ISSUES IN REDUCED ORDER MODELING OF MECHANICAL SYSTEMS

A common modeling technique in investigating the dynamics of mechanical systems is the use of second order matrix differential equations. In such a formulation, the coefficient matrices contain the physical mass, damping, and stiffness parameters of the sys- tem, which in turn affect the vibrational parameters such as the nat- ural frequencies and modeshapes. The construction of a mass-- damping--stiffness model based on material properties and the systemís geometry, as done in finite element analysis, is a relatively straight forward procedure and is widely employed. The identifica- tion of such a model from the measured dynamic response, on the other hand, has proven to be a tough challenge, and it is often re- ferred to as the ëëinverse vibration problem". Recently the authors have presented a solution to this problem based on identified state space realizations. This approach uses measured input and output data in the Observer/Kalman filter IDen- tification (OKID) algorithm to identify the Markov parameters of the system, which are then used in the Eigensystem Realization Algo- rithm (ERA) to realize the discrete time first-order system matrices. This initial state space model is further refined by minimizing the output error between the measured and predicted response using a non-linear optimization approach based on sequential quadratic programming techniques. Once the final form of the state space model is obtained, the physical parameters of the second-order (fi- nite element) model are retrieved from this state space model using complex eigenvalues and properly scaled complex eigenvectors. This solution has proven to be more flexible and general than the 80 JEM,128(3),340-350. previously presented solutions to the problem, and it has been used effectively in estimating the physical parameters of various models. On the other hand, even though the requirements on the number of available sensors and actuators for a full order identification has been improved with the aforementioned solution, the question of obtaining reduced order models in the absence of full instrumenta- tion has not been fully investigated yet. This study represents a first attempt at providing some alternate formulations that address the problem of insufficient instrumentation. The discussions are devot- ed to the review of the full order modeling problem, effects of insuf- ficient instrumentation, possible reduced order modeling schemes, and alternate formulations that can be developed by employing as- sumptions that are often used in the literature, such as having a di- agonal mass matrix and/or a classically damped system.