scholarly journals Bifurcation Analysis of Composite Laminated Piezoelectric Rectangular Plate Structure in the Case of 1:2 Internal Resonance

2019 ◽  
Vol 2019 ◽  
pp. 1-23 ◽  
Author(s):  
Y. H. Guo ◽  
W. Zhang
Keyword(s):  
Rectangular Plate ◽  
Internal Resonance ◽  
Singularity Theory ◽  
Bifurcation Problem ◽  
Universal Unfolding ◽  
Sign Function ◽  
Plate Structure ◽  
Parametric Plane ◽  
Transition Set ◽  
Bifurcation Problems

In this paper, the authors study the bifurcation problems of the composite laminated piezoelectric rectangular plate structure with three bifurcation parameters by singularity theory in the case of 1:2 internal resonance. The sign function is employed to the universal unfolding of bifurcation equations in this system. The proposed approach can ensure the nondegenerate conditions of the universal unfolding of bifurcation equations in this system to be satisfied. The study presents that the proposed system with three bifurcation parameters is a high codimensional bifurcation problem with codimension 4, and 6 forms of universal unfolding are given. Numerical results show that the whole parametric plane can be divided into several persistent regions by the transition set, and the bifurcation diagrams in different persistent regions are obtained.

scholarly journals Singularity Analysis of Composite Laminated Piezoelectric Rectangular Plate Structure with 1 : 2 Internal Resonance

2021 ◽  
Vol 2021 ◽  
pp. 1-22
Author(s):  
Yuhong Guo ◽  
Yuhua Guo ◽  
Wei Zhang ◽  
Ruiping Wen
Keyword(s):  
Rectangular Plate ◽  
Internal Resonance ◽  
Dynamical Behavior ◽  
Singularity Theory ◽  
Singularity Analysis ◽  
Polar Coordinates ◽  
Universal Unfolding ◽  
Averaged Equations ◽  
Transition Set ◽  
The Stability

This study investigates the dynamical behavior of the composite laminated piezoelectric rectangular plate with 1 : 2 internal resonance near the singularity using the extended singularity theory method. Based on the previous four-dimensional averaged equations of polar coordinates where the partial derivative terms are not equal to zero, the universal unfolding with codimension 3 of the proposed system is given. The main material parameters that affect the dynamic behavior of the laminated piezoelectric rectangular composite plate near the singularity under transverse excitation are revealed by the transition set of universal unfolding with codimension 3. In addition, the plots of the transition set in three bifurcation parameters space are discussed. These numerical results can show that the stability near the singularity of the proposed system is better when period ratio is less than zero.

BIFURCATION ANALYSIS ON A SELF-EXCITED HYSTERETIC SYSTEM

2004 ◽  
Vol 14 (08) ◽  
pp. 2825-2842 ◽  
Author(s):  
ZHIQIANG WU ◽  
PEI YU ◽  
KEQI WANG
Keyword(s):  
Mechanical System ◽  
Singularity Theory ◽  
Bifurcation Problem ◽  
Van Der Pol ◽  
Smooth Functions ◽  
Restoring Force ◽  
Symbolic Notation ◽  
Hysteretic Damper ◽  
Bifurcation Problems ◽  
Unconstrained Problems

This paper investigates periodic bifurcation solutions of a mechanical system which involves a van der Pol type damping and a hysteretic damper representing restoring force. This system has recently been studied based on the singularity theory for bifurcations of smooth functions. However, the results do not actually take into account the property of nonsmoothness involved in the system. In particular, the transition varieties due to constraint boundaries were ignored, resulting in failure in finding some important bifurcation solutions. To reveal all possible bifurcation patterns for such systems, a new method is developed in this paper. With this method, a continuous, piecewise smooth bifurcation problem can be transformed into several subbifurcation problems with either single-sided or double-sided constraints. Further, the constrained bifurcation problems are converted to unconstrained problems and then singularity theory is employed to find transition varieties. Explicit formulas are applied to reconsider the mechanical system. Numerical simulations are carried out to verify analytical predictions. Moreover, symbolic notation for a sequence of bifurcations is introduced to easily show the characteristics of bifurcations, and also the comparison of different bifurcations. The method developed in this paper can be easily extended to study bifurcation problems with other types of nonsmoothness.

scholarly journals Research on dynamic characteristics of plate under pedestrian excitation based on Newmark-β

2018 ◽  
Vol 37 (4) ◽  
pp. 682-699
Author(s):  
Xinfang Ge ◽  
Weirong Wang ◽  
Wei Yuan
Keyword(s):  
Rectangular Plate ◽  
Dynamic Characteristics ◽  
Vibration Isolation ◽  
Great Influence ◽  
Interaction Model ◽  
Vertical Direction ◽  
Structural Vibration ◽  
Normal Walking ◽  
Plate Structure ◽  
Pedestrian Excitation

Development of micro and ultra-precision machining, precision instruments and equipment, precision assembly and testing has put forward more and more high requirements to vibration isolation on environmental elements, especially the pedestrian excitation generated by workers' normal walking. Therefore, it is very important to study the pedestrian excitation's influence on vibration characteristics of precision instruments and equipment. In this study, dynamic model including mathematical model of pedestrian excitation, interaction model between pedestrian and rectangular plate structure, the human–plate coupled dynamic equation in vertical direction of pedestrian–plate structure was established. And then we use the Newmark-β method to solve the time-domain step-by-step integration of the first four order modes' dynamic equations and study the influence of the linear notion trajectory along the central axis direction on the dynamic characteristics of the rectangular plate. By simulation, we discussed plate structure response under different conditions, including plate structure displacement and acceleration response under the single person excitation with different velocities, under normal walking velocity with different number of pedestrians and under this case of different distance between two pedestrians. The results show that the structural vibration induced by pedestrian excitation has great influence on dynamic characteristics of plate.

