We describe the key stages of the development of generalized dynamic theories of bending vibrations of the bars, plates, and shells, based on the shear model proposed by S. P. Timoshenko, outstanding Ukrainian scientist, in 1921 (“On the correction for shear of the differential equation for transverse vibrations of prismatic bar,” Phil. Mag., Ser. 6, 41, No. 245, 744–746). We present the mathematical construction of equations of the theory of plates derived as hyperbolic approximations of the elastodynamic problem for a layer. The analytic expression for the coefficient of thickness shear is obtained. Some contemporary investigations are also discussed. As an example, we consider the process of wave propagation along an elastically restrained strip.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. S. Bakhvalov and M. É. Églit, “On the equations of high order of accuracy used to describe vibrations of thin plates,” Prikl. Mat. Mekh., 69, No. 4, 656–675 (2005).
B. F. Vlasov, “On the equations of the theory of bending of plates,” Izv. Akad. Nauk SSSR, Otd. Tekh. Nauk, No. 12, 57–60 (1957).
R. V. Gol’dshtein, “Supplement. On the application of the method of boundary integral equations for the solution of problems of the continuum mechanics,” in: Mechanics: Advances in Foreign Science [in Russian], Issue 15: T. A. Cruse and F. J. Rizzo (editors), Method of Boundary Integral Equations, Mir, Moscow (1978), pp. 183–209.
É. I. Grigolyuk and I. T. Selezov, “Nonclassical theories of the vibration of bars, plates, and shells,” in: Advances in Science and Technology [in Russian], Ser. Mechanics of Deformable Solids, Vol. 5, VINITI, Moscow (1973), pp. 1–272.
N. A. Kil’chevskii, “Generalization of the contemporary theory of shells,” Prikl. Mat. Mekh., 2, No. 4, 427–438 (1939).
A. B. Kudryashov, “Beam finite element based on the S. P. Timoshenko theory,” in: Transactions of the Central Aerohydrodynamic Institute [in Russian], Issue 2664 (2004), pp. 207–212. (04.11–16В.111).
P. F. Ledorezov, “On the analysis of transverse shears and rotary inertia in the vibration bending of a viscoelastic plate-strip,” in: Mechanics of Deformable Media (Saratov State Univ.) [in Russian], Issue 14 (2002), pp. 144–151.
V. V. Nesterenko, “On the theory of transverse vibrations of the Timoshenko beam,” Prikl. Mat. Mekh., 57, No. 4, 83–91 (1993).
S. D. Ponomarev, V. L. Biderman, K. K. Likharev, et al., Foundations of theContemporary Methods of Strength Analysis in Machine Building: Analysis Under Statistical Loads [in Russian], Mashgiz, Moscow (1950).
I. T. Selezov, “Concept of hyperbolicity in the theory of controlled dynamic systems,” Kibern. Vychisl. Tekh., No. 1, 131–137 (1969).
I. T. Selezov and Yu. G. Krivonos, Mathematical Methods in the Problems of Wave Propagation and Diffraction [in Russian], Naukova Dumka, Kiev (2012).
I. T. Selezov, V. A. Tkachenko, and Yu. P. D’yachenko, “Propagation of harmonic waves in the Timoshenko-type plate with elastically restrained edges,” in: Conf. on the Problems of the Dynamics of Interaction of Deformable Media (Goris, Armenia, March, 1989) [in Russian], Izd. Akad. Nauk Arm. SSR, Yerevan (1990), pp. 240–243.
I. T. Selezov, “Investigation of transverse vibrations of the plate,” Prykl. Mekh., 6, No. 3, 319–327 (1960).
Ya. S. Uflyand, “Wave propagation under transverse vibrations of bars and plates,” Prikl. Mat. Mekh., 12, No. 3, 287–300 (1948).
B. M. Barbashov and V. V. Nesterenko, “Continuous symmetries in field theory,” Fortschr. Phys./Prog. Phys., 31, No. 10, 535–567 (1983).
A. L. Cauchy, “Sur l’equilibre et le mouvement d’une lame solide,” Exercices Math., 3, 245–326 (1828).
Author information
Authors and Affiliations
Additional information
Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 56, No. 2, pp. 102–111, April–June, 2013.
Rights and permissions
About this article
Cite this article
Selezov, I.T. On Construction of the Refined Equations of Vibration of Elastic Plates. J Math Sci 203, 123–133 (2014). https://doi.org/10.1007/s10958-014-2095-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-014-2095-5