應用模態疊加法之阻尼器最佳化配置

Optimal Placement of Dampers in Building Structures Using Mode Superposition Method

張耿毓(Keng-Yu Chang)

指導教授 ： 呂良正

國立臺灣大學/工學院/土木工程學研究所/碩士(2013年)

https://doi.org/10.6342/NTU.2013.00859

摘要

阻尼器最佳化配置研究必須對結構物進行動力分析以求得結構物的反應，但加裝阻尼器的結構系統往往會成為非古典阻尼結構系統，對於非古典阻尼系統一般都以直接積分法(Newmark β、差分法等)求得結構物的反應，但直接積分法於自由度大的結構物需要大量計算時間，影響阻尼器最佳化的計算效率。
本研究整理現行文獻中關於三種非古典阻尼動力系統的模態疊加方程式，前兩種利用質量、勁度和阻尼矩陣組成的特徵值問題，以其正交性質求出解耦方程式後再導出其模態疊加式，第三種使用特徵分解的概念，並將狀態空間運動方程式進行Laplace Transform導出模態疊加式。第一種模態疊加式需將過阻尼模態實數特徵值進行配對，求得對應的模態頻率和模態阻尼比後進行疊加，第二、三種模態疊加式則將過阻尼模態一一疊加。本研究證明三種模態疊加式本質上相同，而且經過案例分析模態疊加式位移反應歷時與直接積分法相同。同時可以根據其模態疊加式發展其反應譜法，來估計結構物的最大反應。
本研究將此模態疊加式和其反應譜法以及直接積分法為動力分析方法應用於阻尼器最佳化配置的簡易法，進行平面剪力屋架的阻尼器最佳化配置分析，分析結果為使用模態疊加法的最佳化配置與直接積分法相同，同時模態疊加法擁有較好的計算效率。
為了將阻尼最佳化應用於真實的結構物，本研究分析三維不對稱結構物阻尼器最佳化配置並應用模態疊加法和直接積分法為動力分析方法。首先合理的簡化結構物的自由度讓其整體動力反應以質心自由度描述，再利用結構分析軟體SAP 2000分析出結構物的質量和勁度矩陣，阻尼器矩陣依幾何關係求得。最後將簡易法概念用至三維不對稱結構物上，最佳化結果使用模態疊加法的最佳化配置與直接積分法相同，同時模態疊加法擁有較好的計算效率。

關鍵字

非古典阻尼系統 ； 直接積分法 ； 二次特徵值問題 ； 解耦方程式 ； 模態疊加法 ； 阻尼器最佳化配置 ； 簡化循序搜尋演算法 ； 簡易法 ； 三維不對稱結構物 ； 雙向地表加速度

並列摘要

When optimize placement of dampers for structures, the responses of structures must be obtained. But structures with supplemental dampers are often non-classically damped systems. Generally, these systems can be analyzed by direct integration methods like Newmark beta method or central difference method so that the responses of the structures can be obtained. If the degree-of-freedom of the structures is considerable, the direct integration methods are time consuming. So it affects computational efficiency of optimizing placement of dampers.
This study presents three current researches of mode superposition methods for the non-classically damped systems. The first two methods are developed by the uncoupled equations which base on the eigenvalue problem consists of mass, stiffness and damping matrices. The third method is developed by using the concept of eigen-decomposition and transforming the state-space equation of motion by Laplace Transform. Then the first method must pair the real eigenvalues of overdamped modes with each other and obtain the corresponding modal frequencies and modal damping ratios and finally superimpose these paired overdamped modes. The second and third methods just superimpose all the overdamped modes one by one. This study proves these three mode superposition methods are the same. And these mode superposition methods can develop the corresponding response spectrum methods so that can estimate the maximun response.
This study applies the mode superposition method, the corresponding response spectrum methods and the direct integration method to optimize placement of dampers for planar shear frames. The optimal method use the Simple method proposed by Leu et al. (2010). The optimal results by using the mode superposition method are the same as using the direct integration method. Furthermore, the mode superposition method provide less analysis time than the direct integration method.
For more real structures, this study applies mode superposition method direct integration method to optimize placement of dampers for two-way asymmetric buildings. At first degree-of-freedom of the structures must be simplified resonable, so use degree-of-freedom of the center of mass to describe the dynamic responses of the whole structures. And then use the software SAP 2000 to obtain mass and stiffness matrices of the structures. Damping matrices can be determined by geometric relationship. Finally, use the Simple method for the two-way asymmetric buildings. The optimal results by using the mode superposition method are the same as using the direct integration method. Furthermore, the mode superposition methods provide less analysis time than the direct integration methods.

並列關鍵字

non-classically damped system ； direct integration method ； quadratic eigenvalue problem ； mode superposition method ； optimal placement of dampers ； Simplifie Sequential Search Algorithm ； Simple method ； Asymmetric buildings ； bi-directional seismic ground motion

參考文獻

黃婉婷 (2012)，應用反應譜分析法之阻尼器最佳化配置，國立台灣大學土木工程學研究所碩士論文。

洪意晴 (2008)，平面不對稱建築物樓層勁度與剛心位置之識別，國立成功大學土木工程研究所碩士論文。

呂良正、張仁德 (2011)，簡易法應用於三維不對稱多層房屋結構的阻尼器最佳化配置，結構工程，第二十六卷，第四期，17-30。

Caughey, T. K. and O’Kelly, M. E. J. (1965). “Classical normal modes in damped linear dynamic systems”, Journal of Applied Mechanics, ASME, 32, 583-588.

Lancaster, P. (1966). Lambda-matrices and vibrating systems, Pergamon Press, Oxford, New York.