Dynamics of a linear oscillator connected to a small strongly non-linear hysteretic absorber
aa r X i v : . [ n li n . C D ] J a n Dynamics of a linear oscillator connected to a smallstrongly non-linear hysteretic absorber
D. Laxalde a,b , F. Thouverez a , J.-J. Sinou a ( a ) Laboratoire de Tribologie et Dynamique des Syst`emes (UMR CNRS 5513)´Ecole Centrale de Lyon, 36 avenue Guy de Collongue, 69134 Ecully Cedex, France ( b ) Snecma – Safran group, 77550 Moissy-Cramayel, France
Abstract
The present investigation deals with the dynamics of a two-degrees-of-freedom system whichconsists of a main linear oscillator and a strongly nonlinear absorber with small mass. Thenonlinear oscillator has a softening hysteretic characteristic represented by a Bouc-Wen model.The periodic solutions of this system are studied and their calculation is performed through anaveraging procedure. The study of nonlinear modes and their stability shows, under specificconditions, the existence of localization which is responsible for a passive irreversible energytransfer from the linear oscillator to the nonlinear one. The dissipative effect of the nonlinearityappears to play an important role in the energy transfer phenomenon and some design criteriacan be drawn regarding this parameter among others to optimize this energy transfer. Thefree transient response is investigated and it is shown that the energy transfer appears whenthe energy input is sufficient in accordance with the predictions from the nonlinear modes.Finally, the steady-state forced response of the system is investigated. When the input ofenergy is sufficient, the resonant response (close to nonlinear modes) experiences localization ofthe vibrations in the nonlinear absorber and jump phenomena.
Keywords:
Hysteresis, nonlinear energy sinks, non-linear modes, averaging method
Vibration control of mechanical systems is of permanent interest in the field of engineering andresearch. Dissipation or absorption of the unwanted and often dangerous vibratory energy can beachieved by various ways, using passive or active devices, dissipative materials, coatings, visco-elastic materials, tuned mass dampers or friction . . .Recently, the interest for nonlinear absorbers kept growing. In particular, several works onvibration control through the use of a small passive strongly nonlinear device have been presented.The concept of energy pumping , that is a passive irreversible one-way energy transfer from a main(linear) structure to a strongly nonlinear (non-linearizable) attachment, was introduced and devel-oped [1, 2, 3]. Results on discrete [2, 4] and continuous systems [3] were presented and in most ofstudies, the nonlinear absorber consisted in an essential cubic nonlinearity. Several methodologieswhere investigated and developed to study this phenomenon. The use of asymptotic techniques to1nd approximate solutions was addressed in several works; Vakakis et al. [2] used an averagingmethod, Gendelman et al. [4] used a multiple scale method, Mikhlin and Reshetnikova [3] used anexpansion method in combining with a Mathieu equation comparison to investigate the stability ofperiodic solutions. . . Moreover Vakakis and Rand [5] proposed a method to derive exact solutions forsystems with cubic nonlinearities based on the use of elliptic functions. Experimental results werealso presented [6, 7] in which the nonlinear device is made of a geometric nonlinearity. Applicationshave been proposed in shock isolation or in civil engineering.A common result to explain this phenomenon is that, under specific conditions, some localiza-tion of the vibratory can occur leading to an irreversible passive transfer of the energy from thelinear structure to the absorber. This result was demonstrated using the stability of the periodicsolutions; Vakakis [8, 1] provided a theoretical background on the subject studying periodic orbitsof the associated Hamiltonian system. Several authors, including Vakakis et al. [2] or Mikhlin andReshetnikova [3] used the concept of Nonlinear Normal Modes (NNM) and their stability to explainthis result. The concept of nonlinear normal modes was first introduced by Rosenberg [9] and hasbeen the subject of many investigations in the past years. Several authors [10, 11] demonstratedthat the use of NNM in studying the dynamics of nonlinear (and in particular strongly nonlinear)systems has interesting applications both in free and forced responses. Beside this concept, the ef-fect of localisation of the vibratory energy and motion confinement due to strong nonlinearity wasalso addressed in several investigations (see Bendiksen [12] for a good review on the subject). Amain feature of such phenomena is that they are more energy-dependent than frequency-dependentand a direct consequence is that the energy sinks are efficient in a quite wide range of frequencywhich contrasts with typical linear tuned dampers.In this paper, we focus on a two degrees-of-freedom system, involving a nonlinear absorber withan hysteretic characteristic. Hysteretic nonlinearity requires some particular modelling and theone which is used in this study is the Bouc-Wen differential model [13, 14]. A detailed descriptionof the model is addressed in section 2. In section 3, an asymptotic method for the study ofperiodic responses is used. It consists of a two variables expansion combined with an averagingprocedure The nonlinear modes and their stability, are first studied in section 4 and numericalresults are discussed to highlight to energy pumping phenomenon. A parametric study emphasizethe importance of the dissipation rate of the non-linear absorber and some design criteria are found.Results on transient response show a correct prediction from the nonlinear modes. In section 6,the behaviour of the system in forced response (harmonic excitation) is then addressed and it isshown that, in accordance with the modal prediction, the system experiences some localizationphenomenon along with jump phenomena.
