以有限元素法分析三維旋轉傾斜尤拉梁的穩態變形與自由振動

Abstract

摘要 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）推導旋轉傾斜尤拉梁的運動方程式，探討具任意傾斜角與設定角之等速旋轉傾斜尤拉梁的穩態變形及以該穩態變形為平衡點的自然振動頻率。 本文將旋轉梁的運動方程式建立在一個剛接在其轉軸的總體座標上，本文在梁元素當前的變形位置上建立元素座標，當前的元素座標與總體座標有相同的速度、加速度、角速度、角加速度。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。元素的剛度矩陣是由元素的節點變形力對節點參數的微分求得，元素的向心力剛度矩陣(centripetal stiffness matrix)、質量矩陣(mass matrix)、陀螺矩陣(gyroscopic matrix) 是由元素的節點慣性力分別對節點參數的微分、節點參數對時間之二次微分的微分、節點參數對時間之一次微分的微分求得。為考慮軸向、扭轉及兩個撓曲變形間的耦合，元素的節點變形力中保留節點參數和其微分到二次項以及扭轉率的三次項，因本穩考慮之振動為微小的振動，元素的節點慣性力中僅保留節點參數和其對時間之微分到一次項。 將系統的非線性運動方程式中對時間的微分項去掉即為系統的穩態平衡方程式，將系統的運動方程式用泰勒級數在穩態變形的位置展開，取到一次項，即為旋轉梁微小振動的運動方程式。 本文利用基於牛頓法的增量迭代法求出軸向、扭轉及兩個側向位移的穩態解。旋轉傾斜梁的頻率方程式為一組代數齊次方程式，該組齊次方程式為一個二次特徵值問題，其係數形成之矩陣的行列式為零時的根，即為自然振動頻率，因該組方程式中存在陀螺矩陣，故其自然振動頻率所對應的振動模態為複變數。本文以二分法來求行列式為零時的根。 本研究以無因次化的數值例題，探討不同梁斷面、設定角、傾斜角、無因次旋轉速度以及無因次轉軸半徑對旋轉Euler梁之穩態變形、自然頻率及振態的影響。

Abstract In this paper a co-rotational finite element formulation combined with the floating frame method is proposed to derive the equations of motion for a rotating inclined Euler beam at constant angular velocity. The steady state deformation and natural frequency of the infinitesimal free vibration measured from the position of the corresponding steady state deformation are investigated for rotating inclined Euler beams with arbitrary setting angle and inclination angle. The equations of motion of the rotating beam are defined in a global moving coordinates rigidly tied to the hub of the rotating beam. The element coordinates are constructed at the current configuration of the beam element. The velocity, acceleration, angular velocity, and angular acceleration of the current element coordinates are set to be the same as those of the global coordinates of the rotating beam. The element deformation nodal forces and inertia nodal forces 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 element stiffness matrix may be obtained by differentiating the element deformation nodal forces with respect to the element nodal parameters. The element centripetal stiffness matrix, mass matrix, and gyroscopic matrix may be obtained by differentiating the element inertia nodal forces with respect to the element nodal parameters, the second time derivative of the element nodal parameters and the first time derivative of the element nodal parameters, respectively. In order to include the nonlinear coupling among the bending, torsional, and stretching deformations, the terms up to the second order of deformation parameters and their spatial derivatives, and the third order term of twist rate are retained in element deformation nodal forces. However, only infinitesimal free vibration is considered here; thus only the terms up to the first order of deformation parameters, and their spatial derivatives and time derivatives are retained in element inertia nodal forces. The steady state equilibrium equations may be obtained by dropping the terms of the time derivatives in the equation of motion. The governing equations for linear vibration may be obtained by the first order power series expansion of the equation of motion at the position of the corresponding steady state deformation. The frequency equation for free vibration of rotating inclined beam is a quadratic eigenvalue problem. An incremental-iterative method based on the Newton-Raphson method is employed for the solution of nonlinear steady state equilibrium equations. The natural frequencies are determined by solving the quadratic eigenvalue problem using the bisection method. Numerical examples are studied to investigate the steady state deformations and the natural frequencies of rotating inclined Euler beams with different cross sections, inclined angles, setting angles, angular velocities, radiuses of the hub, and slenderness ratios.