Nonlinear Vibro-impact System Research Articles

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.

Stochastic bifurcations of nonlinear vibroimpact system with time delay and fractional derivative excited by Gaussian white noise

This paper investigates the stochastic dynamics in the nonlinear vibroimpact system with time delay and fractional derivative under Gaussian white noise excitation. Firstly, based on the definition of Caputo-type fractional derivative and the method of non-smooth transformation, the original system is transformed into an equivalent delayed stochastic vibroimpact system without fractional derivative. Then, by using the stochastic averaging method, the stationary density function of the stochastic Itoˆ equation is obtained. The effectiveness of the proposed method was validated by comparing the consistency between the original system and the optimized system without fractional derivative or the term of time delay. At last, we also explore the stochastic P-bifurcation induced by the power spectral density of two uncorrelated noises, time delay, fractional order, and restitution coefficient of the system.

Pipe vibration attenuation through internal damping and optimal design of vibro-impact systems

Pipelines periodically supported by rack structures (PPRs) are common in chemical and petrochemical plants, among others, and conventional tools such as dampers and hysteretic absorbers are commonly used to mitigate large vibrations in these systems. In this study, we explore two alternative strategies: (i) enhancing the attenuation rate of PPR vibrations through structural internal damping, and (ii) using nonlinear vibro-impact systems (VIS) to reduce seismic vibrations in a PPR. To shed light on the first strategy, we develop analytical dispersion relations for a PPR and show how damping can improve the mitigation capabilities of the periodic system. As for the second strategy, we consider a 9-node beam, i.e., a single span (SS) of a PPR equipped with a VIS, and combine the central composite design (CCD) and Kriging metamodelling to maximize dissipation energy and minimize the number of impacts. This multi-objective optimization problem aims to find the most effective design solution for the VIS in terms of gap and coefficient of restitution (COR). Additionally, we consider the stochastic nature of seismic input and the possible chaotic behavior of the VIS. To account for the sensitive variability of the number of impacts in seismic records, we perform incremental dynamic analyses and calculate fragility functions for various engineering demand parameters, including the number of impacts. We define a 3D surface for selecting the optimal gap-COR pair. When impacts occur, transient results can be chaotic, and we compute the largest Lyapunov exponents of a few representative trajectories.

Probabilistic response analysis of nonlinear vibro-impact systems with two correlated Gaussian white noises

The response distributions of a dynamic system under the effect of correlation noise not only demonstrated a Gaussian symmetric distribution, but also exhibited a non-Gaussian skewed distribution. Thus, correlation noise can induce more complex dynamic phenomena in the response of a nonlinear system. The purpose of this study is to explore the stochastic response and bifurcation of nonlinear vibro-impact systems driven by correlated additive and multiplicative Gaussian white noises using the path integral (PI) method. First, we derived an equivalent correlated stochastic system without collision conditions in the new ancillary phase space using the non-smooth Ivanov transform method. We then used Novikov’s theorem to derive an effective Fokker–Planck–Kolmogorov equation for the equivalent correlated stochastic system. Second, we used PI technology, which combines the three-point Gauss–Legendre quadrature rule to solve the effective Fokker–Planck–Kolmogorov equation and obtained the probability density functions (PDFs) of the equivalent correlated stochastic system. Subsequently, we obtained the PDFs of the original vibro-impact stochastic system using the relationship between the initial state space and the new auxiliary phase space. Finally, the proposed method was used to investigate the transient and stationary responses of a Duffing-Van der Pol Vibro-impact system perturbed by additive and multiplicative random noises. In addition, this technique can also assess the effects of restitution coefficients and barrier position on stochastic P-bifurcation. The accuracy of this method was verified by comparing it with a Monte Carlo simulation.

A new technique for the global property of the vibro-impact system at the impact instant

In this paper, a new technique is proposed to study the global properties of the vibro-impact system at the impact instant. A special Poincaré section on the impact surface is chosen, which is composed of the co-dimensional time and the velocity. And a smooth impact-to-impact mapping is constructed as the Poincaré mapping. Consequently, we design a new technique for analyzing the dynamics of the system impact. The new technique specializes in analyzing the global property of the vibro-impact system at the impact instant. It cleverly utilizes the characteristics of the vibro-impact system and avoids the difficulties of non-smooth characteristics caused by impacts. Therefore, the dynamics of the vibro-impact system can be well retained. It is found that our method is valid and widely applicable to vibro-impact systems. A linear vibro-impact system and a nonlinear vibro-impact system are chosen as examples to verify the effectiveness of the new technique. The results show that the vibro-impact system exhibits some interesting dynamical phenomena at the impact instant, such as, the periodic cyclic characteristic and dramatic change of the impact behaviors.

