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.

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.

scholarly journals Normal form approach in the motion planning of space robots: a case study

2021 ◽  
Author(s):  
Krzysztof Tchoń ◽  
Katarzyna Zadarnowska
Keyword(s):  
Normal Form ◽  
Motion Planning ◽  
Normal Forms ◽  
Inverse Dynamics ◽  
Planning Problem ◽  
Inverse Dynamics Problem ◽  
Velocity Level ◽  
Form Approach ◽  
Space Approach

AbstractWe examine applicability of normal forms of non-holonomic robotic systems to the problem of motion planning. A case study is analyzed of a planar, free-floating space robot consisting of a mobile base equipped with an on-board manipulator. It is assumed that during the robot’s motion its conserved angular momentum is zero. The motion planning problem is first solved at velocity level, and then torques at the joints are found as a solution of an inverse dynamics problem. A novelty of this paper lies in using the chained normal form of the robot’s dynamics and corresponding feedback transformations for motion planning at the velocity level. Two basic cases are studied, depending on the position of mounting point of the on-board manipulator. Comprehensive computational results are presented, and compared with the results provided by the Endogenous Configuration Space Approach. Advantages and limitations of applying normal forms for robot motion planning are discussed.

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

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.

The applications of a modified normal form approach

2001 ◽  
Vol 22 (7) ◽  
pp. 802-807
Author(s):  
Zhang Wei-yi ◽  
Koncay Huseyin ◽  
Ye Min
Keyword(s):  
Normal Form ◽  
Modified Normal Form Approach ◽  
Form Approach

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.

METHODS OF REDUCING OF LARGE BOOLEAN FORMULAS REPRESENTED IN A CONJUNCTIVE NORMAL FORM FOR DETERMINING THEIR SATISFIABILITY

2020 ◽  
Author(s):  
N.I. Gdansky ◽  
◽  
A.A. Denisov ◽  
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Conjunctive Normal Form ◽  
Boolean Formulas

The article explores the satisfiability of conjunctive normal forms used in modeling systems.The problems of CNF preprocessing are considered.The analysis of particular methods for reducing this formulas, which have polynomial input complexity is given.

NORMAL FORMS FOR FUZZY LOGIC — AN APPLICATION OF KOLMOGOROV’S THEOREM

1996 ◽  
Vol 04 (04) ◽  
pp. 331-349 ◽  
Author(s):  
VLADIK KREINOVICH ◽  
HUNG T. NGUYEN ◽  
DAVID A. SPRECHER
Keyword(s):  
Neural Networks ◽  
Fuzzy Logic ◽  
Normal Form ◽  
Normal Forms ◽  
Approximation Property ◽  
Universal Approximation ◽  
Approximation Result ◽  
Logical Operations ◽  
Kolmogorov's Theorem

This paper addresses mathematical aspects of fuzzy logic. The main results obtained in this paper are: 1. the introduction of a concept of normal form in fuzzy logic using hedges; 2. using Kolmogorov’s theorem, we prove that all logical operations in fuzzy logic have normal forms; 3. for min-max operators, we obtain an approximation result similar to the universal approximation property of neural networks.

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