"The material presented in this thesis is designed to provide the practicing engineer with a practical, approximate method of solution for the increasingly important problem of plate vibrations. The technique presented herein is basically an application of the use of generalized coordinates to describe the elastic and inertial properties of a finite element representation of a nonuniform rectangular plate. The plate will be idealized by utilizing a modification of the grid work technique developed by Hrennifkoff as a lattice work of interconnecting beams. By employing the elastic displacements as generalized coordinates, the only required compatibility conditions are that the displacements at beam interconnecting points be the same for beams in mutually perpendicular planes. Relationships are then derived which will allow the system kinetic and potential energies to be expressed as functions of linear algebraic equations. the vibratory equations of motion are then obtained by application of Lagrange's equations. The resulting equations are solved by standard matrix manipulation techniques (3, 10) employing a large high speed digital computer. It should be noted that pairs of identical roots will appear for the case of a square plate. this condition limits the practicality of using matrix iterative premultiplication for a solution procedure, as this technique is extremely sensitive to convergence for repeated roots. Therefore, the solution used for this problem was a technique developed by C. G. J. Jacobi, and modified by J. Von Neumann for use on large digital computers, which will converge for any number of repeated roots. Note that although the actual solution performed for this thesis was for a uniform plate, the method may be applied directly to any arbitrary cross-section. To apply this method to a nonuniform plate it is only necessary to obtain an approximation of the mass and stiffness properties at each cross-section (grid point)"--Abstract, page ii-iii.
Basye, C. B.
Cunningham, Floyd M.
Keith, Harold D. (Harold Dean), 1941-
Mechanical and Aerospace Engineering
M.S. in Engineering Mechanics
University of Missouri at Rolla
vii, 87 pages
© 1968 F. D. Utterback, All rights reserved.
Thesis - Open Access
Library of Congress Subject Headings
Frequencies of oscillating systems
Plates (Engineering) -- Vibration
Vibration -- Mathematical models
Print OCLC #
Electronic OCLC #
Link to Catalog Record
Utterback, F. David, "A numerical (matrix) solution for the vibration frequencies and modes of a nonuniform rectangular plate with arbitrary edge conditions" (1968). Masters Theses. 6766.