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Analytic treatment of a system with a vibro-impact nonlinear energy sink
Journal of Sound and Vibration 331 (2012) 4599–4608
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Rapid Communications
Analytic treatment of a system with a vibro-impact nonlinear energy sink O.V. Gendelman n Faculty of Mechanical Engineering, Technion, Israel Institute of Technology, Haifa 32000, Israel
a r t i c l e i n f o
abstract
Article history: Received 2 February 2012 Received in revised form 14 May 2012 Accepted 14 May 2012 Handling Editor: L.N. Virgin Available online 15 June 2012
This Communication reports on an approximate analytic procedure for analysis of transient damped response in a system comprising a primary linear oscillator and vibro-impact nonlinear energy sink (NES). Multiple-scale expansion reduces the problem in the first approximation to low-dimensional vibro-impact system with forcing and damping; the latter problem yields to direct analytic approach. The results computed by this way fairly correlate with direct numeric simulations. The analysis takes advantage of small ratio of the NES and the primary masses, but does not require the restitution coefficient to be close to unity. The proposed approach paves the way for efficient design of systems with vibro-impact NESs. & 2012 Elsevier Ltd. All rights reserved.
1. Introduction The nonlinear energy sink (NES) has been defined as a single-degree-of-freedom (SDOF) structural element with relatively small mass and weak dissipation, attached to a primary structure via essentially nonlinear coupling [1–3]. If the primary structure is excited by a shock whose energy is above a certain critical threshold, the NES can act as broadband passive and adaptive controller by absorbing vibration energy from the primary structure in an almost irreversible manner. This process is referred to as passive targeted energy transfer (TET) [4,5]. The TET normally occurs via transient resonance captures, made possible by the essential (nonlinearizable) stiffness nonlinearity of the NES which prevents a preferential resonance frequency. In previous works it has been shown theoretically, numerically and experimentally that the NES can efficiently protect a primary structure against impulsive excitations [6], harmonic (narrowband) loads [7,8], and seismic excitations [9], and it has also been applied to passively suppress aeroelastic instabilities [10] and drill-string instabilities [11]. A detailed discussion of the concepts of NES, TET and related issues is presented in a recent monograph [11]. Theoretically, the type of nonlinearity involved in the NES may be rather diverse. In the first papers devoted to the subject [1,2] pure cubic nonlinearity was considered. Later more involved types of nonlinear stiffness were incorporated, including general non-polynomial [12] monotonic functions, NES with multiple states of equilibrium [13], as well as nonsmooth and vibro-impact NES [9,14,15]. Experimentally, only a handful of nonlinear stiffness functions were realized and tested in a context of the NES. Nonlinear stiffness close to purely cubic was realized with the help of elastic strings or springs with minimal pre-tension [11,16,17]. The non-smooth NES was realized by combining linear elastic and vibroimpact elements [14,15]. Recently, it was demonstrated that simple eccentric rotator can be efficiently used as the NES [18]. Two latter types of the NES (vibro-impact and rotational) are quite different from the other types, since they do not
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require any nonlinear spring. This drastic simplification makes them arguably the most viable candidates for diverse applications. It should be mentioned that the vibro-impact elements were recognized and implemented long ago as viable engineering solutions for different problems related to vibration absorption and mitigation [19–22]. Commonly, analysis and design of such vibro-impact vibration absorbers involved steady-state responses to external forcing with constant amplitude and frequency spectrum. The TET process is quite different, since it is the transient response of the system—therefore the methods commonly used for the analysis of the steady-state responses are not sufficient. To date, all attempts to describe the dynamics of the vibro-impact NES were concentrated around numeric simulations of conservative and damped dynamics [9,11,23]. For smooth NESs, approximate analytic description of transient damped responses has been achieved by singular perturbation approach, based on averaging and multiple scales analysis [11,12,24]. This approach allowed description and prediction of possible response regimes also in forced systems involving the NESs [25]. As for the vibro-impact systems, such approach is not applicable directly due to the nonsmooth character of the system. Current Communication addresses exactly this issue and suggests the analytic approach tailored for the description of the transient responses in damped systems with the vibro-impact NESs. 2. Description of the model and asymptotic analysis Let us consider the simplest possible design comprising primary linear oscillator with the vibro-impact nonlinear energy sink. The latter is realized as straight cavity in which the impacting particle is allowed to move freely. Sketch of the system is presented in Fig. 1. Without affecting the generality, mass and rigidity of the linear spring in the primary oscillator are adopted to be unit, and length of the cavity for the motion of the vibro-impact particle is adopted to be equal to 2. Mass of the impacting particle is equal to e, a linear damping coefficient of the primary oscillator is equal to el. Variables u(t) and v(t) denote the displacements of the linear oscillator and the NES, respectively, with respect to a laboratory frame. Impact conditions in the cavity can be conveniently written down in terms of a relative displacement: _ j þ 0Þ ¼ kwðt _ j 0Þ, wðt
wðt j Þ ¼ 71
wðtÞ ¼ uðtÞvðtÞ
(1)
here 0 rkr1 is a restitution coefficient, tj is a time instance of the jth impact, tj 7 0 corresponds to the states immediately before and after the impact. Condition of total momentum conservation in the course of each impact is written as: _ j þ 0Þ þ evðt _ j 0Þ ¼ uðt _ j þ 0Þ _ j 0Þ þ evðt uðt
(2)
From Eqs. (1) and (2) it is easy to derive amount of momentum transferred from the impacting particle to the primary oscillator (and vice versa, by virtue of the conservation law) in each impact: _ j þ 0Þuðt _ j 0Þ ¼ DP ¼ uðt
eð1 þ kÞ _ wðt j 0Þ 1þe
(3)
With account of (3), it is possible to write down the equations of motion of the considered system in the following general form: þ kÞ P _ u€ þ elu_ þu þ eð1 wðt0Þdðtt j Þ ¼ 0 1þe j
ð1þ kÞ X € _ v wðt0Þ dðttj Þ ¼ 0 1þe j
(4)
here d(t) is common Dirac delta-function. Physically, it corresponds to infinite force applied to the system in the course of the instantaneous impact. The amount of momentum transferred in the course of the impact is determined by Eq. (3). The summation in Eq. (4) is performed over all impacts. If one introduces a new variable, proportional to the displacement
Fig. 1. Sketch of the dynamical system comprising the linear primary oscillator and the vibro-impact NES.
