Flutter instability and other singularity phenomena in symmetric systems via combination of mass distribution and weak damping

The local dynamic instability of autonomous conservative, lumped-mass (discrete) systems, is thoroughly discussed when negligibly small dissipative forces are included. It is shown that such small forces may change drastically the response of these systems. Hence, existing, widely accepted, findings based on the omission of damping could not be valid if damping, being always present in actual systems, is included. More specifically the conditions under which the above systems may experience dynamic bifurcations associated either with a degenerate or a generic Hopf bifurcation are examined in detail by studying the effect of the structure of the damping matrix on the Jacobian eigenvalues. The case whereby this phenomenon may occur before divergence is discussed in connection with the individual or coupling effect of non-uniform mass and stiffness distribution. Jump phenomena in the critical dynamic loading at a certain mass distribution are also assessed. Numerical results verified by a non-linear dynamic analysis using 2-DOF and 3-DOF models confirm the validity of the theoretical findings as well as the efficiency of the technique proposed herein.     

Finite element analysis of post-buckling dynamics in plates—Part I: An asymptotic approach

Various static and dynamic aspects of post-buckled thin plates, including the transition of buckled patterns, post-buckling dynamics, secondary bifurcation, and dynamic snapping (mode jumping phenomenon), are investigated systematically using asymptotical and non-stationary finite element methods. In part I, the secondary dynamic instability and the local post-secondary buckling behavior of thin rectangular plates under generalized (mechanical and thermal) loading is investigated using an asymptotic numerical method which combines Koiter’s nonlinear instability theory with the finite element technique. A dynamic multi-mode reduction method—similar to its static single-mode counterpart: Liapunov–Schmidt reduction—is developed in this perturbation approach. Post-secondary buckling equilibrium branches are obtained by solving the reduced low-dimensional parametric equations and their stability properties are determined directly by checking the eigenvalues of the resulting Jacobian matrix. Typical post-secondary buckling forms—transcritical, supercritical and subcritical bifurcations are observed according to different combinations of boundary conditions and load types. Geometric imperfection analysis shows that not only the secondary bifurcation load but also changes in the fundamental post-secondary buckling behavior are affected. The post-buckling dynamics and the global analysis of mode jumping of the plates are addressed in part II.     

The Peierls stress in non-local elasticity

The effect of non-locality on the Peierls stress of a dislocation, predicted within the framework of the Peierls-Nabarro model, is investigated. Both the integral formulation of non-local elasticity and the gradient elasticity model are considered. A modification of the non-local kernel of the integral formulation is proposed and its effect on the dislocation core shape and size, and on the Peierls stress are discussed. The new kernel is longer ranged and physically meaningful, improving therefore upon the existing Gaussian-like non-locality kernels. As in the original Peierls-Nabarro model, lattice trapping cannot be captured in the purely continuum non-local formulation and therefore, a semi-discrete framework is used. The constitutive law of the elastic continuum and that of the glide plane are considered both local and non-local in separate models. The major effect is obtained upon rendering non-local the constitutive law of the continuum, while non-locality in the rebound force law of the glide plane has a marginal effect. The Peierls stress is seen to increase with increasing the intrinsic length scale of the non-local formulation, while the core size decreases accordingly. The solution becomes unstable at intrinsic length scales larger than a critical value. Modifications of the rebound force law entail significant changes in the core configuration and critical stress. The discussion provides insight into the issue of internal length scale selection in non-local elasticity models.     

亏损系统模态优化控制Riccati方程的求解方法

讨论亏损重复特征值的Ｒｉｃｃａｔｉ方程的模态控制算法．由于亏损特征值的Ｊｏｒｄａｎ块矩阵的阶数ｍ小于状态方程的阶数ｎ,即ｍ＜＜ｎ;所提出的模态控制算法可极大地减少计算工作量．其中的数值例子说明了该方法的有效性．     

Discontinuous bifurcation analysis of thermodynamically consistent gradient poroplastic materials

