Vibro-impact Oscillator Research Articles

The behavior of bilinear impact oscillators subjected to random forcings

A multistable regime of a bilinear vibro-impact oscillator in the presence of noise has been studied, for which relatively lesser attractors coexist with a well-defined basin boundary structure in the phase space. The effect of adding parametric noise in this model has been investigated and the switching that the system undergoes between the basins of its different attractors has been examined.

Validation of vibro-impact force models by numerical simulation, perturbation methods and experiments

The frequency response of a single degree of freedom vibro-impact oscillator is analyzed using Harmonic Linearization, Averaging and Numeric Simulation, considering three different impact force models: one given by a piecewise-linear function (Kelvin-Voigt model), another by a high-order power function, and a third one combining the advantages of the other two. Experimental validation is carried out using control-based continuation to obtain the experimental frequency response, including its unstable branch.

Periodic solutions of n-dofs autonomous vibro-impact oscillators with one lasting contact phase

In the continuum structural mechanics framework, a unilateral contact condition between two flexible bodies does not generate impulsive contact forces. However, finite-dimensional systems, derived from a finite element semi-discretization in space for instance, and undergoing a unilateral contact condition, require an additional impact law: Unilateral contact occurrences then become impacts of zero duration unless (i) the impact law is purely inelastic, or (ii) the pre-impact velocity is zero. This contribution explores autonomous periodic solutions with one contact phase per period and zero pre-impact velocity [case (ii)], for any n-dof mechanical systems involving linear free-flight dynamics together with a linear unilateral contact constraint. A recent work has shown that such solutions seem to be limits of periodic trajectories with k impacts per period as k increases. Minimal analytic equations governing the existence of such solutions are proposed, and it is proven that, generically, they occur only for discrete values of the period. It is also shown that the graphs of such periodic solutions have two axes of symmetry in time. Results are illustrated on a spring–mass system and on a 4-dof two-dimensional system made of 1D finite elements. Animations of SPPs with up to 30 dofs are provided.

Bifurcation analysis of a stochastically excited vibro-impact Duffing-Van der Pol oscillator with bilateral rigid barriers

This study focuses on the bifurcation analysis of a Duffing-Van der Pol (DVDP) oscillator with rigid barriers, subjected to additive white noise excitations. This involves mapping the non-smooth vibro-impact system on a continuous phase plane through non-smooth variable transformations. The formulation considers the general case of asymmetrically positioned both sided rigid barriers. The probabilistic description of the system dynamics is subsequently defined through the corresponding Fokker-Planck (FP) equation. The stationary probability density function of the response is numerically obtained by solving the FP equation using finite element method. The stochastic stability of the vibro-impact DVPD oscillator is examined both in terms of D- and P-bifurcations.

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.

Energy Exchange and Localization in Essentially Nonlinear Oscillatory Systems: Canonical Formalism

Over recent years, a lot of progress has been achieved in understanding of the relationship between localization and transport of energy in essentially nonlinear oscillatory systems. In this paper, we are going to demonstrate that the structure of the resonance manifold can be conveniently described in terms of canonical action–angle (AA) variables. Such formalism has important theoretical advantages: all resonance manifolds may be described at the same level of complexity, appearance of additional conservation laws on these manifolds is easily proven both in autonomous and nonautonomous settings. The harmonic balance-based complexification approach, used in many previous studies on the subject, is shown to be a particular case of the canonical formalism. Moreover, application of the canonic averaging allows treatment of much broader variety of dynamical models. As an example, energy exchanges in systems of coupled trigonometrical and vibro-impact oscillators are considered.

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.

