Abstract
In this paper, ongoing studies to solve nonlinear differential equations are extended by combining the Newmark-beta integration method and the piecewise linearization approach. The discussed method is illustrated with a practical example. In doing so, the coupled nonlinear differential equations of an impact oscillator, which incorporates the Hertzian contact, are derived. To investigate this problem, an object-oriented computer code, based on the presented method, is written in MATLAB. Furthermore, the discussed problem is solved numerically using the Runge–Kutta commercial code. To verify the calculated results, the contact durations, which are obtained using the discussed methods, are compared with the previous analytical results. In this study, accuracy of solution and the process time (cost) are selected as two main parameters of the solution method. The so-called adequacy factor is presented to combine the two main parameters of solution. Finally, it is shown that in the case of Hertzian contact, the presented method can be more adequate than the Runge–Kutta method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Kerschen G., Worden K., Vakakis A.F., Golinval J.: Past, present and future of nonlinear system identification in structural dynamics. Mech. Syst. Signal Process. 20, 505–592 (2006)
Clough R.W., Penzien J.: Dynamics of Structures. 2nd edn. Computers and Structures, Berkeley (2010)
Stronge W.J.: Impact Mechanics. Cambridge University Press, Cambridge (2000)
Goldsmith W.: Impact: The Theory and Physical Behaviour of Colliding Solids. Dover Publications, Mineola (2002)
Pedersen S.L.: Model of contact between rollers and sprockets in chain-drive systems. Arch. Appl. Mech. 74(7), 489–508 (2005). doi:10.1007/s00419-004-0363-4
Ziegler P., Eberhard P., Schweizer B.: Simulation of impacts in geartrains using different approaches. Arch. Appl. Mech. 76(9), 537–548 (2006). doi:10.1007/s00419-006-0060-6
Dimentberg M.F., Iourtchenko D.V.: Random vibrations with impacts: a review. Nonlinear Dyn. 36(2), 229–254 (2004). doi:10.1023/B:NODY.0000045510.93602.ca
Zhang D.-G., Angeles J.: Impact dynamics of flexible-joint robots. Comput. Struct. 83(1), 25–33 (2005)
Blazejczyk-Okolewska B.: Analysis of an impact damper of vibrations. Chaos Solitons Fractals 12(11), 1983–1988 (2001)
Cheng C.C., Cheng C.C., Cheng C.C.: Free vibration analysis of a resilient impact damper. Int. J. Mech. Sci. 45(4), 589–604 (2003)
Ibrahim R.A.: Vibro-Impact Dynamics : Modeling, Mapping and Applications. Lecture Notes in Applied and Computational Mechanics, vol. 43. Springer, Berlin (2009)
Popplewell N., Liao M.: A simple design procedure for optimum impact dampers. J. Sound Vib. 146(3), 519–526 (1991)
Zinjade P.B., Mallik A.K.: Impact damper for controlling friction-driven oscillations. J. Sound Vib. 306(1–2), 238– 251 (2007)
Son L., Hara S., Yamada K., Matsuhisa H.: Experiment of shock vibration control using active momentum exchange impact damper. J. Vib. Control 16(1), 49–64 (2010). doi:10.1177/1077546309102675
Bapat C.N., Sankar S.: Single unit impact damper in free and forced vibration. J. Sound Vib. 99(1), 85–94 (1985)
Cheng J., Xu H.: Inner mass impact damper for attenuating structure vibration. Int. J. Solids Struct. 43(17), 5355–5369 (2006)
Wouw N.V.D., Bosch H.L.A.V.D., Kraker A.D., Campen D.H.V.: Experimental and numerical analysis of nonlinear phenomena in a stochastically excited beam system with impact. Chaos Solitons Fractals 9(8), 1409–1428 (1998). doi:10.1016/s0960-0779(98)00073-3
Palm W.J.: Mechanical Vibration. Wiley, Hoboken (2007)
Rao S.S.: Vibration of Continuous Systems. Wiley, Hoboken (2007)
Ba’zant Z.P., Cedolin L.: Stability of Structures : Elastic, Inelastic, Fracture and Damage Theories World Scientific edn. World Scientific Pub, Hackensack (2010)
Simitses G.J., Hodges D.H.: Fundamentals of Structural Stability. Elsevier/Butterworth-Heinemann, Amsterdam (2006)
Thomson W.T.: Theory of Vibration with Applications, 4th edn. Prentice Hall, Englewood Cliffs (1993)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Afsharfard, A., Farshidianfar, A. An efficient method to solve the strongly coupled nonlinear differential equations of impact dampers. Arch Appl Mech 82, 977–984 (2012). https://doi.org/10.1007/s00419-011-0605-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-011-0605-1