Exact mathematical solution for nonlinear free transverse vibrations of beams

Document Type : Article


Department of Mechanical Engineering, Bu-Ali Sina University, Hamedan, P.O. Box 65175-4161, Iran


In the present paper, an exact mathematical solution has been obtained for nonlinear free transverse vibration of beams, for the first time. The nonlinear governing partial differential equation in un-deformed coordinates system has been converted in two coupled partial differential equations in deformed coordinates system. A mathematical explanation is obtained for nonlinear mode shapes as well as natural frequencies versus geometrical and material properties of beam. It is shown that as the s th mode of transverse vibration excited, the mode 2s th of in-plane vibration will be developed. The result of present work is compared with those obtained from Galerkin method and the observed agreement confirms the exact mathematical solution. It is shown that governing equation is linear in time domain. As a parameter, the amplitude to length ratio (Λ⁄l) has been proposed to show when the nonlinear terms become dominant in the behavior of structure


Main Subjects

1. Bernoulli, J. "Essai theorique sur les vibrations de plaques elastiques rectangulaires et Libres", Nova Acta Acad. Petropolit., 5, pp. 197-219 (1789).
2. Timoshenko, S.P. "On the correction for the shear of the differential equation for transverse vibration of prismatic bars", Phil. Mag., 41, pp. 744-746 (1921).
3. Timoshenko, S.P. "On the transverse vibration of bars of uniform cross sections", Phil. Mag., 43(6), pp. 125-131 (1922).
4. Reddy, J.N. "A simple higher order theory for laminated composite plates", J. App. Mech., 51, pp. 745- 752 (1984).
5. Pai, P.F. and Nayfeh, A.H. "Nonlinear nonplanar oscillations of cantilever beam under lateral base excitation", Int. J. Non. Linear. Mech., 25, pp. 455-474 (1990).
6. Pai, P.F. and Nayfeh, A.H. "A nonlinear composite beam theory", Nonlinear. Dyn., 3, pp. 431-463 (1992).
7. Pai, P.F., Palazotto, A.N., and Greer, J.M. "Polar decomposition and appropriate strain and stresses for nonlinear structural analysis", Comp. Struct., 66, pp. 823-840 (1998).
8. Hodges, D.H. and Dowell, E.H. "Nonlinear equation of motion for the elastic bending and torsion of twisted non-uniform rotor blades", NASA TN D-7818 (1974).
9. Hodges, D.H. "Nonlinear equations of motion or cantilever rotor blades in hover with pitch link flexibility, twist, precone, droop, sweep, torque offset and blade root offset", NASA TM X-73 (1976).
10. Dowel, E.H., Traybar, J., and Hodges, D.H. "An experimental-theoretical correlation study of nonlinear bending and torsion deformations of a cantilever beam", J. Sound. Vib., 50, pp. 533-544 (1977).
11. Crespo da Silva, M.R.M. and Glynn, C.C. "Nonlinear flexural-flexural-torsional dynamics of in-extensional beams-I. Equations of motion", J. Struct. Mech., 6, pp. 437-448 (1978).
12. Alkire, K. "An analysis of rotor blade twist variables associated with different Euler sequences and pretwist treatments", NASA TM-84394 (1984).
13. Rosen, A. and Rand, O. "Numerical model of the nonlinear model of curved rods", Comp. Struct., 22, pp. 785-799 (1986).
14. Bauchau, O.A. and Hong, C.H. "Large displacement analysis of naturally curved and twisted composite beams", AIAA. J., 25, pp. 1469-1475 (1987).
15. Rosen, A., Loeway, R.G., and Mathew, M.B. "Nonlinear analysis of pretwisted roads using principle curvature transformation. Part I: Theoretical derivation", AIAA. J., 25, pp. 470-478 (1987).
16. Minguet, P. and Dugundji, J. "Experiments and analysis for composite blades under large deflection, Part I. Static behavior", AIAA J., 28, pp. 1573-1579 (1990).
17. Pai, P.F. and Nayfeh, A.H. "A fully nonlinear theory of curved and twisted composite rotor blades accounting for warping and tree-dimensional stress effects", Int. J. Solids. Struct., 31, pp. 1309-1340 (1994).
18. Banan, M.R., Karami, G., and Farshad, M. "Nonlinear theory of elastic spatial rods", Int. J. Solids. Struct., 27, pp. 713-724 (1991).
19. Simo, J.C. and Vu-Quoc, L. "A geometrical exact rod model incorporating shear and torsion-warping deformation", Int. J. Solids. Struct., 27, pp. 371-393 (1991).
