A segmenting scheme for evaluating exact high-order modes of uniform Timoshenko beams

Abstract

Numerical evaluation of exact high-order modes of uniform Timoshenko beams is a classic problem in the field of acoustics and vibration. In the classic method, however, the determinant of the frequency matrix dominated by hyperbolic functions can increase exponentially and quickly reach the precision limitation of a computer. As a result, a bottleneck occurs in the evaluation of high-order modes due to round-off errors. Addressing this bottleneck, this study proposes an enhanced classic method relying on a segmenting scheme for evaluating exact high-order modes of uniform Timoshenko beams. With the segmenting scheme, a uniform Timoshenko beam is uniformly segmented into several segments, whereby the hyperbolic functions involved in the determinant of the frequency matrix become much smaller for the same given frequency. Accordingly, under the fixed precision limitation in a computer, higher-order modes can be obtained. The capacity of this enhanced classic method for evaluating exact high-order modes is validated by scenarios of a uniform Timoshenko beam with different numbers of segments. The results show that the high-order modal frequencies and mode shapes can be properly obtained. The accuracy of the modal frequencies is verified by the well-established exact dynamic stiffness method with the Wittrick-Williams algorithm. The results show that the modal frequencies are highly accurate. Applied to beams of different materials, the artificial segmenting is more suitable than the natural segmenting in evaluating high-order modes.