Thesis (M.S., Mechanical Engineering)--University of Idaho, June 2014 | This thesis demonstrates the practical application of a generalized method to solve for beam deflection. Castigliano’s theorem is augmented through the use of the method of Lagrange Multipliers to solve nearly any beam deflection problem. This novel method, first developed by the late Dr. Ju of the University of New Mexico, is used as the foundation to develop a simple tool to solve a marine propulsion shaft alignment problem. This method proved to be computationally inexpensive and the program can be run on nearly any computer. An optimization code determines the optimal bearing offsets which allow the shaft alignment to satisfy predetermined alignment constraints. Experimental validation of the computer model’s predicted deflection results were promising, with model deflection results within 10% of experimental values.