International Journal of Non-Linear Mechanics 79 (2016) 26–37

Contents lists available at ScienceDirect

International Journal of Non-Linear Mechanics

journal homepage: www.elsevier.com/locate/nlm

Reconstructing the transient, dissipative dynamics of a bistable Duffing oscillator with an enhanced averaging method and Jacobian elliptic functions

Chunlin Zhang a,b, R.L. Harne c,n, Bing Li a, K.W. Wang b a State Key Laboratory for Manufacturing and Systems Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, PR China b Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA c Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH 43210, USA article info abstract

Article history: Systems characterized by the governing equation of the bistable, double-well Duffing oscillator are ever- Received 27 March 2015 present throughout the fields of science and engineering. While the prediction of the transient dynamics Received in revised form of these strongly nonlinear oscillators has been a particular research interest, the sufficiently accurate 4 October 2015 reconstruction of the dissipative behaviors continues to be an unrealized goal. In this study, an enhanced Accepted 4 November 2015 averaging method using Jacobian elliptic functions is presented to faithfully predict the transient, dis- Available online 14 November 2015 sipative dynamics of a bistable Duffing oscillator. The analytical approach is uniquely applied to recon- Keywords: struct the intrawell and interwell dynamic regimes. By relaxing the requirement for small variation of the fi Bistable Duf ng oscillator transient, averaged parameters in the proposed solution formulation, the resulting analytical predictions Transient dynamics are in excellent agreement with exact trajectories of displacement and velocity determined via numerical Averaging method integration of the governing equation. A wide range of system parameters and initial conditions are Jacobian elliptic functions utilized to assess the accuracy and computational efficiency of the analytical method, and the consistent agreement between numerical and analytical results verifies the robustness of the proposed method. Although the analytical formulations are distinct for the two dynamic regimes, it is found that directly splicing the inter- and intrawell predictions facilitates good agreement with the exact dynamics of the full reconstructed, transient trajectory. & 2015 Elsevier Ltd. All rights reserved.

1. Introduction Historically, the transient dynamics of bistable oscillators have been of great interest, for example as relate to excitations that are The governing equation of the bistable, double-well Duffing impulsive in nature such as blast loading [9], plastic impact [10],and oscillator has been widely used to evaluate a large variety of non- thermal shocks [11], which could induce a critical well escape event linear systems including slender aerostructures that may buckle or multiple of the "snap-through" behaviors. Based upon the wide under loads [1], microelectromechanical switches [2],vibration- applicability of the governing equation of the bistable Duffing based energy harvesters [3,4], electrical circuits [5], and optical oscillator, there is great merit to develop an accurate prediction systems [6]. Recent developments in smart/adaptive structures have strategy for the transient, dissipative dynamics of bistable systems. particularly focused on bistable/buckled platforms for performance Using structural terminologies in the study hereafter (e.g., and functionality enhancements [7], while commercial attention is displacement, mass, etc.), the governing equation of the double- fi even lately recognizing the potential for bistability to provide well Duf ng oscillator and initial conditions examined in this favorable new potentials [8]. The characteristic and strongly non- research are expressed using € þ _ þ 3 ¼ ; ðÞj ¼ ; _ðÞj ¼ _ ð Þ linear dynamics of well escape, stochastic resonance, and chaos are mx dx k1x k3x 0 xt t ¼ 0 x0 xt t ¼ 0 x0 1 uniquely captured by the governing equation of the archetype bis- where m is mass, d is damping constant, k ðÞi ¼ 1; 3 are stiffness table Duffing oscillator. Together, these features illustrate the broad i constants. Note that while the above expressions and the proposed applicability of the oscillator model in terms of disciplinary rele- _ approach are applicable for all sets of initial conditions ðÞx0; x0 , for vance as well as for rich, dynamical studies. the purpose of illustration it is assumed that the oscillator is ori- ginally at rest in a statically stable equilibrium with finite initial n _ðÞj ¼ _ Corresponding author. velocity xt t ¼ 0 x0 throughout this paper. This can be viewed as E-mail address: [email protected] (R.L. Harne). an example with the system at static equilibrium for t o0, and http://dx.doi.org/10.1016/j.ijnonlinmec.2015.11.002 0020-7462/& 2015 Elsevier Ltd. All rights reserved. C. Zhang et al. / International Journal of Non-Linear Mechanics 79 (2016) 26–37 27

Fig. 1. (a) Schematic of the bistable, double-well Duffing oscillator illustrating the displacement coordinate xtðÞconvention. (b) Initial velocity imposed on the bistable oscillator originally at rest: interwell vibrations transition into intrawell oscillators as time increases. under impact loading at t ¼ 0 such that the velocity immediately Lakrad and Belhaq [25] adopted the multiple scales method to changes according to the impulse-momentum theorem. Fig. 1 approximate the undamped oscillations. Yuste and Bejarano [14] (a) shows a schematic model of such a bistable oscillator and studied the amplitude decay of a bistable oscillator and reported illustrates the double-well potential energy profile according to satisfactory agreement with directly numerically integrated results the coordinate convention that xtðÞ¼0 is the unstable equilibrium for small degrees of nonlinearity. Yuste [26] extended the har- when k1 40 and k3 40. In the case where k1 o0 and k3 a0, Eq. (1) monic balance method to enable the approximation of several represents the monostable Duffing oscillator which undergoes cycles of oscillation when the nonlinearity was non-trivial. Cveti- motions within the individual well of potential energy. In contrast canin [27] approximated the transient behaviors using complex to such behaviors, the bistable oscillator may exhibit two quali- generating functions and demonstrated two examples that tatively distinct dynamics: intrawell oscillations confined to one showed good agreement with numerical results. These advance- local well of potential energy and interwell motions which cross ments have provided new pathways towards the prediction of the the unstable equilibrium. Fig. 1(b) illustrates these distinct transient behaviors of bistable Duffing oscillators. On the other dynamic regimes for a bistable oscillator excited with finite initial hand, by their implementation of the Jacobian elliptic functions velocity from a starting position of a stable equilibrium. The dark having fixed modulus/phase, the approximations are suitable for trajectory in three dimensions is the phase plane of displacement only a few cycles of dissipative oscillation and may be severely and velocity in time, whereas the projections in displacement– limited in parameter selection for favorable results. time or velocity–time show the unique contributions in those The objective of this research is to develop an enhanced ana- planes. The (light solid curve) mapping on the far plane of dis- lytical prediction strategy for the transient dynamics of bistable placement–velocity indicates the homoclinic orbits which differ- oscillators. The aim is to faithfully reconstruct the motions over a entiate intra- from interwell behaviors according to the instanta- large number of free decay vibration cycles including both intra- neous level of system energy for the undamped oscillator. Due to and interwell behaviors, and with high-fidelity for a significant the amplitude of the imposed initial velocity, interwell dynamics range of system parameters. A promising idea to meet this goal is are first activated which lead to approximately 8 crossings to leverage the averaging method with the Jacobian elliptic func- between the local wells of potential energy. Then, a period occurs tions. Indeed, Coppola and Rand [28,29] employed these tools to such that the dynamics transition from inter- to intrawell on a approximate the transient dynamics of several monostable, non- time scale sufficiently slower than the natural period of the free linear oscillators, which was the strategy likewise followed by oscillations [12]. Finally, the oscillator mass vibrates with small, Belhaq and Lakrad [30]. However, the classical averaging method decaying amplitudes within one of the local wells of potential is based upon a near-identity transformation [15] which assumes energy. The accurate analytical prediction of the system response that the time rate of change of the averaged amplitude of oscil- and characterisitics in these two dynamic regimes is the objective lation is sufficiently small [28]. As a result, the applicability of the of this research. classical method is limited to generating accurate predictions for Numerous methods have been proposed to analytically predict the case of small oscillations near equilibria. Yet, when considering the transient dynamics of the monostable and bistable Duffing bistable Duffing oscillators, the interwell vibrations are char- oscillator equations. Individual trigonometric functions, specifi- acteristically "far from equilibrium" and therefore undergo large cally sine and cosine, are sometimes assumed to be the analytical amplitude decay from initial conditions, prior to transitioning to solutions [13]. Yet, in practice, such selection leads to poor accu- intrawell behaviors which eventually become near-linear motions. racy when the nonlinearity is considerable [14], k3=k1 ZOðÞ1 , due It is clear that the classical averaging method is unsuitable to to the diffusion of energy to many harmonics away from the predict such a significant variation in dynamics. fundamental [15]. For enhanced accuracy, Barkham and Soudack To advance the state of the art and overcome these limitations [16] introduced the Jacobian elliptic functions as generating solu- for an accurate and long-time prediction of the transient dynamics tions for the Duffing equation. Since then many approximate of bistable oscillators, this research develops an enhanced aver- analytical solution strategies have been proposed based on the aging method with the Jacobian elliptic functions building upon properties of Jacobian elliptic functions. A large variety of such the method established by Coppola and Rand [28]. Yet, instead of studies have sought to predict the transient dynamics of the applying near-identity transformations that limit applicability to monostable Duffing oscillator. Notable contributions have oscillations near equilibria, the approach developed here enables employed these functions in the context of approximate analytical slow variation in oscillation amplitude and modulus/phase so that solution strategies including the method of multiple scales [17], the full dynamic range of motions of the bistable system may be the Krylov–Bogoliubov method [18,19], the "C–J" method [20], the accurately reconstructed. The following sections detail this new averaging method [21], the method of harmonic balance [22], and strategy as individually applied to the intrawell and interwell the homotopy analysis method [23,24]. oscillations regimes. Because previous endeavors have only suc- The Jacobian elliptic functions have also been utilized towards ceeded in reconstructing a few of the transient oscillation cycles of the prediction of the dynamics of bistable Duffing oscillators. bistable oscillators or the overall amplitude decay [31], the most 28 C. Zhang et al. / International Journal of Non-Linear Mechanics 79 (2016) 26–37 constructive and comprehensive assessments of the developments Using the identities of the Jacobian elliptic functions [33], the in this research are obtained by comparison of the current analy- initial amplitude and phase are found to be qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ÀÁÀÁ tical predictions to results of numerically integrating the govern- ÀÁ 1=2 = 2 sn φ ; k cn φ ; k ing equations, considered to be the exact dynamics. Such a rigor- D ¼ β 1 2 U αþ αβx2 þ2βx_2 ; d0 ÀÁd d0 d 0 0 0 dn φ ; k ous assessment is then presented, exploring the accuracy of the d0 d proposed approach using a large range of system and excitation x_ ¼ 0 : ð10Þ parameters. Prior to concluding remarks, a discussion is provided ω 2 x0 dkd to summarize and examine the applicability, limitations, and fi potential of the proposed analytical approach. When the initial energy of the bistable oscillator is suf cient to ðÞj 4 induce interwell vibration E t ¼ 0 0 , the exact vibration response can be found by assuming the solution of Eq. (4) in the following form [15] 2. Analytical prediction formulations for the undamped sys- ¼ ðω þφ ; Þ¼ ð ; Þ ð Þ tem: intra- and interwell oscillations xc C0cn ct c0 kc C0cn uc kc C0cn 11

where uc and kc are the argument and modulus of the Jacobian The schematic model illustrated in Fig. 1(a) is analyzed here. elliptic function cnðÞ uc; kc , respectively. The period of cnðÞ uc; kc is From Eq. (1), by setting 4KkðÞc , where KkðÞc is the complete elliptic integral of the first kind γ ¼ d=m; α ¼ k1=m; β ¼ k3=m; ð2Þ with respect to the modulus of kc and is hereafter expressed as Kc for brevity. When damping is absent, ωc and kc are constants, and the governing equation is simplified as φ c0 is the initial argument which is determined by initial condi- € _ 3 _ xþγxαxþβx ¼ 0; xj t ¼ 0 ¼ x0; xj t ¼ 0: ð3Þ tions. Likewise, by taking the derivatives of Eq. (11) and sub- stituting appropriate relations into (4), one finds that Eq. (3) is used throughout the remainder of this research. Note ÀÁ hi fi ω2 ¼ β 2 α; 2 ¼ β 2= ω2 ¼ β 2= ðβ 2 αÞ : ð Þ thatno the pxedffiffiffiffiffiffiffiffiffi points (equilibria) of the system occur at c C0 kc C0 2 c C0 2 C0 12 x ¼ 0; 7 α=β , where the unstable equilibrium is x ¼ 0 and Thus, the unperturbed interwell vibration of Eq. (4) is expres- the latter two, non-zero values are stable equilibria. Additionally, sed by note that the total energy of the system is 0 !1 _2 2 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 EtðÞ¼ 2x 2αx þβx =4, so that the homoclinic orbits (here at βC2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ¼ C cn@ βC2 αUt þφ ; 0 A: ð13Þ ¼ fi _ ¼ 7 α 2 β 4= c 0 0 c0 ðβ 2 αÞ E 0) are de ned by x x x 2. Therefore, intrawell 2 C0 behaviors satisfy Eo0 whereas interwell oscillations occur when E40. From Eq. (13), the argument uc and modulus kc are functions of ÀÁ φ In the absence of damping γ ¼ 0 the amplitude C0. Here, the initial amplitude C0 and argument c0 depend on initial conditions: 3 x€ αxþβx ¼ 0; xj ¼ ¼ x ; x_j ¼ ¼ x_ : ð4Þ t 0 0 t 0 0 ¼ ðφ ; Þ; _ ¼ω ðφ ; ÞU ðφ ; ÞðÞ x0 C0cn c0 kc x0 cC0sn c0 kc dn c0 kc 14 Researchers have shown that the equation of motion for the Using the elliptic function identities [33], the initial amplitude undamped, bistable Duffing oscillator is exactly solved using and phase are found Jacobian elliptic functions [15,32]. In the following, x represents d qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ÀÁ 1=2 the intrawell vibration and xc denotes the interwell vibration. = 2 snðφ ; k Þ C ¼ β 1 2 U αþ αβx2 þ2βx_2 ; c0 c Udnðφ ; k Þ For intrawell vibration, the response is obtained by assuming 0 0 0 ðφ ; Þ c0 c cn c0 kc the solution of Eq. (4) in the form [32] _ ÀÁ ¼ x0 : ð Þ ¼ ζ ω þφ ; ¼ ζ ðÞ; ≡ζ ð Þ 15 xd D0dn dt d0 kd D0dn ud kd D0dn 5 ωcx0 where ζ ¼ 71isdefined in Eq. (8); ud and kd are the argument and modulus of Jacobian elliptic function dnðÞ ud; kd , respectively ðÞ; ðÞ ðÞ [33]. The period of dn ud kd is 2Kkd , where Kkd is the complete 2.1. Constraints on the parameters fi elliptic integral of the rst kind and is hereafter expressed as Kd for ω ðÞ; ðÞ; ðÞ; brevity. In the absence of damping, d and kd are constants, and For the Jacobian elliptic functions sn u k , cn u k and dn u k , φ r 2 r d0 is the initial argument and is determined by initial conditions. the modulus is intrinsically constrained to satisfy 0 k 1. By By taking the derivatives of Eq. (5) and substituting appropriate applying this constraint to the analytical solution of intrawell relations into (4), one finds that vibration, one finds   ω2 ¼ β 2= ; 2 ¼ α= β 2 : ð Þ 0rk2 r1 3 1=2r α= βD2 r1: ð16Þ d D0 2 kd 2 2 D0 6 d 0

