Accurate Calculation of Free Frequencies of Beams and Rectangular Plates

Authors: R .Lassoued, M. Guenfoud

Abstract:

An accurate procedure to determine free vibrations of beams and plates is presented. The natural frequencies are exact solutions of governing vibration equations witch load to a nonlinear homogeny system. The bilinear and linear structures considered simulate a bridge. The dynamic behavior of this one is analyzed by using the theory of the orthotropic plate simply supported on two sides and free on the two others. The plate can be excited by a convoy of constant or harmonic loads. The determination of the dynamic response of the structures considered requires knowledge of the free frequencies and the shape modes of vibrations. Our work is in this context. Indeed, we are interested to develop a self-consistent calculation of the Eigen frequencies. The formulation is based on the determination of the solution of the differential equations of vibrations. The boundary conditions corresponding to the shape modes permit to lead to a homogeneous system. Determination of the noncommonplace solutions of this system led to a nonlinear problem in Eigen frequencies. We thus, develop a computer code for the determination of the eigenvalues. It is based on a method of bisection with interpolation whose precision reaches 10 -12. Moreover, to determine the corresponding modes, the calculation algorithm that we develop uses the method of Gauss with a partial optimization of the "pivots" combined with an inverse power procedure. The Eigen frequencies of a plate simply supported along two opposite sides while considering the two other free sides are thus analyzed. The results could be generalized with the case of a beam by regarding it as a plate with low width. We give, in this paper, some examples of treated cases. The comparison with results presented in the literature is completely satisfactory.

Keywords: Free frequencies, beams, rectangular plates.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1078368

References:

[1] GJ.S. Wu, C.W. Dai, Dynamic responses of multi-span non uniform beam due to moving loads, J. Struct. Eng. 113,1987, pp. 458-74.
[2] F. Moussu, M. Nivoit , Determination of elastic constants of orthotropic plates by a modal anlysis/method of superposition, Journal of sound and vibrations 165 (1), 1993, pp. 149-163.
[3] D.J. Gorman, Highly accurate free vibration eigenvalues for the completely free orthotropic plate, Journal of sound and vibrations, 280, 2005, pp. 1095-1115.
[4] D.J. Gorman, freeing-plane vibration analysis of rectangular plates with elastic support normal to the boundaries, Journal of sound and vibrations, article in press, available online at www.sciencedirect.com .
[5] A.W. Leissa,, Vibrations of plates, NASA SP-160.
[6] A.W. Leissa, The free vibration of rectangular plates, Journal of sound and vibrations, 31,1973, pp. 257-293.
[7] C.W. Lim and all, Numerical aspects for free vibration of thick plates. Part I : Formulation and verification, Computer methods in Applied Mechanics and Engineering, 156, 1998a, pp.15- 29.
[8] N.J. Huffington and W.H. Hoppmann, On the transverse vibrations of rectangular orthotropic plates, Journal of Applied Mechanics ASME, 25, 1958, pp. 389-395.
[9] X.Q. Zhu and S.S. Law, Identification of vehicle axle loads from bridge dynamic responses, Journal of Sound and Vibration , 236 (4), 2000, pp. 705-724.
[10] F.T.K. Au and M.F. Wang, ÔÇ× Sound radiation from forced vibration of rectangular orthotropic plates under moving loads", Journal of Sound and Vibration, 281, (2005), pp. 1057- 1075.