The stochastic bifurcation of the vibro-impact system on the impact surface via a new path integration method.

In this paper, a new path integration algorithm is proposed for the non-autonomous vibro-impact system. It can be used to obtain the stochastic response of the system at the impact instant. The innovation of this algorithm is that the fixed excitation phase at the impact instant is introduced. The fixed excitation phase and the impact surface are combined as the special impact surface, and the algorithm sets up the different evolution methods of the probability density function and the impact completion condition. It can directly capture the stochastic response of the system at the impact instant without any non-smooth approximations. The non-smooth characteristics of the impact can be well preserved. This new path integration algorithm has good applicability. It has no limitation on the restitution coefficient. The design idea of this algorithm may provide a framework for the non-smooth system subject to the periodic excitation. In the end, a linear vibro-impact system and a nonlinear vibro-impact system excited by the external periodic force are used as examples to show how our method could be implemented. The stochastic P-bifurcation phenomena induced by parameters also are considered. The Monte Carlo simulations verify the effectiveness and accuracy of the proposed path integration algorithm.

The response of stochastic vibro-impact system calculated by a new path integration algorithm

In this paper, a new path integration algorithm is proposed specifically for the stochastic vibro-impact system. By introducing the concept of absorption surface and the impact completion condition, this new path integration algorithm can be directly used to study the stochastic response of vibro-impact systems without any non-smooth approximation. The algorithm is suitable for arbitrary recovery coefficients and can retain the non-smooth characteristics of the vibro-impact system. Linear and nonlinear vibro-impact systems excited by the Gaussian white noise are used as examples to show the unique results obtained by our method. Finally, the Monte Carlo method is used to verify the efficiency and accuracy of this new algorithm.

Reliability and control of strongly nonlinear vibro-impact system under external and parametric Gaussian noises

This paper mainly focuses on reliability and the optimal bounded control for maximizing system reliability of a strongly nonlinear vibro-impact system. Firstly, the new stochastic averaging in which the impact condition is converted to the system energy is applied to obtain the averaged Ito stochastic differential equation, by which the associated Backward Kolmogorov (BK) equation and Generalized Pontryagin (GP) equation are derived. Then, the dynamical programming equations are obtained based on the dynamical programming principle, by which the optimal bounded control for maximizing system reliability is devised Finally, the effects of the bounded control and noise intensity on the reliability of the vibro-impact system are discussed in detail; meanwhile, the influence of impact conditions on the system’s reliability is also studied. The feasibility and effectiveness of the proposed analytical method are substantiated by numerical results obtained from Monte-Carlo simulation.

Optimal bounded control of stochastically excited strongly nonlinear vibro-impact system

An optimal bounded control strategy for strongly nonlinear vibro-impact systems under stochastic excitations with actuator saturation is proposed. First, the impact effect is incorporated in an equivalent equation by using a nonsmooth transformation. Under the assumption of light damping and weak random perturbation, the system energy is a slowly varying process. By using the stochastic averaging of envelope for strongly nonlinear systems, the partially averaged Itô stochastic differential equation for system energy can be derived. The optimal control problem is transformed from the original optimal control problem for the state variables to an equivalent optimal control problem for the system energy, which decreases the dimensions of the optimal control problem. Then, based on stochastic maximum principle, an adjoint equation for the adjoint variable and the maximum condition of partially averaged control problem are established. For infinite time-interval ergodic control, the adjoint variable is assumed to be a stationary process and the adjoint equation can be further simplified. Finally, the probability density function of the system energy and other statistics of the optimally controlled system are derived by calculating the associated Fokker–Plank–Kolmogorov equation. For comparison, the bang–bang control is also investigated and the control results are compared to show the advantages of the developed control strategy.

Reliability analysis of nonlinear vibro-impact systems with both randomly fluctuating restoring and damping terms

Abstract In this paper, the reliability analysis of vibro-impact system with both randomly fluctuating restoring and damping terms is studied by a modified path integration (PI) method. Specifically, the Ivanov nonsmooth transformation technique is adopted to transform the vibro-impact system into the system without barrier, then the modified PI application on vibro-impact systems is applied, which is based on the Gaussian closure method and the localized approximation of moment function. According to the first passage theory, the reliability function, the first passage probability density function (PDF) and the mean first passage time are numerically calculated. In the framework of our numerical results, the influences of different random restoring terms, random damping terms and impact conditions on the system’s reliability are discussed. The modified PI results are compared with the Monte Carlo Simulation (MCS) results, which shows that the proposed PI method can not only provide sufficiently accurate results to observe the weak influence of parameters, but also has obvious advantages in computational efficiency.

