Escape of harmonically forced classical particle from an infinite-range potential well. O.V.Gendelman Faculty of Mechanical Engineering, Technion – Israel Institute of Technology Haifa 3200003 Israel e-mail:
[email protected] The paper considers a process of escape of classical particle from a one-dimensional potential well by virtue of an external harmonic forcing. We address a partiular model of the infinite-range potential well that allows independent adjustment of the well depth and of the frequency of small oscillations. The problem can be conveniently reformulated in terms of action-angle variables. Further averaging provides a nontrivial conservation law for the slow flow. Thus, one can consider the problem in terms of averaged dynamics on primary 1:1 resonance manifold. This simplification allows efficient analytic exploration of the escape process, and yields a theoretical prediction for a minimal forcing amplitude required for the escape, as a function of the excitation frequency. This function exhibits a single minimum for certain intermediate frequency value. Numeric simulations are in complete qualitative and reasonable quantitative agreement with the theoretical predictions. Keywords: Potential well, external forcing, action-angle variables, separatrix crossing. 1. Introduction. In majority of common designs in mechanical engineering and related fields, dynamical elements work in linear or weakly nonlinear (quasilinear) regime. This choice is easily understandable, since such regimes are deeply understood, predictable and assessable by welldeveloped methods of analysis [1, 2]. These methods often rely on ideas of averaging, multiplescale expansions and other asymptotic techniques [3-5]. Still, essential nonlinearities occur in mechanical systems due to various reasons, including clearances, impacts, friction, material nonlinearities and plasticity [6-8]. In many applications, such behavior is necessary and
important [6, 7]. In others, it is completely unwanted, but still profoundly reveals itself in dynamics. Cracks in continuous structures are one example of this sort [9-11]. In last two decades it was realized that intentional use of strongly nonlinear elements in broad variety of mechanical systems can bring about significant enhancement of their performance. Among other applications, one can mention targeted energy transfer in essentially nonlinear systems with applications for energy absorption and harvesting [12-24], wave propagation, mitigation, redistribution and arrest in granular crystals, granular media and other systems with “sonic vacuum” (i.e. with zero linear sound velocity) [25-32], as well as wave control with the help of acoustic metamaterials with essentially nonlinear elements [33 - 36]. Analysis of dynamics in essentially nonlinear systems is a major challenge. Complete picture can be obtained only for extremely rare completely integrable systems [3]. For other systems, it is sometimes possible to derive exact periodic solutions – examples of this sort include nonlinear normal modes [37-39] and discrete breathers in selected models [40-42]. Some information on periodic solutions in broad variety of systems may be obtained by approximate and numeric methods [43-45]. However, in the essentially nonlinear systems, the superposition principle is absent, and usually it is hardly possible to apply it even approximately. Despite all valuable insights obtained from the periodic solutions, they are insufficient for understanding the transient dynamics and energy transport in such systems. However, the nonstationary processes are usually the most interesting and important for applications. Major progress in theoretical study of energy transport in essentially nonlinear systems has been achieved, since it was realized the most efficient transport usually occurs in conditions of resonance. This observation allows one to treat the system in the vicinity of the resonance manifold (RM), and to restrict the consideration by averaged equations of motion (usually referred to as slow-flow equations). This crucial simplification often allows reduction of dimensionality and gives rise to conservation laws absent in the complete system beyond the RM. Technically, in many of the works mentioned above, the averaging has been performed with the help of complex variables (complexification-averaging approach, CxA) [46 - 48]. This approach follows back to models with self-trapping [49] and rotating-wave approximation [50] in the lattice dynamics. From mathematical point of view, this approach is equivalent to classical
