Limiting Phase Trajectories and Resonance Energy Transfer in a System of Two Coupled Oscillators

Abstract

We study a problem of energy exchange in a system of two coupled oscillators subject to 1 : 1 resonance. Our results exploit the concept of limiting phase trajectories (LPTs). The LPT, associated with full energy transfer, is, in certain sense, an alternative to nonlinear normal modes characterized by conservation of energy. We consider two benchmark examples. As a first example, we construct an LPT and examine the convergence to stationary oscillations for a Duffing oscillator subjected to resonance harmonic excitation. As a second example, we treat resonance oscillations in a system of two nonlinearly coupled oscillators. We demonstrate the reduction of the equations of motion to an equation of a single oscillator. It is shown that the most intense energy exchange and beating arise when motion of the equivalent oscillator is close to an LPT. Damped beating and the convergence to rest in a system with dissipation are demonstrated.

1. Introduction

The problem of passive irreversible transfer of mechanical energy (referred to as energy pumping) in oscillatory systems has been studied intensively over last decades; see, for example, [1, 2] for recent advances and references. In this case, the key role of transient process is evident, in contrast to great majority of conventional problems of nonlinear dynamics, in which the main attention has been given to nonlinear normal modes (NNMs), characterized by the conservation of energy. Recent studies [3, 4]; have shown that the NNM approach is effective in the case of weak energy exchange, while the concept of the limiting phase trajectories (LPT) can be used to describe intense energy exchange between weakly coupled oscillators or oscillatory chains.

The limiting phase trajectory (LPT) has been introduced [3] as a trajectory corresponding to oscillations with the most intensive energy exchange between weakly coupled oscillators or an oscillator and a source of energy; the transition from energy exchange to energy localization at one of the oscillators is associated with the disappearance of the LPT. Recently, the LPT ideas have been applied to the analysis of resonance-forced vibrations in a 1DOF and 2DOF nondissipative system [3–6]. An explicit expression of the LPT in a single oscillator [5, 6] is prohibitively difficult for practical utility. The purpose of the present paper is to show that explicit asymptotic solutions can be obtained for a class of problems that are associated with the maximum energy exchange, and introduce relevant techniques.

The paper is organized as follows. The first part is concerned with the construction of the LPT for the Duffing oscillator subject to 1 : 1 resonance harmonic excitation. In Section 2 we briefly reproduce the results of [5, 6]. We derive the averaged equations determining the slowly varying envelope and the phase of the nonstationary motion and then construct the LPT for different types of motion. Section 3 introduces the nonsmooth temporal transformations as an effective tool of the nonlinear analysis. Section 4 examines the transformations of the LPT into stationary motion in the weakly dissipated system.

In the remainder of the paper (Section 5), we analyse the dynamics of a 2DOF system. The system consists of a linear oscillator of mass 𝑀 (the source of energy) coupled with a mass 𝑚 (an energy sink) by a nonlinear spring with a weak linear component. Excitation is due to an initial impulse acting upon the mass 𝑀. It is shown that motion of the overall structure can be divided into two stages. The first stage is associated with the maximum energy exchange between the oscillators; here, motion is close to beating in a nondissipative system. At the second stage, trajectories of both masses in the damped system are approaching to rest. The task is to construct an explicit asymptotic solution describing both stages of motion for each oscillator. To this end, we reduce the equations of a 2DOF system to an equation of a single oscillator and then find beating oscillations characterizing the most intense energy exchange in a nondissipative system. A special attention is given to a difference between the dynamics of the Duffing oscillator and an equivalent oscillator in the presence of dissipation. While the Duffing oscillator is subjected to harmonic excitation, the 2DOF system is excited by an initial impulse applied to the linear oscillator. The response of the linear oscillator, exponentially vanishing at 𝑡→∞, stands for an external excitation for the nonlinear energy sink. We examine the transformation of beating to damped oscillations. We show that a solution of the system, linearized near the rest state, is sufficient to describe the second part of the trajectory.

2. Resonance Oscillations of the Duffing Oscillator

2.1. Main Equations

We investigate the transient response of the Duffing oscillator in the presence of resonance 1:1. The dimensionless equation of motion is𝑑²𝑢𝑑𝑡²+2𝜀𝛾𝑑𝑢𝑑𝑡+𝑢+8𝛼𝜀𝑢³=2𝜀𝐹sin(1+𝜀𝑠)𝑡, (2.1) where 𝜀>0 is a small parameter. We recall that maximum energy pumping from the source of excitation into the oscillator takes place if the oscillator is initially at rest; this corresponds to the initial conditions𝑡=0⁺,𝑢=0,𝑑𝑢𝑑𝑡=0. (2.2) An orbit satisfying conditions (2.2) is said to be the limiting phase trajectory [3].

In order to describe the nonlinear dynamics, we invoke a complex-valued transformation [7–9]. Introduce the variables 𝜓 and 𝜓^∗, such that 1𝑢=2𝑖(𝜓−𝜓^∗1),𝑣=2(𝜓+𝜓^∗),𝜓=𝑣+𝑖𝑢,𝜓^∗=𝑣−𝑖𝑢, (2.3) where 𝑖 = √−1; the asterisk denotes complex conjugate. It will be shown that only one complex function is sufficient for a complete description of the dynamics. Inserting 𝜓, 𝜓^∗ from (2.3) into (2.1), a little algebra shows that (2.1) is equivalent to the following (still exact) equation of motion 𝑑𝜓𝑑𝑡−𝑖𝜓+𝜀𝑖𝛼(𝜓−𝜓^∗)³+𝜀𝛾(𝜓+𝜓^∗)−2𝜀𝐹sin(1+𝜀𝑠)𝑡=0. (2.4)

Applying the multiple scales method [10, 11], we construct an approximate solution of (2.4) as an expansion𝜓(𝑡,𝜀)=𝜓₀(𝜏₀,𝜏₁)+𝜀𝜓₁(𝜏₀,𝜏₁𝑑)+⋯,=𝜕𝑑𝑡𝜕𝜏₀𝜕+𝜀𝜕𝜏₁,𝑑²𝑑𝑡²=𝜕²𝜕𝜏²₀𝜕+2𝜀²𝜕𝜏₀𝜕𝜏₁+⋯, (2.5) where 𝜏₀=𝑡 and 𝜏₁=𝜀𝑡 are the fast and slow time-scales, respectively. A similar representation is valid for the function 𝜓^∗. Then we substitute expressions (2.5) in (2.4) and equate the coefficients of like powers of 𝜀. In the leading order approximation, we obtain 𝜕𝜓₀𝜕𝜏₀−𝑖𝜓₀𝜓=0,₀(𝜏₀,𝜏₁)=𝜑₀(𝜏₁)𝑒^𝑖𝜏0 (2.6) A slow function 𝜑₀(𝜏₁) will be found at the next level of approximation. Equating the coefficients of order 𝜀 leads to𝜕𝜓₁𝜕𝜏₀+𝑑𝜑₀𝑑𝜏₁𝑒^𝑖𝜏0−𝑖𝜓₁+𝑖𝛼(𝜓₀−𝜓^∗₀)³+𝛾(𝜓₀+𝜓^∗₀)+𝑖𝐹(𝑒^{𝑖(𝜏0+𝑠𝜏1)}−𝑒^{−𝑖(𝜏0+𝑠𝜏1)})=0, (2.7)

