A new Fourier series solution for free vibration of non-uniform beams, resting on variable elastic foundation

Document Type : Article


1 Department of Civil Engineering, University of Tehran, Tehran, Iran

2 Department of Civil Engineering, Sharif University of Technology, Azadi St., Tehran, Iran

3 Department of Mechanical Engineering, Rice University, Houston, TX, USA


In this research, the combination of Fourier sine series and Fourier cosine series is employed to develop an analytical method for free vibration analysis of an Euler-Bernoulli beam of varying cross- section, fully or partially supported by a variable elastic foundation. The foundation stiffness and cross section of the beam are considered as arbitrary functions in the beam length direction. The idea of the proposed method is to superpose Fourier sine and Fourier cosine series to satisfy general elastically end constraints and therefore no auxiliary functions are required to supplement the Fourier series. This method provides a simple, accurate and flexible solution for various beam problems and is also able to be extended to other cases whose governing differential equations are nonlinear. Moreover, this method is applicable for plate problems with different boundary conditions if two-dimensional Fourier sine and cosine series are taken as displacement function.
Numerical examples are carried out illustrating the accuracy and efficiency of the presented approach.


Main Subjects

1. Jategaonkar, R. and Chehil, D.S. Natural frequencies
of a beam with varying section properties", J. Sound
Vib., 133(2), pp. 303-322 (1989).
2. Katsikadelis, J.T. and Tsiatas, G.C. Non-linear dynamic
analysis of beams with variable sti ness", J.
Sound Vib., 270(4-5), pp. 847-863 (2004).
3. Nikkhah Bahrami, M., Khoshbayani Arani, M., and
Rasekh Saleh, N. Modi ed wave approach for calculation
of natural frequencies and mode shapes in
arbitrary non-uniform beams", Sci. Iran., B, 18(5),
pp. 1088-1094 (2011).
4. Huang, Y. and Li, X-F. A new approach for free
vibration of axially functionally graded beams with
non-uniform cross-section", J. Sound Vib., 329(11),
pp. 2291-2303 (2010).
5. Au, F.T.K., Zheng, D.Y., and Cheung, Y.K. Vibration
and stability of non-uniform beams with abrupt
2978 S.E. Motaghian et al./Scientia Iranica, Transactions A: Civil Engineering 25 (2018) 2967{2979
changes of cross-section by using modi ed beam vibration
functions", Appl. Math. Model., 29(1), pp. 19-34
6. Naguleswaran, S. vibration of an Euler-Bernoulli
beam of constant depth and with linearly varying
breadth", J. Sound Vib., 153(3), pp. 509-522 (1992).
7. Firouz-Abadi, R.D., Haddadpoura, H., and Novinzadehb,
A.B. An asymptotic solution to transverse
free vibrations of variable-section beams", J. Sound
Vib., 304(3-5), pp. 530-54 (2007).
8. Datta, A.K. and Sil, S.N. An analytical of free undamped
vibration of beams of varying cross-section",
Comput. Struct., 59(3), pp. 479-483 (1996).
9. Banerjee, J.R., Su, H., and Jackson, D.R. Free
vibration of rotating tapered beams using the dynamic
sti ness method", J. Sound Vib., 298(4-5), pp. 1034-
1054 (2006).
10. Caruntu, D.I. Dynamic modal characteristics of
transverse vibrations of cantilevers of parabolic thickness",
Mech. Res. Commun., 36(3), pp. 391-404
11. Laura, P.A.A., Gutierrez, R.H., and Rossi, R.E. Free
vibrations of beams of bilinearly varying thickness",
Ocean. Eng., 23(1), pp. 1-6 (1996).
12. Chaudhari, T.D. and Maiti, S.K. Modelling of transverse
vibration of beam of linearly variable depth with
edge crack", Eng. Fract. Mech., 63(4), pp. 425-445
13. Ece, M.C., Aydogdu, M., and Taskin, V. Vibration of
a variable cross-section beam", Mech. Res. Commun.,
34(1), pp. 78-84 (2007).
14. Tanaka, M. and Bercin, A.N. Finite element modelling
of the coupled bending and torsional free vibration
of uniform beams with an arbitrary cross-section",
Appl. Math. Model., 21(6), pp. 339-344 (1997).