harmonic balance with slowly varying amplitudes [51]. Applications of this method to energy exchange in model oscillatory systems were demonstrated in recent works [52, 53]. In the same time, it is clear that applicability of the harmonic balance-based methods is limited. First of all, they can be rigorously justified only for the quasilinear systems or for nonlinear potentials with power close to 2 [54]. The harmonic-balance related methods are also used for quartic and even stronger nonlinearities [55] and even yield reasonable results, but with limited and unknown a priori accuracy. Partial remedy for periodic solutions may be achieved with account of multiple harmonics in the expansion. Still, the attempts to use slowly varying amplitudes for multiple harmonics may lead to a system that mathematically is even more complex than the initial one [16]. Besides, for very interesting and important systems that include clearances, impacts or rotators, these methods have additional limitations. It is possible, however, to devise efficient approaches for exploration of transient responses and targeted energy transfer in some systems that include a single vibro-impact element or rotator [56, 57]. While handling these models, one also invokes the exploration of RMs, but beyond harmonic balance or CxA. The treatment heavily relies on particular simplifications available for pure impacting particle or pure eccentric rotator. Recently, it was demonstrated [58], that the structure of the RM could be most conveniently described in terms of canonical action-angle (AA) variables. The AA variables are famous and widely used instrument in a theory of dynamical systems [3, 59-61]. The AA variables were instrumental in formulation of many prominent results and theories. Among others, one can mention theory of adiabatic invariants [59], formulation and proof of KAM theorem [3, 60, 61], development of canonical perturbation theory [62, 63], explorations on Hamiltonian chaos [64, 65], autoresonant phenomena [66, 67], targeted energy transfer [13] etc. The AA formalism has important theoretical advantages [58] - all RMs may be described at the same level of complexity. Exploration of the dynamics on the RMs in terms of the AA variables reveals all regularities mentioned above (reduction of the dimensionality, additional conservation laws). Current paper treats a problem of particle trapped in a one –dimensional potential well with finite depth and rapidly decaying tails at [59]. This model is relevant for many possible physical settings, one of which is a particle attached to an active center of absorption on a
surface. The physical question addressed here is whether this particle can be released from the trap by external harmonic forcing. Free oscillations of the particle in the vicinity of equilibrium have certain well – defined finite frequency. The escape from the trap formally corresponds to an infinite remotion of the particle. In the case of the free oscillations, minimum energy required for the escape is equal to the depth of the potential well, and corresponding frequency is zero – the phase trajectory is a separatrix between bounded and unbounded motions of the particle. So, in order to escape from the well, the particle should cross the separatrix [68, 69]. This fact makes the description of the RM rather difficult. The paper is devoted exactly to this problem. We are going to demonstrate that the process of escape can be efficiently described on of the RM in terms of the AA variables. Critical amplitude-frequency curve for the external forcing, that describes the escape threshold, is related to a reconnection of special phase trajectories on this RM. Section 2 is devoted to description of the model and theoretical computation of the RM based on the conservation law for the averaged system. In Section 3 we relate the escape from the potential well to modifications of the RM structure. Section 4 comprises numeric validations and illustrations of the analytic findings. Section 5 contains a number of concluding remarks. 2. Description of the model and computation of the resonance manifold. General approach to computation of the RM in terms of the AA variables for the particle under external periodic forcing has been presented in [58]. Main details of this approach are derived below in slightly simplified form, for the sake of completeness. Let us consider the particle in the potential well under action of external periodic forcing. This system can be described by the following equation: q
V F sin q
(1)
Here q(t) is a generalized dimensionless displacement of the particle, V(q) is the external potential, F and Ω are dimensionless amplitude and frequency of the external forcing respectively, τ is a dimensionless time. The dot denotes differentiation with respect to τ. Mass of the particle is set to unity without loss of generality. Hamiltonian of the system described by Equation (1) is written as follows:
H H 0 ( p, q) Fq sin ; H 0
p2 V (q), p q 2
(2)
H 0 ( p, q) is a time-independent component of the Hamiltonian. Then, this component
induces a transformation to the AA variables in accordance with well-known formulas [59-61]:
I (E)
1 2
p(q, E )dq; =
I
q
p(q, I )dq
(3)
0