In order to avoid the secular growth of 𝜓₁(𝜏₀,𝜏₁) in 𝜏₀, that is, avoid a response not uniformly valid with increasing time, we eliminate resonance terms from (2.7). This yields the following equation for 𝜑₀:𝑑𝜑₀𝑑𝜏₁+𝛾𝜑₀−3𝑖𝛼|𝜑₀|²𝜑₀=−𝑖𝐹𝑒^{𝑖𝑠𝜏1},𝜑₀(0)=0. (2.8) Next we introduce the polar representation 𝜑₀=𝑎𝑒^𝑖𝛿, (2.9) where a and 𝛿 represent a real amplitude and a real phase of the process 𝜑₀(𝜏₁). Inserting (2.9) into (2.8) and setting separately the real and imaginary parts of the resulting equations equal to zero, (2.8) is transformed into the system 𝑑𝑎𝑑𝜏₁𝑎+𝛾𝑎=−𝐹sin▵,𝑑Δ𝑑𝜏₁=−𝑠𝑎+3𝛼𝑎³−𝐹cosΔ, (2.10) where a > 0, Δ = δ−𝑠𝜏₁. It now follows from (2.3), (2.9) that𝑢(𝑡,𝜀)=𝑎(𝜏₁)sin(𝑡+Δ(𝜏₁))+𝑂(𝜀). (2.11) This means that the amplitude 𝑎(𝜏₁) and the phase Δ(𝜏₁) completely determine the process u(t,ε) (in the leading-order approximation). Note that a = 0 if the oscillator is not excited.

2.2. Critical Parameters and LPTs of the Undamped Oscillator

In this section, we recall main definitions and results concerning the dynamics of the nondissipative system. In the absence of damping, system (2.10) is rewritten as 𝑑𝑎𝑑𝜏₁𝑎=−𝐹sinΔ,𝑑Δ𝑑𝜏₁=−𝑠𝑎+3𝛼𝑎³−𝐹cosΔ. (2.12)

It is easy to prove that system (2.12) conserves the integral of motion3𝐻=4𝛼𝑎⁴−𝑠𝑎²2−𝐹𝑎cosΔ=𝐻₀, (2.13) where 𝐻₀ depends on initial conditions. In the phase plane, the LPT corresponds to the contour 𝐻=0, as only in this case a system trajectory goes through the point 𝑎=0. Taking 𝐻₀=0, we obtain the following expression:3𝐻=𝑎(4𝛼𝑎³−𝑠2𝑎−𝐹cosΔ)=0. (2.14)

Formula (2.14) implies that the LPT has two branches, the first branch is a = 0; the second branch satisfies the cubic equation34𝛼𝑎³−𝑠2𝑎−𝐹cosΔ=0. (2.15)

Equality (2.15) determines the second initial condition 𝑎(0⁺)=0, cosΔ(0⁺)=0. We suppose that d𝑎/d𝜏1>0 at 𝜏₁=0⁺; under this assumption, Δ(0⁺)=−𝜋/2. Hence the initial conditions for the LPT take the form𝜏₁=0⁺,𝑎(0⁺)=0,Δ(0⁺𝜋)=−2. (2.16) Throughout this paper, we write 0 instead of 0⁺, except as otherwise noted.

Next we determine critical parameters of system (2.12). The steady states of (2.12) satisfies the equations𝑑𝑎𝑑𝜏₁=0,𝑑Δ𝑑𝜏₁=0. (2.17) The second equation is equivalent to the equality−𝑠𝑎+3𝛼𝑎³−𝐹sgn(cosΔ)=0, (2.18) where cosΔ=±1. We analyze the properties of (2.18) considering the properties of its discriminant 𝐷₁𝐷₁=19𝛼²(𝐹²4−𝑠³81𝛼). (2.19) If 𝐷₁<0, (2.18) has 3 different real roots; if 𝐷₁>0, (2.18) has a single real and two complex conjugate roots; if 𝐷₁=0, two real roots will merge; see, for example, [12]. The latter condition gives the first critical value of the parameter α𝛼^∗₁=4𝑠³81𝐹². (2.20)

A straightforward investigation proves that, if 𝛼>𝛼^∗₁ (strong nonlinearity), there exists only a single stable centre 𝐶₊: (0, 𝑎₊) (Figure 1); if 𝛼<𝛼^∗₁ (weak nonlinearity), there exist two stable centres 𝐶₋: (−π, 𝑎₋), C₊: (0, 𝑎₊), and an intermediate unstable hyperbolic point O: (−π, 𝑎₀), see Figures 2 and 3.

(a)

(b)

Figure 1 

Phase portrait (a) and plot of a(τ 1) (b) for s = 0.4, F = 0.13, α = 0.187 = ${\alpha }_1^{*}$.

Figure 2 

Passage from small to large oscillations: α = 0.0935 = ${\alpha }_2^{*}$.

(a)

(b)

Figure 3 

Phase portrait (a) and plot of a(${\tau }_1$) (b) for quasilinear oscillations: s = 0.4, F = 0.13, α = 0.093 $<{\alpha }_2^{*}$.

We now suppose that 𝛼<𝛼^∗₁, that is the system may exhibit both types of oscillations, either near Δ = −π, or near Δ = 0 (Figures 2–4). In both cases, the LPT begins at a = 0, Δ = −π/2 but the run of the LPT depends on the relationship between the parameters. In order to find a critical value 𝛼^∗₂<𝛼^∗₁ ensuring the transition from small to large oscillations, we analyze (2.15). Consider the discriminant of (2.15)𝐷₂=4𝛼²(𝐹²−2𝑠³81𝛼) (2.21) In the critical case α = 𝛼^∗₂, an unstable hyperbolic point coincides with the maximum of the left branch of the LPT at Δ = −π (Figure 2). This means that 𝐷₂ = 0 at 𝛼^∗₂, or 𝛼^∗₂=2𝑠³81𝐹²=𝛼^∗₁2, (2.22) which defines a boundary between small quasilinear (𝛼<𝛼^∗₂) and large nonlinear (𝛼>𝛼^∗₂) oscillations. In particular, for s = 0.4, F = 0.13 we obtain 𝛼^∗₂ = 0.0935 (Figure 2).

