[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Journal of Applied Nonlinear Dynamics
ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Fractional Order PD Control of Friction-Induced Vibrations in a Continuous System

Journal of Applied Nonlinear Dynamics 10(3) (2021) 413--429 | DOI:10.5890/JAND.2021.09.005

Tejas Kokane, Ashesh Saha

Department of Mechanical Engineering, National Institute of Technology Calicut, Kerala - 673601, India

Download Full Text PDF

 

Abstract

Fractional-order PD (PD$^{\lambda })$ control of friction-induced vibrations in a beam-mass model is analysed in this paper. Mathematical modelling of the system with a piezo-patch actuator bonded to the beam surface is presented. Linear stability analysis is performed to determine the stability boundary corresponding to the Hopf bifurcation points. The nature of bifurcation is found to be supercritical from the non-linear analysis by the method of averaging. The role of fractional-order on the effectiveness of the controller in quenching friction-induced vibrations is thoroughly investigated. The efficacy of the controller with varying size and locations of the piezo-patch is also studied.

References

  1. [1]  Chen, F., Tan, C.A., and Quaglia, R.L. (2006), Disc brake squeal: mechanism, analysis, evaluation, and reduction/prevention, Warrendale, Pennsylvania: SAE International.
  2. [2]  Crowther, A., Zhang, N., Liu, D.K., and Jeyakumaran, J.K. (2004), Analysis and simulation of clutch engagement judder and stick-slip in automotive powertrain systems, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 218(12), 1427-1446.
  3. [3]  Vahid-Araghi, O. and Golnaraghi, F. (2011), Friction-induced vibration in lead screw drives, New York: Dordrecht; Heidelberg: Springer-Verlag.
  4. [4]  Thomsen, J.J. (1999), Using fast vibrations to quench friction-induced oscillations, Journal of Sound and Vibration, 218(5), 1079-1102.
  5. [5]  Andreaus, U. and Casini, P. (2001), Dynamics of friction oscillators excited by a moving base and/or driving force, Journal of Sound and Vibration, 245(4), 685-699.
  6. [6]  Oestreich, M., Hinrichs, N., and Popp, K. (1996), Bifurcation and stability analysis for a non-smooth friction oscillator, Archive of Applied Mechanics, 66, 301-314.
  7. [7]  Hinrichs, N., Oestreich, M., and Popp, K. (1998), On the modeling of friction oscillators, Journal of Sound and Vibration, 216(3), 435-459.
  8. [8]  Hetzler, H., Schwarzer, D., and Seemann, W. (2007), Analytical investigation of steady-state stability and Hopf-bifurcations occurring in sliding friction oscillators with application to low-frequency disc brake noise, Communications in Nonlinear Science and Numerical Simulation, 12(1), 83-99.
  9. [9]  Li, Y. and Feng, Z.C. (2004), Bifurcation and chaos in friction-induced vibration, Communications in Nonlinear Science and Numerical Simulation, 9, 633-647.
  10. [10]  Stelter, P. (1992), Nonlinear vibrations of structures induced by dry friction, Nonlinear Dynamics, 3, 329-345.
  11. [11]  Saha, A., Pandey, S.S., Bhattacharya, B., and Wahi, P. (2011), Analysis and control of friction-induced oscillations in a continuous system, Journal of Vibration and Control, 18(3), 467-480.
  12. [12]  Hoffmann, N., Fischer, M., Allgaier, R., and Gaul, L. (2002), A minimal model for studying properties of the mode-coupling type instability in friction induced oscillations, Mechanics research communications, 29, 197-205.
  13. [13]  Hoffmann, N. and Gaul, L. (2003), Effects of damping on mode coupling instability in friction induced oscillations, ZAMM Zeitschrift fur Angewante Mathematik und Mechanik, 83, 524-534.
  14. [14]  Sinou, J.J. and Jezequel, L. (2007), Mode coupling instability in friction-induced vibrations and its dependency on system parameters including damping, European Journal of Mechanics - A/Solids, 256(1), 106-122.
  15. [15]  Von Wagner, U., Hochlenert, D., and Hagedorn, P. (2007), Minimal models for disc brake squeal, Journal of Sound and Vibration, 302, 527-539.
