Computation of Normal Forms for High Dimensional Nonlinear Systems and Application to Nonplanar Motions of a Cantilever Beam

2003 ◽  
Author(s):  
Wei Zhang ◽  
Feng-Xia Wang ◽  
Jean W. Zu
Keyword(s):  
Nonlinear Systems ◽  
Cantilever Beam ◽  
Normal Forms ◽  
Adjoint Operator ◽  
Operator Method ◽  
Computational Method ◽  
High Dimensional ◽  
Polynomial Equations ◽  
Efficient Computation ◽  
Averaged Equations

A new and efficient computation of normal forms is developed in this paper for high dimensional nonlinear systems, and the computational method is applied to nonplanar motion of a cantilever beam. The method is based on the adjoint operator method and has the advantage of directly calculating coefficients of normal forms. Moreover, the new method is easy to apply to engineering applications, and the final partial differential equations of various resonant cases appear in a canonical form whose solutions can be conveniently obtained using polynomial equations. With the aid of the Maple software, a symbolic program for computing the normal forms of high dimensional nonlinear systems is developed. Based on the symbolic program, the normal forms and their coefficients of the averaged equations for nonplanar motions of a cantilever beam are calculated for two resonant cases.

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.

THEORIES OF MULTI-PULSE GLOBAL BIFURCATIONS FOR HIGH-DIMENSIONAL SYSTEMS AND APPLICATION TO CANTILEVER BEAM

2008 ◽  
Vol 22 (24) ◽  
pp. 4089-4141 ◽  
Author(s):  
W. ZHANG ◽  
M. H. YAO
Keyword(s):  
Nonlinear Systems ◽  
Cantilever Beam ◽  
Chaotic Dynamics ◽  
Adjoint Operator ◽  
Operator Method ◽  
Melnikov Method ◽  
Heteroclinic Orbits ◽  
High Dimensional ◽  
Phase Method ◽  
Global Bifurcations

The aim of this survey paper is to illustrate the perspectives on the theories of the single- and multi-pulse global bifurcations and chaotic dynamics of high-dimensional nonlinear systems and applications to several engineering problems in the past two decades. Two main methods for studying the Shilnikov type multi-pulse homoclinic and heteroclinic orbits in high-dimensional nonlinear systems, which are the energy-phase method and generalized Melnikov method, are briefly demonstrated in the theoretical frame. In addition, the theory of normal form and an improved adjoint operator method for high-dimensional nonlinear systems is also applied to describe a reducing procedure to high-dimensional nonlinear systems. The aforementioned methods are utilized to investigate the Shilnikov type multi-pulse homoclinic bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of the cantilever beam subjected to a harmonic axial excitation and two transverse excitations at the free end. How to employ these methods to analyze the Shilnikov type multi-pulse homoclinic and heteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications is demonstrated through this example.

Further Reduction of Normal Forms for High Dimensional Nonlinear Systems and Application to a Composite Laminated Piezoelectric Plate

2013 ◽  
Vol 291-294 ◽  
pp. 2662-2665
Author(s):  
Shu Ping Chen ◽  
Wei Zhang
Keyword(s):  
Nonlinear Systems ◽  
Normal Forms ◽  
Nonlinear Oscillations ◽  
Piezoelectric Plate ◽  
Nonlinear Differential Equations ◽  
High Dimensional ◽  
Computation Method ◽  
Normal Form Theory ◽  
Averaged Equations ◽  
Form Theory

Normal form theory is robust and useful for direct bifurcation and stability analysis of nonlinear differential equations in real engineering problems. This paper develops a new computation method for obtaining a significant refinement of the normal forms for high dimensional nonlinear systems. In the theoretical model for the nonlinear oscillation of a composite laminated piezoelectric plate, the computation method is applied to compute the coefficients of the normal forms for the case of one double zero and a pair of pure imaginary eigenvalues. The algorithm is implemented in Maple V and the normal forms of the averaged equations and their coefficients for nonlinear oscillations of the composite laminated piezoelectric plate under combined parametric and transverse excitations are calculated.

Computation of Normal Forms of Bogdanov-Takens Singularities for High Dimensional Nonlinear Systems

2010 ◽  
Author(s):  
Shuping Chen ◽  
Wei Zhang ◽  
Youhua Qian ◽  
Jane W. Z. Lu ◽  
Andrew Y. T. Leung ◽  
...  
Keyword(s):  
Nonlinear Systems ◽  
Normal Forms ◽  
High Dimensional

On the Normal Forms Associated with High Dimensional Systems

2000 ◽  
Vol 123 (2) ◽  
pp. 157-169 ◽  
Author(s):  
Koncay Huseyin ◽  
Weiyi Zhang
Keyword(s):  
Nonlinear Systems ◽  
Normal Form ◽  
Normal Forms ◽  
High Dimensional ◽  
Engineering Applications ◽  
Modified Normal Form Approach ◽  
Form Approach

In this paper, a modified normal form approach is proposed for the analysis of high dimensional nonlinear systems. Using the modified approach, calculations of normal forms and, in particular, the related coefficients are carried out much more conveniently. Certain high dimensional systems, including systems with inner resonances, are investigated. These systems exist widely in engineering applications. To illustrate the approach, five examples are presented.

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