本文主要討論應用於椼架結構減震之線性摩擦耗能器的平衡化配置設計問題。此 類線性摩擦耗能器係屬於被動式能量耗散器的一種；在週期負載作用下，它會因負載增加或 負載釋放而呈現不同之勁度，以至於有三角形之遲滯迴圈的非線性現象產生。本文在設計此 類線性摩擦耗能器的最佳化擺設配置時，則採用線性模型作為設計基準。首先吾人探討有現 維度系統之能量平衡，其結果顯示 Hankel 奇異值可用於連接系統之可控制性與可觀測性， 同時結合所有 Hankel 奇異值更可表現系統中控制每個狀態時所需之致動器能量以及觀測每 個狀態時所需之感測器靈敏度。因此，吾人採用系統之 Hankel 範數的平方 -- 亦即系統未 來之輸出能量與過去之輸入能量的比值 -- 為配置指標並作為耗能器配置之良窊的標準。針 對具有不同耗能器配置之椼架結構，當結構之配置指標愈小則表示其耗能器配置設計愈佳。 由於此配置指標亦可表成系統 Hankel 奇異值的函數，因此在計算結構體的配置指標之前， 必須先將結構之振動方程式轉化成模態狀態方程式，進一步將之轉成平衡化狀態方程式之表 示式以計算 Hankel 奇異值而求得配置指標。 當結構體之阻尼係數遠小於 1 時， 第Ⅰ個 Hankel 奇異值即為第Ⅰ個振動模態之頻率和模態阻尼係數的比值。 反之，當結構體之阻尼 係數並未遠小於 1 時，吾人提供一演算法則以計算系統之 Hankel 奇異值。 為了驗證使用 此配置指標來篩選椼架結構之線性摩擦耗能器配置設計之可行性，本文針對不同配置之椼架 結構探討在任意外來干擾作用下，耗能器之減振效果。研究結果顯示當椼架結構配置指標愈 小，其減振效果愈佳，換言之，耗能器配置設計愈佳。 This paper is concerned with te placement of linear-friction dissipators for truss structures. This type of passive energy dissipator shows different stiffness in loading and unloading, leading to triangular hysteresis loops under cyclic loading. The linearized model of the dissipator and the energy balancing technique are used to design the best placement problem. After discussing the energy balancing for linear finite dimensional system, the result shows that the Hankel singular values can serve as the joint controllability and observability and we can combine these values to indicate the actuator power and sensitivity of sensor measurement. In order to have a good strategy for dissipator placement, the energy ratio of future system output energy to the pass control input energy is proposed as a placement index, which is a function of the Hankel singular values. The smaller the index, the better damping effect is provided. In the case of very small damping, i.e. the square of the damping coefficient is much less than unity. the Hankel singular values can be solved in the modal state-space repressentation and can be expressed as a combination of modal frequency and modal damping coefficient for different vibration mode, independently. For truss structure with large damping, the Hankel singular values must be solved in state-space representation, and a computational algorithm is provided to compute the design index. Varieties of different dissipator placements of truss structure are discussed based on the criterion given by placement index and these numerical examples show a satisfactory results.
