Smooth Oscillator Research Articles

Subharmonic resonance and fixed-range asymptotic stability of the fractional-order SD oscillator

The smooth and discontinuous (SD) oscillator is a typical system with strong nonlinear characteristics, and it is widely used in low-frequency vibration isolation and energy harvesting. A fractional damping model denoted by the Caputo model is introduced into the SD oscillator to adjust the property of the secondary resonance and evaluate the stability of the system. The influence of the fractional damping model on the one-third subharmonic resonance and the fixed-range asymptotic stability is studied. Residue theory and the Laplace transform are used to solve the fractional damping model. The amplitude–frequency response function and the existence conditions are derived by means of the averaging method. Lyapunov theory is used to determine the stable criteria of steady-state solutions. The cell-mapping method is ameliorated and used to calculate the fixed-range asymptotic stability of the one-third subharmonic resonance. The main results are as follows: a gap in the excitation amplitude occurs in the region of the existence condition of the one-third subharmonic resonance when the smooth parameter is smaller than 1. The generation of one-third subharmonic resonance is totally avoided for all frequencies when the excitation amplitudes are within the gap. The width of the gap, as well as the amplitude of the one-third subharmonic resonance, is affected by the parameters of the fractional damping term. The fixed-range asymptotic stability of the one-third subharmonic resonance is weak when the fractional damping parameters are large, which indicates a low resistance of the one-third subharmonic resonance to the external disturbance. The tuning effects of the fractional damping model on the one-third subharmonic resonance and fixed-range asymptotic stability are beneficial for the applications of SD oscillators.

Fast-slow dynamics analysis in an externally excited smooth and discontinuous oscillator with a pair of irrational nonlinearities

The study of fast-slow oscillations in systems with irrational nonlinearity that may yield abundant dynamical mechanisms is not well developed. This paper aims to investigate the fast-slow dynamics in an excited mass-spring oscillator with a pair of irrational nonlinearities, which can undergo the dynamical transition from smooth to discontinuous characteristics depending on the values of a smoothness parameter. Three different types of fast-slow oscillations are reported in this interesting smooth and discontinuous (SD) oscillator with a pair of irrational nonlinearities. Due to the smooth and discontinuous characteristics of this SD oscillator, we consider its dynamical behaviors under the smooth and discontinuous cases, respectively. Based on the fast-slow analysis and the two-parameter bifurcation analysis, the smooth fast-slow dynamics associated with fold hysteresis and its turnover are revealed. In the discontinuous case, the system can be viewed as a piecewise-smooth dynamical system governed by three different subsystems in different regions divided by two nonsmooth boundaries. In particular, the nonsmooth boundaries can be divided into parts with different dynamical behaviors, including escaping and crossing lines. Unlike the smooth case, there is no change in the stability of the equilibrium in these three subsystems. However, transitions of system trajectory induced by crossing lines can account for the generation of fast-slow oscillations in the piecewise-smooth system. As a result, the smooth and piecewise-smooth fast-slow dynamics in the excited SD oscillator with a pair of irrational nonlinearities are revealed, which deepens the understanding of fast-slow dynamics of the dynamical systems with irrational nonlinearity.

Fractional dynamical behavior of a new nonlinear smooth and discontinuous (SD) oscillator for vibration energy harvesting with nonlinear magnetic coupling

Fractional dynamical behavior of a new nonlinear smooth and discontinuous (SD) oscillator for vibration energy harvesting with nonlinear magnetic coupling