O.V. Gendelman / Journal of Sound and Vibration 331 (2012) 4599–4608
of the center of masses of the system and rescales the time and the linear damping coefficient as follows: pffiffiffiffiffiffiffiffiffiffi l XðtÞ ¼ uðtÞ þ evðtÞ, t ¼ t 1þ e, g ¼ pffiffiffiffiffiffiffiffiffiffi 1þe
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(5)
then System (4) is reduced to the following form: X tt þ X þ egX t þ e2 gwt þ ew ¼ 0 X wt ðt0Þdðttj Þ ¼ 0 wtt þ X þ egX t þ e2 gwt þ ew þ ð1þ kÞ
(6)
j
Summation in the second equation of System (6) is also performed over the successive impacts, but with respect to the new time scale. System (6) is the basis for further analysis. Up to this point, the transformations were exact. Further analysis of System (6) is possible if one supposes that the mass of the impacting particle is small with respect to the overall mass of the system. Then, e{1 can be used as a small parameter in multiple scales expansion. Common procedure of this sort [26] includes multiple time scales and power series expansion of the dependent variables: d @ @ ¼ þe þ dt @t0 @t1 X ¼ X 0 ðt0 , t1 ,:::Þ þ eX 1 ðt0 , t1 ,:::Þ; w ¼ w0 ðt0 , t1 ,:::Þ þ ew1 ðt0 , t1 ,:::Þ:
tk ¼ ek t,k ¼ 0,1,:::;
(7)
In this work only two first time scales are used. Substituting (7) into (6) and keeping only the terms of zero order with respect to e, one obtains: @2 X 0 þX 0 ¼ 0 ) X 0 ¼ Cðt1 Þsinðt0 þ cðt1 ÞÞ @t20 X @w0 @2 w0 þ X 0 þ ð1þ kÞ 9 dðt0 t0j Þ ¼ 0 @t0 t0 0 @t20 j
(8)
here C(t1),c(t1) are slowly varying amplitude and phase of the X variable, respectively. The second equation of System (8), in essence, describes a simple inelastic vibro-impact oscillator subject to harmonic forcing; the amplitude and the phase of the external force are constant with respect to the fast time scale t0. The forced–damped one-dimensional vibro-impact oscillators similar to one described by the second equation of (8) were considered quite widely [27,28]. We look for the solution of this equation in the following form: w0 ¼ Cðt1 Þsinðt0 þ cðt1 ÞÞ þ f ðt0 , t1 Þ
(9)
From (8) and (9), one obtains: X @f @2 f þ ð1 þ kÞ ð 9 þCðt1 Þcosðt0 þ cðt1 ÞÞÞdðt0 t0j Þ ¼ 0 @t0 t0 0 @t20 j
(10)
Solution of Eq. (10) corresponds to a sequence of ‘‘free flights’’ divided by subsequent impacts. The simplest case of the TET involves a capture into 1:1 resonance between the primary system and the NES. Then, in order to study the energy dissipation in the system, we look for the solution of Eq. (10), which corresponds to symmetric motion of the particle with the frequency of the external oscillations. Thus, the transient process leading to a capture into this stable response regime is not considered. With respect to time scale t0, the steady-state response will have constant amplitude, since all variations occur at slower time scale. In other terms, the solution of Eq. (10) is searched in a form [29–31] f ðt0 , t1 Þ ¼
2a
p
arcsinðcosðt0 ZÞÞ
(11)
where Z þ pj, j¼0, 71,72,... are yet unknown time instants of the successive impacts with respect to the fast time scale and a is the amplitude of the function f. Sketch of function f according to Eq. (11) is presented in Fig. 2. Integration of Eq. (10) over the time in a small interval around the time instant corresponding to the impact, t0 ¼ Z yields the condition of the momentum conservation:
4a
p
þ ð1 þkÞð
2a
p
þ CcosðZ þ cÞÞ ¼ 0 )
) CcosðZ þ cÞ ¼ sa, s ¼
2ð1kÞ
pð1 þ kÞ
(12)
The impacts occur as w¼ 71. Then, from (1), (9) and (11) one obtains the second relationship between the parameters of the problem: CsinðZ þ cÞ þ a ¼ 1
(13)
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Fig. 2. Schematic presentation of function f(t0,t1) according to Eq. (11).
