Integral Relations for Disturbance Isolation

Abstract

Introduction P ASSIVE and active dynamic systems of high order are employed for the purpose of disturbance isolation. Such systems’ theory and designmethods could beneŽ t from using the design theory of active electrical networks. The feasibility and the available performanceof such systemscan be evaluatedusingBode integrals. This enables the system engineer to resolve the design tradeoffs without actually designing the system and the subsystems.A Bode integral is applied to estimate the performanceof a disturbance isolation system. Consider the system in Fig. 1a of two bodies connected with an active strut,2 which is a linearmotor. A force disturbancesource F1 is applied to the body M1 (capital letters designate Laplace transforms). The force F3 is applied via the massless active strut to the body M3 . To increase the disturbance isolation, the force division ratio KF D F3/ F1 should be made small. The strut mobility (in some literature called mechanical impedance) Z2 is the ratio of the difference in the velocities at the ends of the strut to the force (because we neglect the strut’s mass, the force is same at the both ends of the strut). Feedback is employed in the active strut to increase jZ2j in order to reduce jKF j. For the purpose of analysiswe use the following electromechanical analogy: power to power, voltage to velocity, current to force, capacitance to mass, and inductance to the inverse of the stiffness coefŽ cient. The electrical equivalentcircuit for the system is shown in Fig. 1b. The current division ratio I3/ I1 is equivalent to KF . The electrical impedance Z2 is the equivalent of the strut mobility. The current division ratio, i.e., the force division ratio, is