A direct approach for simplifying nonlinear systems with external periodic excitation using normal forms

2019 ◽  
Vol 99 (2) ◽  
pp. 1065-1088 ◽  
Author(s):  
Peter M. B. Waswa ◽  
Sangram Redkar
Keyword(s):  
Nonlinear Systems ◽  
Normal Forms ◽  
Direct Approach ◽  
Periodic Excitation

A direct analysis of nonlinear systems with external periodic excitations via normal forms

2008 ◽  
Vol 55 (1-2) ◽  
pp. 79-93 ◽  
Author(s):  
Amit P. Gabale ◽  
S. C. Sinha
Keyword(s):  
Nonlinear Systems ◽  
Normal Forms ◽  
Direct Analysis

A New Approach for Computing the Normal Forms of High Dimensional Nonlinear Systems

2010 ◽  
Author(s):  
Shuping Chen ◽  
Wei Zhang ◽  
Minghui Yao
Keyword(s):  
Nonlinear Systems ◽  
Normal Form ◽  
Cantilever Beam ◽  
Normal Forms ◽  
Nonlinear Differential Equations ◽  
Real Life ◽  
High Dimensional ◽  
Computation Method ◽  
Normal Form Theory ◽  
Averaged Equations

Normal form theory is very useful for direct bifurcation and stability analysis of nonlinear differential equations modeled in real life. This paper develops a new computation method for obtaining a significant refinement of the normal forms for high dimensional nonlinear systems. The method developed here uses the lower order nonlinear terms in the normal form for the simplifications of higher order terms. In the theoretical model for the nonplanar nonlinear oscillation of a cantilever beam, the computation method is applied to compute the coefficients of the normal forms for the case of two non-semisimple double zero eigenvalues. The normal forms of the averaged equations and their coefficients for non-planar non-linear oscillations of the cantilever beam under combined parametric and forcing excitations are calculated.

Order Reduction of Nonlinear Systems Subjected to an External Periodic Excitation

2005 ◽  
Author(s):  
Sangram Redkar ◽  
S. C. Sinha
Keyword(s):  
Nonlinear Systems ◽  
State Space ◽  
Invariant Manifold ◽  
Large Scale ◽  
Order Reduction ◽  
Reduced Order Models ◽  
Periodic Excitation ◽  
External Excitation ◽  
Reduced Order ◽  
Linear Approach

In this work, the basic problem of order reduction nonlinear systems subjected to an external periodic excitation is considered. This problem deserves attention because the modes that interact (linearly or nonlinearly) with the external excitation dominate the response. A linear approach like the Guyan reduction does not always guarantee accurate results, particularly when nonlinear interactions are strong. In order to overcome limitations of the linear approach, a nonlinear order reduction methodology through a generalization of the invariant manifold technique is proposed. Traditionally, the invariant manifold techniques for unforced problems are extended to the forced problems by ‘augmenting’ the state space, i.e., forcing is treated as an additional degree of freedom and an invariant manifold is constructed. However, in the approach suggested here a nonlinear time-dependent relationship between the dominant and the non-dominant states is assumed and the dimension of the state space remains the same. This methodology not only yields accurate reduced order models but also explains the consequences of various ‘primary’ and ‘secondary resonances’ present in the system. Following this approach, various ‘reducibility conditions’ are obtained that show interactions among the eigenvalues, the nonlinearities and the external excitation. One can also recover all ‘resonance conditions’ commonly obtained via perturbation or averaging techniques. These methodologies are applied to some typical problems and results for large-scale and reduced order models are compared. It is anticipated that these techniques will provide a useful tool in the analysis and control of large-scale externally excited nonlinear systems.

Observer normal forms for a class of nonlinear systems by means of coupled auxiliary dynamics

2020 ◽  
Vol 30 (13) ◽  
pp. 4960-4978
Author(s):  
Li‐Fei Wang ◽  
Driss Boutat ◽  
Da‐Yan Liu
Keyword(s):  
Nonlinear Systems ◽  
Normal Forms

scholarly journals Data-driven control of nonlinear systems: An on-line direct approach

Automatica ◽  
2017 ◽  
Vol 75 ◽  
pp. 1-10 ◽  
Author(s):  
Marko Tanaskovic ◽  
Lorenzo Fagiano ◽  
Carlo Novara ◽  
Manfred Morari
Keyword(s):  
Nonlinear Systems ◽  
Direct Approach ◽  
Data Driven ◽  
On Line

Lucid Analysis of Periodically Forced Nonlinear Systems via Normal Forms

2019 ◽  
Author(s):  
Peter M. B. Waswa ◽  
Sangram Redkar
Keyword(s):  
Nonlinear Systems ◽  
Ad Hoc ◽  
Normal Forms ◽  
Numerical Results ◽  
Duffing Equation ◽  
System State ◽  
Periodic Forcing ◽  
Periodic Coefficients ◽  
Duffing’S Equation ◽  
State Augmentation

Abstract This paper presents a straightforward methodology to analyze periodically forced nonlinear systems with constant and periodic coefficients via normal forms. We demonstrate how the intuitive system state augmentation facilitates construction of normal forms by avoiding ad-hoc addition of equation variables, book-keeping parameters and detuning parameters. Moreover, this technique directly connects the periodic forcing terms and periodic coefficients of the nonlinearity with the augmented states — making it applicable to all periodically forced nonlinear systems. Accuracy of this approach is successfully verified via fulfilled compliance between analytical and numerical results of forced Duffing’s equation and Mathieu-Duffing equation.

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