Vibro-impact Oscillator Research Articles

Suppressing homoclinic chaos for a class of vibro-impact oscillators by non-harmonic periodic excitations

Suppressing homoclinic chaos for a class of vibro-impact oscillators by non-harmonic periodic excitations

Homoclinic Bifurcations and Chaotic Dynamics in a Bistable Vibro-Impact SD Oscillator Subject to Gaussian White Noise

This paper studies the effect of Gaussian white noise on homoclinic bifurcations and chaotic dynamics of a bistable, vibro-impact Smooth-and-Discontinuous (SD) oscillator. First, the SD oscillator is reproduced and generalized by installing a slider on a fixed rod, so the slider is connected by a pair of linear springs initially pre-compressed in the vertical direction to achieve bistable vibration characteristics, and two screw nuts are installed on the rod as two adjustable bilateral rigid constraints to generate the vibro-impact. A discontinuous dynamical equation with a map defined on switching boundaries to represent velocity loss during each collision is derived to describe the vibration pattern of the bistable, vibro-impact SD oscillator through studying the persistence of the unique, unperturbed, nonsmooth, homoclinic structure. Second, the general framework of random Melnikov process for a class of bistable, vibro-impact systems contaminated with Gaussian white noise is derived and employed through the corresponding Melnikov function to obtain the necessary parameter thresholds for homoclinic tangency and possible chaos of the bistable, vibro-impact SD oscillator. Third, the effectiveness of a semi-analytical prediction by the Melnikov function is verified using the largest Lyapunov exponent, bifurcation series, and 0–1 test. Finally, the sensitivity to the initial values of chaos is verified by the fractal attractor basins, and the influence of the Gaussian white noise on periodic and chaotic structures is studied through Poincaré mapping to show the rich dynamical geometric structures.

Stochastic dynamics of mechanical systems with impacts via the Step Matrix multiplication based Path Integration method

AbstractIn this work we propose the Step Matrix Multiplication based Path Integration method (SMM-PI) for nonlinear vibro-impact oscillator systems. This method allows the efficient and accurate deterministic computation of the time-dependent response probability density function by transforming the corresponding Chapman–Kolmogorov equation to a matrix–vector multiplication using high-order numerical time-stepping and interpolation methods. Additionally, the SMM-PI approach yields the computation of the joint probability distribution for response and impact velocity, as well as the time between impacts and other important characteristics. The method is applied to a nonlinear oscillator with a pair of impact barriers, and to a linear oscillator with a single barrier, providing relevant densities and analysing energy accumulation and absorption properties. We validate the results with the help of stochastic Monte-Carlo simulations and show the superior ability of the introduced formulation to compute accurate response statistics.

Vibration mitigation and energy harvesting of vibro–impact dielectric elastomer oscillators

Dielectric elastomers (DEs), as a type of intelligent materials, have the characteristics of light weight, high energy density, high conversion efficiency, and damping effect, and thus can achieve the potential dual–use value of concurrent vibration mitigation (VM) and energy harvesting (EH) by combining with mechanical structures. This paper investigates the vibro–impact dielectric elastomer oscillator (VI DEO) and the proposed rolling VI DEO (RVI DEO) to achieve or enhance the potential dual–use value of DE oscillators. First, the schematics of the VI DEO and the RVI DEO are elaborated in detail. Then, the two dynamics analysis models of the linear oscillator (LO)–VI DEO and LO–RVI DEO systems are developed. On this basis, the VM and EH performances of the two DE oscillators under transient and harmonic excitations are numerically studied. It is demonstrated that the two DE oscillators have the most effective VM performance when the LO–VI DEO and LO–RVI DEO systems behave most efficiently in the 1:1 resonance vibro–impact regime with targeted energy transfer (TET). Hence, to achieve this response regime, the influences of the parameters such as the ball's mass, impact distance, pre–stretched ratio and friction coefficient of the two DE oscillators on their VM and EH performances are investigated one by one. The research results show that the most effective VM can only be achieved by adjusting the ball's mass, and the largest output power is generated in the resonance region. Hence, we further perform multi–parameter optimization to find the two DE oscillators with the best performance. Based on the multi–parameter optimization results, the performances of the two optimal oscillators are evaluated under mid–energy level harmonic and variable–frequency excitations with different amplitudes, and it is proved that both optimal DE oscillators not only have good performances, but also the RVI DEO has a better robustness.

