PROBLEMS ON VIBRATIONS OF ANISOTROPIC CIRCULAR PLATES AND SPHERICAL SHELLS

Abstract

The present investigation is a step to satisfy the challenging needs of the field of vibration of elastic bodies. This work is an attempt to study some of the problems on vibration of anisotropic plates and shells. It is divided into two parts. The first part deals with the transverse and longitudinal vibrations of elastic circular plates. Vibrations of elastic spherical shells are discussed in the second part. Chapter I of the thesis is devoted to the introduction. The development of the subject upto the present day has been presented in the graded manner. The importance of the present field and the scope of the further research work is also given. PART ONE of the thesis comprises four chapters. Chapter II deals with the torsionless axisymmetric transverse vibrations of circular plates. Equations governing transverse vibrations of circular plates, having transverse isotropy, are derived from the equations of motion of elasticity by integration over the thickness of the plate. The equations of motion include the secondary effects due to rotatory inertia and transverse shear. The solution is obtained in terms of Bessel functions. Two edge condition^, clamped and free, are considered. Numerical results for the natural frequencies of first four modes are found out. The plots for the relative displacement and the strain component are obtained for each frequency. A comparison is made in the behaviour of an isotropic plate and a transversely isotropic plate. In chapter III the equations governing asymmetric transverse vibrations of transversely isotropic circular plates are solved in terms of Bessel functions. Natural frequencies for first four modes are calculated for free and clamped edge conditions. Two materials are taken for the purpose. A comparative study of the frequencies as predicted by classical theory, shear theory and the general theory, is made. The mode shapes are also shown in each normal mode according to the general theory. The dynamic response of moving forces is studied in chapter IV. We consider that the plate is subjected to a dynamic surface and body force field. The equations for asymmetric transverse vibrations of transversely isotropic circular plates are solved. The auxiliary variables are introduced for the external force field and displacement components. Further variables are introduced to simplify the equations and to get a solution in terms of Bessel functions. The functions representing the system of forces are now taken in the form of infinite series in r. Frequencies for five modes are found for axisymmetric forced vibrations for various values of the amplitude of the dynamic force. Chapter V deals with the equations governing longitudinal vibrations of transversely isotropic circular plates. These equations are derived from the three dimensional equations of elasticity by integration over the plate thickness. XI These equations include the secondary effects due to rotatory inertia, transverse shear and transverse normal strain. A separable solution in terms of Bessel functions is obtained. Frequency equation for a roller supported edge condition is obtained. The cases of torsional and torsionless axisymmetric vibrations are deduced as particular cases. Numerical results for the natural frequencies and mode shapes for first six normal modes for asymmetric and torsionless axisymmetric vibrations are found for two different materials. A comparison of frequencies as predicted by classical theory and the general theory is also given. PART TWO of the thesis deaLs with the vibrations of non shallow and shallow anisotropic spherical shells and comprises five chapters. Chapter VI contains the derivation and discussion of the equations of motion of non shallow orthotropic spherical shells. The equations are derived by Hamilton's principle. These equations include the terms due to rotatory inertia, radial shear, radial normal stress, radial normal strain, surface loads and body forces. Appropriate boundary and initial conditions are derived. Compatibility relations are obtained. Uniqueness of solution is shown. It is seen that the conjugate property is satisfied and the reciprocal theorem holds. The equations governing torsionless axisymmetric vibrations of transversely isotropic non shallow spherical shells are solved in chapter VII. The solution is obtained in terms of Legendre functions. Frequency equations are obtained xii for clamped and free edge conditions. Natural frequencies for various opening angles and various values of thickness of the shell are found. Frequencies for first six modes for hemispherical shells made of two different materials with clamped and free edge conditions are obtained. The relative plots for displacements and strains are shown. In chapter VIII the equations for asymmetric vibrations of transversely isotropic non shallow spherical shells are solved in terms of associated Legendre functions. Natural frequencies of first seven modes of a hemispherical shell with clamped edge condition for various values of thickness are obtained for two different materials. In chapter IX the single equation governing torsional vibrations of an orthotropic spherical shell is solved in terms of associated Legendre functions and Bessel functions. Numerical results are obtained for the natural frequencies of the shells of various opening angles for clamped and free edge conditions. The effect of the thickness of the shell on the frequency is studied. A relative study of the behaviours of the three material shells is also made. Chapter X deals with the vibrations of shallow spherical shells in the presence of dynamic surface and body forces. The equations of motion are derived by integration over the shell thickness. The auxiliary variables for force field and displacements are introduced to obtain a simplified form of equations. Solution is obtained in terms of Bessel functions. xiii She elastoki.netic response of the shell to dynamic surface and body force field is studied. The field is expressed in terms of various power series and the solution is also obtained in power series. Solution for torsionless axisymmetric and torsional axisymmetric vibrations are deduced as particular cases. The numerical work presented in the thesis is done on IBM 1620 Computer at S.E.R. C. , Roorkee and IBM 360 at Delhi School of Economics, Delhi. The material of chapters III and VII was presented by author in 14th Congress of Theoretical and Applied Mechanics held at Kurukshetra in December 1969.