Non-Linear Forced Vibrations of Plates by an Asymptotic–numerical Method

Nonlinear forced vibration of damped plates coupling asymptotic numerical method and reduction models

“…As is well known, in the literature of ANM [2,18], to solve problems (10) and (11) by a classical finite element method (FEM), one returns to a pure displacement formulation. After the discretization by FEM, at each order p we solve a linear system of dimension NT (6) written in the following form:…”

“…P jc cos jwt + P js sin jwt (2) where H is the number of harmonics, ω is the frequency, c and s denote, respectively, the cosines and sinus factors and j is the harmonic number with j = 0, H − 1. Then, we can write the general excitation vector in the form of (2H − 1) harmonic blocs:…”

“…The use of asymptotic numerical method permits one to obtain a large part of the nonlinear solution by solving a sequence of linear problems, which all have the same stiffness matrix. This method has been successively applied in structural mechanics [2] and in fluid mechanics [3]. The ANM permits in one hand to compute the nonlinear solution curves and in the other hand to easily determine instabilities or bifurcation points that can appear in the nonlinear branch [4].…”

Modal analysis of a continuous gyroscopic second-order system with nonlinear constraints

“…In order to represent the response in a compact manner [25] l jÀn ¼ l jn ; w jÀn ¼ w jn (26) for n ¼ 1; 2; . .…”

“…The stable response is calculated after the vibration has reached a steady-state. While this method is well suited to finding the stable branches of the response, another approach, such as an asymptotic method [26], is needed to capture the unstable branches of the response. The properties of the string and guide are representative of magnetic tape systems and are listed in Table 1.…”