ELASTIC VIBRATIONS OF BEAMS, PLATES AND SHELLS

Abstract

'"he present thesis considers soae of the problems on elastic vibrations of solid and hollow beams and sandwich plates and shells. It is divided into three parts, "'"he first part d.eals with the transverse vibrations of solid and hollow beams and comprises chapters 1 ant! II. In chapter I, the transverse vibrations of solid rectangular beams are considered retaining th© secondary effects of rotatory inertia and transverse sho»*r deformation. *hm present investigations improve upon *imoshenko' s theory of bending of beams, "^he equations of motion are derived by iamilton principle and the solutions of these equations are compared with those of ""iraoshenko's equation arid tha exact equation p:iven ^y I am*. In Chapter IIf the transverse vibrations of box type bessss (i.e. a rectangular beam hollow in the middle) are considered. The secondary effects of shear <*orce3 and the rotatory in«r*ia are retained and the sides and top ^5 bottom of the b«ara are taken to bend tjBiiagfl different angles due to shear, "^he equations of motion are derived by Hamilton principle. These equations are solved for the case of a canti lever and the frequency parameter obtained fo» various rao'es is compared with those for Tlmoshenko*s equation. The second part of the thesis deals with *he vibrations of sandwich places ana comprises chapters III and IV. In chapter III, *he torsionlesj nxlsyametric vibrations of circular sandwich plates are considered. The equations of motion are derived by Haailton principle. The present ♦heory is applicable -iieven to cases where the facings have •aarsslaele thicknesses compared to core and **or any ratio of material densities and elastic constants of the core and facings. The solution of these equations is obtained in terns of Vessel functions with the aid of three auxiliary variables. The frequency para-eter is computed for clamped and free edge conditions for first four modes of vibrations. Frequencies of circular uniform plates are also obtained for the aake of comparison by taking the facing thickness aero. In chapter IV, the asymmetric vibrations of circular sandwich plates are studied. The equations of motion are derived by Hsoil ton principle by adopting the aaae analysis aa in the last chapter. *he solution of eqxjations of motion it obtain in terms of Basse! functi ,ns with the aid ©f five auxiliary variables. The frequency parameter is computed for clasped edge conditions for the first four modes of vibrations for each of the circumferential modes 1,2 and 3 for various thicknesses of facings and core. The frequency parameter for circular uniform plates is also computed by putting the facing thickness sero in the equations for sandwich rlates. The third part of *he thesis deals ^ith the vibrations of spherical sandwich shells arid comprises chapters V to VIII. In chapter V, the equations for the asymmetric eiferatlona of aoderatelv thick, non-shallow, spheric-.! sandwich shells are derived by Haailton principle. The secondary effects of transverse shear deformation and rotatory Inertia are taken into consideration both for the core and the facings and no restriction is imposed on the thickness ra*io of the facings to tha* of core. -iii- The solution of equations of motion is obtained in terms of associated Legendre functions with the help of seven auxiliary variables, ^e nature of the various typea of solutions tor a closed spherical sandwich shell hava also been discussed. In chapter VI, the equations governing the torsionleas nxisyametric vibrations of moderately thick, non-shallow, spherical sandwich shells are obtained from the last chapter by considering the notion in the circumferential direction as aero. These equations are solved with the heir of four auxiliary variables in terms of Legendr * functions, "''he frequency parameter is computed for closed shells fo^ various values or thickness of the facings ai*d core and for the open shells close* at the apex for various values of opening an lie. The frequency parameter for r^onoco ;ue shells is also computed By taking the facing thickness as *sro. in chapter VII, the equations governing the torsional vibrations of moderately thick, non-shallow sandwich spherical shells arc obtained from chapter V by considering the motion exclusively in tr;e circumferential direction. These equations are solved with the help of three auxiliary variables in terms of first 4ifferential coefficient of Legendre functions. The frequency parameter is computed for the shell closed at the apex and clamped at the other edge for various values of opening angle and thicknesses of the facings an core, ""he equations for sonocoque shells are alao derived from the equatiuns of sandwich shells by putting facing thickneas aero and thm frequency parameter is computed ,for the s«tke of comparison. In the chapter '/III, the equations governing the vi>?ati©BS -ivof moderately thick, shallow, sandwich spherical shells are derived from the equations of chapter V by taking certain simplifying assumptions. The solutions of the equations for asymmetric, torsionleas axisyometric and torsional vibrations are obtained in terms of Bessel functions with the aid of auxiliary variables, frequency parameter is Computed for torsional vibrations for a shell closed at the apex and clamped at the other edge for various values of opening angle and thicknesses of the core and facings. The equations for monocoque shells are alao derived as a particular case and solved. The extensive numerical work involved in this *hesis was done on IBM 1«0 Computer at ^BHC, Roorkee and on d>C 3*00 Computer at TIFR,Bombay, '"he work presented in this thesis is original research by the author. The aateri&l of chapters I and II has been accepted for publication in Ho.2 o* Vol. 34 (19«p) of the Proceedings of the National Institute of sciences of India (Fart A) and in Vol.10 of the Indian Journal of Mathematics (196P).