Singularity analysis of Duffing-van der Pol system with two bifurcation parameters under multi-frequency excitations

Bifurcation properties of a Duffing-van der Pol system with two parameters under multi-frequency excitations are studied. Three cases are discussed： （1） λ 1 is considered as bifurcation parameter, （2） λ 2 is considered as bifurcation parameter, and （3） λ 1 and λ 2 are both considered as bifurcation parameters. According to the definition of transition sets, the whole parametric space is divided into several different persistent regions by the transition sets for different cases. The bifurcation diagrams in different persistent regions are obtained, which provides a theoretical basis for optimal design of the system.     

More generalized hybrid variational principle and corresponding finite element model

According to recent studies of the generalized variational principle by Professor Chien Weizang, the more generalized hybrid variational principle for finite element method is given, from which a new kind of the generalized hybrid element model is etablished.Using the thin plate bending element with varying thickness as an example, we compare various hybrid elements based on different generalized variational principles.     

The random variational principle in finite deformation of elasticity and finite element method

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The random variational principle and finite element method

In this paper, we introduced the random materials, geometrical shapes, force and displacement boundary condition directly into the functional variational formulations and developed a unified random variational principle and finite element method with the small parameter perturbation method. Numerical examples showed that the methods have the advantages of the simple and convenient program implementation, and are effective for the random mechanics problems.     

A method for establishing generalized variational principle

A method for establishing generalized variational principle is proposed in this paper. It is based on the analysis of mechanical meaning and it can be applied to problems in which the variational principles are needed but no corresponding variational principle is available. In this paper, the Hu-Washizu ’s generalized variational principle and the Hu’s generalized principle of complementary energy are derived from the mechanical meaning instead of from the generalization of the principle of minimum potenlial energy and the correct proofs of these two generaleed variational principles are given. It is also proved that this is wrong if one beleives that σ_ij, e_ij and ui are independent variables each other based on the reason that these three kinds of variables are all contained in these two generalized variational principles. The condition of using these two variational principles in a correct manner is also explained.     

Random perturbations of nonlinear oscillators: Homogenization and large deviations

This paper considers two prototypical strongly nonlinear oscillators with weak dissipation and noise: a pendulum which is acted on by a small noisy torque and opposed by a friction; and the noisy Duffing-van der Pol oscillator. But over a long time the weak dissipative and noise effects can be significant. This paper develops asymptotic techniques based on averaging and large deviations to study the effects of weak noise on the escape from the domain of attraction of stable equilibrium points and limit cycles in phase-space.     

RESPONSE OF PARAMETRICALLY EXCITED DUFFING-VAN DER POL OSCILLATOR WITH DELAYED FEEDBACK

The dynamical behaviour of a parametrically excited Duffing-van der Pol oscillator under linear-plus-nonlinear state feedback control with a time delay is concerned. By means of the method of averaging together with truncation of Taylor expansions, two slow-flow equations on the amplitude and phase of response were derived for the case of principal parametric resonance. It is shown that the stability condition for the trivial solution is only associated with the linear terms in the original systems besides the amplitude and frequency of parametric excitation. And the trivial solution can be stabilized by appreciate choice of gains and time delay in feedback control. Different from the case of the trivial solution, the stability condition for nontrivial solutions is also associated with nonlinear terms besides linear terms in the original system. It is demonstrated that nontrivial steady state responses may lose their stability by saddle-node (SN) or Hopf bifurcation (HB) as parameters vary. The simulations, obtained by numerically integrating the original system, are in good agreement with the analytical results.     

A generalized variational principle in micromorphic thermoelasticity

Recently Nappa obtains a Gurtin-type variational principle for micromorphic thermoelasticity. However, the use of convolutions is of course restricted to the linear case, which sets a limit to the rang of applicability. This paper establishes a classic variational principles for the discussed problem by the semi-inverse method.     

VARIATIONAL PRINCIPLE AND GENERALIZED VARIATIONAL PRINCIPLE IN HYDRODYNAMICS OF A CLASS OF NON－NEWTONIAN FLUID

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A finite element parallel algorithm of the parametric variational principle for elastic contact problems

Basetl on the finite element solution of the parametric varialional principle of elastic con/del problem, a corresponding parallel algorithm has been created bv utilizing the specialities of parallel computer and the architecture of concurrent processing in this paper. In this algorithm. the parallelisms have heen realized in the processes of creation and assembly of stiffness matrix, of the static condensation, of the solution of stresses and in many other aspects. The programme of this algorithm has been realized on ELXSI-6400 parallel computer of Xi'an Jiaotong University. The results of computation show that the computational time can be saved efficiently and it is an effective parallel algorithm for the analyses of contact problems.     

