Symmetry in Piezoelectric Disks and Rings: Modal and Harmonic Analysis with Experimental Comparison

Abstract

Exploiting geometrical symmetry to reduce the computational domain is a well-established strategy in finite element analysis. However, its systematic application to coupled piezoelectric modal problems requires that symmetry conditions be satisfied simultaneously by the mechanical and electrical fields. In this work, a parity-based framework is developed for class 6 mm piezoelectric disks and rings. Starting from coupled piezoelastic constitutive equations, the displacement components and electric potential, together with the associated stresses and electric displacements, are classified into eight electromechanical symmetry classes. The corresponding mechanical and electrical boundary conditions are then derived, allowing each symmetry class to be computed independently using only one eighth of the original geometry. Comparison with full-domain finite element calculations shows that the complete set of modes can be recovered and classified with very small frequency differences, while the reduced models achieve a substantially lower computational cost. The framework is subsequently applied to piezoelectric rings with varying outer diameters to track the evolution and degeneracy of modal families and to identify symmetry-equivalent classes. Finally, selected resonances are experimentally tracked through impedance measurements as the ring diameter is varied and compared with harmonic finite element predictions. The proposed approach therefore provides not merely a domain reduction technique but a physically interpretable classification of coupled electromechanical modes that facilitates efficient modal identification and resonance tracking in piezoelectric disks and rings.

1. Introduction

The field of piezoelectrics is closely related to acoustics, mainly because piezoelectric materials are widely used to generate and sense acoustic vibrations [1]. This has led to the development of numerous piezoelectric devices, including sensors [2,3,4], actuators [5,6,7,8,9,10,11,12], and energy harvesters [13,14,15], which continue to be actively investigated, together with modifications of and improvements in the materials employed in these applications [16,17]. The vibration characteristics and natural modes of elastic plates have also been extensively studied [18,19,20]. In piezoelectricity, several studies have addressed material and geometrical symmetry [21,22,23,24,25,26,27,28], although much of this work has focused on material symmetry. A representative application in which the vibrational behavior of piezoelectric components is particularly relevant is the Langevin transducer, which commonly incorporates cylindrical piezoelectric elements in a sandwiched configuration [29]. Since these devices operate by exciting specific resonances, identifying and understanding the natural modes of vibration of piezoelectric media remains an important aspect of piezoelectric device design.

Finite element analysis is one of the main numerical tools used to determine the natural frequencies and vibration modes of piezoelectric structures. However, the modal analysis of three-dimensional structures can become computationally demanding, particularly when fine meshes are required or when a large number of modes must be calculated and identified. Exploiting geometrical symmetry to reduce the computational domain is a well-established strategy for alleviating this cost. More generally, symmetry-based numerical methods can replace a problem defined over the complete domain with a set of smaller independent problems defined over a reduced symmetry cell [30]. Group-theoretical approaches provide a systematic framework for exploiting structural symmetry and can be used to classify vibration modes and decompose the original eigenvalue problem into smaller independent or decoupled problems associated with different symmetry types [31,32]. For structures consisting of repeated sectors, cyclic-symmetry formulations similarly exploit rotational periodicity to reduce the numerical problem and have been extensively used in vibration and modal analyses [33]. Related periodic formulations based on Bloch–Floquet concepts relate fields on periodic boundaries through phase relationships and are widely used in wave propagation problems. In piezoelectric systems, periodic boundary conditions incorporating such phase relationships have been used, for example, in finite element simulations of periodic surface acoustic wave structures [34].

These established approaches demonstrate that reducing a computational domain through symmetry or periodicity is not, by itself, a new concept. The contribution of the present work instead lies in a direct parity-based formulation of the coupled electromechanical fields of class 6 mm piezoelectric disks and rings. Starting from piezoelastic constitutive equations, the displacement components and electric potential are assigned even or odd parity with respect to the three coordinate planes. The corresponding parity of the stresses and electric displacements then follows from the constitutive equations, providing a systematic procedure for deriving both the mechanical and electrical boundary conditions on the symmetry planes. This procedure yields eight electromechanical symmetry classes that can be independently solved using one eighth of the complete geometry. Thus, symmetry is employed here not only as a means of reducing computational cost but also as a framework for classifying and identifying the coupled electromechanical vibration modes of piezoelectric disks and rings.

