Structural Engineering and Mechanics

  • 제40권5호
  • /
  • Pages.705-718
  • /
  • 2011
  • /
  • 1225-4568(pISSN)
  • /
  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Parametrically excited viscoelastic beam-spring systems: nonlinear dynamics and stability

  • Ghayesh, Mergen H. (Department of Mechanical Engineering, McGill University)
  • 투고 : 2010.11.30
  • 심사 : 2011.10.21
  • 발행 : 2011.12.10
⟨ 이전 논문 다음 논문 ⟩

초록

The aim of the investigation described in this paper is to study the nonlinear parametric vibrations and stability of a simply-supported viscoelastic beam with an intra-span spring. Taking into account a time-dependent tension inside the beam as the main source of parametric excitations, as well as employing a two-parameter rheological model, the equations of motion are derived using Newton's second law of motion. These equations are then solved via a perturbation technique which yields approximate analytical expressions for the frequency-response curves. Regarding the main parametric resonance case, the local stability of limit cycles is analyzed. Moreover, some numerical examples are provided in the last section.

키워드

참고문헌

  1. Birman, V. (1986), "On the effects of nonlinear elastic foundation on free vibration of beams", ASME J. Appl. Mech., 53, 471-473. https://doi.org/10.1115/1.3171790
  2. Chen, L.Q. and Yang, X.D. (2005a), "Steady state response of axially moving viscoelastic beams with pulsating speed: comparison of two nonlinear models", Int. J. Solids. Struct., 42, 37-50. https://doi.org/10.1016/j.ijsolstr.2004.07.003
  3. Chen, L.Q. and Yang, X.D. (2006), "Vibration and stability of an axially moving viscoelastic beam with hybrid supports", Eur. J. Mech., 25, 996-1008. https://doi.org/10.1016/j.euromechsol.2005.11.010
  4. Chen, L.Q., Zhang, N.H. and Zu, J.W. (2002), "Bifurcation and chaos of an axially moving visco-elastic strings", Chaos Soliton. Fract., 29, 81-90.
  5. Chen, L.Q. and Yang, X.D. (2005b), "Stability in parametric resonance of axially moving viscoelastic beams with time-dependent speed", J. Sound. Vib., 284, 879-891. https://doi.org/10.1016/j.jsv.2004.07.024
  6. Chen, L.Q., Tang, Y.Q. and Lim, C.W. (2010), "Dynamic stability in parametric resonance of axially accelerating viscoelastic Timoshenko beams", J. Sound. Vib., 329, 547-565 https://doi.org/10.1016/j.jsv.2009.09.031
  7. Cohen, Y.B. (2001), Electro Active Polymer (EPA) Actuators as Artificial Muscles, Reality, Potential, and Challenges, SPIE Press.
  8. Darabi, M.A., Kazemirad, S. and Ghayesh, M.H. (2011), "Free vibrations of beam-mass-spring systems: Analytical analysis with numerical confirmation", Acta Mecha. Sinica. (in press)
  9. Dowell, E.H. (1980), "Component mode analysis of nonlinear and nonconservative systems", ASME J. Appl. Mech., 47, 172-176. https://doi.org/10.1115/1.3153598
  10. Eisley, J.G. (1964), "Nonlinear vibration of beams and rectangular plates", ZAMP, 15, 167-175. https://doi.org/10.1007/BF01602658
  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, 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. Machine Theory, 44, 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. Nonlin. Mech., 45, 382-394. https://doi.org/10.1016/j.ijnonlinmec.2009.12.011
  14. Ghayesh, M.H. and Balar, S. (2008), "Non-linear parametric vibration and stability of axially moving viscoelastic Rayleigh beams", Int. J. Solids. Struct., 45, 6451-6467. https://doi.org/10.1016/j.ijsolstr.2008.08.002
  15. 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, 2850-2859. https://doi.org/10.1016/j.apm.2009.12.019
  16. Ghayesh, M.H., Yourdkhani, M., Balar, S. and Reid, T. (2010), "Vibrations and stability of axially traveling laminated beams", Appl. Math. Comput., 217, 545-556. https://doi.org/10.1016/j.amc.2010.05.088
  17. 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
  18. Ghayesh, M.H. (2011), "On the natural frequencies, complex mode functions, and critical speeds of axially traveling laminated beams: Parametric study", Acta Mecha. Solida Sinica. (in press)
  19. Ghayesh, M.H. and Paidoussis, M.P. (2010a), "Dynamics of a fluid-conveying cantilevered pipe with intermediate spring support", ASME Conference Proceedings, 2010(54518), 893-902.
  20. Ghayesh, M.H. and Paidoussis, M.P. (2010b), "Three-dimensional dynamics of a cantilevered pipe conveying fluid, additionally supported by an intermediate spring array", Int. J. Nonlin. 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. (2011a), "Three-dimensional dynamics of a fluidconveying 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., Kazemirad, S., Darabi, M.A. and Woo, P. (2011b), "Thermo-mechanical nonlinear vibration analysis of a spring-mass-beam system", Arch. Appl. Mech. (in press)
  23. Ghayesh, M.H., Alijani, F. and Darabi, M.A. (2011c), "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
  24. Ghayesh, M.H., Kazemirad, S. and Darabi, M.A. (2011d), "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
  25. Hu, K. and Kirmser, P.G. (1971), "On the nonlinear vibrations of free-free beams", ASME J. Appl. Mech., 38, 461-466. https://doi.org/10.1115/1.3408798
  26. Karlik, B., Ozkaya, E., Aydin, S. and Pakdemirli, M. (1998), "Vibrations of a beam-mass systems using artificial neural networks", Comput. Struct., 69, 339-347. https://doi.org/10.1016/S0045-7949(98)00126-6
  27. Marynowski, K. and Kapitaniak, T. (2002), "Kelvin-voigt versus burgers internal damping in modeling of axially moving viscoelastic web", Int. J. Nonlin. Mech., 37, 1147-1161. https://doi.org/10.1016/S0020-7462(01)00142-1
  28. Marynowski, K. (2004), "Non-linear vibrations of an axially moving viscoelastic web with time-dependent tension", Chaos Soliton. Fract., 21, 2004, 481-490. https://doi.org/10.1016/j.chaos.2003.12.020
  29. Marynowski, K. (2006), "Two-dimensional rheological element in modelling of axially moving viscoelastic web", Eur. J. Mech. A-Soild., 25, 729-744. https://doi.org/10.1016/j.euromechsol.2005.10.005
  30. Marynowski, K. and Kapitaniak, T. (2007), "Zener internal damping in modeling of axially moving viscoelastic beam with time-dependent tension", Int. J. Nonlin. Mech., 42, 118-131. https://doi.org/10.1016/j.ijnonlinmec.2006.09.006
  31. Marynowski, K. (2010), "Free vibration analysis of the axially moving Levy-type viscoelastic plate", Eur. J. Mech. A-Soild., 29, 879-886. https://doi.org/10.1016/j.euromechsol.2010.03.010
  32. Mockensturm, E.M. and Guo, J. (2005), "Nonlinear vibration of parametrically excited, viscoelastic, axially moving strings", J. Appl. Mech., 72, 374-380. https://doi.org/10.1115/1.1827248
  33. Nayfeh, A.H. (1993), Problems in Perturbation, Wiley, New York, USA.
  34. Nayfeh, A.H. and Mook, D.T. (1979), Nonlinear Oscillations, Wiley, New York, USA.
  35. Ozkaya, E., Pakdemirli, M. and Oz, H.R. (1997), "Non-linear vibrations of a beam-mass system under different boundary conditions", J. Sound. Vib., 199, 679-696. https://doi.org/10.1006/jsvi.1996.0663
  36. Pakdemirli, M. and Boyaci, H. (2003), "Non-linear vibrations of a simple-simple beam with a nonideal support in between", J. Sound. Vib., 268, 331-341. https://doi.org/10.1016/S0022-460X(03)00363-8
  37. Pakdemirli, M. and Nayfeh, A.H. (1994), "Nonlinear Vibrations of a Beam-Spring-mass System", ASME J. Vib. Acoust., 116, 433-439. https://doi.org/10.1115/1.2930446
  38. Srinivasan, A. V. (1965), "Large amplitude-free oscillations of beams and plates", AIAA J., 3, 1951-1953. https://doi.org/10.2514/3.3290
  39. Szemplinska-Stupnicka, W. (1990), The behaviour of nonlinear vibration systems, II., Kluwer, Netherlands.
  40. Thomsen, J.J. (2003), Vibrations and Stability, Advanced Theory, Analysis, and Tools, Springer-Verlag, Berlin, Heidelberg.
  41. Tseng, W.Y. and Dugundji, J. (1971), "Nonlinear vibrations of a buckled beam under harmonic excitation", ASME J. Appl. Mech., 38, 467-472. https://doi.org/10.1115/1.3408799
  42. Wrenn, B.G. and Mayers, J. (1970), "Nonlinear beam vibration with variable axial boundary restraint", AIAA J., 8, 1718-1720. https://doi.org/10.2514/3.5979
  43. Zhang, N.H. and Chen, L.Q. (2005), "Nonlinear dynamical analysis of axially moving viscoelastic string", Chaos Soliton. Fract., 24(4),1065-1074. https://doi.org/10.1016/j.chaos.2004.09.113
  44. Zhang, N.H. (2008), "Dynamic analysis of an axially moving viscoelastic string by the Galerkin method using translating string eigenfunctions", Chaos Soliton. Fract., 35, 291-302. https://doi.org/10.1016/j.chaos.2006.05.021

