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The subharmonic bifurcation solution of nonlinear Mathieu's equation and Euler dynamic buckling problems
Abstract
A new approach is presented in this paper on the basis of dynamic systems theory. This paper presents the form of a generic
classification of stable response diagrams for the nonlinear Mathieu equation. In addition, a general method is presented
for determining the topological type of the response diagram for a given equation. This method has been successfully applied
to Euler dynamic buckling problems. Some new results are obtained.
... For the bifurcation of periodic solutions of polynomial systems, many scholars did a lot of researches and got some results [2][3][4][5][6][7][8][9]. Chen et al. [4] proposed the C-L method to reveal the relationship between the bifurcation periodic solutions and the structural parameters of nonlinear system. ...
... For the bifurcation of periodic solutions of polynomial systems, many scholars did a lot of researches and got some results [2][3][4][5][6][7][8][9]. Chen et al. [4] proposed the C-L method to reveal the relationship between the bifurcation periodic solutions and the structural parameters of nonlinear system. Teixeira et al. [5] studied the conditions of bifurcation of periodic solutions by the averaging theory and obtained the number of periodic solutions of a four-dimensional perturbed system. ...
... The tools for proving these results are the averaging theory and Brouwer degree theory [2,3]. Firstly, we introduce the scale transformations for system (4) ( , , , ) ...
In this paper, we study the bifurcation of periodic solutions for a four-dimensional deployable circular mesh antenna system. The tools for proving these results are the averaging theory and Brouwer degree theory. Based on constructing displacement maps, we study the bifurcation of the periodic solutions of linear center, and to discuss the maximum number of periodic solutions in certain parameter control conditions. The results in this paper are helpful to the study of nonlinear dynamic characteristics and vibration control of deployable circular mesh antenna model.
... Later, Golubitsky and others [304,305] further improved on singularity theory and introduced both group and singularity theory into bifurcation theory, thus launching a new field of research on the analysis of the bifurcation behaviors of dynamic systems. Chen and Langford [306] combined the Lyapunov-Schmidt reduction method with singularity theory to study the 1/2 subharmonic bifurcation behaviors of the nonlinear Mathieu equation. This hybrid concept was used to construct a general method of analyzing the topological structure of the periodic bifurcation solution of a nonautonomous system. ...
... Thus, singularity theory is an effective way to optimize engineering system parameters. The nonlinear Mathieu equation is shown in Eq. (51), and its parameters are described in detail in Ref. [306]. Both Bogoliubov and Mitropolky [307] and Nayfeh and Mook [308] studied the same Eq. ...
... The results in Fig. 19 are not topologically equivalent. Chen and Langford [306] (Fig. 20) unified the results in 1988. These authors found that different bifurcation forms arise in six different parameter regions. ...
... The traditional perturbation theory can only obtain a sort of concrete response curves for the research on periodic solutions of the nonlinear vibration system [14][15] . Chen and Langford [16] proposed a new method to study the periodic bifurcation solution to the nonlinear vibration equation with parametric excitation, providing six typical bifurcation response patterns in different parametric regions. The singularity theory can be applied in various nonlinear systems, e.g., the restricted system and the hysteretic system [1,17] . ...
... 16 x 2 λ 3 + 64D 2 10 ω 2 D 2 16 x 4 − 16λ 4 + 128D 2 10 ω 2 D 16 x 2 λ + 64D 2 10 ω 2 λ 2 = 0. ...
A method for seeking main bifurcation parameters of a class of nonlinear
dynamical systems is proposed. The method is based on the effects of parametric variation of dynamical systems on eigenvalues of the Frechet matrix. The singularity theory is
used to study the engineering unfolding (EU) and the universal unfolding (UU) of an arch
structure model, respectively. Unfolding parameters of EU are combination of concerned
physical parameters in actual engineering, and equivalence of unfolding parameters and
physical parameters is verified. Transient sets and bifurcation behaviors of EU and UU
are compared to illustrate that EU can reflect main bifurcation characteristics of nonlinear systems in engineering. The results improve the understanding and the scope of
applicability of EU in actual engineering systems when UU is difficult to be obtained.
