*3.2. Experimental Validation*

In this section, a test specimen of the radially layered cylindrical piezoceramic/epoxy composite transducer was fabricated by utilizing the mold-filling technique [52], as shown in Figure 4a. Similar to the previous work [52], the specimen fabrication mainly includes 8 steps: (1) mold and piezoceramic rings preparation, (2) epoxy preparation, (3) pouring epoxy into mould, (4) curing, (5) demolding, (6) polishing, (7) silvering and (8) final specimen. A difference is that in the step (3), three different epoxy materials were poured into the mold in this experiment. Three different epoxy materials, shown in Table 2, were prepared by mixing curing agents of 4,4-methylenedianiline and bisphenol-A epoxy resin (E-51 with an epoxy value of 0.51 mol/100 g) at a mass ratio of 15/100, 17/100, 19/100, respectively. The curing agents of 4,4-methylenedianiline was provided by Acros Organics Co. (Geel, Belgium). The E-51 was supplied by Nantong Xingchen Synthetic Material Co., Ltd. (Nantong, China). PZT-5H was selected as the piezoceramic material, shown in Table 1, which was provided by Baoding Hongsheng Acoustics Electron Apparatus Co., Ltd. (Baoding, China). The specimen sizes were same as the given ones. The impedance test system is shown in Figure 4b, which included an Agilent 4294A Precision Impedance Analyzer for measurement and a computer for data acquisition. The electrical parallel connection was realized by using two conductive copper foil tapes. The measured impedance spectra and the phase of the impedance over the frequency range between 10 kHz and 40 kHz are shown in Figure 5. From the spectra, the first resonance and anti-resonance frequencies can be obtained as 23.179 kHz and 23.780 kHz, respectively. These two frequencies are also addressed in Table 3 to compare with the theoretical values. As can be seen, the calculated values are larger than the experimental ones; however, they agree reasonably well with each other. The relative errors between the theoretical values and experimental values for the first resonance frequency and the first anti-resonance frequency are 11.37% and 11.91%, respectively. There are two main factors accounting for the errors. Firstly, the theoretical model was established based on plane stress assumption, which is not the case for the fabricated composite. Secondly, the material parameters provided by the manufacturer were used here and the provided values may not be the exact values of the components used.

**Figure 4.** Experimental setup: (**a**) the fabricated specimen; (**b**) the impedance test system.

**Figure 5.** Measured impedance spectrum and its phase.

#### **4. Results and Discussion**

In this section, the effects of material parameters of epoxy layers, piezoceramic material types, and locations of piezoceramic rings on the electromechanical characteristics will be analyzed and discussed.

#### *4.1. Effect of Material Parameters of Epoxy Layers*

In the above experiment, the transducer with a sequence of material parameters -1 --2 --3 for epoxy layers #1, #2 and #3 was fabricated and tested. Here, the sequence -1 --2 --3 denotes that the material parameters for epoxy layers #1, #2, and #3 are materials -1 , -2 , and -3 , respectively. Keeping the PZT-5H and geometric dimensions of the transducer unchanged, 27 sequences can be formulated according to different material arrangements of these three epoxy layers. Figure 6 plots

the electromechanical characteristics for these 27 sequences. These electromechanical characteristics are the first resonance and anti-resonance frequencies and the corresponding electromechanical coupling factors. It can be seen that these 27 different sequences present 27 sets of electromechanical characteristics, which enable the multi-frequency characteristics of the transducer. In addition, transducer with the sequence -3 --3 --2 has the maximum first resonance and anti-resonance frequencies, while the one with the sequence -1 --1 --1 has the minimum frequencies. The transducer with the sequence -1 --3 --3 has the maximum electromechanical coupling factor, while the one with the sequence -3 --1 --2 has the minimum value. As can be seen from Table 1, since Poisson's ratios are the same for all these three epoxy layers, the Young's modulus and density are the contributing factors to the variation in the electromechanical characteristics. The following analysis will discuss their effects on the electromechanical characteristics in order to distinguish the dominant factor.

