Response enhancement in an oscillator chain

doi:10.1016/j.cnsns.2015.07.003

Communications in Nonlinear Science and Numerical Simulation

Volume 30, Issues 1–3, January 2016, Pages 373-386

https://doi.org/10.1016/j.cnsns.2015.07.003 Get rights and content

Highlights

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An analysis of capture into resonance in an oscillator chain with a nonlinear actuator (the Duffing oscillator) is provided.
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An effect of slow modulation of the natural and external frequencies of the actuator on the occurrence of autoresonance in the entire array is investigated.
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Explicit asymptotic solutions describing capture into resonance and escape from it are derived.
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Numerical simulations prove a good agreement between the analytical and numerical (exact) results.

Abstract

This paper investigates the emergence of autoresonance (AR) oscillations with permanently growing energy in a chain of time-invariant linear oscillators weakly coupled to a nonlinear actuator (the Duffing oscillator) driven by an external force. Two types of forcing are studied: (1) harmonic forcing with constant frequency is applied to the actuator with slowly-varying parameters; (2) harmonic forcing with a slowly increasing frequency is applied to the nonlinear actuator with constant parameters. In both cases, the linear chain is time-invariant, and the system is initially engaged in resonance. It has been proved in earlier works that a slow increase of the forcing frequency or an equivalent decrease of stiffness play a similar role in the emergence of AR in a single Duffing oscillator. This paper shows that in the system of the first type AR the nonlinear oscillator generates oscillations with growing amplitudes in the chain, but in the system of the second type energy transfer from the nonlinear oscillator is insufficient to excite high-energy motion in the attachment. The difference in the dynamical behavior is explained by different resonance properties of the systems. It is also shown that a slow change of stiffness may enhance the response of the nonlinear oscillator and make it sufficient to support oscillations with growing energy in the linear attachment even beyond the linear resonance. Explicit asymptotic approximations of the solutions are obtained. Close proximity of the derived approximations to exact (numerical) results is demonstrated.

Section snippets

Introduction and motivation

It is well known that high-energy resonant oscillations in a linear time-invariant oscillator are generated by an external force whose constant frequency matches the frequency of the oscillator. In this case, the change of the forcing and/or oscillator frequency results in escape from resonance. On the contrary, the frequency of a nonlinear oscillator changes as the amplitude changes, and the oscillator remains in resonance with its drive if the driving frequency and/or other parameters vary

Main equations

Consider the following model of two coupled oscillators: $\begin{matrix} m_{1} \frac{d^{2} u_{1}}{d t^{2}} + c_{1} u_{1} + c_{1, 0} (u_{1} - u_{0}) = 0, \\ m_{0} \frac{d^{2} u_{0}}{d t^{2}} + C (t) u_{0} + γ u_{0}^{3} + c_{0, 1} (u_{0} - u_{1}) = A \sin ω t . \end{matrix}$ In (1), u₀ and u₁ denote absolute displacements of the nonlinear and linear oscillators, respectively; m₀ and m₁ are their masses; c₁ is stiffness of the linear oscillator; c_1,₀ = c_0,1 is the linear coupling coefficient; γ is the coefficient of cubic nonlinearity; C(t) = c₀ − (κ₁ + κ₂t), κ_1,₂ > 0; A and ω are the amplitude and the frequency of the periodic force.

The

Energy localization and exchange in a 2DOF system excited by forcing with slow frequency modulation

In this section we briefly analyze energy transport in a system excited by forcing with slowly increasing frequency. First, the time-invariant system is considered. The equations of motion are reduced to the form similar to (3): $\begin{matrix} \frac{d^{2} u_{1}}{d τ_{0}^{2}} + u_{1} + 2 ɛ λ_{1} (u_{1} - u_{0}) = 0, \\ \frac{d^{2} u_{0}}{d τ_{0}^{2}} + u_{0} + 2 ɛ λ_{0} (u_{0} - u_{1}) + 8 ɛ a u_{0}^{3} = 2 ɛ F \sin (τ_{0} + θ (τ)), \\ \frac{d θ}{d τ} = s ζ_{0} (τ), \end{matrix}$ where τ = εsτ₀, ζ₀(τ) = 1 + βτ; other coefficients are defined by relations (2). Transformations (4)–(6) together with an additional change the following of the leading-order complex

Equations of a multidimensional model

In this section we extend the results obtained for two coupled oscillators to a chain of n linear oscillators weakly coupled to a nonlinear actuator. The equations of the uniaxial motion of the array are given by $\begin{matrix} m_{n} \frac{d^{2} u_{n}}{d t^{2}} & + & c_{n} u_{n} + c_{n, n - 1} (u_{n} - u_{n - 1}) = 0 . \\ m_{r} \frac{d^{2} u_{r}}{d t^{2}} & + & c_{r} u_{r} + c_{r, r - 1} (u_{r} - u_{r - 1}) + c_{r, r + 1} (u_{r} - u_{r + 1}) = 0, 1 \leq r \leq n - 1, \\ m_{0} \frac{d^{2} u_{0}}{d t^{2}} & + & C (t) u_{0} + γ u_{0}^{3} + c_{0, 1} (u_{0} - u_{1}) = A \cos ω t, \end{matrix}$ where u_r represents the absolute displacement of the rth oscillator from the equilibrium position, m_r is its mass; the coefficients c_r and c_r,r₊₁ = c_r_+1,_r

Conclusions

It was shown in early works on particle acceleration that autoresonance could potentially serve as a mechanism to excite and control the required high-energy regime in a single oscillator. This principle was further employed in various fields of applied physics. However, the behavior of the multi-dimensional array can drastically differ from the dynamics of a single oscillator. This paper has investigated the emergence of high-energy oscillations in a model consisting of a chain of

Acknowledgments

Support for this work received from the Russian Foundation for Basic Research (RFBR grant 14-01-00284) and the Russian Academy of Sciences (Programs 1 and 15) is gratefully acknowledged. The author is grateful to L.I. Manevitch for useful discussion. The author would also like to thank the anonymous Reviewers for useful comments and recommendations.