The proposed classification is first verified by comparing the modal response of a full piezoelectric disk with the solutions obtained from the eight symmetry-reduced models. The framework is then applied to piezoelectric rings with varying outer diameters to follow the evolution of their natural frequencies and to investigate relationships and degeneracies among the different symmetry classes. Finally, selected resonances are experimentally tracked through impedance measurements as the outer diameter of a PZT-8 ring is varied and are compared with harmonic finite element predictions. Therefore, the purpose of the proposed formulation is not only to accelerate the numerical calculation of natural frequencies but also to provide a physically interpretable classification that facilitates modal identification and resonance tracking in piezoelectric disks and rings.

In Section 2, a summary of the piezoelasticity theory is presented along with the constitutive equations for a crystal of hexagonal symmetry, which is the crystal class considered throughout this work. With the main theory established, in Section 3, an example of the symmetry parity conditions and the corresponding boundary conditions to be applied to a one-eighth model are derived. Then, this specific symmetry is tested by performing a modal analysis of a full piezoelectric disk and comparing it with the corresponding one-eighth model. Next, in Section 4, the symmetry framework is applied to a piezoelectric ring with varying external diameters, and relevant relationships among the eight symmetry classes are discussed. Section 5 presents the experimental setup, in which a vector network analyzer (VNA) is used to measure the impedance of the piezoelectric ring while varying its external diameter, allowing selected resonances to be tracked. These experimental results are reported in Section 6 and compared with harmonic finite element analyses. Finally, Section 7 summarizes the main findings and limitations of this work.

2. Piezoelasticity Theory for a Crystal of Hexagonal Symmetry (Class 6 mm)

This section summarizes the piezoelasticity equations that model piezoelectric media in general [35], then proceeds to delimit them for the specific case of the PZT8 material, which is a crystal of hexagonal symmetry. It starts with Equations (1)–(3), called the equations of motion, coupled with Equation (4), the electric displacement equation.

$${\displaystyle \frac{\partial {\sigma }_{xx}}{\partial x}}+{\displaystyle \frac{\partial {\sigma }_{xy}}{\partial y}}+{\displaystyle \frac{\partial {\sigma }_{xz}}{\partial z}}=\rho \ddot{u},$$

(1)

$${\displaystyle \frac{\partial {\sigma }_{yx}}{\partial x}}+{\displaystyle \frac{\partial {\sigma }_{yy}}{\partial y}}+{\displaystyle \frac{\partial {\sigma }_{yz}}{\partial z}}=\rho \ddot{v},$$

(2)

$${\displaystyle \frac{\partial {\sigma }_{zx}}{\partial x}}+{\displaystyle \frac{\partial {\sigma }_{zy}}{\partial y}}+{\displaystyle \frac{\partial {\sigma }_{zz}}{\partial z}}=\rho \ddot{w},$$

(3)

$${\displaystyle \frac{\partial D_x}{\partial x}}+{\displaystyle \frac{\partial D_y}{\partial y}}+{\displaystyle \frac{\partial D_z}{\partial z}}=0,$$

(4)

where ${\sigma }_{xx}(x,y,z,t)$, ${\sigma }_{yy}(x,y,z,t)$ and ${\sigma }_{zz}(x,y,z.t)$ are the perpendicular stresses, and ${\sigma }_{yz}(x,y,z,t)$, ${\sigma }_{xz}(x,y,z,t)$, and ${\sigma }_{xy}(x,y,z,t)$ are the tangential stresses. Also, $u(x,y,z,t)$ is the x-displacement; similarly, $v(x,y,z,t)$ is the y-displacement, and $w(x,y,z,t)$ is the z-displacement. In addition, $\phi (x,y,z,t)$ is the electric potential. Furthermore, $D_x(x,y,z,t)$, $D_y(x,y,z,t)$ and $D_z(x,y,z,t)$ are the electric displacements. Lastly, $\rho$ is the density of the material. The constitutive equations of a crystal of hexagonal symmetry, also called class 6mm, which is the crystal studied in this paper, are the following:

