Stochastic stability of parametrically excited random systems

Multidegree-of-freedom dynamic systems subjected to parametric excitation are analyzed for stochastic stability. The variation of excitation intensity with time is described by the sum of a harmonic function and a stationary random process. The stability boundaries are determined by the stochastic averaging method. The effect of random parametric excitation on the stability of trivial solutions of systems of differential equations for the moments of phase variables is studied. It is assumed that the frequency of harmonic component falls within the region of combination resonances. Stability conditions for the first and second moments are obtained. It turns out that additional parametric excitation may have a stabilizing or destabilizing effect, depending on the values of certain parameters of random excitation. As an example, the stability of a beam in plane bending is analyzed.Published in Prikladnaya Mekhanika, Vol. 40, No. 10, pp. 135–144, October 2004.     

Dynamic stability of a viscoelastic rotating shaft under parametric random excitation

This paper deals with dynamic stability of a viscoelastic rotating shaft subjected to a parametric random axial compressive thrust, by using moment Lyapunov exponents and the largest Lyapunov exponents as indicators. The equation of motion for the shaft is derived, which is a system of gyroscopic stochastic differential equations. The method of stochastic averaging is used to decouple the governing equations into Itô equations, from which the moment Lyapunov exponent is obtained by using mathematical transformations only. The largest Lyapunov exponent is obtained through its relation with moment Lyapunov exponents. The effects of various parameters on the stochastic dynamic stability are discussed. The approximate analytical results are confirmed by Monte Carlo simulation.     

Bifurcation in a parametrically excited two-degree-of-freedom nonlinear oscillating system with 1∶2 internal resonance

The nonlinear response of a two-degree-of-freedom nonlinear oscillating system to parametric excitation is examined for the case of 1∶2 internal resonance and, principal parametric resonance with respect to the lower mode. The method of multiple scales is used to derive four first-order autonomous ordinary differential equations for the modulation of the amplitudes and phases. The steadystate solutions of the modulated equations and their stability are investigated. The trivial solutions lose their stability through pitchfork bifurcation giving rise to coupled mode solutions. The Melnikov method is used to study the global bifurcation behavior, the critical parameter is determined at which the dynamical system possesses a Smale horseshoe type of chaos. Project supported by the National Natural Science Foundation of China (19472046)     

An extensive FPGA-based realization study about the Izhikevich neurons and their bio-inspired applications

This study is aimed at analysing damping and gyroscopic effects on the stability of parametrically excited continuous rotor systems, taking into account both external (non-rotating) and internal (rotating) damping distributions. As case-study giving rise to a set of coupled differential Mathieu–Hill equations with both damping and gyroscopic terms, a balanced shaft is considered, modelled as a spinning Timoshenko beam loaded by oscillating axial end thrust and twisting moment, with the possibility of carrying additional inertial elements like discs or flywheels. After discretization of the equations of motion into a set of coupled ordinary differential Mathieu–Hill equations, stability is studied via eigenproblem formulation, obtained by applying the harmonic balance method. The occurrence of simple and combination parametric resonances is analysed introducing the notion of characteristic circle on the complex plane and deriving analytical expressions for critical solutions, including combination parametric resonances, valid for a large class of rotors. A numerical algorithm is then developed for computing global stability thresholds in the presence of both damping and gyroscopic terms, also valid when closed-form expressions of critical solutions do not exist. The influence on stability of damping distributions and gyroscopic actions is then analysed with respect to frequency and amplitude of the external loads on stability charts in the form of Ince–Strutt diagrams.

     

Nonlinear Random Response of Ocean Structures Using First- and Second-Order Stochastic Averaging

This paper deals with the dynamic response of nonlinear elastic structure subjected to random hydrodynamic forces and parametric excitation using a first- and second-order stochastic averaging method. The governing equation of motion is derived by using Hamilton's principle, taking into account inertia and curvature nonlinearities and work done due to hydrodynamic forces. Within the framework of first-order stochastic averaging, the system response statistics and stability boundaries are obtained. Unfortunately, the effects of nonlinear inertia and curvature are not reflected in the final results, since the contribution of these nonlinearities is lost during the averaging process. In the absence of hydrodynamic forces, the method fails to give bounded response statistics, and the analysis yields stability conditions. It is the second-order stochastic averaging which can capture the influence of stiffness and inertia nonlinearities that were lost in the first-order averaging process. The results of the second-order averaging are compared with those predicted by Gaussian and non-Gaussian closures and by Monte Carlo simulation. In the absence of parametric excitation, the non-Gaussian closure solutions are in good agreement with Monte Carlo simulation. On the other hand, in the absence of hydrodynamic forces, second-order averaging gives more reliable results in the neighborhood of stochastic bifurcation. However, under pure parametric random excitation, the stochastic averaging and Monte Carlo simulation predict the on-off intermittency phenomenon near bifurcation point, in addition to stochastic bifurcation in probability.     