Survival probability determination of nonlinear oscillators subject to combined deterministic periodic and non-stationary stochastic loads

This paper presents a technique based on Wiener path integrals (WPI) for computing the survival probability of nonlinear oscillators under combined nonstationary stochastic and periodic excitation. Specifically, the equation of motion is equivalently decomposed into a correlated combination of a nonlinear stochastic differential equation (SDE) and a nonlinear deterministic differential equation. The nonstationary response probability density function (PDF) of the stochastic component is obtained using the Wiener path integral concept in conjunction with stochastic averaging/linearization. The approach involves utilizing stochastic averaging/linearization techniques to transform the nonlinear SDE into a linear one with varying equivalent stiffness and damping. Afterwards, using the WPI method and the concept of the most probable path, a closed-form approximate analytical expression of the joint transition probability density function (PDF) is obtained for small intervals corresponding to the equivalent linear SDE. Additionally, the survival probability of the original nonlinear oscillator subject to combined excitation with fixed barriers can be approximated equivalently by using the survival probability with time-varying barriers of the equivalent linear system with time-varying stiffness and damping subject only to the nonstationary stochastic excitation. Finally, by using the analytical expression of the transition PDF of the equivalent linear system and a discrete version of the Chapman–Kolmogorov (C–K) equation, it is possible to determine the response and survival probability of the original oscillator under combined excitation step by step. Examples involving a Duffing oscillator and a vibro-impact oscillator are analyzed to showcase the precision and efficacy of the suggested technique in contrast to relevant Monte Carlo simulations.

A novel method for solving response of stochastic vibro-impact systems with two stoppers

The research of stochastic steady-state probability density response of vibration systems including bilateral impacts and gaps is of great theoretical and practical significance. Usually, the non-smooth transformation is introduced in the response calculation, while eliminates the non-smooth characteristics of the systems. Therefore, this paper propose a new method to preserve the velocity jump under the bilateral impact without non-smooth transformation. Based on the generalized cell mapping theory, a complete process is regarded as the one-step transition probability that the stochastic trajectories start from one stopper, and return to the original stopper after impacting with another stopper. Then, the intervals of response are established based on the initial stopper to ensure the continuity of the state space. We analyzed the probability density response of a bilateral Rayleigh vibro-impact oscillator and a piezoelectric energy harvesting device with two symmetric stoppers by utilizing the proposed method, respectively. Comparing with Monte Carlo simulations, it is fully demonstrated the effectiveness of this method. At the same time, the cases using our proposed method are found to have wide applicability under different position of stoppers, restitution coefficient, noise excitation and system parameters.

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.

Energy-force interactions in vibroimpact systems

The energy-force interactions in vibroimpact systems are studied according to the developed universal mathematical model, which simulates multi-mass vibroimpact systems, the masses of which are connected to each other and to a fixed base by nonretaining one-sided and retaining two-sided elastic bonds, with the excitation of these systems by inertial debalanced vibrators driven by a limited power The results of the research indicate a higher efficiency of vibrations of vibroimpact systems in comparison with a non-vibroimpact system of similar parameters. The highest energy efficiency of oscillations of vibroimpact systems and, accordingly, their maximum efficiency takes place at the excitation parameters that precede the disruption of vibroimpact oscillations. At the same time, maximum accelerations and reactions in their elastic bonds are excited in the systems. Based on this, it is necessary to accept the maximum permissible acceleration of the elements of their constructions as criteria for choosing rational process parameters in viboimpact systems, which ensures the reliability and durability of their operation.