From Eqs. (12,13) one can easily find the relationship between the parameters a and Z and the unknown parameters C(t1),c(t1) of the solution X0(t0,t1). More specifically, from (12) and (13) one obtains: pffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 7 1 þ s2 C 2 ðt1 ÞC 2min s a¼ , C min ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi (14) 1 þ s2 1 þ s2 and then the value of Z can be found from any of Eqs. (12) or (13). Eq. (14) is rather interesting in its own turn. First of all, it delivers the minimum possible amplitude of X0 which still allows the TET in the regime of 1:1 resonance. Besides, it defines the explicit expression for slow invariant manifold (SIM, [11,24]) of this problem—a pair of attractor and repeller with respect to the time scale t0. More exactly, two branches of solution (14) correspond to stable and unstable symmetric steady-state responses of the damped vibro-impact oscillator with harmonic external forcing. The SIM (14) is presented in Fig. 3. As it was mentioned above, the upper branch of (14) is the stable attractor of System (10) at the fast time scale t0. In order to study the energy dissipation in the system, more refined study of the dynamics at the slow time scale t1 is required. For this sake, solution (11) is rewritten as: f ðt0 , t1 Þ ¼
2a
p
arcsinðcosðt0 ZÞÞ ¼
1 8a X ð1Þk þ 1
p2 k ¼ 1 ð2k1Þ2
cosðð2k1Þðt0 ZÞ
(15)
Multiple-scales expansion of the firs equation of System (6) with respect to the slow time scale t1 is written with account of (8), (9) and (15), as: @2 @2 @ 8a X1 ð1Þk þ 1 X1 þ X1 ¼ X 0 g X 0 X 0 2 cosðð2k1Þðt0 ZÞÞ ¼ 2 k¼1 @t0 @t0 @t1 p @t0 ð2k1Þ2 ¼
@2 ðCsinðt0 þ cÞÞ @ðCsinðt0 þ cÞÞ g Csinðt0 þ cÞ @t0 @t1 @t0 8a X1
ð1Þk þ 1
p2
ð2k1Þ2
k¼1
cosðð2k1Þððt0 þ cÞðZ þ cÞÞ
(16)
Secular terms in (16) will be absent if the following relationships will hold: 2
dC 8a þ gC þ 2 cosðZ þ cÞ ¼ 0; dt1 p
2C
dc 8a þC þ 2 sinðZ þ cÞ ¼ 0 dt1 p
Substituting (12) and (13) into (17), we obtain: dðC 2 Þ d t1
þ gC 2 þ 8sa p2 ¼ 2
dðC 2 Þ dt1
1þ
þ gC 2 þ 8ps2 ð
ffi pffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ s2 C 2 C 2min 2 Þ ¼ 0; 1 þ s2
(17)
O.V. Gendelman / Journal of Sound and Vibration 331 (2012) 4599–4608
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Fig. 3. Slow invariant manifold of the system, the restitution coefficient k¼0.62.
dc 1 8að1aÞ ¼ þ 2 dt1 p2 C 2
(18)
The first equation of System (18) can be solved by explicit (albeit somewhat complicated) quadrature which delivers analytic solution for C(t1). Solution for c(t1) can be obtained from the second equation of (18) by direct integration, once the expression for C(t1) is known. If the linear damping at the primary oscillator is absent (g ¼0), then the integration of the first equation in System (18) is easy. This integration yields the result for C(t1) in the following implicit form: ffi pffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 1 þ s2 C 20 C 2min 4st1 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffip lnð pffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Þ þ pffiffiffiffiffiffiffiffiffi1ffipffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ p2 ð1 (19) þ s2 Þ 2 2 2 2 2 2 1þ
1 þ s2
C ðt1 ÞC min
1þ
1 þ s2
C 0 C min
1þ
1 þ s2
C ðt1 ÞC min
Similar implicit expression for the case ga0 also can be calculated, but is omitted here due to its awkwardness. It is interesting to mention that close to the minimal value of the amplitude, the last term in the parentheses in the first equation of System (18) may be omitted and one obtains: dðC 2 Þ 8s þ ¼0 dt1 p2 ð1 þ s2 Þ2
(20)
Physically, Eq. (20) demonstrates that in the process of the TET the energy is dissipated at almost constant rate. It is rather different from common systems with linear damping, where the rate of the dissipation decreases with the decrease of energy. 3. Numeric results In order to get further insight into the TET process in this vibro-impact system and to verify the analytic results obtained in the previous Section, we simulate numerically the behavior of System (1) with initial conditions corresponding to zero displacement and nonzero velocity of the primary mass; it is supposed that initially the impacting particle is placed in the left end of the cavity (cf. Fig. 1). For the sake of simplicity, linear damping at the primary mass is also omitted; therefore, the only dissipation mechanism in the system is related to the inelastic impacts. Complete set of parameters and
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_ _ Fig. 4. Response of System (1) in the case of small-amplitude impact, k¼0.62, e ¼ 0.05, uð0Þ ¼ 0, uð0Þ ¼ 0:15,vð0Þ ¼ 0:97, vð0Þ ¼ 0, (a) response of the primary mass, u(t); (b) response of the impacting particle, v(t).
O.V. Gendelman / Journal of Sound and Vibration 331 (2012) 4599–4608
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_ _ Fig. 5. Fig. 4. Response of System (1) in the case of large-amplitude impact, k¼0.62, e ¼ 0.05, uð0Þ ¼ 0, uð0Þ ¼ 0:5,vð0Þ ¼ 0:97, vð0Þ ¼ 0, (a) response of the primary mass, u(t); (b) response of the impacting particle, v(t).
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Fig. 6. Relative energy remaining in the system (21) versus time for the cases of small and large amplitude of the initial impacts.
