**1. Introduction**

During the drilling operations for exploration and development of oil and gas, the measurements play an important role in obtaining the physical parameters of geologic formations and the monitoring of downhole conditions [1]. This valuable information can be derived in di fferent ways. For example, the drill bit can be withdrawn from the borehole and a "wireline logging tool" can be lowered into the borehole to take measurements [2]. Obviously, such a wireline logging after drilling will not be advantageous in obtaining the real-time and reliability data, as well as in the convenient operation of measurements and drilling [3]. Alternatively, the approach called measurement while drilling (MWD) or logging while drilling (LWD) with real-time data transmission increasingly attracts the attention in the oil and gas industry [4].

MWD is a technology of obtaining real-time measurement of various data at the bottom of the well during the drilling process [5]. Compared with conventional wireline logging, MWD is helpful to avoid the shortages in application [6]. This benefits from the measurement without stopping drilling. In the process of MWD, the communication and transmission between a downhole drilling assembly and a surface of a well are the key technology. A wireless system for telemetering data from downhole to surface will be an e ffective solution. The available approaches include utilizing such as mud pressure pulse, insulated conductor, electromagnetic wave, or acoustic wave [7]. For example, an acoustic

telemetry system (ATS) by means of acoustic waves propagating along drill strings was commercialized in 2000 [8,9]. This demonstrates the value of acoustic telemetry. In ATS, the electroacoustic transducer is an indispensable apparatus. Its performances play a determinative role in transmitting or receiving the acoustic waves, and its further impact may be related to the e ffect of acoustic telemetry [10]. Obviously, the importance of the transducer cannot be ignored.

In the practical applications, considering the special function and severe downhole situation [11,12], the stringent demands for transducers are derived from confined space to assemble in drill collar, adaptability to acoustic channel along drill strings, high downhole pressure and temperature, strong shock and vibration, etc. These limitations facilitate the transducers to implement the performances of small size, light weight, low frequency, high power, pressure-resisting, heat-resisting, reliability, etc.

However, the acoustic telemetry transducers that fulfill all the requirements are di fficult to develop. There have been some types of transducers to be developed for ATS while drilling, since the use of telemetry by acoustic waves through the drill strings was suggested in the 1940s due to steel tubular such as drill pipe with good and e ffective acoustic propagation [13]. In 1961, a magnetostrictive cylinder was invented by Woodworth for generating and/or receiving acoustic waves, and it had a low impedance of about 4000 ohms at the frequencies of 10 to 20 kHz [14]. In 1982, an acoustic well-logging transmitting and receiving transducer was invented by Dennis, which comprised the stacked piezoceramic rings and a resonating metallic plate, and operated at a desirable frequency lower than 15 kHz [15]. In 1989, a rare earth acoustic transducer was utilized by Liu to provide low-frequency acoustic energy in an acoustic well logging apparatus. That was attributed to the low acoustic velocity of the Terfenol-D alloys, so it was possible to produce a lower frequency below 10 kHz without increasing the transducer length [16]. In 1997, one acoustic transducer using PZT-8 type piezoelectric ceramic was provided by Drumheller for use in an acoustic telemetry system. The transducer was capable of delivering 10 W of acoustic energy at the frequency of 1.2 kHz, and its e fficiency of energy conversion was approximately 40% [17]. In 2015, a transducer comprised of piezoelectric ceramic rings and thin metallic electrodes was tested in the drill strings for the acoustic telemetry by Zhao. The transducer had the performances of broad bandwidth and low frequency below 3.1 kHz [18].

In addition, there are some other types of transducers to be used in ATS. Their di fferent configurations and characteristics were present in references [6,19–24]. All the studies provide good guidance. In respect of function realization, many of them can be used as a low-frequency acoustic source. While in respect of special requirements of ATS, some types of transducers still have much room for improvement, especially in miniaturizing the transducer to assemble in drill collar. As a practical application, for the most common drill pipe with a diameter of 178 mm, the maximum transverse size will be only about 40 mm to be provided for the electroacoustic transducer. Figure 1 illustrates the limitations subject to a drill collar of 178 mm in diameter. Obviously, it is di fficult to develop the low frequency, high power acoustic source in such a restricted size.