(a)

(b)

Figure 4 

Phase portrait (a) and plot of a(${\tau }_1$) (b) for strongly nonlinear oscillations: s = 0.4, F = 0.13, ${\alpha }_2^{*}<\alpha$ = 0.094 $<{\alpha }_1^{*}$.

Figures 3 and 4 are plotted for 𝛼<𝛼^∗₂ and 𝛼>𝛼^∗₂, respectively. In Figure 3(a), one can see the LPT encircling the center 𝐶₋ of relatively small oscillations; Figure 4(a) shows the LPT encircling the centre 𝐶₊ of large oscillations; this case is associated with the maximum energy absorption. Figures 3(b) and 4(b) demonstrate the behavior of the function a(𝜏₁) corresponding to the respective branch of the LPT. Note that both branches of the LPT begin at the same point a = 0, Δ = −π/2.

If α= 𝛼^∗₁ = 2𝛼^∗₂, the above-mentioned coincidence of the stable and unstable points at Δ = −π results in the transformation of the phase portraits (Figure 1) and disappearance of the stable centre C₋. Figure 1(a) demonstrates a single stable fixed point at Δ = 0.

2.3. Reduction to the 2nd-Order Equation

For the further analysis, it is convenient to reduce the equation of the LPT to the second-order form. Using (2.15) to exclude Δ, we obtain the following equation:𝑑²𝑎𝑑𝜏²₁+𝑓(𝑎)=0,𝑎=0,𝑑𝑎𝑑𝜏₁=𝐹at𝜏₁=0, (2.23) where 𝑎𝑓(𝑎)=4(32𝛼𝑎²9−𝑠)(2𝛼𝑎²−𝑠)=𝑑Ψ(𝑎),𝑎𝑑𝑎Ψ(𝑎)=²8(32𝛼𝑎²−𝑠)². (2.24) Note that (2.23) can be treated as the equation of a conservative oscillator with potential Ψ(𝑎), yielding the integral of energy1𝐸=2𝑣²1+Ψ(𝑎)=2𝐹²,𝑣=𝑑𝑎𝑑𝜏₁. (2.25)

Figure 5 depicts the potential Ψ(𝑎) in the small and large scales for system (3.1) with the parameters α = 1/3, s = 1/4. The phase portrait is given in Figure 6.

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(b)

Figure 5 

Potential Ψ(a) in the small (a) and large (b) scales.

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(b)

Figure 6 

Phase portraits of (2.23) in the small (a) and large (b) scales.

The amplitude of oscillations 𝐴₀ is defined by the following equality (see Figure 6(b)): 𝐸=Ψ(𝐴₀1)=2𝐹². (2.26)

From (2.25) we have 𝑣=𝑑𝑎/𝑑𝜏₁=±𝐹²−2Ψ(𝑎). Thus, the half-period of oscillations is defined as𝑇₁(𝐴₀)=^𝐴0₀𝑑𝑎𝐹²−2Ψ(𝑎). (2.27)

It follows from Figures 1 and 6 that a high-energy system is weakly sensitive to the shape of the potential; the orbit is close to the straight line until it reaches the wall of the potential well. This implies that motion of system (2.23) is similar to the dynamics of a particle moving with constant velocity between two motion-limiters. A connection between the smooth and vibro-impact modes of motion in a smooth nonlinear oscillator has been revealed in [13, 14]; a detailed exposition of this approach can be found in [2, Chapter 6].

The vibro-impact hypothesis suggests that the time 𝑇^∗₁ to reach 𝐴₀ is calculated as 𝑇^∗₁=𝐴₀𝐹. (2.28)

Note that 𝑇^∗₁<𝑇₁, as the vibro-impact approximation ignores the deceleration of motion in the vicinity of 𝐴₀, when the velocity falls below the maximum level 𝐹. Formally, 𝑇^∗₁ can be found from (2.27) by letting Ψ(a) = 0.

3. Analysis of the Transient Dynamics

In what follows, we consider the dynamics of a weakly damped oscillator with strong nonlinearity (𝛼>𝛼^∗); it is often the case of particular interest.

As seen in Figure 7, the damped system exhibits strongly nonlinear behavior on the time interval [0,𝑇^∗₁]; an instant 𝑇^∗₁ corresponds to the first maximum of the function a(𝜏₁). After that motion becomes similar to smooth oscillations about the stationary point. This allows separating the transient dynamics into two stages. While on the interval 0 ≤ 𝜏₁ ≤ 𝑇^∗₁ motion is close to the LPT of the undamped system, at the second stage, 𝜏₁ ≥ 𝑇^∗₁, motion is similar to quasilinear oscillations.

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(b)

Figure 7 

Phase portrait (a) and plot of a(${\tau }_1$) (b) for strongly nonlinear oscillations in a dissipative system.

In the remainder of this section we investigate a segment of the trajectory on the interval [0, 𝑇^∗₁]. The task is to calculate an instant 𝑇^∗₁ and the values a(𝑇^∗₁) and Δ(𝑇^∗₁) determining the starting point for the second interval of motion. For simplicity, we assume that dissipation is sufficiently small and may be ignored on the interval [0, 𝑇^∗₁]. This allows one to approximate the first part of the trajectory by a corresponding segment of the LPT of the nondissipative system.

As mentioned above, the dynamics of a strongly nonlinear oscillator is similar to free motion of a particle moving with constant velocity between two motion-limiters. This allows us to employ the method of nonsmooth transformations [1, 2] in the study of strongly nonlinear oscillations.

At the first step, we introduce nonsmooth functions 𝜏(𝜙) and e(ϕ) = dτ/dϕ defined as follows:2𝜏=𝜏(𝜙)=𝜋|arcsin(sin𝜋𝜙2)|,𝑒(𝜙)=1,0<𝜙≤1,𝑒(𝜙)=−1,1<𝜙≤2, (3.1) where ϕ = Ω𝜏₁, the frequency Ω will be found below. Plots of functions (3.1) are given in Figure 8. In a general setting, the solution of (2.12) is constructed in the form 𝑎(𝜏₁)=𝑋₁(𝜏)+𝑒(𝜙)𝑌₁(𝜏),Δ(𝜏₁)=𝑋₂(𝜏)+𝑒(𝜙)𝑌₂𝑑(𝜏),𝑑𝜏₁𝜕=Ω(𝑒+𝜕𝜕𝜏𝜕𝜙). (3.2)

(a)

(b)

Figure 8 

Functions τ(ϕ) and e(ϕ).

