滑動梁之動態分析

Abstract

本研究的主要目的為提出一簡單有效的共旋轉有限元素法及一數值程序，探討滑動梁的幾何非線性動態反應。為了正確的描述及預測滑動梁的動態反應，本研究考慮了在稜柱形導槽內外之梁的運動。在本研究中將梁元素分為二種，第一種是普通梁元素，在稜柱形導槽外時，該元素的運動不受限制，在稜柱形導槽內時，該元素只能在軸方向運動。第二種梁元素為本研究提出的一個特別元素，稱為轉接梁元素，該元素有一部分在稜柱形導槽內，另一部分在稜柱形導槽外。轉接梁元素未變形的長度為一固定長度，但其在稜柱形導槽內的部分變形前的長度則為時間的函數。本研究在梁元素當前的變形位置上建立元素座標，並在元素座標上以正確的變形機制推導普通梁元素及轉接梁元素的節點內力及剛度矩陣。本研究採用基於Newmark直接積分法及Newton-Raphson法的增量迭代法求解非線性動態平衡方程式。本研究以數值例題探討滑動梁結構受不同負荷及端點軸向運動的幾何非線性動態行為並與文獻的結果比較，以說明本研究中提出的方法的準確性及有效性。

A simple and effective consistent co-rotational total Lagrangian finite element formulation and a numerical procedure are proposed to investigate the geometric nonlinear dynamic response of sliding beam. To exactly predict the dynamic response of the sliding beam, the total length of the sliding beam is considered. The motion of the beam element is not restrained when it is outside the prismatic joint. The lateral motion of the beam is fully restrained when it is inside the prismatic joint. The ordinary beam element is used here when it is inside or outside the prismatic joint. A transition beam element developed here when it is partially housed inside the prismatic joint. The total undeformed length of the transition element is constant. However, the undeformed length housed inside the prismatic joint is time dependent. The kinematics, deformations, and equations of motion of the transition beam element are defined in terms of two element coordinate systems constructed at the current configuration of the deformed beam element. The principle of virtual work, d’Alembert principle and the consistent second order linearization of the fully geometrically nonlinear beam theory are used to derive the deformation nodal force and inertia nodal force of the beam element. In element nodal forces, all coupling between bending and stretching deformations of the beam element is considered. An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson method is employed for the solution of nonlinear dynamic equilibrium equations. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.