Vibro-impact System Research Articles

Diversity and transition of periodic motion of a periodically excited soft-impacting machinery

Dynamics of a periodically excited vibro-impact system with soft impacts is investigated. Essential features of period-one multi-impact motion group and correlated transition characteristics in low-frequency range are discussed in detail by the way of two-parameter bifurcation space providing qualitative domains for different periodic motions. The main focus is given to the effect of sensitive parameters including constraint stiffness k 0, clearance threshold b, and damping parameter ζ on the system response. The low-frequency characteristics in the finite-dimensional parameter space are particularly explored. It is found that the increase of k 0 induces multi-type bifurcation of period-one double-impact symmetrical motion, which induces a rich variety of periodic motions, and period-one multi-impact motion group orbit primarily exist in the small-clearance b and low-frequency ω zone. Based on the evolution irreversibility of adjacent period-one multi-impact orbit, the mechanism of singularies appearing in pairs and two different transition zones (hysteresis and liguliform zones) is studied, the result of which provides a theoretical reference value for the common low-frequency vibration instability phenomenon in the field of mechanical engineering. For small-damping coefficient ζ, period-one multi-impact motion has a large quantity, and the main bridge for the transition of adjacent period-one multi-impact motion is liguliform zone, which embraces period-one multi-impact asymmetrical motion and period-n multi-impact subharmonic motion and a certain chaotic zone. For large-damping coefficient ζ, the amount of period-one multi-impact motion group is reduced, and the main bridge for the transition of adjacent period-one multi-impact motion is hysteresis zone, where adjacent period-one multi-impact orbits can coexist according to initial conditions. As designing and renovating impact mechanical equipment, the reasonable matching law of dynamic parameters can be determined through two-parameter bifurcation space, which is conducive to making the system work in stable periodic motion and obtaining larger instantaneous impact velocity.

A data-driven dynamic method of downhole rock characterisation for the vibro-impact drilling system

For the real-time characterisation of an inhomogeneous impact inhibiting constraint such as downhole rock layers, an unconventional method using machine learning (ML) and drill-bit vibrations is investigated. An impact oscillator with one-sided elastic constraint is employed in modelling the bit-rock impact actions. Measurable drill-bit dynamics, such as acceleration, were acquired and processed into features and 2D-images that were later used in developing ML models capable of predicting the stiffness of impacted rock constraint. Explored ML networks include Multilayer Perceptron (MLP), Convolutional Neural Network and Long Short-Term Memory Network. Both simulation and experimental studies have been presented to validate the proposed method while using coefficient of determination (R2) and normalised mean absolute error (NMAE) as the performance metrics of the ML models. Results showed that the feature-based models had better performances for both simulation and experiment compared to the raw signal and 2D-image based models. Aside being simple and computationally less expensive, the feature-based MLP models outperformed other models having R2 values > 0.7 and NMAE values < 0.2 for both simulation and experiment, thus presenting them as the preferred ML model for dynamic downhole rock characterisation. In general, this study presents a new modality to achieving logging-while-drilling during deep-hole drilling operations such as carried out in hydrocarbon, mineral and geothermal exploration.

Analytical researches of a vibrating mill with vibroimpact excitation of a crushing chamber

The scheme of a vibrating mill with a drive from inertial debalanced vibrators and with vibroimpact excitation of a grinding chamber with a technological medium has been developed. Analytical researches of the vibration mill were performed according to the universal mathematical model of vibroimpact systems. It was established that vibrating mills with vibroimpact excitation of the grinding chamber, in comparison with typical vibrating mills, which are made according to a single-mass dynamic scheme, provide: an increase in the vibration parameters of the grinding chamber with an increase in the mass of the technological medium, which to some extent levels out the decrease in the vibration parameters of the technological medium and, accordingly, increases mill productivity; the possibility of realizing significant accelerations of the grinding chamber with the technological medium at relatively insignificant accelerations of the mill frame with vibration exciters, which increases the reliability and durability of the mill’s operation; transfer of insignificant dynamic loads to the base, which do not exceed the static reactions from the force of gravity, which makes it possible to install the mill on a fixed base without its fixation and the need to arrange a massive foundation under its supporting surface.

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.

