粘弹壁板颤振的非线性动力特性

研究粘弹材料壁板在超声速气流作用下颤振时的分岔及混沌等复杂动力学特性。采用von Karman大变形理论及Kelvin粘弹阻尼模型建立壁板的动力学方程,通过线性活塞理论建立气动力模型。利用迦辽金法将壁板颤振模型转化为常微分方程组,并使用Gear的BDF方法进行数值求解。通过数值模拟研究了该系统在粘弹阻尼作用下的动力学行为以及粘弹阻尼的影响。计算结果表明,粘弹壁板颤振系统表现出丰富的动力学行为,其二次分岔特性很复杂。随着粘弹性阻尼的增大,系统的稳定解区域在减小,而静态屈曲解几乎不受影响,同时发现混沌运动区域也随着粘弹阻尼的增大而减小。

Accumulation of Mutant Genes and Its Pathway with Reproductive Isolation

According to the fitness of heterozygote was lower than homozygote among panmictic population,the process of generational accumulate of mutant gene r was considered.Branch point of r＇s frequency by generational evolution which revealed the hereditary incompatibility between R and r,was worked out,and it was found that genetic drift can make r have higher frequency to surpass the branch point to form reproductive isolation.It was not enough to have the three conditions of mutation,genetic drift and natural selection to be the drive of biological evolution;hybrid weakness,the repelling interaction between the genetic background of original population and the new mutation,were also needed.

STUDY ON BIFURCATION BEHAVIOR IN CONTINUOUS FERMENTATION OF ETHANOL

As a typical biochemical reaction, the process of continuous fermentation of ethanol is studied in this paper. An improved model is set forward and in agreement with experiments. Nonlinear oscillations of the process are analyzed with analytical and numerical methods. The Hopf bifurcation region is fixed and further analyses are given.

DOUBLE BIFURCATION OF NONLINEAR DUFFING'S OSCILLATOR

The transition boundaries of period doubling on the physical parameter plane of a Duffing system are obtained by the general Newton′s method, and the motion on different areas divided by transition boundaries is studied in this paper. When the physical parameters transpass the boundaries, the solutions of period T =2π/ω will lose their stability, and the solutions of period T =2π/ω take place. Continuous period doubling bifurcations lead to chaos.

Dynamic analysis of a flexible rotor supported by ball bearings with damping rings based on FEM and lumped mass theory

A dynamic model of a flexible rotor supported by ball bearings with rubber damping rings was proposed by combining the finite element and the mass-centralized method.In the proposed model,the rotor was built with the Timoshenko beam element,while the supports and bearing outer rings were modelled by the mass-centralized method.Meanwhile,the influences of the rotor’s gravity,unbalanced force and nonlinear bearing force were considered.The governing equations were solved by precise integration and the Runge-Kutta hybrid numerical algorithm.To verify the correctness of the modelling method,theoretical and experimental analysis is carried out by a rotor-bearing test platform,where the error rate between the theoretical and experimental studies is less than 10%.Besides that,the influence of the rubber damping ring on the dynamic properties of the rotor-bearing coupling system is also analyzed.The conclusions obtained are in agreement with the real-world deployment.On this basis,the bifurcation and chaos behaviors of the coupling system were carried out with rotational speed and rubber damping ring’s stiffness.The results reveal that as rotational speed increases,the system enters into chaos by routes of crisis,quasi-periodic and intermittent bifurcation.However,the paths of crisis,quasi-periodic bifurcation,and Hopf bifurcation to chaos were detected under the parameter of rubber damping ring’s stiffness.Additionally,the bearing gap affects the rotor system’s dynamic characteristics.Moreover,the excessive bearing gap will make the system’s periodic motion change into chaos,and the rubber damping ring’s stiffness has a substantial impact on the system motion.

Hopf Bifurcation Control of a Hyperchaotic Circuit System

This paper is concerned with the Hopf bifurcation control of a new hyperchaotic circuit system. The stability of the hyperchaotie circuit system depends on a selected control parameter is studied, and the critical value of the system parameter at which Hopf bifurcation occurs is investigated. Theoretical analysis give the stability of the Hopf bifurcation. In particular, washout filter aided feedback controllers are designed for delaying the bifurcation point and ensuring the stability of the bifurcated limit cycles. An important feature of the control laws is that they do not result in any change in the set of equilibria. Computer simulation results are presented to confirm the analytical predictions.