initial conditions for the simulation is presented in figure captions. Time series for variables u(t) and v(t) for the case of relatively low initial impact are presented in Fig. 4a, b. One can easily observe that there is no significant damping in the system. The situation changes drastically if higher values of the initial impact are considered. Time series for this case are presented in Fig. 5a, b. Rapid decay of the amplitude of the primary mass is clearly observed while it is engaged into 1:1 resonance with the impacting particle. These graphs demonstrate the process of targeted energy transfer in the system. To make this point more clear, the time series for relative energy stored in the system are presented. This relative energy is defined as the ratio of instantaneous energy in the system and the initial energy: Er ¼ EðtÞ=Eð0Þ ¼
u_ 2 ðtÞ þ u2 ðtÞ þ ev_ 2 ðtÞ u_ 2 ð0Þ
(21)
Time series for the relative energy for two sets of initial conditions used for Figs. 4, 5 is presented in Fig. 6. It is easy to see a significant enhancement of the energy dissipation due to the TET. In order to compare the numeric results for the envelope with analytic predictions, System (6) is simulated with parameters and initial conditions equivalent to those used for demonstration of the TET in Fig. 5. Comparison of the numeric results for the variable X(t) (we remind that it is proportional to the displacement of the center of masses of the system) and its envelope C(et) computed from (19) with appropriate initial conditions, is presented in Fig. 7. It follows that the analytic approach developed above allows rather exact prediction of the system response. Not less important, it also succeeds to predict the breakdown of the 1:1 resonance at the response amplitude XðtÞ ¼ C min . 4. Concluding remarks The way of treatment presented above allows approximate analytic treatment of the damped responses in the systems with vibro-impact NES. The treatment is quite different from previous works on damped responses in the systems with the NESs [11,24], since no averaging is used on any step of the analysis. In fact, the treatment is based on two simplifying assumptions. The first one is the smallness of the NES mass; this assumption allows the use of the multiple scales expansion. The second simplification is related to direct consideration of the steady-state solution of Eq. (10) instead of detailed analysis of domains of attraction and transients leading to this regime. Due to discontinuous nature of Eq. (10), such analysis would require numeric simulation of discrete nonlinear maps and is beyond the scope of this paper. This simplification is a main source of inaccuracies in the prediction of the envelope in Fig. 7—one can observe that at the initial stage of the process the system does not settle immediately at the envelope predicted by the SIM, but is eventually
O.V. Gendelman / Journal of Sound and Vibration 331 (2012) 4599–4608
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Fig. 7. Time series X(t) (proportional to displacement of the center of masses). Solid line—numeric simulation, dotted line—analytic prediction according to (19).
attracted to it. Despite this inaccuracy, the treatment presented here allows quite accurate description for characteristic amplitude and time scale of the TET process in this system. It should be mentioned that the restitution coefficient used in all simulations and calculations is more or less realistic and by no means close to unity. In computational studies on optimization of the performance of the system with VI NES [9,15,23] the values of the optimal restitution coefficient were close to those used here. The methodology presented above can be used for analysis of damped response in systems with more complicated NES configurations, for instance, including elastic components, one-sided impacts etc.
Acknowledgments The author is very grateful to Dr. Yuli Starosvetsky for useful discussions. The author is very grateful to Binational US—Israel Scientific Foundation (BSF) for financial support of this work (grant 2008055). References [1] O.V. Gendelman, Transition of energy to a nonlinear localized mode in a highly asymmetric system of two oscillators, Nonlinear Dynamics 25 (2001) 237–253. [2] A.F. Vakakis, O.V. Gendelman, Energy pumping in nonlinear mechanical oscillators: Part II—resonance capture, ASME Journal of Applied Mechanics 68 (2001) 42–48. [3] L.I. Manevitch, E. Gourdon, C.H. Lamarque, Towards the design of an optimal energetic sink in a strongly inhomogeneous two-degree-of-freedom system, ASME Journal of Applied Mechanics 74 (2007) 1078–1086. [4] D.D. Quinn, O. Gendelman, G. Kerschen, T.P. Sapsis, L.A. Bergman, A.F. Vakakis, Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures: Part I, Journal of Sound and Vibration 311 (2008) 1228–1248. [5] T.P. Sapsis, A.F. Vakakis, O.V. Gendelman, L.A. Bergman, G. Kerschen, D.D. Quinn, Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures: Part II, analytical study, Journal of Sound and Vibration 325 (2009) 297–320. [6] A.F. Vakakis, Inducing passive nonlinear energy sinks in vibrating systems, ASME Journal of Vibrations and Acoustics 123 (2001) 324–332. [7] Y. Starosvetsky, O.V. Gendelman, Strongly modulated response in forced 2DOF oscillatory system with essential mass and potential asymmetry, Physica D 237 (2008) 1719–1733. [8] Y. Starosvetsky, O.V. Gendelman, Vibration absorption in systems comprising nonlinear energy sink: nonlinear damping, Journal of Sound and Vibration 324 (2009) 916–939. [9] F. Nucera, A.F. Vakakis, D.M. McFarland, L.A. Bergman, G. Kerschen, Targeted energy transfers in vibro-impact oscillators for seismic mitigation, Nonlinear Dynamics 50 (2007) 651–677.
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[10] O.V. Gendelman, A.F. Vakakis, L.A. Bergman, D.M. McFarland, Asymptotic analysis of passive nonlinear suppression of aeroelastic instabilities of a rigid wing in subsonic flow, SIA, Journal of Applied Mathematics 70 (2010) 1655–1677. [11] A.F. Vakakis, O.V. Gendelman, G. Kerschen, L.A. Bergman, D.M. McFarland, Y.S. Lee, Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems, I and II, Springer, Berlin, 2008. [12] O.V. Gendelman, Targeted energy transfer in systems with non-polynomial nonlinearity, Journal of Sound and Vibration 315 (2008) 732–745. [13] O.V. Gendelman, C.H. Lamarque, Dynamics of linear oscillator coupled to strongly nonlinear attachment with multiple states of equilibrium, Chaos, Solitons and Fractals 24 (2005) 501–509. [14] F. Georgiades, A.F. Vakakis, D.M. McFarland, L.A. Bergman, Shock isolation through passive energy pumping caused by non-smooth nonlinearities, International Journal of Bifurcations and Chaos 15 (2005) 1–13. [15] F. Nucera, F. Lo Iacono, D.M. McFarland, L.A. Bergman, A.F. Vakakis, Application of broadband nonlinear targeted energy transfers for seismic mitigation of a shear frame: Part II experimental results, Journal of Sound and Vibration 313 (2008) 57–76. [16] E. Gourdon, N.A. Alexander, C.A. Taylor, C.H. Lamarque, S. Pernot, Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: theoretical and experimental results, Journal of Sound and Vibration 300 (2007) 522–551. [17] R. Bellet, B. Cochelin, P. Herzog, P.