Probabilistic response analysis for a class of nonlinear vibro-impact oscillator with bilateral constraints under colored noise excitation

Abstract In this paper, a new stochastic method combining the stochastic averaging of quasi-conservative and energy loss is developed to analyze the probabilistic response of a class of nonlinear vibro-impact oscillator with bilateral barriers driven by colored noise. According to the relationships between the total energy and the potential energy of impact positions of the nonlinear system, the energy loss of every movement period of the unperturbed nonlinear vibro-impact system with given total energy is analyzed in detail. Based on this analysis, an averaged It o ^ stochastic differential equation is derived through the stochastic method which combining a nonlinear transformation and the stochastic averaging of quasi-conservative, in which the averaged It o ^ equation has the typical characteristics of piecewise smooth. Next, the stationary probability density function (SPDF) is obtained by solving the averaged drift and diffusion terms of the corresponding piecewise smooth Fokker-Planck equation for the averaged It o ^ equation with the Fourier series expansion. Finally, an example is given to validate the effectiveness of the proposed method. The effects of four physical quantities of the stochastic vibro-impact system: the positions of impact, restitution coefficient, noise intensity and correlation time of the colored noise, on the probabilistic response are discussed by the SPDFs in detail, and the results are verified through the Monte Carlo simulation.

Stochastic Bifurcation of a Strongly Non-Linear Vibro-Impact System with Coulomb Friction under Real Noise

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.

Stochastic Responses of Lightly Nonlinear Vibroimpact System with Inelastic Impact Subjected to External Poisson White Noise Excitation

A procedure for analyzing stationary responses of lightly nonlinear vibroimpact system with inelastic impact subjected to external Poisson white noise excitation is proposed. First, the original vibroimpact system is transformed to a new system without velocity jump in terms of the Zhuravlev nonsmooth coordinate transformation and the Dirac delta function. Second, the averaged generalized Fokker-Planck-Kolmogorov (FPK) equation for transformed system under parametric excitation of Poisson white noise is derived by stochastic averaging method. Third, the averaged generalized FPK equation is solved by using the perturbation technique and inverse transformation of the Zhuravlev nonsmooth coordinate transformation to obtain the approximately stationary solutions for response probability density functions of original vibroimpact system. Last, analytical and numerical results for two typical lightly nonlinear vibroimpact systems are presented to assess the effectiveness of the proposed method. It is found that they are in good agreement and the proposed method is quite effective.

Stochastic response of Duffing-Van der Pol vibro-impact system with viscoelastic term under wide-band excitation

This paper aims to investigate stochastic dynamics of nonlinear vibro-impact system with viscoelastic term subjected to wide-band noise. Firstly, the viscoelastic term is approximated as equivalent damping and stiffness forces. Then, the non-smooth transformation is adopted to convert the vibro-impact system into a system without barriers. By using of the stochastic averaging method based on generalized harmonic function, the probability densities of the stationary response are obtained and dynamical influence of viscoelastic parameters and restitution coefficient of the system are discussed as well. Comparison between the analytical simulation and numerical simulation results shows an excellent agreement.

Analysis of chaotic saddles in a nonlinear vibro-impact system

In this paper, a computational investigation of chaotic saddles in a nonlinear vibro-impact system is presented. For a classical Duffing vibro-impact oscillator, we employ the bisection procedure and an improved stagger-and-step method to present evidence of visual chaotic saddles on the fractal basin boundary and in the internal basin, respectively. The results show that the period saddles play an important role in the evolution of chaotic saddle. The dynamics mechanics of three types of bifurcation such as saddle-node bifurcation, chaotic saddle crisis bifurcation and interior chaotic crisis bifurcation are discussed. The results reveal that the period saddle created at saddle-node bifurcation is responsible for the switch of the internal chaotic saddle to the boundary chaotic saddle. At chaotic saddle crisis bifurcation, a large chaotic saddle can divide into two different chaotic saddle connected by a period saddle. The intersection points between stable and unstable manifolds of this period saddle supply access for chaotic orbits from one chaotic saddle to another and eventually induce the coupling of these two chaotic saddle. Interior chaotic crisis bifurcation is associated with the intersection of stable and unstable manifolds of the period saddle connecting two chaotic invariant sets. In addition, the gaps in chaotic saddle is responsible for the fractal structure.