A system of two oscillators linearly connected is considered. The main oscillator, with mass M ,represents an approximation to some continuous elastic system; the small one, with mass m , is theabsorber and is strongly nonlinear. The coupling is assumed to be weak. Also, the system remainstechnologically realistic since the mass ratio about 2% for the numerical applications.The motion of this system is governed by the following system of nonlinear equations:¨ x ( t ) + λ ˙ x ( t ) + ω x ( t ) + ǫ ( x ( t ) − v ( t )) = f ( t ) (1a)2 v ( t ) + λ ˙ v ( t ) + r ( ˙ v ( t ) , t ) + ǫ ( v ( t ) − x ( t )) = 0 (1b)where x and v are respectively the displacements of the main mass and of the absorber, λ and λ are damping ratios, ω is the natural angular frequency of the linear (uncoupled) oscillator, f ( t )represents an external forcing and ǫ and ǫ are coupling ratios such that: ǫ ǫ = mM (2)The term r represents a nonlinear and hysteretic restoring force which means that it dependsof the history of the non-linear motion.Various kind of systems experience hysteretic behaviours in dynamics; some systems inelastic orwith memory may have a restoring force dependent of the history of the deformation, some other,such as rubber or cable isolator are design to dissipate the vibratory energy in the hysteretic loop.The Bouc-Wen differential model, originally proposed by Bouc [13] and reviewed by Wen [14], isone of the most used phenomenological model of hysteresis in mechanics. It is also used for systemidentification of hysteretic systems. The hysteretic force r ( t ) is based on the displacement v ( t ) timehistory and is given by the following differential equation :˙ r ( t ) = A ˙ v ( t ) − ν (cid:0) β | ˙ v ( t ) || r ( t ) | n − r ( t ) − γ ˙ v ( t ) | r ( t ) | n (cid:1) (3)where A , ν , β , γ and n are the loop parameters of the Bouc-Wen model. A proper choice ofthese parameter allows to describe a wide range of hysteresis loops, with softening or hardeningbehaviour, different levels of nonlinearity, with various intermediate states possibilities (smooth orbilinear). In what follows, the parameter n is set to 1. In this section, we will use an averaging method to derive the periodic solutions of a nonlinearsystem such as system (1). Let’s consider the following general dynamical problem:¨ z ( t ) + g ( ˙ z , z , t ) = (4)in which z is a displacement vector and the term g includes along with linear (stiffness) terms, anynonlinear term and excitations.We seek a solution of (4) in the following form: z ( t ) = z ( τ, η ) = a ( η ) cos( τ + ϕ ( η )) (5)where the amplitude a and phase ϕ are slowly varying quantities (time scale η ) with respect tothe fast time scale τ . This transformation from the displacement dependent variables (one variablerespectively) to the amplitudes and phases dependent variables (two variables respectively) allowsus to impose an additional condition. It is usual to choose that the velocity has a similar form tothe linear case:˙ z = d z ( τ, η ) dt = − dτdt a ( η ) sin( τ + ϕ ( η )) (6)3he time scale τ can then be interpreted as the time scale of the periodic motion whereas, the timescale η represents a perturbation time scale. Also, in equation (6), the term: dτdt = ω (7)represents an angular-frequency-like ”variable” assumed to be constant in time. However, thisangular frequency ω can be amplitude-dependent and a priori unknown as in the case of nonlinearmodes (see section 4) or not as in the case of steady-state forced response where it corresponds tothe excitation frequency (see section 