SOME VIBRATION PROBLEMS IN ELASTICITY

Abstract

this thesis Is an attempt to study some of the problems of vibrations of beams and plates on elastic founnation. It is divided IvM ami parts, me first part deals with problems relating to the vibrations of uniform b ams which includes tha effects of rotatory inertia and shear. The equations for vibrations of a uniform free-free beam mooordlng It flmosheako theory resting on elastic foundation have been numerically solvod, Tibrations of beams with internal viscous deap* ing are also studied. Tiaoo-elastle beams on elastic foundation with the standard linear element typo of internal damping are cmsldered. Deflection amplitudes at different points on the beam for the case of forced vibrations of cantilever h«aa of vi*eo*ela*tle material on elastic faun* »tion are calculated. ?he eaaes of beams of elastic and vieoe~slastie material oa vlsee-elastlc fee*'at Ian have also been discussed and deflection amplitudes for forced vibrations of cantilever elastic b*am on visoo-elastie foundation are calculated at diffe rent points cm the beam. In these oisee external damping has not been c neidered. In the second part we have obtained the vibratloas equations of an isotropic elastic plate resting on elastic foundation according to Klailin'a theory, and solved levlea'ly for t a case of a square plate resting on elastic foundations. Vibrations of elastle moderately thick circular plate resting on elastic foundation are considered. ?na equations of vibration are obtained by applying Hamilton's principle. The a lution of the sanations la obtained in terns of Bessel functions with the aid of three auxiliary variables. The frequencies for first five modes are obtained for torsi unless axlsymmetrlc vibrations with frea edge conditions* for sonsymmetric vibrations with free edge conditions, also the frequencies for first five nodes are obtained, Porotd vibrations of a circular plate resting on elastic faun**tie« are considered by taktmg Wading function to be uniform an well as a variable. numeric*! 1 solution for tha cases, wh#n shear and rotatory inertia effects negl ected and when these are included, are givan by expandlag the Vessel function Tmstly, a stiffened plate resting on elastic foundation Is considered. :iers an orthotopic plate in wnloh the offset of rttatory inertia is taken into account is taken to be an equivalent to stiffened plate, Preameoetes with different node -umbers are calculated. Hal onteata of each chapter are cut lined below t- Chnnter X - fhe behaviour of the foundation Is considered and the reason for taking in our thesis a particular type of foundation (i.e. flakier»s type) is explained. Behav iour if viseo-elastie material is also explained. Chanter XI- Tlbratlons of a beam resting on slactic faun*at ion have be*n considered, fccaerieal solution for the deflect on amplitudes is obtained for five different slendemess rutios. In each case three madam of vibrations are calculated. ir III - ?ibrations of a visoo-elactic beam on elastic foundation are considered for the oase of a canti lever, whoae clamped and executes force* lateral oscill ations, flsfleetlem amplitudes at M differs at points ^n the beam for different frequencies are calculated. Ohepter IT- Viorfttioas of a hmmsj on viseo-alaatlo found* ation are considered for t';e cases of the beam of elastic and vieeo-elaetie material. Such types of beans are of practical importance in b£m8 and wells. The solution is given for the oae* of elastic beam on visco-elactic few* ation. >r T- Vibrations of a rectangular plate resting on elastic fountattoa are considered according to Tlndlln's theory. 'Ssre the t henry deludes the effects of shear and rotatory inertia in the sane way as Timoshenko's theory of bare, flumerlcal Walts are worked ant f^r various ratios of ttUUsaMmn to the side. Chapter ?i- Asymmetric vibrations of a circular plate resting on elastic f outlet ion are considered as log Hamilton** prlneipls. dilution in terms of Vessel function using three auxiliary varioles is found sat. Pre que malea for first five modes are calculated. aanatcr TXX- Axiey^metric vibrations af a circular plate reetimg on elastic fc-adstsm are considered, equations of (CV? notion* are derived aalag -Hwilton*s principle, Frequencies for the firct five nodes arc calculated. Chapter Till- forced vibrations of a circular plate rest ing on elastic foundation are emaldered. Here we have taken the equation of vibration of circular plate resting on elastic foundation when the secondary effects of chear and rotatory inertia are neglected and compared when thece effacte are taken Into account. !he loading function is taken to be uniform. Ala^ we have take^ the equation of vib.-atioR of circular plate resting on elastic foundation when aemon*ary effects are taken into account and the loading function is a variable. *e have derived the equat ions for solution using !iankel transform. Chanter I&. Tibrat Ions of stiffened plate resting on elastic foundations are considered, "fere an orthotrople plate in w• Ich effect of rotatory 'martin is taken Into account to be an equivalent stiffened plate. Frequencies for different node numbers are calculated.