旋轉滑動梁之動態分析

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數值計算程序，以探討旋轉滑動梁的幾何非線性動態反應。為了正確的描述及預測旋轉滑動梁的動態反應，本研究考慮了在稜柱形導槽內外梁的運動。 在本研究中將梁元素分為二種。第一種為普通梁元素，其在稜柱形導槽外時，該元素的運動不受限制，而在稜柱形導槽內時，該元素與導槽一起剛體旋轉且相對於導槽在導槽的軸向運動；第二種梁元素為一特別元素，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。梁元素的變形是在元素座標上描述，此座標系統為附著在每一個元素，且建立在該元素當前的變形位置上，並在元素座標上以完整的非線性梁理論及利用虛功原理推導普通梁元素及轉接梁元素的節點內力和剛度矩陣。 為了方便描述系統的運動，本研究在導槽當前的位置上建立一導槽座標系統，並在該座標系統建立旋轉的運動方程式。 本研究使用基於Newmark直接積分法和Newton-Raphson法的增量迭代法來求解非線性運動方程式。本研究最後以文獻上的數值例題驗證本研究所提出之方法的準確性及可行性，並分析受不同既定位移及旋轉的旋轉滑動梁以探討其幾何非線性的動態反應。由本文分析的結果可知，受不同初始條件之旋轉滑動梁的側向動態反應可由其單一自由度之等效系統來預測或解釋。

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 the rotating sliding beam. To exactly predict the dynamic response of the rotating sliding beam, the total length of the rotating 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 coincides with the rotation of the prismatic joint and is restrained relative to the prismatic joint 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 is developed here when it is partially housed inside the prismatic joint. The kinematics, and deformations of the beam element are defined in terms of the element coordinate system 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. To conveniently describe the motion of the system, the equations of motion of the system are defined in terms of the prismatic joint coordinate system constructed at the current configuration of the prismatic joint. 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. From the results of the numerical examples, it seems that the lateral dynamic response of the rotating sliding beam with different initial conditions may be predicted or explained by an equivalent spring-mass system with single degree of freedom.