A new vibro-impact bistable oscillator with an adjustable rigid wall

In this paper, a new vibro-impact bistable oscillator under vicious damping and external harmonic excitation is proposed by adding an adjustable rigid wall to one side of the archetypal smooth and discontinuous (SD) oscillator and an impact mapping is given to describe energy loss during collisions. In the unperturbed ideal case, there exist rich and perfect geometrical structure including grazing phenomena such as periodic orbits and homoclinic orbits tangent to the wall due to its adjustability as a constraint. Perturbed dynamics is studied to show the influence of the constraint on the original SD oscillator through the Lyapunov exponent diagrams, bifurcation diagrams, and the corresponding phase diagrams and attractors. Some new non-smooth chaotic attractors and periodic orbits are found to have grazing or transversal interaction with the unilateral rigid wall. Due to the effect of the introduced impact mapping and different locations of the rigid wall, the finger-like chaotic attractors of the SD oscillator will be suppressed and changed into different non-smooth periodic motions by varying the non-dimensional parameter which reflects the degree of the springs compressed, while the periodic motions of the SD oscillator will become new non-smooth chaotic attractors or different non-smooth periodic motions.

Multistability and Jump in the Harmonically Excited SD Oscillator

Coexisting attractors and the consequent jump in a harmonically excited smooth and discontinuous (SD) oscillator with double potential wells are studied in detail herein. The intra-well periodic solutions in the vicinity of the nontrivial equilibria and the inter-well periodic solutions are generated theoretically. Then, their stability and conditions for local bifurcation are discussed. Furthermore, the point mapping method is utilized to depict the fractal basins of attraction of the attractors intuitively. Complex hidden attractors, such as period-3 responses and chaos, are found. It follows that jumps among multiple attractors can be easily triggered by an increase in the excitation level or a small disturbance of the initial condition. The results offer an opportunity for a more comprehensive understanding and better utilization of the multistability characteristics of the SD oscillator.

Homoclinic–Heteroclinic Bifurcations and Chaos in a Coupled SD Oscillator Subjected to Gaussian Colored Noise

The so-called coupled smooth and discontinuous (SD) oscillator whose stiffness term leads to a transcendental function is a simple mass-spring system constrained to a straight line by two parameters, which are the dimensionless distances to the fixed point. This paper studies the homoclinic–heteroclinic chaos in a coupled SD oscillator subjected to Gaussian colored noise. In order to investigate the chaos thresholds analytically, the piecewise linearization approximation is used to fit the transcendental function. Stochastic nonsmooth Melnikov method with homoclinic–heteroclinic orbits is developed to study chaos thresholds of oscillators with tri-stable potential. Based on stochastic Melnikov process, the mean square criterion and the rate of phase-space flux function theory are used to study the chaotic motions of a coupled SD oscillator under weak noise and strong noise, respectively. The obtained results show that it is effective to use the piecewise linear approximation to analyze chaos in the coupled SD oscillator subjected to Gaussian colored noise. It also lays the foundation for chaos research of other nonsmooth mechanical vibration systems under random excitation.

The Nonlinear Dynamics Characteristics and Snap-Through of an SD Oscillator with Nonlinear Fractional Damping

A smooth and discontinuous (SD) oscillator is a typical multi-stable state system with strong nonlinear properties and has been widely used in many fields. The nonlinear dynamic characteristics of the system have not been thoroughly investigated because the nonlinear restoring force cannot be integrated. In this paper, the nonlinear restoring force is represented by a piecewise nonlinear function. The equivalent coefficients of fractional damping are obtained with an orthogonal function. The influence of fractional damping on the transition set, the amplitude–frequency response and the snap-through of the SD oscillator are analyzed. The conclusions are as follows: The nonlinear piecewise function accurately mimics the nonlinear restoring force and maintains a nonlinearity property. Fractional damping can significantly affect the stiffness and damping property simultaneously. The equivalent coefficients of the fractional damping are variable with regard to the fractional-order power of the excitation frequency. A hysteresis point, a bifurcation point, a frequency island, pitchfork bifurcations and transcritical bifurcations were discovered in the small-amplitude resonant region. In the non-resonant region, the increase in the fractional parameters leads to the probability of snap-through declining by increasing the symmetry of the attraction domain or reducing the number of stable states.

