The Vibration Of Beams And Plates Studied Using Orthogonal Polynomials

The orthogonally generated polynomials proposed by R. B. Bhat as admissible functions for use in the Rayleigh-Ritz method by dynamic and static problems of single-span beams and rectangular plates are generalized for use in the study of various more complicated beam and plate problems. The orthogonal polynomials themselves are discussed in some detail. These functions are then used in the Rayleigh-Ritz method to obtain solutions for the free vibration problems of slender beams, thin rectangular plates, a box-like structure, and thin annular, circular and sectorial plates. Various complicating effects are included.;The approach presented in this thesis is straightforward but more general than the approaches presented previously in the literature. The accuracy and versatility of the approach are demonstrated using various example problems. Numerical results are generated both for new problems and for problems for which comparison results are available in the literature.;For the vibration problem of slender beams, the analysis is presented for beams subject to complicating factors which include the existence of an arbitrary number of concentrated masses (with or without rotary inertia) and/or intermediate simple supports, and subject to any combination of free, simply supported or clamped boundary conditions and/or elastic supports. The effects of constant axial loading (either constant directional or tangential follower force) and variable cross section are also included.;For thin rectangular plates, the analysis is presented for rectangularly orthotropic plates (which include the isotropic case) with any number of intermediate line supports and point supports. (The Lagrangian multiplier method is used for point supported plates.) The analysis is further extended to one type of box-like structure.;The analysis of annular plates includes the effects of polar orthotropy, intermediate concentric ring supports and radially varying thickness. By permitting the inner radius to become very small, circular plates are also treated, including the case with a central point support.;Finally, annular and circular sectorial plates are treated. Again the analysis includes several complicating effects such as polar orthotropy, intermediate simple supports in both radial and circumferential directions and varying thickness in both directions