Coupled systems mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Nonlinear stability and bifurcations of an axially accelerating beam with an intermediate spring-support

  • Ghayesh, Mergen H. (Department of Mechanical Engineering, McGill University) ;
  • Amabili, Marco (Department of Mechanical Engineering, McGill University)
  • Received : 2013.05.12
  • Accepted : 2013.07.11
  • Published : 2013.06.25
⟨ Previous Next ⟩

Abstract

The present work aims at investigating the nonlinear dynamics, bifurcations, and stability of an axially accelerating beam with an intermediate spring-support. The problem of a parametrically excited system is addressed for the gyroscopic system. A geometric nonlinearity due to mid-plane stretching is considered and Hamilton's principle is employed to derive the nonlinear equation of motion. The equation is then reduced into a set of nonlinear ordinary differential equations with coupled terms via Galerkin's method. For the system in the sub-critical speed regime, the pseudo-arclength continuation technique is employed to plot the frequency-response curves. The results are presented for the system with and without a three-to-one internal resonance between the first two transverse modes. Also, the global dynamics of the system is investigated using direct time integration of the discretized equations. The mean axial speed and the amplitude of speed variations are varied as the bifurcation parameters and the bifurcation diagrams of Poincare maps are constructed.

Keywords

References

  1. Ahmadian, M., Nasrabadi, V. and Mohammadi, H. (2010), "Nonlinear transversal vibration of an axially moving viscoelastic string on a viscoelastic guide subjected to mono-frequency excitation", Acta Mech., 214(3), 357-373. https://doi.org/10.1007/s00707-009-0277-x
  2. Bayat, M., Pakar, I. and Bayat, M. (2013), "On the large amplitude free vibrations of axially loaded Euler-Bernoulli beams", Steel Compos. Struct., 14(1), 73-83. https://doi.org/10.12989/scs.2013.14.1.073
  3. Chen, L.Q., Ding, H. and Lim, C.W. (2012), "Principal parametric resonance of axially accelerating viscoelastic beams: multi-scale analysis and differential quadrature verification", Shock Vib., 19(4), 527-543. https://doi.org/10.1155/2012/948459
  4. Chen, L.Q. and Yang, X.D. (2005), "Steady-state response of axially moving viscoelastic beams with pulsating speed: comparison of two nonlinear models", Int. J. Solids Struct., 42(1), 37-50. https://doi.org/10.1016/j.ijsolstr.2004.07.003
  5. Ding, H. and Chen, L.Q. (2010), "Galerkin methods for natural frequencies of high-speed axially moving beams", J. Sound Vib., 329(17), 3484-3494. https://doi.org/10.1016/j.jsv.2010.03.005
  6. Ding, H. and Chen, L. (2009), "Nonlinear dynamics of axially accelerating viscoelastic beams based on differential quadrature", Acta Mech. Solida Sinica, 22(3), 267-275. https://doi.org/10.1016/S0894-9166(09)60274-3
  7. 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.
  8. Ghayesh, M. (2012), "Stability and bifurcations of an axially moving beam with an intermediate spring support", Nonlinear Dynam., 69(1), 193-210. https://doi.org/10.1007/s11071-011-0257-2
  9. Ghayesh, M., Alijani, F. and Darabi, M. (2011), "An analytical solution for nonlinear dynamics of a viscoelastic beam-heavy mass system", J. Mech. Sci. Tech., 25(8), 1915-1923. https://doi.org/10.1007/s12206-011-0519-4
  10. 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
  11. 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
  12. Ghayesh, M.H. (2010), "Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation", Int. J. Nonlinear Mech., 45(4), 382-394. https://doi.org/10.1016/j.ijnonlinmec.2009.12.011
  13. Ghayesh, M.H. (2011), "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
  14. Ghayesh, M.H. (2011), "On the natural frequencies, complex mode functions, and critical speeds of axially traveling laminated beams: parametric study", Acta Mech. Solida Sinica, 24(4), 373-382. https://doi.org/10.1016/S0894-9166(11)60038-4
  15. Ghayesh, M.H., Amabili, M. and Farokhi, H. (2013), "Coupled global dynamics of an axially moving viscoelastic beam", Int. J. Nonlinear Mech., 51(0), 54-74. https://doi.org/10.1016/j.ijnonlinmec.2012.12.008
  16. Ghayesh, M.H. and Balar, S. (2008), "Non-linear parametric vibration and stability of axially moving visco-elastic Rayleigh beams", Int. J. Solids Struct., 45(25-26), 6451-6467. https://doi.org/10.1016/j.ijsolstr.2008.08.002
  17. 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
  18. 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(0), 18-34. https://doi.org/10.1016/j.mechmachtheory.2012.01.008
  19. 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
