Nonlinear Normal Modes in a Two-Stage Isolator Using a Modified Finite-Element Galerkin Method

Cheng Li,Hongguang Li,Giuseppe Ruta

doi:10.1155/2021/6680487

Abstract

A modified Galerkin method is proposed to approximate the nonlinear normal modes in a new type of a two-stage isolator. Besides the displacement of payload and the force transmissibility of this typical nonlinear dynamic system, the nonlinear normal modes defined as invariant manifolds can provide more information about the nonlinear coupling between the system components when periodic motions corresponding to the normal modes of the system occur. The presented approach applies a combination of finite-element discretization and Fourier series expansion for the approximate invariant manifolds. A Galerkin projection of the governing equations for the approximate invariant manifolds yields a set of nonlinear algebraic equations in expansion coefficients, which can be solved numerically with a general choice of zero as initial guess for the cases in this work. The resultant approximate solutions for the invariant manifolds can accurately describe the nonlinear interactions between system components in periodic motions of the specific nonlinear normal modes. In addition, one can solve the invariant manifolds for an annular domain of interest directly by this method, without considering other domain that includes the origin of phase space.

Highlights

Linear vibration isolators can be used when their natural frequencies are well below the excitation frequency
A comprehensive review about nonlinear passive vibration isolators is given by Ibrahim [2]. is kind of geometric design for stiffness nonlinearity is exploited in a class of two-stage nonlinear isolator [3, 4] which can have a desirable small static deflection as well as a smaller transmissibility in the frequency range of isolation compared with the corresponding linear two-stage isolator
Conclusions e proposed modified Galerkin method can be applied to accurately compute the nonlinear normal modes of a discrete dynamic system such as the new type of two-stage nonlinear isolator presented in this work

Summary

Introduction

Linear vibration isolators can be used when their natural frequencies are well below the excitation frequency. Is kind of geometric design for stiffness nonlinearity is exploited in a class of two-stage nonlinear isolator [3, 4] which can have a desirable small static deflection as well as a smaller transmissibility in the frequency range of isolation compared with the corresponding linear two-stage isolator. Two versions of this class of two-stage nonlinear isolator are proposed. With a Galerkin projection method [14], an approximation of the NNMs defined as invariant manifolds can be solved accurately for a wide range of nonlinear vibrating systems. For the concerned NNMs of the new twostage isolator, considering the strong nonlinearity of this system and the resulting strongly coupled algebraic equations from the standard Galerkin procedure, a modified finite-element Galerkin approach is proposed to simplify the nonlinear algebraic equations and to facilitate the solving of the partial differential equations which govern the NNM invariant manifolds

A New Type of a Two-Stage Nonlinear Isolator

The Computation for NNMs as Invariant Manifolds

Nonlinear Normal Modes of Type III Isolator in Different Cases