# Energy Pumping in Nonlinear Mechanical Oscillators: Part I-Dynamics of the Underlying Hamiltonian Systems

Oleg V. Gendelman, Leonid I Manevitch, Alexander F Vakakis, Robert M'Closkey

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Oleg V. Gendelman, Leonid I Manevitch, Alexander F Vakakis, Robert M'Closkey. Energy Pumping in Nonlinear Mechanical Oscillators: Part I-Dynamics of the Underlying Hamiltonian Systems. Journal of Applied Mechanics, 2001, 68 (1), pp.34-41. 10.1115/1.1345524. hal-04077677

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# Energy Pumping in Nonlinear Mechanical Oscillators: Part I—Dynamics of the Underlying Hamiltonian Systems

O. Gendelman, L. I. Manevitch

Institute of Chemical Physics, Russian Academy of Sciences, Moscow, Russia

A. F. Vakakis

Department of Mechanical and Industrial Engineering, University of Illinois, Urbana

R. M'Closkey

Department of Mechanical and Aerospace Engineering, University of California, Los Angeles

The systems considered in this work are composed of weakly coupled, linear and essentially nonlinear (nonlinearizable) components. In Part I of this work we present numerical evidence of energy pumping in coupled nonlinear mechanical oscillators, i.e., of one-way (irreversible) ‘‘channeling’’ of externally imparted energy from the linear to the nonlinear part of the system, provided that the energy is above a critical level. Clearly, no such phenomenon is possible in the linear system. To obtain a better understanding of the energy pumping phenomenon we first analyze the dynamics of the underlying Hamiltonian system (corresponding to zero damping). First we reduce the equations of motion on an isoenergetic manifold of the dynamical flow, and then compute subharmonic orbits by employing nonsmooth transformation of coordinates which lead to nonlinear boundary value problems. It is conjectured that a 1:1 stable subharmonic orbit of the underlying Hamiltonian system is mainly responsible for the energy pumping phenomenon. This orbit cannot be excited at sufficiently low energies. In Part II of this work the energy pumping phenomenon is further analyzed, and it is shown that it is caused by transient resonance capture on a 1:1 resonance manifold of the system.

## 1 Introduction

In this and a companion paper we study nonlinear energy pumping in coupled mechanical oscillators. By this terminology, we denote the controlled spatial transfer of vibrational energy from the point of its initial generation to a different (predetermined) point where it eventually localizes. In essence, the energy pumping phenomenon corresponds to the controlled one-way channeling of the vibrational energy to a passive nonlinear ‘‘sink’’ where it localizes and diminishes in time due to damping dissipation. There exist numerous studies in the literature on ‘‘static’’ mode localization, spatial motion confinement, and on energy transfer due to internal resonances in coupled mechanical oscillators. The nonlinear energy pumping phenomenon discussed herein is a distinct nonlinear mechanism of energy transfer since it is realized through resonance capture ([1]).

Linear and nonlinear passive ‘‘static’’ mode localization and spatial motion confinement in periodic and nonperiodic coupled oscillators have been studied extensively in the literature ([2–9]). In these studies, linear and nonlinear standing wave motions were analyzed in ordered and disordered periodic coupled oscillators, and the existence of spatially localized free and forced standing waves was rigorously proven by means of theoretical, numerical, and experimental techniques. The standing wave localization considered in these previous works can be classified as ‘‘static’’ since it does not involve any controlled spatial transfer (transition) of energy through the system; indeed linear or nonlinear mode localization can be realized through appropriate selection of the initial conditions of the system, and does not involve any spatial ‘‘flow’’ of energy through the system.

Nonlinear transfer of energy between nonlinear modes in inter-nal resonance has also been studied extensively (cf. [10]). In addition, as recently shown by Nayfeh and co-workers, under certain conditions energy transfer from high to low-frequency modes of a weakly nonlinear structure can also occur ([11]). However, these nonlinear energy exchanges are solely due to modal interactions and do not necessarily involve controlled, one-way spatial transfer of energy through the system.

To the authors’ best knowledge the only previous study of the nonlinear energy pumping phenomenon is the one by Gendelman [12]. In that work a system of two weakly coupled oscillators, a linear and an essentially (nonlinearizable) nonlinear one, was considered. Pumping of energy was demonstrated numerically by showing that, under certain conditions, energy initially imparted in the linear oscillator transfers to the essentially nonlinear one, even though this later oscillator is not directly excited. However, no rigorous analysis and explanation of this phenomenon is given in that work.

In Part I of this work we present numerical evidence of energy pumping in two and three-degrees-of-freedom coupled oscillators with essential nonlinearities and weak viscous damping. We then focus (for simplicity) in the two-degrees-of-freedom case, and analyze systematically the bifurcation structure of the free nonlinear periodic orbits of the underlying Hamiltonian system with no damping. We show that the occurrence (or lack of) energy pumping can be explained by considering the 1-1 and higher order resonant orbits of the Hamiltonian system. A direct analysis of the energy pumping phenomenon is carried in Part II by transforming the damped equations of motion using the action-angle variables of the underlying Hamiltonian system. We show that energy
pumping is a resonance capture phenomenon on a 1-1 resonant manifold, and construct analytical approximations of energy pumping.

## 2 Nonlinear Energy Pumping: Numerical Evidence

Consider the following two-degrees-of-freedom system composed of two weakly coupled and weakly damped oscillators:

$$ 
\begin{aligned}
\ddot{y}_1 + \varepsilon\lambda\dot{y}_1 + C y_1^3 + \varepsilon(y_1 - y_2) &= 0 \\
\ddot{y}_2 + \varepsilon\lambda\dot{y}_2 + \omega_2^2 y_2 + \varepsilon(y_2 - y_1) &= 0
\end{aligned}
\quad (1) $$

Weak coupling is assured by requiring that $\varepsilon \ll 1$, and all other variables are assumed to be $O(1)$ quantities; dots denote differentiation with respect to the independent variable $t$ (time). For $\varepsilon=0$ the system decomposes into two uncoupled nonlinear and linear oscillators, labeled “Oscillators 1 and 2,” respectively. We note that oscillator 1 is essentially nonlinear (nonlinearizable).

In Fig. 1 we depict the transient responses of the two oscillators for $\lambda=0.5$, $\omega_1^2=0.9$, $C=5.0$, $\varepsilon=0.1$, and initial conditions $y_1(0) = y_2(0)=0$, $\dot{y}_1(0)=0$, $\dot{y}_2(0)=\sqrt{2}h$, where $h$ (the energy of the system at $t=0+$) varies; these initial conditions correspond to impulsive excitation of oscillator 2 at $t=0$. For $h=0.5$ (cf. Fig. 1(a)) both oscillators perform damped free oscillations and no energy pumping occurs, since most energy is stored in the directly excited oscillator 2. By increasing the initial energy level to $h=0.8$ and 1.1.25 (cf. Figs. 1(b,c)), it is observed that energy transfer from the directly excited oscillator 2 to the unexcited oscillator 1 takes place; indeed, after an initial transient state most of the vibrational energy is irreversibly transferred (“pumped”) to oscillator 1. By further increasing the initial energy level the energy pumping phenomenon becomes less pronounced. This numerical simulation indicates that, for fixed system parameter values, energy pumping in the weakly coupled system takes place, above a specific value of the initial energy level (strength of the excitation).

Similar results are obtained for the three-degrees-of-freedom system governed by

$$ 
\begin{aligned}
\ddot{y}_1 + \varepsilon\lambda\dot{y}_1 + C y_1^3 + \varepsilon(y_1 - y_2) &= 0 \\
\ddot{y}_2 + \varepsilon\lambda\dot{y}_2 + \omega_2^2 y_2 + \varepsilon(y_2 - y_1) + d(y_2 - y_3) &= 0 \\
\ddot{y}_3 + \varepsilon\lambda\dot{y}_3 + \omega_2^2 y_3 + d(y_3 - y_2) &= 0
\end{aligned}
\quad (2) $$

representing two strongly coupled linear oscillators that are weakly attached to an essentially nonlinear oscillator. In Fig. 2 we depict the transient response of this system for $\lambda=0.5$, $\omega_2^2=0.9$, $C=5.0$, $d=1.0$, $\varepsilon=0.1$, and zero initial conditions except $\dot{y}_3(0) \neq 0$. Whereas for low excitation no energy pumping to the nonlinear oscillator occurs (cf. Fig. 2(a)), as $\dot{y}_3(0)$ increases energy pumping takes place (cf. Figs. 2(b,c)). Hence the nonlinear energy pumping phenomenon can also be realized in multi-degree-of-freedom systems.

