Nonlinear forced vibration of rectangular plates by modified multiple scale method

Abstract

In the present study, the nonlinear forced vibration of a rectangular plate is investigated analytically using modified multiple scales method for the first time. The plate is subjected to transversal harmonic excitation, and the boundary condition is assumed to be simply supported. The von Karman nonlinear strain displacement relations are used. The extended Hamilton principle and classical plate theory are applied to derive the partial differential equations of motions. This research focuses on resonance case with 3:1 internal resonance. By applying Galerkin method, the nonlinear partial differential equations are transformed into time dependent nonlinear ordinary differential equations, which are then solved analytically by modified multiple scales method. This proposed approach is very simple and straightforward. The obtained results are then compared with both the traditional multiple scales method and previous studies, and excellent compatibility is noticed. The effect of some of the main parameters of the system is also examined.

Nonlinear Flap-Wise Vibration Characteristics of Wind Turbine Blades Based on Multi-Scale Analysis Method

“…Setting the results of the differential equations of blade vibration as a measure of multiple independent variables, y is considered as a function of t and ςt. Introducing the independent variable T n = ς n t, n = 0, 1, • • • , the derivative concerning time t thus becomes a partial derivative expansion concerning T, i.e., [19].…”

“…Moreover, transition sets made up of bifurcation and hysteresis sets may be built to examine the nonlinear vibration of the blade by including system factors such as singularity theory and van der Pol damping. Multiple Scale Analysis (MSM) refers to the decomposition of the motion of a system into ingredients on different time scales, thus revealing the dynamics of the system [16]. Through multiscale analysis, various dynamic characteristics at different time and spatial scales can be discovered in a two-dimensional nonlinear vibration system, such as periodic vibration, chaotic vibration, and twin phenomena.…”

“…Through multiscale analysis, various dynamic characteristics at different time and spatial scales can be discovered in a two-dimensional nonlinear vibration system, such as periodic vibration, chaotic vibration, and twin phenomena. These characteristics not only help us understand the motion law of the system but also provide important references for the control and optimization of the system [17][18][19]. When studying the stability of the steady-state solution of blade vibration in 1:1 internal resonance using the multiscale method, the Routh-Hurwitz criterion can be used [20].…”