Solutions Of Periodic Motions Research Articles

The periodic rotary motions of a rigid body in a new domain of angular velocity

In the previous works, the limiting case for the motion of a rigid body about a fixed point in a Newtonian force field, which comes from a gravity center lies on Z-axis, is solved. The authors apply the small parameter technique which is achieved giving the body a sufficiently large angular velocity component ro about the fixed z-axis of the body. The periodic solutions of motion are obtained in neighborhood ro tends to infty. In our work, we aim to find periodic solutions to the problem of motion in the neighborhood of r0 tends to 0. So, we give a new assumption that: ro is sufficiently small. Under this assumption, we must achieve a large parameter and search for another technique for solving this problem. This technique is named; a large parameter technique instead of the small one well known previously. We see the advantage of the new technique which appears in saving high energy used to begin the motion and give the solution of the problem in another domain. The obtained solutions by the new technique depend on ro. We consider that the center of mass of this body does not necessarily coincide with the fixed point O. We reduce the six nonlinear differential equations of the body and their three first integrals to a quasilinear autonomous system of two degrees of freedom and one first integral. We solve the rational case when the frequencies of the generating system are rational except (,omega = ,1,,2,1/2,3,1/3, ldots ) under the condition gamma^{primeprime}_{0} = cos theta_{o} approx 0. We use the fourth-order Runge–Kutta method to find the periodic solutions in the closed interval of the time t and to compare the analytical method with the numerical one.

Applying the Large Parameter Technique for Solving a Slow Rotary Motion of a Disc about a Fixed Point

In this paper, the motion of a disk about a fixed point under the influence of a Newtonian force field and gravity one is considered. We modify the large parameter technique which is achieved by giving the body a sufficiently small angular velocity component r0 about the fixed z-axis of the disk. The periodic solutions of motion are obtained in the neighborhood r0 tends to 0. This case of study is excluded from the previous works because of the appearance of a singular point in the denominator of the obtained solutions. Euler-Poison equations of motion are obtained with their first integrals. These equations are reduced to a quasilinear autonomous system of two degrees of freedom and one first integral. The periodic solutions for this system are obtained under the new initial conditions. Computerizing the obtained periodic solutions through a numerical technique for validation of results is done. Two types of analytical and numerical solutions in the new domain of the angular velocity are obtained. Geometric interpretations of motion are presented to show the orientation of the body at any instant of time t.

Experimental and analytical periodic motions in a first-order nonlinear circuit system

In this paper, the analytical solution of a first-order nonlinear circuit is obtained through the generalized harmonic method. The analytical solutions of periodic motions are firstly expressed by finite Fourier series. Both the stability and bifurcations of the nonlinear system with a sinusoidal power-supply are studied by the eigenvalues of the system matrix. In order to verify the analytical solution, the researchers compared the spectrums and waveforms obtained from the analytical model, the simulation model and the circuit. The comparison of the spectrum shows the generalized harmonic balance method can describe the amplitude and phase of the nonlinear circuit more precisely. The comparison of the waveform indicates the output waveform from the simulation model have a good agreement with the results obtained from the analytical model. In the real circuit system, little errors on the output voltage amplitude and response speed maybe due to the nonlinear elements in the circuit and intrinsic voltage attenuation of the chips.

Analytical solutions of periodic motions in a first-order quadratic nonlinear system

In this paper, analytical solutions of period-1 motion of a first-order quadratic nonlinear system with both parametric excitation and external excitation are investigated through the generalized harmonic balance method. The analytical solutions of the system are obtained via solving the coefficients of all harmonic terms at the equilibrium position. The precision of analytical solutions is guaranteed via convergence study of harmonic balance terms. Stability analysis is carried out via eigenvalue analysis. The analytical solutions are different from the perturbation analysis solutions. Moreover, the trajectories of periodic motions obtained from analytical solutions can better explain the dynamics of the system. To verify the accuracy of analytical solutions, numerical simulation are performed and the simulation results are compared with the analytical solutions. The harmonic spectra are also presented to illustrate of the contribution of each harmonic term on a periodic motion.

