Analaytical solutions of traveling string and beam vibration problem

ABSTRACT In this study, vibrations of axially moving continua are investigated. In chapter 2, the equations of motion for axialiy traveling string and beam are derived using Hamilton 's Principle. Dependence of equations on type and dimensions of the material used is eliminated by nondimensionalizing the equations of motion. Then the solutions are obtained for string, slender beam and beam by using these equations. In chapter 3, the string problem is considered. Lie Group Theory is used to solve the equation. Since the string velocity is assumed to be arbitrary, Equivalence Transformation is applied with the Lie Group Theory. The symmetries of the equation are obtained. By using these symmetries exact solutions are found for various velocity functions. Axial velocity is considered in different forms such as arbitrary velocity, constant velocity, constant acceleration velocity, harmonically varying velocity, and exponentially varying velocity. Then, of the exact solutions obtained for various velocity parameters, those that would satisfy the boundary conditions approximately are selected and the boundary conditions are applied to these solutions. Thus approximate solutions satisfying the boundary conditions are obtained. In chapter 4, the flexible beam problem is considered. The equation is obtained by assuming small flexural rigidity. This equation is solved by using the Method of Matched Asymptotic Expansions. First, the outer expansion solution is obtained using the Method of Multiple Scales, a perturbation technique. It is observed that this outer expansion solution does not satisfy some of the boundary conditions for some extreme points. In order to eliminate this problem, inner expansion solutions are obtained by making a second expansion in the neighborhood of extreme points. Then these solutions are combined to obtain composite solutions approximately satisfying all the boundary conditions. The problem is solved for simple- simple and fixed-fixed boundary conditions. Harmonically varying velocity and exponentially varying velocity cases are considered. In chapter 5, axially moving beam problem is considered. Lie Group Theory is also used for the solution of beam equation. Velocity function is assumed to be arbitrary and thus symmetries of the equation are obtained by using the Lie Group Theory combined with Equivalence Transformations. By using these symmetries, exact solutions are found for an axially moving beam. Beam velocity is considered in different forms such as arbitrary velocity, constant velocity, constant acceleration velocity, harmonically varying velocity, and exponentially varying velocity. Then, of the exact solutions obtained, those that would satisfy the boundary conditions are selected and approximate solutions for simply supported beam are obtained. XV