Random Melnikov Method and Induced Chaos in Bistable Vibro-Impact Oscillators with Bounded Noise

In this paper, the Melnikov method for an abstract stochastic nonsmooth hybrid system is derived in detail and employed to study the homoclinic bifurcations and induced chaos in a bistable vibro-impact SD oscillator with bounded noise first proposed by Li et al. 2021, in which the geometric nonlinearity is used to produce the irrational restoring force and an impact map is introduced to describe energy loss during collisions. The initial Poincaré section is selected on the right rigid constraint so that rigorous perturbation analysis can be carried out to avoid the extension of trajectories near the switching manifold. The stochastic Melnikov function with geometrical intuition is obtained to detect the threshold of parameters for homocinic tangency of the perturbed stochastic stable and unstable manifolds by calculating the energy difference of their initial points on the Poincaré section. Finally, the analytical Melnikov analysis combined with numerical simulations is carried out to validate the developed Melnikov method for analyzing the global dynamics of the bistable vibro-impact SD oscillator subject to bounded noise.

A Wiener path integral technique for determining the stochastic response of nonlinear oscillators with fractional derivative elements: A constrained variational formulation with free boundaries

A technique based on the concept of Wiener path integral (WPI) is developed for determining approximately the joint response probability density function (PDF) of nonlinear oscillators endowed with fractional derivative elements. Specifically, first, the dependence of the state of the system on its history due to the fractional derivative terms is accounted for, alternatively, by augmenting the response vector and by considering additional auxiliary state variables and equations. In this regard, the original single-degree-of-freedom (SDOF) nonlinear system with fractional derivative terms is cast, equivalently, into a multi-degree-of-freedom (MDOF) nonlinear system involving integer-order derivatives only. From a mathematics perspective, the equations of motion referring to the latter can be interpreted as constrained. Second, to circumvent the challenge of increased dimensionality of the problem due to the augmentation of the response vector, a WPI formulation with mixed fixed/free boundary conditions is developed for determining directly any lower-dimensional joint PDF corresponding to a subset only of the response vector components. This can be construed as an approximation-free dimension reduction approach that renders the associated computational cost independent of the total number of stochastic dimensions of the problem. Thus, the original SDOF oscillator joint PDF corresponding to the response displacement and velocity is determined efficiently, while circumventing the computationally challenging task of treating directly equations of motion involving fractional derivatives. Two illustrative numerical examples are considered for demonstrating the reliability of the developed technique. These pertain to a nonlinear Duffing and a nonlinear vibro-impact oscillators with fractional derivative elements subjected to combined stochastic and deterministic periodic loading. Note that alternative standard approximate techniques, such as statistical linearization, need to be significantly modified and extended to account for such cases of combined loading. Remarkably, it is shown herein that the WPI technique exhibits the additional advantage of treating such types of excitation in a straightforward manner without the need for any ad hoc modifications. Comparisons with pertinent Monte Carlo simulation data are included as well.

Suppressing Homoclinic Chaos for Vibro-Impact Oscillators

In this paper, theoretical framework and numerical verification for suppressing homoclinic chaos of a class of vibro-impact oscillators are discussed by adding parametric excitations in the form of [Formula: see text] as the control item. The analytical Melnikov method for planar vibro-impact systems is employed to obtain the corresponding thresholds of parameters as sufficient conditions for suppressing chaos. Two typical oscillators are presented to show the effectiveness of theoretical analysis for suppressing homoclinic chaos by tuning the amplitudes, frequencies and phases of the parametric excitations.