The Nonlinear Bifurcation and Chaos of Coupled Heave and Pitch Motions of a Truss Spar Platform

This paper presents the results from a numerical study on the nonlinear dynamic behaviors including bifurcation and chaos of a truss spar platform. In view of the mutual influences between the heave and the pitch modes, the coupled heave and pitch motion equations of the spar platform hull were established in the regular waves. In order to analyze the nonlinear motions of the platform, three-dimensional maximum Lyapunov exponent graphs and the bifurcation graphs were constructed, the Poincare maps and the power spectrums of the platform response were calculated. It was found that the platform motions are sensitive to wave fre- quency. With changing wave frequency, the platform undergoes complicated nonlinear motions, including 1/2 sub-harmonic motion, quasi-periodic motion and chaotic motion. When the wave frequency approaches the natural frequency of the heave mode of the platform, the platform moves with quasi-periodic motion and chaotic motional temately. For a certain range of wave frequencies, the platform moves with totally chaotic motion. The range of wave frequencies which leads to chaotic motion of the platform increases with increasing wave height. The three-dimensional maximum Lyapunov exponent graphs and the bifurcation graphs reveal the nonlinear motions of the spar platform under different wave conditions.

Analytical Hopf Bifurcation and Stability Analysis of T System

Complex dynamics are studied in the T system, a three-dimensional autonomous nonlinear system. In particular, we perform an extended Hopf bifurcation analysis of the system. The periodic orbit immediately following the Hopf bifurcation is constructed analytically for the T system using the method of multiple scales, and the stability of such orbits is analyzed. Such analytical results complement the numerical results present in the literature. The analytical results in the post-bifurcation regime are verified and extended via numerical simulations, as well as by the use of standard power spectra, autocorrelation functions, and fractal dimensions diagnostics. We find that the T system exhibits interesting behaviors in many parameter regimes.

Bifurcation and chaos study on transverse-torsional coupled 2K-H planetary gear train with multiple clearances

A new non-linear transverse-torsional coupled model was proposed for 2K-H planetary gear train, and gear's geometric eccentricity error, comprehensive transmission error, time-varying meshing stiffness, sun-planet and planet-ring gear pair's backlashes and sun gear's bearing clearance were taken into consideration. The solution of differential governing equation of motion was solved by applying variable step-size Runge-Kutta numerical integration method. The system motion state was investigated systematically and qualitatively, and exhibited diverse characteristics of bifurcation and chaos as well as non-linear behavior under different bifurcation parameters including meshing frequency, sun-planet backlash, planet-ring backlash and sun gear's bearing clearance. Analysis results show that the increasing damping could suppress the region of chaotic motion and improve the system's stability significantly. The route of crisis to chaotic motion was observed under the bifurcation parameter of meshing frequency. However, the routes of period doubling and crisis to chaos were identified under the bifurcation parameter of sun-planet backlash; besides, several different types of routes to chaos were observed and coexisted under the bifurcation parameter of planet-ring backlash including period doubling, Hopf bifurcation, 3T-periodic channel and crisis. Additionally, planet-ring backlash generated a strong coupling effect to system's non-linear behavior while the sun gear's bearing clearance produced weak coupling effect. Finally, quasi-periodic motion could be found under all above–mentioned bifurcation parameters and closely associated with the 3T-periodic motion.

Instability and Death of Spiral Wave in a Two-Dimensional Array of Hindmarsh-Rose Neurons

Spiral wave could be observed in the excitable media, the neurons are often excitable within appropriateparameters. The appearance and formation of spiral wave in the cardiac tissue is linked to monomorphic ventriculartachycardia that can denervate into polymorphic tachycardia and ventricular fibrillation. The neuronal system oftenconsists of a large number of neurons with complex connections. In this paper, we theoretically study the transitionfrom spiral wave to spiral turbulence and homogeneous state (death of spiral wave) in two-dimensional array of theHindmarsh-Rose neuron with completely nearest-neighbor connections. In our numerical studies, a stable rotating spiralwave is developed and selected as the initial state, then the bifurcation parameters are changed to different values toobserve the transition from spiral wave to homogeneous state, breakup of spiral wave and weak change of spiral wave,respectively. A statistical factor of synchronization is defined with the mean field theory to analyze the transition fromspiral wave to other spatial states, and the snapshots of the membrane potentials of all neurons and time series of meanmembrane potentials of all neurons are also plotted to discuss the change of spiral wave. It is found that the sharpchanging points in the curve for factor of synchronization vs. bifurcation parameter indicate sudden transition fromspiral wave to other states. And the results are independent of the number of neurons we used.

