應用相平面軌跡積分方法探討非線性系統響應特性

Investigation on Response Characteristics of Nonlinear System Using Phase Portrait Integration Method

鄭正德(Jang-Der Jeng)

指導教授 ： ARRAY(0xc93df50)

中原大學/工學院/機械工程研究所/博士(2009年)

https://doi.org/10.6840/CYCU.2009.00918

摘要

摘 要
應用有效方法分析及判別非線性動態系統之響應型式，一直是學者主要研究目標，本文利用相軌跡積分法分析非線性系統之響應與分歧特性，此非線性系統包含三個部份進行探討：磨蹭轉子系統、受外力作用耦合Duffing系統及van der Pol 系統。此積分法是以相平面上之軌跡與原點之間的距離為積分函數對時間積分，於不同取值時間所得到的積分值為固定常數來判斷系統的響應週期。從非線性系統之數值模擬積分結果比較中可知，以Poincare截面法分析系統響應週期時，若系統響應為高階次諧和振動，則Poincare截面上之採樣點可能會過於接近，如此將造成分析上的困擾，因而誤判系統響應週期，且相軌跡積分法能以有限的實驗量測或數值模擬數據，判斷系統響應型態，並有效降低量測和計算誤差。本文將此方法與Poincare截面法所得的結果進行比較，以驗證此積分法具鑑別系統響應之效率，並針對系統參數變化對非線性系統的影響進行探討，從模擬結果顯示非線性系統之響應呈現非常複雜的週期與渾沌運動型態，且發現增加非線性系統之剛度和阻尼能抑制渾沌振動發生並且降低振幅，此現象能被有效地應用在非線性系統分析並且改善效能。

關鍵字

相軌跡積分法 ； 磨蹭轉子系統 ； Duffing系統 ； van der Pol 系統 ； Poincaré截面

並列摘要

Abstract
Applying effective methods to analyze and distinguish all kinds of response patterns is the major research object of nonlinear dynamics. In this study, we propose a phase portrait integration method to analyze the nonlinear system, including rub-impacting rotor system, forced coupled Duffing system and van der Pol system. This method numerically integrates the distance between state trajectory and the origin with pre-determined starting times of integration and predicted intervals due to excitation periods. This identification process is based on the fact that the integration would be constant if the integration interval is equal to the response period. From the comparison of the results of the nonlinear system, it is inferred that when Poincare section method is used to analyze system response periods, if the system response is a high-order subharmonic vibration, the sampling points on Poincare section might be too close to cause disturbances for analysis. Thus, misjudgments about system response periods will be made. This method can be applied to determine the response patterns, regarding limited simulation results or measuring data. Besides, in analyzing system responses, the experimental measuring or calculating errors can be decreased, and misjudgment of the system response can be avoided. The results of the integration method in numerical simulation are compared with those of the Poincare section method in order to examine the efficacy of the usage of this method. The effects of the parameter changes on the vibration features of the nonlinear system are investigated in this study. From simulation results, it shows that the responses of these nonlinear systems exhibit very complicated types of periodic and chaotic motions. It is also found that the stiffness and damping of the nonlinear system can suppress chaotic vibration and reduce vibration amplitude. The phenomena can be effectively used for analysis of a nonlinear system as well as for improvement of system.

並列關鍵字

phase portrait integration method ； rub-impacting rotor system ； Duffing system ； van der Pol system ； Poincaré section

參考文獻

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