Optimization of a vibro-impact damper design using MATLAB tools

The paper studies the dynamics of a vibro-impact system consisting of a main (primary) structure and a vibro-impact damper coupled to it. A vibro-impact damper is a vibro-impact nonlinear energy sink (VI NES). The optimal damper design should provide the best vibration mitigation for the primary structure. The optimization procedures for finding the optimal NES design are carried out using standard MATLAB tools. We used different MATLAB programs, namely surf program, which graphically shows the ranges of parameter pairs to be optimized; fminsearch and fminconprograms, which search for local minima of the objective function. It is shown that the optimization procedure itself is ambiguous and contains a sufficient amount of arbitrariness. Its result is also ambiguous. It is due to the presence of the many possible sets of damper parameters that can provide maximum mitigation of the main structure vibrations. We do not use the genetic algorithm gabecause it selects random intermediate results and yields randomly selected parameter sets from the optimal parameters manifold. Setting the objective function and its parameters plays a crucial role in the optimization process. We have chosen the maximum total energy of the primary structure as the objective function. Each resulting variant of the damper parameter set should be carefully tested and analyzed. We compared the five obtained optimal designs for dampers with two different masses. When analyzing them, we observed different motion modes, namely periodic modes of different periodicity with different number of impacts per cycle, with different ratio of bodies motion periods: 1:1 resonance with resonance capture and 2:1 resonance; amulti-periodic mode with many impacts per cycle, which turned out to be an amplitude-modulated mode – Amplitude Modulated Signal . The final decision on the optimal damper design may be made taking into account various engineering considerations regarding its mass and other parameters. It should be based on the options obtained as a result of the optimization procedure.

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.

The influence of various optimization procedures on the dynamics and efficiency of nonlinear energy sink with synergistic effect consideration

The vibro-impact system under consideration consists of a primary structure, which is a linear oscillator (LO) and a vibro-impact damper coupled with it. The damper is a vibro-impact nonlinear energy sink (VI NES). The dynamics of this system and damper efficiency in attenuation vibrations of the primary structure are studied using numerical simulation. The paper examines the problem of choosing optimal damper parameters that provide the best mitigation of the primary structure energy. The optimization procedure was carried out in several stages. The multiple parameters optimization manifested a synergistic effect. This made it possible to select the most effective VI NES design. The chosen damper design provided the best energy reduction. Thanks to it, the irregular regimes of the system movement turned into the periodic ones. The damper hits both the primary structure directly and an obstacle rigidly connected to the primary structure.

Moment Lyapunov exponent and stochastic stability of a vibro-impact system driven by non-Gaussian colored noise

Vibro-impact phenomena are prevalent in practical engineering, making research on their stochastic dynamic characteristic of great practical significance. However, the research on stochastic stability and bifurcation of vibro-impact systems is still limited, especially the moment stability. In this paper, based on the pth moment Lyapunov exponent, the stochastic stability of a vibro-impact system driven by non-Gaussian colored noise is investigated. Firstly, the smooth stochastic dynamic system is obtained making use of a non-smooth transformation and the non-Gaussian colored noise is simplified to an Ornstein-Uhlenbeck process by utilizing the path-integral method. Thereafter, through applying the L.Arnold perturbation method, the second-order approximate solution of the pth moment Lyapunov exponent is calculated, which agree well with the simulation results given by the Monte Carlo method. Finally, the effects of the noise parameters, natural frequency, coefficient of restitution, and damping coefficient on the stochastic stability of the vibro-impact are studied. Due to the existence of impact factor, the natural frequency has a direct and significant effect on the stochastic stability of the system.

Non-stationary stochastic response determination of vibro-impact system under combination harmonic and Gaussian white noise excitations

Vibro-impact system (VI-S) have demonstrated their potential to contribute to engineering practice throughout the last few years. Nonetheless, their scope of application are still very limited. One of a primary reason is that dynamic behaviors of VI-S under the combination of deterministic period excitations (deterministic period excitation) and random noises (random noises) are not entirely understood. Toward this, the radial basis function neural network (rbfnn) is used to analyze dynamical behaviors evolution of VI-S under deterministic period excitation combined with random noises. To illustrate the potential of the proposed scheme, two classical examples of Mathieu–Duffing oscillator and Duffing–Rayleigh oscillator are considered. The evolutionary of dynamics behaviors and stochastic P-bifurcation phenomenon are examined through a qualitative change of the probability distribution, which indicates that system parameters and noise intensity, respectively, can be treated as bifurcation-induced parameters. Besides, numerical results are also carried out to confirm the accuracy of semi-analytical prediction.

