Primary Parametric Resonance Research Articles

Flow-induced instability under bounded noise excitation in cross-flow

Flow-induced vibration of a single cylinder in a cross-flow is mainly due to vortex shedding, which is usually considered as a forced vibration problem. It is shown that flow-induced vibration of a cylinder in the lock-in region is a combination of forced resonant vibration and fluid-damping-induced instability, which leads to time-dependent-fluid-damping-induced parametric resonance and constant-negative-damping-induced instability. The time-dependent fluid damping can be modeled as a bounded noise. The dynamic stability of a two-dimensional system under bounded noise excitation with a narrow-band characteristic is studied through the determination of the moment Lyapunov exponent and the Lyapunov exponent. The case when the system is in primary parametric resonance in the absence of noise is considered and the effect of noise on the parametric resonance is investigated. For small amplitudes of the bounded noise, analytical expansions of the moment Lyapunov exponents and Lyapunov exponents are obtained, which are shown to be in excellent agreement with those obtained using Monte Carlo simulation. The theory of stochastic stability is applied to explore the stability of a cylinder in a cross-flow. The analytical and numerical results show that the time-dependent-fluid-damping-induced parametric resonance could occur, which suggests that parametric resonance also contributes to the vibration of the cylinder in the lock-in range.

Lagrange-Maxwell equation and magnetic saturation parametric resonance of generator set

Lagrange-Maxwell’s equation is extended firstly. With the theory of electromechanical analytical dynamics, the magnetic complement energy in air gap of generator is acquired. The torsional vibration differential equations with periodic coefficients of rotor shafting of generator which is in the state of magnetic saturation are established. It is shown that the magnetic saturation may cause double frequency electromagnetic moment. By means of the averaging method, the first approximate solution and corresponding solution of the primary parametric resonance is obtained. The characteristics and laws of the primary parametric resonance excited by the electromagnetism are analyzed and some of new phenomena are revealed.

HIGHER-DIMENSIONAL PERIODIC AND CHAOTIC OSCILLATIONS FOR VISCOELASTIC MOVING BELT WITH MULTIPLE INTERNAL RESONANCES

In this paper, higher-dimensional periodic and chaotic oscillations for a parametrically excited viscoelastic moving belt with multiple internal resonances are investigated for the first time. The external damping and internal damping of the material for the viscoelastic moving belt are considered simultaneously. First, the nonlinear governing equation of planar motion for the viscoelastic moving belt with the external damping is given. Then, the transverse nonlinear oscillations of the viscoelastic moving belt are considered. The method of multiple scales and the Galerkin approach are applied directly to the governing partial differential equation of motion for the viscoelastic moving belt to obtain an eight-dimensional averaged equation for the case of 1:2:3:4 internal resonances for the first-, the second-, the third- and the fourth-order modes and primary parametric resonance of the first-order mode. Finally, numerical method is used to investigate higher-dimensional periodic and chaotic motions of the viscoelastic moving belt. The results of numerical simulation demonstrate that there exist the period, period 2, period 4, multiple period and chaotic motions of the viscoelastic moving belt. The multipulse chaotic motions of the viscoelastic moving belt are observed from numerical simulations.

The capture into parametric autoresonance

In this work, we show that the capture into parametric resonance may be explained as pitchfork bifurcation in the primary parametric resonance equation. We prove that the solution close to the moment of the capture is descibed by the Painleve-2 equation. We obtain connection formulae for the asymptotic solution of the primary parametric resonance equation before and after the capture using the matching of asymptotic expansions.

Periodic and Chaotic Motions of a Rotor-Active Magnetic Bearing with Quadratic and Cubic Terms and Time-Varying Stiffness

In this paper, we use the asymptotic perturbation method to investigate nonlinear oscillations and chaotic dynamics in a rotor-active magnetic bearings (AMB) system with 8-pole legs and the time-varying stiffness. The stiffness in the AMB is considered as the time varying in a periodic form. Because of considering the weight of the rotor, the formulation on the electromagnetic force resultants includes the quadratic and cubic nonlinearities. The resulting dimensionless equations of motion for the rotor-AMB system with the time-varying stiffness in the horizontal and vertical directions are a two-degree-of-freedom nonlinear system with quadratic and cubic nonlinearities and parametric excitation. The asymptotic perturbation method is used to obtain the averaged equations in the case of primary parametric resonance and 1/2 subharmonic resonance. It is found that there exist period-3, period-4, period-6, period-7, period-8, quasiperiodic and chaotic modulated amplitude oscillations in the rotor-AMB system with the time-varying stiffness. It is seen from the numerical results that there are the phenomena of the multiple solutions and the soft-spring type and the hardening-spring type in nonlinear frequency-response curves for the rotor-AMB system. The parametric excitation, or the time-varying stiffness produced by the PD controller is considered to be a controlling force which can control the chaotic response in the rotor-AMB system to a period n motion.

