非線性動態系統外力的閉合係數展開識別法

The closed-form coefficients expansion method to identify external forces of nonlinear dynamical systems

魏蒙(Meng Wei)

指導教授 ： 劉進賢 ； 鍾立來

國立臺灣大學/工學院/土木工程學研究所/碩士(2017年)

https://doi.org/10.6342/NTU201703572

摘要

本論文主要研究問題為非線性動態系統的外力識別問題。首先將含噪音訊號的二階微分問題轉換成二階常微分方程（ODE），如此會將原問題改寫成一個回復未知外力反算問題的特例。本論文利用擬時間積分法的技巧將ODE方程式轉換為一拋物線型的偏微分方程（PDE），如此便可以導入廣義格林第二定理，並再利用伴隨崔維茲（Trefftz）測試函數的概念推導出位移和外力的邊界積分關係式。接著利用弱形式方法的概念，先分別計算出弱形式之一階微分運算子和二階微分運算子，最後透過閉合係數展開識別法的假設，將未知外力展開為級數解的形式。透過量測位移資料並以矩陣形式逆轉換的技巧，在不需要任何迭代的情況下，將未知外力之係數以閉合解的形式解出，最後將其疊加回復成所求外力。最終可以將弱形式方法應用在噪音干擾下，回復長時間作用下各種非線性系統未知外力的反算問題當中。

關鍵字

弱型式微分運算子 ； 非線性反算振動問題 ； 伴隨崔維茲測試函數 ； 廣義格林第二恆等式 ； 擬時間積分法 ； 閉合形式數值微分器

並列摘要

In this thesis, we investigate the external force identification problem in a nonlinear dynamic system. For the recovery of unknown external force in the inverse vibration problems, we consider the second-order derivative of a noisy signal as a second-order linear ordinary differential equation (ODE). Then, it will turn the inverse problem into a special case of the recovery of unknown force in a second-order linear system. After that, we transform the linear ODE of motion into a linear parabolic-type partial differential equation (PDE) by the fictitious time integration method (FTIM), and then use the Green second identity to derive a boundary integral equation in terms of the adjoint Trefftz test functions. Further, we derive a weak-form method to compute the weak-form first-order and second-order differentiators (WFFOD&WFSOD) in terms of series expansion, of which the expansion coefficients can be determined exactly in closed-form without needing of any iterations. Finally, we can recover the external force for nonlinear structures within a large time span and under a large noise.

並列關鍵字

Weak-form differentiator ； Nonlinear inverse vibration problem ； Adjoint Trefftz test functions ； Fictitious time integration method ； Closed-form numerical differentiators

參考文獻

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