摩擦鐘擺型調諧質塊阻尼器之最佳化設計研究

Abstract

本文旨在探討結構加裝單一FPS型TMD之最佳化研究，考慮定摩擦與變摩擦兩種摩擦系統，其中定摩擦系統之摩擦係數為定值，不隨著摩擦面位置不同而改變；變摩擦系統之摩擦係數則為變數，隨著摩擦面位置不同而改變。首先，將動力平衡方程式轉換成一階狀態方程式，以求得離散時間系統之狀態方程式來進行動力分析，在TMD之摩擦力計算方面，以一個方程式囊括兩種滑動行為 (滑動狀態與非滑動狀態) 而求出摩擦力。接著，推導出最佳化理論，於計算流程中採用迭代法並搭配golden section search以求取最佳化參數。再者，針對TMD設計參數，即質量比、頻率比及摩擦參數，進行參數分析 (靈敏度分析)，探討設計參數對減振之影響與效果。最後，以臺北101大樓為案例分析對象，模擬臺北101加裝FPS型TMD之減振效果，驗証FPS型TMD之可行性，並與原來的減振元件 (黏滯型TMD) 做一比較。分析結果顯示，FPS型TMD之擺動在小角度的前提下，最佳化理論確實可行，可達到良好的減振效果；而參數分析部分，頻率比對於結構減振相當靈敏，摩擦參數則較不靈敏；並由臺北101大樓的案例分析可知，只要設計得宜，不論是定摩擦系統、或者變摩擦系統之FPS型TMD，均可達減振目的，使其頂樓加速度峰值降至5 gal以下，而其中變摩擦系統之FPS型TMD減振效果最為顯著。

In this report, optimal design of a structure implemented with a friction pendulum system (FPS) typed tuned mass damper (TMD) is proposed. Two kinds of friction system are considered. One of them is fixed friction system whose friction coefficient is fixed and independent of relative displacement; the other one is variable friction system whose friction coefficient is variable and dependent on relative displacement. At first, by employing the first order state-space equation transferred from the motion equation of a structure implemented with a FPS typed TMD, the discrete-time state-space equation by which the time history analysis can be conducted is derived. In computing friction force, an equation which includes two kinds of motion behaviors (stick and slip states) is proposed. Then, the procedure of optimal theory is proposed to obtain the optimal design parameters of FPS typed TMD, and the iteration method and golden section search are used in the process. Moreover, the effect of design parameters to vibration suppression (sensibility analysis) is studied. Finally, as an example the structural response of Taipei 101 implemented with a FPS typed TMD subjected to designed wind force is simulated, and the feasibility is verified and compared with the original design of the nonlinear viscous TMD for Taipei 101. The result shows that optimal theory works but relies on an assumption: small angle of sliding of FPS typed TMD. In the analysis of parameters, it shows that vibration suppression is very sensitive to frequency ratio, but not to friction parameters. In case study, once the TMD is properly designed, no matter fixed or variable FPS typed TMD, the requirement of comfort for Taipei 101 can be satisfied such that the peak acceleration of roof is reduced to less than 5 gal, and the variable FPS typed TMD is most effective one.