20. Ho, C.H., Scott, R.A., and Eisley, J.G. "Nonplanar, nonlinear oscillations of a beam-I. Forced motion", Int. J. Non. Linear. Mech., 10, pp. 113-127 (1975).
21. Heyliger, P.R. and Reddy, J.N. "A higher order beam finite element for bending and vibration problems", J. Sound. Vib., 126, pp. 309-326 (1988).
22. Sheinman, I. and Adan, M. "The effect of shear deformation on post-bockling behavior of laminated beams", J. App. Mech., 54, pp. 558-562 (1987).
23. Bolotin, V.V., The Dynamic Stability of Elastic Systems, Holden-Day, San Francisco, California, 24 (1964).
24. Moody, P. "The parametric response of imperfect column, in developments in mechanics", Proceeding of the 10-th Midwestern Mechanics Conference, pp. 329- 346 (1967).
25. Crespo da Silva, M.R.M. and Glynn, C.C. "Nonlinear flexural-flexural-torsional dynamics of in-extensional beams-II. Forced motion", J. Struct. Mech., 6, pp. 437-448 (1978).
26. Nayfeh, A.H. and Pai, P.F. "Nonlinear nonplanar parametric responses of in-extensional beam", Int. J. Non. Linear. Mech., 24, pp. 139-158 (1989).
27. Nayfeh, A.H. and Pai, P.F., Linear and Nonlinear Structural Mechanic, John Wiley & Sons, Inc. (2004).
28. Ahmed, A. and Rhali, B. "Geometrically nonlinear transverse vibrations of Bernoulli-Euler beams carrying a finite number of masses and taking into account their rotatory inertia", 6, pp. 489-494 (2017).
29. Wang, Y., Ding, H., and Chen, L. "Nonlinear vibration of axially accelerating hyper-elastic beams", Int. J. Non. Linear. Mech. (2018) (In Press).
30. Seddighi, H. and Eipakchi, H.R. "Dynamic response of an axially moving viscoelastic Timoshenko beam", J. Solid Mech., 8, pp. 78-92 (2016).
31. Casalotti, A., El-Borgi, S., and Lacarbonara, W. "Metamaterial beam with embedded nonlinear vibration absorbers", Int. J. Non. Linear. Mech., 98, pp. 32-42 (2018).
32. Wang, T., Sheng, M., and Qin, Q. "Multi-flexural band gaps in an Euler-Bernoulli beam with lateral local resonators", Phys. Lett. A., 380, pp. 525-529 (2016).
33. Asghari, M., Kahrobaiyan, M.H., and Ahmadian, M.T. "A nonlinear Timoshenko beam formulation based on the modified couple stress theory", Int. J. Eng. Sci., 8, pp. 1749-1761 (2010).
34. Lewandowski, R. and Wielentejczyk, P. "Nonlinear vibration of viscoelastic beams described using fractional order derivatives", J. Sounds. Vib., 11, pp. 228-243 (2017).
35. Wielentejczyk, P. and Lewandowski, R. "Geometrically nonlinear, steady state vibration of viscoelastic beams", Int. J. Non. Linear. Mech., 7, pp. 177-186 (2017).
36. Roozbahani, M.M., Heydarzadeh Arani, N., Moghimi Zand, M., and Mousavi Mashhadi, M. "Analytical solutions to nonlinear oscillations of micro/nano beams using higher-order beam theory", Sci. Iran., Trans. B Mech., 23(5), pp. 2179-2193 (2016).
37. Alipour, A., Zand, M.M., and Daneshpajooh, H. "Analytical solution to nonlinear behavior of electrostatically actuated nano-beams incorporating van derWaals and Casimir forces", Sci. Iran., Trans. B Mech., 22(3), pp. 1322-1329 (2015).
38. Stojancovic, V. "Geometrically nonlinear vibrations of beams supported by a nonlinear elastic foundation with variable discontinuity", Commun. Non. Linear. Sci. Num. Simul., 28, pp. 66-80 (2015).
39. Szilard, R., Theories and Applications of Plate Analysis: Classical, Numerical and Engineering Methods, John Wiley & Sons, Inc. (2004).
40. Amabili, M., Nonlinear Vibrations and Stability of Shells and Plates, Cambridge University Press, New York, USA (2008).
41. Meirovitch, L., Fundamentals of Vibrations, McGraw- Hill (2001).
42. Nayfeh, A.H. and Mook, D.T., Nonlinear Oscillation, John Wiley & Sons, Inc. (1995).
43. Amini, M.H., Soleimani, M., Altafi, A., and Rastgoo, A. "Effects of geometric nonlinearity on free and forced vibration analysis of moderately thick annular functionally graded plate", Mech. Adv. Mater. Struct., 20, pp. 709-720 (2013).
44. Rao, S.S., Vibration of Continuous Systems, John Wiley & Sons, Inc. (2007).
45. Asadi Dalir, M. and Seifi, R. "Direct method for deriving equilibrium equations in solid continuous systems", Eng. Solid. Mech., 2, pp. 321-330 (2014).