Analytical Approximations for Dry Friction-induced Stick-Slip and Pure-Slip Vibration Amplitudes of a Self-Excited SD Oscillator

An analytical and numerical investigation into pure-slip and stick-slip oscillations induced by dry friction between a rigid mass linked by an inclined spring, modeled by the archetypal self-excited smooth and discontinuous (SD) oscillator, and the classical moving rigid belt, is presented. The friction force between surface contacts is modeled in the sense of Stribeck effect to formulate the friction model that the friction force firstly decreases and then increases with increasing relative sliding speed. Some perturbation methods are considered into this system for establishing the approximate analytical expressions of the occurring conditions, vibration amplitudes, and base frequencies of dry friction-induced stick-slip and pure-slip oscillations. For pure-slip oscillations, two different approaches are applied to analyze this self-excited SD oscillator. One of them is the homotopy perturbation method by constructing the nonlinear amplitude and frequency. Based on the multiple-scales homotopy perturbation method, a nonlinear equation for amplitude of the analytical approximate solution is constructed, which containing all parameters of problem. For stick-slip oscillations, the analytical approximations for amplitude and frequency are obtained by perturbation methods for finite time intervals of the stick phase, which is linked to the subsequent slip phase under the conditions of continuity and periodicity. The accuracy of analytical approximations is verified by the comparison between analytical approximations and numerical simulations. These analytical expressions are needed for gaining a deeper understanding of dry friction-induced pure-slip and stick-slip oscillations for the friction system with geometric nonlinearity.

The Effect of Fractional Damping and Time-delayed Feedback on the Stochastic Resonance of Asymmetric SD Oscillator

This paper proposes the stiffness nonlinearities and asymmetric SD (smooth and discontinuous) oscillators under time-delay feedback control with a fractional damping. With the effect of displacement and velocity feedback, the oscillator is adjusted to the desired vibration state and then the stochastic resonance (SR) is achieved. This article discusses the contribution of various system parameters and time-delay feedback to SR, especially which induced by fractional damping. It should be noted that this paper provides effective guidance for fault diagnosis and weak signal detection, energy harvesting, vibration isolation and vibration reduction.

A Novel Dynamic Absorber with Variable Frequency and Damping

Smoothness and discontinuous (SD) oscillator is a nonlinear oscillator with the variable frequency, whose frequency can be varied with the smoothing parameter. However, how to adjust the smoothing parameter has not been solved in the actual device. In this paper, the shape memory alloy (SMA) is introduced into the SD oscillator to form the SMA-SD oscillator to adjust the smoothing parameters. Combining the SMA-SD oscillator with MRF, a nonlinear dynamic vibration absorber (DVA) with variable frequency and damping is designed. The structure and control principle of the designed DVA is studied to achieve the two variable characteristics simultaneously by adjusting the current intensity. Numerical results on a two-degree-of-freedom coupled system show that the proposed DVA can adapt to different working conditions only by adjusting the current intensity.

Threshold of Multiple Stick-Slip Chaos for an Archetypal Self-Excited SD Oscillator Driven by Moving Belt Friction

In this paper, we investigate the multiple stick-slip chaotic motion of an archetypal self-excited smooth and discontinuous (SD) oscillator driven by moving belt friction, which is constructed with the SD oscillator and the classical moving belt. The friction force between the mass and the belt is modeled as a Coulomb friction for this system. The energy introduction or dissipation during the slip and stick modes in the system is analyzed. The analytical expressions of homoclinic orbits of the unperturbed SD oscillator are derived by using a special coordinate transformation without any pronominal truncation to retain the natural characteristics, which allows us to utilize the Melnikov’s method to obtain the chaotic thresholds of the self-excited SD oscillator in the presence of the damping and external excitation. Numerical simulations are carried out to demonstrate the multiple stick-slip dynamics of the system, which show the efficiency of the prediction for stick-slip chaos of the perturbed self-excited system. The results presented herein this paper demonstrate the complicated dynamics of stick-slip periodic solutions, multiple stick-slip chaotic solutions and also coexistence of multiple solutions for the perturbed self-excited SD oscillator.