tric elements and the plate has been firstly discussed, which has given a result which does not attenuate with regard to the order of eigenmodes. It has shown our configuration using piezoelectric elements has an intrinsic advantage compared to other solutions using directly displacement signals. Further calculation has indicated that there are several peaks appearing in the time reversal result of an excitation pulse. The most important peak has been proved that it is at exactly the same position of the original excitation pulse, which has validated the use of time reversal method for position detection.
Pulse creation using responses obtained from the time reversal method. According to results
of the response obtained by applying the time reversal method, if the reversal signal of an excitation pulse is directly used for the creation of a pulse vibration, one main peak and several associated peaks will appear on the plate. However, the theoretical study in this thesis has proved it is possible to combine several responses to cancel out associated peaks and thus create a perfect pulse vibration. We have found out approximate functions to describe these peaks with two types of functions. They have allowed to find out the coefficients for cancelling associated peaks. These original results can finally give conditions for the position arrangement of piezoelectric actuators.
Self-oscillating class-D amplifier as a piezoelectric drive. In our application, due to require-
ments of the system, the drive should be able to provide a high supply voltage and a wide dynamic frequency range for piezoelectric actuators with a great capacitance. It is the first time, a self-oscillating class-D amplifier is used for this case. Thanks to the principle of self- oscillating, the working frequency can reach nearly the limit of switching components. The theoretical modelling has also been established to study its stability. The stability criteria have been proposed, which is useful for the selection of parameters.
Approximate expression of a non-periodic PWM signal in spectral domain. The analytical
models exists for periodic PWM to obtain theoretical spectral values. However, due to the complexity of the non-periodic PWM, we noted a lack of a similar tool to study its theoretical result. In this thesis, we have proposed a method to obtain an approximate expression of the spectrum of a non-periodic PWM. In the case of using a self-oscillating class-D amplifier, this expression can be simplified thanks to its property: the shortest switching period is almost constant. This result can fill the blank for modelling non-periodic PWM signals and is useful to guide the output filter design.
8.2 Outlook and perspectives
According to the results obtained, future works can be highlighted in the following topics either to enhance the working plate or for further applications.
Working plate design. This study focuses on the basic functional principles that can be used
for a working plate. For example, the generation of a pulse wave provides the possibility to generate any kind of surface displacement. However, the desired waveform of vibration
Chapter 8. Conclusion
depends on the study of interactions between parts and plate, which has been discussed in [90]. Together with the other operations using single eigenmode such as general separation and orientation mentioned in [90], a new design of the working plate can be achieved by combining all these knowledges in order to create an optimised solution.
Structure health monitoring applications. One of the applications of the emission source
position detection is about the structure health monitoring. The objective is to locate the position of cracks inside the material. According to the result of our study, the time reversal method requires only one piezoelectric actuator, because it makes use of all eigenmodes information contained within the received signal. Compared to other methods using only arrival delay information of signals, the time reversal method can theoretically give a more precise result. A study on this subject would be interesting to improve their current method.
Tactile applications. If dropping parts are replaced by a finger, the plate becomes a tactile
input device and the vibration created on the plate is equivalent to a feedback for the user. The advantage of the time reversal method is that it allows turning any rigid flat surface (table, window) to a tactile input device if a piezoelectric sensor/actuator is glued. The precision depends on the size of the piezoelectric sensor which can be adjusted according to the application environment. As humain-machine interfaces become more and more important for "smart" devices, it would be interesting to apply our results in this field.
A
Piezoelectric actuator equivalent cir-
cuit parameter expressions
The general method to calculate the model’s parameters have been given in Section 3.5. Supposing the vector Rmand vector Xmare measured data of dimension n £ 1 at frequencies !m, the objective is to find the R(!m) which can satisfy the expression (A.1).
argmin k1,...,kq (Rm° R(!m))T(Rm° R(!m)) (A.1) Rk C0 R1 L1 C1 Zk
(a) Kim’s model
Rp C0 R1 L1 C1 Zp (b) Park’s model Rs C0 Rg p R1 L1 C1 Zg (c) Guan’s model Figure A.1 – Unloaded piezoelectric ceramic models
A.1 Constant resistor model (Kim’s model)
Kim’s model is shown in Fig. A.1a. It contains only one constant resistor in series with the capacitor [54]. We define that V1= [1...1]T is a vector of n constant elements. The optimum
value of its resistance is the average of all the measurement which can be expressed in (A.2)
Rk= kk=
VT
Appendix A. Piezoelectric actuator equivalent circuit parameter expressions