ALMOST-BALANCED STRUCTURAL DYNAMICS

Abstract

Analytical models and experimentally derived models are often represented in the modal co-ordinates. The modal models include natural modes and their displacements. The scaling of the modes is arbitrary, and thus the scaling factor can be freely chosen. By the proper choice of the scaling factor one obtains controllability and observability grammians of a structure that are almost equal, and the system model is almost balanced. This new model is given either in the form of a second order differential equation or in state space representation. The properties of the almost-balanced modes and structures are derived, including its H2, H∞and Hankel norms. The norms of the modes and of the system are expressed in terms of the modal parameters, such as natural frequencies, modal damping, and input and output gains. The properties of the almost balanced structure are used to reduce the model. In particular, the H2, H∞and Hankel system norms are used to evaluate and to minimize the reduction error. Next, the method of placement of actuators or sensors in the almost-balanced co-ordinates is presented. The norm of the almost-balanced mode with a set of actuators (or sensors) is the root-mean-square sum of the norms of this mode for each single actuator (or sensor). Using this property one finds a specified number of actuator or sensor locations such that the system performance evaluated at these locations is close to that of the system with a larger set of candidate locations.