Coupled systems mechanics

  • 제1권3호
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  • Pages.235-245
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  • 2012
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  • 2234-2184(pISSN)
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  • 2234-2192(eISSN)
테크노프레스 (Techno-Press)
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DOI QR코드

DOI QR Code

Internal resonance and nonlinear response of an axially moving beam: two numerical techniques

  • Ghayesh, Mergen H. (Department of Mechanical Engineering, McGill University) ;
  • Amabili, Marco (Department of Mechanical Engineering, McGill University)
  • 투고 : 2012.06.21
  • 심사 : 2012.09.08
  • 발행 : 2012.09.25
⟨ 이전 논문 다음 논문 ⟩

초록

The nonlinear resonant response of an axially moving beam is investigated in this paper via two different numerical techniques: the pseudo-arclength continuation technique and direct time integration. In particular, the response is examined for the system in the neighborhood of a three-to-one internal resonance between the first two modes as well as for the case where it is not. The equation of motion is reduced into a set of nonlinear ordinary differential equation via the Galerkin technique. This set is solved using the pseudo-arclength continuation technique and the results are confirmed through use of direct time integration. Vibration characteristics of the system are presented in the form of frequency-response curves, time histories, phase-plane diagrams, and fast Fourier transforms (FFTs).

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참고문헌

  1. Amabili, M. (2010), "Geometrically nonlinear vibrations of rectangular plates carrying a concentrated mass", J. Sound Vib., 329(21), 4501-4514. https://doi.org/10.1016/j.jsv.2010.04.024
  2. Amabili, M. and Carra, S. (2012), "Experiments and simulations for large-amplitude vibrations of rectangular plates carrying concentrated masses", J. Sound Vib., 331(1), 155-166. https://doi.org/10.1016/j.jsv.2011.08.008
  3. Amabili, M., Pellegrini, M., Righi, F. and Vinci, F. (2006), "Effect of concentrated masses with rotary inertia on vibrations of rectangular plates", J. Sound Vib., 295(1-2), 1-12. https://doi.org/10.1016/j.jsv.2005.11.035
  4. Burak Özhan, B. and Pakdemirli, M. (2010), "A general solution procedure for the forced vibrations of a system with cubic nonlinearities: Three-to-one internal resonances with external excitation", J. Sound Vib., 329(13), 2603-2615. https://doi.org/10.1016/j.jsv.2010.01.010
  5. Chen, L.Q. (2005), "Analysis and control of transverse vibrations of axially moving strings", Appl. Mech. Rev., 58(2), 91-116. https://doi.org/10.1115/1.1849169
  6. Chen, L.Q. and Chen, H. (2010), "Asymptotic analysis on nonlinear vibration of axially accelerating viscoelastic strings with the standard linear solid model", J. Eng. Math., 67(3), 205-218. https://doi.org/10.1007/s10665-009-9316-9
  7. Chen, L.Q. and Ding, H. (2010), "Steady-State transverse response in coupled planar vibration of axially moving viscoelastic beams", J. Vib. Acoust., 132(1), 011009. https://doi.org/10.1115/1.4000468
  8. Darabi, M.A., Kazemirad, S. and Ghayesh, M.H. (2012), "Free vibrations of beam-mass-spring systems: analytical analysis with numerical confirmation", Acta Mech. Sinica, 28 (2), 468-481. https://doi.org/10.1007/s10409-012-0010-1
  9. Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Sandstede, B. and Wang, X. (1998), AUTO 97: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont), Concordia University, Montreal, Canada.
  10. Ghayesh, M.H. (2011), "On the natural frequencies, complex mode functions, and critical speeds of axially traveling laminated beams: Parametric study", Acta Mech. Solida Sin., 24(4), 373-382. https://doi.org/10.1016/S0894-9166(11)60038-4
  11. Ghayesh, M.H. (2008), "Nonlinear transversal vibration and stability of an axially moving viscoelastic string supported by a partial viscoelastic guide", J. Sound Vib., 314(3-5), 757-774. https://doi.org/10.1016/j.jsv.2008.01.030
  12. Ghayesh, M.H. (2009), "Stability characteristics of an axially accelerating string supported by an elastic foundation", Mech. Mach. Theory, 44(10), 1964-1979. https://doi.org/10.1016/j.mechmachtheory.2009.05.004
