The paper presents an overall numerical approach for the simulation and design of smart lightweight structures in order to control vibration and noise. The design process requires an overall virtual computer model, which includes the passive mechanical structure, the acoustic fluid, the active materials, such as piezoelectric patch actuators glued to the structure, as well as the controller. Based on the finite element method such an overall model is presented in the paper. An acoustic box is used to verify the computational approach by comparing simulated and measured results.

The paper presents an overall analysis and design approach for smart lightweight structures to actively reduce vibration and noise. As smart materials, distributed piezoelectric patches are attached to the structure. The basis of the approach is an overall finite element model, which includes the structure itself, the acoustic fluid, the piezoelectric actuators and sensors as well as the controller. As a test example a smart acoustic box is simulated and the simulation results are compared with measured data. Finally, also industrial applications are briefly presented.

Piezoelectric patches are widely used as distributed actuators and sensors for active vibration control. Active control of sound in this way is mainly interesting in the low frequency range, where passive methods are less effective. To simulate such electromechanical-acoustical systems a virtual overall model based on the FEM is presented in the paper. Different approaches for model reduction with respect to controller design purposes are discussed. As a simple test example the numerical model of an acoustic box is studied. Finite Element AnalysisThe derivation of the finite element model is based on the mechanical equilibrium, the electric equilibrium, the linear coupled electromechanical constitutive equations [3] and the linear acoustic wave equation [4]. Small displacements are considered and acoustic response is regarded as small pertubations to an ambient reference state. Using the velocity potential Φ as a nodal degree of freedom of the acoustic fluid the symmetric semi-discrete system of equations of the electro-mechanical-acoustic field problem can be written as where w is the vector of the nodal displacements and ϕ is the vector of the nodal electric potentials. Equation (1) can also be written in a shorter form as Mr + Cṙ + Kr = f. Based on this equation the coupling of acoustic hexahedron elements with active layered SemiLoof -shell elements was realized in our general purpose FEM software package COSAR. Model Truncation TechniquesA finite element model with a large number n of degrees of freedom is infeasible for controller design purposes. Therefore a model reduction of the numerical model is performed. Introducing new coordinates r = Tq, with T = t 1 t 2 . . . t k , the equation (1) can be written asIn this way the number of degrees of freedom is truncated from n to k. To receive the transformation matrix T any set of k orthogonal vectors t i may be choosen. If we have no further knowledge about these vectors their number has to be quiet large in order to receive good results. As well known from structural dynamics the motion of a vibrating structure can be represented as a composition of a few eigenmodes only. This method should also be applied to vibro-acoustic systems using several modes of the interesting frequency range. As far as the acoustic coupling is described by means of the first time derivatives of the nodal degrees of freedom the eigenproblem which arises from equation (1) results in complex eigenvectors. In order to avoid the great effort of solving this coupled eigenvalue problem the so called component mode synthesis [1] can be used. In this method the transformation matrix T consists of the modal matrices Q u and Q a containing a few preselected uncoupled structural and acoustical eigenmodes, respectively,In general it is not possible to decouple the system of equations with this transformation matrix.

The design process of engineering smart structures using piezoelectric patches as distributed sensors and actuators requires a virtual overall model, which includes the main functional parts. In the paper a suitable software tool based on the FEM is presented to simulate coupled electro‐mechanical‐acoustical problems regarding interior noise problems. The modal truncation technique is used to design a LQ‐controller as well as to study the time‐depending behaviour of a clamed smart plate.

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