Thus, the undamped, intrawell oscillations are described using Considering the instantaneous displacement and velocity of ¼ ζ _ ¼ζ ω 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi hi intrawell vibration are xd D0dn and xd D0 dk sncn, 1=2 d ¼ ζ β 2= U þφ ; α= β 2 : ð Þ xd D0dn D0 2 t d0 2 2 D0 7 respectively, the instantaneous system energy then can be derived as ζ  In Eq. (7), the sign indicator , initial amplitude D0, and initial ¼ _2= α 2= þβ 4= ¼ 2 β 2 α = : ð Þ φ ζ Ed xd 2 xd 2 xd 4 D0 D0 2 4 17 argument d0 are all determined by the initial conditions. indi- cates the intrawell position and is expressed as According to Eqs. (16) and (17), when the instantaneous energy ( sgn½ x ; if x a0 is negative, the bistable oscillator undergoes intrawell oscillations ζ ¼ 0 0 : ð Þ ðÞ; _ 8 which are expressible using dn ud kd . On the other hand, for sgn½x0 ; if x0 ¼ 0 interwell vibration, application of the above modulus constraint The initial displacement and velocity when t ¼ 0 satisfy leads to  ÀÁ ÀÁÀÁ ¼ ζ φ ; ; _ ¼ζ ω 2 φ ; φ ; : ð Þ 0rk2 r1 3 0rα= βC2 r1=2: ð18Þ x0 D0dn d0 kd x0 D0 dkdsn d0 kd cn d0 kd 9 d 0 C. Zhang et al. / International Journal of Non-Linear Mechanics 79 (2016) 26–37 29

Considering that the displacement and velocity are expressed integrable so that here the averaging method is utilized to obtain ¼ _ ¼ ω ψ using xc C0cn and xc C0 csndn, the instantaneous energy is analytical approximations of the amplitude D and phase d.  ¼ _2= α 2= þβ 4= ¼ 2 β 2 α = : ð Þ Ec xc 2 xc 2 xc 4 C0 C0 2 4 19 3.1. Approximation of the decaying amplitude D for intrawell vibration Similarly, when the energy is positive, the bistable oscillator undergoes interwell vibration such that the undamped dynamics The oscillation amplitude smoothly decreases as time proceeds are described by cnðu; kÞ. Therefore, βA2 ¼ 2α is the critical con- and is assumed to be nearly constant over one period of the dition demarcating the intra- from the interwell oscillation vibration (i.e., a period of 2K with respect to argument u ). Thus, regimes, in which A is the amplitude corresponding to D and C , d d 0 0 parameters k2, ω , and K which are functions of the amplitude D respectively. d d d are also considered to remain nearly constant over one period. Consequently, instead of being assumed sufficiently small as in the _ classical averaging method, by Eq. (26) the variables D and ψ_ are 3. Approximate prediction of the damped, intrawell vibration d assumed to slowly vary by a finite degree due to the relation to the Jacobian elliptic functions. Therefore, the averaged variables over The intrawell vibration of the damped bistable oscillator, γ 40, one period 2Kd are smooth, local means of the instantaneous is assumed to satisfy the form in Eq. (5) while the amplitude D ¼ variables and are thus termed "averaged instantaneous variables". DtðÞand argument u ¼ 2K ψ ðÞt both slowly vary in time. The d d d The averaged instantaneous amplitude is derived from Eq. (26a) to new variable 2K ψ is used as the argument because it leads to d d yield periodically variational equations that can be averaged. The other  γ 2 γ 2 Z variables are amplitude dependent and satisfy Eq. (6) from the _ Dkd 1 D 2 kd 2Kd ¼ U 1 2 undamped relations. D 2 2 dn dud k k 2Kd 0 ÀÁ d Z d x ðÞ¼t ζDtðÞdn 2K ψ ; k ¼ ζDtðÞdnðÞ u ; k ζDdn ð20aÞ γ 2Kd d d d d d d þ DU 1 4 2 dn dud k 2Kd 0 _ ¼ ζ ω 0 ð Þ d   xd D ddn 20b 2 3 2 2 2 2 k 1 2 2 k 1 d 2 k Ed _ d ¼γD4 þ d U 5 ) D ¼γD4 ¼ ðÞ; 2 ¼ α=β 2; ω2 ¼ β 2= : ð Þ 2 2 2 Kd Kkd kd 2 2 D d D 2 20c 3k 3k Kd 3k #d d d fi Differentiating Eq. (20a), one obtains the rst-order derivative 2k2 E þ d U d ð27Þ of xd as 2 K  3kd d 0 ∂ ψ _ ¼ ζdD þ ψ 0 0 0 þ dn þζd d⋅ 0 ð Þ xd dn 2D K k dn Dk 2DKddn 21 where K and E denote the complete elliptic integrals of the first dt d d d d ∂k dt d d d and second kinds, respectively; and D is very close to D and so that 0 where ðÞU denotes differentiation of the variable with respect to D is replaced by D since the latter is nearly constant over one its argument, thus period of oscillation. Eq. (27) is non-integrable and the right side is an implicit function of D.Tofind an explicit and integrable 0 dKd 0 dkd 0 dωd 0 ∂dnðÞ ud; kd K ; k ; ω ; dn : ð22Þ approximation of Eq. (27), a polynomial fit is employed. Eq. (27) is d dk d dD d dD ∂u d d rewritten as Comparing Eqs. (20b) and (21) one finds  2   2 k 1 2 _ 2 2 d 2 kd Ed dD 0 0 0 0 ∂dn dψ 0 0 ¼γ U ð Þ; ¼ þ U : ð Þ þ ψ þ þ d U ¼ ω : ð Þ D D f kd fkd 2 2 28 dn 2D dKdkddn Dkd 2DKddn D ddn 23 3k 3k Kd dt ∂kd dt d d Considering the values of elliptic integrals E and K are com- From Eq. (20b), the second-order derivative of xd can be derived  d d 2 2 as pletely determined by kd, thus fkd is a pure function depending   ÀÁ 0 on k2. Recalling Eqs. (6) and (16), the mapping correspondence α dD 0 0 0 0 0 ∂dn d  x€ ¼ ζ ω þDω dn þ2Dψ ω K k dn″þDω k 2 2 2 d dt d d d d d d d d ∂k =βD -k is a bijection. Thus fk could also be expressed as a  d d d dψ function of α=βD2 by changing the independent variable from k2 þ d U2DK ω dn″ : ð24Þ  d dt d d α=β 2 2 2 to D . The values of kd and fkd are calculated sequentially on 2 Substituting Eqs. (20) and (24) into (3), one obtains the range of 1=2rα=βD r1 by using Eqs. (6) and (28), respec-  α=β 2- 2 0 tively. Accordingly, the mapping correspondence of D fkd dD ÀÁ 0 0 0 ∂dn ω þDω0 dn0 þ2Dψ ω K k dn″þDω k d d d d d d d d ∂ is numerically established to enable the analytical integration of dt kd Eq. (27). dψ  þ d U ω ″þγ ω 0 α þβ 3 3 ¼ : ð Þ 2 2DKd ddn D ddn Ddn D dn 0 25 It is found that fk is smooth on a large range of 0:55rα=β dt d D2 r1 and is sufficiently approximated via an individual poly- Combining Eqs. (23) and (25), dD=dt and dψ =dt are solved to d nomial fit using yield !λ γ γ hi  α dD D 02 D 2 2 2 4 2 ¼ μ⋅ þυ: ð Þ ¼ dn ¼ k 1þ 2k dn dn ð26aÞ fk p1 29 dt 2 2 d d d βD2 kd kd 8 9 where μ, υ, and λ are undetermined coefficients. A beneficial 0 < 2 ÂÃÀÁ= dψ ω γdn 2k 0 advantage of Eq. (29) is that it provides an integrable approx- d ¼ d þ dn d ZuðÞ; k dn þdn 1dn2 2: 2 2 d d ; dt 2Kd imation of Eq. (27). For different values of λ, μ and υ are calculated 2Kdkd kd 1 kd using a linear fit. The performance of various fitting parameter sets ð26bÞ 2 is estimated using R ¼ 1SSE=SST, where SSE is the sum of ðÞ; 2 where Zud kd is the Jacobian Zeta function. Eq. (26) is non- squared residuals and SST is the sum of squared fkd , so that 30 C. Zhang et al. / International Journal of Non-Linear Mechanics 79 (2016) 26–37