... As a qualitative analysis method, singularity theory has an essential role in determining the stability of dynamic systems, which have been developed by several researchers [4][5][6] and references therein. Chen et al. [7][8] studied the bifurcation characteristics of practical engineering systems with aid of singularity theory. ...
The research gradually highlights vibration and dynamical analysis of symmetric coupled nonlinear oscillators model with clearance. The aim of this paper is the bifurcation analysis of the symmetric coupled nonlinear oscillators modeled by a four-dimensional nonsmooth system. The approximate solution of this system is obtained with aid of averaging method and Krylov-Bogoliubov (KB) transformation presented by new notations of matrices. The bifurcation function is derived to investigate its dynamic behaviour by singularity theory. The results obtained provide guidance for the nonlinear vibration of symmetric coupled nonlinear oscillators model with clearance.
... Martinet [2] discussed the open fold of smooth mapping buds under the strong equivalence and gave various forms of open fold theorem. Chen and Langford [3] gave the C-L method, by combining the L-S method and singularity theory, and studied the bifurcation of the periodic solution for nonlinear dynamical equation undergoing parametric excitation. Jin and Zou [4] studied a restrained pipe conveying fluid and obtained dynamical behaviors by singularity theory in different persistent regions. ...
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.
... Based on this theory, the transition sets on the unfolding parameter plane and the corresponding bifurcation diagrams with different topological structures can be given [30]. Chen and Langford [31] studied a generally non-linear Mathieu equation and obtained six typical bifurcating response curves by setting the system parameters as the unfolding parameters. By using the singularity theory, Qin et al. [23] studied the bifurcation of an elastic cable with 1:1 internal resonance. ...
The bifurcation of the periodic response of a micro-machined gyroscope with cubic supporting stiffness and fractional electrostatic forces is investigated. The pull-in phenomenon is analyzed to show that the system can have a stable periodic response when the detecting voltage is kept within a certain range. The method of averaging and the residue theorem are employed to give the averaging equations for the case of primary resonance and 1:1 internal resonance. Transition sets on the driving/detecting voltage plane that divide the parameter plane into 12 persistent regions and the corresponding bifurcation diagrams are obtained via the singularity theory. The results show that multiple solutions of the resonance curves appear with a large driving voltage and a small detecting voltage, which may lead to an uncertain output of the gyroscope. The effects of driving and detecting voltages on mechanical sensitivity and nonlinearity are analyzed for three persistent regions considering the operation requirements of the micro-machined gyroscope. The results indicate that in the region with a small driving voltage, the mechanical sensitivity is much smaller. In the other two regions, the variations in the mechanical sensitivity and nonlinearity are analogous. It is possible that the system has a maximum mechanical sensitivity and minimum nonlinearity for an appropriate range of detecting voltages.
... Until 1985, Golubitsky and Schaeffer [18] elaborated the mathematical deductions and applications about the singularity theory in their books, based on which the singularity theory became a powerful mathematical tool to investigate bifurcation modes of the nonlinear dynamic system. Chen et al. [19,20] applied the steady-state bifurcation theory of the dynamic system to investigate topological structures of bifurcation modes for a nonlinear dynamic system, and they proposed a new method, namely, C-L method, to study bifurcation modes for the nonlinear dynamic system of a single bifurcation parameter. Furthermore, Qin et al. [21,22] developed the single state-variable singularity method into the two-state-variable singularity method through rigorous mathematical proof and gave mathematical formulas of the transition sets, i.e., the bifurcation set, the hysteresis set, and the double limit set. is research applied the two-state-variable singularity method to investigate bifurcation modes for the nonlinear dual-rotor system. ...
