Self-oscillation of the Froude pendulum (Numerical study)

A numerical study of the self-oscillation of the Froude friction pendulum is presented. For comparison with approximate analytical or graphical solutions, the cubic approximation is used as one of the approximations of the friction characteristic; changes in the case of other approximations are shown.By results of the conducted computational experiment was built characteristics of the amplitude of self-oscillations from dimensionless ratios, complexes of similarity, which showed the convergence of the estimated and actual (obtained by numerical integration) values of the amplitudes of oscillation for small values of friction and slope characteristics; if you increase the moment amplitude is also increased.It is noted that the results of computer modeling will significantly depend on the design, manufacturing technology and operating conditions of the device in question.

Robust Hyperbolic Chaos in Froude Pendulum with Delayed Feedback and Periodic Braking

We indicate a possibility of implementing hyperbolic chaos using a Froude pendulum that is able to produce self-oscillations due to the suspension on a shaft rotating at constant angular velocity, in the presence of time-delay feedback and of periodic braking by the application of additional frictional force. We formulate a mathematical model and carry out its numerical research. In the parameter space we reveal areas of chaotic and regular dynamics using the analysis of Lyapunov exponents and some other diagnostic tools. It is shown that there are regions in the parameter space where the Poincaré stroboscopic map has an attractor, which is a kind of Smale–Williams solenoid embedded in the infinite-dimensional state space. We confirm the hyperbolicity of the attractor by numerical calculations including the analysis of angles of intersections of stable and unstable invariant subspaces of vectors of small perturbations for trajectories on the attractor and verify the absence of tangencies between these subspaces.

Identification of Nonlinear Friction Properties in the Conditions of Self-Excited Vibrations

The paper presents method of the friction coefficient characteristics determination for kinematic pairs in the self-excited vibration conditions occurring in the Froude pendulum. Friction coefficients were calculated by measuring the vibration amplitude of the pendulum. Measurement of this amplitude for kinetic friction coefficient is carried out in the conditions of sliding friction and a static one — when the conditions of stickslip phenomena exist. The proposed method was verified using the LuGre friction model.

SUPPRESSING OR INDUCING CHAOS BY WEAK RESONANT EXCITATIONS IN AN EXTERNALLY-FORCED FROUDE PENDULUM

Based on analytic and numerical investigations of chaotic vibrations and quasiperiodic rotations of the Froude pendulum, we present a sufficient condition for controlling chaos by means of a weak resonant excitation as the initial phase difference Ψ varies. It is shown via the Melnikov function method that the initial phase difference Ψ plays a vital role in suppressing or inducing chaotic motions or quasiperiodic rotations.

VIBRATION OF EXTERNALLY-FORCED FROUDE PENDULUM

Motion of self-excited Froude pendulum under external forcing were analyzed. Differential equation of motion includes the nonlinear damping term of Rayleigh's type. Using multiple time scale method and Lyapunov theory, vibrations, synchronization and stability of the system were examined. Chaotic motion was analyzed here by means of Lyapunov exponent and Melnikov approach.