預測和計算具有間隙和預應力的片段線性系統的模態特性相當具有挑戰性。片段線性剛性引起的非線性特性，限制了使用高效的線性模態分析方法的可能性。目前，計算這些系統的非線性模態的主要方法是使用數值延拓計算技術(numerical continuation framework)。這種技術結合了數值積分、打靶法和擬弧長延拓法，通過跌代計算的方式獲得系統的非線性模態。然而，這種計算方法的成本會隨著系統的非線性複雜度和模型大小的增加而增加。本研究提出了一種混合延拓計算技術，結合了解析和數值方法，能夠有效地計算片段線性系統的非線性模態。這種混合延拓計算技術使用半解析方法進行跌代打靶程序，因此能夠大幅減少傳統數值延拓計算技術的計算負擔。本文在具有間隙的質量彈簧阻尼系統上驗證了此計算技術，並使用傳統數值延拓計算技術進行了驗證比較。本文提出的方法能夠顯著的提升計算效率。

The prediction of the modal properties of piecewise-linear mechanical systems with clearances and prestress presents a computational challenge since the nonlinearity induced by piecewise-linear stiffness eliminates the use of efficient linear modal analysis techniques. The most common approach to obtain these structural systems' nonlinear normal modes (NNMs) is a numerical framework that integrates numerical integration, the shooting method, and the pseudo-arclength continuation scheme. This numerical continuation framework (NCF) computes NNMs through iterative numerical calculations; thus, the computational cost of the nonlinear modal analysis of complex nonlinear systems, or piecewise-linear systems in particular, becomes prohibitively expensive as the model size increases. In this work, a hybrid continuation framework combining analytic and numerical methods is proposed to efficiently compute the NNMs of piecewise-linear non-smooth systems. This new hybrid framework uses a semi-analytic method to conduct the iterative shooting procedure; thus, the computational burden of the numerical continuation can be significantly reduced. The proposed method is demonstrated on a spring-mass oscillator with contact elements, and the NNMs obtained using the proposed method are validated by those computed using the traditional numerical continuation framework. The modal properties of the system can be computed using the proposed framework with significant computational savings.

摘要 I
Abstract II
Acknowledgment III
Content IV
List of Figures V
List of Tables VIII
Nomenclature IX
Chapter 1. Introduction 1
Chapter 2. Methodology 4
2.1. The adapted HSNC method 4
2.2. Shooting method and continuation of periodic solutions 11
Chapter 3. Results and Discussion 15
3.1. Demonstration of NCF 15
3.2. n-DOF spring-mass system with single bilinear stiffness 23
3.3. n-DOF spring-mass system with two bilinear stiffness 33
Chapter 4. Conclusions and Future Work 45
Chapter 5. References 46

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