$${\sigma }_{xx}=C_{11}{\displaystyle \frac{\partial u}{\partial x}}+C_{12}{\displaystyle \frac{\partial v}{\partial y}}+C_{13}{\displaystyle \frac{\partial w}{\partial z}}+e_{13}{\displaystyle \frac{\partial \phi }{\partial z}},$$

(5)

$${\sigma }_{yy}=C_{12}{\displaystyle \frac{\partial u}{\partial x}}+C_{11}{\displaystyle \frac{\partial v}{\partial y}}+C_{13}{\displaystyle \frac{\partial w}{\partial z}}+e_{13}{\displaystyle \frac{\partial \phi }{\partial z}},$$

(6)

$${\sigma }_{zz}=C_{13}\left({\displaystyle \frac{\partial u}{\partial x}}+{\displaystyle \frac{\partial v}{\partial y}}\right)+C_{33}{\displaystyle \frac{\partial w}{\partial z}}+e_{33}{\displaystyle \frac{\partial \phi }{\partial z}},$$

(7)

$${\sigma }_{yz}={\sigma }_{zy}=C_{44}\left({\displaystyle \frac{\partial v}{\partial z}}+{\displaystyle \frac{\partial w}{\partial y}}\right)+e_{51}{\displaystyle \frac{\partial \phi }{\partial y}},$$

(8)

$${\sigma }_{xz}={\sigma }_{zx}=C_{44}\left({\displaystyle \frac{\partial u}{\partial z}}+{\displaystyle \frac{\partial w}{\partial x}}\right)+e_{51}{\displaystyle \frac{\partial \phi }{\partial x}},$$

(9)

$${\sigma }_{xy}={\sigma }_{yx}=C_{66}\left({\displaystyle \frac{\partial v}{\partial x}}+{\displaystyle \frac{\partial u}{\partial y}}\right),$$

(10)

$$D_x=e_{51}{\displaystyle \frac{\partial u}{\partial z}}+e_{51}{\displaystyle \frac{\partial w}{\partial x}}-{\epsilon }_{11}{\displaystyle \frac{\partial \phi }{\partial x}},$$

(11)

$$D_y=e_{51}{\displaystyle \frac{\partial v}{\partial z}}+e_{51}{\displaystyle \frac{\partial w}{\partial y}}-{\epsilon }_{11}{\displaystyle \frac{\partial \phi }{\partial y}},$$

(12)

$$D_z=e_{13}{\displaystyle \frac{\partial u}{\partial x}}+e_{13}{\displaystyle \frac{\partial v}{\partial y}}+e_{33}{\displaystyle \frac{\partial w}{\partial z}}-{\epsilon }_{33}{\displaystyle \frac{\partial \phi }{\partial z}},$$

(13)

where $C_{11}, C_{12}, C_{13}, C_{33}, C_{44}$ and $C_{66}$ are the elastic constants, and $C_{66}=(C_{11}-C_{12})/2$. Also, $e_{13}, e_{33}$ and $e_{51}$ are the piezoelectric constants. Lastly, ${\epsilon }_{11}$ and ${\epsilon }_{33}$ are the dielectric constants. To perform a modal and harmonic analysis, the time dependence of the functions is assumed to be of the form $F(x,y,z,t)=f(x,y,z)e^{-i\omega t}$, which in general allows one to decouple time dependence from Equations (1)–(13); from here on, the functions are assumed to be time-independent. Equations (5)–(13) are the basis of modeling piezoelectric solids of class 6 mm, and with the aid of the Finite Element Method (FEM), this paper can model rings and disks.