We recall that 𝜕e/∂ϕ = δ(ϕ − n), where 𝛿(𝜙−𝑛) is Dirac’s delta-function, n = 1, 2,…. We exclude δ-singularity by requiring𝑌_1,2=0at𝜏=1,2,…. (3.3) This implies that𝑑𝑎𝑑𝜏₁=Ω(𝑒𝜕𝑋₁+𝜕𝜏𝜕𝑌₁𝜕𝜏),𝑑Δ𝑑𝜏₁=Ω(𝑒𝜕𝑋₂+𝜕𝜏𝜕𝑌₂)𝜕𝜏 (3.4) provided 𝑌_1,2=0 at τ = 1, 2,…. To derive the equations for 𝑋_𝑖, 𝑌_𝑖, i = 1, 2, we insert (3.4) into (2.12) and separate the terms with and without e. This yields the set of equations Ω𝜕𝑌₁𝜕𝜏=−𝐹sin𝑋₂cos𝑌₂,Ω𝜕𝑋₁𝜕𝜏=−𝐹sin𝑌₂cos𝑋₂,Ω(𝑋₁𝜕𝑌₂𝜕𝜏+𝑌₁𝜕𝑋₂𝜕𝜏)+𝑠𝑋₁−3𝛼𝑋³₁−9𝛼𝑋₁𝑌²₁=−𝐹cos𝑌₂cos𝑋₂,Ω(𝑋₁𝜕𝑋₂𝜕𝜏+𝑌₁𝜕𝑌₂𝜕𝜏)=𝐹sin𝑌₂sin𝑋₂. (3.5)

It is easy to prove that (3.5) are satisfied by 𝑌₁ = 0, 𝑋₂ = 0. Under these conditions, the variables 𝑋₁ and 𝑌₂ satisfy the equations similar to (2.12)Ω𝜕𝑋₁𝜕𝜏+𝐹sin𝑌₂=0,Ω𝑋₁𝜕𝑌₂𝜕𝜏+𝑠𝑋₁−3𝛼𝑋³₁+𝐹cos𝑌₂=0, (3.6) with the initial conditions 𝑋₁ = 0, 𝑌₂ = −π/2 at τ = 0. System (3.6) is integrable, yielding the integral of motion similar to (2.14)34𝛼𝑋³₁−𝑠2𝑋₁−𝐹cos𝑌₂=0. (3.7)

Then, it follows from (3.3) and (3.6) that𝜕𝑋₁𝜕𝜏=0at𝑌₂=0,𝜏=1. (3.8) It is worth noting that equalities (3.8) have a clear physical meaning: they represent the condition of maximum of 𝑋₁at 𝑌₂ = 0.

For the further analysis, it is convenient to transfer (3.6) into the second-order form. Using (3.7) to exclude 𝑌₂, the resulting equation and the initial conditions are written as Ω₂𝜕²𝑋₁𝜕𝜏²+𝑓(𝑋₁𝑋)=0,₁=0,Ω𝑑𝑋₁𝑑𝜏=𝐹, (3.9) where f(𝑋₁) is defined by (2.24). A precise solution of (3.9), expressed in terms of elliptic functions, is prohibitively difficult for practical utility [6, 13]. In order to highlight the substantial dynamical features, the solution to (3.6), (3.9) is expressed in terms of successive approximations 𝑋₁=𝑥₀+𝑥₁+⋯,𝑌₂=𝑦₀+𝑦₁+⋯,Ω=Ω₀(1+𝜀₁+⋯), (3.10) where it is assumed that |𝑥₁(𝜏|≪|𝑥₀(𝜏)|,|𝑦₁(𝜏|≪|𝑦₀(𝜏)|,𝜀₁≪1 on an interval of interest. The validity of this assumption will be tested below by numerical simulations. Since the vibro-impact approximation is insensitive to the presence of the potential, the function 𝑥₀ is chosen as the solution of the equation𝜕²𝑥₀𝜕𝜏²=0 (3.11) with the initial conditions x₀ = 0, Ω₀∂𝑥₀/∂τ = F at τ = 0. It follows from (3.11) that 𝑥₀(𝜏)=𝐴₀𝜏,𝐴₀Ω₀=𝐹. (3.12) From (3.1) and the maximum condition we haveΩ₀𝑇^∗₁=1,𝑇^∗₁=Ω₀⁻¹=𝐴₀𝐹,𝑥₀(𝜏)=𝐹𝜏₁,0≤𝜏₁≤𝑇^∗₁,𝑥₀(𝜏(𝑇^∗₁))=𝑥₀(1)=𝐴₀. (3.13)

We now recall that system (3.9) is conservative; it possesses the integral of energy 𝐸₁=12𝑉²₁+1Ω²Ψ(𝑋₁1)=2Ω²𝐹²,𝑉₁=𝑑𝑋₁𝑑𝜏, (3.14) where Ψ(X₁) is defined by (2.24). By analogy with (2.26), we obtainΨ(𝑋₁1)=2𝐹²at𝜏=1. (3.15)

Inserting (3.10), (3.12) into (3.15) and ignoring small terms, we then have the equation to determine 𝐴₀Ψ(𝐴₀1)=2𝐹². (3.16) Given 𝐴₀, we obtain from 𝑇^∗₁ = 𝐴₀/F (see (2.28)).

The approximation x₁ is governed by the following equation: Ω²₀𝜕²𝑥₁𝜕𝜏²=−𝑓(𝑥₀𝑥),₁(𝜏)=−Ω₀^−2𝜏₀(𝜏−𝜉)𝑓(𝐴₀𝜉)d𝜉, (3.17) Given 𝐴₀and Ω₀, formula (3.17) yields𝑥₁𝐴(𝜏)=−⁰𝜏³4Ω²₀[9𝛼²56(𝐴₀𝜏)⁴−3𝑠𝛼5(𝐴₀𝜏)²+𝑠²6]. (3.18)

We now find the function 𝑌₂(𝜏). Arguing as above, we construct 𝑌₂=𝑦₀+𝑦₁, where 𝑦₀ can be found from the first equation (3.6), in which we let 𝑋₁ = 𝑥₀. This yields 𝑦₀𝐹=−arcsin(𝐴₀Ω₀𝜋)=−2,0<𝜏<1. (3.19)

The term 𝑦₁(𝜏) is defined by the second equation (3.6). As before, we take 𝑋₁ = 𝑥₀ and exclude cos𝑌₂ by (3.7) to get𝜕𝑦₁=1𝜕𝜏Ω₀𝑠(−2+94𝛼𝐴²₀𝜏²𝑦),₁(𝜏,𝑡₀1)=Ω₀(−𝑠𝜏2+34𝛼𝐴²₀𝜏³). (3.20) As an example, we calculate the LPT for system (2.12) with the parameters 𝛾=0,𝑠=0.2,𝛼=0.333,𝐹=1 (3.21) and compare the results with the numerical solution for system (2.10), in which 𝛾=0.04.

