旋轉三維Timoshenko梁之運動方程式及穩態解

Abstract

本研究主要目的是推導一三維Timoshenko 旋轉梁的運動方程式，並由此運動方程式求得旋轉梁的穩態平衡方程式，並求在不同轉速時，軸向變形與扭轉變形的數值穩態解。本文將旋轉梁分割成若干段，稱每一段為一梁元素，並提出一三維Timoshenko梁的變形機制。利用非線性梁理論的一致線性化、虛功原理、d’Alembert原理及共旋轉法在旋轉元素座標上推導出三維Timoshenko旋轉梁元素正確的運動方程式，並由此運動方程式求得旋轉梁的穩態平衡方程式。 本文利用Galerkin 法求得旋轉梁穩態時轉速與節點變形參數間的穩態非線性平衡方程式，再利用基於牛頓法的增量迭代法求得其解。本文並以數值例題探討不同的轉速時，不同的設定角、不同的斷面、不同的長度對旋轉梁穩態軸向及扭轉變形的影響。

The objective of this paper is to derive the equations of motion and to solve the steady state axial and torsional deformations for the doubly symmetric three dimensional rotating Timoshenko beam. A co-rotational formulation combined with the rotating frame method is used here. The rotating beam is divided into several beam elements. The kinematics of beam element is defined in terms of rotating element coordinates, which are constructed at the current configuration of the beam element. The equations of motion of the beam element are derived by consistent linearization of the fully geometrically nonlinear beam theory using the d’Alembert principle and the virtual work principle in the current rotating element coordinates. The steady state equilibrium equations of the rotating beam element can be obtained from the equations of motion of the beam element. A Galerkin method is applied to the steady state equilibrium equations and an incremental iterative method based on the Newton-Raphson method is used here for the solution of the nonlinear steady state equilibrium equations. Numerical examples are studied to investigate the effect of rotating speed, setting angle, cross section of the beam and length of the beam on the steady state axial and torsional deformations for the three dimensional rotating Timoshenko beam.