  13. Ghayesh, M.H. (2010), "Parametric vibrations and stability of an axially accelerating string guided by a nonlinear elastic foundation", Int. J. Nonlinear Mech., 45(4), 382-394. https://doi.org/10.1016/j.ijnonlinmec.2009.12.011
  14. Ghayesh, M.H. (2011a), "Nonlinear forced dynamics of an axially moving viscoelastic beam with an internal resonance", Int. J. Mech. Sci., 53(11), 1022-1037. https://doi.org/10.1016/j.ijmecsci.2011.08.010
  15. Ghayesh, M.H. (2011b), "On the natural frequencies, complex mode functions, and critical speeds of axially traveling laminated beams: parametric study", Acta Mech. Solida Sin., 24(4), 373-382. https://doi.org/10.1016/S0894-9166(11)60038-4
  16. Ghayesh, M.H. (2011c), "Parametrically excited viscoelastic beam-spring systems: Nonlinear dynamics and stability", Struct. Eng. Mech., 40(5), 705-718. https://doi.org/10.12989/sem.2011.40.5.705
  17. Ghayesh, M.H. (2012a), "Coupled longitudinal-transverse dynamics of an axially accelerating beam", J. Sound Vib., 331(23), 5107-5124. https://doi.org/10.1016/j.jsv.2012.06.018
  18. Ghayesh, M.H. (2012b), "Nonlinear dynamic response of a simply-supported Kelvin-Voigt viscoelastic beam, additionally supported by a nonlinear spring", Nonlinear Anal. Real., 13(3), 1319-1333. https://doi.org/10.1016/j.nonrwa.2011.10.009
  19. Ghayesh, M.H. (2012c), "Stability and bifurcations of an axially moving beam with an intermediate spring support", Nonlinear Dynam., 69(1-2), 193-210. https://doi.org/10.1007/s11071-011-0257-2
  20. Ghayesh, M.H. (2012d), "Subharmonic dynamics of an axially accelerating beam", Arch. Appl. Mech., in press.
  21. Ghayesh, M.H., Alijani, F. and Darabi, M.A. (2011), "An analytical solution for nonlinear dynamics of a viscoelastic beam-heavy mass system", J. Mech. Sci. Technol., 25(8), 1915-1923. https://doi.org/10.1007/s12206-011-0519-4
  22. Ghayesh, M.H. and Amabili, M. (2012), "Steady-state transverse response of an axially moving beam with time dependent axial speed", Int. J. Nonlinear Mech., (in press).
  23. Ghayesh, M.H., Amabili, M. and Païdoussis, M.P. (2012), "Nonlinear vibrations and stability of an axially moving beam with an intermediate spring-support: two-dimensional analysis", Nonlinear Dynam., in press.
  24. Ghayesh, M.H. and Balar, S. (2010), "Non-linear parametric vibration and stability analysis for two dynamic models of axially moving Timoshenko beams", Appl. Math. Model., 34(10), 2850-2859. https://doi.org/10.1016/j.apm.2009.12.019
  25. Ghayesh, M.H., Kafiabad, H.A. and Reid, T. (2012), "Sub- and super-critical nonlinear dynamics of a harmonically excited axially moving beam", Int. J. Solids Struct., 49(1), 227-243. https://doi.org/10.1016/j.ijsolstr.2011.10.007
  26. Ghayesh, M.H., Kazemirad, S. and Amabili, M. (2012), "Coupled longitudinal-transverse dynamics of an axially moving beam with an internal resonance", Mech. Mach. Theory, 52, 18-34. https://doi.org/10.1016/j.mechmachtheory.2012.01.008
  27. Ghayesh, M.H., Kazemirad, S. and Darabi, M.A. (2011), "A general solution procedure for vibrations of systems with cubic nonlinearities and nonlinear/time-dependent internal boundary conditions", J. Sound Vib., 330(22), 5382-5400. https://doi.org/10.1016/j.jsv.2011.06.001
  28. Ghayesh, M.H., Kazemirad, S., Darabi, M.A. and Woo, P. (2012), "Thermo-mechanical nonlinear vibration analysis of a spring-mass-beam system", Arch. Appl. Mech., 82(3), 317-331. https://doi.org/10.1007/s00419-011-0558-4
  29. Ghayesh, M.H., Kazemirad, S. and Reid, T. (2012), "Nonlinear vibrations and stability of parametrically exited systems with cubic nonlinearities and internal boundary conditions: A general solution procedure", Appl. Math. Model., 36, 3299-3311. https://doi.org/10.1016/j.apm.2011.09.084
  30. Ghayesh, M.H. and Khadem, S.E. (2007), "Non-linear vibration and stability analysis of a partially supported conveyor belt by a distributed viscoelastic foundation", Struct. Eng. Mech., 27(1), 17-32. https://doi.org/10.12989/sem.2007.27.1.017
  31. Ghayesh, M.H. and Moradian, N. (2011), "Nonlinear dynamic response of axially moving, stretched viscoelastic strings", Arch. Appl. Mech., 81(6), 781-799. https://doi.org/10.1007/s00419-010-0446-3
  32. Ghayesh, M.H., Païdoussis, M.P. and Amabili, M. (2012), "Subcritical parametric response of an axially accelerating beam", Thin Wall. Struct., 60, 185-193. https://doi.org/10.1016/j.tws.2012.06.012
