Numerical Stability of the Transfer Matrix Method Based on Transcendental Functions for Vibration Analyses of the Structures

This study analyzes the cause of three numerical instabilities for the transfer matrix method (TMM) by controlling the precision of the arbitrary digits in the numerical device as follows: (1) the missing natural frequency, (2) the inaccurate results, and (3) the computational difficulty of the high-order natural frequency. Two types of TMMs are examined. One TMM is composed of the trigonometric and hyperbolic functions (4×4 TMM) for the transverse vibration of a straight beam, and the other is composed of exponential functions (8×8 TMM) for the coupled bending vibrations of a twisted beam. TMMs with numerical stability can obtain the eigenpairs of over 220 up to the overflow (21024) in the numerical device. The explicit range capable of computing the eigenpairs using the 4×4 and 8×8 TMMs without any modifications is proposed in the analyzed results.