Stability of motion state and bifurcation properties of planetary gear train

A nonlinear lateral-torsional coupled vibration model of a planetary gear system was established by taking transmission errors,time varying meshing stiffness and multiple gear backlashes into account.The bifurcation diagram of the system's motion state with rotational speed of sun gear was conducted through four steps.As a bifurcation parameter,the effect of rotational speed on the bifurcation properties of the system was assessed.The study results reveal that periodic motion is the main motion state of planetary gear train in low speed region when ns<2 350 r/min,but chaos motion state is dominant in high speed region when ns>2 350 r/min,The way of periodic motion to chaos is doubling bifurcation.There are two kinds of unstable modes and nine unstable regions in the speed region when 1 000 r/min<ns<3 000 r/min.

Super-harmonic resonance of gear transmission system under stick-slip vibration in high-speed train

This work deals with super-harmonic responses and the stabilities of a gear transmission system of a high-speed train under the stick-slip oscillation of the wheel-set.The dynamic model of the system is developed with consideration on the factors including the time-varying system stiffness,the transmission error,the tooth backlash and the self-excited excitation of the wheel-set.The frequency-response equation of the system at super-harmonic resonance is obtained by the multiple scales method,and the stabilities of the system are analyzed using the perturbation theory.Complex nonlinear behaviors of the system including multi-valued solutions,jump phenomenon,hardening stiffness are found.The effects of the equivalent damping and the loads of the system under the stick-slip oscillation are analyzed.It shows that the change of the load can obviously influence the resonance frequency of the system and have little effect on the steady-state response amplitude of the system.The damping of the system has a negative effect,opposite to the load.The synthetic damping of the system composed of meshing damping and equivalent damping may be less than zero when the wheel-set has a large slippage,and the system loses its stability owing to the Hopf bifurcation.Analytical results are validated by numerical simulations.

Competition of Spatial and Temporal Instabilities under Time Delay near Codimension-Two Turing-Hopf Bifurcations

Competition of spatial and temporal instabilities under time delay near the codimension-two Turing-Hopfbifurcations is studied in a reaction-diffusion equation.The time delay changes remarkably the oscillation frequency,theintrinsic wave vector,and the intensities of both Turing and Hopf modes.The application of appropriate time delaycan control the competition between the Turing and Hopf modes.Analysis shows that individual or both feedbacks canrealize the control of the transformation between the Turing and Hopf patterns.Two-dimensional numerical simulationsvalidate the analytical results.

Control of Codimension-2 Bautin Bifurcation in Chaotic Lü System

In this paper, we design a feedback controller, and analytically determine a control criterion so as to control the codimension-2 Bautin bifurcation in the chaotic Lfi system. According to the control criterion, we determine a potential Bautin bifurcation region （denoted by P） of the controlled system. This region contains the Bautin bifurcation region （denoted by Q） of the uncontrolled system as its proper subregion. The controlled system can exhibit Bautin bifurcation in P or its proper subregion provided the control gains are properly chosen. Particularly, we can control the appearance of Bautin bifurcation at any appointed point in the region P. Due to the relationship between Bantin bifurcation and Hopf bifurcation, the control scheme thereby is also viable for controlling the creation and stability of the Hopf bifurcation. In the controller, there are two terms： a linear term and a nonlinear cubic term. We show that the former determines the location of the Hopf bifurcation while the latter regulates its criticality. We also carry out numerical studies, and the simulation results confirm our analyticai predictions.

Study on the dependence of the two-dimensional Ikeda model on the parameter

Based on the property of solutions of the nonlinear differential equation,this paper focuses on the behavior of solutions to the two-dimensional Ikeda model,especially the dependence of the solutions on the parameter.The dependency relationship of the two-dimensional Ikeda model on the parameter is revealed by a large sample of proper numerical simulations.With the parameter varying from 0 to 1,the numerical solutions change from a point attractor to periodic solutions,then to chaos,and end up with a limit cycle.Furthermore,the route from bifurcation to chaos is shown to be continuous period-doubling bifurcations.The nonlinear structures presented by the solution of the two-dimensional Ikeda model indicate that,by setting different model parameters,one can test a new method that will be adopted to study atmospheric or oceanic predictability and/or stability.The corresponding test results provide some useful information on the ability of the new approach overcoming the impacts of strong nonlinearity.