Periodic motions and limit cycles of linear cable galloping

In this paper, analytical galloping dynamics of linear cables with the uniform airflow of the wind is discussed through a two-degree-of-freedom nonlinear oscillator. The nonlinearity in the two-degree-of-freedom oscillator is only from aero-dynamic forces caused by the uniform airflow. The analytical solutions of periodic motions for linear cable galloping are discussed from the analytical method with the finite Fourier series. The corresponding stability and bifurcation of the periodic motions of the linear cable galloping are determined through the two-degree-of-freedom nonlinear system. Frequency-amplitude analysis of periodic motions and limit cycles of linear cable galloping are presented. Numerical illustrations of trajectories and amplitude spectrums are given for galloping motions of linear cables. From such analytical solutions, galloping phenomenon in flow-induced vibration can be further understood. The galloping dynamics of linear cables is similar to the dynamics of the van der Pol oscillator.

Further analytic solutions for periodic motions in the Duffing oscillator

Harmonic balance method is employed for analytic periodic motions in the Duffing oscillator. Traditionally, one used the perturbation method to discuss the nonlinearity of periodic motions in the Duffing oscillator. For decades, one cannot achieve the appropriate analytical solutions of periodic motions. The harmonic balance method with prescribed-computational accuracy will be used, for a better understanding of nonlinear characteristics of periodic motions in the Duffing oscillator. In this method, the finite Fourier series with coefficient varying with time is adopted to approximately describe periodic motions in nonlinear dynamical system. The dynamical system of the coefficients of the Fourier series is obtained. The equilibrium of such a system gives the constant coefficients in the Fourier series. Thus, the periodic solution is obtained and the stability and bifurcation of the periodic motions can be determined. The analytic periodic motions should be further revisited, since the Duffing oscillator is extensively applied in engineering, and the comprehensive investigation of such periodic motions in the Duffing oscillator is necessary and significant. In this paper, the further analytic solutions of periodic motion in the Duffing oscillator will be investigated, and frequency amplitude characteristics will be discussed, and numerical simulation of periodic motions will be presented. Some new roads from periodic motions to chaos based on generalized harmonic balance method are found, such as independent period-2 motions to chaos and independent period-4 motions to chaos.

On frequency responses of period-1 motions to chaos in a periodically forced, time-delayed quadratic nonlinear system

In this paper, frequency responses of periodic motions to chaos in a periodically forced, damped, quadratic nonlinear oscillator are investigated through the finite Fourier series analysis of discrete solutions of periodic motions. The discrete solutions of periodic motions are obtained from mapping structures of discrete nodes, and the corresponding stability and bifurcation analysis of periodic motions are completed by the eigenvalue analysis of fixed points in discrete nonlinear dynamical systems. The frequency–amplitude characteristics for bifurcation trees of periodic motions to chaos are discussed, and the quantity levels of harmonic amplitudes for different harmonic orders are illustrated clearly. Numerical results of periodic motions are illustrated to show harmonic frequency responses effects on periodic motions.

On Periodic Motions in a Parametric Hardening Duffing Oscillator

In this paper, analytical solutions for periodic motions in a parametric hardening Duffing oscillator are presented using the finite Fourier series expression, and the corresponding stability and bifurcation analysis for such periodic motions are carried out. The frequency-amplitude characteristics of asymmetric period-1 and symmetric period-2 motions are discussed. The hardening Mathieu–Duffing oscillator is also numerically simulated to verify the approximate analytical solutions of periodic motions. Period-1 asymmetric and period-2 symmetric motions are illustrated for a better understanding of periodic motions in the hardening Mathieu–Duffing oscillator.

Nonlinear Vibration of a Continuum Rotor with Transverse Electromagnetic and Bearing Excitations

The nonlinear vibration of a rotor excited by transverse electromagnetic and oil-film forces is presented in this paper. The rotor-bearing system is modeled as a continuum beam which is loaded by a distributed electromagnetic load and is supported by two oil-film bearings. The governing equation of motion is derived and discretized as a group of ordinary differential equations using the Galerkin's method. The stability of the equilibrium of the rotor is analyzed with the Routh-Hurwitz criterion and the occurrence of the Andronov-Hopf bifurcation is pointed out. The approximate solution of periodic motion is obtained using the averaging method. The stability of steady response is analyzed and the amplitude-frequency curve of primary resonance is illustrated. The Runge-Kutta method is adopted to numerically solve transient response of the rotor-bearing system. Comparisons are made to present influences of electromagnetic load, oil-film force and both of them on the nonlinear vibration response. Bifurcation diagrams of the transverse motion versus rotation speed, electromagnetic parameter and bearing parameters are provided to show periodic motion, quasi-periodic motion and period-doubling bifurcations. Diagrams of time history, shaft orbit, the Poincaré section and fast Fourier transformation of the transverse vibration are presented for further understanding of the rotor response.