2 higher values of R indicate improvements in the fitting to the vibration of the damped system in Eq. (3) is assumed to satisfy the 2 fi fi 2 ¼ : solution form in Eqs. (11) and (12) while parameters C ¼ CtðÞand exact fkd . By this procedure, a high- delity tofR 0 9987 is ψ ¼ ψ ðÞt are assumed to slowly vary in time. The variable 4K ψ realized across 0:55rα=βD2 r1 using λ ¼ 2, μ ¼0:15011, and c c c c υ ¼ : is used as the argument. Then, 0 14714. By substituting Eq. (29) into (28) and integrating the ÀÁ ¼ ðÞ ðÞ¼; ðÞ ψ ; ð Þ result, the decaying amplitude of intrawell oscillation is found xc Ctcn uc kc Ctcn 4Kc c kc Ccn 35a hi μ μ 1=4 ¼ α2 þβ2 4 4υγt α2 ⋅β 1=2 ð Þ _ ¼ ω 0 ð Þ D υ D0 e υ 30 xc C ccn 35b ÀÁ 2 2 2 where D0 is the initial amplitude determined by initial conditions, ¼ ðÞ; ¼ β = ω2 ; ω2 ¼ β α: ð Þ Kc Kkc kc C 2 c c C 35c expressed in Eq. (10). Thus, the velocity amplitude of the oscillation is _ The velocity xc is also obtained by differentiating Eq. (35a) given by Eq. (20b) according to the displacement amplitude D.  ∂ ψ _ ¼ dC þ ψ 0 0 0 þ 0 cn þd c U 0 ð Þ xc cn 4C cKckccn Ckc 4CKccn 36 3.2. Time trajectories of intrawell displacement dt ∂kc dt and The instantaneous displacement is expressed in Eq. (20a).To ¼ 0 = ; 0 = ; ω0 ω = ; 0 ∂ ðÞ; =∂ : reconstruct the trajectory in time, the unknown argument ud 2 Kc dKc dkc kc dkc dC c d c dC cn cn uc kc uc ψ ðÞ; Kd d of the Jacobian elliptic function dn ud kd must be approxi- ð37Þ mated. The first order derivative of 2K ψ is d d Comparing Eqs. (35b) and (36) leads to ÀÁ ψ 0 0  _ d dKd dkd dD d d _ u ¼ 2K ψ ¼ 2ψ þ2K ¼ 2ψ K k Dþ2K ψ_ : dC 0 0 0 ∂cn dψ d dt d d d dk dD dt d dt d d d d d cnþ4Cψ K k cn0 þCk þ c U4CK cn0 ¼ Cω cn0: ð38Þ d dt c c c c ∂k dt c c ð31Þ c 0 0 From Eq. (35b), the second order derivative of xc is derived as Then, K and k are expressed by  d d ÀÁ 0 ! dC 0 0 0 0 0 ∂cn 2 x€ ¼ ω þCω cn þ4Cψ ω K k cn″þCω k 0 dK 1 E 0 dk 1 2α 2k 1 c c c c c c c c c d ¼ d ; d ¼ ¼ d U : ð Þ dt ∂kc Kd 2 Kd kd 3 32 dkd kd l dD kd βD kd D ψ d d c þ U4CKcωccn″: ð39Þ Substituting Eqs. (27) and (32) into (31) and averaging the dt result, one obtains Substituting Eqs. (35) and (39) into Eq. (3) one obtains  0  1 0 !2 ÀÁ ∂ 2 2 dC 0 0 0 0 0 cn 2 kd 1 ω þCω cn þ4Cψ ω K k cn″þCω k _ 2 kd Ed @ 2 kd Ed A c c c c c c c c u ¼2γψ K þ þω : ð33Þ dt ∂kc d 2 2 d 2 2 K d kd ld 3kd 3kd d ψ d c 0 3 3 þ U4CKcωccn″þγCωccn αCcnþβC cn ¼ 0: ð40Þ ψ dt Note that d and ud depend on the intrawell oscillation = ψ = amplitude D and thus they slowly vary during one period 2Kd. Combining Eqs. (38) and (40), dC dt and d c dt are found ψ Accordingly, d and ud are approximated by the averaged = ¼γ ðÞ0 2 ð Þ ψ dC dt Ccn 41a instantaneous parameters d and ud, and from Eq. (5) the initial ψ φ = φ  h  values of and ud are 2Kd and , respectively. By aver- ÀÁ d d0 d0 _ ψ = ¼ ω þγ 0 2 ðÞ; 0 þ 2 2 = ½; ψ ¼ ω = d c dt c cn cn 1 2k Zuc kc cn k cn 1 cn aging over one period 0 2Kd , it is found that d d 2Kd. Thus, c c ψ i d and ud can be approximated as Z Z Z 1k2 =4K ð41bÞ t φ t φ t ω c c ψ ðÞ¼ψ ðÞþ ψ_ ffi d0 þ ψ_ ¼ d0 þ d ð Þ d t d 0 ddt ddt dt 34a 0 2Kd 0 2Kd 0 2Kd where ZuðÞ; k is the Jacobian Zeta function and Zc is the notation ( ! Z Z used here to indicate ZuðÞc; kc . Like before, Eq. (41) is not able to be t t 2k2 E ðÞ¼ ðÞþ _ ffiφ þ γψ d d integrated analytically so a similar strategy of averaging and ud t ud 0 uddt d0 2 d 2 2 Kd 0 0 k l fi 0  1 9 d d polynomial tting is employed to develop approximate predic- 2 2 = tions for the averaged instantaneous interwell vibration amplitude 2 kd 1 2k E @ þ d d Aþω dt ð34bÞ and argument. 2 2 K d; 3kd 3kd d 4.1. Approximation of the decaying amplitude C for interwell in which φ is the initial argument determined by initial condi- d0 vibration tions in Eq. (10) which is readily calculated using the Jacobian inverse elliptic functions. Though the integration of Eq. (34) is not The averaged instantaneous amplitude of oscillation is possible analytically, the integrand contains only functions of the obtained using the averaging method over one period½ 0; 4K . amplitude and thus the integration may be numerically deter- c Z "Z ψ 4Kc γ 4Kc mined. As a result, in this study d and ud are calculated using _ 1 02 C 2 C ¼γC cn ðÞuc; kc duc ¼ sn duc trapezoidal integration based upon the analytically-derived 4Kc 0 4Kc 0 amplitude following the approach developed in Section 3.1. Z # 4Kc 2 4 kc sn duc 0 "#"# 4. Approximate prediction of the damped, interwell vibration 2 2 2 2 ¼γ 1 kc þ2kc 1 Ec ) _ ¼γ 1 kc þ2kc 1 Ec C 2 2 C C 2 2 3k 3k Kc 3k 3k Kc The bistable oscillator undergoes interwell vibration when the c c c c ð42Þ transient system energy is positive, EtðÞ40. Using a similar strategy as that employed for the intrawell vibrations in Section 3, where Ec ¼ EkðÞc and Kc ¼ KkðÞc are the complete elliptic integral of the analytical prediction of the transient, dissipative interwell the second and first kinds, respectively; and the averaged instan- dynamics is developed for the bistable oscillator. The interwell taneous amplitude C is smooth local mean of the vibration C. Zhang et al. / International Journal of Non-Linear Mechanics 79 (2016) 26–37 31 amplitude C. Considering that C is very close to C, C is therefore Substituting Eq. (48) into (47) and averaging the result, one replaced by C in Eq. (42). The right side of Eq. (42) is an implicit obtains fi ! 2 ! 3 function of the amplitude C. To enable a polynomial tting pro-  2 _ Ec 2 α cedure for the right-hand side of Eq. (42), the equation may be u ¼ 4γψ K 2k 1 4a þb5þω : ð49Þ c c 2 c c β 2 c written lc C   2 2 Considering that ψ and u are functions of the interwell _ ¼γ U 2 ; 2 ¼ 1 kc þ2kc 1 Ec : ð Þ c c C C hkc hkc 2 2 43 3k 3k Kc vibration amplitude and thus also slowly vary during one period c c ψ 4Kc, c and uc can be approximated by the averaged instantaneous ψ ψ φ = Note that the complete elliptic integrals Ec and Kc are deter- parameters c and uc, having initial values c and uc are c0 4Kc 2 2 2 φ mined by kc , and hkc is a function that depends on argument kc . and c0, respectively, from Eq. (11). By averaging over one period ½; ψ_ ¼ ω = ψ Recalling Eqs. (18) and (35c), it is evident that the mapping cor- 0 4Kc , it is found that c c 4Kc. Thus c and uc are respondence α=βC2-k2 is a bijection. Thus, one may change the approximated by c  Z Z Z 2 α=β 2 2 t φ t φ t ω argument from kc to C , and hkc could be expressed as a ψ ðÞ¼ψ ðÞþ ψ_ ffi c0 þ ψ_ ¼ c0 þ c ð Þ c t c 0 cdt cdt dt 50a 2 2 4K 4K 4K function of α=βC . On the range of 0rα=βC r0:5, the values of 0 c 0 c 0 c ( ! k2 and hk2 are calculated sequentially by Eqs. (35c) and (43), Z Z  c c t t E u ðÞ¼t u ðÞþ0 u_ dt ffiφ þ 4γψ c K 2k2 1 respectively. Accordingly, the mapping correspondence α=βC2-h c c c c0 c 2 c c  0 0 l 2 ! 3 9 c 2 2 = kc is numerically established to enable the analytical integration α Â4 þ 5þω ð Þ of Eq. (42). a b c dt 50b  βC2 ; 2 fi It is found that hkc is suf ciently approximated by an indi- 2 φ vidual polynomial fit across the broad range of 0:05rα=βC r in which, c0 is the initial argument determined by the amplitude 0:45 using and velocity and is calculated via the Jacobian inverse elliptic ! functions. Though the integration of Eq. (50) is not possible ana-  n α lytically, the integrand contains only functions of the amplitude 2 ¼ U þ ð Þ hkc q1 a b 44 βC2 and may be determined numerically. Thus, as for intrawell oscil- lations, the interwell argument uc is determined numerically by where a, b, and n are undetermined coefficients. Using linear trapezoidal integration from the analytically-derived amplitude polynomial fitting,a and b are determined for a range of selections CtðÞ, after which the instantaneous displacement xcðÞt is found by of n where the best fitting performance is obtained at R2 ¼ 0:9962 Eq. (35). when n ¼ 2, a ¼0:78592, and b ¼ 0:32051. By substituting Eq. (44) into (42) and integrating the result, the amplitude of interwell oscillation is found to be 5. Accuracy of the analytical predictions

