Nonlinear Static Analysis and Superharmonic Influence on Nonlinear Forced Vibration of Timoshenko Beams

  • Brajesh PanigrahiEmail author
  • Goutam Pohit
Conference paper
Part of the Lecture Notes on Multidisciplinary Industrial Engineering book series (LNMUINEN)


In the present work, static analysis and subsequently superharmonic influence on the nonlinear dynamic behavior of externally excited thick beams are investigated. Energy equations are derived considering Timoshenko beam theory. For the static analysis, classical Ritz method is followed. Nonlinear load–deflection response is obtained considering various geometric parameters such as length-to-depth ratio and load application points. For the vibration analysis, differential equations are obtained considering the Lagrange’s equation. Subsequently, harmonic balance method is employed for multi-DOF systems, which reduce the differential equations into nonlinear set of algebraic equation. These equations are tackled by enforcing an iterative scheme based on modified direct substitution method. Simple harmonic assumption although provides a very good prediction for small amplitude vibration problem. However, it is inadequate for the system having large amplitude vibration. It is shown that for accurate solution higher-order harmonics must be considered.


Nonlinear static behavior Large amplitude vibration Harmonic balance method Modified direct substitution technique 



Temporal coordinates


Temporal coordinates


Width of beam


Temporal coordinates


Young’s modulus


Amplitude of loading


Depth of beam

K1, K2, K3, K4

Stiffness parameters

k3, k4

Stiffness parameter (dimensionless)


Length of beam

M1, M2, M3

Inertial parameters


Inertial parameter (dimensionless)


Number of polynomial terms


Kinetic energy


Kinetic energy (dimensionless)


Longitudinal displacement


Potential energy


Potential energy (dimensionless)


Axial displacement (dimensionless)


Transverse displacement


Transverse displacement (dimensionless)


Load application point

Greek Symbols


Poisson’s ratio (dimensionless)