Dynamics of a hybrid cubic vibro-impact oscillator and nonlinear energy sink

In the current work, we model the HCVI oscillator by a forced particle in a hybrid quartic-square potential well with infinite depth under periodic external forcing. The slow-flow dynamics of the system is analyzed in the framework of isolated resonance approximation by canonical transformation to action–angle (AA) variables and the corresponding reduced resonance manifold (RM). Two types of bifurcation are examined and described analytically. The former is associated with the SNO-regime and the HCVI-regime and vice versa, and the latter with reaching a chosen maximal transient energy level. Unlike previous studies, three underlying dynamical mechanisms that govern the occurrence of bifurcations are identified: two maximum mechanisms and one saddle mechanism. The first two correspond to a gradual increase in the system’s response amplitude for a proportional increase in the excitation intensity, and the latter corresponds to an abrupt increase in the system’s response and therefore more potentially hazardous when takes place in engineering systems and more effective for energy pumping when takes place in the HCVI-NES. Both mechanism types are universal for systems that undergo escape from a potential well. The maximal transient energy is predicted analytically over the space of excitation parameters and described using iso-energy contours. The response curves of the system are obtained analytically, allowing a full perspective on the HCVI-oscillator dynamical response, and the HCVI-NES performances and design optimization. All theoretical results are supported by numerical simulations.

Melnikov analysis of subharmonic motions for a class of bistable vibro-impact oscillators

Melnikov analysis of subharmonic motions for a class of bistable vibro-impact oscillators

Dynamics of the Vibro-Impact Nonlinear Damped and Forced Oscillator under the Influence of the Electromagnetic Actuation

The main objective of the present work is to find an approximate analytical solution for the nonlinear differential equation of the vibro-impact oscillator under the influence of the electromagnetic actuation near the primary resonance. The trigger of vibro-impact regime is due to Hertzian contact. The optimal auxiliary functions method (OAFM) is utilized to give an analytical approximate solution of the problem. The influences of static normal load and electromagnetic actuation near the primary resonance are completely studied. The main novelties of the proposed procedure are the presence of some new adequate auxiliary functions, the introduction of the convergence-control parameters, the original construction of the initial and of the first iteration, and the freedom to choose the method for determining the optimal values of the convergence-control parameters. All these led to an explicit and accurate analytical solution, which is another novelty proposed in the paper. This technique is very accurate, simple, effective, and easy to apply using only the first iteration. A second objective was to perform an analysis of stability of the model using the multiple scales method and the eigenvalues of the Jacobian matrix.

Resonance and energy transfer in forced vibro-impact systems with linear compliance

We address a non-stationary dynamic of a vibro-impact oscillator with linear compliance under harmonic excitation. The analytic treatment is based on the well-known approximation of isolated resonance. To this end, an appropriate set of action–angle variables is formulated for the single degree of freedom conservative vibro-impact oscillator with linear compliance. At the next stage, the dynamics of the forced system is considered in conditions of a primary 1:1 resonance, common in most applications. The approximation of the isolated resonance reduces the dynamics to a planar resonance manifold with foliation induced by a global invariant derived for the 1:1 isolated resonance flow. This foliation allows for an appropriate description of the steady-state, modulated and transient responses, and their bifurcations. The results are verified with direct numeric simulations.

Dynamics of stochastic vibro-impact oscillator with compliant contact force models

The focus of this study is on numerically investigating the effects of additive noise on the dynamics of vibro-impact oscillators with compliant supports. Various models for the compliant supports have been considered, which include elastic, elasto-plastic, linear and nonlinear dissipative contact force models, and which follow constitutive laws that are discontinuous. It has been shown that noise superimposed on the harmonic excitations alters the dynamical features of the non-smooth systems significantly. Stochastic bifurcation analyses have been carried out to investigate the stability characteristics of the systems with respect to the different compliant force models and a comparison has been presented on the advantages and limitations of the various contact force models in the presence of noisy excitations.