3. Symmetry Conditions

In the first part of this section, an example of a symmetry condition was derived from the constitutive Equations (5)–(13) and denoted as $S_3$. To test $S_3$, we performed a modal analysis using ANSYS APDL 2024 R1 in a full piezoelectric disk, then compared it to 1/8th of the disk, but first, the boundary conditions need to be derived from the symmetry condition.

In Equation (5), let u be odd along x, and let v, w, and $\phi$ be even along x; therefore ${\sigma }_{xx}$ is even along x. From Equation (10), it can be seen that ${\sigma }_{xy}$ is odd along x. Similarly, from Equations (9) and (11), it can be seen that ${\sigma }_{xz}$ and $D_x$ are odd along x. In the same manner, let $u, w, \phi$ be even and v be odd along y; from Equations (6), (8), (10) and (12), it can be seen that ${\sigma }_{yx}, {\sigma }_{yz}, D_y$ are odd along y, and ${\sigma }_{yy}$ is even along y. Also, let $u, v$ be odd and $w, \phi$ be even along z; from Equations (7)–(9) and (13), it can be seen that ${\sigma }_{zx}, {\sigma }_{zy}$ are even and ${\sigma }_{zz}, D_z$ are odd along z. These parity conditions are going to be arbitrarily called symmetry 3 and denoted by $S_3$, and they are summarized in

Table 1. $S_3$ symmetry condition.

Planes	Displacements and Electric Potential	Stresses and Electric Displacement
$x=0$	$u(x,y,z)=-u(-x,y,z)$	${\sigma }_{xx}(x,y,z)={\sigma }_{xx}(-x,y,z)$
	$v(x,y,z)=v(-x,y,z)$	${\sigma }_{xy}(x,y,z)=-{\sigma }_{xy}(-x,y,z)$
	$w(x,y,z)=w(-x,y,z)$	${\sigma }_{xz}(x,y,z)=-{\sigma }_{xz}(-x,y,z)$
	$\phi (x,y,z)=\phi (-x,y,z)$	$D_x(x,y,z)=-D_x(-x,y,z)$
$y=0$	$u(x,y,z)=u(x,-y,z)$	${\sigma }_{yx}(x,y,z)=-{\sigma }_{yx}(x,-y,z)$
	$v(x,y,z)=-v(x,-y,z)$	${\sigma }_{yy}(x,y,z)={\sigma }_{yy}(x,-y,z)$
	$w(x,y,z)=w(x,-y,z)$	${\sigma }_{yz}(x,y,z)=-{\sigma }_{yz}(x,-y,z)$
	$\phi (x,y,z)=\phi (x,-y,z)$	$D_y(x,y,z)=-D_y(x,-y,z)$
$z=0$	$u(x,y,z)=-u(x,y,-z)$	${\sigma }_{zx}(x,y,z)={\sigma }_{zx}(x,y,-z)$
	$v(x,y,z)=-v(x,y,-z)$	${\sigma }_{zy}(x,y,z)={\sigma }_{zy}(x,y,-z)$
	$w(x,y,z)=w(x,y,-z)$	${\sigma }_{zz}(x,y,z)=-{\sigma }_{zz}(x,y,-z)$
	$\phi (x,y,z)=\phi (x,y,-z)$	$D_z(x,y,z)=-D_z(x,y,-z)$

. In the same fashion, the other seven symmetries can be derived from the constitutive Equations (5)–(13); these symmetry conditions are summarized in Appendix A and further tested in piezoelectric rings. These symmetry conditions establish certain behavior along the main axes ($x, y$ and z); for example, if we slice the disk in half in the plane $x=0$ and we know the displacements $u, v, w$ and the electric displacements $\phi$ in one of the halves using the symmetry conditions in Table 1, the displacements and electric displacements in the other half can be calculated. Thus, if we slice the disk along planes $x=0, y=0$ and $z=0$, the model can be reduced to 1/8th. To test $S_3$, first the boundary conditions for 1/8th of the disk need to be derived.