Normalized axial coordinate




Mass density


Polynomial functions


Rotational displacement


Normalized rotational displacement


Frequency of excitation


Normalized frequency


  1. 1.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)zbMATHGoogle Scholar
  2. 2.
    Meirovitch, L.: Methods of Analytical Dynamics. McGraw-Hill, New York (1970)zbMATHGoogle Scholar
  3. 3.
    Noor, A.K., Peters, J.M.: Reduced basis technique for nonlinear analysis of structures. AIAA J. 18, 455–462 (1980)CrossRefGoogle Scholar
  4. 4.
    Desai, Y.M., Popplewell, N., Shah, A.H., Buragohain, D.N.: Geometric nonlinear static analysis of cable supported structures. Comput. Struct. 29, 1001–1009 (1988)CrossRefGoogle Scholar
  5. 5.
    Bathe, K.J., Bolourchi, S.: Large displacement analysis of three-dimensional beam structures. Int. J. Numer. Meth. Eng. 14, 961–986 (1979)CrossRefGoogle Scholar
  6. 6.
    Mata, P., Oller, S., Barbat, A.H.: Static analysis of beam structures under nonlinear geometric and constitutive behavior. Comput. Meth. Appl. Mech. Eng. 196, 4458–4478 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Newmark, N.M.: A method of computation for structural dynamics. ASCE Eng. Mech. Div. 85, 67–94 (1959)Google Scholar
  8. 8.
    Eisley, J.G.: Nonlinear vibration of beams and rectangular plates. Zeitschrift für angewandte Mathematik und Physik ZAMP 15, 167–175 (1964)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chu, H., Herrmann, G.: Influence of large amplitudes on free flexural vibrations of rectangular elastic plates. ASME J. Appl. Mech. 23, 532–540 (1956)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Yamaki, N.: Influence of large amplitudes on flexural vibrations of elastic plates. J. Appl. Math. Mech. 41, 501–510 (1961)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Hsu, C.S.: On the application of elliptic functions in nonlinear forced oscillations. Q. Appl. Math. 17, 393–407 (1960)CrossRefGoogle Scholar
  12. 12.
    Evensen, D.A.: Nonlinear vibrations of beams with various boundary conditions. AIAA J. 6, 370–372 (1968)CrossRefGoogle Scholar
  13. 13.
    Tseng, W.Y., Dugundji, J.: Nonlinear vibrations of a beam under harmonic excitation. J. Appl. Mech. 37, 292–297 (1970)CrossRefGoogle Scholar
  14. 14.
    Tseng, W.Y., Dugundji, J.: Nonlinear vibrations of a buckled beam under harmonic excitation. J. Appl. Mech. 38, 467–476 (1971)CrossRefGoogle Scholar
  15. 15.
    Bennett, J.A., Eisley, J.G.: A multiple degree-of-freedom approach to nonlinear beam vibrations. AIAA J. 8, 734–739 (1970)CrossRefGoogle Scholar
  16. 16.
    Mei, C.: Nonlinear vibration of beams by matrix displacement method. AIAA J. 10, 355–357 (1972)CrossRefGoogle Scholar
  17. 17.
    Stupnicka, W.: A study of main and secondary resonances in nonlinear multi-degree-of-freedom vibrating systems. Int. J. Nonlinear Mech. 10, 289–304 (1975)CrossRefGoogle Scholar
  18. 18.
    Stupnicka, W.: The generalised harmonic balance method for determining the combination resonance in the parametric dynamic systems. J. Sound Vib. 58, 347–361 (1978)CrossRefGoogle Scholar
  19. 19.
    Stupnicka, W.: Nonlinear normal modes and the generalised Ritz method in the problems of vibrations of nonlinear elastic continuous systems. Int. J. Nonlinear Mech. 18, 149–165 (1983)CrossRefGoogle Scholar
  20. 20.
    Rao, G., Raju, K.: Finite element formulation for the large amplitude free vibrations of beams and orthotropic plates. J. Comput. Struct. 6, 169–172 (1976)CrossRefGoogle Scholar
  21. 21.
    Lau, S.L., Cheung, Y.K.: Amplitude incremental variational principle for nonlinear vibration of elastic systems. J. Appl. Mech. 48, 59–964 (1981)zbMATHGoogle Scholar
  22. 22.
    Lau, S.L., Cheung, Y.K., Wu, S.Y.: Incremental harmonic balance method with multiple time scales for aperiodic vibration of nonlinear systems. J. Appl. Mech. 50, 871–876 (1983)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Bennouna, M.M.K., White, R.G.: The effects of large vibration amplitudes on the fundamental mode shape of a clamped-clamped uniform beam. J. Sound Vib. 96, 309–331 (1984)CrossRefGoogle Scholar
  24. 24.
    Bennouna, M.M.K., White, R.G.: The effects of large vibration amplitudes on the dynamic strain response of a clamped-clamped beam with consideration of fatigue life. J. Sound Vib. 96, 281–308 (1984)CrossRefGoogle Scholar
