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Periodically Forced Nonlinear Oscillatory Acoustic Vacuum

Makrina Agaoglou, Michal Feckan, Michal Pospisil, Vassilis M. Rothos, Alexander F. Vakakis

TL;DR

Melnikov-type analysis is applied for the persistence of periodic oscillations of a reduced system of particles coupled by linear springs under distributed harmonic excitation.

Abstract

In this work, we study the in-plane oscillations of a finite lattice of particles coupled by linear springs under distributed harmonic excitation. Melnikov-type analysis is applied for the persistence of periodic oscillations of a reduced system.

Periodically Forced Nonlinear Oscillatory Acoustic Vacuum

TL;DR

Melnikov-type analysis is applied for the persistence of periodic oscillations of a reduced system of particles coupled by linear springs under distributed harmonic excitation.

Abstract

In this work, we study the in-plane oscillations of a finite lattice of particles coupled by linear springs under distributed harmonic excitation. Melnikov-type analysis is applied for the persistence of periodic oscillations of a reduced system.
Paper Structure (5 sections, 2 theorems, 64 equations, 3 figures)

This paper contains 5 sections, 2 theorems, 64 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Model
  3. Two-Mode System
  4. Melnikov Analysis for Periodic Oscillations
  5. Discussion

Key Result

Theorem 1

If there are $\beta_{1,0}\in[0,2\pi)$, $\rho_{0}$ satisfying m4, $\mu_{1,0}$ and $\mu_{2,0}$ with $\mu_{1,0}^2+\mu_{2,0}^2=1$ solving con1, then for any $\mu_1$ near $\mu_{1,0}$ and $\mu_2$ near $\mu_{2,0}$ with $\mu_{1}^2+\mu_{2}^2=1$ and $\epsilon\ne0$ small, there are $\beta_1(\epsilon)$ near $\b

Figures (3)

  • Figure 1: Forced and damped lattice oscillating in the plane (see ZMSBV)
  • Figure 2: Top panel: Graph for $\theta$ for different initial values of $\theta _{0}$. Bottom panel: Graph for $\Delta$ for different initial values of $\theta _{0}$.
  • Figure 3: Orbits in the phase portrait of \ref{['eq.32']}, where $(\theta,\Delta)\in [0,\frac{\pi}{2}]\times [0,\pi]$.

Theorems & Definitions (3)

  • Theorem 1
  • Corollary 1
  • Proof 1