ON SOME COUPLED VIBRATION PROBLEMS OF ROTATING AND HARMONICALLY EXCITED ELASTIC BEAMS

Abstract

The investigation presented in this thesis is an attempt to etady some of the coupled vibration problems of rotating and harmonically sxcitsd clastic beams* The thesis is divided into two parts* The first part deals with the problems relating to the coupled bending end torsional vibrations of rotating beams of different geometry to study the importance of design parameters on the vibration characteristics of turbine blades for further developments of axial turboaaehinss and gas turbines and comprises; six chapters* Ths governing differential equstions of coupled vibrations of a beon of linearly varying oross-seotion In a osntrifugal fores flsld have been obtained* The shsar osntrs of each oroso~seotion of ths beam dose not coincide with the centre of gravity, conse quently torsional and bending oscillations are coupled* The equations are solved by a method based on liaylelgh's quotient for a semi-circular blade taking the shaps function in ths form of a series sach tsrm of which satlaflss sll the boundary conditions. The coupled torsion el vibrations of a beam of linearly varying channel sross-seetlon snd an exponential rod in the omtrifugal force field have been obtained -iiusing ths shape function which satisfies ths geometrical as well as dynamical boundary conditions of ths bsam*Here ths effects of hub radius change, area of erose-eeetion ohangs and setting snglss made by the minor axis of Inertia with the direction of circumferential velocity on the coupled frequencies have been studied* The accuracy of the fundamental mods of coupled vibrations Is investi gated by oonsldsrlng the application of the method to uncoupled vibrations* Also, the general equations for ths coupled bendingbending- torsional vibrations of a pretwisted slander bean mounted on the periphery of a rotating disc have been obtained* Ths effecte of the mount of pretwist snd ths hub radius ohangs on the coupled frequencies have been Investigated* The simple theoretical relations hPve been formulatsd using ths perturbation technique for determining the ohangs of oouplsd frequencies for ths small ohangs in ths hub radius and small ohangs In the oresa-seotion at the frss snd of ths rotating bsam of linearly varying cross-section. Higher order representations of frequencies versus hub radius snd oross-e action ohangs are also obtained and a chart is provided to show ths comparison between linear and the infinite order representation* The second part of the thesis consists of problems on eoupled torsionsl vibratione of a beam under harmenio excitation snd comprises four chapters* The governing diffsiential equations of coupled vibrations of a bsam of linearly varying channel cross-ssetion harmonically excited by the bass motion have been obtained* A method based on Rayleigh's quotient has been used to obtain an uppsr bound for ths fundamsntal frequency and ths accuracy of this frequency is Investigatsd similar as sarllsr for ths case of a rotating beam* The perturbation technique has been used to lnve tlgate the effects of small forcing amplitude change snd area of c rose-sect ion ohangs on ths coupled vibrations of a beam of linearly varying eross-eection harmonically excited by the base motion* Also, the gsnsral equations for the banding-bending torsional vibrations of a prstwistsd slender beam excited by the base motion have been obtained* Ths offsets of amount of pretwist and foroing amplitude change on the coupled frequencies have been studied. Lastly, the nonlinear coupled vibrations of a uniform beam with fixed ends excited by the periodic motion of its supporting base In a direction normal to the beam span is investigated analytically. Ths harmonic balance method has been ussd to obtain simple harmonic, supsrhamonlo and subharmonic motions. Ths stability of ths problem has been discussed with the corresponding variational Hill type equation. —It— fae contents of each chapter are outlined as **'•* PA» -I Chaptsr-I A reasonably good survey of literature on vibration problems of rotating and harmonically exoited clastic beams has been mads from local libraries, National Documentation Centre, Hew Delhi and through personal sfforts by requesting centres of advance studies and authors of papsrs on the subject concerned* The upto-date development of the subject with authors has been presented for the completeness of the work. Chapter-II The differential equations for eoupled vibra tions of a beam of linearly varying eross-seetlon in a oentrifugal force a eld are obtained and a method bassd on RaylsighU quotient le used taking a series solution for obtaining an upper bound for the fundamental frequency of vibration, lbs material is to be published in KPVsnbsr (1974) In the Journal of the Aeronautical Society of India. Chapter IXI(a) The differential equations for coupled vibrations of a beam of linearly varying channel crosssecticn in a centrifugal force field are obtained. A method bassd on Raylsigh*s quotlsnt is used to obtain en upper bound for the fundamental frequency and ths accuracy of this nppsr bound is invsstlgatsd by considering the application Of the method to uncoupled vibration* the- contents of the snTtff14rn?H9fi fffft bOQ«FW for PI Proct of Brtione^. faUWf 0* 3c*t» 111(b) The differential equations for eoupled vibrations of a bsam of exponentially varying oroe -action in a centrifugal force field are obtained* Fundamental mode of vibration for different parametric constants ars obtained at two setting anglss mads by the minor axis of inertia with the direction of the circumferential velocity. Ths accuracy of fundamental mode Is also investigated. Chapter-IV The general equations for eoupled vibrations of a prstwistsd slender beam in a centrifugal force field ars obtainsd* The effects of ohangs of various parameters liks prstwist , hub radius and set tint of angle X on the eoupled frequencies have been studied* The Material of ths chapter is accepted for publication in, the Journal of Aeronautical Soojfftr 9* i&U* Chapter -V The offeot of small hub radius ohangs on frequencies of coupled vibrations of a beam of linearly varying e resa-seotion in a oentrifugal force Held are obtainsd by using perturbation technique* The first and higher order relatione are obtained and a chart le provided to show the comparison between the linear and Infinite order representations* -vi— Chnptsr-VI Ths effect of small cross-emotion ohangs on frequencies of coupled vibrations of a bans of linearly varying cross-section in a centrifugal force field ars obtainsd. Using psrturbation technique the solutions upto second order are obtained snd the constants are determined from the known paramstsrs. PA1T-II Chapter-VII The differential equations for coupled vibrations of a beam of linearly varying channel crosssection harmonically exolted by the base action ars obtained end fundamental mode of vibration has been calculated by Raylsigh's quotient method* The accuracy of ths fundamental mods is also investigated* Chapter-VIIZ The perturbation technique is ussd to invest!- gate the effect of email change In forcing amplituds and small area of oross-seotion ohangs on frequencies of coupled vibrations of a beam of linearly varying crosssection excited by ths periodic motion of its supporting Chaptsr-IX General stunt ions of coupled vibrations of a prstwistsd tl-nder beam under harmonic excitation are obtainsd* The effects of amount of pretwist and forcing amplituds change on the frequencies have been tudied. the chapter, are accepted for publication —vii— 9f fteUaa^ toUWi- of sol, Chaptsr-X Nonlinear coupled vibrations of a besa with fixsd ends excited by the periodic motion of its supporting base in a direction normal to the beam span have been investigated* The haxmonie balance method has been ussd to obtain simple harmonic, euperharmonie and subharmonic motions.