94 P. S. Varoto et al. Output Voltage FRF on Horizontal Beam - With Tip Mass on Top Position (a) Output Voltage FRF on Vertical Beam # 1 Beam - With Tip Mass on Top Position Output Voltage FRF on Vertical Beam # 2 Beam - With Tip Mass on Top Position (b) (c) Frequency [Hz] 0 50 100 150 200 250 300 Frequency [Hz] 0 50 100 150 200 250 300 Frequency [Hz] 0 50 100 150 200 250 300 10 1 10 0 10 -1 10 -2 10 -3 10 -4 Magnitude [V/g] 10 1 10 0 10 -1 10 -2 10 -3 10 -4 Magnitude [V/g] 10 1 10 0 10 -1 10 -2 10 -3 10 -4 Magnitude [V/g] Fig. 10.13 Transmissibility FRF with tip mass added (solid line: α =90◦; dashed line: α =60◦; dashed-dotted: α =0◦). (a) Horizontal beam (b) Bottom vertical beam (c) Top vertical beam 0 10 20 30 40 50 60 70 80 90 [deg] 0 20 40 60 80 100 120 140 160 180 200 Freq. [Hz] (a) (b) Fig. 10.14 Natural frequency of the PVEH: numerical (lines) and experimental (markers). (a) Without tip mass (b) With tip mass The voltage transmissibility FRF in Fig. 10.12 show clear changes in natural frequency values between the configurations. The first, third and fourth mode shapes are almost unaffected by the rotation angle and always present in the FRFs. The second peak show up only in the transmissibility FRF of the second vertical beam with rotation angle α = 0◦. This because the second vibration mode of the structure without tip mass is the first bending mode of the vertical top beam, therefore it creates a strain only in the piezo-patch related to this beam. It is visible only at zero rotation angle because the general output voltage in this configuration is very low, and increasing the rotation angle this peak is hide below the other higher voltage modes. The fifth mode is substantially affected by the rotation angle α it moves of more than 70 Hz between the configurations. Its peak is really visible in the configuration α =0◦ and α =90◦ while it is really damped in the configuration α =60◦. This effect is given by the nature of the mode shapes, that is a coupled bending-torsional mode, and it emphasizes the bending behaviour when the rotation angle is higher. Finally another mode at around 200 Hz visibly moves changing the rotation angle. The same considerations can be derived for the structure with the tip mass, with the exception that the natural frequencies are lower and the output voltage is generally higher. The comparison between the numerical computed natural frequencies of the system changing the rotation angle α and the experimental natural frequencies are shown in Fig. 10.14a, b respectively for the configuration without and with the tip mass. The voltage transmissibility FRFs were curve-fitted using the fraction rational polynomial method. The first model of the system already shows good correlation with the experimental data. Numerical mode 6 and 7 are not experimentally identified in both configuration shown in Fig. 10.14. This because they are mainly beams bending in the direction parallel to the piezo-patches and therefore can not be detected using the piezo-patches as sensors.
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