Chaos in Non-ideal Mechanical System with Clearance

Abstract

In this paper a non-ideal mechanical system with clearance is considered. The mechanical model of the system is an oscillator connected with an unbalanced motor. Due to the existence of clearance the connecting force between motor and the fixed part of the system is discontinuous but linear. The mathematical model of the system is represented by two coupled second-order differential equations. The transient and steady-state motion and also the stability of the system are analyzed. The Sommerfeld effect is detected. For certain values of the system parameters the motion is chaotic. This is caused by the period doubling bifurcation. The existence of chaos is proved with maximal Lyapunov exponent. A new chaos control method based on the known energy analysis is introduced and the chaotic motion is transformed into a periodic one.

The dynamic behavior of a cantilever beam coupled to a non-ideal unbalanced motor through numerical and experimental analysis

“…The computation of the basins follows closely the procedure described in [19], adapted to take the advantage of parallel computation techniques currently available. A grid of initial conditions on the x-_ ϕ plane is integrated using the Runge-Kutta method of order (4,5), and the asymptotic behavior of the system is identified numerically. The limit-cycles are represented by the Poincaré section, which in this case is a single point.…”

“…The results showed that jump phenomena and chaos are present for certain values of the parameters in the resonant regime. Zukovic and Cvetićanin [3,4] detected the Sommerfeld effect and chaotic regimes on the dynamics of a non-ideal system comprised an oscillator connected with an unbalanced motor with clearance.…”

“…Simultaneously, the amplitude of oscillations of the excited system jumps to a much lower value. Before the jump, the non-ideal oscillating system cannot pass through the resonance frequency of the system, or requires an intensive interaction between the vibrating system and the energy source to be able to do so [1,3,4].…”

Dynamics of the cutting mechanism with flexible support and non-ideal forcing

“…(17) the following system of two coupled nonlinear differential equations is obtained (20) are non-linear. Besides, as the driving torque is a function of the crank rotation, the mechanical system examined with the equations is put into the class of non-ideal dynamical systems [6][7][8][9][10][11], with consequences that will be encountered and discussed in the following sections. Before solving the complete set of equations of motion (Eqs.…”

“…Direct integration technique gives the description of the run-up and close-down response of the mechanism. Steady-state periodic Mechanism and Machine Theory 58 (2012) [1][2][3][4][5][6][7][8][9][10][11][12] motions are obtained numerically, but also analytically for some special cases. Motion is compared with that for the case of ideal forcing when the angular velocity of the input element is constant.…”