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).