Relative Rotation Nonlinear Dynamic System Research Articles

Chaos Analysis and Control of Relative Rotation System with Mathieu-Duffing Oscillator

Chaos analysis and control of relative rotation nonlinear dynamic system with Mathieu-Duffing oscillator are investigated. By using Lagrange equation, the dynamics equation of relative rotation system has been established. Melnikov’s method is applied to predict the chaotic behavior of this system. Moreover, the chaotic dynamical behavior can be controlled by adding the Gaussian white noise to the proposed system for the sake of changing chaos state into stable state. Through numerical calculation, the Poincaré map analysis and phase portraits are carried out to confirm main results.

Hopf bifurcation control for a coupled nonlinear relative rotation system with time-delay feedbacks

This paper investigates the Hopf bifurcations resulting from time delay in a coupled relative-rotation system with time-delay feedbacks. Firstly, considering external excitation, the dynamical equation of relative rotation nonlinear dynamical system with primary resonance and 1:1 internal resonance under time-delay feedbacks is deduced. Secondly, the averaging equation is obtained by the multiple scales method. The periodic solution in a closed form is presented by a perturbation approach. At last, numerical simulations confirm that time-delay theoretical analyses have influence on the Hopf bifurcation point and the stability of periodic solution.

The periodic solution problem of a relative rotation nonlinear system with nonlinear elastic force and generalized damping force

The periodic solution problem of a relative rotation nonlinear system is considered. Firstly, the relative rotation nonlinear dynamic system is established, which contains nonlinear elastic force, commonly damped force and forcing periodic force. Secondly, the result about the nonexistence of periodic solution of the corresponding autonomous system is obtained, and some results about the existence of periodic solutions of the system are obtained by using the continuation theorem of coincidence degree theory. The significance is that we generalize the existing results of the literature. Finally an example is given to illustrate that our results are right.

Bifurcation and chaos of some strongly nonlinear relative rotation system with time-varying clearance

The dynamic equation for the relative rotation nonlinear dynamic system with time-varying clearance is investigated. Firstly, transformation parameter is deduced by using the method of MLP; the bifurcation response equations of 1/2 harmonic resonance then are generated by the method of multiple scales, while singularity analysis is employed to obtain the transition set of steady motion; further more the bifurcation characteristic and the bifurcation of the system under the situation of non-autonomy are analyzed. Finally, numerical simulation exhibits many different motions, such as periodic motion, period-doubling motion, and chaos. It is shown that the change of clearance and damp parameters may influence the motion state of the system.

The periodic solution problem of a relative rotation nonlinear dynamic system with time-varying stiffness

Firstly, the relative rotation nonlinear dynamic system is established, which contains time-varying stiffness, commonly damping force and compulsive periodic force. Secondly, some results of the existence and uniqueness of periodic solutions of the system are obtained by using the continuation theorem of coincidence degree theory. The significance is that we generalize the results published in the literature. Furthermore, an example is given to illustrate that our results are new.

Bifurcation and chaos in some relative rotation systems with Mathieu-Duffing oscillator

The dynamic equation of relative rotation nonlinear dynamic system with Mathieu-Duffing oscillator is investigated. Firstly, the bifurcation response align of the relative rotation system under primary resonance-basic parameters condition is deduced using the method of multiple scales, and a singularity analysis is employed to obtain the transition set of steady motion. Secondly, a global bifurcation of the system, some probable routes leading to chaos and multiple times leading to chaos with parametric and external excitation amplitude changes have been discussed by using Melnikov method, and the necessary condition for chaotic motion of the system is presented. Finally, a numerical method is employed to further prove the effectiveness of the theoretical research.

Lyapunov-Schmidt reduction and singularity analysis of a high-dimensional relative-rotation nonlinear dynamical system

The dimensionality reduction and bifurcation of some high-dimensional relative-rotation nonlinear dynamical system are studied. Considering the nonlinear influence factor of a relative-rotation nonlinear dynamic system, the high-dimensional relative-rotation torsional vibration global dynamical equation is established based on Lagrange equation. The equivalent low-dimensional bifurcation equation, which can reveal the low-dimensional equivalent bifurcation equation between the nonlinear dynamics and parameters, can be obtained by reducing the dimensionality system using the method of Lyapunov-Schmidt reduction. On this basis, the bifurcation characteristic is analyzed by taking universal unfolding on the bifurcation equation through using the singularity theory. The simulation is carried out with actual parameters. The parameter region of torsional vibration and the effect of the parameters on the vibration are discussed.

