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- PublicationTravelling waves in boundary-controlled, non-uniform, cascaded lumped systemsA 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.
- PublicationBoundary-controlled travelling and standing waves in cascaded lumped systemsThis paper shows how pure travelling waves in cascaded, lumped, uniform, mass-spring systems can be defined, established, and maintained, by controlling two boundary actuators, one at each end. In most cases the control system for each actuator requires identifying and measuring the notional component waves propagating in opposite directions at the actuator-system interfaces. These measured component waves are then used to form the control inputs to the actuators. The paper also shows how the boundaries can be actively controlled to establish and maintain standing waves of arbitrary standing wave ratio, including those corresponding to the classical modes of vibration of such systems with textbook boundary conditions. These vibration modes are achieved and maintained by controlled reflection of the pure travelling wave components. The proposed control systems are also robust to system disturbances: they react to overcome external disturbances quickly and so to re-establish the desired steady motion.