Exploit the study of the scenarios for the control of the response of single-degree-of-freedom systems with bumpers

In earthquake-prone areas, seismic isolation systems are designed to preserve structural safety and prevent occupants’ injury and properties’ damage. However, seismically isolated structures can be subject to large displacements relative to the ground under earthquakes, which lead to define two problems, the first problem concerns to excessive displacements, whereas the second problem relates to excessive accelerations that occur when the displacements are limited by an obstacle with high relative stiffness. An arrangement that favors the solution of these problems is an appropriate interposition between the isolated structure and the obstacle of the shock absorbers or bumpers. The aim of this paper is to investigate, via numerical parametric analyses, the possibility to exploit the occurrence of impact in a two-sided vibro-impact isolated single-degree-of-freedom (SDOF) system under harmonic base excitation with beneficial effects. Through the analysis of the response scenarios which can occur varying the obstacles’ parameters, that is the width of the gap and the mechanical (stiffness and damping) properties, the target is to reduce both the maximum value of the displacement and of the acceleration of the mass, compared to the free flight condition, without possibly reducing the vibration isolation frequency range. This research path leads to the definition of vibro-impacting control systems and to the free and constrained single and multi-objective optimal design of the control system consisting of the integrated assembly of the isolation system, bumpers and gaps. The main results obtained consist in having (i) found a unique optimality condition, (ii) calculated the dissipated energy and given an energetic interpretation to the optimality condition (iii) built maps, that allow to advance behavior predictions, provide indications on the design and allow to make evaluations of the performance of the vibro-impacting system as a function of the stiffness and damping parameters and of the width of the gap. The stiffness ratio and the damping ratio, which define the mechanical properties of the bumpers, are such that the dimensionless relaxation time is about 1, (iv) finally the results are compared to multi-objective optimal design. Ultimately, the advantages of this new isolation system are: (1) provide a physical limit switch to the isolation system in order to limit the maximum displacement of the mass; (2) reduce the effects of amplification dynamics in the resonance zone; (3) reduce the static displacement, all without possibly modifying the dynamic response in the isolation zone.

Response analysis of the vibro-impact system under fractional-order joint random excitation

As a kind of good damping material, viscoelastic material is widely used in machinery, civil engineering, and other fields. In this paper, the viscoelasticity of the system is described by fractional differentiation. The dynamic response of a unilateral vibro-impact system with a viscoelastic oscillator under joint random excitation is studied, in which joint random excitation is composed of additive and multiplicative white noise. The fractional-order derivative was calculated based on Caputo’s definition, and the fractional derivative was equivalent to the corresponding linear damping force and linear restoring force. As a result, a new random system without fractional-order terms was obtained. A non-smooth transformation was introduced, which was equivalent to the original system to a new system without a velocity jump. The steady-state probability density functions of fractional-order vibro-impact systems under joint random excitation are solved by using the random average method and non-smooth transformation. In addition, the effects of parameters on the steady-state response of the system are analyzed.

Nonlinear characteristics identification of an impact oscillator with a one-sided elastic constraint

Impacting systems are widely used in many engineering applications, such as self-propelled robots, energy harvesting and percussive drilling, which exhibit rich and complex nonlinear phenomena. Among these applications, predicting nonlinearities and estimating system parameters are of great interest of nonlinear dynamics research community. Backbone curve is an analytical tool that captures the frequency–amplitude dependence of nonlinear systems. In this paper, we estimate the impacting stiffness of a single-degree-of-freedom non-smooth dynamical system qualitatively by using the backbone curve. It was found that an increase of the impacting stiffness may lead to lowering the backbone curve. An adaptive differential evolution algorithm with the Metropolis criterion is proposed to identify the parameters of the impacting system quantitatively using experimental data, which are consistent with our theoretical predictions. Finally, the identified parameters are verified, and the limitations of the backbone curve are drawn. The nonlinear characteristics identification method studied in this paper can be extended to other vibro-impact systems and is potentially applicable to structural health monitoring and robotic sensing.

