Reduced Models for the Active Vibration Control of Light Structures

The vibration control of mechanical structures is an important issue for the preservation of their structural integrity, especially in the case of light structures such as electronic cards aboard flying vehicles. One of the most effective methods in this domain is active vibration control, which requires an observation of the structure and an adequate command. Therefore, one must use a satisfactory numerical model of the structure. One way of achieving such a model consists in improving a given initial model. The governing principle of the updating method used in this paper, based on the modified constitutive relation error principle [1, 2], consists in dividing the equations of the problem into two groups: a reliable group and an unreliable group. Then, among the mechanical fields which verify the equations of the reliable group exactly, one seeks the field which minimizes a norm containing the residuals of the unreliable group and some distance between the observed experimental data and the numerical results. Like many other widely used updating techniques [3, 4], this is an optimization-based method whose objective function has a very high calculation cost. In-flight updating requires rapid calculations; therefore, the cost function must be approximated. In order to do that, we use vector spaces over a field of truncated rational functions of multiple variables. Tensor manipulations on such vector spaces lead to very-low-cost, rational objective functions. Furthermore, this technique enables one to calculate the gradient and higher-order derivatives of the cost function very inexpensively, leading to very efficient minimization algorithms. In this paper, we illustrate the method by updating a beam. This is a simple problem, but one which is representative of the updating of electronic cards. The beam is instrumented with piezoelectric sensors and actuators, which are taken into account in the numerical model. The experimental data are simulated by applying perturbations to the stiffness, damping and/or mass parameters in the beam itself, in the piezoelectric components or at the boundaries.