244 G. Matten et al. Fig. 24.1 Geometry of the system An eigenvalue study of the system is performed. Specifically, bend, the eigenvalue associated with the first bending mode of the beam is computed. Following the method presented in [4], the optimal shunt has a resistance of 1463 and a capacitance of 4:918nF. The lower face of the transducer is electrically grounded. The shunt is modeled by a weak contribution of a voltage, calculated as a function of the electric charge, and is applied on the upper face of the transducer. The following mathematical equation is verified: u D q 1 RC C (24.1) with Rand C the resistance and capacitance of the shunt, q the electric charge on the upper face of the transducer and the eigenvalue variable. The steel substrate is modeled by a linear elastic material, the transducer by a piezoelectric material, bothwith a 0:1% isotropic loss factor. The glue between these two parts is modeled with shell elements and a linear elastic material. All the three domains share the same displacement field u. The end of the substrate (see the grey surface on Fig. 24.1) is fixed (u D0). The electrodynamical problem is solved by computations of a multiphysical finite element model. The sensitivity analysis is performed using an in-house analysis software developped in matlab which controls finite element computations performed with Comsol©. After a convergence validation, a 3,338 elements with displacement and voltage degrees of freedom mesh has been chosen. The computation time is about half a minute. 24.3 Sensitivity Analysis 24.3.1 Model Parameters The objective of this study is to analyse the sensitivity of the model presented in the previous section. Especially the geometry, the material properties and the shunt paramters have been chosen to have a sufficient knowledge of the system behavior to design an experiment. The beam parameters are: Parameter L_beam b_beam e_beam Nu_beam Rho_beam E_beam Description Length Width Thickness Poisson ratio Density Young modulus Nominal value 5.5 cm 1 cm 1 mm 0.3 1,500kg.m 3 200Gpa Parameter deviation ˙10% ˙10% ˙10% ˙10% ˙10% ˙10% The shunted transducer parameters are: Parameter e_piezo L_piezo p_piezo R_shunt C_shunt Description Thickness Length Position Shunt resistance Shunt capacitance Nominal value 500 m 2 cm 2 mm 1,463 4:918nF Parameter deviation ˙10% ˙10% ˙10% ˙10% ˙10%
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