開口薄壁梁在軸向負載作用下之自然振動分析

Abstract

本研究使用共旋轉全拉格朗日法探討點對稱的三維開口薄壁梁承受合力通過斷面形心的軸向負載作用下之幾何非線性靜態行為及在靜態平衡位置的微小振動，並且考慮因軸向負載所引起的雙力矩。 本研究採用的梁元素有兩個節點，每個節點有七個自由度，元素節點定於梁元素兩端斷面的剪心，並以剪心軸當元素的參考軸。本研究描述梁元素的運動是在元素當前的變形位置上所建立的元素座標系統，將元素座標系統視為一固定的局部座標，因此梁元素在元素座標系統的速度及加速度即為絕對速度與絕對加速度。梁元素的節點變形力、節點慣性力是利用非線性梁理論、D’Alembert原理和虛功原理及一致性二階線性化在當前的元素座標上求得。 將非線性運動方程式的慣性項去掉，即為靜態的非線性平衡方程式。將運動方程式在靜態平衡位置用泰勒級數展開取到一次項即為在靜態平衡位置下之線性振動的統御方程式。本研究採用結合牛頓-拉福森法與弧長法的增量迭代法求解非線性平衡方程式。本研究採用次空間法(Subspace Iteration Method)求解梁結構的自然頻率及振動模態。 本研究以數值例題探討軸向負載及因軸向負載產生的雙力矩對不同長度、邊界條件的Z形斷面梁的臨界狀態、臨界負荷及自然頻率的影響。本研究的目的是分析固定軸向負載所引起的雙力矩對點對稱開口薄壁斷面梁之自然運動的影響，利用Vlasov’s理論的運動微分方程式之通解可推導出梁在各種邊界條件下的自由振動解析解。為了研究雙力矩對自然頻率的影響，數值例題將以對稱Z形對面梁為例，將研究結果使用ANSYS有限元素模型驗證，證明雙力矩對扭轉自然頻率的評估有很大的影響。本研究考慮的合力通過特定斷面形心並且與斷面垂直且合力的方向與結構的長軸平行。

The geometrical nonlinear static behavior and infinitesimal free vibration around the static equilibrium position are studied using total Lagrangian finite element method for three dimensional thin-walled beams with point-symmetric open section subjected to axial load with its resultant passing through the centroid of beam cross section. The bimoment induced by axial load is considered in this study. The element employed here has two nodes with seven degrees of freedom per node. The element nodes are chosen to be located at the shear center of the end cross sections of the beam element and the shear center axis is chosen to be the reference axis. The kinematics of the beam element is described in the current element coordinate system constructed at the current configuration of the element. The current element coordinate system is regarded as an inertial local coordinate system. Thus, the first and the second time derivative of the position vector defined in the element coordinates are the absolute velocity and absolute acceleration. The element deformation nodal forces and inertia nodal forces are systematically derived by the d'Alembert principle, the virtual work principle and consistent second order linearization in the current element coordinates. The equilibrium equations may be obtained by dropping the terms of the inertia forces in the equation of motion. The governing equations for linear vibration around the static equilibrium position are obtained by the first order Taylor series expansion of the equation of motion at the static equilibrium position. 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 the nonlinear equilibrium equations. The subspace iterative method is used for the solution of natural frequencies and vibration modes for the free vibration. Numerical examples are studied to investigate the effects of the axial load and bimoment induced by axial load on the critical state, critical load and the natural frequencies of z cross section beams with different lengths and boundary conditions under axial loading. The objective of the paper is to analyze the influence of bimoment induced by constant axial loads on the free motion of thin-walled beams with point-symmetric open cross- section. For various boundary conditions, a closed-form solution for natural frequencies of free harmonic vibrations was derived by using a general solution of governing differential equations of motion based on Vlasov’s theory. In order to investigate the effect of the bimoment on natural frequencies, the numerical examples with symmetric Z cross-section are given. The obtained results, verified using an ANSYS finite element model, demonstrate that the influence of the bimoment is important in the assessment of torsional natural frequencies. A force with its resultant passing through the centroid of a particular section and being perpendicular to the plane of the section. A force in a direction parallel to the long axis of the structure