  16. [16]  Flint, J. and Hulten, J. (2002), Lining-deformation-induced modal coupling as squeal generator in a distributed parameter disc brake model, Journal of Sound and Vibration, 254(1), 1-21.
  17. [17]  Joe, Y.G., Cha, B.G., Sim, H.J., Lee, H.J., and Oh, J.E. (2008), Analysis of disc brake instability due to friction-induced vibration using a distributed parameter model, International Journal of Automotive Technology, 9(2), 161-171.
  18. [18]  Ibrahim, R.A. (1994), Friction-induced vibration, chatter, squeal, and chaos: Part II: dynamics and modeling, ASME Applied Mechanics Review 47(7), 227-253.
  19. [19]  Saha, A. and Wahi, P. (2014), An analytical study of time-delayed control of friction-induced vibrations in a system with a dynamic friction model, International Journal of Non-linear Mechanics, 63, 60-70.
  20. [20]  Dahl, P.R. (1976), Solid friction damping of mechanical vibrations, AIAA Journal, 14(12), 1675-1682.
  21. [21]  Canudas de Wit, C., Olsson, H., {\AA}str\"{o}m, K.J. and Lischinsky, P. (1995), A new model for control of systems with friction, IEEE Transactions on Automatic Control, 40(3), 419--425.
  22. [22]  Lampaert, V., Al-Bender, F., and Swevers, J. (2003), A generalized Maxwell-Slip friction model appropriate for control purpose, Proceedings IEEE International Conference of Physics and Control, 4, 1170 -- 1177.
  23. [23]  Zinjade, P.B. and Mallik, A.K. (2007), Impact damper for controlling friction-driven oscillations, Journal of Vibration and Control, 306, 238-251.
  24. [24]  Heckl, M.A. and Abrahams, I.D. (1996), Active control of friction driven oscillations, Journal of Sound and Vibration, 193(1), 417-426.
  25. [25]  Saha, A. (2018), Time-delayed control of friction-induced vibrations in a continuous system, International Journal of Mechanical and Production Engineering Research and Development, Special issue, June: 111-118.
  26. [26]  Neubauer, M. and Oleskiewicz, R. (2008), Brake squeal control with shunted piezoceramics efficient modelling and experiments, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 222(7), 1141-1151.
  27. [27]  Bhisikar, K.K., Vyawahare, V.A., and Joshi, M.M. (2014), Design of fractional-order PD controller for unstable and integrating systems, Proceeding of the 11$^{th}$ World Congress on Intelligent Control and Automation, Shenyang, China, June 29 - July 4.
  28. [28]  Podlubny, I. (1999), Fractional-order systems and PI$^{\lambda }D^{\mu }$-controllers, IEEE Transactions on Automatic Control, 44(1), 208-214.
  29. [29]  Mondal, R., Dey, J., and Halder, S. (2018), Design of fractional order controller for non-minimum phase unstable plant, International Journal of Mechanical and Production Engineering Research and Development, Special issue, June: 148-155.
  30. [30]  Shen, Y., Yang, S., Xing, H., and Ma, H. (2012), Primary resonance of Duffing oscillator with two kinds of fractional-order derivatives, International Journal of Non-Linear Mechanics, 47(9), 975-983.
  31. [31]  Shen, Y., Wei, P., and Yang, S.P. (2014), Primary resonance of fractional-order van der Pol oscillator, Nonlinear Dynamics 77(4), 1629-1642.
  32. [32]  Kavyanpoor, M. and Shokrollahi, S. (2017), Dynamic behaviors of a fractional order nonlinear oscillator, Journal of King Saud University -- Science, 31(1), 14-20.
  33. [33]  Moheimani, S.O.R. and Fleming, A.J. (2006), Piezoelectric transducers for vibration control and damping (Advances in industrial control), London: Springer-Verlag.
  34. [34]  Das, S. (2011), Functional fractional calculus, Berlin: Springer Science & Business Media.
  35. [35]  Mathai, A.M. and Haubold, H.J. (2017), Fractional and Multivariable Calculus, Cham, Switzerland: Springer.
  36. [36]  Garrappa, R. (2018), Numerical solutions of fractional differential equations: A survey and a software tutorial. Mathematics, 6(2), 16.

Copyright © 2011-2024 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates