The numerical treatment of coupled elastic structures subjected to random excitations

Abstract

Aerospace structures can be subjected to severe random excitations. These excitations can be classified into different categories (mechanical, acoustical). At the same time, the dynamic behavior of aerospace components can be strongly affected by the surrounding fluid medium so that coupled (i.e. fluid-structure interaction) effects have to be accounted for. The numerical treatment of such random excitations acting on coupled elastic structures is the subject of the present paper. Random excitations are assimilated to stationary random processes. A special attention is devoted to the handling of a diffuse acoustic field. The discrete model relies on an hybrid FEM-BEM technique. A displacement-based FEM model is selected for the structure while a BEM model is used for the acoustic fluid medium. The BEM model relies on an indirect boundary integral representation and a variational solution scheme which allows to preserve the symmetry of the related coupled system. An efficient solution scheme is proposed in order to handle the random response. An application to a plate structure subjected to random mechanical loading is presented. Introduction The characteristics of the dynamic response of a mechanical structure depend on one hand on the characteristics of the load (e.g. frequency content) and on the other hand on structural properties (stiffness, mass and damping). In general, time dependent loads show statistical variations and consequently the response is not deterministic. Since time dependency is involved, a simple characterization of the load as a random variable is not sufficient. For this reason, the concept of random process has been introduced'. At the same time, the studied mechanical structures can interact with the surrounding fluid in a way such that the dynamical response is perturbed. The handling of such kinematical and mechanical coupling effects requires the selection of appropriate numerical models. The unbounded character of the fluid domain calls also for the use of a boundary integral formulation which is able to handle exactly the Sommerfeld radiation condition at infinite distance. The coupled model presented here is precisely based on the selection of a displacementbased FEM model for the structure while a BEM model is selected for the acoustic fluid. This BEM model relies on an indirect boundary integral representation particularly well suited for thin structures (where the thickness is assumed to be small versus the acoustic wavelength). An efficient solution procedure is proposed in order to compute random mechanical and acoustical responses. The manuscript is organized as follows. The first section is devoted to the characterization of random excitations as stationary random processes. A special attention is devoted to the diffuse acoustic field usually encountered in reverberant rooms. The second section is related to the coupled discrete model used for setting up the transfer function. The third section shows how to proceed for getting the random response of the coupled system. A numerical example related to a plate structure is presented in the last section. Random excitations Characterization of random excitations The considered excitations can be classified into two main categories: (1) mechanical and (2) acoustical. The mechanical excitations refer to prescribed mechanical loads (F) and/or prescribed displacements (u) or accelerations (a) at support points. Acoustical excitations can be induced by sources with a random amplitude (A). It is assumed that each random excitation gives rise, in this discrete model context, to a load vector which appears as the product of a deterministic load vector (called a load pattern) by the related parametrized random variable (F, u, a or A). The particular case of a diffuse field (as produced in a reverberant room) deserves some attention since it implies (at least in principle) the consideration of a great number of acoustic sources. This could be impractical for some applications so that an alternative procedure based on the analytical investigation of an infinite number of plane waves could be used. This approach is addressed at the end of this section. Whatever the key characteristic of the random excitation (F, or u, or a, or A,) is, his random nature could be defined in the same way. Let us denote by x, such an excitation. The random character of x ((t) can be described by referring to weakly stationary random processes. The key characteristics of any weakly stationary process x,(t) are the mean (which is constant) and the autocorrelation function R x (t) given by: