Chaotic oscillations in a piecewise linear spring-mass system

Chaotic oscillations are useful in assessing the health of a structure. Hence, simple chaotic systems which can easily be realized mechanically or electro-mechanically are highly desired. We study a new piecewise linear spring-mass system. The chaotic behaviour in this system is characterized using bifurcation diagrams and the invariant parameters of the dynamics. We also show that there exists a stochastic analogue of this system, which mimics the dynamical features of its deterministic counterpart. This allows a greater flexibility in practical designs as the chaotic oscillations are obtained either deterministically or stochastically. Also, the oscillations are low dimensional, which reduces the computational resources needed for obtaining the invariant parameters of this system.     

Turnover of hysteresis determines novel bursting in Duffing system with multiple-frequency external forcings

This paper investigates bursting dynamics of the Duffing system with multiple-frequency external forcings, in which novel bursting patterns (i.e., the so-called turnover-of-hysteresis-induced bursting patterns) can be observed. Typically, distinct oscillations are observed in the quasi-static processes of these bursting patterns. We show that the oscillations appear in the quasi-static processes because the equilibrium hysteresis curve of the fast subsystem becomes the one with twists and turns, and this forms a new route to bursting, which we call turnover of hysteresis. Besides, we investigate the effects of forcing frequencies and amplitudes on the turnover-of-hysteresis-induced bursting. Our study shows that the three frequency components observed in the bursting are decided by the forcing frequencies, while the transition of the bursting depends on the forcing amplitudes.     

Chaotic thresholds for the piecewise linear discontinuous system with multiple well potentials

In this paper we investigate the bifurcations and the chaos of a piecewise linear discontinuous (PWLD) system based upon a rig-coupled SD oscillator, which can be smooth or discontinuous (SD) depending on the value of a system parameter, proposed in [18], showing the equilibrium bifurcations and the transitions between single, double and triple well dynamics for smooth regions. All solutions of the perturbed PWLD system, including equilibria, periodic orbits and homoclinic-like and heteroclinic-like orbits, are obtained and also the chaotic solutions are given analytically for this system. This allows us to employ the Melnikov method to detect the chaotic criterion analytically from the breaking of the homoclinic-like and heteroclinic-like orbits in the presence of viscous damping and an external harmonic driving force. The results presented here in this paper show the complicated dynamics for PWLD system of the subharmonic solutions, chaotic solutions and the coexistence of multiple solutions for the single well system, double well system and the triple well dynamics.     

Non-linear dynamics of a mechanical system with a frictional unilateral constraint

We analyze the dynamics of a two-dimensional system constituted by two masses subjected to elastic, gravitational and viscous forces and constrained by a moving frictional mono-lateral surface. The model exhibits a time-varying dynamics capable of reproducing the hopping phenomenon, an unwanted phenomenon observed in many applications such as the motion of a robotic arm on a surface or that of a wiper on a windscreen. The system dynamics, besides being affected by geometrical non-linearities, has a non-smooth nature due to the impact and friction laws involved in the model. The complexity of the resulting equations and of the transition conditions require the problem to be solved numerically. Various periodic motions are found and the effect of varying the system parameters, in particular the friction coefficient, is investigated. Finally, simulations are used to gain some insight the behavior of the windscreen wiper.     

基于控制响应的时变系统模态参数辨识的改进子空间方法

提出了一种基于系统控制信号激发的响应数据来辨识时变系统模态参数的改进子空间方法。该方法以系统控制响应信号建立系统的状态空间输出方程并构造了一个广义Hankel矩阵,通过对该矩阵做奇异值分解(SVD),用广义能观阵的估计代替输出矩阵,然后利用奇异值矩阵的正交性,有效地降低了噪声敏感性和计算量,从而容易地辨识出等效状态下的系统矩阵,最后采用转换矩阵辨识出时变系统的模态参数。通过理论分析、仿真和实验,讨论了不同信噪比对辨识结果的影响,验证了该方法的有效性。     

Principal parametric resonances of non-linear mechanical system with two-frequency and self-excitations

The principal parametric resonance of a single-degree-of-freedom system with non-linear two-frequency parametric and self-excitations is investigated. In particular, the case in which the parametric excitation terms with close frequencies is examined. The method of multiple scales is used to determine the equations that describe to first-order the modulation of the amplitude and phase. Qualitative analysis and asymptotic expansion techniques are employed to predict the existence of steady state responses. Stability is investigated. The effect of damping, magnitudes of non-linear excitation and self-excitation are analyzed.     

用具有负定矩阵的梯度系统构造稳定的变质量力学系统

随着科学技术的发展,对喷气飞机、火箭等变质量系统动力学的研究显得越来越重要, 并且总是希望变质量系统的解是稳定的或渐近稳定的. 而通用的研究稳定性的Lyapunov直接法有很大难度, 因为直接从微分方程出发构造Lyapunov函数往往很难实现. 本文给出一种研究稳定性的间接方法, 即梯度系统方法. 该方法不但能揭示动力学系统的内在结构, 而且有助于探索系统的稳定性、渐进性和分岔等动力学行为. 梯度系统的函数V通常取为Lyapunov函数, 因此梯度系统比较适合用Lyapunov函数来研究. 列写出变质量完整力学系统的运动方程,在系统非奇异情形下,求得所有广义加速度. 提出一类具有负定矩阵的梯度系统, 并研究该梯度系统解的稳定性. 把这类梯度系统和变质量力学系统有机结合,给出变质量力学系统的解可以是稳定的或渐近稳定的条件, 进一步利用矩阵为负定非对称的梯度系统构造出一些解为稳定或渐近稳定的变质量力学系统. 通过具体例子,研究了变质量系统的单自由度运动,在怎样的质量变化规律、微粒分离速度和加力下,其解是稳定的或渐近稳定的. 本文的构造方法也适合其它类型的动力学系统.      

Modeling particle sedimentation in viscous fluids with a coupled immersed boundary method and discrete element method

Numerical techniques have increasingly been used to model fluid–particle two-phase flows. Coupling the immersed boundary method (IBM) and discrete element method (DEM) is one promising approach for modeling particulate flows. In this study, IBM was coupled with DEM to improve the reliability and accuracy of IBM for determining the positions of particles during the sedimentation process within viscous fluids. The required ratio of the particle diameter to the grid size (D/dx) was determined by comparing the simulation results with the analytical solution and experimental data. A dynamic mesh refinement model was utilised in the IBM model to refine the computational fluid dynamics grid near the particles. In addition, an optimum coupling interval between the IBM and DEM models was determined based on the experimental results of a single particle sedimentation within silicon oil at a Reynolds number of 1.5. The experimental results and the analytical solution were then utilised to validate the IBM–DEM model at Reynolds numbers of 4.1, 11.6, and 31.9. Finally, the validated model was utilised to investigate the sedimentation process for more than one particle by modeling the drafting-kissing-tumbling process and the Boycott phenomenon. Benchmark tests showed that the IBM–DEM technique preserves the advantages of DEM for tracking a group of particles, while the IBM provides a reliable and accurate approach for modeling the particle–fluid interaction.     

Uniform convergence for the difference scheme in the conservation form of ordinary differential equation with a small parameter

In this paper, we consider a singular perturbation boundary problem for a self-adjoint ordinary differential equaiton. We construct a class of difference schemes with fitted factors, and give the sufficient conditions under which the solution of difference scheme converges uniformly to the solution of differential equation. From this we propose several specific schemes under weaker conditions, and give much higher order of uniform convergence.     

Stabilizing effect of in-plane angular misalignment in 2DOF sliding system with in-plane anisotropic stiffness

Recently, it has been theoretically shown that in a 1DOF sliding system, the in-plane angular misalignment (referred to as the yaw angle misalignment (YAM)) has a stabilizing effect to suppress the self-excited vibration induced by the velocity-weakening friction. The YAM theory has been supported qualitatively and quantitatively by some experiments and numerical simulations. However, in some other experiments with another type of apparatuses, the suppression condition was qualitatively different from the theoretical prediction. Based on the above, in this study, the YAM theory has been extended to a 2DOF sliding system with in-plane anisotropic stiffness. Numerical simulation and eigenvalue analysis revealed that the YAM around 45° had a damping effect to suppress the self-excited vibration induced by the velocity-weakening friction, with no supplementary mechanical devices (such as dampers or actuators) to suppress the vibration, which was consistent with the previous experimental results.     

Stability in critical cases of a holonomic mechanical system with two degrees of freedom subject to impacts

Stability conditions for a holonomic impulsive mechanical system with two degrees of freedom in the critical case of one pair of complex-conjugate multipliers are established. An analog of the first Lyapunov exponent is calculated __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 6, pp. 105–117, June 2008.     

Constructing a Poincaré map for a holonomic mechanical system with two degrees of freedom subject to impacts

An approach to the construction of Poincaré maps for a nonlinear system with impulsive effect is proposed. The approach is based on linear change of variables that brings the Poincaré map into the simplest form __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 5, pp. 115–122, May 2008.     

Improvement of parameter estimation for non-linear hysteretic systems with slip by a fast Bayesian bootstrap filter

Modeling and identification of non-linear hysteretic systems are widely encountered in the structural dynamics field, especially for the hysteresis with slip. A model, called SL model, which can describe the pinching of most practical hysteresis loops perfectly was proposed by Baber and Noori (J. Eng. Mech. 111 (1985) 1010). A method of estimating the parameters of SL model on the basis of input-output data based on bootstrap filter was proposed by the writers. Bootstrap filter is a filtering method based on Bayesian state estimation and Monte Carlo method, which has the great advantage of being able to handle any functional non-linearity and system and/or measurement noise of any distribution. The standard bootstrap filter, however, is not time efficient, i.e., it is very time consuming and is not suitable for real-time applications. In this paper, previous work by the writers is extended to do the parameter estimation of SL model by a fast Bayesian bootstrap filtering technique. Simulation results are presented to demonstrate the performance of the algorithm.     