Subharmonic Response of Linear Vibroimpact System with Fractional Derivative Damping to a Randomly Disrobed Periodic Excitation

The subharmonic response of single-degree-of-freedom vibroimpact oscillator with fractional derivative damping and one-sided barrier under narrow-band random excitation is investigated. With the help of a special Zhuravlev transformation, the system is reduced to one without impacts, thereby permitting the applications of asymptotic averaging over the period for slowly varying random process. The analytical expression of the response amplitude is obtained in the case without random disorder, while only the approximate analytical expressions for the steady-state moments of the response amplitude are obtained in the case with random disorder. The effects of the fractional order derivative term, damping term, random disorder, and the coefficient of restitution and other system parameters on the system response are discussed. Theoretical analyses and numerical simulations show that fractional derivative makes both the system damping and stiffness coefficients increase, such that it changes the system parameters region at which the response amplitude reaches the maximum. The system energy loss in collision is equivalent to increasing the damping coefficient of the system. System response amplitude will increase when the excitation frequency is close to the resonant frequency and will decay rapidly when the excitation frequency gradually deviates from the resonance frequency.

Multi-valued responses of a nonlinear vibro-impact system excited by random narrow-band noise

We investigate the multi-valued responses of a non-linear vibro-impact oscillator with a one-sided barrier subject to random narrow-band excitation. The frequency response of the system is obtained using the Krylov-Bogoliubov averaging method. Meanwhile, the backbone curve and the critical equation of unstable region are also derived for the deterministic case. Then, the method of moment is applied to obtain the iterative calculation equation for the mean-square response amplitude under the stochastic case. Excitation frequency, nonlinearity intensity, damping parameters, especially the distance between the system’s static equilibrium position and the barrier can lead to triple-valued response under certain case. In some conditions the impact system may have two or four steady-state solutions, which is an interesting phenomenon for impact system. The unstable region is one uniform part while under smaller nonlinearity intensity it is divided into two parts. Moreover, we also find that as random noise intensity increases, the pervasion of the phase trajectories strengthens, and then destroys the topological property of the phase trajectories.

Recent advances in vibro-impact dynamics and collision of ocean vessels

The treatment of ship impacts and collisions takes different approaches depending on the emphasis of each discipline. For example, dynamicists, physicist, and mathematicians are dealing with developing analytical models and mappings of vibro-impact systems. On the other hand, naval architects and ship designers are interested in developing design codes and structural assessments due to slamming loads, liquid sloshing impact loads in liquefied natural gas tanks and ship grounding accidents. The purpose of this review is to highlight the main differences of the two disciplines. It begins with a brief account of the theory of vibro-impact dynamics based on modeling and mapping of systems experiencing discontinuous changes in their state of motion due to collision. The main techniques used in modeling include power-law phenomenological modeling, Hertzian modeling, and non-smooth coordinate transformations originally developed by Zhuravlev and Ivanov. In view of their effectiveness, both Zhuravlev and Ivanov non-smooth coordinate transformations will be described and assessed for the case of ship roll dynamics experiencing impact with rigid barriers. These transformations have the advantage of converting the vibro-impact oscillator into an oscillator without barriers such that the corresponding equation of motion does not contain any impact term. One of the recent results dealing with the coefficient of restitution is that its value monotonically decreases with the impact velocity and not unique but random in nature. Slamming loads and grounding events of ocean waves acting on the bottom of high speed vessels will be assessed with reference to the ship structural damage. It will be noticed that naval architects and marine engineers are treating these problems using different approaches from those used by dynamicists. The problem of sloshing impact in liquefied natural gas cargo and related problems will be assessed based on the numerical and experimental results. It is important for vessel designers to determine the capacity of ships to resist random slamming loads, sloshing loading impact, grounding accidents and ships collisions.

Probabilistic Solution of Vibro-Impact Systems Under Additive Gaussian White Noise

This paper presents a solution procedure for the stationary probability density function (PDF) of the response of vibro-impact systems under additive Gaussian white noise. The constraint is a unilateral zero-offset barrier. The vibro-impact system is first converted into a system without barriers using the Zhuravlev nonsmooth coordinate transformation. The stationary PDF of the converted system is governed by the Fokker–Planck equation which is solved by the exponential-polynomial closure (EPC) method. A vibro-impact Duffing oscillator with either elastic or lightly inelastic impacts is considered in a numerical analysis. Meanwhile, the level of nonlinearity in displacement is also examined in this study as well as the case of negative linear stiffness. Comparison with the simulated results shows that the EPC method can present a satisfactory PDF for displacement and velocity when the polynomial order is taken as 4 in the investigated cases. The tail of the PDF also works well with the simulated result.