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Autoresonance in weakly dissipative Klein–Gordon chains
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The work [15] suggested a different approach for studying resonant oscillations starting at the initial rest state and corresponding to the maximum energy transfer from the source of energy to the oscillator. This approach, based on the direct application of the multiple time-scales method [16], was effectively applied to the analysis of autoresonance in the undamped single Duffing oscillator [17] and the pair of oscillators [18,19], as well as to the examination of autoresonance in quasi-linear and strongly nonlinear undamped chains starting at rest and subjected to harmonic forcing with a slowly-varying frequency [20–22]. In the present paper, this approach is extended to the analysis of autoresonance in a one-dimensional dissipative chain being initially at rest and governed by harmonic forcing with a slowly-varying frequency.
The problem of response enhancement (autoresonance) in a weakly dissipative Klein–Gordon chain subjected to harmonic forcing with a slowly-varying frequency may be of interest for both a fundamental standpoint as well as practical applications, with special attention to the problems of vibration control in macro-, micro- and nano-structures. Previous investigations have given main attention to the emergence and stability of autoresonance in non-dissipative quasi-linear and strongly-nonlinear chains. In this work, we extend recent results to analytical and numerical studies of the influence of dissipation on autoresonance in a weakly dissipative quasi-linear array. Analytical approximations and numerical examples demonstrate that weak dissipation permits the emergence of autoresonance on a bounded time interval but an increase of dissipation leads to successive failure of resonance of all particles in the chain starting from the ones most distant from the source of energy. The developed asymptotic procedures give rise to simple quasi-steady autoresonant solutions and allow estimating the duration of autoresonance, together with the quantitative assessment of critical dissipation. It is shown that the system dynamics agrees with the theoretical predictions for the entirely autoresonant and entirely non-resonant chains but the intermediate cases of autoresonant energy localization in a part of the chain should be examined separately.
Energy transfer in autoresonant Klein–Gordon chains
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In most of previous studies, AR in the forced oscillator was considered as an effective tool for exciting high-energy oscillations in the entire array. However, recent results [12,13] have shown that this principle is not universal because capture into resonance of a multi-particle chain is a much more complicated phenomenon than a similar effect for a single oscillator [14] and the behavior of each element in the chain may differ from the dynamics of a single oscillator. This effect was recently analyzed for oscillator arrays, which comprise a chain of time-invariant linear oscillators weakly coupled to a nonlinear actuator [12,13].
In this work we examine autoresonant oscillations in a Klein–Gordon chain of finite length. The chain is subjected to an external periodic forcing with a slowly varying frequency applied at one edge of the chain. Explicit asymptotic equations describing the amplitudes and the phases of oscillations are derived. These equations demonstrate that, in contrast to the chains with linear attachments, the nonlinear chain can be entirely captured into resonance provided that its structural and excitation parameters exceed their critical thresholds. It is shown that at large times the amplitudes of AR oscillations converge to a monotonically growing mean amplitude that is equal for all oscillators. The threshold values of the structural and excitation parameters, which allow the emergence of autoresonance in the entire chain, are determined. The derived analytic results are in good agreement with numerical simulation.
Asymptotic Analysis of Autoresonant Oscillator Chains
2016, Procedia IUTAM
This paper investigates the emergence of autoresonance (AR) with growing energy in a chain of time-invariant linear oscillators weakly coupled to a nonlinear actuator (the Duffing oscillator) driven by an external periodic force. Two types of forcing are studied: (1) harmonic forcing with constant frequency is applied to an actuator with slowly-varying parameters; (2) harmonic forcing with a slowly increasing frequency is applied to an actuator with constant parameters. In both cases, the linear attachment is time-invariant, and the system is initially engaged in resonance. It is shown that in case (1) AR the nonlinear oscillator generates oscillations with growing amplitudes in the attached chain, while in case (2) energy transfer from the nonlinear oscillator is insufficient to excite high-energy motion in the attachment. The difference in the dynamical behavior is explained by different resonance properties of the systems. It is also shown that a slow change of stiffness may enhance the response of the nonlinear oscillator and make it sufficient to support oscillations with growing energy in the linear attachment even beyond the linear resonance. Explicit asymptotic approximations of the solutions are obtained.
Autoresonance in a strongly nonlinear chain driven at one end
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View full text

Response enhancement in an oscillator chain

Highlights

Abstract

Section snippets

Introduction and motivation

Main equations

Energy localization and exchange in a 2DOF system excited by forcing with slow frequency modulation

Equations of a multidimensional model

Conclusions

Acknowledgments

J Appl Math Mech

Phys Nonlinear Phenom

Commun Nonlinear Sci Numer Simulat

Commun Nonlinear Sci Numer Simulat

A new method of accelerating relativistic particles

Comp Rend (Dokaldy) Acad Sci URSS

The synchrotron – a proposed high-energy particle accelerator

Phys Rev

High-energy particle accelerators

On the origin of commensurabilities amongst satellites of Saturn

Mon Not R Astron Soc

Evolution of satellite resonances by tidal dissipation

Astron J

Random aspects of beam physics and laser-plasma interactions

Autoresonance in stimulated Raman scattering

Charge and energy transfer dynamics in molecular systems

Mathematical aspects of classical and celestial mechanics

Capture into resonance and escape from it in a forced nonlinear pendulum

Regul Chaot Dyn