Optimal Topology Design of Piezoelectric Actuators for Vibrating Plane Structures

This paper applies structural topological des ign methodologies to the positioning of patches of piezoelectric (PZT) actuators on vibrating plane structures in order to reduce the total radiated sound. The continuous design variables are the thicknesses of and the voltages applied to the piezoelectric patches and the optimization problem is solved using the Method of Moving Asymptotes (MMA). In order to drive the solution to a discrete 0/1 distribution of thicknesses a new constraint type is introduced for the PZT layer instead of using a familiar SIMP -type material. This new constraint, called SRV, is a constraint on the Sum of the Reciprocal Variables (thicknesses in our case) which in conjunction with a constant material constraint drives the solution to a discrete design. In a first stage a vibratin g clamped beam problem was solved followed by a grid of beam and finally a vibrating plate was studied. Several examples of vibrating beams, grid of beams and plates demonstrate the potential of the proposed approach. I. Introduction This paper propose s a numerical method for the optimal positioning of patches of piezoelectric (PZT) material bonded to vibrating surfaces in order to reduce the radiated noise. Due to their mechanical properties, piezoelectric actuators are often used to reduce the total r adiated sound transmitted from a vibrating structure, by applying suitable voltage to the patches. Several authors have studied the design and behavior of PZT actuated vibrating structures. Ref. 1 used gradient -based optimization methods and a finite eleme nt (FE) discretization to reduce the radiated sound power from a structure, by finding the material’s optimal properties (density, Young’s modulus). Ref. 2 addressed the same issue by using the element’s thickness as design variables. Ref. 3 presented nume rous optimization methods for ‘quiet’ structures, and a survey of different objective functions for acoustical problems concerning vibrating plates. In Ref. 4 several piezoelectric actuators were connected at prescribed locations to a simply supported plat e to reduce the radiated sound. For every actuators pattern an optimization over the voltage was performed. Ref. 5 The dynamic response of a clamped vibrating plate excited by a harmonic force (caused by a piezoelectric actuator), having actuators bonded t o its upper and lower surfaces, was measured theoretically and experimentally. In another work 6 noise reduction was obtained by using a single actuator, which was represented by four local moments, at different locations. In both cases good agreement was a chieved between the theoretical and experimental results. Ref. 7 performed a simultaneous optimization, with respect to the structural topology and the actuators’ locations, in order to achieve the best structural shape for reducing the bending and torsion al vibrations caused by an external force. Ref. 8 used genetic algorithms for finding optimal locations and voltages to reduce the radiated sound from a simply supported plate. In an additional paper by the same group 9 , a comparison was performed for deter mining the relative significance of several parameters, such as: location, size and number of actuators. It was found, for instance, that actuator locations have a crucial importance while their size is the least significant. A clear demonstration of the m ethod of using piezoelectric actuators and sensors is presented in a series of papers, investigating the case of baffled plate excited by harmonic pressure, at close -to -resonance frequencies. Ref. 10 presented the FE formulation. Ref. 11 studied the influenc e of a single piezoelectric actuator located at the center of the plate’s upper surface while optimizing its size and the applied voltage. An optimization of a single disc -shaped actuator (optimal size, location and applied voltage) is presented in Ref. 12 along with the development of an automatic mesh generator program to support the need for a new FE analysis in each