Application of Distributed Parameter Concepts to Dynamic Analysis and Control of Bending Vibrations

As an alternative to the normal vibration mode approach to dynamic analysis and control of bending vibrations, such vibrations are studied in terms of the distributed parameter concepts of propagation, reflection, and characteristic termination. In particular, the dynamics of lateral vibration of a thin uniform beam are factored into a form that separates the process of propagation from boundary effects. This allows the effects of various terminal impedance matrices to be described in terms of a reflection matrix, which is a generalization of the concept of reflection coefficient for the wave equation. It is shown that the reflection matrix can be nulled by terminating a beam in its characteristic impedance matrix. Several special cases of terminal impedance matrix are considered in detail. The reflection matrices are derived for these cases, and the response of control systems incorporating these terminal impedance matrices is studied by analog simulation.