Nonlinear dynamics of dry friction oscillator subjected to combined harmonic and random excitations

Nonlinear Dynamics - The dynamics of a nonlinear single degree freedom oscillator on a moving belt subjected to combined harmonic and random excitations is numerically investigated. The dynamics is...     

Resonant response of a non-linear vibro-impact system to combined deterministic harmonic and random excitations

The resonant resonance response of a single-degree-of-freedom non-linear vibro-impact oscillator, with cubic non-linearity items, to combined deterministic harmonic and random excitations is investigated. The method of multiple scales is used to derive the equations of modulation of amplitude and phase. The effects of damping, detuning, and intensity of random excitations are analyzed by means of perturbation and stochastic averaging method. The theoretical analyses verified by numerical simulations show that when the intensity of the random excitation increases, the non-trivial steady-state solution may change from a limit cycle to a diffused limit cycle. Under certain conditions, impact system may have two steady-state responses. One is a non-impact response, and the other is either an impact one or a non-impact one.     

Bifurcation and stability analysis for a non-smooth friction oscillator

Summary Friction-induced self-sustained oscillations, also known as stick-slip vibrations, occur in mechanical systems as well as in everyday life. On the basis of a one-dimensional map, the bifurcation behaviour including unstable branches is investigated for a friction oscillator with simultaneous self-and external excitation. The chosen way of mapping also allows a simple determination of Lyapunov exponents.Dedicated to Prof. Dr.-Ing. Dr.-Ing. E.h. Dr. h.c. mult. Erwin Stein on the occasion of his 65th birthday.     

Asymptotic analysis of the mathematical expectation of the total energy of a harmonic oscillator under random pulse perturbation

We study the behavior of the mathematical expectation of the total energy of a harmonic frictionless oscillator as t → ∞ under external Poisson pulse perturbation.     

Nonlinear response of SDOF systems under combined deterministic and random excitations

Dynamical systems subjected to random excitations exhibit non-linear behavior for sufficiently large motion. The multiple time scale method has been extensively utilized in the framework of non-linear deterministic analysis to obtain two averaged first-order differential equations describing the slow time scale modulation of amplitude and phase response. In this paper the multiple time scale method, opportunely modified to take properly into account the correlation structure of the stochastic input process, is adopted to derive a stochastic frequency-response relationship involving the response amplitude statistics and the input power spectral density. A low-intensity noise is assumed to separate the strong mean motion from its weak fluctuations. The moment differential equations of phase and amplitude are derived and a linearization technique applied to evaluate the second order statistics. The theory is validated through digital simulations on a nonlinear single degree of freedom model for the transversal oscillation of a cantilever beam with tip force and to a Duffing-Rayleigh oscillator, to analyze non-linear damping effects.     

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.     

Rocking response of rectangular rigid blocks under random noise base excitations

The response of a rigid rectangular block resting on a rigid foundation and acted upon simultaneously by a horizontal and a vertical random white-noise excitation is considered. In the equation of motion, the energy dissipation is modeled through a viscous damping term. Under the assumption that the body does not topple, the steady-state joint probability density function of the rotation and the rotational velocity is obtained using the Fokker-Planck equation approach. Closed form solution is obtained for a specific combination of system parameters. A more general but approximate solution to the joint probability density function based on the method of equivalent non-linearization is also presented. Further, the problem of overturning of the block is approached in the framework of the diffusion methods for first passage failure studies. The overturning of the block is deemed incipient when the response trajectories in the phase plane cross the separatrix of the conservative unforced system. Expressions for the moments of first passage time are obtained via a series solution to the governing generalized Pontriagin-Vitt equations. Numerical results illustra- tive of the theoretical solutions are presented and their validity is examined through limited amount of digital simulations.     

Stochastic jump and bifurcation of Duffing oscillator with fractional derivative damping under combined harmonic and white noise excitations

The stochastic jump and bifurcation of Duffing oscillator with fractional derivative damping of order α (0<α<1) under combined harmonic and white noise excitations are studied. First, the system state is approximately represented by two-dimensional time-homogeneous diffusive Markov process of amplitude and phase difference using the stochastic averaging method. Then, the method of reduced Fokker–Plank–Kolmogorov (FPK) equation is used to predict the stationary response of the original system. The phenomenon of stochastic jump and bifurcation as the fractional orders' change is examined.     

Analytical parametric study of bi-linear hysteretic model of dry friction under harmonic,impulse and random excitations

Hysteretic behavior due to some nonlinear sources is a common phenomenon in many dynamical systems. One of the sources of this behavior in mechanical systems is dry friction. Dry friction in bolted or riveted joints of mechanical structures makes their dynamic behavior hysteretic. Bi-linear hysteresis is one of the models that can be used to study these systems which is used in this paper. A SDOF system containing a bi-linear hysteretic element called Jenkins element under harmonic, impulse and random excitations is considered. For all three types of excitations, the effects of system and excitation parameters on the defined equivalent system parameters and the response specifications are studied. Harmonic balance method is employed for harmonic excitation studies, and optimum friction threshold for minimizing response amplitude is obtained versus other system parameters and response amplitude. Energy balance method is used for impulse excitation through which the desired decaying ratio can be achieved by tuning the friction threshold, depending on stiffness ratio. System under random excitation is investigated by equivalent linearization technique in two steps. At the first step, equivalent properties are obtained versus instantaneous amplitude of response. In this step, the paper contains the parametric study of system in which the variations of equivalent parameters are described when physical parameters of system or input intensity vary. Overall variance of system response is determined in the second step, and optimum sliding threshold is obtained to have minimum overall variance of system response.     