**Figure 6.** Electromechanical characteristics of the transducers with different material sequences of the epoxy layers.

Keeping other parameters unchanged, Figure 7 presents the effects of variation in density on the electromechanical characteristics. We had one reference group and three comparison groups. Here, the -1 --2 --3 combination was selected as the reference group, and three special cases with the same density within the group were selected as the comparison groups. From Figure 7, it can be observed that all of the first resonance and anti-resonance frequencies, as well as the corresponding electromechanical coupling factors, are very close to each other. A maximum relative error is −0.14%, which indicates that the effect of density on the electromechanical characteristics is very small and even negligible.

Similarly, Figure 8 shows the effects of variation in Young's modulus on the electromechanical characteristics of the transducer. For this case, three special cases with the same Young's modulus within each group were selected as the comparison groups. The differences between the electromechanical characteristics of these examples can be seen from Figure 8, where the maximum relative error is −3.24%. It is worth noting that this maximum error is 23 times more than that for density, which proves that the Young's modulus is the dominant factor for the electromechanical characteristics of the transducer. These results can serve as a good reference for designing the transducer.

**Figure 7.** Influence of density of the epoxy layers on the electromechanical characteristics of the transducer. Error = (Reference group − Comparison group)/Reference group.

**Figure 8.** Influence of Young's modulus of the epoxy layers on the electromechanical characteristics of the transducer. Error = (Reference group − Comparison group)/Reference group.

Further, keeping the density of all epoxy layers as 1186 kg/m3, Figure 9 plots the variation of the electromechanical characteristics of the transducer when Young's modulus of epoxy layers changes from 2300 × 10<sup>6</sup> N/m<sup>2</sup> to 2940 × 10<sup>6</sup> N/m2. Four cases are presented, i.e., the case of changing all epoxy layers, the case with only epoxy disk #1 changing, the case with only epoxy ring #2 changing, and the case with only epoxy ring #3 changing. It can be seen that the first resonance and anti-resonance frequencies increase with the increase of the Young's modulus of the epoxy layers. This is because larger Young's modulus will increase the stiffness of the transducer, which leads to higher resonant frequencies. Furthermore, it can be seen that changing Young's modulus of epoxy disk #1 and epoxy ring #3 has negligible effects on these two frequencies as compared to the case of changing the Young's modulus of epoxy ring #2. Therefore, in the transducer design, adjusting the Young's modulus of epoxy ring #2 can only realize frequency control of the proposed radial layered cylindrical piezoceramic/epoxy composite transducer. From Figure 9, it can also be found that for every case, the variation of Young's modulus of the epoxy layers has almost no effect on the corresponding electromechanical coupling factors. Here, it should be pointed out that the Poisson's ratio also greatly influences the electromechanical characteristics of the piezoelectric composites, which has been proved by the previous works [59–61]. However, in the present work, the main focus is to design a type of new transducers controlled by Young's modulus of the epoxy layers. Therefore, three different epoxy materials were chosen for the epoxy layers, which have the same Poisson's ratio, approximately equal

**Figure 9.** Electromechanical characteristics versus Young's moduli of the epoxy layers: (**a**) all epoxy layers; (**b**) only epoxy disk #1; (**c**) only epoxy ring #2; (**d**) only epoxy ring #3.