  20. Ghayesh, M.H. and Paidoussis, M.P. (2010), "Three-dimensional dynamics of a cantilevered pipe conveying fluid, additionally supported by an intermediate spring array", Int. J. Nonlinear Mech., 45(5), 507-524. https://doi.org/10.1016/j.ijnonlinmec.2010.02.001
  21. Ghayesh, M.H., Paidoussis, M.P. and Modarres-Sadeghi, Y. (2011), "Three-dimensional dynamics of a fluid-conveying cantilevered pipe fitted with an additional spring-support and an end-mass", J. Sound Vib., 330(12), 2869-2899. https://doi.org/10.1016/j.jsv.2010.12.023
  22. 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
  23. 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
  24. Kural, S. and O zkaya, E. (2012), "Vibrations of an axially accelerating, multiple supported flexible beam", Struct. Eng. Mech., 44(4).
  25. Marynowski, K. (2004), "Non-linear vibrations of an axially moving viscoelastic web with time-dependent tension", Chaos Soliton. Fract., 21(2), 481-490. https://doi.org/10.1016/j.chaos.2003.12.020
  26. Marynowski, K. and Kapitaniak, T. (2002), "Kelvin-Voigt versus Burgers 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
  27. Movahedian, B. (2012), "Dynamic stiffness matrix method for axially moving micro-beam", Int. Multiscale Mech., 5(4).
  28. O z, H.R. and Pakdemirli, M. (1999), "Vibrations of an axially moving beam with time-dependent velocity", J. Sound Vib., 227(2), 239-257. https://doi.org/10.1006/jsvi.1999.2247
  29. Oz, H.R., Pakdemirli, M. and Boyacı, 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
  30. O z, H.R., Pakdemirli, M. and O zkaya, E. (1998), "Transition behaviour from string to beam for an axially accelerating material", J. Sound Vib., 215(3), 571-576. https://doi.org/10.1006/jsvi.1998.1572
  31. O zkaya, E. and Pakdemirli, M. (2000), "Vibrations of an axially accelerating beam with small flexural stiffness", J. Sound Vib., 234(3), 521-535. https://doi.org/10.1006/jsvi.2000.2890
  32. Pakdemirli, M. and O z, H.R. (2008), "Infinite mode analysis and truncation to resonant modes of axially accelerated beam vibrations", J. Sound Vib., 311(3-5), 1052-1074. https://doi.org/10.1016/j.jsv.2007.10.003
  33. 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
  34. 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
  35. Pellicano, F. and Vestroni, F. (2000), "Nonlinear dynamics and bifurcations of an axially moving beam", J. Vib. Acoust., 122(1), 21-30. https://doi.org/10.1115/1.568433
  36. Ravindra, B. and Zhu, W.D. (1998), "Low-dimensional chaotic response of axially accelerating continuum in the supercritical regime", Arch. Appl. Mech., 68(3-4), 195-205. https://doi.org/10.1007/s004190050157
  37. Riedel, C.H. and Tan, C.A. (2002), "Coupled, forced response of an axially moving strip with internal resonance", Int. J. Nonlinear Mech., 37(1), 101-116. https://doi.org/10.1016/S0020-7462(00)00100-1
  38. Saffari, H., Mohammadnejad, M. and Bagheripour, M.H. (2012), "Free vibration analysis of non-prismatic beams under variable axial forces", Struct. Eng. Mech., 43(50).
  39. Sahebkar, S.M., Ghazavi, M.R., Khadem, S.E. and Ghayesh, M.H. (2011), "Nonlinear vibration analysis of an axially moving drillstring system with time dependent axial load and axial velocity in inclined well", Mech. Mach. Theory, 46(5), 743-760. https://doi.org/10.1016/j.mechmachtheory.2010.12.003
  40. Song, Z., Li, W. and Liu, G. (2012), "Stability and non-stationary vibration analysis of beams subjected to periodic axial forces using discrete singular convolution", Struct. Eng. Mech., 44(4), 487-499. https://doi.org/10.12989/sem.2012.44.4.487
  41. Suweken, G. and Van Horssen, W.T. (2003), "On the transversal vibrations of a conveyor belt with a low and time-varying velocity. Part I: the string-like case", J. Sound Vib., 264(1), 117-133. https://doi.org/10.1016/S0022-460X(02)01168-9
  42. 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
  43. Tang, Y.Q., Chen, L.Q. and Yang, X.D. (2008), "Natural frequencies, modes and critical speeds of axially moving Timoshenko beams with different boundary conditions", Int. J. Mech. Sci., 50(10-11), 1448-1458. https://doi.org/10.1016/j.ijmecsci.2008.09.001
  44. Yang, X.D. and Chen, L.Q. (2005), "Bifurcation and chaos of an axially accelerating viscoelastic beam", Chaos Soliton. Fract., 23(1), 249-258. https://doi.org/10.1016/j.chaos.2004.04.008

Cited by

  1. Influence of roll-to-roll system’s dynamics on axially moving web vibration vol.21, pp.3, 2013, https://doi.org/10.21595/jve.2018.19872
  2. Dynamic Behavior Analysis of an Axially Loaded Beam Supported by a Nonlinear Spring-Mass System vol.21, pp.11, 2021, https://doi.org/10.1142/s0219455421501522