The boundary conditions for the whole disk were free, which implies two conditions: mechanically free and charge-free. The first means that the stresses in the surface are 0, and the latter means that the electric displacement normal to the surface is 0. These conditions are also known as natural conditions; they are by default applied in ANSYS APDL when modeling a free solid. These free boundary conditions give us the conditions for the top, the bottom, and the perimeter of the disk, but to model 1/8th of the disk, we also need boundary conditions for the following planes: $x=0, y=0$ and $z=0$. For example, from the $S_3$ symmetry conditions in Table 1, it can be seen that $u, {\sigma }_{xy}, {\sigma }_{xz}$, and $D_x$ are odd along x and therefore 0 in the plane $x=0$. Similarly, ${\sigma }_{yx}, v, {\sigma }_{yz}, D_y$ are odd along y and therefore 0 in the plane $y=0$. Lastly, $u, v, {\sigma }_{zz}, D_z$ are odd along z and therefore 0 in the plane $z=0$. These results can be summarized as in

Table 2. $S_3$ boundary conditions of 1/8th of disk.

Planes	Boundary Conditions
$x=0$	$u=0,$   ${\sigma }_{xy}=0,$   ${\sigma }_{xz}=0,$   $D_x=0$
$y=0$	${\sigma }_{yx}=0,$   $v=0,$   ${\sigma }_{yz}=0,$   $D_y=0$
$z=0$	$u=0,$   $v=0,$   ${\sigma }_{zz}=0,$   $D_z=0$

. These boundary conditions were applied by setting the corresponding displacements and electric displacements to 0 in their respective planes. The rest of the boundary conditions for 1/8th of the disk for the other seven symmetries are obtained similarly and described in Appendix B. With these boundary conditions, what follows is to test them in a modal analysis.

To test these symmetry conditions, a modal analysis of a piezoelectric disk of PZT-8 material was performed using ANSYS APDL. The element used was a 10-node tetrahedral, coupled-field solid; it is designed for modeling irregular meshes. The mesh convergence study can be found in Appendix C. The dimensions of the disk were 45 mm in diameter and 5 mm in thickness. A comparison between the whole disk and 1/8th, but applying the corresponding boundary conditions for each symmetry, was made, yielding

Table 3. A modal analysis comparison of the first 15 modes: full disk vs. 1/8th, applying symmetry. The relative percentage error ${\epsilon }_{r_i}$ was calculated as follows: ${\epsilon }_{r_i}=\left|{\displaystyle \frac{f_{i_{\mathrm{Full}\mathrm{disk}}}-f_{i_{1/8\mathrm{th}\mathrm{disk}}}}{f_{i_{\mathrm{Full}\mathrm{disk}}}}}\right|\times 100\%$ where $i=\{1,\dots ,15\}$.

Mode	Full Disk Frequency (kHz)	1/8th Disk Frequency (kHz)	Symmetry	${\mathit{\epsilon }}_{{\mathit{r}}_{\mathit{i}}}$ (%)
1	8.244907029	8.244504183	$S_5$	0.004885998090495
2	8.245022873	8.244501652	$S_3$	0.006321644075803
3	15.79232589	15.7914604	$S_3$	0.005480446680427
4	18.33666786	18.334746	$S_7$	0.010480966414797
5	18.33691553	18.33478875	$S_1$	0.011598351950294
6	30.56963027	30.56560189	$S_5$	0.013177719077468
7	30.56984097	30.56562276	$S_3$	0.013798599751110
8	32.34379639	32.33860997	$S_1$	0.016035285213472
9	32.3439833	32.33860942	$S_7$	0.016614774841293
10	34.55706892	34.55706798	$S_4$	0.000002720138105
11	34.55707018	34.55706881	$S_6$	0.000003964456456
12	41.66292497	41.66293442	$S_2$	0.000022682036864
13	41.66292636	41.66292874	$S_8$	0.000005712512792
14	44.32679759	44.32155998	$S_7$	0.011815899827549
15	44.32783761	44.32171719	$S_1$	0.013807170234312

.

