Nonlinear Vibration Analysis of Timoshenko Beams Using the Differential Quadrature Method

This paper addresses the large-amplitude free vibration of simplysupported Timoshenko beams with immovable ends. Various nonlineareffects are taken into account in the present formulation and thegoverning differential equations are established based on theHamilton Principle. The differential quadrature method (DQM) isemployed to solve the nonlinear differential equations. Theeffects of nonlinear terms on the frequency of the Timoshenkobeams are discussed in detail. Comparison is made with otheravailable results of the Bernoulli–Euler beams and Timoshenkobeams. It is concluded that the nonlinear term of the axial forceis the dominant factor in the nonlinear vibration of Timoshenkobeams and the nonlinear shear deformation term cannot be neglectedfor short beams, especially for large-amplitude vibrations.

[1]  I. Etsion.  Literature Review : A Review of Mechanical Face Seal Dynamics , 1982 .

[2]  W. Chen,et al.  The Study on the Nonlinear Computations of the DQ and DC Methods , 1999, ArXiv.

[3]  Chang Shu,et al.  ON OPTIMAL SELECTION OF INTERIOR POINTS FOR APPLYING DISCRETIZED BOUNDARY CONDITIONS IN DQ VIBRATION ANALYSIS OF BEAMS AND PLATES , 1999 .

[4]  C. Shu.  Differential Quadrature and Its Application in Engineering , 2000 .

[5]  W. Chen,et al.  The application of special matrix product to differential quadrature solution of geometrically nonlinear bending of orthotropic rectangular plates , 1999, ArXiv.

[6]  T. C. Huang,et al.  New Tables of Eigenfunctions Representing Normal Modes of Vibration of Timoshenko Beams , 1995 .

[7]  M. Sathyamoorthy,et al.  Literature Review : Nonlinear Analysis of Beams Part Ii: Finite Element Methods , 1982 .

[8]  G. Venkateswara Rao,et al.  Large-amplitude free vibrations of beams—A discussion on various formulations and assumptions , 1990 .

[9]  C. Bert,et al.  Differential Quadrature Method in Computational Mechanics: A Review , 1996 .

[10]  Gangan Prathap,et al.  Galerkin finite element method for non-linear beam vibrations , 1980 .

[11]  R. Bellman,et al.  DIFFERENTIAL QUADRATURE AND LONG-TERM INTEGRATION , 1971 .

[12]  M. Sathyamoorthy,et al.  Literature Review : Nonlinear Analysis of Beams Part I: A Survey of Recent Advances , 1982 .

[13]  Wei Chen,et al.  ON THE DQ ANALYSIS OF GEOMETRICALLY NON-LINEAR VIBRATION OF IMMOVABLY SIMPLY-SUPPORTED BEAMS , 1997 .

[14]  G. V. Rao,et al.  Nonlinear vibrations of beams considering shear deformation and rotary inertia , 1976 .

[15]  P.A.A. Laura,et al.  Analysis of Vibrating Timoshenko Beams Using the Method of Differential Quadrature , 1993 .

[16]  R. Bellman,et al.  DIFFERENTIAL QUADRATURE: A TECHNIQUE FOR THE RAPID SOLUTION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS , 1972 .

[17]  C. Shu,et al.  APPLICATION OF GENERALIZED DIFFERENTIAL QUADRATURE TO SOLVE TWO-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS , 1992 .

[18]  D. Evensen.  Nonlinear vibrations of beams with various boundary conditions. , 1968 .

[19]  C. Dolph.  On the Timoshenko theory of transverse beam vibrations , 1954 .

[20]  T. Huang,et al.  The Effect of Rotatory Inertia and of Shear Deformation on the Frequency and Normal Mode Equations of Uniform Beams With Simple End Conditions , 1961 .

[21]  C. Bert,et al.  Application of the quadrature method to flexural vibration analysis of a geometrically nonlinear beam , 1992 .