-O. Mattei, Experimental study of targeted energy transfer from an acoustic system to a nonlinear membrane absorber, Journal of Sound and Vibration 329 (2010) 2768–2791. [18] O.V. Gendelman, G. Sigalov, L.I. Manevitch, M. Mane, A.F. Vakakis, L.A. Bergman, Dynamics of an eccentric rotational nonlinear energy sink, Journal of Applied Mechanics, Transactions of the ASME 79 (011012) (2012) 1–9. [19] I.G. Rusakov, A.A. Kharkevich, Forced vibrations of a system that is striking a stop (in Russian), Journal of Technical Physics 12 (1942) 715–721. [20] R.N. Arnold, Response of an impact vibration absorber to forced vibration, Actes IX Congress on Applied Mechanics, Brussels, 1957. [21] V.I. Babitsky, A.M. Veprik, Universal bumpered vibration isolator for severe environment, Journal of Sound and Vibration 218 (1998) 269–292. [22] V.I. Babitsky, Theory of Vibro-Impact Systems, Nauka, Moscow, 1978 (Revised English translation from Russian: 1998 Berlin, Heidelbnerg, New York: Springer). [23] I. Karayannis, A.F. Vakakis, F. Georgiades, Vibro-impact attachments as shock absorbers, Proceedings IMechE, Journal of Mechanical Engineering Science 222 (2008) 1899–1908. [24] O.V. Gendelman, Bifurcations of nonlinear normal modes of linear oscillator with strongly nonlinear damped attachment, Nonlinear Dynamics 37 (2004) 115–128. [25] O.V Gendelman, Targeted energy transfer in systems with external and self-excitation, Proceedings of the Institution of Mechanical Engineers Part C, Journal of Mechanical Engineering Science, Invited Review 225 (2011) 2007–2043. [26] A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations, Wiley, NY, 1995. [27] S.W. Shaw, P.J. Holmes, A periodically forced impact oscillator with large dissipation, ASME Journal of Applied Mechanics 50 (1983) 849–857. [28] S.W. Shaw, R.H. Rand, The transition to chaos in a simple mechanical system, International Journal of Non-Linear Mechanics 24 (1989) 41–56. [29] V.N. Pilipchuk, Impact modes in discrete vibrating systems with rigid barriers, International Journal of Non-linear Mechanics 36 (2001) 999–1012. [30] V.N. Pilipchuk, Some remarks on non-smooth transformations of space and time for vibrating systems with rigid barriers, Journal of Applied Mathematics and Mechanics 66 (2002) 31–37. [31] V.N. Pilipchuk, Analytical study of vibrating systems with strong non-linearities by employing saw-tooth time transformations, ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik 75 (1996) 389–390.
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Rapid Communications
Analytic treatment of a system with a vibro-impact nonlinear energy sink O.V. Gendelman n Faculty of Mechanical Engineering, Technion, Israel Institute of Technology, Haifa 32000, Israel
a r t i c l e i n f o
abstract
Article history: Received 2 February 2012 Received in revised form 14 May 2012 Accepted 14 May 2012 Handling Editor: L.N. Virgin Available online 15 June 2012
This Communication reports on an approximate analytic procedure for analysis of transient damped response in a system comprising a primary linear oscillator and vibro-impact nonlinear energy sink (NES). Multiple-scale expansion reduces the problem in the first approximation to low-dimensional vibro-impact system with forcing and damping; the latter problem yields to direct analytic approach. The results computed by this way fairly correlate with direct numeric simulations. The analysis takes advantage of small ratio of the NES and the primary masses, but does not require the restitution coefficient to be close to unity. The proposed approach paves the way for efficient design of systems with vibro-impact NESs. & 2012 Elsevier Ltd. All rights reserved.
1. Introduction The nonlinear energy sink (NES) has been defined as a single-degree-of-freedom (SDOF) structural element with relatively small mass and weak dissipation, attached to a primary structure via essentially nonlinear coupling [1–3]. If the primary structure is excited by a shock whose energy is above a certain critical threshold, the NES can act as broadband passive and adaptive controller by absorbing vibration energy from the primary structure in an almost irreversible manner. This process is referred to as passive targeted energy transfer (TET) [4,5]. The TET normally occurs via transient resonance captures, made possible by the essential (nonlinearizable) stiffness nonlinearity of the NES which prevents a preferential resonance frequency. In previous works it has been shown theoretically, numerically and experimentally that the NES can efficiently protect a primary structure against impulsive excitations [6], harmonic (narrowband) loads [7,8], and seismic excitations [9], and it has also been applied to passively suppress aeroelastic instabilities [10] and drill-string instabilities [11]. A detailed discussion of the concepts of NES, TET and related issues is presented in a recent monograph [11]. Theoretically, the type of nonlinearity involved in the NES may be rather diverse. In the first papers devoted to the subject [1,2] pure cubic nonlinearity was considered. Later more involved types of nonlinear stiffness were incorporated, including general non-polynomial [12] monotonic functions, NES with multiple states of equilibrium [13], as well as nonsmooth and vibro-impact NES [9,14,15]. Experimentally, only a handful of nonlinear stiffness functions were realized and tested in a context of the NES. Nonlinear stiffness close to purely cubic was realized with the help of elastic strings or springs with minimal pre-tension [11,16,17]. The non-smooth NES was realized by combining linear elastic and vibroimpact elements [14,15]. Recently, it was demonstrated that simple eccentric rotator can be efficiently used as the NES [18]. Two latter types of the NES (vibro-impact and rotational) are quite different from the other types, since they do not