6).Equation (6) now becomes:˙ z = − ω a ( η ) sin( τ + ϕ ( η )) (8)Differentiating (5) with respect to time and equating the result with (8), we find: a ′ cos( τ + ϕ ) − a ϕ ′ sin( τ + ϕ ) = (9)where . ′ denotes derivatives with respect to the slow time scale η .We have obtained the first equation governing the evolution of the slow flow variables a and ϕ .The second one can be obtained by first differentiating (8) with respect to t , which yields:¨ z = − ω a ′ sin( τ + ϕ ) − ω a ϕ ′ cos( τ + ϕ ) − ω a cos( τ + ϕ ) (10)and substituting ¨ z into (4): − ω a ′ sin( τ + ϕ ) − ω a ϕ ′ cos( τ + ϕ ) − ω a cos( τ + ϕ ) + g ( a cos( τ + ϕ ) , − ω a sin( τ + ϕ )) = (11)Finally, solving equations (9) and (11) for slow flow variables variations a ′ and ϕ ′ we have thefollowing system: a ′ = sin( τ + ϕ ) ω (cid:0) G ( a , ϕ, τ ) − ω a cos( τ + ϕ ) (cid:1) (12a) a ϕ ′ = cos( τ + ϕ ) ω (cid:0) G ( a , ϕ, τ ) − ω a cos( τ + ϕ ) (cid:1) (12b)where G ( a , ϕ ) is obtained by substituting z and ˙ z in g .The system (12) is now in standard form and can be averaged over the fast time scales τ withthe slow flow variables a and ϕ being taken as constants. a ′ = 1 ω π Z π G ( a , ϕ, ψ ) sin ψdψ (13a) a ϕ ′ = 1 ω (cid:20) π Z π G ( a , ϕ, ψ ) cos ψdψ − ω a (cid:21) (13b)The averaged problem defined by the differential system (13) provides approximate solutions withthe particular form defined by (5). The approximation is made on the form of the nonlinear termsin g which are assumed (due to averaging) to be proportional to the harmonic functions cos ψ andsin ψ . 4his formalism will be used in the following section to derive particular solutions to differentproblems, including free vibrations or forced vibrations, by substituting the proper function g inequations (13).In the system subject of this paper, the general term g ( z , ˙ z , t ) includes, along with linear terms,nonlinear dissipative terms defined by the differential Bouc-Wen model. As these terms are difficultto average in close form, we have used a numeric approach. However, in order to proceed with theanalytical calculations, we introduce a similar notation to equation (5) for the averaged nonlinearrestoring force r of the Bouc-Wen model. The resulting amplitude a r and phase ϕ r are derivednumerically by substituting v ( t ) = a v cos( τ + ϕ v ) in relation (3) and performing an averaging onthe fast time scale (as in equation (13)): a r e iϕ r = 12 π Z π r ( ˙ v, t ) e − iψ dψ (14)This step introduce no additional approximation or simplification to the averaging procedure ofequation (13). Details on the variations of the averaged hysteretic restoring force with the non-linear displacement v are provided in section 4.3 where the equivalent stiffness and damping arestudied. In section, we perform a modal analysis of the system (1) unforced and undamped. Modal studiesof non-linear systems are numerous in the literature [9, 11, 10, 15, 16] and the theoretical concept ofnon-linear modes is found to provide interesting dynamical descriptions as well as a valuable designtool. The derivation of non-linear modes is first performed analytically. To do so, we first applythe averaging method described in section 3 to the initial equations of motion (1) (without forcingand damping); then, the study of the fixed points of the resulting system leads to the definitionof an eigenvalue problem. In a second time, numerical results are presented for several types ofBouc-Wen hysteretic restoring forces and to possibility of localization phenomenon and possibleenergy pumping are demonstrated.