A stochastic averaging method for analyzing vibro-impact systems under Gaussian white noise excitations

A new stochastic averaging method for predicting the response of vibro-impact (VI) systems to random perturbations is proposed. First, the free VI system (without damping and random perturbation) is analyzed. The impact condition for the displacement is transformed to that for the system energy. Thus, the motion of the free VI systems is divided into periodic motion without impact and quasi-periodic motion with impact according to the level of system energy. The energy loss during each impact is found to be related to the restitution factor and the energy level before impact. Under the assumption of lightly damping and weakly random perturbation, the system energy is a slowly varying process and an averaged Itô stochastic differential equation for system energy can be derived. The drift and diffusion coefficients of the averaged Itô equation for system energy without impact are the functions of the damping and the random excitations, and those for system energy with impact are the functions of the damping, the random excitations and the impact energy loss. Finally, the averaged Fokker–Plank–Kolmogorov (FPK) equation associated with the averaged Itô equation is derived and solved to yield the stationary probability density of system energy. Numerical results for a nonlinear VI oscillator are obtained to illustrate the proposed stochastic averaging method. Monte-Carlo simulation (MCS) is also conducted to show that the proposed stochastic averaging method is quite effective. • A new stochastic averaging method was proposed. • The impact condition for displacement is transformed to that for system energy. • A direct way to deal with the energy loss due to impact is proposed. • The analytical results agree well with the MCS results.

2C24 Optimal sampled-data control by vibration manipulation function for one mass-spring system obtained from three vibro-impact oscillators under Grover algorithm(The 12th International Conference on Motion and Vibration Control)

2C24 Optimal sampled-data control by vibration manipulation function for one mass-spring system obtained from three vibro-impact oscillators under Grover algorithm(The 12th International Conference on Motion and Vibration Control)

S1620205 振動操作関数によるHDD磁気ヘッドのサンプル値制振制御([S162-02]情報機器メカニクスとヘッドディスクインターフェイス/情報・精密機器のサーボ・スマート制御(2))

Rotational vibration manipulation function for enforced angular displacement is applied to suppress residual vibration of swing arm of HDD device during seeking control. The function is derived from a three-body vibro-impact rotational oscillator under Grover algorithm. Since it can sampled-data control angle and angular velocity of the magnetic head in every natural period, robust control can be achieved without changing long or short-distance seeking mode. Because of the properties of rotational vibration manipulation function, the proposed method requires that the natural period of swing arm should be integral multiplication of the sampling time of servo-information.

Bifurcation of Safe Basins and Chaos in Nonlinear Vibroimpact Oscillator under Harmonic and Bounded Noise Excitations

The erosion of the safe basins and chaotic motions of a nonlinear vibroimpact oscillator under both harmonic and bounded random noise is studied. Using the Melnikov method, the system’s Melnikov integral is computed and the parametric threshold for chaotic motions is obtained. Using the Monte-Carlo and Runge-Kutta methods, the erosion of the safe basins is also discussed. The sudden change in the character of the stochastic safe basins when the bifurcation parameter of the system passes through a critical value may be defined as an alternative stochastic bifurcation. It is founded that random noise may destroy the integrity of the safe basins, bring forward the occurrence of the stochastic bifurcation, and make the parametric threshold for motions vary in a larger region, hence making the system become more unsafely and chaotic motions may occur more easily.

Stochastic response of vibro-impact Duffing oscillators under external and parametric Gaussian white noises

Abstract This study presents a solution procedure for the stationary probability density function (PDF) of the response of vibro-impact Duffing oscillators under external and parametric Gaussian white noises. First the Zhuravlev non-smooth coordinate transformation is adopted to convert a vibro-impact oscillator into an oscillator without barriers. The stationary PDF of the converted oscillator is governed by the Fokker–Planck (FP) equation. The FP equation is solved by the exponential-polynomial closure (EPC) method. Illustrative examples are presented with vibro-impact Duffing oscillators under external and parametric Gaussian white noises to show the effectiveness of the solution procedure. The parametric excitation is acting in displacement and the constraint is a unilateral zero-offset barrier. The restitution coefficient of impacts is taken as 0.90. Comparison with the simulated results shows that the proposed solution procedure can provide good approximate PDFs for displacement and velocity although a little difference exists in the tail of these PDFs. This difference may be due to the weak approximation on the response of the vibro-impact oscillators using a continuous Markov process when the restitution coefficient is not very close to unity.