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require any nonlinear spring. This drastic simplification makes them arguably the most viable candidates for diverse applications. It should be mentioned that the vibro-impact elements were recognized and implemented long ago as viable engineering solutions for different problems related to vibration absorption and mitigation [19–22]. Commonly, analysis and design of such vibro-impact vibration absorbers involved steady-state responses to external forcing with constant amplitude and frequency spectrum. The TET process is quite different, since it is the transient response of the system—therefore the methods commonly used for the analysis of the steady-state responses are not sufficient. To date, all attempts to describe the dynamics of the vibro-impact NES were concentrated around numeric simulations of conservative and damped dynamics [9,11,23]. For smooth NESs, approximate analytic description of transient damped responses has been achieved by singular perturbation approach, based on averaging and multiple scales analysis [11,12,24]. This approach allowed description and prediction of possible response regimes also in forced systems involving the NESs [25]. As for the vibro-impact systems, such approach is not applicable directly due to the nonsmooth character of the system. Current Communication addresses exactly this issue and suggests the analytic approach tailored for the description of the transient responses in damped systems with the vibro-impact NESs. 2. Description of the model and asymptotic analysis Let us consider the simplest possible design comprising primary linear oscillator with the vibro-impact nonlinear energy sink. The latter is realized as straight cavity in which the impacting particle is allowed to move freely. Sketch of the system is presented in Fig. 1. Without affecting the generality, mass and rigidity of the linear spring in the primary oscillator are adopted to be unit, and length of the cavity for the motion of the vibro-impact particle is adopted to be equal to 2. Mass of the impacting particle is equal to e, a linear damping coefficient of the primary oscillator is equal to el. Variables u(t) and v(t) denote the displacements of the linear oscillator and the NES, respectively, with respect to a laboratory frame. Impact conditions in the cavity can be conveniently written down in terms of a relative displacement: _ j þ 0Þ ¼ kwðt _ j 0Þ, wðt
wðt j Þ ¼ 71
wðtÞ ¼ uðtÞvðtÞ
(1)
here 0 rkr1 is a restitution coefficient, tj is a time instance of the jth impact, tj 7 0 corresponds to the states immediately before and after the impact. Condition of total momentum conservation in the course of each impact is written as: _ j þ 0Þ þ evðt _ j 0Þ ¼ uðt _ j þ 0Þ _ j 0Þ þ evðt uðt
(2)
From Eqs. (1) and (2) it is easy to derive amount of momentum transferred from the impacting particle to the primary oscillator (and vice versa, by virtue of the conservation law) in each impact: _ j þ 0Þuðt _ j 0Þ ¼ DP ¼ uðt
eð1 þ kÞ _ wðt j 0Þ 1þe
(3)
With account of (3), it is possible to write down the equations of motion of the considered system in the following general form: þ kÞ P _ u€ þ elu_ þu þ eð1 wðt0Þdðtt j Þ ¼ 0 1þe j
ð1þ kÞ X € _ v wðt0Þ dðttj Þ ¼ 0 1þe j
(4)
here d(t) is common Dirac delta-function. Physically, it corresponds to infinite force applied to the system in the course of the instantaneous impact. The amount of momentum transferred in the course of the impact is determined by Eq. (3). The summation in Eq. (4) is performed over all impacts. If one introduces a new variable, proportional to the displacement
Fig. 1. Sketch of the dynamical system comprising the linear primary oscillator and the vibro-impact NES.
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of the center of masses of the system and rescales the time and the linear damping coefficient as follows: pffiffiffiffiffiffiffiffiffiffi l XðtÞ ¼ uðtÞ þ evðtÞ, t ¼ t 1þ e, g ¼ pffiffiffiffiffiffiffiffiffiffi 1þe
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(5)
then System (4) is reduced to the following form: X tt þ X þ egX t þ e2 gwt þ ew ¼ 0 X wt ðt0Þdðttj Þ ¼ 0 wtt þ X þ egX t þ e2 gwt þ ew þ ð1þ kÞ
(6)
j
Summation in the second equation of System (6) is also performed over the successive impacts, but with respect to the new time scale. System (6) is the basis for further analysis. Up to this point, the transformations were exact. Further analysis of System (6) is possible if one supposes that the mass of the impacting particle is small with respect to the overall mass of the system. Then, e{1 can be used as a small parameter in multiple scales expansion. Common procedure of this sort [26] includes multiple time scales and power series expansion of the dependent variables: d @ @ ¼ þe þ dt @t0 @t1 X ¼ X 0 ðt0 , t1 ,:::Þ þ eX 1 ðt0 , t1 ,:::Þ; w ¼ w0 ðt0 , t1 ,:::Þ þ ew1 ðt0 , t1 ,:::Þ:
tk ¼ ek t,k ¼ 0,1,:::;
(7)
In this work only two first time scales are used. Substituting (7) into (6) and keeping only the terms of zero order with respect to e, one obtains: @2 X 0 þX 0 ¼ 0 ) X 0 ¼ Cðt1 Þsinðt0 þ cðt1 ÞÞ @t20 X @w0 @2 w0 þ X 0 þ ð1þ kÞ 9 dðt0 t0j Þ ¼ 0 @t0 t0 0 @t20 j
(8)
here C(t1),c(t1) are slowly varying amplitude and phase of the X variable, respectively. The second equation of System (8), in essence, describes a simple inelastic vibro-impact oscillator subject to harmonic forcing; the amplitude and the phase of the external force are constant with respect to the fast time scale t0. The forced–damped one-dimensional vibro-impact oscillators similar to one described by the second equation of (8) were considered quite widely [27,28]. We look for the solution of this equation in the following form: w0 ¼ Cðt1 Þsinðt0 þ cðt1 ÞÞ þ f ðt0 , t1 Þ
(9)
From (8) and (9), one obtains: X @f @2 f þ ð1 þ kÞ ð 9 þCðt1 Þcosðt0 þ cðt1 ÞÞÞdðt0 t0j Þ ¼ 0 @t0 t0 0 @t20 j
(10)
Solution of Eq. (10) corresponds to a sequence of ‘‘free flights’’ divided by subsequent impacts. The simplest case of the TET involves a capture into 1:1 resonance between the primary system and the NES. Then, in order to study the energy dissipation in the system, we look for the solution of Eq. (10), which corresponds to symmetric motion of the particle with the frequency of the external oscillations. Thus, the transient process leading to a capture into this stable response regime is not considered. With respect to time scale t0, the steady-state response will have constant amplitude, since all variations occur at slower time scale. In other terms, the solution of Eq. (10) is searched in a form [29–31] f ðt0 , t1 Þ ¼
2a
p
arcsinðcosðt0 ZÞÞ
(11)
where Z þ pj, j¼0, 71,72,... are yet unknown time instants of the successive impacts with respect to the fast time scale and a is the amplitude of the function f. Sketch of function f according to Eq. (11) is presented in Fig. 2. Integration of Eq. (10) over the time in a small interval around the time instant corresponding to the impact, t0 ¼ Z yields the condition of the momentum conservation:
4a
p
þ ð1 þkÞð
2a
p
þ CcosðZ þ cÞÞ ¼ 0 )
) CcosðZ þ cÞ ¼ sa, s ¼
2ð1kÞ
pð1 þ kÞ
(12)
The impacts occur as w¼ 71. Then, from (1), (9) and (11) one obtains the second relationship between the parameters of the problem: CsinðZ þ cÞ þ a ¼ 1
(13)
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Fig. 2. Schematic presentation of function f(t0,t1) according to Eq. (11).
