In this work, a novel class of bilateral vibro-impact model is proposed. Unlike the traditional impact model, the energy dissipation of the novel bilateral system is measured by the model of the restitution coefficient dependent with velocity. Then the random vibration of the system is investigated in the presence of Gaussian white noise excitations. The motions of the unperturbed impact system are firstly considered and grouped into two categories. Then, the mean drift and diffusion coefficients of the two kinds of motion are calculated with the stochastic averaging methodology for energy envelope under the assumed condition that the impact vibration system is quasi-conservative. Subsequently, the probability density functions of stationary responses are computed with solving the averaged Fokker-Plank-Kolmogorov equation. Finally, two illustrations are chosen to demonstrate the reliability of the presented technique. And, the validation of analytical results is verified by the simulation data generated by Monte Carlo.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
The code that support the findings of this study are available from the corresponding author upon reasonable request.
Dimentberg MF, Iourtchenko DV (1999) Towards incorporating impact losses into random vibration analyses: a model problem. Probabilistic Eng Mech 14:323–328
Ibrahim RA (2009) Vibro-impact dynamics: modeling, mapping and applications. Springer, Berlin
Kecik K, Brzeski P, Perlikowski P (2016) Non-linear dynamics and optimization of a harvester-absorber system. Int J Struct Stab Dyn 17:1740001
Liu WB, Dai HL et al (2017) Suppressing wind-induced oscillations of prismatic structures by dynamic vibration absorbers. Int J Struct Stab Dyn 17:1750056
Pratiknyo YB, Setiawan R, Suweca IW (2019) Experimental and theoretical investigation of combined expansion tube-axial splitting as impact energy absorbers. Int J Struct Stab Dyn 20:2050021
Zhuravlev VF (1976) A method for analyzing vibration-impact systems by means of special functions. Mech Solids 11:23–27
Dimentberg M, Menyailov A (1979) Response of a single-mass vibroimpact system to white-noise random excitation. ZAMM - J Appl Math Mech 59:709–716
Ivanov AP (1994) Impact oscillations: linear theory of stability and bifurcations. J Sound Vib 178:361–378
Xu W, Li C, Yue X, Rong H (2016) Stochastic responses of a vibro-impact system with additive and multiplicative colored noise excitations. Int J Dyn Control 4:393–399
Dimentberg MF, Iourtchenko DV (2004) Random vibrations with impacts: a review. Nonlinear Dyn 36:229–254
Er G-K (1998) An improved closure method for analysis of nonlinear stochastic systems. Nonlinear Dyn 17:285–297
Zhu HT (2015) Stochastic response of a vibro-impact Duffing system under external Poisson impulses. Nonlinear Dyn 82:1001–1013
Zhu HT (2015) Stochastic response of a parametrically excited vibro-impact system with a nonzero offset constraint. Int J Dyn Control 4:180–194
Chen L, Qian J, Zhu H, Sun J (2019) The closed-form stationary probability distribution of the stochastically excited vibro-impact oscillators. J Sound Vib 439:260–270
Yurchenko D, Song L (2006) Numerical investigation of a response probability density function of stochastic vibroimpact systems with inelastic impacts. Int J Non Linear Mech 41:447–455
Wang L, Ma S, Sun C et al (2018) The stochastic response of a class of impact systems calculated by a new strategy based on generalized cell mapping method. J Appl Mech 85:54502
Kecskemethy A, Lüder J (1996) Rigid and elastic approaches for the modeling of collisions with friction in multibody systems. Zeitschrift für Angew Math und Mech 76:243–244
Wang Y, Mason MT (1992) Two-Dimensional Rigid-Body Collisions With Friction. J Appl Mech 59:635
Johnson KL (1985) Contact Mechanics. In: Cambridge, UK: Cambridge University Press
Stronge WJ (1995) Theoretical coefficient of restitution for planar impact of rough elasto-plastic bodies. Am Soc Mech Eng Appl Mech Div AMD 205:351–362
Thornton C (1997) Coefficient of restitution for collinear collisions of elastic-perfectly plastic spheres. J Appl Mech Asme 64:383–386
Mesarovic SD, Fleck NA (1999) Spherical indentation of elastic-plastic solids. Proc Math Phys Eng ences 455:2707–2728
Vu-Quoc L, Zhang X (2000) A normal force-displacement model for contacting spheres accounting for plastic deformation: force-driven formulation. J Appl Mech 67:363–371
Kogut L, Etsion I (2002) Elastic-plastic contact analysis of a sphere and a rigid flat. J Appl Mech 69:657–662
Zhang X, Vu-Quoc L (2002) Modeling the dependence of the coefficient of restitution on the impact velocity in elasto-plastic collisions. Int J Impact Eng 27:317–341
Weir G, Mcgavin P (2008) The coefficient of restitution for the idealized impact of a spherical, nano-scale particle on a rigid plane. Proc R Soc A Math 464:1364–5021
Etsion I, Kligerman Y, Kadin Y (2005) Unloading of an elastic–plastic loaded spherical contact. Int J Solids Struct 42:3716–3729
Kharaz AH, Gorham DA (2000) A study of the restitution coefficient in elastic-plastic impact. Philos Mag Lett 80:549–559
Minamoto H, Kawamura S (2011) Moderately high speed impact of two identical spheres. Int J Impact Eng 38:123–129
Ma D, Liu C (2015) Contact law and coefficient of restitution in elastoplastic spheres. J Appl Mech 82:121006–121015
Liu D, Li J, Meng Y (2019) Probabilistic response analysis for a class of nonlinear vibro-impact oscillator with bilateral constraints under colored noise excitation. Chaos Solitons Fractals 122:179–188
Kumar P, Narayanan S, Gupta S (2016) Bifurcation analysis of a stochastically excited vibro-impact Duffing-Van der Pol oscillator with bilateral rigid barriers. Int J Mech ences 127:103–117
Yang G, Xu W, Gu X, Huang D (2016) Response analysis for a vibroimpact Duffing system with bilateral barriers under external and parametric Gaussian white noises. Chaos Solitons Fractals 87:125–135
Chen L, Zhu H, Sun JQ (2019) Novel method for random vibration analysis of single-degree-of-freedom vibroimpact systems with bilateral barriers. Appl Math Mech 40:1759–1776
Su M, Xu W, Zhang Y, Yang G (2021) Response of a vibro-impact energy harvesting system with bilateral rigid stoppers under Gaussian white noise. Appl Math Model 89:991–1003
Qian J, Chen L (2021) Random vibration of SDOF vibro-impact oscillators with restitution factor related to velocity under wide-band noise excitations. Mech Syst Signal Process 147:107082
Zhu W, Lin YK (1905) stochastic averaging of energy envelope. J Eng Mech 117:1890–1905
The National Natural Science Foundation of China (No. 11672111, No. 12072118), the Program for New Century Excellent Talents in Fujian Province University, the Natural Science Foundation of Fujian Province of China (No. 2019J01049).
Conflicts of interest
The authors declare that they have no conflict of interest.
Availability of data and material
The data and material that support the findings of this study are available from the corresponding author upon reasonable request.
About this article
Cite this article
Qian, J., Chen, L. & Liu, S. A new type of bilateral vibro-impact model: random vibration analysis. Int. J. Dynam. Control (2021). https://doi.org/10.1007/s40435-021-00759-7
- Bilateral vibro-impact systems
- Velocity-dependent restitution coefficient
- Stochastic averaging
- Random vibration