From Table 3, it can be seen that by applying the respective boundary conditions to 1/8th of the model, all the modes of the full model were obtained and classified into their respective symmetries. Also, from Figure 1, it can be seen that the 1/8th model proved to be up to 69 times faster. The time used by the 1/8th model increases a lot slower than that for the full model, which allows for more element density without sacrificing time and accuracy. To further test $S_3$ symmetry, plots of the Degrees Of Freedom (DOFs) at the first standalone mode (no doublets; doublets are also commonly known as degenerate modes) along lines parallel to the principal axis were made, as shown in Figure 2, Figure 3 and Figure 4. These graphs are taken from the full disk model. It is important to note that they follow $S_3$ parity conditions from Table 1 even though the $S_3$ boundary conditions from Table 2 were not applied to this model.

Lastly, to graphically compare the 1/8th model with the full disk model, contour plots of the DOFs were made, as shown in Figure 5. It can be seen that the 1/8th model closely resembles the full model. Similar analyses were made for the other seven symmetries by applying their respective boundary conditions shown in Appendix D, which allowed us to classify the modes of vibration. To further explore these symmetries, in the following section, they were applied to a piezoelectric ring to see how varying the external diameter has an impact on their natural frequencies.

4. Symmetry Applied to Piezoelectric Rings with a Varying External Diameter

The aim of this section, besides testing the symmetries in another common geometry, is to test how varying the external diameter affects natural frequencies. To achieve this first, we performed several modal analyses of a piezoelectric ring of PZT-8 material, with a fixed internal diameter of 15 mm and a thickness of 5 mm (these measures were chosen because they model the real ring discussed in the following sections) at an external diameter varying from 20 mm to 45 mm. Then, we proceeded to do the same for 1/8th of the disk but applied each of the eight symmetry conditions and compared them. Lastly, a comparison between the symmetries, highlighting the most important findings, is presented.

First, we can see a comparison between the full ring and 1/8th of the ring in Figure 6. Here it is important to note that every mode of the full ring is followed by a mode in 1/8th of the model. A plot classifying each of the eight symmetries is presented in Figure 7. These plots were made by performing modal analyses in 1/8th of the whole ring and applying the respective boundary conditions.

Then, from Figure 7, we isolate $S_1$ and $S_7$ and compare them in Figure 8, as well as $S_2$ and $S_8$ in Figure 9. From these plots, one can infer that $S_1=S_7$ and $S_2=S_8$, and this is indeed true. One can prove this by looking at a crystal of hexagonal symmetry, as it has the same constitutive properties in the x and y directions. So, as long as the geometry is symmetrical in the x and y directions, one can interchange x with y and u with v at the same time, yielding the same system. This explains one of the many reasons why this system has several doublets. Hence, it can be said that the system has eight total symmetries that can be reduced to six. For example, in the case of an ellipsoidal prism, these eight symmetries cannot be reduced, and the doublets due to $S_1$ and $S_7$ separate, as well as the doublets due to $S_2$ and $S_8$.

Lastly, from Figure 7, we isolate $S_3$ and $S_5$ and compare them in Figure 10, as well as $S_4$ and $S_6$ in Figure 11. Here, in both figures, it can be seen that they have several modes in common (doublets), but they also present unique modes corresponding to each symmetry. So, in this symmetry, they cannot be reduced.

5. Experimental Setup

A VNA was used to measure the impedance of the piezoelectric rings while reducing their diameter to see how it affects certain resonance frequencies. We started with a piezoelectric ring of PZT-8 material, as seen in Figure 12a, with an initial outer diameter of 45 mm, an inner diameter of 15 mm and a thickness of 5 mm. Then, we proceeded to reduce its outer diameter by approximately 1 mm each time up to 31 mm and measure its impedance while following resonances.

The VNA used was the Bode 100 from OMICRON Lab, shown in Figure 12b. It uses a sinusoidal waveform with a frequency range of 1 Hz to 50 MHz. The output level was set to 13 dB to reduce external noise as much as possible, which translates to 1 Vrms, as was used with the 50 $\Omega$ load. Impedance analysis was performed in the range of 10 kHz to 500 kHz with a step of 10 Hz, with a receiver bandwidth of 300 Hz. To get more accurate results, we measured the impedance six times and averaged the measurements; this helped reduce the noise in the measurements. As the aim is to measure the piezoelectric ring free of any external load (mechanical or electrical), we laid the piezoelectric ring between two copper plates, only ensuring contact, Figure 13.