Here H 0 ( p, q) E const defines a constant energy level. By inverting expressions (3), one obtains explicit formulas for the canonical change of variables p( I , ), q( I , ) and for the conservative component of the Hamiltonian: H0 H 0 ( I ) E( I ) . The canonical transformation outlined above does not include the explicit time dependence; therefore the Hamiltonian of transformed system (1) is written in the following form: H H0 ( I ) Fq( I , )sin
(4)
Due to 2π-periodicity of the angle variable, the Hamiltonian (4) can be rewritten in terms of Fourier series [64]: H H0 (I )
iF qm ( I ) exp i(m ) exp i (m ) ; qm q* m 2 m
(5)
Hamilton equations will take the form: H F mqm ( I ) exp i(m ) exp i(m ) 2 m H H 0 iF qm ( I ) exp i(m ) exp i(m ) I I 2 m I I
(6)
We consider the most profound primary 1:1 resonance. To consider this regime, one assumes slow evolution of the phase variable t ; all other phase combinations in Equations (6) should be considered as fast. Averaging over these fast phase variables yields the following system of the slow-flow equations:
F q1 ( J )ei q1* ( J )ei 2 H ( J ) iF q1 ( J ) i q1* ( J ) i 0 e e J 2 J J J
(7)
Here J (t ) I (t ) is the average of the action variable over fast phases. It is easy to see by direct differentiation that System (2) possesses the following first integral: C H0 (J )
iF q1 ( J )ei q1* ( J )ei J const 2
(8)
Expression (8) defines a family of 1:1 RMs for the considered problem. The value of constant C is determined by initial conditions – the values of the action and slow phase, at which the system is captured by the RM. The first integral (8) is a particular case of general conservation law for the RMs of the single-DOF systems with periodically time-dependent Hamiltonian [58]. For particular problem considered in the current paper – the escape the forced particle from the potential well – we adopt a well-known model potential [59, 61]: V ( x)
V0 cosh 2 x
(9)
This potential describes the potential well with depth V0 , characteristic width 1 and exponential decay of the attractive force at x ; x is a physical displacement of the particle. In terms of physical variables, the motion of particle under harmonic forcing in the well with potential (9) is described by the following equation: m
d 2 x 2Vo sinh x A sin t dt 2 cosh 3 x
(10)
Here m is the mass of the particle, A and ω are physical forcing amplitude and frequency respectively. Transition to the non-dimensional variables is performed as: 0t , q x, 0
2V0 A , = , F m 0 2V0
Then, one obtains the equation in the form (1): q
sinh q F sin . cosh 3 q
(11)
1 2
Here the potential V (q) cosh 2 q . Transition to the action-angle variables for this potential is well-known [59, 61] and yields invertible expressions in terms of elementary functions: 2I I 2 1 H 0 ( I ) (1 I )2 , q( I , ) arcsinh sin , 0 I 1 1 I 2
(12)
For the sake of completeness, the details of derivation of expressions (12) are presented in the Appendix. For the unforced system, the value I 0 corresponds to the bottom of the potential well, and I 1 - to the separatrix between the bounded and unbounded motions. Expressions (12) allow immediate computation of the 1:1 RMs for the considered problem in accordance with Equation (8): 1 2F C ( J , ) (1 J )2 (K (k ) E(k ))cos J const, k 2 J J 2 2 k
(13)
Here K (k ) and E(k ) are complete elliptic integrals of the first and the second kind, respectively, k is the modulus of the elliptic integrals. Details of derivation of Expression (13) are presented in the Appendix. 3. Structure of the resonance manifold and the escape threshold. The next step is to explore the qualitative structure (phase portrait) of the RM (13). It is easy to see that the fixed points of this phase portrait correspond to solutions of the following algebraic equations: C ( J , ) C ( J , ) 0 sin 0; 0 J
(14)
The first equation yields that the fixed points may appear only at lines e 0, . The second equation is awkward and hardly solvable; then, one should rely on a combination of
numeric and analytic methods. Since the primary purpose of exploration is the possibility of the escape form potential well under the harmonic forcing, we are mainly interested in the behavior of the special orbit on the RM, which corresponds to zero initial conditions of the particle. In recent literature [46-48, 52, 53] such orbits are referred to as limiting phase trajectories (LPTs). This convention will be followed below. From mathematical point of view, however, the LPT does not have any special properties – other (nonzero) initial conditions are legitimate as well. It turns out that the qualitative structure of the RM is different in the cases of small and large forcing frequencies Ω. It is easy to identify the LPT, since it corresponds to the initial condition J 0 . Then, it should satisfy the following equation: C ( J , ) 1/ 2
(15)
Typical transformation of the phase portrait in the case of relatively small forcing frequencies is presented in Figure 1. All phase portraits below are plane views of the phase cylinder ( J , ); 0 J 1, 0 2 .
