# Application of exponential functions in weighted residuals method in structural mechanics. Part II: static and vibration analysis of rectangular plate

## Authors

• Yulia Bai Igor Sikorsky Kyiv Polytechnic Institute, Ukraine
• Igor Orynyak Igor Sikorsky Kyiv Polytechnic Institute, Ukraine

## Keywords:

rectangular plate; clamped – clamped plate; Galerkin method; weighted residual method; free vibration; natural frequecies and modes; weight functions.

## Abstract

The paper is continuation of our efforts on application of the properly constructed sets of exponential functions as the trial (basic) functions in weighted residuals method, WRM, on example of classical tasks of structural mechanics. The purpose of this paper is justification of new method’s efficiency as opposed to getting new results. So, static deformation and free vibration of isotropic thin – walled plate are considered here. Another peculiarity of paper is choice of weight (test) functions, where three options are investigated: it is the same as trial one (Galerkin method); it is taken as results of application of differential operator to trial function (least square method); it equals to the second derivative of trial function with respect to both x and y coordinate (moment method). Solution is considered as product of two independent sets of functions with respect to x or y coordinates. Each set is the combination of five consequent exponential functions, where coefficient at first function is equal to one, and four other coefficients are to satisfy two boundary conditions at each opposite boundary. The only arbitrary value in this method is the scaling factor at exponents, the reasonable range of which was carefully investigated and was shown to have a negligible impact on results.

Static deformation was investigated on example of simple supported plate when outer loading is either symmetrical and concentrated near the center or is shifted to any corner point. It was demonstrated that results converge to correct solution much quickly than in classical Navier method, while moment method seems to be a best choice. Then method was applied to free vibration analysis, and again the accuracy of results on frequencies and mode shape were excellent even at small number of terms. At last the vibration of relatively complicated case of clamped – clamped plate was analyzed and very encouraged results as to efficiency and accuracy were achieved.

## References

I.V. Orynyak, Y.P. Bai, “Application of exponential functions in weighted residuals method in structural mechanics on example of axisymmetrical shell problem”, Mechanics and Advanced Technologies, No. 3 (90), pp. 19–28, 2020. https://doi.org/10.20535/2521-1943.2020.0.209618

C.A.J. Fletcher, Computational Galerkin methods. New-York, USA: Springer-Verlag, 1984. https://doi.org/10.1007/978-3-642-85949-6

K.-J. Bathe, Finite Element Procedures. 2nd ed, Watertown, New-York, USA, 2016.

M.J. Gander and G. Wanner, “From Euler, Ritz, and Galerkin to modern computing”, SIAM Review, Vol. 54, No. 4, pp. 627–666, 2012. https://doi.org/10.1137/100804036

A.W Leissa, “The historical bases of the Rayleigh and Ritz methods”, Journal of Sound and Vibration, Vol. 287, No 4–5, pp. 961–978, 2005. https://doi.org/10.1016/j.jsv.2004.12.021

A.S. Sayyad, “On the free vibration analysis of laminated composite and sandwich plates: A review of recent literature with some numerical results”, Composite Structures, Vol. 129, pp. 177–201, 2015. https://doi.org/10.1016/j.compstruct.2015.04.007

S.P. Timoshenko and S. Woinowsky-Krieger. Theory of plates and shells. New York, USA: McGraw-Hill, 1959.

V. Meleshko. “Bending of an elastic rectangular clamped plate: exact versus “engineering” solutions”, Journal of Elas-ticity, Vol. 48, pp. 1–50, 1997. https://doi.org/10.1023/A:1007472709175

W. Voigt, “Bemerkungen zu dem problem der transversalen schwingungen rechteckiger platten“, Nachr. Ges. Wiss., No. 6, pp. 225–230, 1893.

A.W. Leissa, “The free vibration of rectangular plates”, Journal of Sound and Vibration, Vol. 31, No. 3, pp. 257–293, 1973. https://doi.org/10.1016/S0022-460X(73)80371-2

D.J. Gorman and S.D. Yu, “A review of the superposition method for computing free vibration eigenvalues of elastic structures”, Computers Structures, Vol. 104–105, pp. 27–37, 2012. https://doi.org/10.1016/j.compstruc.2012.02.018

R. Banerjee et al., “Dynamic stiffness matrix of a rectangular plate for the general case, Journal of Sound and Vibration, Vol. 342, pp. 177–199, 2015. https://doi.org/10.1016/j.jsv.2014.12.031

G.B. Warburton, “The vibration of rectangular plates”, Proc. Inst. Mech. Engrs., Vol. 168, pp. 371–384, 1954. https://doi.org/10.1243/PIME_PROC_1954_168_040_02

Moreno-García P. et al., “A review and study on Ritz method admissible functions with emphasis on buckling and free vibration of isotropic and anisotropic beams and plates, Archives of Computational Methods in Engineering, Vol. 25, pp. 785–815, 2018. https://doi.org/10.1007/s11831-017-9214-7

D. Young, “Vibration of rectangular plates by the Ritz method”, J Appl Mech, Vol. 17, No. 4, pp. 448–453, 1950.

J.R. Gartner “Improved numerical computation of uniform beam characteristic values and characteristic functions”, J. Sound Vib, Vol. 84, No. 2, pp. 481–489, 1982. https://doi.org/10.1016/S0022-460X(82)80029-1

R.B. Bhat,” Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh-Ritz meth-od”, J. Sound Vib., Vol. 102, No. 4, pp. 493–499, 1985. https://doi.org/10.1016/S0022-460X(85)80109-7

C.S. Kim et al., “On the flexural vibration of rectangular plates approached by using simple polynomials in the Rayleigh-Ritz method”, Journal of Sound and Vibration, Vol. 143, No. 3, pp. 379 – 394, 1990. https://doi.org/10.1016/0022-460X(90)90730-N

G.B. Chai, “Free vibration of rectangular isotropic plates with and without a concentrated mass”, Comput Struct, Vol. 48, No. 3, pp. 529–533, 1993. https://doi.org/10.1016/0045-7949(93)90331-7

D. Zhou, “Natural frequencies of rectangular plates using a set of static beam functions in Rayleigh-Ritz method”, Jour-nal of Sound and Vibration, Vol. 189, No. 1, pp. 81–87, 1996. https://doi.org/10.1006/jsvi.1996.0006

M. El-Gamel et al., “Sinc-Galerkin solution to the clamped plate eigenvalue problem”, SeMA Journal, Vol. 74, pp. 165–180, 2017. https://doi.org/10.1007/s40324-016-0086-9

R.D. Blevins, “Formulas for dynamics, acoustics and vibration”, in Natural frequency of plates and shells, New Jersey, USA: John Wiley and Sons, 2016, ch.5. https://doi.org/10.1002/9781119038122.ch5

S. Durvasula, “Natural frequencies and modes of clamped skew plates”, AIAA Journal, Vol. 7, pp. 1164–1167, 1969. https://doi.org/10.2514/3.5296

D.J. Gorman, Free vibration analysis of rectangular plates. Amsterdam: Elsevier, 1982. https://doi.org/10.1115/1.3162564