Theory for the estimation of structural vibration parameters from incomplete data

Abstract

N system identification and modal analysis, the investigator is often faced with the problem of having to construct a model of finite order to represent a continuous vibrating structure. In other words, he is charged with obtaining a reduced-order model which, by definition, must be limited in its representation of the continuous system to a finite range of frequencies. The quality of the reduced-order model will be contingent upon the completeness of the measured data: a common cause of incompletenes s occurs when the number of modes excited in the test is fewer than the number of measurement stations (i.e., less than the order of the identified model). When it is required to construct a structural model (in terms of mass, stiffness, and damping parameters) from measured data, it is vital that such a model should in some sense be physically meaningful. The problems associated with the construction of reduced-order structural models from incomplete data have been highlighted by Berman and Flannelly1 and Herman.2 It is obvious that a reduced-order model cannot possibly reproduce the behavior of a continuous structure across an infinite range of frequencies. In the real structure the mass is distributed, whereas in the model the mass is discretized. In the finite-element method, consistent mass and stiffness matrices may be formulated by the minimization of an energy functional and, for this reason, such matrices may be considered to possess physical meaning. In reality, finite-element models do not exactly represent the dynamics of vibrating structures because they are incapable of dealing with boundary conditions such as imperfect hinges or flexibility in welded joints. It is then necessary to modify the finite-element model slightly in order that it should replicate the observed dynamics of the vibrating structure in a required range of frequencies. Methods have been developed for the estimation of structural parameters based on measured modal data. Chen and Garba3 presented a least-squares technique that may be applied iteratively together with matrix perturbation (for computation of the Jacobian matrix and reanalysis of the eigenvalues and eigenvectors) until the measured and computed eigendata are sufficiently in agreement. However, modal methods are known to have difficulty in treating closely spaced modes and can only deal with out-of-range modes by the addition of residual flexibilities. In this Note, a method is presented that does not depend on a modal decomposition but uses measured frequency responses to construct a structural model that varies minimally from an initial finite-element representation. A model constructed by this method is thus physically meaningful in the sense that a finite-element model is meaningful. A simple problem is contrived to illustrate an extreme case of incomplete data, which is solved using a continuous-fre quency- domain, least-squares filter.4 Using this approach, the Jacobian matrix is a constant matrix. Other problems associated with measure