Multidimensional models of traveling waves and nonlinear modes in cylindrical shells

Abstract

The nonlinear dynamic deformation of a cylindrical shell is modeled taking into account several conjugate vibration modes. Two types of motion are considered: nonlinear normal modes and traveling waves Introduction. The geometrically nonlinear deformation vibrations of cylindrical shells attract the attention of both researchers and engineers [1,2,5,10]. This is because of the wide use of cylindrical shells in aerospace, power, and mechanical engineering. The available data on the nonlinear vibrations of cylindrical shells are reviewed in [6,8].The models used to analyze the nonlinear vibrations of cylindrical shells may be divided into two large groups: (i) low-dimensional models with two or three degrees of freedom [1, 2, 5, 10] and (ii) multidimensional models obtained and studied numerically [8].There are many cylindrical shells with a crowded spectrum of natural frequencies. In describing their geometrically nonlinear dynamic deformation, it is necessary to account for three to four natural modes. It is such shells that are addressed here.1. Equations of Vibrations. Consider a hinged thin shell with shear and longitudinal inertia neglected. The shell undergoes geometrically nonlinear deformation that causes small strains, moderate displacements, and nonlinear strain-displacement relationship. The stresses and strains are related by Hooke's law. The shell is perfectly cylindrical and its vibrations are described by the Donnell-Mushtari-Vlasov equations

Review of Applications of Nonlinear Normal Modes for Vibrating Mechanical Systems

“…Interaction of eigenmodes with close frequencies is investigated in Refs. [28,29]. Donnell equations with respect to displacements are used to analyze shell vibrations.…”

“…Initial imperfections and the dynamical deflection are chosen in the form of Eqs. (29) and (40), respectively. Analysis of the four-degree-of-freedom dynamical system shows that NNMs are observed in the flutter region.…”

Non-linear vibrations of shells: A literature review from 2003 to 2013

“…A simple approximate transverse displacement expansion including the driven mode and the axisymmetric modes (neglecting the com panion modes and using the squared sin axisymmetric term) was used for the minimization of the energy functional and it was shown that the shell exhibits a softening behavior. The method of harmonic balance was also successfully used by Avramov [65,66] in obtaining the non linear normal modes and traveling waves of simply supported circular cylindrical shells.…”

Nonlinears Normal Modes for Vibrating Mechanical Systems. Review of Theoretical Developments

“…Such vibrations take place in structures and machines. NNMs of free vibrations of cylindrical shells are considered in papers [33,38]. NNMs of shallow shells with geometrical nonlinearity interacting with a fluid are analyzed in Ref.…”

“…NNMs are used to analyze dynamics of pretwisted beam with geometrical nonlinearity in Refs. [38,47]. NNMs of beams parametric vibrations are analyzed in the paper [26].…”