Optimality conditions for maximizing the tip velocity of a cantilever beam

The calculus of variations is a powerful tool for establishing the necessity and sufficiency of the optimality conditions of a class of complex optimization problems. In this study, calculus of variations is applied to a beam problem to maximize its dynamic response. More specifically, the research reported seeks to maximize the tip velocity of a cantilever beam and is delimited to adjusting the height of the beam with the other parameter being held constant. An equivalent problem is defined, and using the Lagrange multiplier method, the Euler equations along with the natural boundary condition which govern the state of solutions are derived.