In this work, analytical and numerical solutions of the condition for discontinuous bifurcation of thermodynamically consistent gradient-based poroplastic materials are obtained and evaluated. The main aim is the analysis of the potentials for localized failure modes in the form of discontinuous bifurcation in partially saturated gradient-based poroplastic materials as well as the dependence of these potentials on the current hydraulic and stress conditions. Also the main differences with the localization conditions of the related local theory for poroplastic materials are evaluated to perfectly understand the regularization capabilities of the non-local gradient-based one. Firstly, the condition for discontinuous bifurcation is formulated from wave propagation analyses in poroplastic media. The material formulation employed in this work for the spectral properties evaluation of the discontinuous bifurcation condition is the thermodynamically consistent, gradient-based modified Cam Clay model for partially saturated porous media previously proposed by the authors. The main and novel feature of this constitutive theory is the inclusion of a gradient internal length of the porous phase which, together with the characteristic length of the solid skeleton, comprehensively defined the non-local characteristics of the represented porous material. After presenting the fundamental equations of the thermodynamically consistent gradient based poroplastic constitutive model, the analytical expressions of the critical hardening/softening modulus for discontinuous bifurcation under both drained and undrained conditions are obtained. As a particular case, the related local constitutive model is also evaluated from the discontinuous bifurcation condition stand point. Then, the localization analysis of the thermodynamically consistent non-local and local poroplastic Cam Clay theories is performed. The results demonstrate, on the one hand and related to the local poroplastic material, the decisive role of the pore pressure and of the volumetric non-associativity degree on the location of the transition point between ductile and brittle failure regimes in the stress space. On the other hand, the results demonstrate as well the regularization capabilities of the non-local gradient-based poroplastic theory, with exception of a particular stress condition which is also evaluated in this work. Finally, it is also shown that, due to dependence of the characteristic lengths for the pore and skeleton phases on the hydraulic and stress conditions, the non-local theory is able to reproduce the strong reduction of failure diffusion that takes place under both, low confinement and low pore pressure of partially saturated porous materials, without loosing, however, the ellipticity of the related differential equations.     

Hopf分岔的代数判据及其在车辆动力学中的应用

利用Hurwitz行列式,给出平衡点失稳而发生Hopf分岔的代数判定准则和计算方法,这一方法将Hopf分岔点的求解转化为一个非线性方程的求解问题,从而克服了以前方法在计算Hopf分岔点时,对于参数的每一次变化通过求特征根并判定特征根的实部是否为零的庞大工作量。应用这一方法,我们进行了非线性车辆系统蛇行运动稳定性的研究,得到了轮对系统发生蛇行运动的临界速度的解析表达式。     

Instability of a Longitudinal Movement along the Cylindrical Surface for a Thermoelastic Web Simulated by Stretched and Heated String

On the basis of classical methods for mathematical physics and mechanics, the stability problem of a thermoelastic web moving at a constant speed without friction along a cylindrical surface is investigated. The web is modeled by a stretched and heated string. At a sufficiently high speed and heating of a string, a loss ofmotion stability and the stringmovement in a direction normal to the cylindrical surface occur. To study the instability, a static method based on the consideration of stationary nontrivial modes of stability loss, that is, on the study of the problem for bifurcation of solutions (eigenvalue problem) for the corresponding differential equations is used. The case of the web motion along the circular cylinder is separately considered and an expression for the critical velocity leading to the instability is found.     

Stability and bifurcation for a flexible beam under a large linear motion with a combination parametric resonance

Stability and bifurcation behaviors for a model of a flexible beam undergoing a large linear motion with a combination parametric resonance are studied by means of a combination of analytical and numerical methods. Three types of critical points for the bifurcation equations near the combination resonance in the presence of internal resonance are considered, which are characterized by a double zero and two negative eigenvalues, a double zero and a pair of purely imaginary eigenvalues, and two pairs of purely imaginary eigenvalues in nonresonant case, respectively. The stability regions of the initial equilibrium solution and the critical bifurcation curves are obtained in terms of the system parameters. Especially, for the third case, the explicit expressions of the critical bifurcation curves leading to incipient and secondary bifurcations are obtained with the aid of normal form theory. Bifurcations leading to Hopf bifurcations and 2-D tori and their stability conditions are also investigated. Some new dynamical behaviors are presented for this system. A time integration scheme is used to find the numerical solutions for these bifurcation cases, and numerical results agree with the analytic ones.     

Fractional diffusion equations for open quantum system

To describe non-local interactions of quantum systems with environment we consider a fractional generalization of the quantum Markovian equation. Quantum analogs of fractional Laplacian operator for coordinate and momentum spaces are suggested. In phase-space form of quantum mechanics we obtain fractional equations for Wigner distribution function, where fractional Laplacian operators of the Grünvald–Letnikov type are used.     

