In this paper, vibrations and bifurcation of a damped system consists of a mass grounded by linear and nonlinear springs and a nonlinear damper is studied. Nonlinear equation of motion is derived using Newton’s equations. Approximate analytical solutions are obtained using multiple time scales (MTS) method. For free vibration, the approximate analytical results are compared with the numerical integration results. Forced vibrations of the system in primary and secondary resonant cases are studied and the effects of different parameters on the frequency-responses are investigated. Moreover, bifurcation of the system is studied considering different control parameters. تفاصيل المقالة

In this paper, a continuum model based on the nonlocal elasticity theory is developed for vibration analysis of embedded orthotropic circular and elliptical micro/nano-plates. The nano-plate is bounded by a Pasternak foundation. Governing vibration equation of the nonlocal nano-plate is derived using Nonlocal Classical Plate Theory (NCPT). The weighted residual statement and the Galerkin method are applied to obtain a Quadratic Functional. The Ritz functions are used to form an assumed expression for transverse displacement which satisfies the kinematic boundary conditions. The Ritz functions eliminate the need for mesh generation and thus large degrees of freedom arising in discretization methods such as Finite Element Method (FEM). Effects of nonlocal parameter, lengths of nano-plate, aspect ratio, mode number, material properties and foundation parameters on the nano-plate natural frequencies are investigated. It is shown that the natural frequencies depend on the non-locality of the micro/nano-plate, especially at small dimensions. تفاصيل المقالة