Stability and time-delayed feedback control of a relative-rotation nonlinear dynamical system under quasic-periodic parametric excitation

The dynamical equation of a relative-rotation nonlinear dynamic system, which contains quasi-periodic parametric excitation and time delays, is established. Bifurcation response equation of 1/2 subharmonic primary parametric resonance is obtained by the method of multiple scales, and the stability of the system is analyzed. By solving the steady state solutions of the uncontrolled system, the effect of quasi-periodic parametric excitation on system response is studied through discussing the dynamics of the system. Time-delay feedback control method is used to control the bifurcation and limit cycle(region). Numerical results show that the bifurcation and the stability of the limit cycle(region) are controlled effectively by changing the time-delay parameters.

Chaos and chaotic control in a relative rotation nonlinear dynamical system under parametric excitation

This paper studies the chaotic behaviours of a relative rotation nonlinear dynamical system under parametric excitation and its control. The dynamical equation of relative rotation nonlinear dynamical system under parametric excitation is deduced by using the dissipation Lagrange equation. The criterion of existence of chaos under parametric excitation is given by using the Melnikov theory. The chaotic behaviours are detected by numerical simulations including bifurcation diagrams, Poincaré map and maximal Lyapunov exponent. Furthermore, it implements chaotic control using non-feedback method. It obtains the parameter condition of chaotic control by the Melnikov theory. Numerical simulation results show the consistence with the theoretical analysis. The chaotic motions can be controlled to period-motions by adding an excitation term.

Precise periodic solutions and uniqueness of periodic solutions of some relative rotation nonlinear dynamic system

The precise periodic solution and uniqueness of periodic solutions of some relative rotation nonlinear dynamical system possessing linear rigidity and nonlinear damping force and forcing periodic force is investigated. Firstly, the stability and uniqueness of limit cycles of a kind of autonomous nonlinear dynamical system are discussed. Secondly, the necessary condition of uniqueness of periodic solutions of the system is presented by using qualitative analysis method. The precise periodic solution of the system is obtained under certain conditions.

Stability and approximate solution of a relative-rotation nonlinear dynamical system with coupled terms

The dynamic behaviors of a relative-rotation nonlinear dynamic system with cubic coupled terms are studied. First, the dynamic equation of coupled system with nonlinear elastic force and generalized friction and harmonic excitation is deduced. The approximate solution of coupled unautonomous equation under harmonic excitation is obtained by the method of multiple scales. The effect of coupled terms on system resonance is analyzed in respect of principal resonance and internal resonance. The singularity stability of bifurcation function of principal resonance is studied by singularity theory, and the transfer concourse and topological structure of bifurcation function are obtained.

Chaotic motion of some relative rotation nonlinear dynamic system

The chaotic motion of a relative rotation nonlinear dynamic system possessing both homoclinic and heteroclinic orbits is investigated. Firstly, the dynamics equation of relative rotation nonlinear dynamics system with nonlinear stiffness and nonlinear damping and forcing excitation is deduced. Secondly, a global bifurcation of the system and a probable route leading to chaos have been discussed by using Melnikov method, and the necessary condition of chaotic motion of system is presented. The chaotic motion of system is complemented by top Lyapunov exponents maps, bifurcation maps, Poincare maps and phase plane plots.

Stability and approximate solution of a relative-rotation nonlinear dynamical system under harmonic excitation

The dynamical equation of a relative-rotation nonlinear dynamical system with nonlinear elastic force and common friction and harmonic excitation is deduced. The singularity stability of the autonomous system is studied by constructing the Lyapunov function. The approximate solution of unautonomous equation with different resonance response under harmonic excitation is obtained by the method of multiple scales, and the stability of main resonance stable state of motion is studied.

Stability of nonlinear dynamical system of relative rotation and approximate solution under forced excitation

The stability of nonlinear dynamical system of relative rotation is studied. Firstly, the dynamics equation of relative rotation autonomous nonlinear dynamical system with commonly damped force and forced excitation is deduced. Secondly, the stability of relative rotation nonlinear dynamical system is studied. For the nonlinear dynamical system, it is proved that the closed orbit bifurcation can occur under some conditions. Finally, The approximate solution of the equation under forced excitation is obtained by the method of multiple scales.