Passive chaos suppression for the planar slider-crank mechanism with a clearance joint by attached vibro-impact oscillator

Chaotic dynamic behavior in clearance joints is the prominent nonlinear factor for multibody systems. This study presents a passive chaos suppression method to stabilize the planar slider-crank mechanism with a clearance joint. The strategy resorts to a vibro-impact oscillator attached to the slider. Primarily, the impact principle is employed to characterize the dynamics in clearance joints. The mathematical model of the integral system is conducted subsequently. The chaotic system behavior is evaluated by bifurcation analysis varying clearance size and crank speed. When the controller is activated, parametric optimizations of the stiffness, clearance coefficient, and particle mass are conducted to achieve optimum chaos elimination. Phase portraits, the largest Lyapunov exponent, and time-frequency distribution are evaluated. Numerical results illuminate that an adjusted oscillator can suppress the chaotic high-frequency composition of system response and help to maintain continuous contact between the joint elements. The results suggest applying the identical clearance size in the revolute joint and vibro-impact oscillator for fast optimization. Moreover, the method is still technology valid when considering different contact principles. Ultimately, a test platform is established to achieve some experimental validations.

Transitions in a noisy birhythmic vibro-impact oscillator with improved memory damping regime

Transitions in a noisy birhythmic vibro-impact oscillator with improved memory damping regime

Stochastic Bifurcations of a Fractional-Order Vibro-Impact Oscillator Subjected to Colored Noise Excitation

A stochastic vibro-impact system has triggered a consistent body of research work aimed at understanding its complex dynamics involving noise and nonsmoothness. Among these works, most focus is on integer-order systems with Gaussian white noise. There is no report yet on response analysis for fractional-order vibro-impact systems subject to colored noise, which is presented in this paper. The biggest challenge for analyzing such systems is how to deal with the fractional derivative of absolute value functions after applying nonsmooth transformation. This problem is solved by introducing the Fourier transformation and deriving the approximate probabilistic solution of the fractional-order vibro-impact oscillator subject to colored noise. The reliability of the developed technique is assessed by numerical solutions. Based on the theoretical result, we also present the critical conditions of stochastic bifurcation induced by system parameters and show bifurcation diagrams in two-parameter planes. In addition, we provide a stochastic bifurcation with respect to joint probability density functions. We find that fractional order, coefficient of restitution factor and correlation time of colored noise excitation can induce stochastic bifurcations.

Dynamics of a hybrid vibro-impact oscillator: canonical formalism

Hybrid vibro-impact (HVI) oscillations is a strongly nonlinear dynamical regime that involves both linear oscillations and collisions under periodic, impulsive, or stochastic excitation. This regime arises in various engineering systems, such as mechanical components under tight rigid constraints, seismic-induced sloshing in partially-filled liquid storage tanks, and more. The adaptive nonlinearity of the HVI oscillator is used by the HVI-nonlinear energy sink as an effective vibration mitigation solution for broad energy and frequency range. Due to the extreme nonlinearity of this regime, traditional analytical methods are inapplicable for the description of its transient dynamics. In the current work, we model the HVI oscillator by a forced particle in a truncated quadratic potential well with infinite depth. The slow flow dynamics of the system in the vicinity of primary resonance is described by canonical transformation to action-angle (AA) variables and the corresponding reduced resonance manifold (RM). Two types of bifurcation are examined. The former is associated with transition between linear oscillations and the HVI-regime and vice versa, and the latter with reaching a chosen maximal transient energy level. The transition boundaries on the forcing parameters plane associated with both bifurcation types are obtained analytically. The maximal transient energy level obtained for any given set of forcing parameters is described analytically as well. The energy jumps associated with the bifurcation of type I and crossing the corresponding transition boundary are obtained. Two underlying dynamical mechanisms that govern the occurrence of bifurcations are identified. They correspond to two distinct scenarios: in the first scenario, the energy of the slow flow gradually reaches the threshold energy level and is thus referred to as the ”maximum” mechanism. The second, potentially more dangerous scenario, involves abrupt transitions of the system’s energy response from a relatively small value to the threshold energy level. This pattern is related to the passage of the slow-flow phase trajectory through the saddle point of the RM, and thus is referred to as the “saddle” mechanism. Both mechanisms are universal for systems that undergo escape from a potential well. All theoretical results are in complete agreement with full-scale numerical simulations.