We now focus exclusively in the two-degrees-of-freedom system (1). Considering the transient responses depicted in Fig. 1 we note that, when energy pumping occurs the motion can be divided into two phases: In the initial phase energy is rigorously pumped from oscillator 2 to oscillator 1 in a one-way (irreversible) transfer, until oscillator 1 reaches a certain amplitude of oscillation; in the second phase of the motion, both oscillators perform decaying oscillations due to damping dissipation with oscillator 1 retaining most of the vibrational energy. Moreover, during the initial energy pumping phase (defined approximately for $0<t<40$ in Fig. 1(b), and $0<t<60$ for Fig. 1(c)), the motion of oscillator 1 is composed of a “fast” oscillation with frequency nearly identical to the natural frequency of oscillator 2, and a “slow” envelope oscillation. This strongly suggests that a 1-1 internal resonance between oscillators 1 and 2 plays an important role in the energy pumping phase, although this still does not explain the one way energy pumping from oscillator 2 to oscillator 1.

<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 1 Numerical transient responses $y_1(t)$ and $y_2(t)$ of system (1) for (a) $h=0.5$, (b) $h=0.8$, (c) $h=1.125$; — oscillator 1, — oscillator 2</figcaption></figure>

Motivated by these observations we now proceed to examine the periodic orbits of the underlying Hamiltonian system by eliminating damping from Eqs. (1). Since system (1) is weakly damped, one expects that, at least at the initial stages of the motion, the dynamics will be greatly influenced by the dynamics of the corresponding (undamped) Hamiltonian system. In turn, the undamped dynamics are dominated by periodic orbits. As a result, we expect that the topological structure of the periodic orbits (and their bifurcations) of the Hamiltonian system, will play a dominant role in the energy pumping phenomenon.
<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 2 Numerical transient response of the three-degrees-of-freedom system for (a) $\dot{y}_3(0)=2.0$, (b) $\dot{y}_3(0)=3.0$, and (c) $\dot{y}_3(0)=4.0$; — oscillator 1, · · · · oscillator 2, - - - oscillator 3</figcaption></figure>

3 Periodic Orbits of the Underlying Hamiltonian System

The underlying Hamiltonian two-degrees-of-freedom system is obtained by setting $\lambda=0$ in (1). At a fixed level of energy (Hamiltonian) we employ the reduction method outlined in [13] to reduce the undamped system (1) to a single-degree-of-freedom nonautonomous oscillator with periodic forcing. This is a standard reduction process by which an $(n+1)$-degree-of-freedom Hamiltonian system with symmetry (symmetry of time translations) is reduced to an $n$-degree-of-freedom nonautonomous system with no symmetry.

Introducing the action-angle variables $(I_2, \theta_2) \in (R^+ \times S^1)$ for oscillator 2 defined by the relations $y_2 = \sqrt{2I_2/\omega_2} \sin \theta_2$, $v_2 = \sqrt{2I_2/\omega_2} \cos \theta_2$, the Hamiltonian of the undamped system is expressed as

$$H^e = F(y_1, v_1) + G(I_2) + \frac{\varepsilon}{2} H^1(y_1, v_1, \theta_2, I_2) \quad (3)$$

where

$$F(y_1, v_1) = (v_1^2/2) + (Cy_1^4/4), \quad G(I_2) = \omega_2 I_2, \\ H^1(y_1, v_1, \theta_2, I_2) = (y_1 - \sqrt{2I_2/\omega_2} \sin \theta_2)^2.$$

The equations of motion can then be placed in the following form:

$$\begin{aligned} \dot{y}_1 &= \frac{\partial F}{\partial v_1} + \frac{\varepsilon}{2} \frac{\partial H^1}{\partial v_1}, & \dot{v}_1 &= -\frac{\partial F}{\partial y_1} - \frac{\varepsilon}{2} \frac{\partial H^1}{\partial y_1}, \\ \dot{\theta}_2 &= \omega_2 + \frac{\varepsilon}{2} \frac{\partial H^1}{\partial I_2}, & \dot{I}_2 &= -\frac{\varepsilon}{2} \frac{\partial H^1}{\partial \theta_2} \end{aligned} \quad (4)$$

