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This manuscript investigated the response of a strongly non-linear vibro-impact (VI) system with Coulomb friction. The impact model is used with classical impact. The excitation is modelled by real noise. First, the VI system is converted into a simplified system without any barrier by non-smooth transformation (symmetric transformation). The stochastic averaging method is adopted to obtain the theoretical stationary probability function of the VI system. Next, the Duffing Van der Pol VI system with Coulomb friction is used to verify the validity of the proposed theoretical method compared with numerical simulations. Moreover, the influence of bandwidth, noise intensity, and friction amplitude are further analyzed in detail on the probability density function (PDF) of distribution of the VI system. The P-bifurcation is studied by a qualitative change of friction amplitude and restitution coefficient on the stationary probability distribution, which indicated that these parameters can arouse the emergence of stochastic P-bifurcation. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamical response and vibrational resonance (VR) of a piecewise electromechanically coupled tri-stable energy harvester (TEH), which is driven by dual-frequency harmonic excitations. To achieve a stable DC output, the TEH is interfaced with a standard rectifier circuit. Using the harmonic balance method combined with the separation of fast and slow variables, a steady-state response together with the analytical expressions of displacement and harvested power is derived. The multi-solution feature in the amplitude-frequency response is observed and can improve the harvesting performance of the TEH under a low-frequency environment. There is an optimal time constant ratio and electromechanical coupled coefficient to maximize the harvested DC power. Meanwhile, the VR phenomenon of the TEH is explored through the response amplitude of the low-frequency input signal, which implies that an appropriate combination can induce the occurrence of VR and improve the rectified voltage. Similarly, the nonlinear stiffness coefficients can be adjusted by changing the magnet distance to induce the appearance of VR. The theoretical solutions are well supported by numerical simulation and experimental verification. Specifically, the theoretical analysis and experimental evidence illustrate that the harvested power under the VR effect is much higher than that without VR.

As an intelligent material, a shape memory alloy has many unique mechanical properties, such as shape memory effect and pseudoelasticity, which have been used in many fields. In this paper, the stochastic bifurcation of the shape memory alloy supporter system subject to harmonic and bounded noise excitations is studied in detail by an extended composite cell coordinate system method. By analyzing the influence of the bounded noise amplitude on stochastic bifurcation, it can be found that there exist three kinds of bifurcation phenomena, including stochastic merging crisis, stochastic boundary crisis, and stochastic interior crisis, which are caused by the collision between an attractor and a saddle within the basin of attraction or the basin boundary. The transfer probability is computed for the first time on the basin boundary and the escaped points within the basin of attraction, which quantitatively explain the phase space structure of the system with stochastic bifurcation. These results demonstrate that the extended composite cell coordinate system method is a powerful tool to reveal the mechanism of stochastic bifurcation and provide a theoretical basis for the operation and design of the shape memory alloy supporter system.

This paper is concerned with the response of a fractional-order stochastic system. The short memory principle is introduced to ensure that the response of the system is a Markov process. The generalized cell mapping method is applied to display the global dynamics of the noise-free system, such as attractors, basins of attraction, basin boundary, saddle, and invariant manifolds. The stochastic generalized cell mapping method is employed to obtain the evolutionary process of probability density functions of the response. The fractional-order ϕ 6 oscillator and the fractional-order smooth and discontinuous oscillator are taken as examples to give the implementations of our strategies. Studies have shown that the evolutionary direction of the probability density function of the fractional-order stochastic system is consistent with the unstable manifold. The effectiveness of the method is confirmed using Monte Carlo results.

We study the transport of a particle subjected to a Lévy noise in a rough ratchet potential which is constructed by superimposing a fast oscillating trigonometric function on a common ratchet background. Due to the superposition of roughness, the transport process exhibits significantly different properties under the excitation of Lévy noises compared to smooth cases. The influence of the roughness on the directional motion is explored by calculating the mean velocities with respect to the Lévy stable index α and the spatial asymmetry parameter q of the ratchet. Variations in the splitting probability have been analyzed to illustrate how roughness affects the transport. In addition, we have examined the influences of roughness on the mean first passage time to know when it accelerates or slows down the first passage process. We find that the roughness can lead to a fast reduction of the absolute value of the mean velocity for small α, however the influence is small for large α. We have illustrated that the ladder-like roughness on the potential wall increases the possibility for particles to cross the gentle side of the ratchet, which results in an increase of the splitting probability to right for the right-skewed ratchet potential. Although the roughness increases the corresponding probability, it does not accelerate the mean first passage process to the right adjacent well. Our results show that the influences of roughness on the mean first passage time are sensitive to the combination of q and α. Hence, the proper q and α can speed up the passage process, otherwise it will slow down it.