簡諧負載響應關聯法於隨機振動力學解析研究

Harmonic Load-Response Correlation Method for Analytical Mechanics Investigation of Random Vibrations

林昀(Yun Lin)

指導教授 ： 王建凱

國立臺灣大學/工學院/機械工程學研究所/碩士(2022年)

https://doi.org/10.6342/NTU202202498

摘要

於機械工程與振動力學研究領域中，對於系統因環境外力與人為操作產生之長時強迫振動變形，常透過其峰值響應來評估整體機械性能，以估計系統的載荷能力或其使用極限，此整體振動峰值響應，不僅關聯於系統材料配置和幾何設計，亦取決於長期過程不確定環境荷載之空間分佈方式與對應時間統計特徵。本論文研究以負載響應關聯法（Load-Response Correlation）做為系統於不確定擬靜態負載過程之極值響應解析基礎，以及以有限元素分析（Finite Element Analysis）做為強迫簡諧振動（Harmonic Vibrations）之響應訊號求解核心，提出「簡諧負載響應關聯法」，為解析系統由隨機週期性荷載所致穩態振動峰值響應之新穎力學理論；基於上述新式解析力學理論，本論文研究更實作了相關的計算與數值方法，包含：商用工程與多物理場分析軟體應用、多自由度動力有限元素計算程式開發與對應多重負載之隨機訊號產生與處理等。
論文內容方面：第一章回顧不確定負載響應關聯法的研究發展背景與相關歷程，用以評估系統於擬靜態隨機外力過程之極值響應；第二章詳細介紹負載響應關聯法之理論基礎，亦以數值計算例，確認由此理論所得之極值響應解析解，具有高度的準確性；第三章實現多自由度簡諧振動之分析與計算，比較數個具不同幾何、材料、邊界條件與負載模式之系統，其多自由度動力有限元素計算與簡諧振動分析結果之差異，並強調簡諧振動分析為本論文提出簡諧負載響應關聯法之關鍵求解核心；第四章提出簡諧負載響應關聯法之詳細理論推導，提出含統計特徵之不確定性週期動態負載引致的穩態響應極值解析式，透過一系列的計算例，系統性地探討此新式理論的準確性，且說明如何以系統穩態響應對應各頻率域傅立葉係數之解析統計特徵值，重建出時域振動峰值響應之準確估計；第五章為論文結論與未來研究展望。

關鍵字

隨機振動 ； 不確定週期負載 ； 峰值振動響應 ； 負載響應關聯 ； 簡諧響應分析

並列摘要

In the research fields of mechanical engineering and vibration mechanics, the overall performance of a system is often estimated as its peak responses regarded as the loading capacity or serviceability of the system for the long-term forced vibrations caused by environmental loads and human operations.Such vibration peak responses intrinsically depend on not only the system material configurations related to geometric designs but the spatial distribution and temporal statistical characteristics of the environmental loads during the long-term uncertainty. This thesis research presents a harmonic load response correlation method, a novel solid mechanics theory for analytically determining system peak responses of steady-state vibrations caused by random and periodic loads.It is essential that the load-response correlation method, giving a remarkably accurate approximation of system extreme value responses during temporally correlated quasi-static load processes, is adopted in the proposed theory. In addition, the system matrice commonly used in finite element analysis is utilized to closely represent actual arrangements of material configurations in a system, including mass, damping, and stiffness contributions. According to the aforementioned mechanics theories, this research implements corresponding computations and numerical methods for validation studies, including applications of commercial engineering and multiphysics modeling software,development of multi-degree-of-freedom dynamic finite element analysis, and processing techniques such as random signal examination, modification, and further synthesis.
The thesis is organized as follows: The background and research history of the uncertain load response correlation method for evaluating the peak response of quasi-static random force processes to systems are thoroughly reviewed in Chapter 1. Chapter 2 introduces the fundamental concepts of the loadresponse correlation method. Numerical examples of linear elastic systems are also given for validation study of this theory. Chapter 3 covers the harmonic vibration analysis for systems with multiple degrees of freedom, which is the kernel of the harmonic load-response correlation method proposed in this thesis. Chapter 4 is the core of this study. First, the theoretical derivations of the proposed harmonic load response correlation method are presented. Afterward, in a series of computational examples, the estimation accuracy of the extreme value of the steady-state vibration responses is shown during uncertain periodic loading processes with statistical characteristics in the time domain. Chapter 5 wraps the conclusion and future work of the thesis.

並列關鍵字

Random vibrations ； Uncertain periodic loads ； Peak vibration responses ； Load-response correlations ； Harmonic response analysis

參考文獻

1. Guest, J.K. and T. Igusa, Structural optimization under uncertain loads and nodal locations. Computer Methods in Applied Mechanics and Engineering, 2008. 198(1): p. 116-124.

2. Kang, B., W. Choi, and G. Park, Structural optimization under equivalent static loads transformed from dynamic loads based on displacement. Computers Structures, 2001. 79(2): p. 145-154.

3. Choi, W.-S. and G.-J. Park, Structural optimization using equivalent static loads at all time intervals. Computer methods in applied mechanics and engineering, 2002. 191(19-20): p. 2105-2122.

4. Pierre, C., A. Ferri, and E. Dowell, Multi-harmonic analysis of dry friction damped systems using an incremental harmonic balance method. 1985.

5. Cameron, T. and J.H. Griffin, An alternating frequency/time domain method for calculating the steady-state response of nonlinear dynamic systems. 1989.