The free vibrations problem of the two degrees of freedom system is defined by:¨ x ( t ) + ω x ( t ) + ǫ ( x ( t ) − v ( t )) = 0 (15a)¨ v ( t ) + r ( ˙ v ( t ) , t ) + ǫ ( v ( t ) − x ( t )) = 0 (15b)Using the averaging method described in section 3, the two displacements variables take theform of (5), ie : x = a x ( η ) cos( τ + ϕ x ( η )) (16a)and v = a v ( η ) cos( τ + ϕ v ( η )) (16b)5here τ = ωt , and ω is the angular frequency of the oscillations.The term G ( a, ϕ, τ ) in equations (12) can be simply derived and introduced in equations (13),which leads to:2 ωa ′ x = − ǫ a v sin( ϕ x − ϕ v ) (17a)2 ωa ′ v = ǫ a x sin( ϕ x − ϕ v ) − m a r sin( ϕ r − ϕ v ) (17b)2 ωa x ϕ ′ x = ( ω + ǫ − ω ) a x − ǫ a v cos( ϕ x − ϕ v ) (17c)2 ωa v ϕ ′ v = ( ǫ − ω ) a v − ǫ a x cos( ϕ x − ϕ v ) + 1 m a r cos( ϕ r − ϕ v ) (17d)As the system (17) is in standard first-order form, we can study its fixed points by makingall time derivatives zero (left hand side). Then one obtains a system of nonlinear equations (withamplitudes a j and phases ϕ j along with the angular frequency ω as unknowns) which can beinterpreted as a non-linear eigenvalue problem to find an approximation of the non-linear modes.However, from the first equation of system (17), one can notice that the fixed points necessaryverify, ϕ x − ϕ v = 0 mod ( π ) (18)Carrying this result in the second equation, one finds that a r sin( ϕ v − ϕ r ) = 0 (19)which is absurd because first a r cannot be zero and second because the hysteretic and dissipativecharacteristics of the nonlinearity impose a non-zero phase difference between the restoring force r and the displacement v .In order to avoid this, the general form of the solutions (given by equations (5) or (16)) needsto be adapted. The formalism was inspired by the complex mode definition for linear systems andis given by: z ( τ, η ) = c ( η ) e − ζη cos( τ + ϕ ( η )) (20)where ζ represents the modal damping and c is the new amplitude variable. The exponential decayallows a proper modelling of the dissipative aspect of the motion and can be related with a complexnatural frequency (by analogy with linear complex modes), λ = − ζ ± iω (21)It was also assumed that this exponential decay is slowly varying.Now, introducing the formalism of complex modes, defined by equation (20), in system (17)and noting a i ( η ) = c i ( η ) e − ζη for i = x, v , one obtains: c ′ x = ζc x − ǫ ω c v sin( ϕ x − ϕ v ) (22a)6 ′ v = ζc v + ǫ ω c x sin( ϕ x − ϕ v ) + 12 ω c r sin( ϕ v − ϕ r ) (22b) c x ϕ ′ x = ω + ǫ − ω ω c x − ǫ ω c v cos( ϕ x − ϕ v ) (22c) c v ϕ ′ v = ǫ − ω ω c v − ǫ ω c x cos( ϕ x − ϕ v ) + 12 ω c r cos( ϕ v − ϕ r ) (22d)It should be noticed that along with the assumptions made on the form of the displacements,equation (20), the nonlinear term a r ( η ) is assumed to have the same exponential decay that thedisplacement. This assumption can be justified by looking at the expression of the averaged nonlin-ear term. In equation (14), if we substitute for ˙ v the new complex form (20), the exponential decayis reported outside the summation symbol and as a consequence, equation (14) can be updatedwith the complex form defined by (20).If we finally combine equations (22c) and (22d) and introduce the phase difference variables ϕ ij = ϕ i − ϕ j , with i, j = x, v or r , we obtain the following system: c ′ x = ζc x − ǫ ω c v sin ϕ xv (23a) c ′ v = ζc v + ǫ ω c x sin ϕ xv + 12 ω c r sin ϕ vr (23b) c x c v ϕ ′ xv = ω + ǫ − ǫ ω c x c v − ω (cid:0) ǫ c v − ǫ c x (cid:1) cos ϕ xv − ω c r cos ϕ vr (23c)This unforced system is in (averaged) standard form and its fixed points are the main approxi-mation of the nonlinear modes.Along with this, we have to introduce an additional relation between the coordinates in orderto normalize the modes. To do so, we define H the global energy of the system by: H = T + U − W d (24)where T , U are respectively the global kinetic and potential energies of the system; W d is the energydissipated by the hysteretic force during one cycle of the motion.Also note that, if the non-linear modes can be approximated by the fixed points of equa-tions (23), these will depend on the value of the angular frequency ω which is an unknown of thefree dynamical problem. This angular frequency ω , which represents the eigenfrequency of themodes, is determined using the additionnal energy relation (24).This formalism introduces the notion of