The numerical analysis of bifurcation problems with application to fluid mechanics

Acta Numerica ◽  
2000 ◽  
Vol 9 ◽  
pp. 39-131 ◽  
Author(s):  
K. A. Cliffe ◽  
A. Spence ◽  
S. J. Tavener
Keyword(s):  
Numerical Analysis ◽  
Fluid Mechanics ◽  
Stokes Equations ◽  
Mixed Finite Element ◽  
Finite Difference Methods ◽  
Singularity Theory ◽  
Mixed Finite Element Method ◽  
Convergence Theory ◽  
Navier Stokes ◽  
Bifurcation Problems

In this review we discuss bifurcation theory in a Banach space setting using the singularity theory developed by Golubitsky and Schaeffer to classify bifurcation points. The numerical analysis of bifurcation problems is discussed and the convergence theory for several important bifurcations is described for both projection and finite difference methods. These results are used to provide a convergence theory for the mixed finite element method applied to the steady incompressible Navier–Stokes equations. Numerical methods for the calculation of several common bifurcations are described and the performance of these methods is illustrated by application to several problems in fluid mechanics. A detailed description of the Taylor–Couette problem is given, and extensive numerical and experimental results are provided for comparison and discussion.

Bifurcations near a multiple eigenvalue of the rectangular plate problem

1986 ◽  
Vol 103 (1-2) ◽  
pp. 35-62
Author(s):  
Zoltán Sadovský
Keyword(s):  
Aspect Ratio ◽  
Rectangular Plate ◽  
Small Perturbation ◽  
Perturbation Parameter ◽  
Multiple Eigenvalue ◽  
Bifurcation Problem ◽  
Operator Equations ◽  
Double Eigenvalue ◽  
Elastic Rectangular Plate ◽  
Degenerate Cases

SynopsisWe consider the bifurcation problem of the Föppl–Kármán equations for a thin elastic rectangular plate near a multiple eigenvalue allowing for a small perturbation parameter related to the aspect ratio of the plate. The first step in the study is to introduce equivalent operator equations in the energy spaces of the problem which explicitly contain the perturbation parameter. By dealing partially with a general formulation, we obtain the main results for the double eigenvalue and Z2 ⊓ Z2 symmetry of bifurcation equations. We are chiefly interested in the degenerate cases of bifurcation equations.

scholarly journals NONDEGENERATE UMBILICS, THE PATH FORMULATION AND GRADIENT BIFURCATION PROBLEMS

2009 ◽  
Vol 19 (09) ◽  
pp. 2965-2977 ◽  
Author(s):  
JACQUES-ELIE FURTER ◽  
ANGELA MARIA SITTA
Keyword(s):  
Cylindrical Panel ◽  
Point Of View ◽  
Singular Behavior ◽  
Universal Unfolding ◽  
Buckling Modes ◽  
The Core ◽  
Bifurcation Problems ◽  
Multiparameter Bifurcation ◽  
Special Parameter

Parametrized contact-equivalence is a successful theory for the understanding and classification of the qualitative local behavior of bifurcation diagrams and their perturbations. Path formulation is an alternative point of view making explicit the singular behavior due to the core of the bifurcation germ (when the parameters vanish) from the effects of the way parameters enter. We show how to use path formulation to classify and structure efficiently multiparameter bifurcation problems in corank 2 problems. In particular, the nondegenerate umbilics singularities are the generic cores in four situations: the general or gradient problems, with or without ℤ2 symmetry where ℤ2 acts on the second component of ℝ2 via κ(x,y) = (x,-y). The universal unfolding of the umbilic singularities have an interesting "Russian doll" type of structure of miniversal unfoldings in all those categories. With the path formulation approach we can handle one, or more, parameter situations using the same framework. We can even consider some special parameter structure (for instance, some internal hierarchy of parameters). We classify the generic bifurcations with 1, 2 or 3 parameters that occur in those cases. Some results are known with one bifurcation parameter, but the others are new. We discuss some applications to the bifurcation of a loaded cylindrical panel. This problem has many natural parameters that provide concrete examples of our generic diagrams around the first interaction of the buckling modes.

The double cusp has five minima

1978 ◽  
Vol 84 (3) ◽  
pp. 537-538 ◽  
Author(s):  
J. Callahan
Keyword(s):  
Critical Points ◽  
Standard Form ◽  
Singularity Theory ◽  
The Other ◽  
Universal Unfolding ◽  
The Real ◽  
Level Curves ◽  
Intersection Points ◽  
The One ◽  
Image Position

The double cusp is the real, compact, unimodal singularitysee (2), (4). Functions in a universal unfolding of the double cusp can have nine non-degenerate critical points near the origin, but no more. Index considerations show that precisely four of the nine are saddles, and it has long been part of the folklore of singularity theory that one of the other five must be a maximum. Indeed, a standard form of the unfolded double cusp (1), (3) is a function having a pair of intersecting ellipses as one of its level curves; see Fig. 1(a). There are saddles at the four intersection points, a maximum inside the central quadrilateral, and a minimum inside each of the other four finite regions bounded by the ellipses. The rest of Fig. 1 suggests, however, that a deformation of this function (in which one of the saddles drops below the level of the other three) might turn the maximum into a fifth minimum. The following proposition shows that a function similar to the one in Fig. 1(d) can be realized in an unfolding of the double cusp.

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