Skip to main content
SpringerLink
Log in

Advances in nonlinear vibration analysis of structures. Part-I. Beams

  • Published:
Sadhana Aims and scope Submit manuscript

Abstract

The development of nonlinear vibration formulations for beams in the literature can be seen to have gone through distinct phases — earlier continuum solutions, development of appropriate forms, extra-variational simplifications, debate and discussions, variationally correct formulations and finally applications. A review of work in each of these phases is very necessary in order to have a complete understanding of the process of evolution of this field. This paper attempts to achieve precisely this objective.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Bhashyam G R, Prathap G 1980 Galerkin finite element method for nonlinear beam vibrations.J. Sound Vib. 72: 191–203

    Article  MATH  Google Scholar 

  • Burgreen D 1951 Free vibrations of a pin-ended column with constant distance between pin ends.J. Appl. Mech., ASME 18: 135–139

    Google Scholar 

  • Chen J S, Huang T 1986 Appropriate forms in nonlinear analysis.J. Eng. Mech. Div., ASCE 111: 1215–1226

    Google Scholar 

  • Dumir P C, Bhaskar A 1988 Some erroneous finite element formulations of nonlinear vibrations of beams and plates.J. Sound Vib. 123: 517–527

    Article  Google Scholar 

  • Evensen D A 1968 Nonlinear vibrations of beams with various boundary conditions.AIAA J. 6: 370–372

    MATH  Google Scholar 

  • Heyliger P R, Reddy J N 1988 A higher order beam finite element for bending and vibration problems.J. Sound Vib. 126: 309–326

    Article  Google Scholar 

  • Kapania R K, Raciti S 1989a Recent advances in analysis of laminated beams and plates, Part I: Shear effects and buckling.AIAA J. 27: 923–934

    MATH  MathSciNet  Google Scholar 

  • Kapania R K, Raciti S 1989b Recent advances in analysis of laminated beams and plates, Part II: Vibrations and wave propagation.AIAA J.27: 935–946

    MATH  MathSciNet  Google Scholar 

  • Kapania R K, Raciti S 1989c Nonlinear vibrations of unsymmetrically laminated beams.AIAA J. 27: 201–210

    MATH  Google Scholar 

  • Kreiger S W 1950 The effect of an axial force on the vibration of hinged bars.J. Appl. Mech., ASME 17: 35–36

    Google Scholar 

  • Lou C L, Sikarskie D L 1975 Nonlinear vibration of beams using a form-function approximation.J. Appl. Mech., ASME 42: 209–214

    MATH  Google Scholar 

  • Mallett R H, Marcal P V 1968 Finite element analysis of nonlinear structures.J. Struct. Div., ASCE 94: 2081–2105

    Google Scholar 

  • Mei C 1972 Nonlinear vibrations of beams by matrix displacement method.AIAA J. 10: 355–357

    Google Scholar 

  • Mei C 1973a Finite element displacement method for large amplitude free flexural vibrations of beams and plates.Comput. Struct. 3: 163–174

    Article  Google Scholar 

  • Mei C 1973b Finite element analysis of nonlinear vibrations of beam columns.AIAA J. 11: 115–117

    Google Scholar 

  • Mei C 1984 Comments on “Lagrange-type formulation for finite element analysis of nonlinear beam vibrations”J. Sound Vib. 94: 445–452

    Article  Google Scholar 

  • Mei C 1986 Discussions of finite element formulations of nonlinear beam vibrations.Comput. Struct. 22: 83–85

    Article  Google Scholar 

  • Padovan J 1980 Nonlinear vibrations of general structures.J. Sound Vib. 72: 427–441

    Article  MATH  Google Scholar 

  • Pandalai K A V, Sathyamoorthy M 1973 On the modal equations of large amplitude flexural vibration of beams, plates, rings and shells.Int. J. Nonlinear Mech. 8: 213–218

    Article  MATH  Google Scholar 

  • Pillai S R R, Rao B N 1992 On nonlinear free vibrations of simply supported uniform beams.J. Sound Vib. 159:527–531

    Article  MATH  Google Scholar 

  • Prathap G 1977 Comments on “Effect on longitudinal or inplane deformation and inertia on the large amplitude flexural vibrations of slender beams and thin plates”.J. Sound Vib.55: 308–311

    Article  Google Scholar 

  • Prathap G 1980 A discussion on “A hybrid, finite element-finite difference approach to simplified large deflection analysis of structures” by Rudolph Szilard.Comput. Struct.11: 251–253

    Article  MATH  Google Scholar 

  • Prathap G, Bhashyam G R 1980 Comments on nonlinear vibrations of immovably supported beams by finite-element method.AIAA J.18: 733–734

    Google Scholar 

  • Prathap G, Varadan T K 1978 The large amplitude vibration of hinged beams.Comput. Struct. 9: 219–222