complex nonlinear modes which correspond (as in thelinear domain) to special solutions of the free vibrations problem where the system’s degrees offreedom oscillate with the same frequency but, in contrast with non-linear normal modes, witha phase difference between them. The dissipative terms introduce a phase difference between thecoordinates and a modal damping ζ which, as the natural frequencies and deformed shapes dependson the system’s energy. It may also be note that, in contrast with classical nonlinear normal modes,these complex modes as defined here are not necessary normal to the iso-energetic curves. Withthis complex modal description, one can handle the dissipative effects of the nonlinearity directly7ithout any assumption regarding their importance. This particular feature will be used in thefollowing numerical applications to determine design rules on the dissipation rates. This is a majorconcern, in engineering applications, when dealing with very weakly damped (linear) structure; inthese situations, the nonlinear oscillator has to ensure the resonance capture phenomenon as wellas the dissipation of the vibratory energy. The stability of the nonlinear modes can be determined using the system (23), which is in standardform, and studying the eigenvalues of the Jacobian matrix D z F ( z , M ) at the equilibrium points. D z F ( z , M ) = ζ − ǫ ω sin ϕ xv − ǫ ω c v cos ϕ xv ǫ ω sin ϕ xv ζ + 12 ω ∂ ( c r sin ϕ vr ) ∂c v ǫ ω c x cos ϕ xv + 12 ω ∂ ( c r sin ϕ vr ) ∂ϕ xv ω + ǫ − ǫ ω c v + ǫ ω c x cos ϕ xv ω + ǫ − ǫ ω c v − ǫ ω c v cos ϕ xv − ω ∂c r cos ϕ vr ∂c v ω (cid:0) ǫ c v − ǫ c x (cid:1) sin ϕ xv − ω ∂ ( c r cos ϕ vr ) ∂ϕ xv (25) The hysteretic Bouc-Wen model is quite versatile and a large variety of hysteretic loops can bemodelled. In this study, we focused on the class of softening hysteretic nonlinearity ( i.e. γ = − . ω and at normalizing the modes. Since ourmotivation is to investigated the vibration control of weakly damped structures, the main oscillatoris assumed to be conservative and the only source of dissipation is the absorber. Example 1.
We first study the case of a weakly dissipative hysteretic cycle which build using the Bouc-Wenparameters β = 10 − and γ = − .
5. The modal quantities are depicted in figures 3. The modal8 r : r e s t o r i ng f o r c e (a) da m p i ng s i ff ne ss (b) Figure 1: Example n ° γ = 10 − ; (a) hysteretic cycle, (b) equivalent stiffness and damping.amplitudes of the linear and nonlinear degrees of freedom, c x and c v , natural frequency ω , andmodal damping ζ are plotted versus the global energy H of the system.There are two main branches (in solid and dashed lines) and an unstable region appears inthe solid line branch symbolized by crosses. We can discriminate several regions with distinctdynamical behaviours: Low energies region: in the solid line branch, the motion appears to be nearly localized inthe linear oscillator which modal curve is a straight line: the nonlinear system’s behaviour islinearizable. Note that, in this region, the natural frequency of the linear oscillator is constant.The dashed line branch shows, on the other hand, a prominent motion in the absorber and adecreasing natural frequency due to the softening characteristic of the nonlinearity.
High energies region: there, the situation is quite reversed, the solid line branch shows a stronglocalization in the nonlinear oscillator with a decreasing natural frequency; and the dashedline branch displays a prominent motion in the linear system.
Intermediate energy region: between the previous asymptotic states, the system experiencesa bifurcation phenomenon of its modes. When the global energy increases, the motion ofthe linear oscillator jumps from a high to a low level; on the other hand, the motion of thenonlinear oscillator jumps from low to high level. Note that this phenomenon appears whenthe nonlinear natural frequency joins the linear one, the system of two coupled oscillatorsenters an internal resonance [17].
Example 2.
The second example involves an hysteretic nonlinearity with a rather high level of dissipation (seefigures 2) which contrasts with the first example where the damping ratio was quite small.The modal quantities are represented in figures 4. There are several differences in this example9 r : r e s t o r i ng f o r c e (a) s i ff ne ss da m p i ng (b) Figure 2: Example n ° γ = 0 .