  25. 25.
    Benamar, R., Bennouna, M.M.K., White, R.G.: The effects of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures. Part I: simply supported and clamped-clamped beams. J. Sound Vib. 149, 179–195 (1991)CrossRefGoogle Scholar
  26. 26.
    Benamar, R., Bennouna, M.M.K., White, R.G.: The effects of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures, part III: fully clamped rectangular isotropic plates-measurements of the mode shape amplitude dependence and the spatial distribution of harmonic distortion. J. Sound Vib. 175, 377–424 (1994)CrossRefGoogle Scholar
  27. 27.
    Azrar, L., Benamar, R., White, R.G.: Semi-analytical approach to the nonlinear dynamic response problem of S–S and C–C beams at large vibration amplitudes, part I: general theory and application to the single mode approach to free and forced vibration analysis. J. Sound Vib. 224(2), 183–207 (1999)CrossRefGoogle Scholar
  28. 28.
    El Kadiri, M., Benamar, R., White, R.G.: The non-linear free vibration of fully clamped rectangular plates: second nonlinear mode for various plate aspect ratios. J. Sound Vib. 228, 333–358 (1999)CrossRefGoogle Scholar
  29. 29.
    Azrar, L., Benamar, R., White, R.G.: A semi-analytical approach to the nonlinear dynamic response problem of beams at large vibration amplitudes, part II: multimode approach to the steady state forced periodic response. Journal Sound Vib. 255(1), 1–41 (2002)CrossRefGoogle Scholar
  30. 30.
    Harras, B., Benamar, R., White, R.G.: Geometrically nonlinear free vibration of fully clamped symmetrically laminated rectangular composite plates. J. Sound Vib. 251, 579–619 (2002)CrossRefGoogle Scholar
  31. 31.
    El Kadiri, M., Benamar, R., White, R.G.: Improvement of the semi-analytical method for determining the geometrically nonlinear response of thin straight structures Part I: application to clamped-clamped and simply supported-clamped beams. J. Sound Vib. 249, 263–305 (2002)CrossRefGoogle Scholar
  32. 32.
    Qaisi, M.I.: Application of the harmonic balance principle to the nonlinear free vibration of beams. Appl. Acoust. 40, 141–151 (1993)CrossRefGoogle Scholar
  33. 33.
    Ribeiro, P., Petyt, M.: Non-linear vibration of beams with internal resonance by the hierarchical finite element method. J. Sound Vib. 224, 591–624 (1999)CrossRefGoogle Scholar
  34. 34.
    Ribeiro, P.: Non-linear forced vibrations of thin/thick beams and plates by the finite element and shooting methods. Comput. Struct. 82, 1413–1423 (2004)CrossRefGoogle Scholar
  35. 35.
    Chen, S.H., Haung, J.L., Sze, K.Y.: Multidimensional Lindstedt–Poincare´ method for nonlinear vibration of axially moving beams. J. Sound Vib. 306, 1–11 (2007)CrossRefGoogle Scholar
  36. 36.
    Cheung, Y.K., Chen, S.H.: Application of the incremental harmonic balance method to cubic non-linearity systems. J. Sound Vib. 140, 273–286 (1990)CrossRefGoogle Scholar
  37. 37.
    Ramezani, A., Alasty, A., Akbari, J.: Effects of rotary inertia and shear deformation on nonlinear free vibration of micro beams. J. Vib. Acoust. 128, 611–615 (2006)CrossRefGoogle Scholar
  38. 38.
    Luongo, A., Rega, G., Vestroni, F.: On nonlinear dynamics of planar shear indeformable beams. J. Appl. Mech. 53, 619–624 (1986)CrossRefGoogle Scholar
  39. 39.
    Lenci, S., Clementi, F., Rega, G.: A comprehensive analysis of hardening/softening behaviour of shearable planar beams with whatever axial boundary constraint. Meccanica 51, 2589–2606 (2016)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Clementi, F., Lenci, S., Rega, G.: Cross-checking asymptotics and numerics in the hardening/softening behaviour of Timoshenko beams with axial end spring and variable slenderness. Arch. Appl. Mech. 87, 865–880 (2017)CrossRefGoogle Scholar
  41. 41.
    Gupta, R.K., Babu, G.J., Janardhan, G.R., Venkateswara Rao, G.: Relatively simple finite element formulation for the large amplitude free vibrations of uniform beams. Finite Elem. Anal. Des. 45, 624–631 (2009)CrossRefGoogle Scholar
  42. 42.
    Kitipornchai, S., Ke, L.L., Yang, J., Xiang, Y.: Non-linear vibration of edge cracked functionally graded Timoshenko beams. J. Sound Vib. 324, 962–982 (2009)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringJadavpur UniversityKolkataIndia

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