Abstract

Guided wave propagation and its dispersion phenomenon in infinite solid elastic rods are encountered in several applications including, mechanical and civil engineering fields. A robust root finding method based on the Padé approximation is presented for studying the dispersion characteristics of guided wave motion in a solid cylindrical rod with a circular cross-section. The theoretical analysis is conducted in the context of the theory of elasticity for deriving the simplified solution for the classic Pochhammer frequency equation. In this research work, dispersion curves of fundamental modes, namely, torsional $T(0,m),$ longitudinal $L(0,m)$ and flexural $F(n,m)$ modes, are studied adequately. Based on the Padé approximants, the solution strategy for the original Pochhammer frequency equation is improved. The calculation results herein show that this method is stable and efficient under the condition of guaranteeing an absolute wavenumber calculation accuracy, which can avoid the leakage of its close root in the curve. The dispersion curves obtained from the root-finding method by Padé approximation are compared with the theoretical ones obtained by solving the corresponding Pochhammer frequency equation. It is observed that they coincide well with each other even in the region where the slope changes sharply, meaning that the proposed rooting strategy can be favorable for the rapid convergence of the theoretical solution. It is also extended to the solution of the wavenumber-frequency curve of the complex wavenumber domain in the viscoelastic waveguide. Henceforth, the proposed rooting approach can be employed to determine the dispersion characteristics of multiple modes in a cylindrical waveguide.

摘要

无限固体弹性棒中的导波传播及其色散现象在包括机械和土木工程领域在内的多种应用中遇到。提出了一种基于 Padé 近似的稳健寻根方法，用于研究圆形横截面实心圆柱杆中导波运动的色散特性。理论分析是在弹性理论的背景下进行的，以推导经典 Pochhammer 频率方程的简化解。在这项研究工作中，基本模式的频散曲线，即扭转$吨(0,米),$纵$\mathrm{大号}(0,米)$和弯曲的$F(n,米)$模式，进行了充分的研究。基于Padé近似，改进了原始Pochhammer频率方程的求解策略。本文的计算结果表明，该方法在保证绝对波数计算精度的情况下是稳定高效的，可以避免其近根在曲线中的泄漏。将通过Padé逼近求根法得到的频散曲线与求解相应Pochhammer频率方程得到的理论频散曲线进行比较。可以观察到，即使在斜率急剧变化的区域，它们也能很好地吻合，这意味着所提出的生根策略有利于理论解的快速收敛。还推广到了粘弹性波导中复波数域的波数-频率曲线的解。此后，所提出的生根方法可用于确定圆柱形波导中多模的色散特性。