Bifurcation scenarios of the noisy duffing-van der pol oscillator

This paper presents a numerical study of the bifurcation behavior of the noisy Duffing-van der Pol oscillator% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4baFfea0dXde9vqpa0lb9% cq0dXdb9IqFHe9FjuP0-iq0dXdbba9pe0lb9hs0dXda91qaq-xfr-x% fj-hmeGabaqaciGacaGaaeqabaWaaeaaeaaakeaatCvAUfKttLeary% qr1ngBPrgaiuGacuWF4baEgaWaaiaaiccacqWF9aqpcaaIGaGaaiik% aerbtLhBMfwzUbacgiGaa4xSdiaaiccacqGHRaWkcaaIGaGaeq4Wdm% 3ccaaIXaGcceqGxbGbaiaaliaaigdakiGacMcacqWF4baEcaaIGaGa% ci4kaiaaiccacqaHYoGycuWF4baEgaGaaiaaiccacqGHsislcaaIGa% Gae8hEaG3aaWbaaSqabeaacaaIZaaaaOGaaGiiaiabgkHiTiaaicca% cqWF4baEdaahaaWcbeqaaiaaikdaaaGccuWF4baEgaGaaiaaiccaci% GGRaGaaGiiaiabeo8aZTGaaGOmaOGabe4vayaacaGaaeOmaiaabYca% aaa!5F62!\[\ddot x = (\alpha + \sigma 1{\rm{\dot W}}1)x + \beta \dot x - x^3 - x^2 \dot x + \sigma 2{\rm{\dot W2,}}\]where , are bifurcation parameters, % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4baFfea0dXde9vqpa0lb9% cq0dXdb9IqFHe9FjuP0-iq0dXdbba9pe0lb9hs0dXda91qaq-xfr-x% fj-hmeGabaqaciGacaGaaeqabaWaaeaaeaaakeaaceqGxbGbaiaali% aaigdakiqabEfagaGaaSGaciOmaaaa!35B4!\[{\rm{\dot W}}1{\rm{\dot W}}2\] are independent white noise processes, and ₁, ₂ are intensity parameters. A stochastic bifurcation here means (a) the qualitative change of stationary measures or (b) the change of stability of invariant measures and the occurrence of new invariant measures for the random dynamical system generated by (1). The first type of bifurcation can be observed when studying the solution of the Fokker-Planck equation, this stationary measure is a quantity corresponding to the one-point motion. More generally, if one is interested in the simultaneous motion of n points (n1) forward and backward in time, then the second type of bifurcation arises naturally, capturing all the stochastic dynamics of (1). Based on the numerical results, we propose definitions of the stochastic pitchfork and Hopf bifurcations.     

PARAMETRIC VARIATIONAL PRINCIPLE BASED ELASTIC-PLASTIC ANALYSIS OF HETEROGENEOUS MATERIALS WITH VORONOI FINITE ELEMENT METHOD

The Voronoi cell finite element method (VCFEM) is adopted to overcome the limitations of the classic displacement based finite element method in the numerical simulation of heterogeneous materials. The parametric variational principle and quadratic programming method are developed for elastic-plastic Voronoi finite element analysis of two-dimensional problems. Finite element formulations are derived and a standard quadratic programming model is deduced from the elastic-plastic equations. Influence of microscopic heterogeneities on the overall mechanical response of heterogeneous materials is studied in detail. The overall properties of heterogeneous materials depend mostly on the size, shape and distribution of the material phases of the microstructure. Numerical examples are presented to demonstrate the validity and effectiveness of the method developed.     

A generalized variational principle of composite shallow shells and its application to the folded shell

In this paper, a generalized variational principle of elastodynamics in composite shallow shells with edge beams is presented, and its equivalence to corresponding basic equations, ridge conditions and boundary conditions is proved. Then this variational principle is applied to the folded shell structure. By means of double series, the approximate analytical solutions for statics and dynamics under common boundary conditions are obtained. The comparison of our results with FEM computations and experiments shows the analytical solutions have good convergence and their accuracy is quite satisfactory.     

The application of the variational principle in the constrained control system

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The variational principles and application of nonlinear numerical manifold method

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The optimal control variational principle and finite elements analysis for viscoplastic dynamics

This paper presents the optimal control variational principle for Perzyna model which is one of the main constitutive relation of viscoplasticity in dynamics. And it could also be transformed to solve the parametric quadratic programming problem. The FEM form of this problem and its implementation have also been discussed in the paper.     

Approximate analytical solution in slow-fast system based on modified multi-scale method

A simple, yet accurate modified multi-scale method (MMSM) for an approximately analytical solution in nonlinear oscillators with two time scales under forced harmonic excitation is proposed. This method depends on the classical multi-scale method (MSM) and the method of variation of parameters. Assuming that the forced excitation is a constant, one could easily obtain the approximate analytical solution of the simplified system based on the traditional MSM. Then, this solution for the oscillator under forced harmonic excitation could be established after replacing the harmonic excitation by the constant excitation. To certify the correctness and precision of the proposed analytical method, the van der Pol system with two scales subject to slowly periodic excitation is investigated; this system presents rich dynamical phenomena such as spiking (SP), spiking-quiescence (SP-QS), and quiescence (QS) responses. The approximate analytical expressions of the three types of responses are given by the MMSM, and it can be found that the precision of the new analytical method is higher than that of the classical MSM and better than that of the harmonic balance method (HBM). The results obtained by the present method are considerably better than those obtained by traditional methods, quantitatively and qualitatively, particularly when the excitation frequency is far less than the natural frequency of the system.     

The embedded theorem for family of quasi metric spaces in Menger space and its application

In this paper the notion of embedding for family of quasi metric spaces in Menger spaces is introduced and its properties are investigated. A common fixed point theorem for sequence of continuous mappings in Menger spaces is proved. These mappings are assumed to satisfy some generalizations of the contraction condition. The proving technique herein seems to be new even for mappings in Menger spaces.     

结构动力分析的随机变分原理及随机有限元法

将结构动力系统的参数及激励的随机性直接引入结构的动力泛函变分表达式中，基于瞬时最小势能原理，应用小参数摄动法，建立了随机结构动力分析的随机变分列式及相应的确机有限元法。算例表明，应用此法分析随机结构动力响应，具有程序实施简便，计算效率高的优点。     

索网天线找形分析的参变量变分有限元法

针对柔性空间索网天线的非线性特点,建立了基于参变量变分描述索网拉压非线性和共旋列式描述几何非线性的有限元控制方程,应用Lernke与改进牛顿法进行求解.通过对索网预张力平衡计算,证明改进牛顿法比Newton-Raphson法具有更强的收敛能力.进一步将力密度法迭代原理与有限元法结合应用于索网天线的非线性找形分析中,获得了理想的索网构型.本文的索网找形方法可广泛应用于空间索网天线结构的设计.