This paper aims to classify bifurcation modes for two interrelated primary resonances of a simple dual-rotor system under double frequency excitations. The four degree-of-freedom (4DOF) dynamic equations of the system considering the nonlinearity of the intershaft bearing can be obtained by using the assumed mode method (AMM) and Lagrange’s equation. A simplified method for dynamic equations is developed due to the symmetry of rotors, based on which the amplitude frequency equations for two interrelated primary resonances are obtained by using the multiple scales method. Furthermore, the validity of the simplified method for dynamic equations and the amplitude frequency equations solved by the multiple scales method are confirmed by numerical verification. Afterwards, the bifurcation analysis for two interrelated primary resonances is carried out according to the two-state-variable singularity method. There exist a total of three different types of bifurcation modes because of double frequency excitations of the dual-rotor system and the nonlinearity of the intershaft bearing. The second primary resonance is more prone to have nonlinear dynamic characteristics than the first primary resonance. This discovery indicates that two interrelated primary resonances of the dual-rotor system may have different bifurcation modes under the same dynamic parameters.
... Different from , the linear stiffness term is considered here. An analytical method, proposed by Chen and Langford [1988] and extended by Qin and Chen [2010] and Li and Chen [2012], namely a twostate variable singularity method, is applied in this paper, which enables us to obtain the bifurcation modes and the transition sets of the nonlinear system. Through the bifurcation investigations, it was found that there are a total of six different types of bifurcation modes due to the effects of the constant force and the magnitude of the harmonic excitation, which extends the findings in that the frequency-response curves can have three markedly different groups of shapes or bifurcation types. ...
This paper focuses on the classification of the bifurcation modes of a Duffing system under the combined excitations of constant force and harmonic excitation. The Harmonic Balance method combined with the arc-length continuation is used to obtain the periodic solutions of the system, and the Floquet theory is employed to analyze the stability of the corresponding solutions. Accordingly, the frequency-response curves affected respectively by the constant force and the magnitude of the harmonic excitation are analyzed to show the basic dynamical properties of the system. Afterwards, the bifurcation investigations are carried out with the aid of the two-state variable singularity method. It is derived that there are a total of six different types of bifurcation modes due to the effects of the constant force and the magnitude of the harmonic excitation. At last, the effects of the nonlinearity parameter and the damping ratio on the bifurcation modes of the system are also discussed. The results obtained in this paper extend the findings in reference that the system can have markedly three types of frequency-response curves: with only one solution, or with maximum three or five solutions for a certain excitation frequency, and contribute to a better understanding of the significant influence of the constant force.
... Lavassani et al. [7] studied equivariant multiple parameter problems by singularity theorem and gave finite definite theorem and normal form of more parameters problem. Chen and Colleagues [8,9] firstly proposed the C-L method to study the bifurcation of the periodic solution of nonlinear dynamical equation undergoing parameter excitation by combining the L-S method and the singularity theory. Seyranian and Mailybaev [10] gave multiple parameter stability theory and discussed the influence of system parameters on stability when parameters change. ...
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.
This book has been written in a frankly partisian spirit-we believe that singularity theory offers an extremely useful approach to bifurcation prob lems and we hope to convert the reader to this view. In this preface we will discuss what we feel are the strengths of the singularity theory approach. This discussion then Ieads naturally into a discussion of the contents of the book and the prerequisites for reading it. Let us emphasize that our principal contribution in this area has been to apply pre-existing techniques from singularity theory, especially unfolding theory and classification theory, to bifurcation problems. Many ofthe ideas in this part of singularity theory were originally proposed by Rene Thom; the subject was then developed rigorously by John Matherand extended by V. I. Arnold. In applying this material to bifurcation problems, we were greatly encouraged by how weil the mathematical ideas of singularity theory meshed with the questions addressed by bifurcation theory. Concerning our title, Singularities and Groups in Bifurcation Theory, it should be mentioned that the present text is the first volume in a two-volume sequence. In this volume our emphasis is on singularity theory, with group theory playing a subordinate role. In Volume II the emphasis will be more balanced. Having made these remarks, Iet us set the context for the discussion of the strengths of the singularity theory approach to bifurcation. As we use the term, bifurcation theory is the study of equations with multiple solutions.
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This book is devoted to the approximate asymptotic methods of solving the problems in the theory of nonlinear oscillations met in many fields of physics and engineering. It is intended for the wide circle of engineering-technical and scientific workers who are concerned with oscillatory processes. Contents include the following: Natural oscillations in quasi-linear systems; The method of the phase plane; The influence of external periodic forces; The method of the mean; Justification of the asymptotic methods.