hi 1=4 ðÞ¼ β2 4 þaα2 4bγt aα2 ⋅β 1=2 ð Þ This section assesses the accuracy of the proposed analytical Ct C0 e 45 b b solution strategy towards predicting the transient, dissipative where C0 is the initial amplitude determined by initial conditions, dynamics of the bistable oscillator. The prediction capabilities for Eq. (15). By Eq. (35b), the velocity amplitude is calculated from the intra- and interwell dynamics are individually evaluated. For decaying amplitude of displacement. Considering the criteria consistency throughout, the value of linear stiffness is α ¼p1ffiffiffiffiffiffiffiffiffi and 2 ¼þ α=β which demarcates intra- from interwell oscillations, βA ¼ 2α, the the initial displacement is the stable equilibrium x0 . time at which the analytical prediction estimates the interwell Thus, to initiate the dynamics away from equilibrium, a variety of oscillations to cease is given by initial velocities are imposed, which, by the constraints described  in Section 2.1, accordingly induce either intra- or interwell oscil- ÀÁ 1 ¼ þ = α2 U β2 4 þα2 = = γ: ð Þ lations. The exact transient dynamics are presumed to be those tend ln 4 a b C0 a b 4b 46 solutions obtained by direct numerical integration of the govern- ing Eq. (3) with a fourth-order Runge–Kutta algorithm conducted in the software MATLAB where the relative integration tolerance is 4.2. Time trajectories of interwell displacement set to be three orders of magnitude more refined than the default tolerance (further refined tolerances did not affect results). In The instantaneous interwell displacement is expressed in Eq. addition to several examples providing qualitative comparisons ¼ ψ (35a). Prior to its determination, the argument uc 4Kc c of the between time/velocity trajectories as predicted by the analysis and ðÞ; fi Jacobian elliptic function cn uc kc must be estimated. The rst by numerical integration, a quantitative evaluation of the analy- order derivative of the argument is obtained as tical prediction accuracy is achieved by computing the correlat- fi ρ ðÞ; U½ðÞ; U ðÞ; 1=2 A½; ÀÁ ψ ion coef cient a;n cov xa xn cov xa xa cov xn xn 0 1 _ duc d dKc dkc dC d c 0 0 _ _ uc ¼ 4Kcψ ¼ 4ψ þ4Kc ¼ 4ψ K k C þ4Kcψ : between the analytical x and numerical x displacement results. dt dt c c dk dC dt dt c c c c a n c This measure has the merit that it simultaneously evaluates ð47Þ the accuracy of both amplitude and phase properties of the 0 0 Then, Kc and kc are expressed by trajectories.  ! 5.1. Transient intrawell oscillations 0 ≡dKc ¼ Kc Ec 2 ¼ 1 Ec ; Kc 2 lc 2 Kc dkc k l Kc kc l c c c For initial qualitative assessments of the analysis against the  0 dk βCα 1 k numerical computations, the dynamics of a bistable oscillator k ≡ c ¼⋅ ¼ 2k2 1 ⋅ c: ð48Þ c dC 2 2k c C having parameters γ ¼ 0:034427 and β ¼ 0:98237 are considered βC2 α c in the regime in which initial conditions lead to only intrawell 32 C. Zhang et al. / International Journal of Non-Linear Mechanics 79 (2016) 26–37

pffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi ¼þ α=β _ ¼ : Uα= β _ ¼ : Uα= β _ ¼ Fig. 2. Transientpffiffiffiffiffiffi intrawell dynamicspffiffiffiffiffiffi of the bistable oscillator for x0 . Displacement when (a) x0 0 92 2 and (b) x0 0 99 2 . Velocity when (c) x0 _ 0:92Uα= 2β and (d) x0 ¼ 0:99Uα= 2β.

