應用射線及模態展開法解析Timoshenko樑的暫態波傳

Abstract

樑的動態行為是工程領域中的一個重要問題。在眾多的樑理論假設中，古典樑理論(Bernoulli-Euler beam)因其簡單、提供合理工程近似等特點，較為常用，但其有高估共振頻及波速無上限的缺陷；提摩盛科樑理論(Timoshenko beam theory)雖較為複雜，但不僅波速有上限，且其穩態反應與精確樑理論(exact theory)有不錯的一致性，因此在動態分析上，提摩盛科樑理論較為合適。本篇論文將探討四個不同的暫態樑問題。 本文主要以兩種不同的解析方法-射線及模態展開法處理提摩盛科樑的動態問題。利用射線法準確、適宜計算短時間等特點，作模態展開法的標竿，來處理較長時間的反應，並提出動態和靜態結果對照，以及頻率域下的特性；另一方面，也以模態展開法處理古典樑來與提摩盛科樑理論比較，依此提出適用古典理論的樑尺寸。此外，本文使用疊加法和接觸理論以模態展開法解析鋼珠撞擊簡支樑的問題。而在懸臂樑方面，亦以模態疊加法導出數學的封閉解。

The topic of dynamics of beams is important in engineering. Among different beam theories, Bernoulli-Euler beam is most widely used owing to simplicity and reasonability. However, for analyzing dynamic problems, the Timoshenko beam is more appropriate. This thesis applies two approaches – ray and normal mode method to deal with the transient response of Timoshenko beam. Ray solution is most accurate and suitable for predicting short time responses and the normal mode method can treat long time responses. Thus, we use the result obtained by ray solution as a standard for the normal mode method to calculate the long time response. Furthermore, the comparison of dynamic and static results is proposed and the frequency responses are discussed as well. On the other hand, the normal mode solution of Bernoulli-Euler beam is demonstrated and we compare its results with Timoshenko beam. The suitable slender ratio of Bernoulli-Euler beam for the analyzing transient displacement response is also presented. In addition, we analyze the responses of the simply supported beam subjected to impact of a steel ball problem. The normal mode solution of the cantilever beam subjected to constant impact force problem is also derived in this study.