^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

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.

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, [

The limiting phase trajectory (LPT) has been introduced [

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

In the remainder of the paper (Section

We investigate the transient response of the Duffing oscillator in the presence of resonance 1:1. The dimensionless equation of motion is

In order to describe the nonlinear dynamics, we invoke a complex-valued transformation [

Applying the multiple scales method [

In order to avoid the secular growth of

In this section, we recall main definitions and results concerning the dynamics of the nondissipative system. In the absence of damping, system (

It is easy to prove that system (

Formula (

Equality (^{+}, except as otherwise noted.

Next we determine critical parameters of system (

A straightforward investigation proves that, if _{+}: (0,

Phase portrait (a) and plot of _{1}

Passage from small to large oscillations:

Phase portrait (a) and plot of

We now suppose that

Phase portrait (a) and plot of

Figures

If _{−}. Figure

For the further analysis, it is convenient to reduce the equation of the LPT to the second-order form. Using (

Figure

Potential

Phase portraits of (

The amplitude of oscillations

From (

It follows from Figures

The vibro-impact hypothesis suggests that the time

Note that

In what follows, we consider the dynamics of a weakly damped oscillator with strong nonlinearity (

As seen in Figure

Phase portrait (a) and plot of

In the remainder of this section we investigate a segment of the trajectory on the interval [0,

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 [

At the first step, we introduce nonsmooth functions

Functions

We recall that

It is easy to prove that (

Then, it follows from (

For the further analysis, it is convenient to transfer (_{0} = 0,

We now recall that system (_{1}) is defined by (

Inserting (

The approximation _{1} is governed by the following equation:

We now find the function

The term

Calculations by formulas (

LPT of system (

It is easy to check by a straightforward calculation that the correction _{1} is negligible. In a similar way, one can evaluate the small term

In this section, we examine quasilinear oscillations on the second interval of motion,

Let

We suppose that the contribution of nonlinear force in oscillations near

Figure

Transient dynamics of system (

Arguing as above, one can obtain the solution in case

We now correlate numerical and analytic results. As seen in Figure

Numerical integration of (

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

Here

In what follows we assume that

In addition, we consider the relative displacement _{ε}

Formula (

To study the system subject to 1:1 resonance, we rewrite (

As in Section

For the resonance effect to be considered in a proper way, the leading-order equation and its solution should be

The function

To avoid secularity, we separate the resonance terms including

Ignoring

Then we insert the polar representation

By analogy with (

In the absence of dissipation (

We now compare analytic and numerical results. We recall that numerical and experimental studies [

The LPT (Figure

LPT of system (

Plots of

We now calculate the critical parameter

Since the accepted value

Here we consider the resonance dynamics of a weakly dissipated system. We recall that, if

We note that the contribution of nonlinear force in oscillations near

In addition, we impose the matching constraints at

Formally, one can reproduce above transformations and reduce system (

By definition,

Plots of

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.

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.