Asymptotic integration of weakly nonlinear systems with slowly varying coefficients

1984 ◽  
Vol 24 (2) ◽  
pp. 125-130 ◽  
Author(s):  
A. Krylovas ◽  
A. Štaras
Keyword(s):  
Nonlinear Systems ◽  
Asymptotic Integration ◽  
Weakly Nonlinear ◽  
Varying Coefficients

On asymptotic integration of weakly nonlinear systems

1977 ◽  
Vol 28 (4) ◽  
pp. 372-386
Author(s):  
Yu. A. Mitropol'skii ◽  
A. M. Samoilenko
Keyword(s):  
Nonlinear Systems ◽  
Asymptotic Integration ◽  
Weakly Nonlinear

Asymptotic integration of weakly nonlinear systems with impulses

1986 ◽  
Vol 37 (3) ◽  
pp. 288-291
Author(s):  
V. V. Ishchuk ◽  
N. A. Perestyuk
Keyword(s):  
Nonlinear Systems ◽  
Asymptotic Integration ◽  
Weakly Nonlinear

scholarly journals Spectral analysis of vibration in weakly non-linear systems

2000 ◽  
Vol 22 (3) ◽  
pp. 181-192
Author(s):  
Nguyen Tien Khiem
Keyword(s):  
Nonlinear Systems ◽  
Spectral Density ◽  
Density Function ◽  
Asymptotic Methods ◽  
Random Excitation ◽  
Spectral Density Function ◽  
Kolmogorov Equation ◽  
Spectral Structure ◽  
Random Load ◽  
Weakly Nonlinear

The weakly nonlinear systems subjected to deterministic excitations have been fully and deeply studied by use of the well developed asymptotic methods [1-4]. The systems excited by a random load have been investigated mostly using the Fokker-Plank-Kolmogorov equation technique combined with the asymptotic methods [5-8]. However, the last approach in most successful cases allows to obtain only a stationary single point probability density function, that contains no information about the correlation nor' consequently, the spectral structure of the response. The linearization technique [9, 10] in general permits the spectral density of the response to be determined, but the spectral function obtained by this method because of the linearization eliminates the effect of the nonlinearity. Thus, spectral structure of response of weakly nonlinear systems to random excitation, to the author's knowledge, has not been studied enough. This paper deals with the above mentioned problem. The main idea of this work is the use of an analytical simulation of random excitation given by its spectral density function and afterward application of the well known procedure of the asymptotic method to obtain an asymptotic expression of the response spectral density function. The obtained spectral relationship covers the linear system case and especially emphasizes the nonlinear effect on the spectral density of response. The theory will be illustrated by an example and at the end of this paper there will be a discussion about the obtained results.  

A Wavelet Based Procedure to Study Vibrations and Stability

1999 ◽  
Author(s):  
S. Pernot ◽  
C. H. Lamarque
Keyword(s):  
Stability Analysis ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Periodic Solutions ◽  
Time Varying ◽  
Differential Systems ◽  
Varying Coefficients ◽  
Linear Differential Systems ◽  
Time Varying Coefficients ◽  
Linear Differential

Abstract A Wavelet-Galerkin procedure is introduced in order to obtain periodic solutions of multidegrees-of-freedom dynamical systems with periodic time-varying coefficients. The procedure is then used to study the vibrations of parametrically excited mechanical systems. As problems of stability analysis of nonlinear systems are often reduced after linearization to problems involving linear differential systems with time-varying coefficients, we demonstrate the method provides efficient practical computations of Floquet exponents and consequently allows to give estimators for stability/instability levels. A few academic examples illustrate the relevance of the method.

On the Equivalence of Normal Form Theory and Multiple Time Scale Method

2007 ◽  
Author(s):  
Fengxia Wang ◽  
Anil K. Bajaj
Keyword(s):  
Nonlinear Systems ◽  
Normal Form ◽  
Differential Equations ◽  
Time Scales ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Perturbation Scheme ◽  
Normal Form Theory ◽  
Weakly Nonlinear ◽  
Form Theory

Multiple time scales technique has long been an important method for the analysis of weakly nonlinear systems. In this technique, a set of multiple time scales are introduced that serve as the independent variables. The evolution of state variables at slower time scales is then determined so as to make the expansions for solutions in a perturbation scheme uniform in natural and slower times. Normal form theory has also recently been used to approximate the dynamics of weakly nonlinear systems. This theory provides a way of finding a coordinate system in which the dynamical system takes the “simplest” form. This is achieved by constructing a series of near-identity nonlinear transformations that make the nonlinear systems as simple as possible. The “simplest” differential equations obtained by the normal form theory are topologically equivalent to the original systems. Both methods can be interpreted as nonlinear perturbations of linear differential equations. In this work, the equivalence of these two methods for constructing periodic solutions is proven, and it is explained why some studies have found the results obtained by the two techniques to be inconsistent.

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