A novel line-symmetric Goldberg 6R linkage with bifurcation property

This paper proposes a novel reconfigurable Goldberg 6R linkage,conformed to the construction of variant serial Goldberg 6R linkage,while simultaneously satisfying the line-symmetric Bricard qualifications.The isomeric mechanism of this novel reconfigurable mechanism is obtained in combination with the isomerization method.The geometrically constrained conditions result in variable motion branches of the mechanism.Based on the singular value decomposition of the Jacobian matrix,the motion branches and branch bifurcation characteristics are analyzed,and the schematics of bifurcations in joint space is derived.This novel 6R linkage features one Goldberg 6R motion branch,two line-symmetric Bricard 6R motion branches,and one Bennett motion branch.With regards to the line-symmetric Bricard 6R motion branches,a similar function for the disassembly and recombination process can be achieved by reconstructing an intermediate configuration through bifurcation.Then,the isomerized generalized variant Goldberg 6R linkage is explicated in a similar way.Acting as a bridge,reconfigurability connects two families of overconstrained mechanisms.

Simplest Normal Forms of Generalized Neimark-Sacker Bifurcation

The normal forms of generalized Neimark-Sacker bifurcation are extensively studied using normal form theory of dynamic system. It is well known that if the normal forms of the generalized Neimark-Sacker bifurcation are expressed in polar coordinates, then all odd order terms must, in general, remain in the normal forms. In this paper, five theorems are presented to show that the conventional Neimark-Sacker bifurcation can be further simplified. The simplest normal forms of generalized Neimark-Sacker bifurcation are calculated. Based on the conventional normal form, using appropriate nonlinear transformations, it is found that the generalized Neimark-Sacker bifurcation has at most two nonlinear terms remaining in the amplitude equations of the simplest normal forms up to any order. There are two kinds of simplest normal forms. Their algebraic expression formulas of the simplest normal forms in terms of the coefficients of the generalized Neimark-Sacker bifurcation systems are given.

Non-linearity in the dynamic world of human movement

Non-linear dynamics,fractals,periodic oscillations,bifurcations,chaos,and other terminologies have been used to describe human biological systems in the literature for a few decades.The eight manuscripts included in this special issue discussed the historical background,

Bifurcation control and eigenstructure assignment in continuous solution polymerization of vinyl acetate

The major difficulty in achieving good performance of industrial polymerization reactors lies in the lack of understanding of their nonlinear dynamics and the lack of well-developed techniques for the control of nonlinear processes, which are usually accompanied with bifurcation phenomenon. This work aims at investigating the nonlinear behavior of the parameterized nonlinear system of vinyl acetate polymerization and further modifying the bifurcation characteristics of this process via a washout filter-aid controller, with all the original steady state equilibria preserved. Advantages and possible extensions of the proposed methodology are discussed to provide scientific guide for further controller design and operation improvement.

Theoretical and experimental study on non-linear vibration characteristic of gear transmission system

In order to investigate the vibration of gear transmission system with clearance, a vibratory test-bed of the gear transmission system was designed. The non-linear dynamic model of the system was presented, with consideration of the effects of nonlinear dynamic gear mesh excitation, flexible rotors and bearings. Integration method was used to investigate the non-linear dynamic response of the system. The results imply that when the mesh frequency is near the natural frequency of gear pair, it is the first primary resonance, the bifurcation appears, and the vibration becomes to be chaotic motion rapidly. When the speed is close to the natural frequency of the first-order bending vibration, it is the second primary resonance, the periodic motion changes to chaos by period doubling bifurcation. The vibratory measurement of test-bed of the gear transmission system was performed. Accelerometers were employed to measure the high frequency vibration. Experimental results show that the vibration acceleration of the gear transmission system includes mesh frequency and sideband. The numerical calculation results of low speed can be validated by experimental results basically. It means that the presented non-linear dynamic model of the gear transmission system is right.