Transient response of nonlinear vibro-impact system under Gaussian white noise excitation through complex fractional moments

The approximate transient response of a nonlinear vibro-impact system under Gaussian white noise excitation is investigated by the methods of stochastic averaging and the Mellin transform. The Zhuravlev nonsmooth transformation is utilized to convert the nonlinear vibro-impact system into an equivalent system without velocity jumps by introducing an impulsive damping term. The Ito stochastic differential equation with respect to the amplitude response and the related Fokker–Plank–Kolmogorov (FPK) equation governing the amplitude response probability density of the vibro-impact system are derived with the stochastic averaging method. The Mellin transform is introduced to solve the FPK equation. The differential relations of complex fractional moments are obtained. The probability density function for transient response of this system constructed by solving a set of differential equations yields complex fractional moments. Two illustrative examples are examined to evaluate the effectiveness of the proposed solution procedure. The effects of restitution factors are investigated on the transient probability density distribution of the vibro-impact systems. At the same time, the convergence and the error analyses for different restitution factors are illustrated. Influences of the truncate term on the convergence and the error are further illustrated. The results obtained from the proposed procedure agree well with the results from Monte Carlo simulations.

Numerical Bifurcation Analysis of Discontinuous 2-DOF Vibroimpact System. Part 2: Frequency-Amplitude Response

This paper discusses bifurcations in nonlinear vibroimpact system. It is a discontinuous dynamical system. We were studying stability and bifurcations in specific two-body two-degree-of-freedom vibroimpact system by numerical parameter continuation method. We adapted parameter continuation technique for this system. Theoretical basis for dynamic characteristics composition was presented in [1]. The instability zones and bifurcation points for loading curves were determined in [1] under excitation amplitude variation. In this paper we investigate the instability zones and bifurcation points under variation of excitation frequency when we consider the frequency-amplitude response. We have observed phenomena unique for nonsmooth systems with discontinuous right-hand side: discontinuous bifurcation points where set-valued Floquet multipliers cross the unit circle by jump. At these points monodromy matrix is changed by jump too. We also have observed chattering regimes leading to chaos.

Response analysis of nonlinear vibro-impact system coupled with viscoelastic force under colored noise excitations

This paper is mainly dealing with the stochastic responses of nonlinear vibro-impact (VI) system coupled with viscoelastic force excited by colored noise. By the aid of approximate conversion for the viscoelastic force, the original stochastic VI system is transformed into an equivalent stochastic system without viscoelastic term. Then, the equations of the converted system are simplified by non-smooth transformation, and the stochastic averaging method is employed to solve the above simplified system. A Van der Pol VI oscillator coupled with viscoelastic force is worked out in detail to illustrate the application of the mentioned method, and therewith the analytical solutions fit the numerical simulation results based on the original system. Therefore, the present analytical means of investigating this system is proved to be feasible. Additionally, the exploration of stochastic P-bifurcation by two different ways is also demonstrated in this paper through varying the value of the certain system parameters. Besides, it shows a noteworthy fact that assigning zero or a positive value to the magnitude of viscoelastic force can also lead to the bimodal shape of different degrees in the process of stochastic bifurcations.

Response of strongly nonlinear vibro-impact system with fractional derivative damping under Gaussian white noise excitation

This paper aims to investigate the strongly nonlinear vibro-impact dynamic system with fractional derivative damping under Gaussian white noise excitation, and this system considers both nonlinear factors and non-smooth factors. With the help of non-smooth transformation, the original system is rewritten as a smooth system, which is an easier handled form. For the altered function, we obtain the approximate stationary solutions analytically by generalized stochastic averaging method. To prove the validity of the approximate analytical methods, an efficient scheme with high accuracy is adopt to simulate the fractional derivative, then the fourth-order Runge–Kutta approach is used to obtain the numerically response statistics. Meanwhile the analytical solutions are verified by the numerical simulation solutions. At last, we research the responses of this system. In this context, we can consider the influences to this system caused by the fractional order, the restitution coefficient and the noise intensity through changing the values of corresponding parameters. The observation and investigation state that fractional derivative term, impact conditions and stochastic excitation can influence the responses of the fractional vibro-impact dynamic system.

Stochastic bifurcations in the nonlinear vibroimpact system with fractional derivative under random excitation

This paper aims to investigate the stochastic bifurcations in the nonlinear vibroimpact system with fractional derivative under random excitation. Firstly, the original stochastic vibroimpact system with fractional derivative is transformed into equivalent stochastic vibroimpact system without fractional derivative. Then, the non-smooth transformation and stochastic averaging method are used to obtain the analytical solutions of the equivalent stochastic system. At last, in order to verify the effectiveness of the above mentioned approach, the van der Pol vibroimpact system with fractional derivative is worked out in detail. A very satisfactory agreement can be found between the analytical results and the numerical results. An interesting phenomenon we found in this paper is that the fractional order and fractional coefficient of the stochastic van der Pol vibroimpact system can induce the occurrence of stochastic P-bifurcation. To the best of authors’ knowledge, the stochastic P-bifurcation phenomena induced by fractional order and fractional coefficient have not been found in the present available literature which studies the dynamical behaviors of stochastic system with fractional derivative under Gaussian white noise excitation.