#### *4.2. Effect of Piezoceramic Material Types*

Selecting the material parameters of epoxy layers as the sequence -1 --1 --1 and keeping geometric dimensions of the transducer unchanged, Figure 10 gives the effect of combinations of five commonly used piezoceramic materials on the electromechanical characteristics. These piezoceramic materials include PZT-5H, PZT-4, EC-64, PZT-5A and BaTiO3, of which material parameters are shown in Table 3. The piezoceramic material types of PZT ring #1 are marked in the abscissa. The piezoceramic material types of PZT ring #2 are listed in the graph. From Figure 10, it can be found that when the PZT ring #2 is chosen as PZT-5A, the transducer has the minimum first resonance and anti-resonance frequencies, but the maximum first electromechanical coupling factor. When the PZT ring #2 is chosen as BaTiO3, the transducer has the maximum first resonance and anti-resonance frequencies, but the minimum first electromechanical coupling factor. When the PZT ring #2 are chosen as PZT-5H, PZT-4, EC-64, the transducer has the similar first resonance and anti-resonance frequencies. The reasons are as follows. For a piezoelectric circular ring in radial vibration, when keeping its geometric sizes unchanged, its resonance frequency depends on the radial sound speed *VrP* = \$*cE*11/*ρ<sup>P</sup>* [36]. The radial sound speed reflects its stiffness–mass ratio, of which values are listed in Table 1. A larger *VrP* for PZT ring #2 means its stiffness is enhanced, which further induces the stiffness increase of the transducer. In addition, its electromechanical coupling effect depends on the plane coupling factor *kP*(*p*) = \$2*<sup>d</sup>*231/[*κσ*33(*sE*11 + *<sup>s</sup>E*12) [36], as shown in Table 1. A larger *kP*(*p*) for PZT ring #2 means its electromechanical coupling effect is better, which further improves the whole

coupling effect. The PZT-5A has the minimum *VrP* and maximum *kP*(*p*); therefore, the transducer with PZT-5A ring #2 has the smallest resonance frequency and best electromechanical coupling effect than the other types.

**Figure 10.** Electromechanical characteristics: (**a**) resonance frequency *fr*; (**b**) anti-resonance frequency *fa*; (**c**) electromechanical coupling factor *kd* versus piezoceramic material types.

#### *4.3. Effect of Locations of Piezoceramic Rings*

Selecting the material parameters of epoxy layers as the sequence -1 --1 --1 , piezoceramic material types of two PZT rings as PZT-5H, and keeping the area of one pizeoceramic ring unchanged, Figure 11 shows the relations between the electromechanical characteristics and locations of the other piezoceramic rings. Here, the inner radii *R*1 and *R*3 of the PZT rings #1 and #2 are used to denote their locations, respectively. The corresponding outer radii *R*2 and *R*4 of the PZT rings #1 and #2 also need to be changed to maintain the same areas, which are defined as *R*2 = \$*S*1/*π* + *R*21 and *R*4 = \$*S*2/*π* + *R*23, respectively. Symbols *S*1 and *S*2 are the areas of the PZT rings #1 and #2, respectively. When fixing the location of PZT ring #1 and varying the location of PZT ring #2, the inner and outer radii *R*3 and *R*4 of the PZT ring #2, and the areas of the epoxy rings #2 and #3 also vary. When fixing the location of PZT ring #2 and varying the location of PZT ring #1, the inner and outer radii *R*1 and *R*2 of the PZT ring #1, and the areas of the epoxy disk #1 and ring #2 also vary. From Figure 11, it is indicated that when the location of PZT ring #1 is fixed, the first resonance and anti-resonance frequencies, as well as the first electromechanical coupling factor, decrease with the increase of location of PZT ring #2. When the location of PZT ring #2 is fixed, the first resonance and anti-resonance frequencies, as well as the first electromechanical coupling factor, firstly increase to the maximum values, and then decrease. That is because different locations of one piezoelectric ring relative to the other will change the geometric sizes itself and those of the adjacent epoxy layers. When the material parameters are unchanged, these variations in the geometric sizes will vary their stiffness and mass, which lead to the change in the electromechanical coupling effect of the transducer. In Figure 11b, three maximum values are *fr* = 25.685 kHz, *fa* = 26.578 kHz, *kd* = 0.26, respectively. The corresponding locations are *R*1 = 12 mm, 12.5 mm, and 14 mm, respectively. This rule can be used to design the improved transducer that has the maximum first resonance and anti-resonance frequencies as well as the first electromechanical coupling factor.

**Figure 11.** Electromechanical characteristics versus locations of piezoceramic rings: (**a**) location of PZT ring #2; (**b**) location of PZT ring #1.