Calculations by formulas (3.13), (3.16) give 𝐴₀ = 1.67, 𝑇^∗₁ = Ω₀⁻¹=167. Thus we have the maximum M = 𝑋₁(𝑇^∗₁) ≈ 1.67 at 𝑇^∗₁≈ 1.67 in the leading-order approximation and 𝑀₁≈ 1.67 at 𝑇^∗₁≈ 2 for the numerical solution a(𝜏₁) (Figure 9(a)); for system (2.10) with 𝛾=0.05 the numerical solution gives the maximum M ≈ 1.56 at 𝑇^∗₁≈ 2.1. This confirms that small dissipation can be ignored over the interval 0 ≤ τ≤ 𝑇^∗.

(a)

(b)

Figure 9 

LPT of system (2.12) (a) and solution a(${\tau }_1$) of system (2.10) (b) solid line—numerical solution; dashed line—the leading-order approximation $x_0$; dot-and-dash line—the first order approximation $x_0$ + $x_1$.

It is easy to check by a straightforward calculation that the correction x₁ is negligible. In a similar way, one can evaluate the small term 𝑦₁.

4. Quasilinear Oscillations

In this section, we examine quasilinear oscillations on the second interval of motion, τ > 1. It is easy to see that an orbit of the dissipative system tends to its steady state as τ→ ∞. The steady state O: (𝑎₀, Δ₀) for system (2.10) is determined by the equality𝑎²[(𝑠−3𝛼𝑎²)²+𝛾²]=𝐹², (4.1) or, for sufficiently small γ,𝛾𝑎₀=−𝐹sinΔ₀,𝑠𝑎₀−3𝛼𝑎³₀=−𝐹cosΔ₀,Δ₀≈−𝛾𝑎₀𝐹+𝑂(𝛾³),𝑎₀(𝑠−3𝛼𝑎²₀)=−𝐹+𝑂(𝛾²). (4.2)

Let 𝜉=𝑎−𝑎₀,𝛽=Δ−Δ₀ denote deviations from the steady state. In addition, we must impose the matching conditions𝑎₀+𝜉=𝑥^∗₀,𝑑𝜉𝑑𝜏₁=0at𝜏₁=𝑇^∗₁, (4.3) where 𝑥^∗₀=𝑥₀(𝑇^∗₁), 𝑇^∗₁ is determined by (2.28).

We suppose that the contribution of nonlinear force in oscillations near 𝑂 is relatively small. Under this assumption, one can consider the system linearized near 𝑂𝑑𝜉𝑑𝜏₁+𝐹𝛽=−𝛾𝜉,𝑑𝛽𝑑𝜏₁−𝑘₁𝑎₀𝜉=−𝛾𝛽, (4.4) where 𝑘₁=9𝛼𝑎²₀−𝑠. If 𝑘₁>0, the solution of system (4.4) takes the form 𝜉(𝑧)=𝑐₀𝑒^{−𝛾(𝜏1−𝑇∗1)}cos𝜅(𝜏₁−𝑇^∗₁),𝛽(𝑧)=𝑟𝑐₀𝑒^{−𝛾(𝜏1−𝑇∗1)}sin𝜅(𝜏₁−𝑇^∗₁),𝜏₁−𝑇^∗₁>0, (4.5) where we denote 𝑐₀=𝑥^∗₀−𝑎₀, 𝜅²=𝐹𝑘₁/𝑎₀>0, 𝑟=𝜅/𝐹. In particular, taking the parameters (3.21) we find 𝑥^∗₀=1.46, 𝑎₀=1.065, Δ₀=0.1, 𝑘₁=3.2, and, therefore, 𝑐₀=0.395, 𝜅=√3.

Figure 10 demonstrates a good agreement between a numerical solution of (2.10) with parameters (3.21) (solid line) and an approximate solution found by matching the segment (3.12) (dot-and-dash) with the solution (4.5) of the linearized systems (dash) at the point 𝑇^∗₁. Despite a certain discrepancy in the initial interval of motion, the numerical and analytic solutions approach closely as 𝜏₁ increases. This implies that a simplified model (3.12), (3.16) matched with solution (4.5) suffices to describe a complicated near-resonance dynamics.

Figure 10 

Transient dynamics of system (2.10): numerics (solid); segment (3.12) (dot-and-dash); solution (4.5) (dash).

Arguing as above, one can obtain the solution in case 𝑘₁<0. Denoting 𝑘₂=𝐹|𝑘₁|/𝑎₀ and assuming 𝛾≪𝑘, we find a solution similar to (4.5) with cosh(k(𝜏₁−𝑇^∗₁)) and sinh(k(𝜏₁−𝑇^∗₁)) in place of cos(κ(𝜏₁−𝑇^∗₁)) and sin(κ(𝜏₁−𝑇^∗₁)), respectively.

We now correlate numerical and analytic results. As seen in Figure 11, the first maximum of the slowly varying envelope of the process u(t,ε) equals 𝑀₁≈ 1.5; it is reached at the instant 𝑡^∗≈ 20, or 𝑇^∗₁≈ 2; the second maximum 𝑀₂≈ 1.4 is at 𝑡^∗≈ 60, 𝑇^∗₁≈ 6, the third maximum 𝑀₃≈ 1.3 is at 𝑡^∗≈ 100, 𝑇^∗₁≈ 10, and so forth. When these results are compared with that of Figure 10, it is apparent that the numerically constructed envelope is in a good agreement with the asymptotic approximations of the function 𝑎(𝜏₁).

Figure 11 

Numerical integration of (2.1): ε = 0.1, s = 0.2, α = 0.333, F = 1, γ = 0.05.

5. Dynamics of a 2DOF System

5.1. Reduction of a 2DOF System to a Single Oscillator

In this section we present a reduction of the dynamical equations of a 2DOF system to an equation of a single oscillator. The system consists of a linear oscillator of mass 𝑀 (the source of energy) coupled with a mass 𝑚 (an energy sink) by a nonlinear spring with a weak linear component. For brevity, we consider the nonlinear spring with cubic nonlinearity. In this case, the equations of motion and the initial conditions have the following form:𝑀𝑑²𝑥₁𝑑𝑡²+𝛾𝑑𝑥₁𝑑𝑡+𝑘₁𝑥₁+𝑘₃(𝑥₁−𝑥₂)³+𝐷(𝑥₁−𝑥₂)+𝜂(𝑑𝑥₁−𝑑𝑡𝑑𝑥₂𝑚𝑑𝑑𝑡)=0,₂𝑥₂𝑑𝑡²−𝑘₃(𝑥₁−𝑥₂)³−𝐷(𝑥₁−𝑥₂)−𝜂(𝑑𝑥₁−𝑑𝑡𝑑𝑥₂𝑑𝑡)=0, (5.1) with the initial conditions 𝑡=0∶𝑥₁=𝑥₂=0∶𝑑𝑥₁𝑑𝑡=𝑉₀>0,𝑑𝑥₂𝑑𝑡=0. (5.2)

Here 𝑥₁ and 𝑥₂ are the displacements of the masses M and m, respectively; 𝑘₁>0 is the stiffness of linear spring; 𝑘₃>0 is the coefficient of nonlinear coupling between the linear oscillator and the sink; D is the coefficient of linear coupling (𝐷<0 corresponds to a system with multiple states of equilibrium); the coefficients 𝛾and 𝜂 characterize dissipation in the linear oscillator and the coupling, respectively. For simplicity, we let 𝛾= 0. Note that energy transfer cannot be activated in a nonexited system. In the absence of external forcing, it requires nonzero initial conditions for at least a single unit.