Stochastic P-Bifurcation Analysis of Fractional Smooth and Discontinuous Oscillator with an Extended Fast Method

In this paper, stochastic bifurcations of a fractional-order smooth and discontinuous (SD) oscillator composed of different viscoelastic materials are studied. As a widely applicable algorithm for various fractional-orders cases, an extended fast algorithm is introduced to obtain the statistics of the response, where the fractional derivative is separated into a history part and a local part with a predetermined memory length. The local part is approximated by a highly accurate algorithm while the history part is computed by an efficient convolution algorithm. Through this accurate and fast method, effects of the system parameters on the dynamic behaviors, such as the fractional order, smoothness parameter, and frequency of harmonic force, are thus successfully investigated. Abundant stochastic P-bifurcation phenomena are discussed in detail. Further, it is found that only when the damping material shows nearly elastic behaviors, the probability density functions of the system exhibit the crater shape. Experiments show that the fast algorithm is accurate for different fractional orders.

The discontinuous limit case of an archetypal oscillator with a constant excitation and van der Pol damping

We investigate global dynamics of the discontinuous limit case of an archetypal oscillator with a constant excitation that is a model of an arch bridge with viscous damping subjected to a sinusoidally varying central load. The discontinuous dynamical system can exhibit three limit cycles and rich nonlinear phenomena, including boundary equilibrium bifurcation, Hopf bifurcation, grazing bifurcation, pseudo saddle–node loop bifurcation and double crossing limit cycle bifurcation. Because of destroyed symmetry for this system, it exhibits more complex and abundant dynamics than the smooth oscillator with van der Pol damping and can appear new dynamical phenomena, such as coexistence of a crossing limit cycle and a pseudo homoclinic loop, coexistence of two crossing limit cycles and the existence of pseudo saddle–node loops. The bifurcation diagram of the discontinuous system is shown in parameter space and all global phase portraits in the Poincaré disk are sketched in the phase plane.

An Archetypal Vibration Isolator with Quasi-zero Stiffness in Multiple Directions

To avoid the failure under the longitudinal, shear, and mixed wave from earthquake, an archetypal vibration isolator based on a smooth and discontinuous (SD) oscillator is proposed. This model comprises a lumped mass mounted by a vertical spring and a horizontal elastic continuum, which separately provides positive and the negative stiffness forming the stable quasi-zero stiffness (SQZS) in all vertical and horizontal directions. The equation of motion is formulated by employing the Lagrange equation, and the SQZS condition is obtained by optimizing the geometrical parameters of the system. The analysis shows that the system admits remarkable performance in vibration isolation with low resonant frequency and a large stroke of SQZS interval. The results also demonstrate the system has improved aseismic behaviour under the complex excitation of seismic wave significantly.

Frequency and Amplitude Identification of Weak Signal Based on the Limit System of Smooth and Discontinuous Oscillator

In this paper, a new method is proposed to identify the frequency and amplitude of weak signals by using a non-smooth system. The variable scale limit system of smooth and discontinuous (SD) oscillator without considering the phase is adopted as the identification system. By using the non-smooth stochastic subharmonic-like Melnikov method, an analytical expression of chaotic threshold under Gaussian white noise is given. Based on the phase diagram and Poincaré section diagram, the influence of noise intensity on the recognition system is studied. According to the non-smooth variable scale-convex-peak frequency identification method, the frequency of the signal to be detected can be accurately identified. Using the numerical simulation, the amplitude of the signal to be measured is identified according to the amplitude bifurcation diagram of the reference signal. The optimal value range of the parameters of the identification system is determined. Through an example of early fault of wheelset bearing of high-speed train, the frequency and amplitude of the early weak fault signal can be identified and the fault location can be determined, which verifies the effectiveness of the above method. The results show that the non-smooth system can identify the frequency and amplitude of the weak signal submerged in strong noise, and it has a wider application and higher accuracy than the continuous system.

