Discrete mechanics for 1-dimensional distributed parameter mechanical systems under free boundary conditions and its application to vibration suppression control of free-fixed strings

Abstract

This study develops discrete mechanics for 1-dimensional distributed parameter mechanical systems under free boundary conditions, and a control method for the systems based on a blending method of discrete mechanics and nonlinear optimization. First, under the assumption on free boundary conditions, discrete Euler-Lagrange equations and boundary equations are derived by discrete Hamilton's principle. Next, for the discrete Euler-Lagrange equations with control inputs and boundary equations, a nonlinear optimal control problem with constraints by setting an objective function, and initial and boundary conditions is formulated. Then, a vibration suppression control problem for a free-fixed string is considered as a physical example. As a result, it turns out that vibration of the string is suppressed and the whole of the system is stabilized by the proposed method.