From Eqs. (12,13) one can easily find the relationship between the parameters a and Z and the unknown parameters C(t1),c(t1) of the solution X0(t0,t1). More specifically, from (12) and (13) one obtains: pffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 7 1 þ s2 C 2 ðt1 ÞC 2min s a¼ , C min ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi (14) 1 þ s2 1 þ s2 and then the value of Z can be found from any of Eqs. (12) or (13). Eq. (14) is rather interesting in its own turn. First of all, it delivers the minimum possible amplitude of X0 which still allows the TET in the regime of 1:1 resonance. Besides, it defines the explicit expression for slow invariant manifold (SIM, [11,24]) of this problem—a pair of attractor and repeller with respect to the time scale t0. More exactly, two branches of solution (14) correspond to stable and unstable symmetric steady-state responses of the damped vibro-impact oscillator with harmonic external forcing. The SIM (14) is presented in Fig. 3. As it was mentioned above, the upper branch of (14) is the stable attractor of System (10) at the fast time scale t0. In order to study the energy dissipation in the system, more refined study of the dynamics at the slow time scale t1 is required. For this sake, solution (11) is rewritten as: f ðt0 , t1 Þ ¼
2a
p
arcsinðcosðt0 ZÞÞ ¼
1 8a X ð1Þk þ 1
p2 k ¼ 1 ð2k1Þ2
cosðð2k1Þðt0 ZÞ
(15)
Multiple-scales expansion of the firs equation of System (6) with respect to the slow time scale t1 is written with account of (8), (9) and (15), as: @2 @2 @ 8a X1 ð1Þk þ 1 X1 þ X1 ¼ X 0 g X 0 X 0 2 cosðð2k1Þðt0 ZÞÞ ¼ 2 k¼1 @t0 @t0 @t1 p @t0 ð2k1Þ2 ¼
@2 ðCsinðt0 þ cÞÞ @ðCsinðt0 þ cÞÞ g Csinðt0 þ cÞ @t0 @t1 @t0 8a X1
ð1Þk þ 1
p2
ð2k1Þ2
k¼1
cosðð2k1Þððt0 þ cÞðZ þ cÞÞ
(16)
Secular terms in (16) will be absent if the following relationships will hold: 2
dC 8a þ gC þ 2 cosðZ þ cÞ ¼ 0; dt1 p
2C
dc 8a þC þ 2 sinðZ þ cÞ ¼ 0 dt1 p
Substituting (12) and (13) into (17), we obtain: dðC 2 Þ d t1
þ gC 2 þ 8sa p2 ¼ 2
dðC 2 Þ dt1
1þ
þ gC 2 þ 8ps2 ð
ffi pffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ s2 C 2 C 2min 2 Þ ¼ 0; 1 þ s2
(17)
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Fig. 3. Slow invariant manifold of the system, the restitution coefficient k¼0.62.
dc 1 8að1aÞ ¼ þ 2 dt1 p2 C 2
(18)
The first equation of System (18) can be solved by explicit (albeit somewhat complicated) quadrature which delivers analytic solution for C(t1). Solution for c(t1) can be obtained from the second equation of (18) by direct integration, once the expression for C(t1) is known. If the linear damping at the primary oscillator is absent (g ¼0), then the integration of the first equation in System (18) is easy. This integration yields the result for C(t1) in the following implicit form: ffi pffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 1 þ s2 C 20 C 2min 4st1 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffip lnð pffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Þ þ pffiffiffiffiffiffiffiffiffi1ffipffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ p2 ð1 (19) þ s2 Þ 2 2 2 2 2 2 1þ
1 þ s2
C ðt1 ÞC min
1þ
1 þ s2
C 0 C min
1þ
1 þ s2
C ðt1 ÞC min
Similar implicit expression for the case ga0 also can be calculated, but is omitted here due to its awkwardness. It is interesting to mention that close to the minimal value of the amplitude, the last term in the parentheses in the first equation of System (18) may be omitted and one obtains: dðC 2 Þ 8s þ ¼0 dt1 p2 ð1 þ s2 Þ2
(20)
Physically, Eq. (20) demonstrates that in the process of the TET the energy is dissipated at almost constant rate. It is rather different from common systems with linear damping, where the rate of the dissipation decreases with the decrease of energy. 3. Numeric results In order to get further insight into the TET process in this vibro-impact system and to verify the analytic results obtained in the previous Section, we simulate numerically the behavior of System (1) with initial conditions corresponding to zero displacement and nonzero velocity of the primary mass; it is supposed that initially the impacting particle is placed in the left end of the cavity (cf. Fig. 1). For the sake of simplicity, linear damping at the primary mass is also omitted; therefore, the only dissipation mechanism in the system is related to the inelastic impacts. Complete set of parameters and
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_ _ Fig. 4. Response of System (1) in the case of small-amplitude impact, k¼0.62, e ¼ 0.05, uð0Þ ¼ 0, uð0Þ ¼ 0:15,vð0Þ ¼ 0:97, vð0Þ ¼ 0, (a) response of the primary mass, u(t); (b) response of the impacting particle, v(t).
