Theoretical analyses are presented for nonlinear vibrations of a clamped rectangular plate under a uniformly distributed periodic load, with the effect of both initial deflection and initial edge displacement taken into consideration The dynamic analog of the Marguerre equations is used and the steady-state solutions are obtained by first applying the Galerkin method and then the harmonic balance method Actual calculations are carried out for the square plate under the assumption of the three degrees-of-freedom system and the frequency response characteristics and typical waveforms are determined under various initial edge displacements including initially buckled cases Effects of both initial deflection and initial edge displacement on the static deflection as well as the lower natural frequencies are also clarified

Nonlinear vibrations of a clamped rectangular plate with initial deflection and initial edge displacement— Part II: Experiment

Nondestructive experimental methods to calculate the buckling load of imperfection sensitive thin-walled structures are one of the most important techniques for the validation of new structures and numerical models of large scale aerospace structures. Vibration correlation technique (VCT) allows determining equivalent boundary conditions and buckling load for several types of structures without reaching the instability point. VCT is already widely used for beam structures, but the technique is still under development for thin-walled plates and shells. This paper intends to explain the capabilities and current limitations of this technique applied to two types of structures under buckling conditions: flat plates and cylindrical shells prone to buckling. Experimental results for a flat plate and a cylindrical shell are presented together with reliable finite element models for both cases. Preliminary results showed that the VCT can be used to determine the realistic boundary conditions of a given test setup, providing valuable data for the estimation of the buckling load by finite element models. Also numerical results herein presented show that VCT can be used as a nondestructive tool to estimate the buckling load of unstiffened cylindrical shells. Experimental tests are currently under development to further validate the approach proposed herein.

Vibration Correlation Technique for the Estimation of Real Boundary Conditions and Buckling Load of Unstiffened Plates and Cylindrical Shells

Non-linear vibrations of a clamped circular plate with initial deflection and initial edge displacement, part I: Theory

Vibration frequency measurements are sometimes used to nondestructively predict buckling loads. Various powers of the fundamental frequency may be plotted as a function of the load. A curve-fitting procedure may be applied to obtain estimates of the buckling load, which occurs when the fundamental frequency decreases to zero. Often such predictions lead to a value that is higher than the actual buckling load. It would be useful if both lower and upper bounds on the buckling load could be obtained from the vibration data. In this technical note, the convexity properties of the curves of load versus a power of the frequency are utilized to predict such bounds. These ideas are demonstrated on typical curves associated with buckling at a limit point and a bifurcation point. The curves are derived from an example of a shallow elastic arch with pinned ends and a sinusoidal initial shape, which is subjected to a sinusoidal distributed load.

J.E.M. Jubb

Papers

Interrelation of structural stability, stiffness, residual stress and natural frequency

The effect of distortion on the lowest natural frequency of a rectangular steel plate