: rα=β 2 r fi oscillations. To satisfy the constraint 0 5 D0 1 for intrawell limitations for one polynomial, a nite deviation exists between 2 vibration, the initialp velocitiesffiffiffiffiffiffi considered for the intrawell beha- the fiptffiffiffiffiffiffiffiffiffi at the extreme of the range where α=βD ¼ 1 such that D _ viors satisfy x0 rα= 2β. ¼þ α=β when t-1. In fact, according to the parameters used Fig. 2 illustrates two examples of (a,b) displacement and the for the fit, after a long time of oscillation, the analytically predicted corresponding (c,d) velocity trajectories for twop initialffiffiffiffiffiffi velocities. final displacement amplitude, Eq. (30), is found to be _ ¼ : Uα= β  rffiffiffiffi rffiffiffiffi Fig. 2(a) and (c) shows results for x0 0 92 pffiffiffiffiffiffi2 ; the results μ 1=4 α α _ ðÞ¼ U : : ð Þ plotted in Fig. 2(b) and (d) are for x0 ¼ 0:99Uα= 2β which indi- lim Dt 1 005 51 t-1 υ β β cates the initial velocity is only 1% less than that required to activate interwell oscillations. In the figures, the dark/solid curves In other words, the final displacement amplitude predicted by are the exact, numerically-integrated results, while the light/ the analytical method (according to the current fit parameters dashed curves are the analytical predictions. For the smaller initial λ; μ; υ) deviates from the exact value with a relative error near velocity, the analytical predictions shown in Fig. 2(a) and (c) are in 0:5%, such that the dissipative dynamics continue to oscillate even excellent agreement with the numerical findings. On the other after an infinite time elapses (albeit, oscillate with equal positive hand, for the larger initial velocity, the instantaneous displace- and negative values around the stable equilibrium). Fig.p 3ffiffiffiffiffiffiillus- _ ¼ : Uα= β ment and velocity phases exhibit greater deviation with respect to trates this interestingpffiffiffiffiffiffi result for the cases (a) x0 0 92 2 and _ the numerical, exact results, Fig. 2(b) and (d). This is due to the (b) x0 ¼ 0:99Uα= 2β, also verifying that the relative errors remain fi fi α=β 2  reduced delity of the polynomial t for the regimes D0 near small, 0.5%. Since the focus of this analytical approach is on the 0.5. Recalling Section 3.1, the fit was made across 0:55rα=βD2 r1 transient dynamics – thus primarily on those motions which occur with good success (very large R2), although intrawell oscillations while the system dissipates considerable energy – the accuracy 2 may also be obtained for α=βD as low as 0.5 (see Sectionpffiffiffiffiffiffi 2.3). For achieved for intrawell vibrations close to the initial time, as shown _ the larger initial velocity condition of x0 ¼ 0:99Uα= 2β, one finds in Fig. 2, is of greater interest than the small, persistent intrawell α=β 2 ¼ oscillations occurring at large times. that at the start of the transient oscillation prediction D0 0:5025 which is outside of the polynomialfitting regime. Thus, the For a quantitative evaluation of the analytical prediction accu- 2 fi racy for intrawell oscillations across a large range of system approximated form of the function fkd has reduced delity for parameters, the correlation coefficients between the analytical and the case having larger initial velocity, which is the source of the numerical results are shown in Fig. 4 for three different initial deviation between the numerical and analytical results shown in pffiffiffiffiffiffi pffiffiffiffiffiffi velocities x_ ¼ (a) 0:92α= 2β, (b) 0:955α= 2β, and (c) Fig. 2(b) and (d). On the other hand, the analytically-predicted pffiffiffiffiffiffi 0 0:99α= 2β. The damping factor γ is chosen from a logarithmically time-varying amplitudes of displacement and velocity are still in spaced range of 300 values spanning½ 0:007079; 0:1 , while the excellent agreement in spite of the instantaneous deviation of nonlinearity β is chosen from a linearly spaced range of 320 values phases with respect to the exact, numerical results. hi over the range α=βD2; 2α=βD2 to satisfy the constraint 0:5rα The dissipative intrawell oscillationspffiffiffiffiffiffiffiffiffi should ultimately settle to 0 0 j ¼ ¼þ α=β fi =β 2 r the stable equilibrium x t-1 x . Yet, due to the tting D0 1 according to the initial velocity considered, and thus the C. Zhang et al. / International Journal of Non-Linear Mechanics 79 (2016) 26–37 33

pffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi _ _ Fig. 3. Persistent intrawell oscillations at large times. x0 ¼þ α=β. Displacement when (a) x0 ¼ 0:92Uα= 2β and (b) x0 ¼ 0:99Uα= 2β.

ÀÁ fi 3 U ρ _ ¼ Fig. 4.pCorrelationffiffiffiffiffiffi coefpcientffiffiffiffiffiffi between thep numericalffiffiffiffiffiffi and analytical results of intrawell dynamics (using 10 log 10 a;n ) with different initial velocities: x0 (a) 0:92α= 2β; (b) 0:955α= 2β and (c) 0:99α= 2β. Lighter contour shadings indicate greater accuracy in the analytical prediction.

progressively increasing initial velocity, from Fig. 4(b) and (c), the Table 1 fidelity of the analytical result reduces from the exact, num- Computation efficiency comparison for the intrawell vibration regime. erically-integrated values, but primarily only for large nonlinearity Initial velocity Computation time [min] Analytical efficiency strengths and smaller damping factors, as evidenced by the darker _ [m/s], x0 enhancement [%], CT CT shading of the contours in those parameter regions. Yet, overall, Analytical, CTa Numerical, CTn 100U n a CTa for a substantial range of nonlinearity and damping parameters, pffiffiffiffiffiffi : Uα= β 8.6090 45.4986 428 the proposed analytical method yields exceptionally accurate 0 92 p2ffiffiffiffiffiffi : Uα= β 8.5924 45.6261 431 results with respect to the exact intrawell dynamics, even when 0 955 pffiffiffiffiffiffi2 0:99Uα= 2β 8.5717 57.4610 570 the initial velocity is only marginally smaller than that sufficient to activate interwell behaviors. Taking another perspective, Table 1 presents the findings of the initial displacement amplitude D via Eq. (10). The correlation 0 computation time required by the analytical and numerical coefficient is computed over the time during which the difference approaches to reconstruct the intrawell trajectories, such that the in intrawell vibration amplitude from cycle-to-cycle is ⩾0.1%. In results of Fig. 4 may be determined. The computations are carried effect, this takes the computation of ρ out to very long times a;n out on a computer with 4 GB of RAM memory and a 3.00 GHz Intel sufficient such that the instantaneous energy in the oscillators are Core 2 Duo processor. For these results, the evaluations for dif- negligible fractions of the initial input. fi ferent damping factors γ and nonlinearity β parameters are over a The results are plotted in Fig. 4, where the correlationÀÁ coef - 3 U ρ grid 100 by 120, respectively, across the range of values shown in cients are shaded in the contours using 10 log 10 a;n : dark shadings indicate large coefficient logarithms which indicates poor Fig. 4. Table 1 shows that the analytical approach enhances the fi correlation and thus greater error between the analytical and computational ef ciency of the large parametric evaluation of intrawell dynamic trajectories reconstruction, on average, by about exact numerical displacement trajectories.ÀÁ In other words, a 3 U ρ ¼ 476%. In other words, to reconstruct to the transient intrawell shading value corresponding to 10 log 10 a;n 0 indicates perfect agreement (the lightest shading level employed in the dynamics to a high degree of accuracy (see Fig. 4), the analytical figure). As seen in Fig. 4(a), when the initial velocity is only 8% less method developed herein provides a valuable computational _ ¼ : α= fi thanpffiffiffiffiffiffi that required to activate interwell oscillations (x0 0 92 ef ciency improvement for those designing and operating struc- 2β), the accuracy of the analytical prediction is very good for a tures which are bistable or buckled in such a way that the Duffing significant range of the nonlinearity and damping parameters. For oscillator model is applicable. 34 C. Zhang et al. / International Journal of Non-Linear Mechanics 79 (2016) 26–37

pffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi ¼þ α=β _ ¼ : Uα= β _ ¼ : Uα= β _ ¼ : Uα= β Fig. 5. Transient interwell dynamicspffiffiffiffiffiffi of the bistablepffiffiffiffiffiffi oscillator for x0 pffiffiffiffiffiffi . Displacement when (a) x0 2 5 2 , (b) x0 3 5 2 , and (b) x0 4 5 2 . _ _ _ Velocity when (d) x0 ¼ 2:5Uα= 2β, (e) x0 ¼ 3:5Uα= 2β, and (f) x0 ¼ 4:5Uα= 2β.

5.2. Transient interwell oscillations The examples in Fig. 5 demonstrate that the analytical predic- tions are in excellent qualitative agreement with the exact, Although intrawell dynamics may be predicted with high numerical dynamics. A small degree of phase error is observed to accuracy, the dissipative, interwell oscillations – which are far accumulate over the prediction times which is more apparent for from equilibrium and indeed cross between both of them! – are larger values of the initial velocity such as that in Fig. 5(c) and (f). historically challenging to analytically predict. To examine the An interesting feature is evident in Fig. 5(a) and (d) where the last accuracy of the proposed analytical method towards reconstruct- analytically-predicted values of displacement and velocity, ing the interwell dynamics, a first set of qualitative assessments is respectively, undergo a jump. Following closehi examination of such γ ¼ : γψ = 2 shown in Fig. 5 where the system parameters are 0 054317 cases, it is seen that the integrand term 4 c Ec lc Kc for the β ¼ : – and 0 70128. Displacement predictions are shown in Fig. 5(a) elliptic function argument uc in Eq. (50b) increases asymptotically (c) while the corresponding velocity trajectories are given in Fig. 5 at the final times of analytical prediction prior to EtðÞ¼0, where (d)–(f); the solid/dark curves indicate the numerical integration the rate of increase is dependent upon system parameters and the fi ndings whereas light/dashed curves are the analyticalpffiffiffiffiffiffi predic- initial velocity. Since the argument uc (i.e., instantaneous total _ tions. The initial velocities are chosen such that x0 Zα= 2β which phase) determines the output of the Jacobian elliptic function is derived from Eq. (15) to satisfy the constraint for interwell evaluation via Eq. (35), an argument which varies dramatically can 2 vibration: 0rα=βC r0:5. Fig. 5 pprovidesffiffiffiffiffiffi examplesp forffiffiffiffiffiffi which the induce a switch of sign between incremental evaluations of the _ : Uα= β : Uα= β initial velocitypffiffiffiffiffiffi x0 is (a,d) 2 5 2 , (b,e) 3 5 2 , and (c,f) elliptic function. The consequence is a potential for sudden 4:5Uα= 2β and plots out each pair of analytical and numerical deviation in value for the predicted displacement/velocity trajec- predictions to the time at which the instantaneous energy of the tories at the last sample of time prior to EtðÞ¼0. Although this respective result is EtðÞ¼0. feature is found to occur somewhat at random based upon the C. Zhang et al. / International Journal of Non-Linear Mechanics 79 (2016) 26–37 35