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_ _ Fig. 5. Fig. 4. Response of System (1) in the case of large-amplitude impact, k¼0.62, e ¼ 0.05, uð0Þ ¼ 0, uð0Þ ¼ 0:5,vð0Þ ¼ 0:97, vð0Þ ¼ 0, (a) response of the primary mass, u(t); (b) response of the impacting particle, v(t).
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Fig. 6. Relative energy remaining in the system (21) versus time for the cases of small and large amplitude of the initial impacts.
initial conditions for the simulation is presented in figure captions. Time series for variables u(t) and v(t) for the case of relatively low initial impact are presented in Fig. 4a, b. One can easily observe that there is no significant damping in the system. The situation changes drastically if higher values of the initial impact are considered. Time series for this case are presented in Fig. 5a, b. Rapid decay of the amplitude of the primary mass is clearly observed while it is engaged into 1:1 resonance with the impacting particle. These graphs demonstrate the process of targeted energy transfer in the system. To make this point more clear, the time series for relative energy stored in the system are presented. This relative energy is defined as the ratio of instantaneous energy in the system and the initial energy: Er ¼ EðtÞ=Eð0Þ ¼
u_ 2 ðtÞ þ u2 ðtÞ þ ev_ 2 ðtÞ u_ 2 ð0Þ
(21)
Time series for the relative energy for two sets of initial conditions used for Figs. 4, 5 is presented in Fig. 6. It is easy to see a significant enhancement of the energy dissipation due to the TET. In order to compare the numeric results for the envelope with analytic predictions, System (6) is simulated with parameters and initial conditions equivalent to those used for demonstration of the TET in Fig. 5. Comparison of the numeric results for the variable X(t) (we remind that it is proportional to the displacement of the center of masses of the system) and its envelope C(et) computed from (19) with appropriate initial conditions, is presented in Fig. 7. It follows that the analytic approach developed above allows rather exact prediction of the system response. Not less important, it also succeeds to predict the breakdown of the 1:1 resonance at the response amplitude XðtÞ ¼ C min . 4. Concluding remarks The way of treatment presented above allows approximate analytic treatment of the damped responses in the systems with vibro-impact NES. The treatment is quite different from previous works on damped responses in the systems with the NESs [11,24], since no averaging is used on any step of the analysis. In fact, the treatment is based on two simplifying assumptions. The first one is the smallness of the NES mass; this assumption allows the use of the multiple scales expansion. The second simplification is related to direct consideration of the steady-state solution of Eq. (10) instead of detailed analysis of domains of attraction and transients leading to this regime. Due to discontinuous nature of Eq. (10), such analysis would require numeric simulation of discrete nonlinear maps and is beyond the scope of this paper. This simplification is a main source of inaccuracies in the prediction of the envelope in Fig. 7—one can observe that at the initial stage of the process the system does not settle immediately at the envelope predicted by the SIM, but is eventually
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Fig. 7. Time series X(t) (proportional to displacement of the center of masses). Solid line—numeric simulation, dotted line—analytic prediction according to (19).
attracted to it. Despite this inaccuracy, the treatment presented here allows quite accurate description for characteristic amplitude and time scale of the TET process in this system. It should be mentioned that the restitution coefficient used in all simulations and calculations is more or less realistic and by no means close to unity. In computational studies on optimization of the performance of the system with VI NES [9,15,23] the values of the optimal restitution coefficient were close to those used here. The methodology presented above can be used for analysis of damped response in systems with more complicated NES configurations, for instance, including elastic components, one-sided impacts etc.
Acknowledgments The author is very grateful to Dr. Yuli Starosvetsky for useful discussions. The author is very grateful to Binational US—Israel Scientific Foundation (BSF) for financial support of this work (grant 2008055). References [1] O.V. Gendelman, Transition of energy to a nonlinear localized mode in a highly asymmetric system of two oscillators, Nonlinear Dynamics 25 (2001) 237–253. [2] A.F. Vakakis, O.V. Gendelman, Energy pumping in nonlinear mechanical oscillators: Part II—resonance capture, ASME Journal of Applied Mechanics 68 (2001) 42–48. [3] L.I. Manevitch, E. Gourdon, C.H. Lamarque, Towards the design of an optimal energetic sink in a strongly inhomogeneous two-degree-of-freedom system, ASME Journal of Applied Mechanics 74 (2007) 1078–1086. [4] D.D. Quinn, O. Gendelman, G. Kerschen, T.P. Sapsis, L.A. Bergman, A.F. Vakakis, Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures: Part I, Journal of Sound and Vibration 311 (2008) 1228–1248. [5] T.P. Sapsis, A.F. Vakakis, O.V. Gendelman, L.A. Bergman, G. Kerschen, D.D. Quinn, Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures: Part II, analytical study, Journal of Sound and Vibration 325 (2009) 297–320. [6] A.F. Vakakis, Inducing passive nonlinear energy sinks in vibrating systems, ASME Journal of Vibrations and Acoustics 123 (2001) 324–332. [7] Y. Starosvetsky, O.V. Gendelman, Strongly modulated response in forced 2DOF oscillatory system with essential mass and potential asymmetry, Physica D 237 (2008) 1719–1733. [8] Y. Starosvetsky, O.V. Gendelman, Vibration absorption in systems comprising nonlinear energy sink: nonlinear damping, Journal of Sound and Vibration 324 (2009) 916–939. [9] F. Nucera, A.F. Vakakis, D.M. McFarland, L.A. Bergman, G. Kerschen, Targeted energy transfers in vibro-impact oscillators for seismic mitigation, Nonlinear Dynamics 50 (2007) 651–677.
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