In what follows we assume that 𝑚/𝑀=𝜀²≪1. Then, we introduce the dimensionless time variable 𝜏₀ = 𝜔₀t, where 𝜔₀ = 𝑘₁/𝑀. In these notations, system (5.1) becomes 𝑑²𝑥₁𝑑𝜏²₀+𝑥₁+𝜀²𝑐(𝑥₁−𝑥₂)³+𝜀³𝑑(𝑥₁−𝑥₂)+𝜀³𝜂(𝑑𝑥₁𝑑𝜏₀−𝑑𝑥₂𝑑𝜏₀𝑑)=0,₂𝑥₂𝑑𝜏²₀−𝑐(𝑥₁−𝑥₂)³−𝜀𝑑(𝑥₁−𝑥₂)−𝜀𝜂(𝑑𝑥₁𝑑𝜏₀−𝑑𝑥₂𝑑𝜏₀𝜏)=0,₀=0∶𝑥₁=𝑥₂=0;𝑣₁=𝜀𝑣₀,𝑣₂=0, (5.3) where we denote𝜀²𝑘𝑐=³𝑘₁,𝜀³𝐷𝑑=𝑘₁,𝜀³𝜂𝜂=𝑘₁𝑀,𝜀𝑣₀=𝑉₀𝜔₀,𝑣𝑖=𝑑𝑥_𝑖𝑑𝜏₀. (5.4)

In addition, we consider the relative displacement u = 𝑥₂ – 𝑥₁, 𝑑𝑢/𝑑𝜏₀ = v satisfying the equation𝑑²𝑢𝑑𝜏²₀+𝑑²𝑥₁𝑑𝜏²₀+𝜀𝜂𝑑𝑢𝑑𝜏₀+𝑐𝑢³+𝜀𝑑𝑢=0, (5.5) with the initial conditions 𝜏₀ = 0: u = 0, v = −ε𝑣₀. Using (5.3), (5.5), the variable 𝑥₁ can be excluded. We recall that oscillations in the damped system vanish at rest 𝑂₁: (𝑥₁ = 𝑥₂ = 0, 𝑣₁ = 𝑣₂ = 0) as 𝜏₀→∞. This implies that the effect of dissipation, whatever small it might be, must be considered in the approximate solution; otherwise, the convergence to 𝑂₁is ignored. Hence the solution of the first equation (5.3) should be written as𝑥₁(𝜏₀)=𝜀𝑣₀ℎ_𝜀(𝜏₀)sin𝜏₀−𝜀²𝑐^𝜏0₀ℎ_𝜀(𝜏₀−𝑠)sin(𝜏₀−𝑠)𝑢³(𝑠)𝑑𝑠+𝜀³ℎ…,_𝜀(𝜏₀)=𝑒^{−(𝜀3𝜂/2)𝜏0}, (5.6) and, by (5.6),𝑑²𝑥₁𝑑𝜏²₀=−𝜀𝑣₀ℎ_𝜀(𝜏₀)sin𝜏₀−𝜀²𝑐𝑢³(𝜏₀)+𝜀²𝑐^𝜏0₀ℎ_𝜀(𝜏₀−𝑠)sin(𝜏₀−𝑠)𝑢³(𝑠)𝑑𝑠+𝜀³⋯. (5.7) Next, we insert (5.7) into (5.5) and ignore small terms insubstantial for the asymptotic analysis. As a result, we obtain the following equation:𝑑²𝑢𝑑𝜏²₀+𝜀𝜂𝑑𝑢𝑑𝜏₀+𝑐𝑢³+𝜀𝑑𝑢=𝜀𝑣₀ℎ_𝜀(𝜏₀)sin𝜏₀−𝜀𝑐_𝜀(𝜏₀𝑐),_𝜀(𝜏₀)=𝜀𝑐𝐼_𝜀,𝐼_𝜀=^𝜏0₀ℎ_𝜀(𝜏₀−𝑠)sin(𝜏₀−𝑠)𝑢³(𝑠)𝑑𝑠, (5.8) with the initial conditions u(0) = 0, v(0) = −ε𝑣₀. We will show that, under conditions of 1:1 resonance, 𝐼_𝜀(𝜏₀) ~ 𝜀⁻¹, c_ε(𝜏₀) ~ 1; this means that the integral term should be taken into consideration in the asymptotic analysis.

Formula (5.8) represents a nonhomogeneous integro-differential equation with respect to u, that is above transformations reduce the original 2DOF system to a single oscillator of a more complicated structure. The initial condition v(0) = −ε𝑣₀ implies that initially the system is close to rest, and the trajectory of system approaches the LPT of (5.8). Thus the task is to construct the LPT for the integro-differential equation (5.8).

5.2. Equations of the Resonance Dynamics

To study the system subject to 1:1 resonance, we rewrite (5.8) in the form𝑑𝑢𝑑𝜏₀−𝑣=0,𝑑𝑣𝑑𝜏₀+(1+2𝜀𝜎)𝑢+𝜀𝜇(𝑐𝑢³−𝑢)+𝜀(𝜂𝑣−𝑣₀ℎ_𝜀(𝜏₀)sin𝜏₀)+𝜀𝑐_𝜀(𝜏₀)=0,𝑢(0)=0,𝑣(0)=−𝜀𝑣₀. (5.9) In (5.9), we denote 𝜇=1/𝜀, 𝜎=𝑑/2. The resonance conditions imply that the parenthetical expression with factor 𝜀𝜇 is relatively small compared to all other terms of order 1.

As in Section 2, we use complex-valued transformations (2.5) and the multiple time-scale method. Inserting (2.5) into (5.9), we then have 𝑑𝜓𝑑𝜏₀𝑐−𝑖(1+𝜀𝜎)𝜓+𝜀𝜇𝑖[8(𝜓−𝜓^∗)³+12𝜓−𝜓^∗]𝜂+𝜀(2(𝜓+𝜓^∗)−𝑖𝜎𝜓^∗−𝑣₀ℎ_𝜀(𝜏₀)sin𝜏₀)−𝜀𝐶_𝜀(𝜏₀𝐶)=0,_𝜀(𝜏₀𝑐)=𝜀𝑖8^𝜏0₀ℎ_𝜀(𝜏₀−𝑠)sin(𝜏₀−𝑠)(𝜓−𝜓^∗)³𝑑𝑠 (5.10) Then we construct an approximate solution of (5.10) in terms of expansions (2.5) with the slow and fast tine scales 𝜏₀ = t, 𝜏₁ = εt, respectively.