Bifurcation Equations Through Multiple-Scales Analysis for a Continuous Model of a Planar Beam

The Multiple-Scale Method is applied directly to a one-dimensional continuous model to derive the equations governing the asymptotic dynamic of the system around a bifurcation point. The theory is illustrated with reference to a specific example, namely an internally constrained planar beam, equipped with a lumped viscoelastic device and loaded by a follower force. Nonlinear, integro-differential equations of motion are derived and expanded up to cubic terms in the transversal displacements and velocities of the beam. They are put in an operator form incorporating the mechanical boundary conditions, which account for the lumped viscoelastic device; the problem is thus governed by mixed algebraic-integro-differential operators. The linear stability of the trivial equilibrium is first studied. It reveals the existence of divergence, Hopf and double-zero bifurcations. The spectral properties of the linear operator and its adjoint are studied at the bifurcation points by obtaining closed-form expressions. Notably, the system is defective at the double-zero point, thus entailing the need to find a generalized eigenvector. A multiple-scale analysis is then performed for the three bifurcations and the relevant bifurcation equations are derived directly in their normal forms. Preliminary numerical results are illustrated for the double-zero bifurcation.     

On stability boundary of linear multi-parameter Hamiltonian systems

In this paper an approximate equation is derived to describe smooth parts of the stability boundary for linear Hamiltonian systems, depending on arbitrary number of parameters. With this equation, we can obtain parameters corresponding to the stability boundary, as well as to the stability and instability domains, provided that one point on the stability boundary is known. Then differential equations describing the evolution of eigenvalues and eigenvectors along a curve on the stability boundary surface are derived. These equations also allow us to obtain curves belonging to the stability boundary. Applications to linear gyroscopic systems are considered and studied with examples. The project supported by the National Science Foundations of Russia and China (10072012)     

On stability, persistence, and Hopf bifurcation in fractional order dynamical systems

This is a preliminary study about the bifurcation phenomenon in fractional order dynamical systems. Persistence of some continuous time fractional order differential equations is studied. A numerical example for Hopf-type bifurcation in a fractional order system is given, hence we propose a modification of the conditions of Hopf bifurcation. Local stability of some biologically motivated functional equations is investigated.     

Stability and Hamiltonian Hopf bifurcation for a nonlinear symmetric bladed rotor

The modal interaction which leads to Hamiltonian Hopf bifurcation is studied for a nonlinear rotating bladed-disk system. The model, which is discussed in the paper, is a Jeffcott rotor carrying a number of planar blades which bend in the plane of the motion. The rigid rotating disk is supported on nonlinear bearings. It is supposed that this dynamical system is a Hamiltonian system which is perturbed by small dissipative and nonlinear forces. Krein’s theorem is employed for obtaining a stability criterion. The nonlinear eigenvalue equations on the stability boundary are turned into ordinary differential equations (ODEs) by differentiating them over the rotating speed. By solving these ODEs, the eigenmodes and the eigenvalues on the stability boundary are obtained. The bifurcation analysis is performed by applying multiple scales method around the boundary. The rotor nonlinear behavior and damping effects are studied for different conditions on the rotating speed and nonlinearity type by the bifurcation equation. It is shown that the damping distribution between the blades and bearings may shift the unstable mode. Depending on the nonlinearity type, subcritical and supercritical Hopf bifurcation are possible.     

Bifurcations and chaotic dynamics in suspended cables under simultaneous parametric and external excitations

The bifurcations and chaotic dynamics of parametrically and externally excited suspended cables are investigated in this paper. The equations of motion governing such systems contain quadratic and cubic nonlinearities, which may result in two-to-one and one-to-one internal resonances. The Galerkin procedure is introduced to simplify the governing equations of motion to ordinary differential equations with two-degree-of-freedom. The case of one-to-one internal resonance between the modes of suspended cables, primary resonant excitation, and principal parametric excitation of suspended cables is considered. Using the method of multiple scales, a parametrically and externally excited system is transformed to the averaged equations. A pseudo arclength scheme is used to trace the branches of the equilibrium solutions and an investigation of the eigenvalues of the Jacobian matrix is used to assess their stability. The equilibrium solutions experience pitchfork, saddle-node, and Hopf bifurcations. A detailed bifurcation analysis of the dynamic (periodic and chaotic) solutions of the averaged equations is presented. Five branches of dynamic solutions are found. Three of these branches that emerge from two Hopf bifurcations and the other two are isolated. The two Hopf bifurcation points, one is supercritical Hopf bifurcation point and another is primary Hopf bifurcation point. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging, boundary crises. Simultaneous occurrence of the limit cycle and chaotic attractors, homoclinic orbits, homoclinic explosions and hyperchaos are also observed.     

On non-local and non-homogeneous elastic continua

A thermodynamic framework endowed with the concept of non-locality residual is adopted to derive non-local models of integral-type for non-homogeneous linear elastic materials. Two expressions of the free energy are considered: the former yields a one-component non-local stress, the latter leads to a two-component local–non-local stress since the stress is expressed as the sum of the classical local stress and of a non-local component identically vanishing in the case of constant strains. The attenuation effects are accounted for by a symmetric space weight function which guarantees the constant strain requirement as well as the dual constant stress condition everywhere in the body. The non-local and non-homogeneous elastic structural boundary-value problem under quasi-static loads is addressed in a geometrically linear range. The complete set of variational formulations for the structural problem is then provided in a unitary framework. The solution uniqueness of the non-local structural model is proved and the non-local FEM is addressed starting from the non-local counterpart of the total potential energy. Numerical applications are provided with reference to a non-homogeneous bar in tension using the Fredholm integral equation and the non-local FEM. The solutions show no pathological features such as numerical instability and mesh sensitivity for degraded bar conditions.     

