Travelling waves in boundary-controlled, non-uniform, cascaded lumped systems

A companion paper considers travelling and standing waves in cascaded, lumped, mass-spring systems, controlled by two boundary actuators, one at each end, when the system is uniform. It first proposes definitions of waves in finite lumped systems. It then shows how to control the actuators to establish desired waves from rest, and to maintain them despite disturbances. The present paper extends this work to the more general, non-uniform case, when mass and spring values can be arbitrary. A special “bi-uniform” case is first studied, consisting of two different uniform cascaded systems in series, with an obvious, uncontrolled, impedance mismatch where they meet. The paper shows how boundary actuator control systems can be designed to establish, and robustly maintain, apparently pure travelling waves of constant amplitude in either the first or the second uniform section, in each case with an appropriate, partial, standing wave pattern in the other section. Then a more general non-uniform case is studied. A definition of a “pure travelling wave” in non-uniform systems is proposed. Curiously, it does not imply constant amplitude motion. It does however yield maximum power transfer between boundary actuators. The definition, and its implementation in a control system, involves extending the notions of “pure” travelling waves, of standing waves, and of input and output impedances of sources and loads, when applied to non-uniform lumped systems. Practical, robust control strategies are presented for all cases. ► We define travelling waves in cascaded, lumped, non-uniform, mass-spring strings.► We control two boundary actuators to achieve travelling waves in non-uniform systems. ► We define wave power and wave impedance for arbitrarily non-uniform systems. ► We postulate the conditions for maximum power transfer from boundary to boundary. ► We show how practical, boundary-actuator control systems should be implemented.