The second harmonic and the validity of Duffing’s equation for vibration of beams with large displacements

Abstract Most analytical and numerical investigations on the transverse vibration of beams with large displacements only find odd harmonics in the response. However, experimental research indicates that the second harmonic is also present in the response and in this paper an explanation for this is sought. Two possibilities are explored. First, the in-plane inertia is not neglected and, therefore, the equations of motion have quadratic and cubic non-linearities, i.e., they are not equations of Duffing type. Secondly, small perturbations, characteristic of experimental work, are included in the excitation and the ensuing vibration is analysed.

[1]  Roman Lewandowski,et al.  Non-Linear Free Vibrations of Beams By the Finite Element and Continuation Methods , 1994 .

[2]  C. Chia.  Nonlinear analysis of plates , 1980 .

[3]  Darren Redfern,et al.  The Maple handbook , 1994 .

[4]  R. White,et al.  The effects of large vibration amplitudes on the fundamental mode shape of a clamped-clamped uniform beam , 1984 .

[5]  R. Seydel.  From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis , 1988 .

[6]  R. Lewandowski.  Computational formulation for periodic vibration of geometrically nonlinear structures—part 2: Numerical strategy and examples , 1997 .

[7]  Roman Lewandowski,et al.  Computational formulation for periodic vibration of geometrically nonlinear structures—part 1: Theoretical background , 1997 .

[8]  Maurice Petyt,et al.  NON-LINEAR VIBRATION OF BEAMS WITH INTERNAL RESONANCE BY THE HIERARCHICAL FINITE-ELEMENT METHOD , 1999 .

[9]  R. White,et al.  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 , 1991 .

[10]  T. K. Varadan,et al.  Ritz finite element approach to nonlinear vibrations of beams , 1984 .

[11]  B. Lazan.  Damping of materials and members in structural mechanics , 1968 .

[12]  C. Mei,et al.  A Finite Element Method for Nonlinear Forced Vibrations of Beams , 1985 .

[13]  C. Mei,et al.  A FINITE ELEMENT TIME DOMAIN MODAL FORMULATION FOR LARGE AMPLITUDE FREE VIBRATIONS OF BEAMS AND PLATES , 1996 .