  33. Ghayesh, M.H., Yourdkhani, M., Balar, S. and Reid, T. (2010), "Vibrations and stability of axially traveling laminated beams", Appl. Math. Comput., 217(2), 545-556. https://doi.org/10.1016/j.amc.2010.05.088
  34. Holmes, P.J. (1978), "Pipes supported at both ends cannot flutter", J. Appl. Mech., 45(3), 619-622. https://doi.org/10.1115/1.3424371
  35. Huang, J.L., Su, R.K.L., Li, W.H. and Chen, S.H. (2011), "Stability and bifurcation of an axially moving beam tuned to three-to-one internal resonances", J. Sound Vib., 330(3), 471-485. https://doi.org/10.1016/j.jsv.2010.04.037
  36. Kazemirad, S., Ghayesh, M.H. and Amabili, M. (2012), "Thermo-mechanical nonlinear dynamics of a buckled axially moving beam", Arch. Appl. Mech., in press.
  37. Marynowski, K. and Kapitaniak, T. (2002), "Kelvin-Voigt versus Bürgers internal damping in modeling of axially moving viscoelastic web", Int. J. Nonlinear Mech., 37(7), 1147-1161. https://doi.org/10.1016/S0020-7462(01)00142-1
  38. Marynowski, K. and Kapitaniak, T. (2007), "Zener internal damping in modelling of axially moving viscoelastic beam with time-dependent tension", Int. J. Nonlinear Mech., 42(1), 118-131. https://doi.org/10.1016/j.ijnonlinmec.2006.09.006
  39. Naguleswaran, S. and Williams, C.J.H. (1968), "Lateral vibration of band-saw blades, pulley belts and the like", Int. J. Mech. Sci., 10(4), 239-250. https://doi.org/10.1016/0020-7403(68)90009-X
  40. Nguyen, Q.C. and Hong, K.S. (2011), "Stabilization of an axially moving web via regulation of axial velocity", J. Sound Vib., 330(20), 4676-4688. https://doi.org/10.1016/j.jsv.2011.04.029
  41. Oz, H.R., Pakdemirli, M. and Boyaci, H. (2001), "Non-linear vibrations and stability of an axially moving beam with time-dependent velocity", Int. J. Nonlinear Mech., 36(1), 107-115. https://doi.org/10.1016/S0020-7462(99)00090-6
  42. Pakdemirli, M. and BoyacI, H. (2003), "Non-linear vibrations of a simple-simple beam with a non-ideal support in between", J. Sound Vib., 268(2), 331-341. https://doi.org/10.1016/S0022-460X(03)00363-8
  43. Pakdemirli, M. and Ozkaya, E. (1998), "Approximate boundary layer solution of a moving beam problem", Math. Comput. Appl., 3, 93-100.
  44. Pakdemirli, M. and Ulsoy, A.G. (1997), "Stability analysis of an axially accelerating string", J. Sound Vib., 203(5), 815-832. https://doi.org/10.1006/jsvi.1996.0935
  45. Pakdemirli, M., Ulsoy, A.G. and Ceranoglu, A. (1994), "Transverse vibration of an axially accelerating string", J. Sound Vib., 169(2), 179-196. https://doi.org/10.1006/jsvi.1994.1012
  46. Pellicano, F. and Vestroni, F. (2002), "Complex dynamics of high-speed axially moving systems", J. Sound Vib., 258(1), 31-44. https://doi.org/10.1006/jsvi.2002.5070
  47. Shih, L.Y. (1971), "Three-dimensional non-linear vibration of a traveling string", Int. J. Nonlinear. Mech., 6(4), 427-434. https://doi.org/10.1016/0020-7462(71)90041-2
  48. Simpson, A. (1973), "Transverse modes and frequencies of beams translating between fixed end supports", J. Mech. Eng. Sci,, 15, 159-164. https://doi.org/10.1243/JMES_JOUR_1973_015_031_02
  49. Stylianou, M. and Tabarrok, B. (1994), "Finite element analysis of an axially moving beam, Part I: time integration", J. Sound Vib., 178(4), 433-453. https://doi.org/10.1006/jsvi.1994.1497
  50. Suweken, G. and Van Horssen, W.T. (2003), "On the weakly nonlinear, transversal vibrations of a conveyor belt with a low and time-varying velocity", Nonlinear Dynam., 31(2), 197-223. https://doi.org/10.1023/A:1022053131286
  51. Tang, Y.Q., Chen, L.Q. and Yang, X.D. (2009), "Parametric resonance of axially moving Timoshenko beams with time-dependent speed", Nonlinear Dynam., 58(4), 715-724. https://doi.org/10.1007/s11071-009-9512-1
  52. Thurman, A.L. and Mote, C.D.J. (1969), "Free, periodic, nonlinear oscillation of an axially moving strip", J. Appl. Mech. - ASME, 36(1), 83-91. https://doi.org/10.1115/1.3564591
  53. Wickert, J.A. and Mote, C.D., Jr. (1988), "Current research on the vibration and stability of moving materials", Shock Vib., 20(5), 3-13. https://doi.org/10.1177/058310248802000503
  54. Yang, X.D., Zhang, W., Chen, L.Q. and Yao, M.H. (2012), "Dynamical analysis of axially moving plate by finite difference method", Nonlinear Dynam., 67(2), 997-1006. https://doi.org/10.1007/s11071-011-0042-2

피인용 문헌

  1. Coupled global dynamics of an axially moving viscoelastic beam vol.51, 2013, https://doi.org/10.1016/j.ijnonlinmec.2012.12.008