where $v_1 = \dot{y}_1$. By fixing the Hamiltonian (total energy) to a constant level $h$, we can express the action $I_2$ in terms of the other variables of the system as follows:

$$\begin{aligned} H^e &= F(y_1, v_1) + G(I_2) + \frac{\varepsilon}{2} H^1(y_1, v_1, \theta_2, I_2) \\ &= h \Rightarrow I_2 = L^e(y_1, v_1, \theta_2, h) \end{aligned} \quad (5)$$

where $L^e$ is a complicated expression. As a reviewer pointed out, the inversion (5) is only possible if the system is nonsingular, i.e., if the condition $\partial H^e / \partial I_2 \neq 0$ is satisfied; clearly, this is the case in our problem. Taking into account (5), eliminating the time variable from (4), and combining the resulting first-order expressions into a single second-order one we obtain the reduced oscillator

$$\begin{aligned} y_1'' + (C/\omega_2^2)y_1^3 &= \varepsilon \frac{1}{2\omega_2^4} \left\{ \begin{aligned} & -2\omega_2^2 y_1 + 4Cy_1^3 \sin^2 \theta_2 \\ & + \frac{\omega_2(4h - 5Cy_1^4)\sin \theta_2}{\sqrt{2h - \frac{Cy_1^4}{2} - \omega_2^2 y_1'^2}} \\ & + \frac{\omega_2 y_1 \cos \theta_2}{\sqrt{2h - \frac{Cy_1^4}{2} - \omega_2^2 y_1'^2}} \end{aligned} \right\} \\ &= \varepsilon g(y_1, y_1', \theta_2) + O(\varepsilon^2). \end{aligned} \quad (6)$$

We note that the derived expression is approximate since it neglects $O(\varepsilon^2)$ terms; this approximation was imposed by the impossibility of finding an exact expression for $L^e$ in (5). As a result, the following analysis is valid only for undamped systems (1) with sufficiently weak coupling. In (6), $y_1 = y_1(\theta_2)$, and primes denote differentiation with respect to $\theta_2$. In addition, the “forcing term” on the right-hand side is $2\pi$-periodic in $\theta_2$. This completes the reduction process.

Employing the previous analysis, the problem of computing the periodic orbits of the undamped system (1) is equivalent to the problem of computing the periodic solutions of the reduced system (6). This equivalence holds since a periodic motion in $y_1$ and $v_1$ under a periodic change in $\theta_2$ leads to a periodic motion for $I_2$ as well. Since the reduced system is essentially nonlinear we resort to an analytical/numerical technique to compute the periodic orbits and their bifurcations. In particular, we introduce a nonsmooth transformation of variables to transform the problem to a
set of nonlinear boundary value problems over finite domains. This technique was first developed by Pilipchuck [14,15] and then applied to smooth and nonsmooth problems in dynamics in a series of works ([16,17]). We refer the reader to these works for more technical details of the method.

We now compute the periodic solutions of (6) with period $T = 4a$ (yet undetermined). We express the solution in the following form:

$$y_1(\theta_2) = X(\tau(\phi)) + e(\phi)Y(\tau(\phi)), \quad \phi = \theta_2/a \quad (7)$$

where the new independent variables $\tau(\phi)$ and $e(\phi)$ are bounded nonsmooth functions of their argument $\phi$:

$$\tau(\phi) = \frac{2}{\pi} \arcsin \left[ \sin \left( \frac{\pi \phi}{2} \right) \right], \quad e(\phi) = \tau'(\phi). \quad (8)$$

The derivative in (8) should be understood in the context of the theory of distributions. Both nonsmooth variables are periodic in $\phi$ with (normalized) period equal to 4. We note that by (7) the solution is expressed in terms of two components. $X$ depends only on $\tau$ and is termed the $R$-component of the solution. $Y$ also depends solely on $\tau$ and is multiplied by $e$; it is termed the $I$-component of the solution. Interestingly, expression (7) has a phenomenological resemblance to complex variable representation with $e$ playing the role of the imaginary constant $j$ (note that $e^2 = -j^2 = 1$).

Employing the transformation (7) we express the derivatives and powers of $y_1$ in (6) in terms of $X$ and $Y$, and set separately the $R$ and $I$-components of the resulting expression equal to zero. We then obtain the following two subproblems governing the $R$ and $I$-components of the solution:

**Subproblem 1.**

$$Y'' + (Ca^2/\omega_2^2)Y^3 = \varepsilon a^2 g(y_1 = Y, y_1' = Y'/a, \theta_2 = m\pi\tau) + O(\varepsilon^2), \\ a = m\pi, \quad X = 0 \quad Y(\pm 1) = 0, \quad m = 1,2,3,\dots \quad (9)$$

**Subproblem 2.**

$$X'' + (Ca^2/\omega_2^2)X^3 = \varepsilon a^2 g \left( y_1 = X, y_1' = X'/a, \theta_2 \right. \\ \left. = \frac{(2n-1)\pi\tau}{2} \right) + O(\varepsilon^2), \\ a = \frac{(2n-1)\pi}{2}, \quad Y = 0 \quad X'(\pm 1) = 0, \quad n = 1,2,3,\dots \quad (10)$$

We note that the above subproblems were obtained by setting either the $R$ or $I$-component of the solution equal to zero. Then, the solutions of each subproblem provide a distinct class of subharmonic motions of the problem. In general, the problem obtained by applying the previous method leads to a coupled system of equations in $X$ and $Y$, however, this case will not be considered here.

Since no analytical solution exists for these nonlinear boundary value problems we need to resort to a numerical method to solve them. Before we perform this numerical computation, however, we make the following remarks concerning the method of nonsmooth transformations. The boundary conditions in (9) and (10) impose smoothness on the transformed derivatives of $y_1$; these boundary conditions define the domain of the solutions of the nonlinear boundary value problems $\tau \in [-1,1]$. In addition, the quarter-period of the solution, $a$, for each subproblem is allowed a countable infinity of values. Taking into account that the period of the periodic solution is $T=4a$, and that the nonhomogeneous term in (6) is $2\pi$-periodic in $\theta_2$, we conclude that Subproblem 1 computes the $2m:1, m=1,2,3,\dots$ subharmonic orbits of (6), whereas Subproblem 2 computes the $(2n-1):1, n=1,2,3,\dots$ subharmonic orbits. A subharmonic orbit of order $p:1$ is a periodic orbit with period equal to $2p\pi$, i.e., $p$-times the period of the nonhomogeneous term $g$. Finally, we note that the periodic solution $y_1(t)$ is obtained from the solutions of the nonlinear boundary value problems either as $y_1(\theta_2) = e(\theta_2/a)Y(\tau(\theta_2/a))$ (Subproblem 1), or as $y_1(\theta_2) = X(\tau(\theta_2/a))$ (Subproblem 2), where $\theta_2 = \omega_2 t + \theta_{20} + O(\varepsilon)$. Now, the nonlinear boundary value problems above provide the solution only in the normalized half-period $\tau \in [-1,1]$. To extend the result over a full normalized period (equal to 4) we need to add the component of the solution in the interval $\tau \in [1,3]$; to perform this we take into account the symmetry properties of the nonsmooth variables $\tau$ and $e$, and add either the antisymmetric image of the solution about the point $(Y, \tau) = (0,1)$ (for Subproblem 1, cf. Fig. 3(a)), or the mirror image of the solution about the line $\tau=1$ (for Subproblem 2, cf. Fig. 3(b)).

$$y_1(\theta_2) = e(\theta_2/a)Y(\tau(\theta_2/a)) \quad (\text{Subproblem 1})$$

<figure><img src="image_8.png" /></figure>

$$y_1(\theta_2) = X(\tau(\theta_2/a)) \quad (\text{Subproblem 2})$$

<figure><img src="image_9.png" /><figcaption>Fig. 3 Construction of the solution $y_1$ over an entire normalized period (equal to 4) from the half-normalized period solutions (a) $eY(\tau)$, and (b) $X(\tau)$</figcaption></figure>

The nonlinear boundary value problems (9) and (10) were solved using a single-point numerical shooting method. In Figs. 4 and 5 we depict the leading low-order subharmonic orbits and their bifurcations for the undamped two-degrees-of-freedom system with $\omega_2^2=0.9, C=5.0, \varepsilon=0.1$ and varying values of the total energy $h$. In these figures we also present one-period representations of a number of subharmonic orbits. In the bifurcation plot of Fig. 4 we depict the values of $Y'(-1)$ at the subharmonic orbits as functions of $h$ (recall that the solution domain of the above nonlinear boundary value problems is $-1 \le \tau \le 1$); in physical terms, each point denotes the initial slope $ay_1'$ of the subharmonic orbit, corresponding to zero initial displacement, $y_1'=0$. In Fig. 5 we plot $X(-1)$ as function of $h$; in physical terms, each point denotes the initial displacement $y_1$ of the subharmonic orbit, corresponding zero initial slope, $y_1'=0$. These plots depict only the responses of the unexcited oscillator 1; the corresponding responses of oscillator 2 are computed using relation (5) derived in
<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 4 Leading 2m:1 subharmonic orbits as functions of h: —Y'(-1) for m=1, —Y'(-1) for m=2, —Y'(-1) for m=3</figcaption></figure>

the course of the reduction process, and the relations $y_2 = \sqrt{2}I_2/\omega_2 \sin \theta_2$, $\theta_2 = \omega_2 t + \theta_{20} + O(\varepsilon^2)$. Finally, we emphasize that, since the reduced system (6) neglects $O(\varepsilon^2)$ terms, the results presented in Figs. 4 and 5 are approximate and valid only for sufficiently small values of $\varepsilon$. Of special interest are the 1:1 subharmonic orbits labeled A-D in Fig. 5. These orbits dominate the dynamics as shown below.

The domain of attraction and the stability of the 1:1 subharmonic orbits were determined by numerical Poincaré maps. These were constructed by considering the original undamped Eqs. (1). First, the four-dimensional phase space of the solutions of (1) was reduced to a three-dimensional isoenergetic manifold $N$ by fixing the total energy to a constant level, $H^e(y_1, \dot{y}_1, y_2, \dot{y}_2) = h$; $N$ was then ‘cut’ by the Poincaré section $\Sigma = \{(y_1, \dot{y}_1, y_2, \dot{y}_2) \in N/y_2 = 0, \dot{y}_2 > 0\}$. The Poincaré map $P^e$ was defined as, $P^e: \Sigma \to \Sigma$, i.e., as a mapping of points on $\Sigma$ to their images under the flow of the dynamical system on $\Sigma$, under orientation preserving restrictions. Stable periodic orbits of (1) appear as centers in the Poincaré map, whereas, unstable periodic orbits appear as saddle points.

In Fig. 6 we depict the Poincaré maps of the undamped system (1) with $C=5.0$, $\varepsilon=0.1$ and varying values of the energy $h$. The 1:1 subharmonic orbits labeled A-D correspond to the ones of the approximate bifurcation plot of Fig. 5. A common feature of all these plots is a large region of regular motion (smooth quasi-periodic orbits) in the upper regions of the plots, surrounding the stable 1:1 subharmonic orbit A. When damping is added to the

system, the large regular region surrounding orbit A is expected to become a large region of attraction, with that orbit becoming an attractor. Additional stable and unstable 1:1 subharmonic orbits of the system are indicated in the Poincaré maps, confirming the approximate asymptotic results of Fig. 5. At the small energy level $h=0.05$ there are two stable subharmonic orbits; both orbits correspond to localized motions, with orbit A localizing in oscillator 1 and orbit B in oscillator 2. At higher values of $h$ the low-energy bifurcation of 1:1 subharmonic orbits (predicted in the plot of Fig. 5) has occurred and there exist four orbits, three stable and one unstable. Note that as $h$ increases orbit A gradually delocalizes from oscillator 1 and localizes in oscillator 2. The regions of chaotic motion (the ‘stochastic sea’) in the maps is a well-documented feature in the dynamics of such strongly nonlinear systems.

