Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Vibrations of an axially accelerating, multiple supported flexible beam

  • Kural, S. (Department of Mechanical Engineering, Celal Bayar University) ;
  • Ozkaya, E. (Department of Mechanical Engineering, Celal Bayar University)
  • Received : 2012.04.02
  • Accepted : 2012.10.27
  • Published : 2012.11.25
⟨ Previous Next ⟩

Abstract

In this study, the transverse vibrations of an axially moving flexible beams resting on multiple supports are investigated. The time-dependent velocity is assumed to vary harmonically about a constant mean velocity. Simple-simple, fixed-fixed, simple-simple-simple and fixed-simple-fixed boundary conditions are considered. The equation of motion becomes independent from geometry and material properties and boundary conditions, since equation is expressed in terms of dimensionless quantities. Then the equation is obtained by assuming small flexural rigidity. For this case, the fourth order spatial derivative multiplies a small parameter; the mathematical model converts to a boundary layer type of problem. Perturbation techniques (The Method of Multiple Scales and The Method of Matched Asymptotic Expansions) are applied to the equation of motion to obtain approximate analytical solutions. Outer expansion solution is obtained by using MMS (The Method of Multiple Scales) and it is observed that this solution does not satisfy the boundary conditions for moment and incline. In order to eliminate this problem, inner solutions are obtained by employing a second expansion near the both ends of the flexible beam. Then the outer and the inner expansion solutions are combined to obtain composite solution which approximately satisfying all the boundary conditions. Effects of axial speed and flexural rigidity on first and second natural frequency of system are investigated. And obtained results are compared with older studies.

Keywords

Acknowledgement

Supported by : Scientific and Technical Research Council of Turkey (TUBITAK)

