Exact mathematical solution for nonlinear free transverse vibrations of beams

Document Type : Article

Author

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

Abstract

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

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1. Bernoulli, J. Essai th_eorique 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 di_erential 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 exibility,  twist, precone, droop, sweep, torque o_set and blade  root o_set", 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  exural-exural-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 di_erent 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 deection, 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 e_ects", 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  _nite element for bending and vibration problems", J.  Sound. Vib., 126, pp. 309{326 (1988).  22. Sheinman, I. and Adan, M. The e_ect 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  exural-exural-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 _nite 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-exural  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 modi_ed couple stress theory", Int. J. Eng. Sci.,  8, pp. 1749{1761 (2010).  M. Asadi-Dalir/Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 1290{1301 1301  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., Alta_, A., and Rastgoo,  A. E_ects 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 Sei_, R. Direct method for deriving  equilibrium equations in solid continuous systems",  Eng. Solid. Mech., 2, pp. 321{330 (2014).