Energy time derivative for constrained linearized systems with applications in Lyapunov-based control schemes

A straightforward and systematic method is presented for developing an expression for the total time derivative of the total mechanical energy for systems governed by equations of motion that have been linearized in some or all of the dependent variables and/or are subject to kinematical constraints. While not guaranteeing that all quadratic terms that would arise from a series expansion of a totally nonlinear expression for the energy time derivative are retained, the method does provide a convenient and consistent approach to developing an expression for the energy time derivative for linearized and/or constrained systems whose governing equations of motion are so complex as to preclude, for practical purposes, the formulation of the energy time derivative by other means. The method is applied to the development of Lyapunov-based control schemes for three illustrative systems, including an axially loaded simply supported beam, a longitudinally accelerated cantilever beam, and a longitudinally accelerated free-free beam traversing two bilateral supports, and results indicate that it produces efficacious as well as practically realizable controllers