**Figure 1.** The schematic diagram of a drill collar with a groove to assemble the transducer.

Many types of transducer, such as Tonpilz transducers, flextensional transducers, flexural transducers, Piezoelectric rings, magnetostrictive transducers, piezoelectric ultrasonic transducers [25,26], are not suitable to MWD application because of the requirements of configurations, vibration modes, radiation field, acoustic impedance matching layers, etc. [27,28]. Under these restrictions of MWD application, the available types of transducer used in an acoustic telemetry system while drilling are few. If just considered as a receiver, MEMS may be an available alternative for detecting the acoustic signals propagated in drill strings. While as a transmitter with high power, the Tonpilz transducer is comparatively suitable because of its longitudinal vibration mode [29]. Generally, Tonpilz transducer can work e ffectively in the frequency range above 10 kHz. According to the advantages of the acoustic channel in drill strings, low frequency of the transducer will be desired in order to achieve a long-distance transmission, especially below 10 kHz and even lower [30]. In order to overcome the limitations in the ATS application, a type of novel longitudinal-flexural complex-mode low-frequency transducers will be presented.

#### **2. Cascade-Connected Transducer with Coupled Longitudinal and Flexural Vibrations**

The novel type of low-frequency piezoelectric transducer is called "cascade-connected transducer", which couples the longitudinal and flexural vibrations. Its longitudinal vibration mode is along the direction of drill strings, which is just in accordance with drill collar. Therefore, the acoustic energy can be transmitted forward along drill strings in the close match between the radiating head of the transducer and the drill collar. This characteristic will o ffer possibilities to take full advantage of the acoustic channel in drill strings. Its flexural vibration mode plays an important role in lowering the resonance frequency of the transducer. Hence, the above coupling makes it possible to implement the characteristics of low frequency, miniaturization, and high power.

The essential framework of the cascade-connected transducer is the multi-segment cascade-connected arrangemen<sup>t</sup> connected by the cross-beams, based on the conventional longitudinal transducer. The roles of the beam lie in not only its connection purpose as a structural part but also its flexural vibration as a functional component. Therefore, the Tonpilz configuration is fundamental, the multi-segment cascade-connected arrangemen<sup>t</sup> is essential, while the bending beam is the key. Figure 2 illustrates the configuration of the cascade-connected transducer. Its main components include piezoelectric stacks, bending beams, radiating head, tail mass, prestressed bolts, and other appurtenances. The radiating head is the region of transmitting the acoustic energy. Generally, its material should be similar to the drill collar, because the more approximate the characteristic impedances of materials are, the more smoothly the acoustic energy will propagate [31]. Typically, the tail mass should be comparatively heavy metal. A large tail-to-head mass ratio is desirable because it will yield a large head velocity. As the derivation, the more acoustic energy will be transmitted from the head. Every piezoelectric stack is glued closely together in series and wired in parallel. The configuration of multi-segment in a cascade-connected arrangemen<sup>t</sup> will be designed to match the length of the groove in the drill collar. There are eight segments in Figure 2. Every segmen<sup>t</sup> includes two columns of piezoelectric stacks, fastened to the bending beams alongside each other by a prestressed bolt. A type of cross-beam will be accepted to ensure the cross-connection between the adjacent segments. Compared with the other types of bending beams, this type of cross-beam will provide su fficient benefits to optimize the performances of the cascade-connected transducer, especially to lower the resonance frequency.

**Figure 2.** The cascade-connected transducer with eight segments.

#### **3. Finite Element Analysis of the Cascade-Connected Transducer**

The models and methods used in transducer analysis and design are always desired. However, it is di fficult to develop an ideal solution because the piezoelectric transducer is an integrated system, which needs to be described in three di fferent domains. A piezoelectric transducer is part acoustical at its moving surface in contact with the acoustic medium, part mechanical as a moving body controlled by forces, and part electrical as a current controlled by voltage [32]. So far, the equivalent circuit method (ECM) and finite element method (FEM) are comparatively comprehensive and e ffective, especially FEM is the prevailing method in the engineering development of transducer.

As a numerical method, the mathematical fundamentals of FEM are variational principle, subdivision, and interpolation [33]. After the whole piezoelectric transducer system is divided into finite elements connected at nodes, the matrix equations of the whole system will be formed. The key to these matrix equations are the governing equations, which can describe the behavior of piezoelectric coupled field. The computer solution of the whole system will be obtained, and the response of any position in the system will be calculated by interpolating. Then, a comprehensive explanation of how the system acts as a whole will be provided. Generally, FEM can model a complicated transducer without large-scale assumptions [34]. The other advantages of convenient modeling, rapid solution, accuracy result, and intuitive illustration are also attractive. Anyway, FEM has become one of the most effective methods to design or simulate the piezoelecteic transducers, especially for the sophisticated configurations, or the complex boundary conditions.