To precisely reduce the outer diameter, a lathe was used by carefully sanding down each piezoelectric ring until the desired dimensions were reached. To ensure it was correctly smoothed down, a digital vernier of a standard precision of 0.02 mm was used, yielding the measurements shown in

Table 4. Measured outer diameter.

Diameter Variable	Measured Outer Diameter (mm)
$D_1$	30.92
$D_2$	31.92
$D_3$	32.92
$D_4$	34.04
$D_5$	34.80
$D_6$	35.86
$D_7$	36.92
$D_8$	37.90
$D_9$	38.92
$D_{10}$	40.02
$D_{11}$	40.82
$D_{12}$	42.02
$D_{13}$	42.70
$D_{14}$	43.96
$D_{15}$	45.00

.

6. Experimental and Numerical Results

Based on the measured outer diameters in Table 4, we performed a harmonic analysis using the same frequency range of 10 kHz to 500 kHz with a step of 10 Hz. Then, we compared the results, which can be seen in Figure 14, Figure 15 and Figure 16. We tracked highlighted resonances while reducing the diameter, yielding

Table 5. Experimental and theoretical resonances.

Diameter	${\mathit{f}}_1$ (Hz)		${\mathit{f}}_2$ (Hz)		${\mathit{f}}_3$ (Hz)		${\mathit{f}}_4$ (Hz)		${\mathit{f}}_5$ (Hz)		${\mathit{f}}_6$ (Hz)
	Exp.	Theor.	Exp.	Theor.	Exp.	Theor.	Exp.	Theor.	Exp.	Theor.	Exp.	Theor.
$D_1$	55,080	48,890	220,210	214,590	297,750	303,260	312,280	319,450	414,890	427,340	482,590	514,900
$D_2$	50,680	48,010	210,690	204,430	300,220	301,970	312,930	315,900	407,150	420,050	466,670	494,500
$D_3$	49,940	47,180	203,220	195,240	299,930	299,070	309,700	313,410	399,420	412,040	456,150	471,060
$D_4$	48,840	46,280	191,870	185,820	295,980	294,620	305,560	312,280	387,940	406,240	442,400	457,420
$D_5$	48,080	45,680	184,430	179,830	289,300	290,430	300,110	311,340	377,340	398,480	432,410	450,520
$D_6$	47,310	44,890	177,120	172,160	279,880	283,060	301,910	308,870	368,350	384,830	429,580	437,900
$D_7$	46,690	44,130	170,330	165,180	279,380	275,900	301,200	308,080	358,320	373,940	425,120	432,280
$D_8$	45,900	43,450	163,980	159,160	270,760	269,110	298,740	307,990	347,760	367,890	418,800	428,380
$D_9$	46,710	42,750	157,930	153,330	264,750	261,850	298,840	306,770	340,630	358,450	415,490	424,640
$D_{10}$	44,980	42,050	152,250	147,620	255,810	254,280	295,560	305,970	330,540	348,020	410,340	420,670
$D_{11}$	44,140	41,530	148,300	143,660	251,790	248,800	296,070	303,850	327,760	343,930	406,000	420,420
$D_{12}$	42,780	40,800	142,310	138,200	243,170	241,010	292,970	301,260	316,800	337,990	403,430	417,930
$D_{13}$	41,350	40,390	139,350	135,280	237,940	236,710	287,130	299,190	312,180	334,080	399,010	416,760
$D_{14}$	40,500	39,670	134,670	130,260	230,950	228,970	290,270	294,830	313,300	328,540	395,740	414,350
$D_{15}$	40,740	39,100	130,720	126,450	227,210	222,780	290,680	290,230	307,630	322,630	392,170	411,640

. Then, we proceeded to plot them vs. the outer diameter, resulting in Figure 17. From Table 5, the relative percentage error was calculated for each resonance, yielding

Table 6. Relative error by resonance; the error was calculated as follows: ${\epsilon }_{f_i}=\left|{\displaystyle \frac{f_{i_{\mathrm{Theor}.}}-f_{i_{\mathrm{Exp}.}}}{f_{i_{\mathrm{Theor}.}}}}\right|\times 100\%$ where $i=\{1,\dots ,6\}$. The frequencies were taken from Table 5.