Dynamics of structures coupled to particle-chain impact dampers: A parametric numerical analysis

Particle impact dampers are dissipative devices that can easily be coupled to structures in order to reduce vibrations over a wide frequency range and in adverse environmental conditions. In this work, a multiple-sphere chain impact damper is coupled to a cantilever beam in order to attenuate its flexural motions. This simple configuration was chosen to better understand the energy dissipation phenomena by the device when the beam is subjected to random excitations. In addition, a parametric study of the coupled system is conducted in order to assert optimal working conditions of the damper. This study is mostly based on a numerical model of the vibro-impacting system, which was calibrated based on an experimental rig. Firstly, the computational modal basis of the bare beam was experimentally identified, and then theoretically modified taking into account the mass of the container. Secondly, calibration of the impact model was achieved by fitting the parameters of the contact formulation with respect to sphere-dropping experiments. Finally, extensive parametric computations are presented and discussed. The vibratory responses are computed using a frequency banded Gaussian random excitation, for several values of the excitation amplitude, for an increasing number of spheres and for several values of the damper clearance. The numerical results concern the r.m.s. motion of the spheres and the beam, the r.m.s. spheres/beam contact forces, the average mechanical and dissipated energies by the beam modes and the damper, as well as the effective beam modal parameters when fitted with the damper, which are essential results for understanding these damping devices. Finally, the numerical results are reduced into relevant dimensionless parameters, which prove reasonably effective in collapsing the dynamical data from such a highly nonlinear system. The present study thus highlights interesting essential features on the dynamics of structures coupled to multiple particle vibro-impact dampers under random excitation.

Моделювання коливальних процесів віброударних систем

In order to improve the efficiency of methods and means of mathematical modeling of vibro-impact systems, a generalized function of the periodic mode of movement of the executive body has been developed. It is presented in the form of the dependence of the shock impulse on the ratio of the angular velocities of the linear conservative system and its own. When obtaining this function, the Heaviside integral jump function and the periodic Green's function were used. The function of the dependence of the oscillation frequency on the impact impulse is determined from the impact conditions for the function of the system's response to a periodic sequence of impulses. The design model of a vibroimpact system is considered, both with one impact element and a motion limiter, and with a double-sided impact pair with alternate impact interactions with the limiters. In the intervals between impacts, there is a linear force interaction. When developing the mathematical model, a stereomechanical impact model was used, which is characterized by the velocity recovery coefficient after the impact. The analysis of the dependence function of the oscillation frequency on the shock impulse made it possible to obtain skeletal diagrams of resonant and quasi-resonant oscillations of vibro-impact systems with one and many degrees of freedom. Based on the obtained phase diagrams of the state of vibro-impact systems, it was determined: in a system with a gap, an increase in the impact speed increases the oscillation frequency, and the vibro-impact nonlinearity is «hard»; in a system with tension, with an increase in the value of the shock impulse, the oscillation frequency decreases (nonlinearity is «soft»). In the absence of a gap, the system is isochronous. Depending on the initial energy reserve and the location of the limiters in an asymmetric oscillatory system, with one degree of freedom, there can be vibro-impact modes with both one (closer located) and both limiters. In a linear conservative system with several degrees of freedom, a single-impact T-periodic regime is realized. If the dissipation during motion and impact is very small, then a regime close to resonant can exist in the system. In this case, periodic oscillations are supported by a weak external periodic force. The developed mathematical model makes it possible to fully describe the process of changing the relative coordinate of the movement of the working body, both in transient and in the established modes of movement of the system.

Numerical steady-state and transient responses of a SDOF system constrained by two optimally designed bumpers

Seismically isolated structures can be subjected to large horizontal displacements relative to the ground, especially in Near-Fault earthquakes, which are characterized by one or more intense pulses of velocity and displacement of long period. One strategy to mitigate the problem of large displacements, which occurs in linearly isolated structures, is the use of deformable and dissipative devices (bumpers). The impact between the structure and the bumpers, if the bumpers are appropriately designed, can produce beneficial effects on the dynamic response of the system, both on displacements and accelerations. In this paper the response obtained from a numerical model of isolated single-degree-of-freedom (SDOF) systems constrained by two bumpers, arranged symmetrically on both sides of the mass of the system with an initial gap, subjected to base harmonic excitation, is studied. This model is called Vibro-Impact Isolation System (V-IIS). The objective of this work is to define a methodology for choosing the design parameter defining the V-IIS (mechanical characteristics of the bumpers, gap and isolation frequency of the system) by observing both steady-state and transient responses of both the system and the bumpers. The study of the transient response is compared with that obtained in the steady-state to assess how representative the latter is of the V-IIS transient response. From the definition of the methodology for choosing the parameters of the V-IIS, through optimal design, the only design parameters are the gap and the isolation frequency of the system. Therefore, an appropriate choice of the gap makes it possible to bring frequency-selective viscous damping in V-IISs, introducing two advantages over linear systems: the reduction of the peak intensity of the responses in the resonance range (both displacement and acceleration) and the reduction of the static displacement of the system, but keeping the dynamic response with which the system is designed unchanged.

