Abstract
This chapter introduces and discusses practically important concept of non-smooth dynamical systems, which are very common in engineering applications. Mathematically, such systems can be considered as piecewise smooth and therefore their global solutions are obtained by stitching local solutions, which are easy to develop by standard methods. If a dynamical system is piecewise linear then an implicit global analytical solution can be given, however the times when non-smoothness occurs have to be determined first. This leads to a set of nonlinear algebraic equations. To illustrate the non-smooth dynamical systems and the methodology of solving them, three mechanical engineering problems were studied. Firstly, a vibro-impact system in a form of moling device was modelled and analysed to understand how the progression rates can be maximised. For this system, periodic trajectories can be reconstructed as they go through three linear subspaces (no contact, contact with progression and contact without progression), and using combination of analytical and numerical methods the optimal range of the system parameters can be identified. In the second application the influence of opening and closing of a fatigue crack on the system dynamics was investigated. Specifically, a novel apparatus to induce aperiodic loading to a specimen with a fatigue crack was studied. It was shown experimentally that fatigue life can be reduced few times if the sample is loaded aperiodically. The analysis of the developed mathematical model shown that as a crack grows linearly before reaching its critical value, the response of the system remains periodic. When its size exceeds the critical value, the system behaviour becomes chaotic and then the crack growth increases exponentially. This phenomenon can be used in structural health monitoring. The last problem comes from rotordynamics, where nonlinear interactions between the rotor and the snubber ring were studied. The influence of the preloading of the snubber ring on the system behaviour was investigated and the range of the system parameters where chaotic vibrations occur was identified. The results obtained from the developed mathematical model confronted with the experiments shown a good degree of correlation.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
Abraham, O.N.L., Brandon, J.: A piecewise linear approach for the modelling of a breathing crack. In: 17th International Seminar on Modal Analysis, Leuven, Belgium, pp. 417–431 (1992)
Actis, R.L., Dimarogonas, A.D.: Non-linear effects due to closing cracks in vibration beams. In: 12th ASME Conference on Mechanical Engineering, Vibration and Noise, Montreal, Canada, pp. 99–104 (1989)
Anderson, T.L.: Fracture mechanics - Fundamental and applications. CRC Press, Boca Raton (1994)
Childs, D.W.: Fractional-frequency rotor motion due to nonsymmetric clearance effects. Trans. ASME, Journal of Engineering for Power 104(3), 533–541 (1982)
Choy, F.K., Padovan, J.: Non-linear transient analysis of rotor-casing rub events. Journal of Sound and Vibration 113(3), 529–545 (1987)
Chu, Y.C., Shen, M.-M.H.: Analysis of forced bilinear oscillators and the application to cracked beam dynamics. AIAA Journal 30, 2512–2519 (1992)
Chu, F., Zhang, Z.: Periodic, quasi-periodic and chaotic vibrations of a rub-impact rotor system supported on oil film bearing. International Journal of Engineering Sciences 35, 963–973 (1997)
Chu, F., Zhang, Z.: Bifurcation and chaos in a rub-impact jeffcott rotor system. J. Sound Vibr. 210, 1–18 (1998)
Collins, K.R., Plaut, R.H., Wauer, J.: Detection of cracks in rotating Timoshenko shafts using axial impulses. Journal of Vibration, Acoustics, Stress, and Reliability in Design, Trans. ASME 113, 74–78 (1991)
Collins, K.R., Plaut, R.H., Wauer, J.: Free and forced longitudinal vibrations of a cantilvered bar with a crack. Journal of Vibration, Acoustics, Stress, and Reliability in Design, Trans. ASME 114, 171–177 (1992)
Ehrich, F.F.: Spontaneous sidebanding in high-speed rotordynamics. Trans. ASME, J. Vibr. Acoust. 114, 498–505 (1992)
Ebrahimi, S., Eberhard, P.: Rigid-elastic modeling of meshing gear wheels in multibody systems. Multibody System Dynamics 16(1), 55–71 (2006)
