Numerical Evaluation of High-Order Modes for Stepped Beam

The numerical evaluation of high-order modes of a uniform Euler–Bernoulli beam has been studied by reformatting the classical expression of mode shapes. That method, however, is inapplicable to a stepped beam due to the nonuniform expressions of the mode shape for each beam segment. Given that concern, this study develops an alternative method for the numerical evaluation of high-order modes for stepped beams. This method effectively expands the space of high-order modal solutions by introducing local coordinate systems to replace the conventional global coordinate system. This set of local coordinate systems can significantly simplify the frequency determinant of vibration equations of a stepped beam, in turn, largely eliminating numerical round-off errors and conducive to high-order mode evaluation. The efficacy of the proposed scheme is validated using various models of Euler–Bernoulli stepped beams. The principle of the method has the potential for extension to other types of Euler–Bernoulli beams with discontinuities in material and geometry. (The Matlab code for the numerical evaluation of high-order modes for stepped beams can be provided by the corresponding author upon request.)