使用移動四自由度的彈簧－阻尼－質量元素來進行承受移動負荷之長方形板的振動分析

VIBRATION OF A RECTANGULAR PLATE DUE TO MOVING LOADS BY USING MOVING FOUR-DEGREE-OF- FREEDOM SPRING-DAMPER-MASS ELEMENT

本文提出移動四自由度彈簧－阻尼－質量元素（由一個集結質量m_(zv)、二個滾輪質量m_(zri) (i = 1,2)、二個彈簧k_(zi) (i = 1,2)與二個阻尼c_(zi) (i = 1,2)所組成）的理論來進行承受移動負荷之長方形板的振動分析，以便將移動負荷的旋轉慣性效應、平移慣性效應、彈簧效應與阻尼效應納入考慮。在本研究中，上述移動四自由度彈簧－阻尼－質量元素的元素特性矩陣是由移動彈簧－阻尼－質量(SDM)系統的動態平衡方程式推導而得。由於彈簧－阻尼－質量系統與支撐長方形板間的交互作用效應已利用形狀函數將其考慮在上述元素特性矩陣內部，因此，元素特性矩陣將隨其瞬時位置而變化，而整體振動系統的特性矩陣也將隨著變化。最後，本文將利用Newmark直接積分法來求解整個結構系統的運動方程式，並計算整體結構系統的動態反應。一些與本研究相關的重要參數（例如：移動SDM系統的速度、彈簧常數、集中質量的中心位置）將加以探討，由本文所提出的數值結果可以發現，上述參數對移動SDM系統與長方形板的影響相當大。

To take the pitching, inertia, spring and damping effects of moving load into account, the moving four-degree-of-freedom spring-damper-mass element, which consists of a lumped mass m_(zv), two roller masses m_(zri) (i = 1,2), two springs k_(zi) (i = 1,2) and two dampers c_(zi) (i = 1,2), is presented in this paper. Based on the dynamic equilibrium equations of the four-degree-of-freedom (dof) moving spring-damper-mass (SDM) system, the element property matrices of the presented element are derived. The interactions between the moving SDM system and the plate are considered, by means of shape functions, in the last matrices. Because the element property matrices of the presented element vary with its instantaneous position on the plate, they are time-dependent matrices, so are the overall property matrices of the entire vibrating system. Dynamic responses of the structural system are calculated by solving the equations of motion of the entire vibrating system with Newmark integration method. Some factors, such as moving speed, spring constants and the position for centre of gravity of lumped mass of the SDM system, closely relating to the title problem are investigated. Numerical results show that the influences of the foregoing parameters on the dynamic responses of the moving subsystem and the plate are considerable.