References

  1. Bagdatli, S.M., Ozkaya, E., Ozyigit, H.A. and Tekin, A. (2009), "Nonlinear vibrations of stepped beam systems using artificial neural networks", Struct. Eng. Mech., 33(1), 15-30. https://doi.org/10.12989/sem.2009.33.1.015
  2. Bagdatli, S.M., Oz, H.R. and Ozkaya, E. (2011), "Dynamics of axially accelerating beams with an intermediate support", J. Vib. Acoust., 133(3), Article Number: 031013.
  3. Bagdatli, S.M., Oz, H.R. and Ozkaya, E. (2011), "Non-linear transverse vibrations and 3:1 internal resonances of a tensioned beam on multiple supports", Math. Comput. Appl., 16(1), 203-215.
  4. Boyaci, H. (2005), "Beam vibrations with non-ideal boundary conditions the seventh international conference on vibration problems", ICOVP-2005, Turkey Springer Proceedings in Physics, 111, 97-102, Istanbul, September.
  5. Boyaci, H. (2006), "Vibrations of stretched damped beams under non-ideal boundary conditions", Sadhana, 31(1), 1-8. https://doi.org/10.1007/BF02703795
  6. Cetin, D. and Simsek, M. (2011), "Free vibration of an axially functionally graded pile with pinned ends embedded in Winkler-Pasternak elastic medium", Struct. Eng. Mech., 40(4), 583-594. https://doi.org/10.12989/sem.2011.40.4.583
  7. Chen, L.Q. and Zhao, W.J. (2005), "A numerical method for simulating transverse vibrations of an axially moving string", Appl. Math. Comput., 160(2), 411-422. https://doi.org/10.1016/j.amc.2003.11.012
  8. Chen, L.Q., Zhang, W. and Zu, J.W. (2009), "Nonlinear dynamics for transverse motion of axially moving strings", Chaos, Soliton. Fract., 40, 78-90. https://doi.org/10.1016/j.chaos.2007.07.023
  9. Chung, J., Han, C.S. and Yi, K. (2001), "Vibration of an axially moving string with geometric non-linearity and translating acceleration", J. Sound Vib., 240(4), 733-746. https://doi.org/10.1006/jsvi.2000.3241
  10. 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
  11. Ghayesh, M.H. (2011), "Nonlinear forced dynamics of an axially moving viscoelastic beam with an internal resonance", Int. J. Mech. Sci., 53, 1022-1037. https://doi.org/10.1016/j.ijmecsci.2011.08.010
  12. 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, 227-243. https://doi.org/10.1016/j.ijsolstr.2011.10.007
  13. 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, 471-485. https://doi.org/10.1016/j.jsv.2010.04.037
  14. Miranker, W.L. (1960), "The wave equation in a medium in motion", IBM J. Res. Development, 4, 36-42. https://doi.org/10.1147/rd.41.0036
  15. Mockenstrum, E.M., Perkins, N.C. and Ulsoy, A.G. (1994), "Stability and limit cycles of parametrically excited, axially moving strings", J. Vib. Acoust., ASME 118, 346-350.
  16. Nahfeh, A.H. (1981), Introduction to Perturbation Techniques, John Wiley, New York.
  17. Nayfeh, A.H., Nayfeh, J.F. and Mook, D.T. (1992), "On methods for continuous systems with quadratic and cubic nonlinearities", Nonlin. Dyn., 3, 145-162. https://doi.org/10.1007/BF00118990
  18. Nguyen, Q.C. and Hong, K.S. (2010), "Asymptotic stabilization of a nonlinear axially moving string by adaptive boundary control", J. Sound Vib., 329, 4588-4603. https://doi.org/10.1016/j.jsv.2010.05.021
  19. Oz, H.R., Pakdemirli, M. and Ozkaya, 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
  20. Oz, 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
  21. Oz, H.R. (2001), "On the vibrations of an axially travelling beam on fixed supports with variable velocity", J. Sound Vib., 239(3), 556-564. https://doi.org/10.1006/jsvi.2000.3077
  22. Ozkaya, 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
  23. Ozkaya, E. and Pakdemirli, M. (2000), "Lie Group theory and analytical solutions for the axially accelerating string problem", J. Sound Vib., 230(4), 729-742. https://doi.org/10.1006/jsvi.1999.2651
  24. Ozkaya, E. (2001), "Linear transverse vibrations of a simply supported beam carrying concentrated masses", Math. Comput. Appl., 6(3), 147-151.
  25. Ozkaya, E. and Oz, H.R. (2002), "Determination of stability regions of axially moving beams using artificial neural networks method", J. Sound Vib., 252(4), 782-789. https://doi.org/10.1006/jsvi.2001.3991
  26. Ozkaya, E. (2002), "Non-linear transverse vibrations of a simply supported beam carriying concentrated masses", J. Sound Vib., 257(3), 413-424. https://doi.org/10.1006/jsvi.2002.5042
  27. Ozkaya, E., Ba datl , S.M. and Oz, H.R. (2008), "Non-linear transverse vibrations and 3:1 internal resonances of a beam with multiple supports", J. Vib. Acous., 130(2), Article Number: 021013.
  28. Parker, R.G. and Kong, L. (2004), "Approximate eigensolutions of axially moving beams with small flexural stiffness", J. Sound Vib., 276, 459-469. https://doi.org/10.1016/j.jsv.2003.11.027
  29. Pakdemirli, M. and Batan, H. (1993), "Dynamic stability of a constantly accelerating strip", J. Sound Vib., 168, 371-378. https://doi.org/10.1006/jsvi.1993.1379
  30. Pakdemirli, M., Ulsoy, A.G. and Ceranoglu, A. (1994), "Transverse vibration of an axially accelerating string", J. Sound Vib., 169, 179-196. https://doi.org/10.1006/jsvi.1994.1012
  31. Pakdemirli, M. (1994), "A comparison of two perturbation methods for vibrations of systems with quadratic and cubic nonlinearities", Mech. Res. Comm., 21, 203-208. https://doi.org/10.1016/0093-6413(94)90093-0
  32. Pakdemirli, M., Nayfeh, S.A. and Nayfeh, A.H. (1995), "Analysis of one-to-one autoparametric resonances in cables-discretization vs direct treatment", Nonlin. Dyn., 8, 65-83. https://doi.org/10.1088/0951-7715/8/1/005
  33. Pakdemirli, M. and Boyaci, H. (1995), "Comparison of direct-perturbation methods with discretizationperturbation methods for nonlinear vibrations", J. Sound Vib., 186, 837-845. https://doi.org/10.1006/jsvi.1995.0491
  34. 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
  35. Pakdemirli, M. and Boyaci, H. (1997), "The direct-perturbation method versus the discretization-perturbation method: Linear systems", J. Sound Vib., 199(5), 825-832. https://doi.org/10.1006/jsvi.1996.0643
  36. Pakdemirli, M. and Ozkaya E. (1998), "Approximate boundary layer solution of a moving beam problem", Math. Comput. Appl., 2(3), 93-100.
  37. Pakdemirli, M. and Boyaci, H. (2001), "Vibrations of a stretched beam with non-ideal boundary conditions", Math. Comput. Appl., 6(3), 217-220.
  38. Pakdemirli, M. and Boyaci, H. (2002), "Effect of non-ideal boundary conditions on the vibrations of continuous systems", J. Sound Vib., 249(4), 815-823. https://doi.org/10.1006/jsvi.2001.3760
  39. Pakdemirli, M. and Boyaci, H. (2003), "Nonlinear vibrations of a simple-simple beam with a non-ideal support in between", J. Sound Vib., 268, 331-341. https://doi.org/10.1016/S0022-460X(03)00363-8
  40. Ponomareva, S.V. and van Horssen, W.T. (2009), "On the transversal vibrations of an axially moving continuum with a time-varying velocity: Transient from string to beam behavior", J. Sound Vib., 325, 959-973. https://doi.org/10.1016/j.jsv.2009.03.038
  41. Tekin, A., Ozkaya, E. and Bagdatli, S.M. (2009), "Three-to one internal resonances in multi stepped beam systems", Appl. Math. Mech.-English Edition, 30(9), 1131-1142. https://doi.org/10.1007/s10483-009-0907-x
  42. Ulsoy, A.G., Mote, C.D. Jr. and Syzmani, R. (1978), "Principal developments in band saw vibration and stability research", Holz als Roh- und Werkstoff, 36, 273-280. https://doi.org/10.1007/BF02610748
  43. Wickert, J.A. and Mote, C.D. Jr. (1988), "Current research on the vibration and stability of axially moving materials", Shock Vib. Dig., 20(5), 3-13. https://doi.org/10.1177/058310248802000503
  44. Wickert, J.A. and Mote, C.D. Jr. (1990), "Classical vibration analysis of axially moving continua", J. Appl. Mech., ASME, 57, 738-744. https://doi.org/10.1115/1.2897085
  45. Wickert, J.A. (1992), "Non-linear vibration of a traveling tensioned beam", Int. J. Nonlin. Mech., 27, 503-517. https://doi.org/10.1016/0020-7462(92)90016-Z

