Nonlinear forced vibrations of a cylindrical shell with two internal resonances

Abstract

Forced vibrations of cylindrical shells described by a system of three ordinary differential equations are studied. There are two internal resonances. Standing and traveling waves in the shells are described by a system of six modulation equations derived using the multiple-scales method. These waves are analyzed for stability Keywords: cylindrical shell, two internal resonances, standing and traveling waves, stability of waves Introduction. Much effort was made to study the nonlinear vibrations of shells. However, as follows from the reviews [10,12], this division of mechanics is far from being fully understood. The monograph [4] studies the vibrations of cylindrical shells with allowance for one or two vibration modes. The book by Leissa [11] was apparently the first in the English-language literature to address the nonlinear vibrations of cylindrical shells. He pointed out in [11] that shells have soft amplitude-frequency characteristics (AFCs). The nonlinear vibrations of cylindrical shells were studied in [5-7] taking into account three vibration modes and using asymptotic methods. Free and forced nonlinear vibrations of cylindrical shells were also addressed in [15,16]. Damping of free vibrations in linear systems and the dynamic behavior of systems with nonlinear vibration damping were studied in [14,17].We will study forced nonlinear vibrations of a cylindrical shell without initial imperfections and with two internal resonances. The associated vibration equations have been derived in the monograph [7], but the vibrations themselves with two internal resonances have not been examined therein. The authors of [7] used a model with three degrees of freedom: two describe conjugate flexural modes, and the third one characterizes an axisymmetric mode. The amplitude of the axisymmetric mode is assumed much less than those of the conjugate modes. In this case, vibrations can be described by a system with two degrees of freedom. We will examine the case where an internal resonance occurs between the frequency of the conjugate modes and the frequency of the axisymmetric mode. In this case, the conjugate modes and the axisymmetric mode exchange energy, which results in high amplitudes of the axisymmetric mode. Therefore, two degrees of freedom are not enough here, and we will use a model with three degrees of freedom.

Vibrations of a single-disk rotor on nonlinear supports

“…To analyze the dynamics of system (6), we will modify the multiple-scales method [8,9]. The motion of the system is described bỹ …”

“…The approach proposed here is essentially different from the standard procedure of the multiple-scales method [8,9] due to the presence of gyroscopic terms in (6).…”

“…Substituting Eqs. (8a) into (8c) and equating the secular terms to zero [8,9], we get two complex modulation equations for the variables ( , ) A A …”

Multidimensional models of traveling waves and nonlinear modes in cylindrical shells

“…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.…”

Effect of the length of a rotating drillstring on the stability of its quasistatic equilibrium

“…This is also promoted by longitudinal gravity, torque, gyroscopic and centrifugal inertial forces of rotary motion, viscous friction, and gyroscopic and relative inertial forces of internal and external fluid flows. Various aspects of these forces were examined in [9][10][11][12][13][14][15][16][17]. …”