Analytical Approximations for Dry Friction-Induced Stick–Slip and Pure-Slip Vibration Amplitudes of a Self-Excited Smooth and Discontinuous Oscillator

Abstract 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 first 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 recent advances for an archetypal smooth and discontinuous oscillator

The so called archetypal smooth and discontinuous (SD) oscillator with irrational nonlinearity is a simple mass-spring system constrained to a straight line by a geometrical parameter α which is the dimensionless distance to the fixed point. The typical phenomenon of this oscillator is the dynamics transition from smooth to discontinuous depending on the smooth changing of the geometrical parameter, which has attracted a quite mount of investigations on the complex nonlinear behaviours of the SD oscillator since it was firstly proposed and appeared in Physical ReviewE74 (4)(2006)046218. The present work provides a comprehensive review of state-of-the-art researches on the SD oscillator begining with the complex nonlinear dynamics under the smooth and discontinuous cases, including the fundamental dynamical characteristics of the unperturbed system, perturbed bifurcations, chaotic motions and also the coexistence of multiple atrractors. Then, the work lists several extended oscillators with irrational type of nonlinear restoring forces based upon the SD oscillator. Finally, the work details the applications of the SD oscillator in engineering fields especially in vibration isolation and energy harvesting by means of the features of negative stiffness and the multiple stabilities. This review work shows the importance of irrational nonlinear restoring forces controlled by geometrical parameters in the engineering structures designing. The concluding remarks suggest further promising directions, such as the dynamics near local and global bifurcation with high co-dimension caused by the increasing geometrical parameters, the construction of universal unfoldings and irrational elliptic function for the situation of multiple geometrical parameters, the design of novel model with geometrical nonlinearity for engineering application and the improvement of experimental method for oscillators with irrational nonlinearity.

Wada basin boundaries and generalized basin cells in a smooth and discontinuous oscillator

Wada basin boundaries and generalized basin cells in a smooth and discontinuous oscillator

An improved cell mapping method based on dimension-extension for fractional systems.

Global analysis of fractional systems is a challenging topic due to the memory property. Without the Markov assumption, the cell mapping method cannot be directly applied to investigate the global dynamics of such systems. In this paper, an improved cell mapping method based on dimension-extension is developed to study the global dynamics of fractional systems. The evolution process is calculated by introducing additional auxiliary variables. Through this treatment, the nonlocal problem is localized in a higher dimension space. Thus, the one-step mappings are successfully described by Markov chains. Global dynamics of fractional systems can be obtained through the proposed method without memory losses. Simulations of the point mapping show great accuracy and efficiency of the method. Abundant global dynamics behaviors are found in the fractional smooth and discontinuous oscillator.

An approximate technique to test chaotic region in a rotating pendulum system with bistable characteristics

This paper is concerned with the chaotic dynamics of a rotating pendulum system with bistable characteristics subjected to a viscous damping and a harmonic forcing. As a prototype of the single-degree-of-freedom system with bistable characteristics, this pendulum system exhibits a transition from smooth to discontinuous dynamics by changing a geometrical parameter. The dynamic behaviors of the unperturbed system with irrational nonlinearity bear significant similarities to the coupling of a simple pendulum and the smooth and discontinuous (SD) oscillator with the coexistence of the standard homoclinic orbits of Duffing type and pendulum type and the coexistence of the nonstandard homoclinic orbits of SD type and pendulum type in the smooth and discontinuous case, respectively. For the perturbed smooth system, we present an approximate technique to analytically obtain the lower bound line for horseshoes chaos arising from the homoclinic orbits of Duffing-type and Pendulum-type tangling, which overcomes the natural difficulties of solving the analytical expression of the homoclinic orbits and calculating the complicated Melnikov integrals. The chaotic thresholds of the perturbed discontinuous system are calculated by applying the numerical technique due to its non-smooth feature. Numerical simulations are carried out to certify the chaotic thresholds, which show the efficiency of the proposed techniques and demonstrate the predicated chaotic motions. Finally, different types of chaotic motions are illustrated via the cylindrical phase portraits. The contribution of this study is also helpful for exploring the dynamical behaviors of the complex nonlinear dynamical system containing the standard homoclinic or heteroclinic orbit in terms of the quantitative calculation.