15. Malekzadeh, P. and Karami, G. A mixed di erential
quadrature and nite element free vibration and buckling
analysis of thick beams on two-parameter elastic
foundations", Appl. Math. Model., 32(7), pp. 1381-
1394 (2008).
16. Laura, P.A.A. and Valerga De Greco, B. Numerical
experiments on free and forced vibrations of beams of
non-uniform cross-section", J. Sound Vib., 120(3), pp.
587-596 (1998).
17. Zohoor, H. and Kakavand, F. Vibration of Euler-
Bernoulli and Timoshenko beams in large overall
motion on
ying support using nite element method",
Sci. Iran., B, 19(4), pp. 1105-1116 (2012).
18. Daeichi, M. and Ahmadian, M.T. Application of variational
iteration method to large vibration analysis of
slenderness beams considering mid-plane stretching",
Sci. Iran., B, 22(5), pp. 1911-1917 (2015).
19. Baghani, M., Asgarshamsi, A., and Goharkhaha, M.
Analytical solution for large amplitude vibrations of
microbeams actuated by an electro-static force", Sci.
Iran., B, 20(5), pp. 1499-1507 (2013).
20. Lee, S.Y., Ke, H.Y., and Kuo, Y.H. Analysis of nonuniform
beam vibration", J. Sound Vib., 142(1), pp.
15-29 (1990).
21. Chen, W.Q., Lu, C.F., and Bian Z.G. A mixed
method for bending and free vibration of beams resting
on a Pasternak elastic foundation", Appl. Math.
Model., 28(10), pp. 877-890 (2004).
22. Thambiratnam, D. and Zuge, Y. Free vibration analysis
of beams on elastic foundation", Comput. Struct.,
60(6), pp. 971-980 (1996).
23. Ding, Z. A general solution to vibrations of beams on
variable Winkler elastic foundation", Comput. Struct.,
47(1), pp. 83-90 (1993).
24. Eisenberger, M. Vibration frequencies for beams on
variable one and two-parameter elastic foundations",
J. Sound Vib., 176(5), pp. 577-584 (1994).
25. Eisenberger, M. and Clastornik, J. Vibrations and
buckling of a beam on a variable Winkler elastic foundation",
J. Sound Vib., 115(2), pp. 233-241 (1987).
26. Soldatos, K.P. and Selvadurai, A.P.S. Flexure of
beams resting on hyperbolic elastic foundations", Int.
J. Solids. Struct., 21(4), pp. 373-388 (1985).
27. Pradhan, S.C. and Murmu, T. Thermo-mechanical vibration
of FGM sandwich beam under variable elastic
foundations using di erential quadrature method", J.
Sound Vib., 321(1-2), pp. 342-362 (2009).
28. Payam, A.F. Sensitivity analysis of vibration modes
of rectangular cantilever beams immersed in
uid to
surface sti ness variations", Sci. Iran., B, 20(4), pp.
1221-1227 (2013).
29. Mutman, U. and Coskun, S.B. Free vibration analysis
of non-uniform Euler beams on elastic foundation
via homotopy perturbation method", Int. J. Mech.
Aerosp. Ind. Mechatron. Eng., 7(7), pp. 432-437
30. Ho, S.H. and Chen, C. Analysis of general elastically
end restrained non-uniform beams using di erential
transform", Appl. Math. Model., 22(4-5), pp. 219-234
31. Catal, S. Solution of free vibration equations of
beam on elastic soil by using difb00erential transform
method", Appl. Math. Model., 32(9), pp. 1744-1757
32. Ebrahimzadeh Hassanabadi, M., Nikkhoo, A., Vaseghi
Amiri, J., and Mehri, B. A new orthonormal polynomial
series expansion method in vibration analysis of
thin beams with non-uniform thickness", Appl. Math.
Model., 37(18-19), pp. 8543-8556 (2013).
33. Wang, J.T.-S. and Lin, C.-C. Dynamic analysis of
generally supported beams using Fourier series", J.
Sound Vib., 196(3), pp. 285-293 (1996).
34. Li, W.L. Free vibrations of beams with general
boundary conditions", J. Sound Vib., 237(4), pp. 709-
725 (2000).
S.E. Motaghian et al./Scientia Iranica, Transactions A: Civil Engineering 25 (2018) 2967{2979 2979
35. Motaghian, S.E., Mo d, M., and Alanjari, P. Exact
solution to free vibration of beams partially supported
by an elastic foundation", Sci. Iran., A, 18(4), pp.
861-866 (2011).
36. Motaghian, S.E., Mo d, M., and Akin J.E. On the
free vibration response of rectangular plates, partially
supported on elastic foundation", Appl. Math. Model.,
36(9), pp. 4473-4482 (2012).