Steady Collision State in One, Two or Three-Body Vibro-Impact Oscillator with External Force or Enforced Displacement

By adding impulse excitation and energy dispersion operations, we obtain steady states of three-body vibro-impact system with two small oscillators coupled to one large oscillator under wave algorithm. From the steady collision conditions of the three-body system, we deduce a trajectory and a function of external force or enforced displacement for steady collision conditions of two kinds of two-body or single-body vibro-impact oscillator. By using either function, mass of the vibro-impact oscillator steadily collies with arbitrary impact positions and velocities. Compared with these functions, steady collisions were not achieved with harmonic external forces or enforced displacements. These functions could be favorable for steady impact process.

Global analysis of crises in a Duffing vibro-impact oscillator with non-viscously damping

以一类含非黏滞阻尼的Duffing单边碰撞系统为研究对象, 运用复合胞坐标系方法, 分析了该系统的全局分岔特性. 对于非黏滞阻尼模型而言, 它与物体运动速度的时间历程相关, 能更真实地反映出结构材料的能量耗散现象. 研究发现, 随着阻尼系数、松弛参数及恢复系数的变化, 系统发生两类激变现象: 一种是混沌吸引子与其吸引域内的混沌鞍发生碰撞而产生的内部激变, 另一种是混沌吸引子与吸引域边界上的周期鞍(混沌鞍)发生碰撞而产生的常规边界激变(混沌边界激变), 这两类激变都使得混沌吸引子的形状发生突然改变.

Vibro-impact forced oscillations of a heavy mass particle along a rough circle excited by a single-frequency force

The first part of the article presents a motion analysis of a vibro-impact system based on an oscillator moving along a rough circle line in a vertical plane, under the action of an external single-frequency force. Non-ideality of the bond originates of sliding Coulomb’s type friction force with coefficient μ = tgα0. The oscillator consists of one heavy mass particle whose forced motion is limited by one stabile angular elongation limiter. The differential equation of motion of the analyzed vibro-impact system cannot be solved explicitly (in closed form). For its approximately solving, the software package WOLFRAM Mathematica 7 is used. The results are checked by using the software package MATLAB R2008a. The combination of the analytical–numerical results for the defined parameters of the analyzed vibro-impact system, is base for the motion analysis visualization, as a first part of this analytic research. In the second part, the influence of parameters (amplitude and circle frequency) of the external single-frequency force to the system motion, based on real geometry and kinetic parameters of the analyzed vibro-impact system, is investigated. As a result of this study in the second part of this article, there is a conclusion that the motion of the analyzed vibro-impact system in or out of the resonant area can be influenced, and this is significant for engineering practice.

Wave Algorithm Analysis of Damping Energy Transfer between Two Globally Coupled Vibro-Impact Oscillators with Proportional Viscous Damping (In Case of Perfect Elastic Collision at Balanced Position)

In this paper, we study periodic energy transfer in multiple vibro-impact oscillators under proportional viscous damping by using wave algorithm. Internal resonance frequency is obtained by the condition that some off-diagonal elements become to 0 in state transition matrix. Under equivalent modal damping rate, we derive the condition of periodic damped and periodic energy transfer that contains sub-harmonic resonance which depends on mass ratio between the impact and non-impact small oscillators. Under non-equivalent modal damping rate, modulation of the frequency in imperfect damped energy transfer is observed. We obtain a state transition matrix of non-equivalent modal damping rate as the perturbation of that in equivalent. The modulation of periodic damped energy transfer depends on the decay of the modal damping rate in the small impact oscillator.