弹性压应力波下直杆动力失稳的机理的判据

基于应力波理论和失稳瞬间能量的转换和守恒,导出了一个直杆动力分岔失稳的准则：（1）直杆在发生分岔失稳的瞬间所释放出的压缩变形能等于屈曲所需变形能与屈曲动能之和;（2）在上述能量转换过程中,能量对时间的变化率服从守恒定律。应用临界条件（1）推导出的直杆动力失稳的控制方程和杆端边界条件以及连续条件,与应用哈密顿原理推导的结果完全相同,但不足以构成求解直杆动力失稳问题的完备定解条件,导出包含两个特征参数的一对特征方程。从而建立了求解直杆动力失稳模态和两个特征参数（临界力参数和失稳惯性项指数参数即动力特征参数）的较严密理论方法。     

Nonlinear stability analysis of a disk brake model

It has become commonly accepted by scientists and engineers that brake squeal is generated by friction-induced self-excited vibrations of the brake system. The noise-free configuration of the brake system loses stability through a flutter-type instability and the system starts oscillating in a limit cycle. Usually, the stability analysis of disk brake models, both analytical as well as finite element based, investigates the linearized models, i.e. the eigenvalues of the linearized equations of motion. However, there are experimentally observed effects not covered by these analyses, even though the full nonlinear models include these effects in principle. The present paper describes the nonlinear stability analysis of a realistic disk brake model with 12 degrees of freedom. Using center manifold theory and artificially increasing the degree of degeneracy of the occurring bifurcation, an analytical expression for the turning points in the bifurcation diagram of the subcritical Hopf bifurcations is calculated. The parameter combination corresponding to the turning points is considered as the practical stability boundary of the system. Basic phenomena known from the operating experience of brake systems tending to squeal problems can be explained on the basis of the practical stability boundary.     

Stability and bifurcation for limit cycle oscillations of an airfoil with external store

In this paper, stability and local bifurcation behaviors for the nonlinear aeroelastic model of an airfoil with external store are investigated using both analytical and numerical methods. Three kinds of degenerated equilibrium points of bifurcation response equations are considered. They are characterized as (1) one pair of purely imaginary eigenvalues and two pairs of conjugate complex roots with negative real parts; (2) two pairs of purely imaginary eigenvalues in nonresonant case and one pair of conjugate complex roots with negative real parts; (3) three pairs of purely imaginary eigenvalues in nonresonant case. With the aid of Maple software and normal form theory, the stability regions of the initial equilibrium point and the explicit expressions of the critical bifurcation curves are obtained, which can lead to static bifurcation and Hopf bifurcation. Under certain conditions, 2-D tori motion may occur. The complex dynamical motions are considered in this paper. Finally, the numerical solutions achieved by the fourth-order Runge–Kutta method agree with the analytic results.     

Nonlinear three-dimensional oscillations of elastically constrained fluid conveying viscoelastic tubes with perfect and broken O(2)-symmetry

The loss of the stability of the trivial downhanging equilibrium position of a slender circular tube conveying incompressible fluid flow is studied. The tube is clamped at its upper end and is free at its lower end. Inbetween, the three-dimensional transversal motion is constrained by an elastic support considered to be rotationally symmetric. Tube equations valid for large displacement but small strain based on Kirchhoff's rod theory and the Kelvin-Voigt viscoelastic law are used.The stability analysis is performed by making use of the methods of the equivariant bifurcation theory; that is, but using the symmetry properties of the original system to drrive the amplitude equations of the critical modes. Two different types of results are given: First, for the perfect O(2)-symmetric system all three generic coincident eigenvalue cases of loss of stability in two-parameter families. Second, for the system with broken O(2)-symmetry due to imperfections, three special cases of loss of stability at simple eigenvalues.     

Second-Order Constitutive Equations

Studies of strain localization, wave dispersion, instability, and plasticity of material promote the use of gradient-dependent constitutive equations. Some of their applications are summarized in a table. Such a group of constitutive equations is called second-order constitutive equations. This paper studies whether second-order constitutive equations can be used to describe some dynamic wave properties. Assumptions and conditions are set up for constitutive equations to study second-order constitutive equations