The bifurcation diagrams and Poincaré maps of the 1:1 subharmonic orbits lead to a preliminary qualitative explanation of the energy pumping phenomenon, which, as shown in the previous section, occurs only above a certain level of the initial energy $h$. When energy pumping occurs, an initial transfer of energy occurs from the directly excited (linear) oscillator 2 to the unexcited (nonlinear) oscillator 1; moreover, the ‘fast’ oscillation during this initial phase of the motion has a frequency nearly identical to the linearized natural frequency $\omega_2$. Hence, it is logical to conclude that the 1:1 subharmonic orbit A existing over the entire range of $h$ and having a large domain of attraction, is mainly
<figure><img src="image_12.png" /><figcaption>Fig. 5 Leading (2n-1):1 subharmonic orbits as functions of h: X(-1) for n=1, -X(-1) for n=2, -X(-1) for n=3; XXXX unstable 1:1 subharmonic orbits</figcaption></figure>

responsible for the energy pumping phenomenon. However, this family of orbits can not be directly excited at $t=0$ since it cannot satisfy pointwise the initial condition $((y_1(0), \dot{y}(0)))=(0,0)$ (this is the initial state of oscillator 1 when the energy pumping phenomenon is initiated); as a result, a transient "bridging" orbit must be initially excited, satisfying zero initial conditions and ultimately "connecting" with the 1:1 subharmonic orbit A. Under these conditions energy pumping occurs. Noting that the amplitude of the 1:1 orbit has a lower bound of approximately 0.47 (cf. Fig. 5) and considering the initial transients of the numerical simulations of Fig. 1, we conjecture that if the initial transient

transients of oscillator 1 can not attain sufficiently large amplitudes, they cannot act as "bridging" orbits to excite the 1:1 subharmonic orbit A and no energy pumping can take place. Thus, for sufficiently low values of $h$ no energy pumping is possible. For higher values of $h$ the initial transients for oscillator 1 attain sufficiently large amplitudes to excite the 1:1 orbit and energy pumping can occur. This conjecture explains the lack of energy pumping for $h=0.5$ in the simulations of Fig. 1. Clearly, the previous arguments form merely a conjecture, but the more rigorous analysis in Part II of this work ([1]) validates these arguments.
<figure><img src="image_13.png" /><figcaption>Fig. 6 Poincaré maps of the dynamics of the undamped system (1) at varying energy levels for $\omega_2^2=0.9$, $C=5.0$, $\epsilon=0.1$: (a) $h=0.05$, (b) $h=0.2$, (c) $h=0.8$, (d) $h=2.0$</figcaption></figure>

## 4 Analytical Approximations

As a final note, we now present an analytical technique to approximate the transient responses of the Hamiltonian system; in contrast to most standard techniques which are based on the assumption of weak nonlinearity, the method used here deals with the strong (nonlinearizable) nonlinearity of oscillator 1. An extension of this technique for the damped system in Part II of this work ([1]) will enable us to analytically approximate the transient responses during the initial phase of energy pumping of Fig. 1.

To this end, we express system (1) in the following form:

$$ 
\begin{aligned}
\ddot{y}_1 + \varepsilon y_1 + Cy_1^3 - \varepsilon y_2 &= 0 \\
\ddot{y}_2 + \omega^2 y_2 - \varepsilon y_1 &= 0
\end{aligned}
\quad (11) $$

where $\omega^2 = \omega_2^2 + \varepsilon$. A transformation to complex variables is now introduced,

$$ \psi_1 = \dot{y}_1 + j\omega y_1, \quad \psi_2 = \dot{y}_2 + j\omega y_2, \quad (12) $$

and (12) are rewritten as

$$ 
\begin{aligned}
& \dot{\psi}_1 - \frac{j\omega}{2}(\psi_1 + \psi_1^*) - \frac{j\varepsilon}{2\omega}(\psi_1 - \psi_1^*) + \frac{jC}{8\omega^3}(\psi_1 - \psi_1^*)^3 \\
& \qquad + \frac{j\varepsilon}{2\omega}(\psi_2 - \psi_2^*) = 0 \\
& \dot{\psi}_2 - j\omega\psi_2 + \frac{j\varepsilon}{2\omega}(\psi_1 - \psi_1^*) = 0
\end{aligned}
\quad (13) $$

where the star denotes complex conjugate.

