# Studies on a Piezoelectric Cylindrical Transducer for Borehole Dipole Acoustic Measurements
## 2 Principles and Numerical Analyses
### Transducer Structures and Its Vibration Principles  
For the traditional piezoelectric dipole source in borehole acoustic measurement, due to the limited strain of the piezoelectric ceramic material itself, it is not easy to generate large displacement at low frequencies [10; 11]. Therefore, it is hard to enhance the transmitting voltage response at low frequencies, especially at frequencies around less than 1 kHz. In addition, the frequency band of traditional source is not wide enough that it can measure shear wave velocity in soft formations. Therefore, in order to improve its radiation performance, especially in the low frequency range, and to extend the bandwidth of the dipole transducer, we proposed a new transducer design that we have patented in ref [16], which is shown in Figure 3.  
The schematic diagram and the sectional view along the vertical axis of the piezoelectric cylindrical transducer with its frame is shown in Figure 3. The rectangular coordinate system is set up in the acoustic center of the transducer. The \(y\)-axis and \(z\)-axis are shown in Figure 3, and the \(x\)-axis is in the inward direction perpendicular to the paper. Figure 3: Schematic diagram of the piezoelectric cylindrical transducer and its frame. Figure 2: Flexural vibration mode of a bender bar: (**a**) First order of bending mode; (**b**) Third order of bending mode. The dipole transducer consists of an internal piezoelectric actuator and external radiation structures. The internal actuator consists of three parallel piezoelectric benders, one of which is longer in the center, and the other two are shorter and of a same size. Each bender consists of two layers of piezoelectric elements and one layer of metal base element, which is commonly used in practice [8]. As shown in Figure 3, three layers are bonded at the two interfaces. The two ends of three parallel positioned benders are fixed to the endcaps bonded with the cylindrical shell. The external radiation structures consist of a cylindrical shell, two endcaps, and two U-shaped springs. The chamber, formed by the cylindrical shell and two endcaps, is filled with air. The cylindrical shell is supported by two U-shaped springs with the end faces fixed to the outside frame of a logging tool. The entire system is immersed in the fluid of the borehole. For numerical modeling, the whole structure is set to be immersed in air or water. During the operation, when a voltage is applied to the electrodes of the two piezoelectric elements, one piezoelectric element elongates and the other one contracts, causing the piezoelectric bender to bend. This makes the cylindrical shell oscillate out of the phase with the inner piezoelectric actuator. The cylindrical shell, acting as the radiation surface of the transducer, is driven by the piezoelectric actuator to produce a pure dipole signal. Because the vibration of the piezoelectric actuator is confined in the space of cylindrical housing, it will not be harmful to the force imparted to the borehole wall [17]. Suppose that a force \(F\) is applied to a borehole wall through borehole fluid by a force \(F_{I}\) from the inner piezoelectric actuator to the cylindrical radiation surface. According to the [PERSON]'s third law, a force \(F_{2}\), equivalent in magnitude but opposite in direction to the force \(F_{I}\) is applied to the piezoelectric actuator, and an anti-reaction force harmful to the force \(F\) would not be imparted to the side wall of the borehole because the inner actuator is fixed in the cylindrical chamber and separated from the borehole fluid. Thus, any waves due to the reaction are not radiated toward the borehole wall [17]. On the other hand, for the conventional bender-type transducers, together with their frame, the opening angle of the sonic window is about 60 degrees due to the limited radiation surface.