4 B. Götz et al. For all experimental vibration attenuation investigations, the piezoelectric transducer P1 excites the beam in y-direction via a controlled voltage signal U1.t/. The transducer P3, either shunted to the RL-shunt or the RLC-shunt, attenuates the vibration acceleration a.t/ at the sensor location xs while the transducers P2 and P4 are operated with short circuited (sc) electrodes. Varying axial tensile and compressive loads 1000N Fx 1500N, with compressive loads in positive xdirection, are applied to the beam at x D0via a spindle-type lifting gear and measured by a force sensor. 1.3 The Beam’s First Mode Eigenfrequency and Coupling Coefficient for Varying Axial Loads The vibration attenuation capability bears on the piezoelectric transducers P3 that is either shunted to a tuned RL- or a tuned RLC-shunt. Varying axial beam loads may change the beam’s first mode resonance angular frequency !sc with short circuited (sc) transducer electrodes and the beam’s first mode general coupling coefficient K33 of transducer P3 iny-direction. Changes in both, !sc and K33 may influence the vibration attenuation with shunted transducers. To investigate the influence of varying loads Fx on!sc andK33, the mathematical receptance model of the transducer P3 is derived in frequency domain. The values of !sc and K33 for axial loads Fx DŒ 1000; 500; 0; 500; 1000; 1500 N are extracted from a least squares fit of the receptance model to the experimental data in frequency domain. As already shown by Kozlowski et al. [10], obtaining !sc and K33 from a curve fitting of the transducer receptance model results in a smaller error since the calculation of both parameters is less influenced by the used frequency resolution in the measurement. 1.3.1 Transducer Receptance Model Figure 1.3 shows the electrical network representation of the piezoelectric transducer P3 connected to the beam. The transducer P3 is described by a gyrator-like two-port transducer network with its electrical capacitance Cp, a internal series resistance Rp and its transducer constant Y [12]. The vibration behavior of the beam’s first mode with short circuited transducer electrodes is modelled by the modal mass m, the modal stiffness k and the assumed hysteretic damping with loss factor resulting in the complex stiffness k0 Dk.1 i /. The complex network receptance seen from the terminals 1and 2 in Fig. 1.3 is obtained by ˛.!/ D 1 i !Z.!/ (1.1) with excitation frequency !. The impedance Z.!/ D U3.!/ I3.!/ D Zp.!/Z1.!/ Zp.!/ CZ1.!/ (1.2) results from the parallel connection of the structural impedance Z1.!/ of the first mode seen from the terminals 3 and 4 the transducer impedance Zp.!/ DRp i !Cp : (1.3) Fig. 1.3 Electrical network model of the piezoelectric transducers and the beam’s first mode k m Cp Rp U3 I3 Y 2 1 3 4
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