Limit Cycle Oscillations of Parametrically Excited Second-Order Nonlinear Systems

1975 ◽  
Vol 42 (1) ◽  
pp. 176-182 ◽  
Author(s):  
C. S. Hsu
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
Asymptotic Analysis ◽  
Limit Cycle ◽  
Parametric Excitation ◽  
Exponential Growth ◽  
Nonlinear Damping ◽  
Second Order ◽  
Instability Regions ◽  
Direct Numerical Integration

A second-order nonlinear system subjected to parametric excitation is investigated. The nonlinear factors included are nonlinear damping and a cubic term in displacement. The primary purpose of the paper is to study the limiting effects of these nonlinear factors on the growth of motion for those systems which are otherwise unstable and have an exponential growth. Through an asymptotic analysis formulas are found for evaluating the limit cycle response amplitude in the first and second instability regions of the Ince-Strutt chart. Some results calculated from these formulas for the important case of velocity square damping are compared against those obtained by direct numerical integration in order to assess their accuracy.

scholarly journals On the problem of stability of a motion of essentially nonlinear systems

2021 ◽  
pp. 3-12
Author(s):  
A.A. Martynyuk ◽  
V.O. Chernienko
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Second Order ◽  
Systems Of Equations ◽  
Affine Systems ◽  
The Stability ◽  
Second Order Systems

This article discusses essentially nonlinear systems. Following the approach of applying the pseudolinear inequalitiesdeveloped in a number of works, new estimates for the variation of Lyapunov functions along solutionsof the considered systems of equations are obtained. Based on these estimates, we obtain sufficient conditionsfor the equiboundedness of solutions of second-order systems and sufficient conditions for the stability of anessentially nonlinear system under large initial perturbations. Conditions for the stability of affine systems arealso obtained.

scholarly journals Interaction between the forced and parametric excitations with different degrees of smallness

1998 ◽  
Vol 20 (2) ◽  
pp. 11-17
Author(s):  
Nguyen Van Dao
Keyword(s):  
Nonlinear System ◽  
Parametric Excitation ◽  
Asymptotic Method ◽  
Special Interest ◽  
Second Order ◽  
First Order ◽  
First Approximation ◽  
Forced Excitation

The nonlinear system under consideration in this paper has a specification which can be stated as an interaction between the first order of smallness no resonance parametric excitation and the second order of smallness resonance forced excitation. In the first approximation these excitations have no effect. However, they do interact one with another in the second approximation.The equations for the amplitude and phase of oscillation are found by means of the asymptotic method. The stationary oscillations and their stability are of special interest.

Exact Stationary Response Solution for Second Order Nonlinear Systems Under Parametric and External White Noise Excitations: Part II

1988 ◽  
Vol 55 (3) ◽  
pp. 702-705 ◽  
Author(s):  
Y. K. Lin ◽  
Guoqiang Cai
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
Probability Distribution ◽  
White Noise ◽  
Probability Density ◽  
Detailed Balance ◽  
Stationary Probability ◽  
Probability Flow ◽  
Parametric And External Excitations ◽  
White Noises

A systematic procedure is developed to obtain the stationary probability density for the response of a nonlinear system under parametric and external excitations of Gaussian white noises. The procedure is devised by separating the circulatory portion of the probability flow from the noncirculatory flow, thus obtaining two sets of equations that must be satisfied by the probability potential. It is shown that these equations are identical to two of the conditions established previously under the assumption of detailed balance; therefore, one remaining condition for detailed balance is superfluous. Three examples are given for illustration, one of which is capable of exhibiting limit cycle and bifurcation behaviors, while another is selected to show that two different systems under two differents sets of excitations may result in the same probability distribution for their responses.

scholarly journals Online Health Management for Complex Nonlinear Systems Based on Hidden Semi-Markov Model Using Sequential Monte Carlo Methods

2012 ◽  
Vol 2012 ◽  
pp. 1-22
Author(s):  
Qinming Liu ◽  
Ming Dong
Keyword(s):  
Monte Carlo ◽  
Nonlinear System ◽  
Nonlinear Systems ◽  
Markov Model ◽  
Sequential Monte Carlo ◽  
Health Management ◽  
Health State ◽  
Health States ◽  
Complex Nonlinear System ◽  
Complex Nonlinear Systems

Health management for a complex nonlinear system is becoming more important for condition-based maintenance and minimizing the related risks and costs over its entire life. However, a complex nonlinear system often operates under dynamically operational and environmental conditions, and it subjects to high levels of uncertainty and unpredictability so that effective methods for online health management are still few now. This paper combines hidden semi-Markov model (HSMM) with sequential Monte Carlo (SMC) methods. HSMM is used to obtain the transition probabilities among health states and health state durations of a complex nonlinear system, while the SMC method is adopted to decrease the computational and space complexity, and describe the probability relationships between multiple health states and monitored observations of a complex nonlinear system. This paper proposes a novel method of multisteps ahead health recognition based on joint probability distribution for health management of a complex nonlinear system. Moreover, a new online health prognostic method is developed. A real case study is used to demonstrate the implementation and potential applications of the proposed methods for online health management of complex nonlinear systems.

scholarly journals Second Approximate Solution of Duffing Equation with Strong Nonlinearity by Homotopy Perturbation Method

1970 ◽  
Vol 30 ◽  
pp. 59-75
Author(s):  
M Alhaz Uddin ◽  
M Abdus Sattar
Keyword(s):  
Approximate Solution ◽  
Nonlinear Systems ◽  
Perturbation Method ◽  
Differential System ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Second Order ◽  
Strong Nonlinearity ◽  
Analytical Technique ◽  
Homotopy Perturbation

 In this paper, the second order approximate solution of a general second order nonlinear ordinary differential system, modeling damped oscillatory process is considered. The new analytical technique based on the work of He’s homotopy perturbation method is developed to find the periodic solution of a second order ordinary nonlinear differential system with damping effects. Usually the second or higher order approximate solutions are able to give better results than the first order approximate solutions. The results show that the analytical approximate solutions obtained by homotopy perturbation method are uniformly valid on the whole solutions domain and they are suitable not only for strongly nonlinear systems, but also for weakly nonlinear systems. Another advantage of this new analytical technique is that it also works for strongly damped, weakly damped and undamped systems. Figures are provided to show the comparison between the analytical and the numerical solutions. Keywords: Homotopy perturbation method; damped oscillation; nonlinear equation; strong nonlinearity. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 30 (2010) 59-75  DOI: http://dx.doi.org/10.3329/ganit.v30i0.8504

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