Nonlinear Vibration of Orthotropic Rectangular Membrane Structures Including Modal Coupling

Abstract

The membrane structure has been applied throughout different fields such as civil engineering, biology, and aeronautics, among others. In many applications, large deflections negate linearizing assumptions, and linear modes begin to interact due to the nonlinearity. This paper considers the coupling effect between vibration modes and develops the theoretical analysis of the free vibration problem for orthotropic rectangular membrane structures. Von Kármán theory is applied to model the nonlinear dynamics of these membrane structures with sufficiently large deformation. The transverse displacement fields are expanded with both symmetric and asymmetric modes, and the stress function form is built with these coupled modes. Then, a reduced model with a set of coupled equations may be obtained by the Galerkin technique, which is then solved numerically by the fourth-order Runge–Kutta method. The model is validated by means of an experimental study. The proposed model can be used to quantitatively predict the softening behavior of amplitude–frequency, confirm the asymmetric characters of mode space distribution, and reveal the influence of various geometric and material parameters on the nonlinear dynamics.

Theoretical and Numerical Studies on Damped Nonlinear Vibration of Orthotropic Saddle Membrane Structures Excited by Hailstone Impact Load

“…Assume that the membrane does not bear an external load apart from the impact load; then, when the point of impact lies in the coordinates on (x 0 , y 0 ) (as shown in Figure 1), the equation becomes [10]…”

“…e effects of prestrain, size, elastic ratio, density, relative amplitude, and dead load on the nonlinearity of square pretensioned membrane were studied. Zheng et al [8], Liu et al [9], and Li et al [10] investigated the large deflection nonlinear free vibration of orthotropic rectangular membrane structure by applying von Kármán's large amplitude theory, D'Alembert's principle, Bubnov-Galerkin approximate method, and Lindstedt-Poincaré perturbation method and obtained the approximate analytical solution in power series of the nonlinear vibration frequency function and the displacement function of the rectangular membrane with four edges fixed. However, they only studied the free vibration of planar rectangular membrane structures.…”

Analytical Solutions for Stochastic Vibration of Orthotropic Membrane under Random Impact Load

“…By substituting Equations (34), (36), and (37) into Equation (29), the governing Equation (29) can be decoupled by the Galerkin method [ 22 ] and expressed as …”

Dynamic response of rectangular membrane excited by heavy rainfall

“…During the procedure of discretizing partial differential equations with the Galerkin method, the symmetric mode shapes were assumed. Subsequently, Li et al. (2018) considered the modal coupling effect and introduced the symmetric and asymmetric modes to study the nonlinear vibration of the membrane structure.…”