Guidelines for the excitation pulse position detection of a beam

As predicted by the theoretical analysis, while using the time reversal method to find out excitation pulse positions, simulations result in multiple peaks. Normally, it is sufficient to check the maximum peak, but, in case of multiple excitation sources, it is necessary to consider all the peaks information.

Examining the source position requires firstly the following parameters: position of the piezo

xP and size of the piezoelectric element a. In case of single excitation source, supposing a

maximal value is found at position xM, then xM is the position of the original excitation. Of

course, there will be associated peaks but with less important amplitudes.

In case of multiple excitations at the same time, supposing we detected multiple local maximal at positions x1...xn. The positions of excitation should satisfy that a linear combination of the

excitation pulse responses (shown in Fig. 4.6) can restore these peaks (or in the meaning of least mean squares).

4.6. Conclusion

4.6 Conclusion

The whole process of using the time reversal method has been studied analytically in this chapter. The interactive relation between a plate and piezoelectric sensor/actuator has been modelled, which introduced a second spatial derivative term to the amplitude result and will reduce the angular frequency on the denominator. The fact of using piezoelectric actuators allows having a response more similar to the original response. Thus it is a better way for the application of the time reversal method compared to previous methods using directly the measured displacement.

Then a mathematical method has been proposed to solve the problem of calculating the amplitude on the plate. If the eigenmodes can satisfy all hypotheses, a distribution of the eigenmode amplitude values can be found out for each position. With these distributions, it is easy to calculate the average normalised amplitude at each position.

An application of the proposed method has been given for an ideal beam as an example. The calculation has been detailed for each step: solution of eigenmodes, verification of hypotheses, calculation of distribution functions and finally amplitudes of each position. The results have shown that for an ideal beam, the restored amplitude by the time reversal method will theoretically have four peaks on the beam. The greatest one is exactly at the same position where the original excitation is applied. An associated one is of half amplitude of the main peak and its position is at xX2= 2xP° xAwhere xPis the position of the piezoelectric element

and xAis the position of original excitation. If 2xP° xA< 0, at xX2= °(2xP° xA), two peaks

will be found around this position.Similarly, two peaks will be found around the positions

xX3= 2xP+ xAor one peak at xX3= 2 ° (2xP+ xA). At the end, a short guideline is given for the

detection of the excitation position on a beam based on the pattern of amplitudes.

Relative publications to the chapter

• X. Liu, Y. Civet and Y. Perriard, "Piezoelectric tactile device feedback generator using acoustic time reversal method," 2016 19th International Conference on Electrical Machines and Systems

5

Pulse generation on a beam

Contents

5.1 Introduction . . . 74 5.2 Amplitude pattern and approximate function . . . 74 5.3 Cancellation of associated peaks . . . 77

5.3.1 Cancellation of first kind of associated peaks . . . 77 5.3.2 Cancellation of second kind of associated peaks . . . 78 5.3.3 Cancellation with a single piezoelectric actuator . . . 80 5.3.4 Cancellation with multiple piezoelectric actuators . . . 82

5.4 Damping effect . . . 83 5.5 Conclusion . . . 84

This chapter discusses the necessary steps for creating a spatial pulse vibration on a beam as example. Firstly, the peak on the beam according to time reversal method can be modelled by two functions. It allows the theoretical analysis for the reduction of associated peaks. Then, it can be proved that with one or multiple piezoelectric actuator and if their positions can satisfy certain conditions, all the associated peaks can be cancelled out. An optimised actuator arrangement can be then realised. Finally, The damping factor is taken into account. The amplitude coefficient should be adjusted mode by mode in order to restore a perfect pulse vibration.

Chapter 5. Pulse generation on a beam

5.1 Introduction

An ideal generated pulse should be in form of a Dirac function, which can be used then to construct any arbitrary functions. According to the response of an excitation pulse using the time reversal method, one can obtain a vibration with one main peak and several associated peaks. It is sufficient to detect the source position, but not enough for the pulse creation. This is why we should find a method to remove all associated peaks to make the time reversal method applicable for the wave creation.

The idea to improve a time reversal response is to combine multiple responses in order to cancel out the associated peaks. In the previous chapter, we only have demonstrated the existence (positions) of the peaks. If we want to remove them by themselves, it is necessary to represent them mathematically. Fortunately, the function of the peaks can be found in form of a Sinc function, which permits to analytically study where and how to combine different responses to cancel out associated peaks.

The position of the piezoelectric actuator is also proved to be important for the creation of pulses on a beam. The general rule can be obtained form the position solution of associated peaks. The arrangement of multiple piezoelectric actuator will also be intruduced.

As noticed in (4.24), the final amplitude not only depends on eigenmodes related terms but also on a damping term. All the analytical results are obtained while there is only eigenmodes related terms. Thus it is necessary to weight all damping term to a "unitary one" for each mode. The detail will be given at the end of this chapter.