Feeny, B.: A non-smooth Coulomb friction oscillator. Physica D 59, 25–38 (1992)
Filippov, A.F.: Differential equations with discontinuous right-hand side. American Mathematical Society Translations 42(2), 199–231 (1978)
Foong, C.H.: Influence of fatigue crack growth on the dynamics of engineering components and structures, PhD thesis, University of Aberdeen (2004)
Foong, C.H., Wiercigroch, M., Deans, W.F.: Novel dynamic fatigue-testing device: Design and measurements. Measurement Science and Technology 17, 2218–2226 (2006)
Foong, C.H., Jakšić, N., Wiercigroch, M., Boltežar, M.: Parameter identification of the fatigue-testing rig. International Journal of Mechanical Sciences 50, 1142–1152 (2008)
Foong, C.H., Pavlovskaia, E., Wiercigroch, M., Deans, W.F.: Chaos caused by fatigue crack growth. Chaos, Solitons and Fractals 16, 651–659 (2003)
Foong, C.H., Wiercigroch, M., Pavlovskaia, E., Deans, W.F.: Nonlinear vibration caused by fatigue. Journal of Sound and Vibration 303, 58–77 (2007)
Friswell, M.I., Penny, J.E.T.: A Simple Nonlinear Model of a Cracked Beam. In: 10th International Modal Analysis Conference, San Diego, California, USA, pp. 516–521 (1992)
Ganesan, R.: Dynamic response and stability of a rotor-support system with non-symmetric bearing clearances. Mechanism Machine Theory 31, 781–798 (1996)
Gonsalves, H.D., Neilson, R.D., Barr, A.D.S.: A study of the response of a discontinuously nonlinear rotor system. Nonlinear Dynamics 7, 451–470 (1995)
Grabec, I.: Chaotic dynamics of the cutting process. International Journal of Machine Tools and Manufacture 28, 19–32 (1988)
Gudmundson, P.: The dynamic behaviour of slender structures with cross-sectional cracks. Journal of the Mechanics and Physics of Solids 31, 329–345 (1983)
Guinea, G.V., Pastor, J.Y., Elices, M.: Stress intensity factor, compliance and CMOD for a general three-point-bend beam. International Journal of Fracture 89, 103–116 (1998)
Hsu, C.S.: Cell-To-Cell Mapping: A Method of Global Analysis for Nonlinear Systems. Springer, Heidelberg (1987)
Ibrahim, A., Ismail, F., Martin, H.R.: Modelling of the dynamics of a continuous beam including nonlinear fatigue crack. International Journal of Analytical and Experimental Modal Analysis 2, 76–82 (1987)
Ismail, F., Ibrahim, A., Martin, H.R.: Identification of fatigue cracks from vibration testing. Journal of Sound and Vibration 140, 305–317 (1990)
Kahraman, A., Singh, R.: Non-linear dynamics of a spur gear pair. Journal of Sound and Vibration 142(1), 49–75 (1990)
Karpenko, E.V.: Nonlinear dynamics of a Jeffcott rotor with imperfections, PhD Thesis, University of Aberdeen (2003)
Karpenko, E.V., Pavlovskaia, E.E., Wiercigroch, M.: Bifurcation analysis of a preloaded Jeffcott rotor. Chaos, Solitons and Fractals 15, 407–416 (2003)
Karpenko, E.V., Wiercigroch, M., Cartmell, M.P.: Regular and chaotic dynamics of a discontinuously nonlinear rotor system. Chaos, Solitons and Fractals 13, 1231–1242 (2002)
Karpenko, E., Wiercigroch, M., Pavlovskaia, E.E., Cartmell, M.P.: Piecewise approximate solutions for a Jeffcott rotor with a snubber ring. International Journal of Mechanical Sciences 44, 475–488 (2002)
Karpenko, E.V., Wiercigroch, M., Pavlovskaia, E.E., Neilson, R.D.: Experimental verification of Jeffcott rotor model with preloaded snubber ring. Journal of Sound and Vibration 298, 907–917 (2006)
Krivtsov, A.M., Wiercigroch, M.: Dry friction model of percussive drilling. Meccanica 34, 425–435 (1999)
Krivtsov, A.M., Wiercigroch, M.: Penetration rate prediction for percussive drilling via dry friction model. Chaos, Solitons and Fractals 11, 2479–2485 (2000)
Kunze, M.: Non-Smooth Dynamical Systems. Springer, New York (2000)
Litak, G., Friswell, M.I.: Dynamics of a gear system with faults in meshing stiffness. Nonlinear Dynamics 41, 415–421 (2005)
Lok, H.P., Neilson, R.D., Rodger, A.A.: Computer-based model of vibro-impact driving. In: Proceedings of ASME DETC: Symposium on Nonlinear Dynamics in Engineering Systems, Las Vegas (1999)
Muszynska, A., Goldman, P.: Chaotic responses of unbalanced rotor/bearing/stator systems with looseness or rubs. Chaos, Solitons and Fractals 5(9), 1683–1704 (1995)
Natsiavas, S.: Periodic response and stability of oscillators with symmetric trilinear restoring force. Journal of Sound and Vibration 134(2), 313–331 (1989)
