On the nonlinear normal modes of free vibration of piecewise linear systems

Abstract

International audienceA modification of the Shaw–Pierre nonlinear normal modes is suggested in order to analyze the vibrations of a piecewise linear mechanical systems with finite degrees of freedom. The use of this approach allows one to reduce to twice the dimension of the nonlinear algebraic equations system for nonlinear normal modes calculations in comparison with systems obtained by previous researchers. Two degrees of freedom and fifteen degrees of freedom nonlinear dynamical systems are investigated numerically by using nonlinear normal modes

Nonlinear modes of piecewise linear systems forced vibrations close to superharmonic resonances

“…Thus, the nonautonomous dynamical system is transformed into the pseudo-autonomous one. The Shaw–Pierre NNM for the piecewise linear systems 38 is used to analyze the pseudo-autonomous dynamical system.…”

“…The motions on the nonlinear mode of the dynamical system (11) are described by the following two ordinary differential equations 38 …”

“…NNMs of the piecewise linear systems free vibrations are analyzed in the literature. 36–38 NNMs of the piecewise linear systems forced vibrations close to the principle resonances are treated by Uspensky and Avramov. 39,40 The rotor vibrations are analyzed by means of NNMs by Avramov and Borysiuk.…”

Non-iterative Rauscher method for 1-DOF system: a new approach to studying non-autonomous system via equivalent autonomous one

“…They are shown on Figure 4,a. When expressions (28) are known, Rauscher expansions can be built via formulae (19). Their correctness may be checked in the following way.…”

“…It also should be noted that different forms and various applications of the Rauscher method are discussed in [11,17]. Among recent works where the Rauscher method is used one can find the following ones: [18,19]. In the present paper only 1-DOF mechanical system is considered:…”