Cited by

  1. Vibration analysis of high nonlinear oscillators using accurate approximate methods vol.46, pp.1, 2013, https://doi.org/10.12989/sem.2013.46.1.137
  2. Free vibration analysis of axially moving beam under non-ideal conditions vol.54, pp.3, 2015, https://doi.org/10.12989/sem.2015.54.3.597
  3. Dynamics of axially accelerating beams with multiple supports vol.74, pp.1-2, 2013, https://doi.org/10.1007/s11071-013-0961-1
  4. Nonlinear vibrations of spring-supported axially moving string vol.81, pp.3, 2015, https://doi.org/10.1007/s11071-015-2086-1
  5. Size-dependent vibrations of a micro beam conveying fluid and resting on an elastic foundation vol.23, pp.7, 2017, https://doi.org/10.1177/1077546315589666
  6. Vibration of axially moving beam supported by viscoelastic foundation vol.38, pp.2, 2017, https://doi.org/10.1007/s10483-017-2170-9
  7. Tailoring the second mode of Euler-Bernoulli beams: an analytical approach vol.51, pp.5, 2014, https://doi.org/10.12989/sem.2014.51.5.773
  8. Non-linear transverse vibrations of tensioned nanobeams using nonlocal beam theory vol.55, pp.2, 2015, https://doi.org/10.12989/sem.2015.55.2.281
  9. Free and forced nonlinear vibration of a transporting belt with pulley support ends vol.92, pp.4, 2018, https://doi.org/10.1007/s11071-018-4179-0
  10. Free vibrations of fluid conveying microbeams under non-ideal boundary conditions vol.24, pp.2, 2012, https://doi.org/10.12989/scs.2017.24.2.141
  11. Influence of roll-to-roll system’s dynamics on axially moving web vibration vol.21, pp.3, 2012, https://doi.org/10.21595/jve.2018.19872