Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Non linear vibrations of stepped beam system under different boundary conditions

  • Ozkaya, E. (Dept. of Mechanical Engineering, Celal Bayar University) ;
  • Tekin, A. (Celal Bayar University)
  • Received : 2006.05.08
  • Accepted : 2007.04.03
  • Published : 2007.10.20
⟨ Previous Next ⟩

Abstract

In this study, the nonlinear vibrations of stepped beams having different boundary conditions were investigated. The equations of motions were obtained using Hamilton's principle and made non dimensional. The stretching effect induced non-linear terms to the equations. Forcing and damping terms were also included in the equations. The dimensionless equations were solved for six different set of boundary conditions. A perturbation method was applied to the equations of motions. The first terms of the perturbation series lead to the linear problem. Natural frequencies for the linear problem were calculated exactly for different boundary conditions. Second order non-linear terms of the perturbation series behave as corrections to the linear problem. Amplitude and phase modulation equations were obtained. Non-linear free and forced vibrations were investigated in detail. The effects of the position and magnitude of the step, as well as effects of different boundary conditions on the vibrations, were determined.

Keywords

Acknowledgement

Supported by : Technical Research Council of Turkey (TUBITAK)

References

  1. Aldraihem, O.J. and Baz, A. (2002), 'Dynamic stability of stepped beams under moving loads', J. Sound Vib., 250(5), 2, 835-848 https://doi.org/10.1006/jsvi.2001.3976
  2. Aydogdu, M. and Taskin, V. (2006), 'Free vibration analysis of functionally graded beams with simply supported edges', Mater. Design
  3. Balasubramanian, T.S., Subramanian, G. and Ramani, T.S. (1990), 'Significance and use of very high order derivatives as nodal degrees of freedom in stepped beam vibration analysis', J. Sound Vib., 137, 353-356 https://doi.org/10.1016/0022-460X(90)90803-8
  4. Hou, J. W. and Yuan, J.Z. (1998), 'Calculation of eigenvalue and eigenvector derivatives for nonlinear beam vibrations', Am. Ins. Aeronaut. Astron. J., 26, 872-880
  5. Jang, S.K. and Bert, C.W. (1989), 'Free Vibration of stepped beams: Exact and numerical solutions', J. Sound Vib., 130(2), 342-346 https://doi.org/10.1016/0022-460X(89)90561-0
  6. Jang, S.K. and Bert, C.W. (1989), 'Free vibrations of stepped beams: Higher mode frequencies and effects of steps on frequency', J. Sound Vib., 132, 164-168 https://doi.org/10.1016/0022-460X(89)90882-1
  7. Krishnan, A., George, G. and Malathi, P. (1998), 'Use of finite difference method in the study of stepped beams', Int. J. Mech. Eng. Education, 26, 11-24 https://doi.org/10.1177/030641909802600103
  8. Kwon, H.D. and Park, Y.P. (2002), 'Dynamic characteristics of stepped cantilever beams connected with a rigid body', J. Sound Vib., 255(4), 701-717 https://doi.org/10.1006/jsvi.2001.4185
  9. McDonald, P.H. (1991), 'Nonlinear dynamics of a beam', Comput. Struct., 40, 1315-1320 https://doi.org/10.1016/0045-7949(91)90401-7
  10. Naguleswaran, S. (2002), 'Natural frequencies, sensitivity and mode shape details of an Euler-Bernoulli beam with one-step change in cross-section and with ends on classical supports', J. Sound Vib., 252, 751-767 https://doi.org/10.1006/jsvi.2001.3743
  11. Naguleswaran, S. (2002), 'Vibration of an Euler-Bernoulli beam on elastic end supports and with up to three step changes in cross section', Int. J. Mech. Sci., 44, 2541-2555 https://doi.org/10.1016/S0020-7403(02)00190-X
  12. Naguleswaran, S. (2003), 'Vibration and stability of an Euler-Bernoulli beam with up to three step changes in cross section and axial force', Int. J. Mech. Sci., 45, 1563-1579 https://doi.org/10.1016/j.ijmecsci.2003.09.001
  13. Pakdemirli, M. and Boyaci, H. (1995), 'Comparison of direct-perturbation methods with discretization-perturbation methods for non-linear vibrations', J. Sound Vib., 186, 837-845 https://doi.org/10.1006/jsvi.1995.0491
  14. Pakdemirli, M. and Boyaci, H. (2001), Vibrations of a stretched beam with non-ideal boundary conditions', Math. Comput. Appl., 6(3), 217-220
  15. Pakdemirli, M. and Nyfeh, A. (1994), 'Nonlinear vibration of a beam-spring-mass system', J. Vib. Acoustics, 166, 433-438
  16. Qaisi, M.I. (1997), 'A power series solution for the nonlinear vibrations of beams', J. Sound Vib., 199, 587-594 https://doi.org/10.1006/jsvi.1996.0696
  17. Oz, H.R., Pakdemirli, M., Ozkaya, E. and Yilmaz, M. (1998), 'Non-linear vibrations of a slight curved beam resting on a non-linear elastic foundation', J. Sound Vib., 221(3), 295-309
  18. Ozkaya, E. (2002), 'Non-linear transverse vibrations of a simply supported beam carrying concentrated masses', J. Sound Vib., 257, 413-424 https://doi.org/10.1006/jsvi.2002.5042
  19. Ozkaya, E., Pakdemirli, M. and Oz, H.R. (1997), 'Nonlinear vibrations of a beam-mass system under different boundary conditions', J. Sound Vib., 199, 679-696 https://doi.org/10.1006/jsvi.1996.0663

Cited by

  1. Nonlinear vibrations of spring-supported axially moving string vol.81, pp.3, 2015, https://doi.org/10.1007/s11071-015-2086-1
  2. Nonlinear vibration of micromachined asymmetric resonators vol.329, pp.13, 2010, https://doi.org/10.1016/j.jsv.2009.10.033
  3. Three-to-one internal resonance in multiple stepped beam systems vol.30, pp.9, 2009, https://doi.org/10.1007/s10483-009-0907-x
  4. A Semianalytical Method for Nonlinear Vibration of Euler-Bernoulli Beams with General Boundary Conditions vol.2010, 2010, https://doi.org/10.1155/2010/591786
  5. Research development of silicon MEMS gyroscopes: a review vol.21, pp.10, 2015, https://doi.org/10.1007/s00542-015-2645-x
  6. Nonlinear Transverse Vibrations and 3:1 Internal Resonances of a Beam With Multiple Supports vol.130, pp.2, 2008, https://doi.org/10.1115/1.2775508
  7. Non linear vibrations of stepped beam systems using artificial neural networks vol.33, pp.1, 2007, https://doi.org/10.12989/sem.2009.33.1.015
  8. Vibration analysis of a beam on a nonlinear elastic foundation vol.62, pp.2, 2007, https://doi.org/10.12989/sem.2017.62.2.171