For the resonance effect to be considered in a proper way, the leading-order equation and its solution should be𝜕𝜓₀𝜕𝜏₀−𝑖(1+𝜀𝜎)𝜓₀𝜓=0,₀(𝜏₀,𝜏₁)=𝜑₀(𝜏₁)𝑒^{𝑖(1+𝜀𝜎)𝜏0}. (5.11)

The function 𝜑₀(𝜏₁) can be found from the equation𝜕𝜓₁𝜕𝜏₀+𝑑𝜑₀𝑑𝜏₁𝑒^{𝑖(1+𝜀𝜎)𝜏0}−𝑖(1+𝜀𝜎)𝜓₁𝑐+𝑖𝜇[8(𝜓₀−𝜓^∗₀)³+12𝜓₀−𝜓^∗₀]+𝜂𝜓+𝜓^∗2−𝑖𝜎𝜓^∗−𝑣₀ℎ_𝜀(𝜏₀)sin𝜏₀+𝐶_0𝜀(𝜏₀)=0, (5.12) where 𝐶_0𝜀(𝜏₀)=𝜀𝑖(𝑐/8)∫^𝜏0₀ℎ_𝜀(𝜏₀−𝑠)sin(𝜏₀−𝑠)(𝜓₀−𝜓^∗₀)³𝑑𝑠.

To avoid secularity, we separate the resonance terms including𝑒^{𝑖(1+𝜀𝜎)𝜏0}and then equate the sum to zero. First, we evaluate 𝐶_0𝜀(𝜏₀). To do so, we present the cubic term as (𝜓₀−𝜓^∗₀)³ = −3|𝜑₀|²𝜑₀𝑒^{𝑖(1+𝜀𝜎)𝜏0} + nonresonance terms, and then write 𝐶_0𝜀(𝜏₀) in the form 𝐶_0𝜀(𝜏₀)=−𝜀3𝑐16|𝜑₀(𝜏₁)|²𝜑₀(𝜏₁)^𝜏0₀𝑒^{−(𝜀3𝜂/2)(𝜏0−𝑠)}(𝑒^{𝑖(𝜏0−𝑠)}−𝑒^{−𝑖(𝜏0−𝑠)})𝑒^{𝑖(1+𝜀𝜎)𝑠}𝑑𝑠+⋯=−𝜀3𝑐16|𝜑₀(𝜏₁)|²𝜑₀(𝜏₁)[𝑆_1𝜀(𝜏₀)−𝑆_2𝜀(𝜏₀)]+⋯, (5.13) where the nonresonance terms are omitted. Here we denote 𝑆_1𝜀(𝜏₀)=^𝜏0₀𝑒^{−(𝜀3𝜂/2)(𝜏0−𝑠)}𝑒^{𝑖(𝜏0−𝑠)}𝑒^{𝑖(1+𝜀𝜎)𝑠}𝑒𝑑𝑠=^{𝑖(1+𝜀𝜎)𝜏0}𝜀(𝑖𝜎+𝜀²𝜂)(1−𝑒^{−[𝑖𝜎+(𝜀3𝜂/2)]𝜏0}𝑆),_2𝜀(𝜏₀)=^𝜏0₀𝑒^{−(𝜀3𝜂/2)(𝜏0−𝑠)}𝑒^{−𝑖(𝜏0−𝑠)}𝑒^{𝑖(1+𝜀𝜎)𝑠}𝑑𝑠=𝑒^{[𝑖−(𝜀3𝜂/2)]𝜏0𝜏0}₀𝑒^{𝜀[𝑖(2+𝜎)+𝜀2𝜂/2]𝑠}𝑑𝑠∼𝑂(1). (5.14)

Ignoring 𝑆₂ compared with 𝑆₁, we calculate𝐶_0𝜀(𝜏₀)=−𝑖(3𝑐16𝜎)|𝜑₀(𝜏₁)|²𝜑₀(𝜏₁)(1−ℎ_1𝜀(𝜏₁)𝑒^{−𝑖𝜎𝜏1})𝑒^{𝑖(1+𝜀𝜎)𝜏0}ℎ+nonresonanceterms,_1𝜀(𝜏₁)=𝑒^{−(𝜀2𝜂/2)𝜏1}. (5.15)(5.16) If we sum the resonance constituents in all other terms of (5.12) and then equate the total sum to zero, we obtain the equation𝜕𝜑₀𝜕𝜏₁+𝑖𝜇(−3𝑐8|𝜑₀|²𝜑₀+12𝜑₀𝜂)+2𝜑₀+𝑖2𝑣₀ℎ_1𝜀(𝜏₁)𝑒^{−𝑖𝜎𝜏1}−𝑖3𝑐16𝜎|𝜑₀(𝜏₁)|²𝜑₀(𝜏₁)(1−ℎ_1𝜀(𝜏₁)𝑒^{−𝑖𝜎𝜏1})=0. (5.17)

Then we insert the polar representation 𝜑₀=𝑎𝑒^𝑖𝛿 into (5.17) and set separately the real and imaginary parts of the resulting equations equal to zero. In these transformations, the last term in (5.17) can be omitted if 2μσ = μd >1. Under this assumption, we obtain𝑑𝑎𝑑𝜏₁=−𝛾𝑎−ℎ_1𝜀(𝜏₁𝑎)𝐹sinΔ,𝑑Δ𝑑𝜏₁=𝜇𝑎(−𝑠+𝛼𝑎²)−ℎ_1𝜀(𝜏₁)𝐹cosΔ, (5.18) where Δ = δ+σ𝜏₁, and σ = 𝑑/2, γ= 𝜂/2, α =3𝑐/8, s =(1/2)(1 − εd), F = 𝑣₀/2.

By analogy with (2.16), we accept the initial conditions𝜋𝑎(0)=0,Δ(0)=−2. (5.19)

In the absence of dissipation (η = 0), system (5.18) takes the form𝑑𝑎𝑑𝜏₁𝑎=−𝐹sinΔ,𝑑Δ𝑑𝜏₁=𝜇𝑎(−𝑠+𝛼𝑎²)−𝐹cosΔ, (5.20) which is very similar to (2.10). Critical parameters, stable centers, and the LPT for (5.20) are derived in the same way as in Section 3. However, as mentioned above, steady state positions of dissipated systems (2.10) and system (5.18) are different. This reflects the fact that, while (2.10) is subjected to persistent harmonic excitation, the effect of an initial impulse exponentially decreases with time. Therefore, the first segment of the trajectory (5.18) can be approximated by a corresponding solution of (5.20) but the second segments convergences to zero as 𝜏₁ = ∞.