Diameter	${\mathit{\epsilon }}_{{\mathit{f}}_1}$ (%)	${\mathit{\epsilon }}_{{\mathit{f}}_2}$ (%)	${\mathit{\epsilon }}_{{\mathit{f}}_3}$ (%)	${\mathit{\epsilon }}_{{\mathit{f}}_4}$ (%)	${\mathit{\epsilon }}_{{\mathit{f}}_5}$ (%)	${\mathit{\epsilon }}_{{\mathit{f}}_6}$ (%)
$D_1$	12.66	2.62	1.82	2.24	2.91	6.28
$D_2$	5.56	3.06	0.58	0.94	3.07	5.63
$D_3$	5.85	4.09	0.29	1.18	3.06	3.17
$D_4$	5.53	3.26	0.46	2.15	4.50	3.28
$D_5$	5.25	2.56	0.39	3.61	5.31	4.02
$D_6$	5.39	2.88	1.12	2.25	4.28	1.90
$D_7$	5.80	3.12	1.26	2.23	4.18	1.66
$D_8$	5.64	3.03	0.61	3.00	5.47	2.24
$D_9$	9.26	3.00	1.11	2.58	4.97	2.15
$D_{10}$	6.97	3.14	0.60	3.40	5.02	2.46
$D_{11}$	6.28	3.23	1.20	2.56	4.70	3.43
$D_{12}$	4.85	2.97	0.90	2.75	6.27	3.47
$D_{13}$	2.38	3.01	0.52	4.03	6.56	4.26
$D_{14}$	2.09	3.39	0.86	1.55	4.64	4.49
$D_{15}$	4.19	3.38	1.99	0.16	4.65	4.73

.

From Figure 17, it can be seen that the theoretical model was able to capture the general trends in the highlighted resonances, but as the frequency gets bigger, so does the error, and this model would prove unreliable after 500 kHz. From Table 5, it can be observed that after $f_3$, the model tends to underestimate resonances. Also according to Table 6, $f_3$ seems to be the resonance at which in general the relative error ${\epsilon }_{f_i}$ tends to be minimal.

7. Conclusions

All in all, this work was able to prove that using symmetry in cylindrical geometries is not only feasible but desirable for computing efficiency. This text was able to provide a simple mathematical approach to symmetry in piezoelectric rings of class 6 mm based on function parities. The symmetries helped us model the behavior of piezoelectric rings with a varying diameter, and similar behavior appeared when performing experiments. When performing modal analysis on the piezoelectric ring with a varying diameter, two important facts stood out, $S_1=S_7$ and $S_2=S_8$, which allowed us to further reduce the symmetries. One of the main limitations of this work can be found in symmetries $S_3$ and $S_5$ as well as in $S_4$ and $S_6$; they share natural frequencies but are not exactly the same, which suggests a further reduction in the symmetries or a possibility for a more concise symmetry description that can limit the doublets further and consequently decrease the computational cost. The other main limitation can be found in the experimental vs. numerical results when performing harmonic analysis. The numerical model represents the general behavior in terms of the shift in resonances, but this numerical model can and should be further improved to closely model the experimental data. Even though in our experiments, the temperature was not measured, it is well known that one of the causes of the shifting in frequencies is temperature. Furthermore, the elastic, piezoelectric, and dielectric properties of the PZT-8 material are reported with an error; this can also shift frequency. For this reason, future work should aim to perform optimization to adjust the experimental and numerical results. This will allow us to characterize piezoelectrics and thus further improve the model.