Stochastic sensitivity analysis of coupled acoustic-structural systems under non-stationary random excitations

This paper presents a new sensitivity analysis method for coupled acoustic–structural systems subjected to non-stationary random excitations. The integral of the response power spectrum density (PSD) of the coupled system is taken as the objective function. The thickness of each structural element is used as a design variable. A time-domain algorithm integrating the pseudo excitation method (PEM), direct differentiation method (DDM) and high precision direct (HPD) integration method is proposed for the sensitivity analysis of the objective function with respect to design variables. Firstly, the PEM is adopted to transform the sensitivity analysis under non-stationary random excitations into the sensitivity analysis under pseudo transient excitations. Then, the sensitivity analysis equation of the coupled system under pseudo transient excitations is derived based on the DDM. Moreover, the HPD integration method is used to efficiently solve the sensitivity analysis equation under pseudo transient excitations in a reduced-order modal space. Numerical examples are presented to demonstrate the validity of the proposed method.     

转子系统的平稳／非平稳随机地震响应分析

应用虚拟激励法结合精细时程积分计算了转子系统受平稳／非平稳随机地震激励的动力响应。采用虚拟激励分析将平稳随机激励转化为稳态简谐激励，将非平稳随机激励转化为瞬态确定性激励，即使对于非对称的油膜刚度阵和阻尼阵，算法仍然简单高效，并得到精确的结果。     

Nonlinearly damped systems under simultaneous broad-band and harmonic excitations

The probability distribution of the response of a nonlinearly damped system subjected to both broad-band and harmonic excitations is investigated. The broad-band excitation is additive, and the harmonic excitations can be either additive or multiplicative. The frequency of a harmonic excitation can be either near or far from a resonance frequency of the system. The stochastic averaging method is applied to obtain the Itô type stochastic differential equations for an averaged system described by a set of slowly varying variables, which are approximated as components of a Markov vector. Then, a procedure based on the concept of stationary potential is used to obtain the exact stationary probability density for a class of such averaged systems. For those systems not belonging to this class, approximate solutions are obtained using the method of weighted residuals. Application of the exact and approximate solution procedures are illustrated in two specific cases, and the results are compared with those obtained from Monte Carlo simulations.     

PRINCIPAL RESPONSE OF VAN DER POL-DUFFING OSCILLATOR UNDER COMBINED DETERMINISTIC AND RANDOM PARAMETRIC EXCITATION

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Reliability of a class of nonlinear systems under switching random excitations

Systems subjected to switching random excitations are practically significant because they include many safety-critical systems such as power plants and communication networks. In this paper, the reliability of multi-degree, nonlinear, non-integrable Hamiltonian systems subjected to switching random excitations is investigated. Such a system is formulated as a continuous–discrete hybrid based upon the Markov jump theory. Stochastic averaging is applied to suppress the rapidly varying parameters of the Markov jump process in order to generate a probability-weighted diffusion equation. The associated backward Kolmogorov equation is then set up, from which the approximate reliability function and probability density of first passage time are obtained. The utility and accuracy of this approximate procedure are demonstrated by two examples.     

ON DOUBLE PEAK PROBABILITY DENSITY FUNCTIONS OF DUFFING OSCILLATOR TO COMBINED DETERMINISTIC AND RANDOM EXCITATIONS

The principal resonance of Duffing oscillator to combined deterministic and random external excitation was investigated. The random excitation was taken to be white noise or harmonic with separable random amplitude and phase. The method of multiple scales was used to determine the equations of modulation of amplitude and phase. The one peak probability density function of each of the two stable stationary solutions was calculated by the linearization method. These two one-peak-density functions were combined using the probability of realization of the two stable stationary solutions to obtain the double peak probability density function. The theoretical analysis are verified by numerical results.     

谐和与宽带噪声联合激励下含分数导数型阻尼的Duffing振子的平稳响应

用广义谐和函数导出了含分数导数型阻尼的Duffing振子在谐和与宽带噪声联合作用下的随机平均方程,然后用有限差分法求解了平均FPK方程得到系统的随机稳态响应,研究了分数阶数对随机跳跃分岔影响。最后,通过与原方程MonteCarlo数字模拟结果的比较,验证了所得结果的正确性。     

Response of nonlinear oscillator under narrow-band random excitation

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

Suppressing homoclinic chaos for a weak periodically excited non-smooth oscillator

Nonlinear Dynamics - We follow our interest in a nonautonomous (2+1)-dimensional coupled nonlinear Schrödinger equation with partially nonlocal nonlinear effect and a linear potential, and get...