피인용 문헌

  1. Nonlinear vibrations and stability of an axially moving beam with an intermediate spring support: two-dimensional analysis vol.70, pp.1, 2012, https://doi.org/10.1007/s11071-012-0458-3
  2. Nonlinear dynamics of axially moving viscoelastic beams over the buckled state vol.112-113, 2012, https://doi.org/10.1016/j.compstruc.2012.09.005
  3. Analysis of a quintic system with fractional damping in the presence of vibrational resonance vol.321, 2018, https://doi.org/10.1016/j.amc.2017.11.028
  4. Steady-state transverse response of an axially moving beam with time-dependent axial speed vol.49, 2013, https://doi.org/10.1016/j.ijnonlinmec.2012.08.003
  5. Subcritical parametric response of an axially accelerating beam vol.60, 2012, https://doi.org/10.1016/j.tws.2012.06.012
  6. Internal resonance and nonlinear response of an axially moving beam: two numerical techniques vol.1, pp.3, 2012, https://doi.org/10.12989/csm.2012.1.3.235
  7. A novel two-dimensional approach to modelling functionally graded beams resting on a soil medium vol.51, pp.5, 2014, https://doi.org/10.12989/sem.2014.51.5.727
  8. Nonlinear dynamic response of a simply-supported Kelvin-Voigt viscoelastic beam, additionally supported by a nonlinear spring vol.13, pp.3, 2011, https://doi.org/10.1016/j.nonrwa.2011.10.009
  9. Parametric resonance of fractional multiple-degree-of-freedom damped beam systems vol.232, pp.12, 2011, https://doi.org/10.1007/s00707-021-03087-1