High frequency flexural vibration of thick rectangular bars and plates

Abstract

With the Timoshenko equation for the bar and the Mindlin equation for the plate Bolotin's solution fitting technique is employed to provide estimates of natural frequencies and approximate modal shapes for finite bars and plates. Unlike Bolotin's boundary region solutions, which satisfy only the Euler-Bernoulli equation in the case of the beam and the Lagrange equation in the case of the plate, the solutions adopted here satisfy the Timoshenko equation for the bar and the Mindlin equation for the plate. Computed natural frequencies of thick bars show excellent agreement with known exact solutions for thick beams in which the effect of shear and rotary inertia are taken into account. Exact solutions for thick plates could be found only for plates with simply supported edges. As expected excellent agreement is obtained with known solutions for this case, but comparisons with finite layer solutions for plates with other boundaries show some discrepancies, which seem to arise from anomalies in the finite layer solutions.