5; (a) hysteretic cycle, (b) equivalent stiffness and damping.with reference to the previous one. First note that there is no instability region and no internalresonance. Then the modal damping curve is quite different from the previous example since thedashed line branch reach higher values of damping especially in the region where the two frequencycurves could interact. In effect, figure 2 shows a particular feature of this nonlinearity which isthat the level of nonlinearity (in this case, the drop of stiffness) is directly related to the levelof damping. As a consequence, a too high value of nonlinear damping can inhibit the resonancecapture and no energy pumping occurs and two oscillators behave quite independently.This comparison clearly shows this importance of the level of dissipation in the nonlinearity onthe efficiency of the energy pumping phenomenon. However, remember that, in the energy pumpingphenomenon, as the vibratory energy gets transferred to the absorber it has to be dissipated. Theoptimal absorber would then be the one with the highest damping ratio allowing the resonancecapture to take place and leading a fast energy dissipation. Example 2 displays a case where a highdissipation rate inhibits the energy transfer whereas example 1 shows an optimal design, in term ofdissipation, which ensures that both the energy transfer and the energy dissipation are achieved.
In this section, we provide examples of transient free responses which highlight the energy pumpingphenomenon and underline the efficiency of the previous modal analysis predictions. We focuson the first example nonlinearity ( γ = 10 − ) which appears to be more efficient according to thenon-linear modes predictions. The exact system of equations (1) (with f ( t ) = 0) was used with theBouc-Wen restoring force defined by equation (3); we impose the initial following initial conditions:˙ x (0) = r H M , x (0) = 0 , ˙ v (0) = 0 and v (0) = 0 (26)10
10 20 30 40 50 60−2−1.5−1−0.500.511.52 h: energy c x : li nea r do f (a) Linear oscillator c v : ab s o r be r (b) Absorber E i gen f r equen cy ( H z ) (c) Eigenfrequency ζ m oda l da m p i ng (d) Modal damping Figure 3: Example 1: Nonlinear modes: (—, - - -), stable; (-x-x-), unstable.11
20 40 60 80 10000.511.522.533.5 h: energy c x : li nea r do f (a) Linear oscillator c v : ab s o r be r (b) Absorber E i gen f r equen cy ( H z ) (c) Eigenfrequency ζ m oda l da m p i ng (d) Modal damping Figure 4: Example 2: Nonlinear modes12here H is the initial energy of the system. These initial conditions simulate an impact on the mainmass. The results of two simulations with different initial energy input are depicted in figures 5to 8. Beside from the displacements x and v history, the history of the instantaneous frequencyis also plotted for both oscillators. The instantaneous frequency was calculated using an Hilberttransform [18]. x : li nea r do f (a) I n s t. f r equen cy ( H z ) (b) Figure 5: Transient response for H = 15 – Linear oscillator; (a) time history, (b) instantaneousfrequency. v : ab s o r be r (a) I n s t. f r equen cy ( H z ) (b) Figure 6: Transient response for H = 15 – Absorber; (a) time history, (b) instantaneous frequency.For H = 15 (figures 5 and 6), the input of energy is smaller than the bifurcation characteristiclevel. Hence, as predicted by the nonlinear modes (see figure 3) and in accordance with the initialconditions, only the solid line branch of the modes is realizable. This means that the vibratoryenergy remains in the main oscillator: no energy pumping occurs.For H = 60 (figures 7 and 8), the initial energy is greater than the critical bifurcation level.As a consequence, the dashed line branch in figures 3 is feasible. The energy is rapidly transferredfrom the main mass to the absorber. To confirm the predictions of the nonlinear modes, we canwatch the history of the instantaneous frequency: in the early moments (after a short transientperiod), the instantaneous frequencies from the linear and nonlinear oscillator join, then (as theglobal energy decreases) both of them increase. This is in accordance with the evolution of thenonlinear natural frequencies of the modes. 13
10 20 30 40 50 60−4−2024 time x : li nea r do f (a) I n s t. f r equen cy ( H z ) (b) Figure 7: Transient response for H = 60 – Linear oscillator; (a) time history, (b) instantaneousfrequency. v : ab s o r be r (a) I n s t. f r equen cy ( H z ) (b) Figure 8: Transient response for H = 60 – Absorber; (a) time history, (b) instantaneous frequency. This final part is dedicated to forced resonance phenomena. We were, in particular, interested inperiodically forced regimes. In contrast with transient phenomena, in steady-state forced vibration,failures generally occur because of high cycle fatigue effects. Therefore, control of forced vibrationsis of primary importance in many fields of mechanical engineering. It will be demonstrated thatthe energy pumping phenomenon can achieve this control function.
