估算含有接觸或碰撞元件的非平滑振動子(non-smooth oscillator)之共振頻率對於各種機械、航太結構的設計、控制、及健康監測至關重要。其中由間歇性接觸引起的非平滑振動系統通常無法使用有效率的線性方法予以分析，因此預測這些系統的振動特性相當具有挑戰性。為了能夠有效率地預測非平滑振動系統的動態特性，Shaw及Holmes首先提出了一個經典的非線性共振頻率運算法：bilinear frequency approximation (BFA)法。有別於其它的非線性數值方法，BFA法使用雙線性動態系統中個別子系統的線性特性來估算非平滑振動子的共振頻率，因此可以避免繁重的數值運算並且快速地分析系統的動態特性。然而傳統的BFA方法只適用於接觸元件中間隙尺寸為零的系統，因此大幅地限制了其應用範圍。本文中提出一種廣義的generalized BFA法，使BFA法能夠應用於接觸元件中存在間隙或是預應力的非平滑振動系統之分析。應用此方法能夠相當快速地建構非平滑振動系統的背骨曲線，以預測這些系統的非線性振動特性。此計算方法能夠應用於分析各種具有接觸非線性的機械、航太、及土木系統之振動響應。本文分別以數值方法以及實驗方法驗證generalized BFA法之準確性。

Estimating the resonant frequency of non-smooth oscillators with contacting elements is critical for design, control, and health monitoring of a variety of mechanical and aerospace structures. The nonlinearity caused by the intermittent contact typically excludes the use of efficient linear methods; hence the prediction of vibration characteristics of these system becomes computationally challenging. In order to facilitate the analysis of the dynamic property of non-smooth oscillators, a classical technique referred to as the bilinear frequency approximation (BFA) method was first developed by Shaw and Holmes. The BFA method estimates the resonant frequency of bilinear oscillators using the linear characteristics of underlying linear systems and hence expensive numerical simulations can be avoided. However, the traditional BFA method is only accurate for single degree-of-freedom (DOF) systems whose gap size between the contacting elements is zero. In this thesis, a generalized BFA method is proposed to extend the capability of the traditional BFA method to capture the resonant frequency of multiple DOF non-smooth systems where either gaps or prestress exist. The generalized BFA method employs the linear properties, including the natural frequencies and mode shapes, of the subsystems and the gap size at the contacting DOF to approximate the resonant frequency of general non-smooth oscillators. The backbone curve of the non-smooth oscillators can be efficiently constructed using the proposed method. The generalized BFA method is demonstrated on a few mechanical oscillators and is validated both numerically and experimentally.

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