    Article  MATH  Google Scholar 

  • Rajasekaran S, Murray D W 1973 Incremental finite element matrices.J. Struct. Div., ASCE 99: 2423–2437

    Google Scholar 

  • Raju I S, Rao G V, Raju K K 1976 Effect of longitudinal or inplane deformation and inertia on the large amplitude flexural vibrations of slender beams and thin plates.J. Sound Vib. 49: 415–422

    Article  MATH  Google Scholar 

  • Raju K K, Rao G V 1984 A note on large amplitude vibrations.Comput. Struct. 18: 1189–1191

    Article  Google Scholar 

  • Rao G V, Raju K K, Raju IS 1976a Finite element formulation for the large amplitude free vibrations of beams and orthotropic plates.Comput. Struct. 6: 169–172

    Article  MATH  Google Scholar 

  • Rao G V, Raju I S, Raju K K 1976b Nonlinear vibrations of beams considering shear deformation and rotary inertia.AIAA J. 14: 685–687

    Article  MATH  Google Scholar 

  • Ray J D, Bert C W 1969 Nonlinear vibrations of a beam with pinned ends.J. Eng. Ind., ASME 91: 977–1004

    Google Scholar 

  • Reddy J N 1979 Finite element modelling of structural vibrations: A review of recent advances.Shock Vib. Dig. 11:25–39

    Article  Google Scholar 

  • Reddy J N, Singh IR 1981 Large deflections and large-amplitude free vibrations of straight and curved beams.Int. J. Numer. Methods Eng. 17: 829–852

    Article  MATH  Google Scholar 

  • Rehfield L W 1975 A simple, approximate method for analysing nonlinear free vibrations of elastic structures.J. Appl. Mech., ASME 42: 509–511

    Google Scholar 

  • Sarma B S, Varadan TK 1982 Certain discussions in the finite element formulation of nonlinear vibration analysis.Comput. Struct. 15: 643–646

    Article  MATH  Google Scholar 

  • Sarma B S, Varadan T K 1983 Lagrange-type formulation for finite element analysis of nonlinear beam vibrations.J. Sound Vib.86: 61–70

    Article  Google Scholar 

  • Sarma B S, Varadan T K 1984 Ritz finite element approach to nonlinear vibrations of beams.Int. J. Numer. Methods Eng.20: 353–367

    Article  MATH  Google Scholar 

  • Sarma B S, Varadan T K 1985 Ritz finite element approach to nonlinear vibrations of a Timoshenko beam.Commun. Appl. Numer. Methods 1: 23–32

    Article  MATH  Google Scholar 

  • Sarma B S, Varadan T K, Prathap G 1988 On various formulations of large amplitude free vibrations of beams.Comput. Struct.29: 959–966

    Article  MATH  Google Scholar 

  • Sathyamoorthy M 1982a Nonlinear analysis of beams, Part-I: A survey of recent advances.Shock Vib. Dig 14: 19–35

    Article  Google Scholar 

  • Sathyamoorthy M 1982b Nonlinear analysis of beams, Part-II: Finite-element methods.Shock Vib. Dig. 14: 7–18

    Article  Google Scholar 

  • Shi Y, Mei C 1996 A finite element time domain model formulation for large amplitude free vibrations of beams and plates.J. Sound Vib. 193: 453–465

    Article  Google Scholar 

  • Singh G, Sharma A K, Rao G V 1990a Large-amplitude free vibrations of beams — A discussion on various formulations and assumptions.J. Sound Vib. 142: 77–85

    Article  Google Scholar 

  • Singh G, Rao G V, Iyengar N G R 1990b Reinvestigation of large amplitude free vibrations of beams using finite elements.J. Sound Vib. 143: 351–355

    Article  Google Scholar 

  • Singh G, Rao G V, Iyengar N G R 1991 Analysis of the nonlinear vibrations of unsymmetrically laminated composite beams.AIAA J. 29: 1727–1735

    MATH  Google Scholar 

  • Srinivasan A V 1965 Large amplitude free oscillations of beams and plates.AIAA J. 3: 1951–1953

    Google Scholar 

  • Srinivasan A V 1966 Nonlinear vibrations of beams and plates.Int. J. Nonlinear Mech. 1: 179–191

    Article  Google Scholar 

  • Srirangarajan H R 1994 Nonlinear free vibrations of uniform beams.J. Sound Vib. 175: 425–427

    Article  MATH  Google Scholar 

  • Szilard R 1978 A hybrid, finite element-finite difference approach to simplified large deflection analysis of structures.Comput. Struct. 9: 341–350

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CSS Foundation, B4/4, Vaigai Nagar, 625 016, Madurai, India

    Sudhakar R. Marur

Authors
  1. Sudhakar R. Marur
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Marur, S.R. Advances in nonlinear vibration analysis of structures. Part-I. Beams. Sadhana 26, 243–249 (2001). https://doi.org/10.1007/BF02703386

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02703386

Keywords

Search

Navigation