We consider an harmonic excitation, f ( t ) = P cos ωt in the system (1). The derivation of periodicsolution uses the averaging procedure of section 3, in which the two displacements variables takethe form of (5), ie : x ( τ, η ) = a x ( η ) cos( τ + ϕ x ( η )) and v ( τ, η ) = a v ( η ) cos( τ + ϕ v ( η )) (27)where τ = ωt . 14he term G ( a, ϕ, τ ) in equations (12) and (13) is respectively for x and v : G x ( a x , ϕ x , τ ) = − λ ωa x sin( τ + ϕ x ) + ω a x cos( τ + ϕ x ) − ǫ a v cos( τ + ϕ v ) − P cos τ (28a) G v ( a v , ϕ v , τ ) = − λ ωa v sin( τ + ϕ v )+ ω a v cos( τ + ϕ v ) − ǫ a x sin( τ + ϕ x )+ a r ( a v , ϕ v ) cos( τ + ϕ r ( a v , ϕ v ))(28b)Then applying the relations (13) to the present example, one obtains: a ′ x = − ǫ ω a v sin( ϕ x − ϕ v ) − λ a x − P ω sin ϕ x (29a) a ′ v = ǫ ω a x sin( ϕ x − ϕ v ) − λ a v + 12 ω a r sin( ϕ v − ϕ r ) (29b) a x ϕ ′ x = ω − ω ω − ǫ ω a v cos( ϕ x − ϕ v ) − P ω cos ϕ x (29c) a v ϕ ′ v = ω − ω ω − ǫ ω a x cos( ϕ x − ϕ v ) + 12 ω a r cos( ϕ v − ϕ r ) (29d)As for the free response, the fixed points of system (29) are to the main approximation to the forcedresponse. Next, these forced response will be investigated for several values of load amplitude P . The following results were obtained by numerically solving the nonlinear system (29) using Newton-like solver combine with an arc-length continuation method [19]. We used the first example Bouc-Wen hysteretic cycle ( γ = 10 − , figure 2).The nonlinear frequency response are compared with the frequency responses of the linearsystem alone; the backbone curves have also been represented in order to see how the prediction ofthe nonlinear modes are in accordance with the forced response.In the first example (figure 9), the forcing level is quite small and we can see in both the linearand nonlinear oscillator have a linearizable behaviour.As the level of the excitation is increased, the system’s behaviour differs from the linear case andseveral interesting phenomena appear. An example of nonlinear response is plotted in figure 10.In this case, the vibratory energy is strongly localized in the nonlinear oscillator (absorber) in thevicinity of the resonance peak. The absorber appears to be efficient. One can also notices thatthe nonlinear response remains quite close to the backbone curves which attests the quality of theprediction of the nonlinear modes. The results of an investigation on the dynamics of a small nonlinear oscillator weakly coupled witha linear oscillator were presented. This investigation focused on hysteretic nonlinearity using aBouc-Wen differential model. It was shown that the absorber can act as an energy sink when it is15 .88 3.89 3.9 3.91 3.92 3.93 3.940.20.40.60.811.21.41.61.822.2 x : li nea r do f Excitation frequency (Hz) (a) Linear oscillator v : ab s o r be r (b) Absorber Figure 9: Forced response – Low energy: (——), nonlinear response, ( − · −· ), linear response (noabsorber), ( − − − ), backbone curves x : li nea r do f Excitation frequency (Hz) (a) Linear oscillator v : ab s o r be r Excitation frequency (Hz) (b) Absorber
Figure 10: Forced response – High energy: (——), nonlinear response, ( − · −· ), linear response (noabsorber), ( − − − ), backbone curves 16roperly designed; in particular it appears that the level of nonlinearity and the level of dampingare important factor for the efficiency of the device. In order to derive approximate solutionsto the nonlinear problem, an averaging strategy was used. Investigations of the free and forcedresponses are presented and, in both cases, it appears that when the energy of the system issufficient some localization of the vibratory energy in the nonlinear absorber appears. With theexamples presented in this paper, we have seen that the nonlinear modes were representative of thebehaviour of nonlinear dissipative system in free response as in forced response.
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