谐和与窄带随机噪声联合作用下Duffing系统的参数主共振

研究了Ｄｕｆｉｎｇ振子在谐和与窄带随机噪声联合激励下的参数主共振响应和稳定性问题．用多尺度法分离了系统的快变项，并求出了系统的最大Ｌｙａｐｕｎｏｖ指数．本文还分析了失稳及跳跃现象，及系统的阻尼项、非线性项、随机项、确定性参激强度对系统响应的影响．数值模拟表明本文提出的方法是有效的．     

Nonlinear study of the dynamic behavior of a string-beam coupled system under combined excitations

In this paper,the nonlinear dynamic behavior of a string-beam coupled system subjected to external,parametric and tuned excitations is presented.The governing equations of motion are obtained for the nonlinear transverse vibrations of the string-beam coupled system which are described by a set of ordinary differential equations with two degrees of freedom.The case of 1:1 internal resonance between the modes of the beam and string,and the primary and combined resonance for the beam is considered.The method of multiple scales is utilized to analyze the nonlinear responses of the string-beam coupled system and obtain approximate solutions up to and including the second-order approximations.All resonance cases are extracted and investigated.Stability of the system is studied using frequency response equations and the phase-plane method.Numerical solutions are carried out and the results are presented graphically and discussed.The effects of the different parameters on both response and stability of the system are investigated.The reported results are compared to the available published work.     

随机窄带参激下二自由度非线性系统

研究了带平方二自由度非线性系统在随机窄带参数激励下，用多尺度法分离了系统的快变项，讨论了系统的各参数对响应的影响。在一定条件下，系统具有两个均方响应值，具有跳跃现象和饱和现象，数值模拟表明提出的方法是有效的。     

窄带噪声作用下二自由度非线性系统的响应

研究二自由度非线性系统在窄带随机噪声激励下的主共振响应,用多尺度法分离了系统的快变项,讨论了系统的阻尼项、随机项等对系统响应的影响。在一定条件下,系统具有两个均方响应值和跳跃现象,饱和现象也存在。数值模拟表明文中所提出的方法是有效的。     

Stability and response of a nonlinear coupled pitch-roll ship model under parametric and harmonic excitations

The present study deals with the response of a two-degree-of-freedom (2DOF) system with quadratic coupling under parametric and harmonic excitations. The method of multiple scale perturbation technique is applied to solve the nonlinear differential equations and obtain approximate solutions up to and including the second order approximations. All resonance cases are extracted and investigated. Stability of the system is studied using frequency response equations and phase-plane method. Numerical solutions are carried out and the results are presented graphically and discussed. The effects of the different parameters on both response and stability of the system are investigated. The reported results are compared to the available published work.     

复合材料层合板1:1参数共振的分岔研究

针对复合材料对称铺设各向异性矩形层合板的物理模型,在同时考虑了材料、阻尼和几何等非线性因素后,建立了二自由度非线性参数振动系统动力学控制方程,并应用多尺度法求得基本参数共振下的近似解析解,利用数值模拟分析了系统的分岔和混沌运动.指出了伽辽金截断对系统动力学分析的影响,以及系统进入混沌的途径.     

Response of nonlinear oscillator under narrow-band random excitation

IntroductionThestudyoftheresponseofnonlinearsystemstonarrow_bandrandomexcitationofconsiderableimportance.Forexample ,theexcitationofsecondarysystemwouldbeanarrow_bandrandomprocessiftheprimarysystemcouldbemodeledasasingle_degree_of_freedomsystemwithlightdampingsubjecttowide_bandexcitation .Inthetheoryofnonlinearrandomvibration ,mostresultsobtainedsofarareattributedtotheresponseofnonlinearoscillatorstowide_bandrandomexcitation .Incomparison ,resultsontheeffectofnarrow_bandexcitationonnonlinearos…     

Dynamic stability of weak equations of continuous systems

The stability analysis method is developed for distributed dynamic problems with relaxed assumptions imposed on solutions. The problem is motivated by structural vibrations with external time-dependent parametric excitations which are controlled using surface-mounted or -embedded actuators and sensors. The strong form of equations involves irregularities which lead to computational difficulties for estimation and control problems. In order to avoid irregular terms resulting from differentiation of the force and moment terms the dynamics equations are written in a weak form. The weak form of dynamics equations of linear mechanical structures is obtained using variational calculus. The study of stability of stochastic weak system is based on examining properties of Liapunov functional along a weak solution.     

Stochastic stability of quasi-partially integrable and non-resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises

The asymptotic Lyapunov stability with probability one of multi-degree-of freedom quasi-partially integrable and non-resonant Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises is studied. First, the averaged stochastic differential equations for quasi partially integrable and non-resonant Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises are derived by means of the stochastic averaging method and the stochastic jump-diffusion chain rule. Then, the expression of the largest Lyapunov exponent of the averaged system is obtained by using a procedure similar to that due to Khasminskii and the properties of stochastic integro-differential equations. Finally, the stochastic stability of the original quasi-partially integrable and non-resonant Hamiltonian systems is determined approximately by using the largest Lyapunov exponent. An example is worked out in detail to illustrate the application of the proposed method. The good agreement between the analytical results and those from digital simulation show that the proposed method is effective.     

Stability of elastic systems under a stochastic parametric excitation

An efficient method to investigate the stability of elastic systems subjected to the parametric force in the form of a random stationary colored noise is suggested. The method is based on the simulation of stochastic processes, numerical solution of differential equations, describing the perturbed motion of the system, and the calculation of top Liapunov exponents. The method results in the estimation of the almost sure stability and the stability with respect to statistical moments of different orders. Since the closed system of equations for moments of desired quantities y _j (t) cannot be obtained, the statistical data processing is applied. The estimation of moments at the instant t _n is obtained by statistical average of derived from the solution of equations for the large number of realizations. This approach allows us to evaluate the influence of different characteristics of random stationary loads on top Liapunov exponents and on the stability of system. The important point is that results found for filtered processes, are principally different from those corresponding to stochastic processes in the form of Gaussian white noises.     

Maximal Lyapunov exponent and almost-sure sample stability for second-order linear stochastic system

The principal resonance of second-order system to random parametric excitation is investigated. The method of multiple scales is used to determine the equations of modulation of amplitude and phase. The effects of damping, detuning, bandwidth, and magnitudes of random excitation are analyzed. The explicit asymptotic formulas for the maximum Lyapunov exponent is obtained. The almost-sure stability or instability of the stochastic Mathieu system depends on the sign of the maximum Lyapunov exponent.     

Bifurcation Analyses in the Parametrically Excited Vibrations of Cylindrical Panels

We consider parametrically excited vibrations of shallow cylindrical panels. The governing system of two coupled nonlinear partial differential equations is discretized by using the Bubnov–Galerkin method. The computations are simplified significantly by the application of computer algebra, and as a result low dimensional models of shell vibrations are readily obtained. After applying numerical continuation techniques and ideas from dynamical systems theory, complete bifurcation diagrams are constructed. Our principal aim is to investigate the interaction between different modes of shell vibrations under parametric excitation. Results for system models with four of the lowest modes are reported. We essentially investigate periodic solutions, their stability and bifurcations within the range of excitation frequency that corresponds to the parametric resonances at the lowest mode of vibration.     

Asymptotic Lyapunov stability with probability one of quasi linear systems subject to multi-time-delayed feedback control and wide-band parametric random excitation

The asymptotic Lyapunov stability with probability one of multi-degree-of-freedom quasi linear systems subject to multi-time-delayed feedback control and multiplicative (parametric) excitation of wide-band random process is studied. First, the multi-time-delayed feedback control forces are expressed approximately in terms of the system state variables without time delay and the system is converted into ordinary quasi linear system. Then, the averaged Itô stochastic differential equations are derived by using the stochastic averaging method for quasi linear systems and the expression for the largest Lyapunov exponent of the linearized averaged Itô equations is derived. Finally, the necessary and sufficient condition for the asymptotic Lyapunov stability with probability one of the trivial solution of the original system is obtained approximately by letting the largest Lyapunov exponent to be negative. An example is worked out in detail to illustrate the application and validity of the proposed procedure and to show the effect of the time delay in feedback control on the largest Lyapunov exponent and the stability of system.     

Solutions for a system of nonlinear random integral and differential equations under weak topology

I.IntroductionTheexistenceandcomparisonresultsofsolutionsfornonlinearVolterraintegralequationsinstrongtopologyofBanachspaceshavebeenobtainedbyVaughnI"'"],LakshmikanthamIl3]andLakshmikantham-Leela114l.TheexistenceresultsofweaksolutionsfortheCauchyproblemof…     

Vibration reduction of a nonlinear spring pendulum under multi external and parametric excitations via a longitudinal absorber

The vibration of a ship pitch-roll motion described by a non-linear spring pendulum system (two degrees of freedom) subjected to multi external and parametric excitations can be reduced using a longitudinal absorber. The method of multiple scale perturbation technique (MSPT) is applied to analyze the response of this system near the simultaneous primary, sub-harmonic and internal resonance. The steady state solution near this resonance case is determined and studied applying Lyapunov’s first method. The stability of the system is investigated using frequency response equations. Numerical simulations are extensive investigations to illustrate the effects of the absorber and some system parameters at selected values on the vibrating system. The simulation results are achieved using MATLAB 7.0 programs. Results are compared to previously published work.