Response Analysis of the Three-Degree-of-Freedom Vibroimpact System with an Uncertain Parameter.

The inherent non-smoothness of the vibroimpact system leads to complex behaviors and a strong sensitivity to parameter changes. Unfortunately, uncertainties and errors in system parameters are inevitable in mechanical engineering. Therefore, investigations of dynamical behaviors for vibroimpact systems with stochastic parameters are highly essential. The present study aims to analyze the dynamical characteristics of the three-degree-of-freedom vibroimpact system with an uncertain parameter by means of the Chebyshev polynomial approximation method. Specifically, the vibroimpact system model considered is one with unilateral constraint. Firstly, the three-degree-of-freedom vibroimpact system with an uncertain parameter is transformed into an equivalent deterministic form using the Chebyshev orthogonal approximation. Then, the ensemble means responses of the stochastic vibroimpact system are derived. Numerical simulations are performed to verify the effectiveness of the approximation method. Furthermore, the periodic and chaos motions under different system parameters are investigated, and the bifurcations of the vibroimpact system are analyzed with the Poincaré map. The results demonstrate that both the restitution coefficient and the random factor can induce the appearance of the periodic bifurcation. It is worth noting that the bifurcations fundamentally differ between the stochastic and deterministic systems. The former has a bifurcation interval, while the latter occurs at a critical point.

The synergistic effect of the multiple parameters of vibro-impact nonlinear energy sink

This article studies the dynamics and efficiency of a vibro-impact damper (single-sided vibro-impact nonlinear energy sink—SSVI NES) depending on the exciting force parameters. The damper is coupled with a linear oscillator—the primary structure. It is shown that the damper is quite effective in a wide range of the exciting force amplitude and in the range of its frequency, which are higher than the resonant frequency; damper efficiency in these regions is fairly stable. The dynamics of the vibro-impact system “primary structure—SSVI NES” is rich and complex, which, however, does not impair the damper efficiency. In complex oscillatory regimes, the damper makes bilateral impacts: it hits both an obstacle and directly against the primary structure, which actually turns a single-sided NES into a double‐sided one. The optimization procedure and the choice of optimal damper parameters play a very important role in damper design. Optimizing multiple damper parameters instead of three shows a synergistic effect and provides better results.

Nonlinear dynamics, bifurcations, and multi-stability in a vibro-impact system with geometric and multi-segmented freeplay nonlinearities

Nonlinear dynamics, bifurcations, and multi-stability in a vibro-impact system with geometric and multi-segmented freeplay nonlinearities

Vibro-impact dynamics of an experimental rig with two-sided constraint and bidirectional drift

This paper demonstrates a theoretical and experimental study on a universal vibro-impact system with bidirectional drift. With a dedicated experimental rig that is able to verify numerical simulations, the system has revealed various bifurcations from a piecewise-smooth dynamical model involving impact and dry friction. This experimental rig with a two-sided constraint is designed with adjustable configurations which can be used for experimental testing of versatile piecewise-smooth dynamical systems, e.g., the impact oscillator, the drifting oscillator and the capsule system. It is able to be customised to alter different system parameters, such as friction, number of constraints, mass ratio, impact gap and constraint stiffness, for developing and testing various control strategies. In this paper we particularly studied several types of bifurcations from our preliminary experimental results and discussed the dynamical responses of the system under different excitation frequencies and amplitudes. It was found that with a higher excitation frequency and amplitude, the vibro-impact dynamics on both ends of the constraints could result in a large amount of uncertainties, thus bi-stable and chaotic motions are likely to be observed. While being unstable, the system runs with a relatively high energy efficiency in this configuration.

Stochastic analysis of vibro-impact bistable energy harvester system under colored noise

Stochastic analysis of vibro-impact bistable energy harvester system under colored noise