挫屈梁之靜態與動態分析

Abstract

本研究主要利用共旋轉有限元素法結合浮動框架法（floating frame method）推導兩端點具旋轉角之挫屈梁受到基座簡諧振動的運動方程式，本文中提出一個數值程序，決定挫屈梁之初始形狀、挫屈梁受到各種負荷時的非線性靜態行為，以及受到基座簡諧振動與非對稱簡諧外力時的非線性動態行為。 本文將挫屈梁的運動方程式建立在一個與其基座有相同速度及加速度的總體座標上，本文在梁元素當前的變形位置上建立元素座標，元素座標與總體座標有相同速度及加速度。本文利用非線性梁理論的一致線性化、d’Alembert原理和虛功原理，在當前的元素座標上，推導梁元素的節點變形力、節點慣性力及剛度矩陣。本文將基座的絕對加速度造成的慣性力視為等效外力。 本文使用基於Newton-Raphson法及弧長控制法的增量迭代法來求解非線性平衡方程式，及採用基於Newmark直接積分法與Newton-Raphson法的增量迭代法求解非線性運動方程式。本研究先以數值例題探討兩端具相同旋轉角之挫屈梁的非線性靜態行為，本研究發現當兩端具旋轉角之挫屈梁受均佈側向載重時，其主要平衡路徑上之分歧點與極限點的力負荷參會隨著端點轉角的增加先大幅增加再減小。本研究再以數值例題探討挫屈梁受到基座簡諧振動與非對稱簡諧外力時的主共振現象、超諧共振現象及跳躍現象，本研究觀察到挫屈梁受到均勻的基座簡諧振動時，會引起挫屈梁的非對稱振動，並找出在不同基座振動頻率，產生跳躍現象所需的最小等效外力。

In this paper a co-rotational finite element formulation combined with the floating frame method is used to derive the equations of motion for a clamped-clamped buckled beam with end rotations subjected to base excitation. A numerical procedure is proposed for the determination of the initial shape of the buckled beam, nonlinear static behavior of the buckled beam subjected to lateral loads, and nonlinear dynamic behavior of the buckled beam subjected to uniform sinusoidal base excitation and asymmetric sinusoidal lateral load. The equations of motion of the buckled beam are defined in a global moving coordinates. The element coordinates are constructed at the current configuration of the beam element. The velocity and acceleration of the global coordinates and the current element coordinates are chosen to be the velocity and acceleration of the base of the buckled beam. The element deformation nodal forces, inertia nodal forces, stiffness matrix and mass matrix are systematically derived by consistent linearization of the fully geometrically non-linear beam theory using the d’Alembert principle and the virtual work principle in the current element coordinates. The inertia force of the buckled beam corresponding to the absolute acceleration of the base is regarded as an equivalent external load for the buckled beam. An incremental-iterative method based on the Newton-Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson method is employed here for the solution of the nonlinear equations of motion. Numerical examples are studied to demonstrate the accuracy and efficiency of the proposed method and to investigate the effect of end rotation on the nonlinear static behavior of buckled beam. When the buckled beam is subjected to uniform lateral load, it is observed that the loading parameters corresponding to the bifurcation point and limit point on the equilibrium path increase remarked and then decrease with the increase of the end rotation of the buckled beam. Numerical examples are also studied to investigate the primary resonance, superharmonic resonance and snap-through phenomenon for the buckled beam subjected to uniform sinusoidal base excitation and asymmetric sinusoidal lateral load. The asymmetric vibration induced by symmetric load is observed. The minimum loading parameters of uniform sinusoidal lateral load at different frequencies required to initiate the dynamic snap-through motion are determined.