Pavlovskaia, E., Wiercigroch, M.: Periodic solutions finder for vibro-impact oscillator with a drift. Journal of Sound and Vibration 267, 893–911 (2003)
Pavlovskaia, E., Wiercigroch, M.: Analytical drift reconstruction for impact oscillator with drift. Chaos, Solitons and Fractals 19(1), 151–161 (2004)
Pavlovskaia, E.E., Wiercigroch, M.: Low dimensional maps for piecewise smooth oscillators. Journal of Sound and Vibration 305(4-5), 750–771 (2007)
Pavlovskaia, E., Karpenko, E.V., Wiercigroch, M.: Nonlinear dynamic interactions of a Jeffcott rotor with a preloaded snubber ring. Journal of Sound and Vibration 276, 361–379 (2004)
Pavlovskaia, E., Wiercigroch, M., Grebogi, C.: Modeling of an impact system with a drift. Physical Review E 64, 56224 (2001)
Pavlovskaia, E., Wiercigroch, M., Grebogi, C.: Two dimensional map for impact oscillator with drift. Physical Review E 70, 36201 (2004)
Pavlovskaia, E.E., Wiercigroch, M., Woo, K.-C., Rodger, A.A.: Modelling of ground moling dynamics by an impact oscillator with a frictional slider. Meccanica 38, 85–97 (2003)
Peterka, F., Vacik, J.: Transition to chaotic motion in mechanical systems with impacts. Journal of Sound and Vibration 154(1), 95–115 (1992)
Rodger, A.A., Littlejohn, G.S.: A study of vibratory driving in granular soils. Geotechnique 30, 269–293 (1980)
Shaw, S.W., Holmes, P.J.: A periodically forced piecewise linear oscillator. Journal of Sound and Vibration 90(1), 129–155 (1983)
Shen, M.-H.H., Chu, Y.C.: Vibrations of beams with a fatigue crack. Computers & Structures 45, 79–93 (1992)
Sin, V.M.T., Wiercigroch, M.: A symmetrically piecewise linear oscillator: design and measurement. Proceeding of the Insitute of Mechanical Engineers Part C 213, 241–249 (1999)
Spektor, M.: Principles of Soil Tool Interaction. Journal of Terramechanics 18, 51–65 (1981)
Wiercigroch, M.: Comments on the study of a harmonically excited linear oscillator with a Coulomb damper. Journal of Sound and Vibration 167, 560–563 (1993)
Wiercigroch, M.: Dynamics of discrete mechanical system with discontinuities, p. 127. Silesian Universtity Press, Gliwice (in Polish)
Wiercigroch, M.: Chaotic vibrations of a simple model of the machine tool-cutting process system. Journal of Vibration and Acoustics Trans. ASME 119(3), 468–475 (1997)
Wiercigroch, M.: Modelling of dynamical systems with motion dependent discontinuities. Chaos, Solitons and Fractals 11(15), 2429–2442 (2000)
Wiercigroch, M.: Applied nonlinear dynamics of non-smooth mechanical systems. Journal of the Brazilian Society of Mechanical Sciences and Engineering XXVIII (4), 519–526 (2006)
Wiercigroch, M., de Kraker, B. (eds.): Applied nonlinear dynamics and chaos of mechanical systems with discontinuities. Nonlinear Science Series A. World Scientific, Singapore (2000)
Wiercigroch, M., Pavlovskaia, E.E.: Nonlinear dynamics of vibro-impact systems: theory and experiments. In: Glasgow, Cartmell, M. (eds.) Proceedings of the 5th International Conference on Modern Practice in Stress and Vibration Analysis, September 9-11, pp. 513–520 (2003)
Wiercigroch, M., Sin, V.M.T.: Experimental study of base excited symmetrically piecewise linear oscillator. Journal of Applied Mechanics, Trans. ASME 65, 657–663 (1998)
Wiercigroch, M., Sin, W.T.V., Li, K.: Measurement of chaotic vibration in symmetrically piecewise linear oscillator. Chaos, Solitons and Fractals 9(1-2), 209–220 (1998)
Wiesenfeld, K., Tufillaro, N.B.: Suppression of period doubling in the dynamics of a bouncing ball. Physica D 26, 321–335 (1987)
Yorke, J.A., Nusse, H.E.: Dynamics. Springer, New York (1998)
Zastrau, B.: Vibration of cracked structures. Archive of Mechanics 37, 736–743 (1985)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Wiercigroch, M., Pavlovskaia, E. (2012). Engineering Applications of Non-smooth Dynamics. In: Warminski, J., Lenci, S., Cartmell, M.P., Rega, G., Wiercigroch, M. (eds) Nonlinear Dynamic Phenomena in Mechanics. Solid Mechanics and Its Applications, vol 181. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2473-0_5
Download citation
DOI: https://doi.org/10.1007/978-94-007-2473-0_5
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-2472-3
Online ISBN: 978-94-007-2473-0
eBook Packages: EngineeringEngineering (R0)