Stick-slip vibrations of a self-excited SD oscillator with Coulomb friction

In this paper, the stick-slip vibrations of an archetypal self-excited smooth and discontinuous (SD) oscillator are investigated. The mathematical model of the self-excited SD oscillator is established by employing Coulomb’s law to formulate the friction between the surfaces of the mass and the moving belt. Complex dynamical behaviors are demonstrated by equilibrium analysis including stability analysis and supercritical pitchfork bifurcations of the system. Closed-form solutions for both stick-slip motions and pure slip motions of the system can be derived and utilized to examine the influence of the belt speed on the steady-state of the system by using Hamilton function. The evolution of sliding regions and the collision of the trajectories with the sliding region are presented for the forced self-excited system resorting to the numerical simulations. The results obtained here offer an opportunity for us to understand the conversion mechanism between the stick and the slip motions for the friction systems with geometric nonlinearity in engineering.

Nonlinear dynamics of a classical rotating pendulum system with multiple excitations**Project supported by the National Natural Science Foundation of China (Grant Nos. 11702078 and 11771115), the Natural Science Foundation of Hebei Province, China (Grant No. A2018201227) and the High-Level Talent Introduction Project of Hebei University, China (Grant No. 801260201111).

We report an attempt to reveal the nonlinear dynamic behavior of a classical rotating pendulum system subjected to combined excitations of constant force and periodic excitation. The unperturbed system characterized by strong irrational nonlinearity bears significant similarities to the coupling of a simple pendulum and a smooth and discontinuous (SD) oscillator, especially the phase trajectory with coexistence of Duffing-type and pendulum-type homoclinic orbits. In order to learn the effect of constant force on this pendulum system, all types of phase portraits are displayed by means of the Hamiltonian function with large constant excitation especially the transitions of complex singular closed orbits. Under sufficiently small perturbations of the viscous damping and constant excitation, the Melnikov method is used to analyze the global structure of the phase space and the feature of trajectories. It is shown, both theoretically and numerically, that this system undergoes a homoclinic bifurcation and then bifurcates a unique attracting rotating limit cycle. Finally, the estimation of the chaotic threshold of the rotating pendulum system with multiple excitations is calculated and the predicted periodic and chaotic motions can be shown by applying numerical simulations.

A multi-directional multi-stable device: Modeling, experiment verification and applications

Multi-stable vibration energy harvester based on spring-mass system has received widespread attention from many researchers, because the spring-mass multi-stable system has the advantages of low natural frequency, simple structure and high output power. This work considers such a energy harvesting system, but with the additional novelty of using the link mechanism to improve the bistable smooth and discontinuous (SD) oscillator, and a multi-directional multi-stable device (MMD) is proposed. Equilibrium analysis demonstrates the multi-stable mechanism and the bifurcation conditions for the unperturbed system. While for the perturbed system, the Extended Averaging Method is carried out to demonstrate the multi-stable behaviors, transitions between different potential wells, and their large amplitude inter-well motions. The experimental results are compared to theoretical results. Both theoretical and experimental data suggest that the MMD produces a large amplitude response at ultra-low frequencies, enabling motion throughout all equilibrium positions. The designed prototype can be easily adopted for efficient energy harvesting from ultra-low frequency vibration sources.

An oscillator with two discontinuous lines and Van der Pol damping

In this paper we investigate the discontinuous limit case of a smooth oscillator with a Van der Pol damping, which is a Filippov system with two discontinuous lines. The qualitative properties of all equilibria including that at infinity are obtained for this discontinuous piecewise smooth oscillator. By applying qualitative theory for smooth systems and for nonsmooth systems, we give necessary and sufficient conditions for the existence of limit cycles and grazing cycles. Particularly, it is demonstrated that this oscillator has at most two large limit cycles, two small limit cycles, one large double limit cycle and three classes of grazing cycles in different parameter regions. We present completely the bifurcation diagram and all global phase portraits of this oscillator model.