5.3. Dynamical Analysis of the Oscillator

5.3.1. Beating in a Nondissipative System

We now compare analytic and numerical results. We recall that numerical and experimental studies [3, 4] have shown the effective energy exchange for sets of parameters. Following [3, 4], we choose 𝜀=0.316(𝜀²=0.1), c = 0.8, d = 0.632 (εd = 0.2), 𝑣₀ = 2.215 (𝜀𝑣₀ = 0.7). Note that in this case μd = 2, and integral terms in (5.17) can be ignored.

The LPT (Figure 12(a)) is calculated as a solution of (5.20) with initial conditions a(0) = 0, Δ(0) = −π/2. One can see that the LPT approximates the envelope of the process 𝑢₂(𝜏₀) (Figure 13) with an error about 10%. Thus we may conclude that the observed intense energy exchange is associated with motion over the LPT.

(a)

(b)

Figure 12 

LPT of system (5.20) (a) and beating solution u(${\tau }_0$) of (5.8).

Figure 13 

Plots of $x_1$(${\tau }_0$) (solid) and $x_2$(${\tau }_0$) (dash).

We now calculate the critical parameter 𝛼^∗₁. Here we have 𝜇² = 10, F = 1.1075, εd = 0.2, s = 0.4, and, by (2.20)𝛼^∗₁=38,𝑐^∗₁=4𝜇²𝑠³27𝐹²,𝑐^∗₁=32𝜇²𝑠³81𝐹²=0.205. (5.21)

Since the accepted value 𝑐0.8>𝑐^∗₁=, the resonance exchange takes place in the domain of large oscillations demonstrated in Figure 1.

5.3.2. The Transient Dynamics

Here we consider the resonance dynamics of a weakly dissipated system. We recall that, if 𝜏₁→ ∞, the transient process in system (5.18) does not turn into a stationary process; it vanishes at rest O: {𝑎=0}. To describe transient oscillations, we use the same arguments as in Section 3. The trajectory is separated into two parts; motion is assumed to be close to strongly nonlinear undamped oscillations over the interval [0, 𝜏^∗₁]; then the orbit approaches the point O with an exponentially decreasing amplitude of oscillations.

We note that the contribution of nonlinear force in oscillations near O is negligible. In particular, this implies that the resonance construction of Section 5 is incorrect if 𝜏₁ is large enough. In order to present the linearised dynamics, one can consider system (5.3) linearized near the point 𝑂₁: (𝑥₁ = 𝑥₂ = 0, 𝑣₁ = 𝑣₂ = 0). Let 𝜉₁ and 𝜉₂ denote small deviations of 𝑥₁, 𝑥₂ from 𝑂₁. A corresponding linearized system takes the form 𝑑²𝜉₁𝑑𝜏²₀+𝜉₁+𝜀³[𝑑(𝜉₁−𝜉₂)+𝜂(𝑑𝜉₁−𝑑𝑡𝑑𝜉₂𝑑𝑑𝑡)]=0,₂𝜉₂𝑑𝜏²₀−𝜀𝑑(𝜉₁−𝜉₂)−𝜀𝜂(𝑑𝜉₁−𝑑𝑡𝑑𝜉₂𝑑𝑡)=0. (5.22)

In addition, we impose the matching constraints at 𝜏₀ = 𝜏^∗₀ = 𝜏^∗₁/𝜀, namely, 𝜉_𝑖=𝑥_𝑖,𝑑𝜉_𝑖𝑑𝜏₀=𝑑𝑥_𝑖𝑑𝜏₀, (5.23) where 𝑥_𝑖 denotes the corresponding solution over [0, 𝜏^∗₀].

Formally, one can reproduce above transformations and reduce system (5.22) to a single integro-differential equation. However, a system of low dimensionality enables a straightforward investigation.

By definition, 𝑥₁(𝜏^∗₀) ≈ 0 but |𝑥₂(𝜏^∗₀)| is close to maximum, that is, d𝑥₂/d𝜏₀ = d𝜉₂/d𝜏₀≈ 0 at 𝜏^∗₀. The solution can be presented in the form𝜉₁(𝜏₀)=𝜀𝑣₀𝑒^{−𝜀3𝜂(𝜏0−𝜏∗0)/2}sin(𝜏₀−𝜏^∗₀𝜉),₂(𝜏₀)=𝐶₀𝑒^{−𝜀𝜂(𝜏0−𝜏∗0)/2}cos[√𝜀𝑑(𝜏₀−𝜏^∗₀)]+𝜀…, (5.24) where 𝜉₁(𝜏₀) is defined by (5.6) with the negligible integral term (see discussion after (5.17)); 𝜉₂(𝜏₀) is obtained as a solution of the second equation in (5.22) provided 𝜉₁(𝜏₀)∼𝑂(𝜀). It is seen from (5.24) that the amplitude of 𝜉₁(𝜏₀) is much lesser than the amplitude of 𝜉₂(𝜏₀) but the period and the rate of decay for 𝜉₂(𝜏₀) vastly exceed similar parameters for 𝜉₁(𝜏₀)). Omitting detailed calculation, we note that the results of numerical simulation (Figure 14) agree with the conclusions following from (5.23), (5.24). The parameters of simulations are 𝜀² = 0.1, c = 0.8, εd = 0.2, εη = 0.2, 𝜀𝑣₀ = 0.7.

Figure 14 

Plots of $x_1$(t) (solid) and $x_2$(t) (dash) in the damped system.

6. Conclusion

In this paper, we have extended the concept of the limiting phase trajectories (LPTs) to dissipative oscillatory systems. Using this concept, we have constructed an approximate solution describing the maximum energy exchange between coupled oscillators. The solution consists of two parts: on an initial interval, the trajectory is close to the LPT of the undamped system; then motion becomes similar to quasilinear oscillations, approaching an asymptotically stable state. We have demonstrated a good agreement between numerical and approximate analytical solutions for two typical examples, namely, the Duffing oscillator with harmonic excitation and a system of two coupled oscillators excited by an initial impulse. In addition, we have developed a procedure for reducing a 2DOF system to a single oscillator. This allows us to obtain an approximate analytic solution describing the energy exchange and beating with complete energy transfer in a 2DOF system.

Acknowledgments

Partial financial support from the Russian Foundation for Basic Research (Grants 08-01-00068), the U.S. Civilian Research & Development Found (grant CGP no. 2920), and the Russian Academy of Sciences (Program 4/OX-08) is acknowledged with thanks. The authors are grateful to D.S. Shepelev for his help in numerical simulations.