**4. Cross-Beams**

The bending beams are one of the main differences between the cascade-connected transducer and the conventional Tonpilz transducer. The beams should be helpful for enhancing the longitudinal vibration, and also for lowering the resonance frequency of the transducer. Therefore, the bending beams need to be designed and optimized. The cross-beam is a good alternative. The material and the thickness of the beam are the major factors to control the flexural mode of cross-beam. The flexural mode, as an indispensable vibration of the whole, plays an important role in the cascade-connected transducer. Hence, its performances need to be comprehended.

For the cascade-connected transducer with twelve segments, there need to be eleven cross-beams. The serial number of each beam is shown in Figure 8, where the beam near the tail is assigned as No. 1 and the beam near the head is assigned as No. 11. According to the above analysis, a vector illustration of vibration of cross-beams is shown in Figure 8, which is separated from the whole system in Figure 4. The result shows that the move of the cross-beams near the ends of the transducer is significant, while the move of the middle cross-beam is low. The vibrations of both ends are opposite. All of the performances agree with the longitudinal vibration of the cascade-connected transducer. Although the longitudinal move of the cross-beam is real, it disguises the fact of flexural vibration. The reason is that each cross-beam is attached to a constant move of longitudinal vibration, which is provided by the whole system. This constant longitudinal move of each beam is different, but every one is larger than the scope of its other vibrations. If the constant longitudinal move of each beam is separated, only the flexural vibration will be left.

**Figure 8.** Vector illustration of vibration at 957.5 Hz (cross-beams).

Figure 9 illustrates that the clear flexural vibration happens to each cross-beam. Their distinctions are just the vibration intensity. Unlike the before-mentioned longitudinal move of the cross-beams, the flexural vibrations of the cross-beams near the ends of the transducer are weak, while those of the middle cross-beam are strong. The strongest flexural vibration occurs at No. 5 beam. Figure 9 illustrates its mode shape. If the undeformed edge is assigned as the reference, it is clear that the two orthogonal branches of the cross-beam vibrate respectively along opposite directions. This mode shape is propitious to enhance the longitudinal vibration of the cascade-connected transducer, and also to achieve the low resonance frequency. This is the advantage of cross-beam.

**Figure 9.** Vector illustration of flexural vibration at 957.5 Hz (cross-beams).

If the maximum relative displacement between the two orthogonal branches of each cross-beam is defined as *Di*, shown in Figure 9, and the subscript "*i*" stands for the serial number of each cross-beam, the value of *D*5 is maximum. Figure 10 illustrates the relative displacements of all the cross-beams. Its *Y*-axis label is the normalized relative displacement *Di D*5 , and its *X*-axis is the serial number of beams. *Di* of the beams at the two ends are dissymmetric because the masses of the head and tail are different.

**Figure 10.** Normalized relative displacements of all the cross-beams.

#### **5. Test and Discussion**

Some cascade-connected transducer prototypes have been manufactured according to the above designs. Two kinds of cascade-connected transducer have the external size of 38 × 38 mm. The prototype with twelve segments is 310 mm long and weighs 1.23 kg, while the one with ten segments is 266 mm long and weighs 0.88 kg. Figure 11 shows the prototypes of cascade-connected transducers, which are encapsulated in polyurethane rubber.

**Figure 11.** Prototypes of cascade-connected transducers.

The admittance curves are obtained from the precision impedance analyzer Agilent 4294A (Agilent, Santa Clara, CA, USA). Figure 12 shows the curves of the cascade-connected transducer with twelve segments. The curves show that the resonance frequency is near 985 Hz, which is slightly higher than the result of FEM. The admittance curves of the cascade-connected transducer with ten segments are shown in Figure 13. Its resonance frequency is near 1112 Hz. The comparison between the measurements and the simulations illustrated respectively in Figure 5 or Figure 7 shows consistency. In practice, the two kinds of cascade-connected transducers can be wired together in parallel to broaden the bandwidth. Figure 14 illustrates the admittance curves of four wired cascade-connected transducers (two for twelve segments and two for ten segments, shown in Figure 11). There are two adjacent peaks. Their coupling can optimize the performances of transducers, which will provide higher power and wider bandwidth to achieve a better effect in the acoustic telemetry system while drilling.

Based on the measurements of resonance frequency, *fr*, and anti-resonance frequency, *fa*, the transducer's effective electromechanical coupling coefficient, *ke*ff, can be calculated. The two kinds of cascade-connected transducer have *ke*ff of 10.11% and 11.93%, respectively.

For a transmitting transducer, the performance of high power is always desired. The power consumption of two kinds of cascade-connected transducers is monitored from a small excitation signal to high driving voltage. The increasing response is basically linear. A safe operating state is as follows. The power consumption of single cascade-connected transducer with twelve segments is 30.9 watts at the frequency of 970 Hz, where the driving voltage is 1188 V p–p (peak to peak), the current is 243 mA p–p, and the current leads the voltage by 64.6 degrees. The power consumption of a single cascade-connected transducer with ten segments is 31.3 Watts at the frequency of 1110 Hz, where the driving voltage is 1188 V p–p, the current is 234 mA p–p, and the current leads the voltage by 63.2 degrees. These is not the maximum power consumption for the transducers because of the limitation of the power amplifier.

**Figure 12.** Admittance curves of cascade-connected transducers with twelve segments (obtained from Agilent 4294A).

**Figure 13.** Admittance curves of cascade-connected transducers with ten segments (obtained from Agilent 4294A).

**Figure 14.** Admittance curves of four wired cascade-connected transducers (two for twelve segments and two for ten segments).
