Simple model of bouncing ball dynamics: displacement of the table assumed as quadratic function of time

Nonlinear dynamics of a bouncing ball moving in gravitational field and colliding with a moving limiter is considered. Displacement of the limiter is a quadratic function of time. Several dynamical modes, such as fixed points, 2 - cycles and chaotic bands are studied analytically and numerically. It is shown that chaotic bands appear due to homoclinic structures created from unstable 2 - cycles in a corner-type bifurcation.Comment: 11 pages, 6 figure

Simple model of bouncing ball dynamics. Displacement of the limiter assumed as a cubic function of time

Nonlinear dynamics of a bouncing ball moving vertically in a gravitational field and colliding with a moving limiter is considered and the Poincare map, describing evolution from an impact to the next impact, is described. Displacement of the limiter is assumed as periodic, cubic function of time. Due to simplicity of this function analytical computations are possible. Several dynamical modes, such as fixed points, 2 - cycles and chaotic bands are studied analytically and numerically. It is shown that chaotic bands are created from fixed points after first period doubling in a corner-type bifurcation. Equation for the time of the next impact is solved exactly for the case of two subsequent impacts occurring in the same period of limiter's motion making analysis of chattering possible.Comment: 8 pages, 1 figure, presented at the DSTA 2011 conference, Lodz, Polan

Analytical periodic motions in a parametrically excited, nonlinear rotating blade

The stability and bifurcation analyses of periodic motions in a rotating blade subject to a torsional excitation are investigated. For high speed rotations, cubic geometric nonlinearity and gyroscopic effects of the rotating blade are considered. From the Galerkin method, the partial differential equation of the nonlinear rotating blade is simplified to the ordinary differential equations, and periodic motions and stability of the rotating blade are studied by the generalized harmonic balance method. The analytical and numerical results of periodic solutions are compared. The rich dynamics and co-existing periodic solutions of the nonlinear rotating blades are investigated

Steady state performance of a nonlinear vibration absorber on vibration reduction of a harmonically forced oscillator

In this paper, the periodic steady-state responses of a nonlinear vibration absorber are investigated using the general harmonically balanced method. The periodic steady-state responses are presented via the frequency-amplitude curves. The bifurcation and stability are investigated through the eigenvalue analysis of a developed dynamical system of coefficients of the finite Fourier series. Numerical simulations of stable symmetric and asymmetric periodic steady-state responses are completed. The harmonic amplitude spectrums show harmonic effects on steady-state periodic motions, and the corresponding accuracy of approximate analytical solutions can be prescribed specifically

Asymmetric responses of rotating, thin disks experiencing large deflections

AbstractAn analytical nonlinear solution for the asymmetric mode vibration of rotating disks is given in this paper through a recently developed, accurate plate theory instead of the von Karman model. The nonlinear solution can reduce to the linear one when nonlinear effects vanish. The symmetrical response is also recovered when the nodal diameter vanishes. The natural frequency varying with rotation speed and deflection amplitude is investigated through a 3.5-inch diameter computer memory disk. From this investigation, it is found that the softening of rotating disks may occur for larger nodal-diameter numbers. The methodology given in this paper can be applied to nonlinear responses in structures such as rotating shafts and traveling plates