Diana V. Bambill

Dynamic stiffening of circular plates and determination of their buckling, in-plane pressure

Abstract This paper presents some numerical experiments performed on transverse vibrations and the elastic stability of homogeneous, isotropic, clamped circular plates of discontinuously varying thickness. The purpose of the study is threefold: (1) to determine dynamic stiffening situations; (2) to experiment on the effect of optimization with respect to an exponential parameter (non-integer version of the Rayleigh-Ritz method or the ‘optimized Rayleigh-Ritz method’) as opposed to the classical version of the energy method; and (3) to ascertain the relative accuracy of the optimized Rayleigh-Ritz method by comparing eigenvalues determined using this technique and those calculated using a very efficient standard finite element code.

Free Vibration Analysis of Centrifugally Stiffened Non Uniform Timoshenko Beams

Rotating beams – like structures are widely used in many engineering fields and are of great interest as they can be used to model blades of wind turbines, helicopter rotors, robotic manipulators, turbo-machinery and aircraft propellers. The governing differential equations of motion in free vibration of a non-uniform rotating Timoshenko beam, with general elastic restraints at the ends are solved using the differential quadrature method, (Bellman & Roth, 1986; Felix et al., 2008, 2009). The equations of motion are derived to include the effects of shear deformation, rotary inertia, hub radius, ends elastically restrained and non-uniform variation of the cross-sectional area of the beam. The presence of a centrifugal force due to the rotational motion is considered as Banerjee has developed, using Hamilton’s principle to capture the centrifugal stiffening arising in fast rotating structures, (Banerjee, 2001). With the proposed model, a great number of different situations are admitted to be solved. Particular cases with classical restraints can be deduced for limiting values of the rigidities. Also step changes in cross-section are considered (Naguleswaran, 2004).

Coupling between Transverse Vibrations and Instability Phenomena of Plates Subjected to In-Plane Loading

As it is known, the problems of free transverse vibrations and instability under in-plane loads of a plate are two different technological situations that have similarities in their approach to elastic solution. In fact, they are two eigenvalue problems in which we analyze the equilibrium situation of the plate in configurations which differ very slightly from the original, undeformed configuration. They are coupled in the event where in-plane forces are applied to the edges of the transversely vibrating plate. The presence of forces can have a significant effect on structural and mechanical performance and should be taken into account in the formulation of the dynamic problem. In this study, distributed forces of linear variation are considered and their influence on the natural frequencies and corresponding normal modes of transverse vibration is analyzed. It also analyzes their impact for the case of vibration control. The forces magnitude is varied and the first natural frequencies of transverse vibration of rectangular thin plates with different combinations of edge conditions are obtained. The critical values of the forces which cause instability are also obtained. Due to the analytical complexity of the problem under study, the Ritz method is employed. Some numerical examples are presented.