a)
b)
c)
Figure 1. Reconnection of the LPT for the case of relatively small forcing frequencies; Ω=0.4, a) F=0.28, b) Fcrit=0.2934, c) F=0.3. Thick red line denotes the LPT (15). We observe that for relatively small forcing the branches of LPT defined by Equation (15) are not connected. Thus, if the particle starts from the zero initial conditions at the RM, the value of J remains relatively small, and the particle does not escape from the potential well (Figure 1a). For larger values of forcing, the LPT has only one branch, which approaches J 1 (Figure 1c). Physically, one should identify this situation with the escape from the potential well. The marginal state that distinguishes between the two is depicted in Figure 1b. Here the LPT
branches connect in the saddle point at e 0 . It is natural to identify the value of Fcrit that yields Figure 1b with the minimal forcing amplitude required for the escape of the particle from the potential well, with given excitation frequency. We refer to this special transition as the LPT reconnection For the case of relatively large excitation amplitudes, one observes the other scenario of the LPT reconnection, as illustrated in Figure 2.
a)
b)
c)
Figure 2. Reconnection of the LPT for the case of relatively large excitation frequencies; Ω=1, a) F=0.24, b) Fcrit=0.25057, c) F=0.26. Thick red line denotes the LPT. In this case, the reconnection occurs in the saddle point at e . Thus, one can identify two possible scenarios of the LPT reconnection. Each point of the LPT reconnection determines the pair ( Fcrit , ) , where Fcrit is the minimal theoretical forcing amplitude required for the escape. Two scenarios pf the LPT reconnection produce two curves in parametric plane ( F , ) . It seems impossible to derive explicit analytic dependence Fcrit () , but implicit computation is relatively easy. To perform that, it is enough to recall that the reconnection occurs at the saddle points. Therefore, the values of Fcrit for given values of Ω should satisfy the following equations: C 0 (1 J ) Fcrit G1 ( J ) 0 J 1 J2 C ( J , ) J Fcrit G0 ( J ) J 0 (16) 2 2 G ( J ) 2(1 J ) E( k ) (1 k 2 )K ( k ) 2 2 G0 ( J ) (K (k ) E(k )), G1 ( J ) 0 , k 2J J k J k 3 (1 k 2 )
The first equation of (16) stems from the requirement to pass through the fixed point; the second one defines the LPT. Positive and negative signs correspond to e and e 0 respectively. The fixed point in question must be the saddle, since it is the only generic possibility for the phase trajectory to pass through it. Equations (16) can be easily solved, and yield the following parametric expressions for Fcrit ( J ) and ( J ) : Fcrit ( J )
G ( J )( J 2 2 J ) 2G0 ( J )( J 1) J2 ; ( J ) 1 2(G0 ( J ) JG1 (G)) 2(G0 ( J ) JG1 (G))
(17)
The curves defined by Equations (17) in physically meaningful part of the parametric plane Fcrit 0, 0 are presented in Figure 3.
Figure 3. The LPT reconnection points in the (F,Ω) plane. The ascending and descending curves correspond to the positive and negative signs in (17) respectively. The scheme near each curve describes the LPT reconnection scenario that corresponds to this curve. The lower scheme corresponds to the intersection point. One can observe that every scenario of the LPT reconnection exists in certain range of the excitation frequencies. The parts of the curves in Figure 3 below the intersection point still correspond to the LPT reconnections, but do not lead to the escape – the second saddle point
prevents it. The intersection point of the curves in Figure 3 is of substantial physical interest – it corresponds to a minimal theoretical amplitude of the forcing, at which the escape from the well is still possible. Numeric solution yields the following coordinates for this point: * * 0.73395, Fcrit 0.09659
(18)
For these special values of the frequency and the forcing amplitude, the RM phase portrait has a peculiar shape presented in Figure 4.
Figure 4. Double reconnection of the LPT for special values of parameters (18). One observes that the LPT coincides with heteroclinic orbit that connects two saddle points. The theoretical exploration presented above predicts the non-monotonous dependence of the minimal forcing amplitude required for the escape, on the excitation frequency – the graph in Figure 3 exhibits a minimum associated with the double LPT reconnection. In order to validate these predictions, we perform direct numeric simulations of System (11) 4. Numeric validations and illustrations. First, we present a typical escape scenario for values of parameters defined in the figure caption (Figure 5).
a)
b)
Figure 5. Time series for solution of equation (11) with zero initial conditions and 0.7, F 0.11 ; a) time series q(t ) ; b) solution in (q, q) plane superimposed on the phase portrait of the unforced system. Separatrix of the unforced system is denoted by dashed line, other phase trajectories – by dotted lines. Solid thick line corresponds to solution of the forced system (q(t ), q(t )) .
The escape process presented in Figure 5 starts from clear resonance pattern (0
