Nonlinear Vibration Analysis of Clamped Skew Plates by a Variational Method

Abstract

Nonlinear free vibration analysis of clamped isotropic skew plates has been presented. The large amplitude of vibration is imparted statically by subjecting the plate under uniform transverse pressure. The mathematical formulation is based on the variational principle in which the displacement fields are assumed as a combination of orthogonal polynomial or transcendental functions, each satisfying the corresponding boundary conditions of the plate. The large amplitude dynamic problem is addressed by solving the corresponding static problem first and subsequently with the resultant static displacement field, the dynamic problem is formulated. The vibration frequencies are obtained from the solution of a standard eigenvalue problem. Entire computational work is carried out in a normalized square domain obtained through an appropriate domain mapping technique. Results of the reduced problem revealed excellent agreement with other studies and a typical comparison of the actual problem is also carried out successfully. Results are furnished in dimensionless amplitude-frequency plane, in the form of backbone curves, and pictorial representations of some vibration mode shapes are made.

A variational analysis for large deflection of skew plates under uniformly distributed load through domain mapping technique

“…Singh et al (2006) considered the vibration behaviour of thermally stressed composite plates whereas Ashour (2009) analyzed free vibration behaviour of symmetrically laminated clamped skew plates. Das et al (2008Das et al ( , 2009) used a variational method to study large amplitude dynamic problem of skew plates under simply supported and clamped boundary conditions. Differential quadrature method has also been used for non-linear vibration problems of laminated composite skew plates (Malekzadeh, 2007(Malekzadeh, , 2008Malekzadeh and Karami, 2006).…”

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“…Large-amplitude free-vibration analysis of a nonlinear system is essentially the same as free-vibration analysis of the system subjected to a static load producing such large-amplitude deflection [Crandall 1956]. Here, the system is assumed to execute small-amplitude vibration about the statically deflected equilibrium configuration as opposed to large-amplitude vibration about its undeformed equilibrium position.…”

“…However, in this case the higher-order functions are generated from the start functions following a two-dimensional implementation of the Gram-Schmidt orthogonalization scheme. The Gram-Schmidt orthogonalization principle and its two-dimensional implementation have been discussed in detail in [Saha et al 2004;Das et al 2009]. It should be noted that the generated functions are for the total domain and they need to be broken down in terms of the subdomains.…”

“…As a result the system of governing equations presented in (6) becomes nonlinear in nature and cannot be solved directly. Hence, the solution methodology specially adopted to solve the set of nonlinear equations employs an iterative procedure utilizing a direct substitution technique with a successive relaxation scheme [Cook et al 1989]. At the start of the solution procedure necessary parameters that cater to the iterative scheme, for example, the relaxation parameter (λ), error limit (ε), etc., are chosen along with a starting load value.…”

“…Such decomposed domains with modified grids corresponding to different types of plates having different stiffener orientations and positions are shown in Figure 2. In nonlinear vibration, frequency is a function of the vibration amplitude and the nature of the deflected profile [Bikri et al 2003;Das et al 2009]; moreover, the amplitude of vibration at a point within a vibrating continuous system depends on the location. In studying the effect of vibration amplitude on the dynamic behavior of stiffened plates, the maximum amplitude of vibration is taken as representative of the amplitude.…”