ÀÁ fi 3 U ρ _ ¼ Fig. 6.pCorrelationffiffiffiffiffiffi coefpffiffiffiffiffifficient between thepffiffiffiffiffiffi numerical and analytical results of interwell dynamics (using 10 log 10 a;n ) with different initial velocities: x0 (a) 2:5Uα= 2β; (b) 3:5Uα= 2β and (c) 4:5Uα= 2β. Lighter contour shadings indicate greater accuracy in the analytical prediction. system parameters and initial velocity selections, it ultimately only Table 2 influences the final displacement/velocity point predictions, and Computation efficiency comparison for the interwell vibration dynamic regime. plays a small role as relates to the accuracy computed via the Initial velocity Computation time [min] Analytical efficiency correlation coefficient. _ [m/s], x0 enhancement [%], CT CT A comprehensive assessment of the accuracy of the analytical Analytical, CTa Numerical, CTn U n a 100 CT method is given in Fig. 6 where a large range of nonlinearity and a fi pffiffiffiffiffiffi damping parametersÀÁ are employed and the correlation coef cient, 2:5Uα= 2β 8.2771 31.1330 276 3 pffiffiffiffiffiffi as 10 Ulog ρ , is plotted by the contour shading. To keep the 3:5Uα= 2β 8.7314 37.2208 340 10 a;n pffiffiffiffiffiffi computation consistent and meaningful, the correlation coefficient 4:5Uα= 2β 9.0337 41.8872 364 is taken between the numerical and analytical displacement data series from the initial time to the time that first results in EtðÞ¼0, whether this condition occurs numerically or by the analytical method also leads to exceptionally high accuracy of predictions for _ prediction. Threep differentffiffiffiffiffiffi initial velocitiespffiffiffiffiffiffi x0 are consideredpffiffiffiffiffiffi in the interwell dynamics. Fig. 6, (a) 2:5Uα= 2β, (b) 3:5Uα= 2β, (c) and 4:5Uα= 2β. The Table 2 presents the corresponding computational time and damping factor γ is chosen on the range of½ 0:007079; 0:1 , while efficiency assessment results for the interwell dynamic regime β βZ α= 2 the nonlinearity strength meets 2 C0 to satisfy the con- evaluation shown in Fig. 6. By using the analytical method straint 0rα=βC2 r0:5. developed in this research, the data in Table 2 indicate that the The results in Fig. 6 show that across this large span of system computation time to obtain accurate transient dynamics recon- fi parametersÀÁ a very high correlation coef cient (light shading of struction is reduced on average by more than a factor of 3: thus, 3 U ρ fi 10 log 10 a;n ) is obtained which indicates high accuracy from the ef ciency enhancement averages about 327%. These results the analytical predictions. Several trends are worth noting. First, a suggest that for engineering applications, such as understanding banded or striped feature is observed in the contours of Fig. 6. This the number of snap-through cycles that might occur for buckled is found to result from small changes in system parameters which and impulse-excited aircraft panels, able to be modeled using the induce one more or one less well-crossing event. Such cases show bistable Duffing oscillator governing equation, the analytical vulnerability in the analytical approach since these examples method may greatly enhance the predictive capabilities of the fi exhibit a signi cant slowing-down of dynamics prior to ultimately engineer and designer via providing an accurate and computa- undergoing one more cross-well dynamic or returning to the local tionally efficient trajectory reconstruction scheme. well of potential energy. On the other hand, the accuracy of ana- lytical predictions is still very high as attested by the large corre- lation coefficients in the banded regions. Second, local, dark con- tour shadings are evident (almost dot-like) which indicate 6. Discussion reductions in the prediction accuracy. These features result from fi the local parametric selections which induce the poor final time In this section, nal remarks are provided to summarize the predictions and thus jump in the displacement trajectories, such as capabilities and limitations of the analytical prediction strategy the example shown in Fig. 5(a) and elucidated above. Finally, for developed in this research. First, although this investigation has the results obtained using the larger initial velocity, Fig. 6(c), an examined the sensitivity of the analytical strategy using different increase in the error is shown via darker shading when the non- initial velocities while the initial displacements have remained fi linearity values β are larger, regardless of the damping factor. It is xed, the authors note from a large variety of case studies not seen that this trend results from a small accumulation of phase presented here that the strategy is found to be similarly accurate error between the analytical predictions and numerical results in predicting the transient dynamics for any combination of initial since the larger initial velocities induce longer periods of interwell conditions. On the other hand, while providing high overall oscillation for the same system parameters. On the other hand, the accuracy according to the correlation coefficient criterion, as examination of specific examples amongst these parameters observed in Figs. 2 and 5 the analytical reconstruction of trajec- reveal that the final amplitude and final time at which the inter- tories may induce an accumulation of phase error for certain initial well oscillations are predicted to occur do not significantly deviate conditions. In practice, this may be unfavorable should one be from the exact, numerical values in spite of the phase error. Thus, interested in the instantaneous oscillator phase, for example to for a large range of parameters and initial velocities, the analytical prescribe active controls to the bistable system. The planned, 36 C. Zhang et al. / International Journal of Non-Linear Mechanics 79 (2016) 26–37

the intrawell prediction. It is seen that this strategy results in overall good agreement in many cases. For example, the example shown in Fig. 7 plots the corresponding analytical results with the numerical data previously presented in Fig. 1(b). Albeit imperfect, the agreement is nevertheless very good and represents a major step forward in the accurate prediction of such strongly nonlinear, transient dynamics.