An approximate solution of (13) is sought, based on the assumption of fast oscillations at frequency $\omega$:

$$ \psi_1 = \varphi_1 e^{j\omega t}, \quad \psi_2 = \varphi_2 e^{j\omega t}. \quad (14) $$

Relations (14) indicate the presence of 1:1 internal resonance in the fast dynamics of oscillators 1 and 2. Substituting (14) into

(13), and averaging over the fast periodic terms $e^{j\omega t}$ we obtain the following set of averaged equations governing the (complex) amplitudes $\varphi_i$, $i=1,2$:

$$ 
\begin{aligned}
\dot{\varphi}_1 + \frac{j}{2} \left( \omega - \frac{\varepsilon}{\omega} \right) \varphi_1 - \frac{3jC}{8\omega^3} |\varphi_1|^2 \varphi_1 + \frac{j\varepsilon}{2\omega} \varphi_2 &= 0 \\
\dot{\varphi}_2 + \frac{j\varepsilon}{2\omega} \varphi_1 &= 0.
\end{aligned}
\quad (15) $$

Interestingly, in contrast to (11), the transformed system (15) is completely integrable, with the following two first integrals of motion:

$$ 
\begin{aligned}
& |\varphi_1|^2 + |\varphi_2|^2 = N^2, \\
& \frac{j\omega}{2} |\varphi_1|^2 - \frac{3jC}{16\omega^3} |\varphi_1|^4 + \frac{j\varepsilon}{2\omega} (\varphi_1\varphi_2^* + \varphi_1^*\varphi_2) = H.
\end{aligned}
\quad (16) $$

Employing these results, the complex amplitudes are expressed as

$$ \varphi_1 = N \sin \theta e^{j\delta_1}, \quad \varphi_2 = N \cos \theta e^{j\delta_2}. \quad (17) $$

Substituting (17) into (15), we obtain the final set of equations on the 2-Torus governing the angle-variables $\theta$ and $\delta = \delta_1 - \delta_2$:

$$ 
\begin{aligned}
& \dot{\delta} + \frac{\omega}{2} - \frac{3CN}{8\omega^3} \sin^2 \theta + \frac{\varepsilon}{\omega} \cot 2\theta \cos \delta = 0 \\
& \dot{\theta} + \frac{\varepsilon}{2\omega} \sin \delta = 0.
\end{aligned}
\quad (18) $$

We note that the orbits of (18) can be analytically computed in terms of pseudo-elliptic quadratures by employing the change of variables (17) in (16), and then integrating the second of Eqs. (18) by quadratures.

By numerically integrating (18) we can study transient (nonperiodic) orbits in the neighborhoods of the 1:1 subharmonic orbits of the underlying Hamiltonian system. In Fig. 7 we present the phase plots of (18) for varying values of the energy-like first integral $N$, confirming the bifurcations of the 1:1 subharmonic orbits depicted in the analytical approximations of Fig. 5 and the numerical results of Fig. 6.
<figure><img src="image_14.png" /><figcaption>Fig. 7 Phase plots of the system of Eqs. (18) for $\omega_2^2=0.9$, $C=5.0$, $\varepsilon=0.1$, and varying values of the first integral of motion $N: (a) N=0.4, (b) N=0.8, (c) N=1.9$</figcaption></figure>

## 5 Discussion

We presented numerical evidence of energy pumping in non-linear mechanical oscillators. The systems considered herein were composed of weakly coupled, linear, and essentially nonlinear (nonlinearizable) parts. In such systems it is possible to induce one-way (irreversible) “channeling” of vibrational energy from the linear to the nonlinear part, provided that the imparted energy

is above a critical level. Hence, energy can be “pumped” to a predetermined part of the system (the nonlinear oscillator), which, in essence acts as a passive nonlinear sink. Clearly, no such phenomenon is possible in linear systems.

A 1:1 stable subharmonic orbit of the underlying Hamiltonian system (obtained by setting damping equal to zero) was conjectured to be responsible for the energy pumping phenomenon. We conjecture that the reason for lack of energy pumping at low energies is due to the fact that the 1:1 subharmonic orbit can not be excited unless the energy of the system is above a critical level. The energy pumping phenomenon will be further studied in a companion paper ([1]) where it will be shown that it is caused by transient resonance capture on a 1:1 resonance manifold of the system.

We remark that the energy pumping phenomenon, which in this work was purely passive, could be enhanced using active control. Utilizing this phenomenon one can introduce passive or active nonlinear sinks in predominantly linear extended periodic structures where externally imparted energy is directed and locally eliminated. This can lead to enhanced vibration and shock isolation designs of extended mechanical systems.

## Acknowledgments

This work was supported in part by NSF Young Investigator Award CMS-94-57750 (Dr. Allison Flatau is the Grant Monitor), and by an International Supplement to this Award (Ms. Cassandra Dudca is the Grant Monitor).

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