Structural dynamic validation of equivalent modeling
5.3 Validations of EFM and ESM models
5.3.2 Dynamic response validation
Both the collocated (measurement at the driving point) and non-collocated (measurement not at the driving point) frequency response measurements are used in this validation. The collocated measurement is more sensitive to the static contributions of the unconsidered dynamics of the system. The frequency response validations are limited to a frequency range of [20Hz−1,000Hz]. Two kinds of FRFs are considered: velocity output on the plate over voltage input to the transducers (inverse piezoelectric effect) and voltage output from the transducers over force input on the plate (direct piezoelectric effect). Hence, the numerical models can be fully validated for both actuation and sensing of the MFC transducers. The validated locations on the plate are given in Figure 5.1.
Inverse piezoelectric effect validation
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Figure 5.12: Inverse piezoelectric frequency response validation between 00 and 74
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Figure 5.13: Inverse piezoelectric frequency response validation between 00 and 00
VALIDATIONS OF EFM AND ESM MODELS 97
The inverse piezoelectric frequency response validations are shown in Figures 5.12 to 5.15. Both the EFM and ESM models can well predict the dynamic responses on the plate caused by the integrated transducers. Meanwhile, the structural complexity of the system is well retained in the low-order models. We can observe from the mode shapes in Figure 5.11 that the transducer placed at 00 has very low controllability for the 4thand 5thmodes. Also, the observability of the two modes at 74 is also very low. Thus, the measurement has a quite large uncertainty around 110Hzin Figure 5.12. Meanwhile, the experimental measurements are sensitive to external disturbances because of the free-free boundary conditions. The laser vibrometers can easily catch the influences of wind and other disturbances on the plate. The fluctuation of the measured FRF in Figure 5.14 is a typical example.
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Figure 5.14: Inverse piezoelectric frequency response validation between 26 and 22
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Figure 5.15: Inverse piezoelectric frequency response validation between 26 and 26
Direct piezoelectric effect validation
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Figure 5.16: Direct piezoelectric frequency response validation between 00 and 74
VALIDATIONS OF EFM AND ESM MODELS 99
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Figure 5.17: Direct piezoelectric frequency response validation between 00 and 00
The direct piezoelectric effect is validated by voltage over force FRFs. From Figures 5.16 to 5.19, we can observe that the predicted FRFs agree well with the experimental measurements too.
For the integrated MFC transducers, low controllability also actually means low observability because the inverse piezoelectric effect is the reversible process of the direct piezoelectric effect. Therefore, it is expected that there are amplitude deviation around the 2nd, 4th and 5th modes in Figures 5.16 and 5.17. The integrated transducer at 26 has better observability than the one at 00 so that high-quality FRFs are measured. The numerical models predicted very good results as shown in Figures 5.18 and 5.19.
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Figure 5.18: Direct piezoelectric frequency response validation between 26 and 22
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Figure 5.19: Direct piezoelectric frequency response validation between 26 and 26
VALIDATIONS OF EFM AND ESM MODELS 101
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Figure 5.20: Mechanical influence of the MFC transducer to the inverse piezoelectric frequency response between 00 and 74 in ESM model
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Figure 5.21: Mechanical influence of the MFC transducer to the direct piezoelectric frequency response between 00 and 00 in ESM model
A similar conclusion can be drawn for the ESM approach. However, the variations of the zeros on the FRFs are less significant than the EFM approach.
That is mostly due to the mechanical contribution of the transducer is weakened by the dynamic condensation techniques when we generate the low-order transducer model.
Reciprocity between two MFC transducers
The experimental data demonstrated that the reciprocity between two MFC transducers retains. As both the inverse and direct piezoelectric couplings of the integrated transducers are strain-based, the reciprocity between the two transducers can be theoretically proven. However, we used the numerical models to demonstrate the reciprocity in this section.
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Figure 5.22: Reciprocity validation between two MFC transducer (High operational voltage)
Figure 5.22 shows the reciprocity between the two transducers. The experimental reference is a voltage over voltage FRF between them. We can observe that the modes of the host plate are well captured in the predicted FRFs by the ESM models. However, there is a large amplitude deviation between the numerical and experimental results. That is mainly due to the hysteresis effect of the MFC transducers in the experimental measurements. The larger the operational voltage on the MFC transducers, the stronger the hysteresis effect [101]. When a high operational voltage drives one of the MFC transducers, the output voltage from the other MFC transducer is also high. Then the hysteresis effect
PIEZOELECTRIC RECIPROCAL RELATION 103
gives a more significant influences on the measured FRFs. The hysteresis effect can be mitigated by giving a lower excitation level according to the hysteresis characteristics of the MFC transducer. Therefore, the predicted FRFs are closer to the measurement obtained from a lower level operational voltage, as shown in Figure 5.23.
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Figure 5.23: Reciprocity validation between two MFC transducer (Low operational voltage)