7. Conclusion

This research developed a new and enhanced averaging method using the Jacobian elliptic functions to analytically Fig. 7. Comparing the fully-reconstructed analytical prediction (light dashed reconstruct the transient, dissipative dynamics of a bistable Duff- curves) with the numerically-integrated, exact results (dark solid curves). The light solid curve on the far displacement–velocity plane is the homoclinic orbit. ing oscillator. By relaxing the requirement for small variation of the averaged parameters as employed classically, the proposed future efforts of the authors include a comprehensive study on the analytical strategy enables an accurate, long-time prediction of the role of initial conditions on the dynamical reconstruction accuracy. transient dynamics, whether of the intra- or interwell dynamic Next, although good agreement is obtained for a vast range of regimes. Comprehensive assessments find that the analytical system parameters of nonlinearity and damping in terms of pre- predictions realize excellent agreement in the decaying ampli- dicting the dynamic response amplitudes DtðÞand CtðÞ– which are tudes of oscillation when compared to the exact results deter- derived in a purely analytical closed-form, see Eqs. (30) and (45) – mined by numerical integration. The instantaneous phase exhibits the reconstruction of the instantaneous displacement and velocity small deviation with respect to the exact results for some system trajectories necessitate a pseudo-analytical determination. This is parameters and initial conditions. The time-varying amplitudes of because such trajectories rely on the time-varying amplitudes as displacement and velocity are derived in an analytical closed-form, well as on the time-varying arguments and moduli, the former while the reconstruction of the corresponding instantaneous tra- which must be numerically evaluated, see Eqs. (34) and (50).On jectories requires a pseudo-analytical approach that is found to be the other hand, as summarized via Tables 1 and 2, the analytical much more computationally efficient, for a high degree of accu- method leads to appreciable computational performance racy, than a conventional numerical integration scheme. Overall, improvements when compared to direct numerical integration by the considerable accuracy and efficiency of the proposed predic- a fourth-order Runge–Kutta algorithm. Thus, despite the pseudo- tion method open the door to the implementation of the method analytical nature of the transient displacement and velocity tra- in settings where such transient behaviors are of importance, jectories, the analytical prediction tools developed in this study are including understanding the shock vulnerability in structural accurate as well as computationally efficient. Moreover, the design and the effective development of bistable vibration-based designer or engineer that requires knowledge only of the dynamic energy harvesters under impulsive excitations. regime time extents or the instantaneous response amplitude or frequency may take full advantage of the closed-form analytical approximations of these characteristics for expedited develop- Acknowledgments ment and design iteration of systems that incorporate bistable or buckled components suitable to be modeled using the bistable This work is supported in part by the National Natural Science Duffing oscillator governing equation of motion. Foundation of China (No. 51475356), the Chinese Scholarship Finally, one may ask why the solution strategy developed here Council (CSC), and the University of Michigan Collegiate does not integrate the interwell and intrawell dynamics predic- Professorship. tions when the initial velocity is great enough to induce cross-well behaviors. In other words, why does the strategy reconstruct the dynamics in pieces rather than continuously like that shown in References Fig. 1(b)? In fact, several efforts to achieve this ideal were under- taken in this research, but none were fruitful. A number of chal- [1] L.N. Virgin, Vibration of Axially Loaded Structures, Cambridge University Press, lenges could be elucidated from such experiences, but a more Cambridge, 2007. meaningful justification for the difficulties may be given by a [2] J. Qiu, J.H. Lang, A.H. Slocum, A curved-beam bistable mechanism, J. Micro- – qualitative commentary. Consider that the transition from inter- to electromech. Syst. 13 (2004) 137 146. [3] T.J. Kazmierski, S. Beeby, Energy Harvesting Systems: Principles, Modeling and intrawell oscillations represents a bifurcation in the overall Applications, Springer, New York, 2011. dynamics: the trajectory crosses the homoclinic orbit, see Fig. 1(b), [4] R.L. Harne, K.W. Wang, A review of the recent research on vibration energy and the global dynamical behavior of the oscillations change sig- harvesting via bistable systems, Smart Mater. Struct. 22 (2013) 023001. fi [5] G. Debnath, T. Zhou, F. Moss, Remarks on stochastic resonance, Phys. Rev. A 39 ni cantly. Under such conditions, the bifurcating dynamics occur (1989) 4323–4326. on a time scale sufficiently slower than the natural periods of free [6] M.I. Dykman, A.L. Velikovich, G.P. Golubev, D.G. Luchinskiĭ, S.V. Tsuprikov, decaying oscillation [12]. As a result, the effective mathematical Stochastic resonance in an all-optical passive bistable system, Sov. Phys. JETP Lett. 53 (1991) 193–197. treatment and reconstruction of such dynamics transition requires [7] N. Hu, R. Burgueño, Buckling-induced smart applications: recent advances and an assessment of the system when reduced to a more appropriate trends, Smart Mater. Struct. 24 (2015) 063001. equation form, namely a normal form [34,35]. This is not to sug- [8] Michelin. Michelin Tweel. [Online]. 〈http://www.michelintweel.com/〉,2015. fi [9] J.S. Humphreys, On dynamic snap buckling of shallow arches, AIAA J. 4 (1966) gest that a uni ed prediction approach is impossible but rather 878–886. that alternative methods may be needed to establish such a fra- [10] A.N. Kounadis, J. Raftoyiannis, J. Mallis, Dynamic buckling of an arch model mework. Nevertheless, a direct splicing of inter- to intrawell under impact loading, J. Sound Vib. 134 (1989) 193–202. fi [11] M.I. Dykman, D.G. Luchinsky, R. Mannella, P.V.E. McClintock, N.D. Stein, N. dynamics predictions may be effected by using the nal states G. Stocks, Stochastic resonance in perspective, IL Nuovo Cimento 17 (1995) determined for the interwell dynamics as the initial conditions for 661–683. C. Zhang et al. / International Journal of Non-Linear Mechanics 79 (2016) 26–37 37

[12] R. Seydel, Practical Bifurcation and Stability Analysis, Springer, New York, [25] F. Lakrad, M. Belhaq, Periodic solutions of strongly non-linear oscillators by 2010. the multiple scales method, J. Sound Vib. 258 (2002) 677–700. [13] R.E. Mickens, A generalization of the method of harmonic balance, J. Sound [26] S.B. Yuste, "Cubication" of non-linear oscillators using the principle of har- Vib. 111 (1986) 515–518. monic balance, Int. J. Non-Linear Mech. 27 (1992) 347–356. [14] S.B. Yuste, J.D. Bejarano, Improvement of a Krylov–Bogoliubov method that [27] L. Cveticanin, An approximate solution for a system of two coupled differential uses Jacobi elliptic functions, J. Sound Vib. 139 (1990) 151–163. equations, J. Sound Vib. 152 (1992) 375–380. [15] I. Kovacic, M.J. Brennan, The Duffing Equation: Nonlinear Oscillators and Their [28] V.T. Coppola, R.H. Rand, Averaging using elliptic functions: approximation of Behaviour, John Wiley & Sons, Chichester, 2011. limit cycles, Acta Mech. 81 (1990) 125–142. [16] P.G.D. Barkham, A.C. Soudack, An extension to the method of Kryloff and [29] V.T. Coppola, R.H. Rand, Macsyma program to implement averaging using Bogoliuboff, Int. J. Control 10 (1969) 377–392. elliptic functions, in: K.R. Meyer, D.S. Schmidt (Eds.), Computer Aided Proofs in [17] M. Belhaq, F. Lakrad, The elliptic multiple scales method for a class of auton- Analysis, Springer, New York, 1991, pp. 71–89. omous strongly non-linear oscillators, J. Sound Vib. 234 (2000) 547–553. [30] M. Belhaq, F. Lakrad, Prediction of homoclinic bifurcation: the elliptic aver- [18] A.C. Soudack, P.G.D. Barkham, On the transient solution of the unforced aging method, Chaos Solitons Fractals 11 (2000) 2251–2258. Duffing equation with large damping, Int. J. Control 13 (1971) 767–770. [31] M.A. Al-Shudeifat, Amplitudes decay in different kinds of nonlinear oscillators, [19] P.G.D. Barkham, A.C. Soudack, Approximate solutions of non-linear non- J. Vib. Acoust. 137 (2015) 031012. autonomous second-order differential equations, Int. J. Control 11 (1970) [32] T. Okabe, T. Kondou, J. Ohnishi, Elliptic averaging methods using the sum of 101–114. Jacobian elliptic delta and zeta functions as the generating solution, Int. J. [20] P.A.T. Christopher, A. Brocklehurst, A generalized form of an approximate Non-Linear Mech. 46 (2011) 159–169. solution to a strongly non-linear, second-order, differential equation, Int. J. [33] P.F. Byrd, M.D. Friedman, Handbook of Elliptic Integrals for Engineers and Control 19 (1974) 831–839. Scientists, Springer-Verlag, Berlin, Heidelberg, New York, 1971. [21] R.E. Mickens, Truly Nonlinear Oscillators: Harmonic Balance, Parameter [34] J. Guckenheimer, P.J. Holmes, Nonlinear Oscillations, Dynamical Systems, and Expansions, Iteration, and Averaging Methods, World Scientific, London, 2010. Bifurcations of Vector Fields, Springer, New York, 1983. [22] S.B. Yuste, Comments on the method of harmonic balance in which Jacobi [35] Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York, elliptic functions are used, J. Sound Vib. 145 (1991) 381–390. 1998. [23] L. Cveticanin, Homotopy-perturbation method for pure nonlinear differential equation, Chaos Solitons Fractals 30 (2006) 1221–1230. [24] J.H. He, Comparison of homotopy perturbation method and homotopy analysis method, Appl. Math. Comput. 156 (2004) 527–539.