Singular perturbation analysis of dynamic fields in a thermoelastic solid with a small surface-breaking crack

An asymptotic analysis is presented for a dynamic problem of a semi-infinite isotropic thermoelastic solid with a small surface breaking crack. The exterior surface of the solid is subjected to a series of short thermal pulses. The crack surface is traction free and an ideal thermal contact is assumed across the crack. The stress intensity factor is asymptotically evaluated as a function of the crack depth and time. The effect of a boundary layer associated with the diffusive term is identified. The theoretical model is supplied with numerical simulations.     

Mathematical modeling of methane flow in a borehole coal mining system

Safety in coal mining is greatly increased by the drainage of the methane content of coal seams through boreholes, simultaneously producing significant energy. The design of suitable drainage technology is based on the mathematical modeling of methane flow in coal seams. In the calculation of the methane pressure, the new mathematical model presented in this paper considers both the sorption phenomenon of methane depending upon the methane pressure and the fact that the variation in methane pressure can create a change in the stress condition of the rock and, as a consequence of this, a change in the permeability of the coal. The new mathematical model can be used for the numerical simulation of the flow processes in coal seams and methane drainage technology can be designed more accurately.     

The application of a transformational zone method to the calculation of radiation heat transfer inside a piston-cylinder system

A radiation heat transfer analysis has been formulated for the space inside an internal combustion engine cylinder using the principles of the Hottel zone and transformational zone methods. The space analyzed has a variable length resulting from the piston movement. The analysis presented here is for radiation in piston-cylinder systems of known temperatures of radiation gas, and internal surfaces of cylinder, head, and piston. Radiative emissions of the radiation gas in the shape of rings were determined on the basis of Stasiek's principle of surface transformation. The results obtained demonstrate the present approach satisfactorily, both qualitatively and quantitatively. Moreover, the division of the analyzed volume into infinitely small elements along the axial direction has significantly simplified the calculations in comparison with the conventionally employed classical Hottel zone method.     

Global dynamical behavior of a three-body system with flexible connection in the gravitational field

Summary In this paper, the global behavior of relative equilibrium states of a three-body satellite with flexible connection under the action of the gravitational torque is studied. With geometric method, the conditions of existence of nontrivial solutions to the relative equilibrium equations are determined. By using reduction method and singularity theory, the conditions of occurrence of bifurcation from trivial solutions are derived, which agree with the existence conditions of nontrivial solutions, and the bifurcation is proved to be pitchfork-bifurcation. The Liapunov stability of each equilibrium state is considered, and a stability diagram in terms of system parameters is presented. Received 10 March 1998; accepted for publication 21 July 1998     

Global bifurcations and chaos in a Van der Pol-Duffing-Mathieu system with three-well potential oscillator

Semi-analytical and semi-numerical method is used to investigate the global bifurcations and chaos in the nonlinear system of a Van der Pol-Duffing-Mathieu oscillator. Semi-analytical and semi-numerical method means that the autonomous system, called Van der Pol-Duffing system, is analytically studied to draw all global bifurcations diagrams in parameter space. These diagrams are called basic bifurcation diagrams. Then fixing parameter in every space and taking parametrically excited amplitude as a bifurcation parameter, we can observe the evolution from a basic bifurcation diagram to chaotic pattern by numerical methods. The project supported by the National Natural Science Foundation of China     

Nonlinear analysis of a rub-impact rotor-bearing system with initial permanent rotor bow

A general model of a rub-impact rotor-bearing system with initial permanent bow is set up and the corresponding governing motion equation is given. The nonlinear oil-film forces from the journal bearing are obtained under the short bearing theory. The rubbing model is assumed to consist of the radial elastic impact and the tangential Coulomb type of friction. Through numerical calculation, rotating speeds, initial permanent bow lengths and phase angles between the mass eccentricity direction and the rotor permanent bow direction are used as control parameters to investigate their effect on the rub-impact rotor-bearing system with the help of bifurcation diagrams, Lyapunov exponents, Poincaré maps, frequency spectrums and orbit maps. Complicated motions, such as periodic, quasi-periodic even chaotic vibrations, are observed. Under the influence of the initial permanent bow, different routes to chaos are found and the speed when the rub happens is changed greatly. Corresponding results can be used to diagnose the rub-impact fault in this kind of rotor systems and this study may contribute to a further understanding of the nonlinear dynamics of such a rub-impact rotor-bearing system with initial permanent bow.     

Nonlinear dynamic analysis of a quarter vehicle system with external periodic excitation

In this paper, a nonlinear dynamic model of a quarter vehicle with nonlinear spring and damping is established. The dynamic characteristics of the vehicle system with external periodic excitation are theoretically investigated by the incremental harmonic balance method and Newmark method, and the accuracy of the incremental harmonic balance method is verified by comparing with the result of Newmark method. The influences of the damping coefficient, excitation amplitude and excitation frequency on the dynamic responses are analyzed. The results show that the vibration behaviors of the vehicle system can be control by adjusting appropriately system parameters with the damping coefficient, excitation amplitude and excitation frequency. The multi-valued properties, spur-harmonic response and hardening type nonlinear behavior are revealed in the presented amplitude-frequency curves. With the changing parameters, the transformation of chaotic motion, quasi-periodic motion and periodic motion is also observed. The conclusions can provide some available evidences for the design and improvement of the vehicle system.