具設定角、傾斜角及預錐角之三維旋轉雙對稱開口薄壁梁在等角速度下的穩態變形及自由振動分析

Abstract

本研究為兩年期計畫。本研究主要利用共旋轉有限元素法結合結合浮動框架法（floating frame method）推導具傾斜角、設定角及預錐角之旋轉雙對稱開口薄壁梁的運動方程式， 探討具任意預錐角、傾斜角與設定角之等速旋轉梁的穩態變形及以該穩態變形為平衡點 的自然振動頻率。 本研究將以 Euler 梁正確的變形機制及一致性共旋轉法推導一個二節點十四個自由度的 雙對稱開口薄壁旋轉梁元素之運動方程式，本研究擬將旋轉梁的運動方程式建立在一個 以等角速度旋轉的總體座標上。旋轉梁的運動包含軸向位移、軸向扭轉及兩個側向位移 的穩態變形及以穩態變形為平衡點的自由振動，本研究考慮的自由振動為微小的振動。 第一年，本研究擬利用 D’Alembert 原理、虛功原理及完全非線性梁理論的一致二階線 性化推導元素節點變形力及慣性力，再將元素的節點變形力對節點參數的微分求得元素 的剛度矩陣，將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)。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將保留節點參數和其微分到二次項以 及扭轉率的三次項，因本研究考慮之振動為微小的振動，元素的節點慣性力中將僅保留 節點參數和其對時間之微分到一次項。為了探討上的方便，本研究擬將元素節點變形 力、慣性力及元素剛度矩陣、各種慣性矩陣無因次化。 本研究擬將系統的非線性運動方程式中對時間的微分項去掉得到系統的穩態平衡方程 式，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 第二年，本研究擬再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，求得旋轉梁微小振動的運動方程式。旋轉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。本文擬以二分法來求行列式為零時的根。本研究擬以無因次化的數值例題，探 討不同梁斷面、預錐角、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 梁之穩態變形、自然頻率及振態的影響。

This study is a two-year project. The beam considered is rigidly tied to a rotating hub at constant angular velocity. A co-rotational finite element formulation combined with the floating frame method will be proposed to derive the equations of motion for a rotating doubly symmetric thin-walled open-section beam with precone angle, setting angle and inclination angle at constant angular velocity. A two-node doubly symmetric thin-walled open-section beam element with seven degrees of freedom per node will be developed. The steady state deformation and natural frequency of the infinitesimal free vibration measured from the position of the corresponding steady state deformation will be investigated for the rotating beam. 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, gyroscopic matrix, and mass matrix may be obtained by differentiating the element inertia nodal forces with respect to the element nodal parameters, the first time derivative of the element nodal parameters and the second 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. Dimensionless numerical examples will be studied to investigate the steady state deformations and the natural frequencies of rotating beams with different cross sections, precone angle, inclined angles, setting angles, angular velocities, radiuses of the hub, and slenderness ratios.