Multi-pulse orbits and chaotic dynamics in motion of parametrically excited viscoelastic moving belt

Abstract In this paper, the Shilnikov type multi-pulse orbits and chaotic dynamics of parametrically excited viscoelastic moving belt are studied in detail. Using Kelvin-type viscoelastic constitutive law, the equations of motion for viscoelastic moving belt with the external damping and parametric excitation are given. The four-dimensional averaged equation under the case of primary parametric resonance is obtained by directly using the method of multiple scales and Galerkin’s approach to the partial differential governing equation of viscoelastic moving belt. From the averaged equations obtained here, the theory of normal form is used to give the explicit expressions of normal form with a double zero and a pair of pure imaginary eigenvalues. Based on normal form, the energy-phrase method is employed to analyze the global bifurcations and chaotic dynamics in parametrically excited viscoelastic moving belt. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Silnikov type multi-pulse homoclinic orbits in the averaged equation. The results obtained above mean the existence of the chaos for the Smale horseshoe sense in parametrically excited viscoelastic moving belt. The chaotic motions of viscoelastic moving belts are also found by using numerical simulation. A new phenomenon on the multi-pulse jumping orbits is observed from three-dimensional phase space.

Multi-pulse chaotic motions of a rotor-active magnetic bearing system with time-varying stiffness

Abstract In this paper, we investigate the Shilnikov type multi-pulse chaotic dynamics for a rotor-active magnetic bearings (AMB) system with 8-pole legs and the time-varying stiffness. The stiffness in the AMB is considered as the time-varying in a periodic form. The dimensionless equation of motion for the rotor-AMB system with the time-varying stiffness in the horizontal and vertical directions is a two-degree-of-freedom nonlinear system with quadratic and cubic nonlinearities and parametric excitation. The asymptotic perturbation method is used to obtain the averaged equations in the case of primary parametric resonance and 1/2 subharmonic resonance. It is found from the numerical results that there are the phenomena of the Shilnikov type multi-pulse chaotic motions for the rotor-AMB system. A new jumping phenomenon is discovered in the rotor-AMB system with the time-varying stiffness.

Parametric Resonance of a Two-Dimensional System Under Bounded Noise Excitation

The dynamic stability of a two-dimensional system under bounded noise excitation with a narrow band characteristic is studied through the determination of the pth moment Lyapunov exponent and the Lyapunov exponent. The case when the system is in primary parametric resonance in the absence of noise is considered and the effect of noise on the parametric resonance is investigated. The partial differential eigenvalue problem governing the moment Lyapunov exponent is established. For small amplitudes of the bounded noise, a method of singular perturbation is applied to determine analytical expansions of the moment Lyapunov exponents and Lyapunov exponents, which are shown to be in excellent agreement with those obtained using numerical approaches.

Asymptotic solutions for a damped non-linear quasi-periodic Mathieu equation

Quasi-periodic (QP) solutions of a weakly damped non-linear QP Mathieu equation are investigated near a double primary parametric resonance. A double multiple scales method is applied to reduce the original QP oscillator to an autonomous system performing two successive reduction. The problem for approximating QP solutions of the original system is then transformed to the study of stationary regimes of the induced autonomous system. Explicit analytical approximations to QP oscillations are obtained and comparisons to numerical integration of the original QP oscillator are provided.

EHL film thickness, additives and gear surface fatigue : Townsend, D.T. and Shimski, J. Gear Technol. (1995) 12 (3), 26–31

This paper investigates the bifurcation and resonant hysteresis of varying compliance (VC) vibrations in a rigid-rotor ball bearing system with Hertzian contact and radial internal bearing clearance. With the aid of arc-length continuation, the harmonic balance and alternating frequency time (HB-AFT) technique is employed to trace the branches of periodic responses, for which the stability is also investigated using Floquet theory. It is found that the soft resonant hysteresis resulted from the Hertzian contact resonance dominates the primary parametric resonance. In addition, this paper also demonstrates that the mechanism of the period doubling in the case of small bearing clearances is arisen from one to two parametrical internal resonance in the primary resonant area, which can lead nonlinear responses such as quasi-period and chaotic motions to the system, and the evolvements of these complex behaviors are also explored.

The modulated double diffusive system: Chaotic behaviour near the convection threshold

We show that chaos through period doubling can be induced in a parametrically modulated double diffusive system near its oscillatory (of frequency ω 0) convection threshold. This happens on modulation with the primary parametric resonance frequency (ω ≅ 2ω 0) at a critical amplitude of modulation. The onset of period doubling is preceded by a symmetry-breaking bifurcation. Convection experiments on binary fluids should reveal this effect.