For the piezoelectric finite elements of the transducer model, their inclusion is the governing equation, which can describe the electric structure coupled field problem, written as [35]

$$
\begin{bmatrix}
\mathbf{M} & \mathbf{0} \\
0 & 0
\end{bmatrix}
\cdot
\begin{bmatrix}
\ddot{\mathbf{k}} \\
\ddot{\mathbf{U}}
\end{bmatrix} + \begin{bmatrix}
\mathbf{C} & 0 \\
0 & 0
\end{bmatrix}
\cdot
\begin{bmatrix}
\dot{\mathbf{k}} \\
\dot{\mathbf{U}}
\end{bmatrix} + \begin{bmatrix}
\mathbf{K} & -\mathbf{K}^{Z} \\
\mathbf{K}^{Z} & \mathbf{K}^{\mathbf{d}}
\end{bmatrix}
\cdot
\begin{bmatrix}
\ddot{\mathbf{t}} \\
\mathbf{U}
\end{bmatrix} = \begin{bmatrix}
\mathbf{F} \\
\mathbf{q}
\end{bmatrix} \tag{1}
$$

where [ξ] is vector of nodal displacements, [ **U**] is the vector of nodal electrical potential, [ **M**] is the mass matrix, [ **C**] is the damping matrix, [ **K**] is the sti ffness matrix, [ **K<sup>Z</sup>**] is the piezoelectric coupling matrix, [**K<sup>d</sup>**]isthedielectricconductivitymatrix,[**F**]isthenodalforcevector,[**q**]istheelectricalloadvector.

 For the finite element program, the complete material properties should be specified. Table 1 lists the details as follows.

During the typical process of FEM solution, it is the most important step to build the finite element model. In some ways, the closer to the transducer prototype the model is, the more accurate the solution will be. However, in practice, in order to reduce the modeling di fficulties or save the calculation time, simplifying the model without influencing the accuracy of the solution will be helpful. For our cascade-connected transducer, a 1/4 symmetrical finite element model will be built according to some assumptions. The following will present two results of finite element analysis, based on two di fferent configurations of cascade-connected transducers including twelve segments and ten segments respectively.


#### *3.1. Cascade-Connected Transducer with Twelve Segments*

Figure 3 illustrates the finite element model of the cascade-connected transducer, which is symmetrical to the XOZ and YOZ plane. The model includes 45,499 nodes and 22,519 elements. The different colors show the different components of the transducer. Its detailed sizes are as follows. The radiating head is 38 × 38 × 15 mm, the tail mass is 38 × 38 × 30 mm, the thickness of cross-beam is 5 mm and the piezoelectric ceramic ring is Φ 14 × 4 mm with a hole of Φ 6 mm. The cascade-connected transducer includes twelve segments, every segmen<sup>t</sup> includes two columns of piezoelectric stacks, and every piezoelectric stack includes four piezoelectric ceramic rings.

**Figure 3.** Finite element model of cascade-connected transducers with twelve segments.

When the finite element model is assumed to be free at both the head and tail end, Figure 4 illustrates the mode of vibration. The modal frequency is near 957 Hz, which is the resonance frequency of the cascade-connected transducer with twelve segments. On the whole viewpoint, the mode shape is longitudinal. Essentially, the piezoelectric stacks, the head, the tail, and the bolts are all longitudinal, while the cross-beams are flexural (their mode shape will be shown in Section 4). The obvious vibrations occur at the ends of the transducer, and comparatively the radiating head section is stronger than the tail mass section. This indicates that more energy is transmitted from the head.

**Figure 4.** Vector illustration of vibration at 957.5 Hz (cascade-connected transducer with twelve segments).

Figure 5 presents the admittance performances of the cascade-connected transducer. The curves show that the peak of the conductance curve appears at 957.5 Hz, which is near the resonance frequency derived from the coupled longitudinal and flexural vibrations. The sharp peak of the conductance curve also predicts that the bandwidth of the cascade-connected transducer is narrow. Therefore, this type of transducer cannot be used as a broadband transducer individually.

**Figure 5.** Admittance curves of the cascade-connected transducer with twelve segments.

#### *3.2. Cascade-Connected Transducers with Ten Segments*

A similar cascade-connected transducer, which includes ten segments, is also presented. All the configurations are the same except the number of segments. The finite element model of the cascade-connected transducer with ten segments is the same as the above. The results are also similar. Figure 6 shows the mode of vibration and Figure 7 shows the admittance curves. They predict that the resonance frequency is near 1092 Hz.

**Figure 6.** Vector illustration of vibration at 1092 Hz (cascade-connected transducer with ten segments).

**Figure 7.** Admittance curves of the cascade-connected transducer with ten segments.
