si)2

dG′i

dC= D0i

{(x2

ci− x2

si)

dxsidC

− 2xcixsi

dxcidC

}(x2

ci+ x2

si)2

527

F Gandhi and I Chopra

dG′′i

dC= D0i

{(x2

ci− x2

si(

dxsidC

− 2xcixsi

dxcidC

}(x2

ci+ x2

si)2

dG′′i

dC= D0i

{(x2

ci− x2

si)

dxcidC

− 2xcixsi

dxsidC

}(x2

ci+ x2

si)2

.

In the above expressions,dxsidK

,dxcidK

,dxsidC

,dxcidC

, xsiandxci

aredefined as follows

dxsi

dK= D0i

(C�)2 − (K)2

[(K)2 + (C�)2]2

dxci

dK= D0i

2(C�)(K)

[(K)2 + (C�)2]2

dxsi

dC= −D0i

�2(C�)(K)

[(K)2 + (C�)2]2

dxci

dC= D0i

�(C�)2 − (K)2

[(K)2 + (C�)2]2

xsi= K2D0i

K22(C2�)2

+ c1D0i+ 8

3πc2D

20i

+ 3

4c3D

30i

+ 32

15πc2D

40i

xci= −(C2�)D0i

K22 + (C2�)2

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