陀螺應用於結構制振的評估

Abstract

都市化的發展使人群往城市集中，為了解決空間不足的問題，都市大樓越蓋越高。但越高的建築物對地震、陣風等外來刺激越脆弱，必需配備能減少建築物振動的裝置。高速自轉的陀螺因陀螺儀效應，當自轉軸傾斜時會產生可觀的反制力矩抵抗晃動，有成為建築物制振器的潛力。本論文探討陀螺應用於結構減振的可能與效果。 本研究的陀螺在受激振前為直立，這個姿態以常用的3-2-3尤拉角描述為奇異點，無法進行數值積分。為了避開奇異點，本論文改採1-2-3尤拉角來推導陀螺的運動方程式。首先驗證不同尤拉角系統間轉換關係的正確性，接著探討陀螺轉速對陀螺運動的影響。 在充分了解陀螺運動後，設計出兩種陀螺用於結構減振的簡易模型，分別是建築只能平面運動的單層模型，與模擬建築主軸因振動而傾斜的雙層模型。由Lagrange方程式求得運動方程式，再以數值積分模擬結果，探討脈衝激振與簡諧激振下，提升陀螺轉速的減振效果。 本論文討論兩種簡易的建築模型，分別是建築只能在平面運動的單層模型，與建築主軸可以傾斜的雙層模型。由Lagrange方程式推導建築模型配備陀螺系統的運動方程式。探討建築系統在兩種基底激振，分別是脈衝激振和簡諧激振，下的反應。利用數值方法討論陀螺轉速對建築制振效果影響。

Urbanization attracts people to cities and results in high population density. To provide enough living space for the residents, high-rise buildings are becoming increasing popular. A high-rise building suffers from the sway induced by wind and seismic activities and should be equipped with vibration suppression devices. Due to the gyroscopic effect, a high speed gyroscope can provide a strong reacting moment to resist the change of rotation direction due to external disturbances. As a consequence, a high speed gyroscope has the potential to be an effective device for the suppression of structure vibrations. This thesis aims to evaluate the ability of a gyroscope for structural vibration suppression. The gyroscope stands vertically in the undisturbed configuration, which coincides with the singular point of the conventional 3-2-3 Euler angles. To avoid this singularity, the 1-2-3 Euler angles are used to describe the configuration of the gyroscope. We first derive the transformation between these two sets of Euler angles and then discuss the effects of spin speed on the behavior of the gyroscope. This thesis discusses two simple structural models: one is a single floor model which is restricted to move in the horizontal plane; the other is a two-floor model which incorporates the bending deformation effects of a real structure. Lagrange equations are used to derive the governing equations of the structure-gyroscope system. The response of the structure with and without the gyroscope under impulsive and sinusoidal base excitations are determined via numerical integration. In this case, the effects of the spin rate of the gyroscope on the structure vibrations are investigated thoroughly.