WU Jinghui，ZHAO Shengquan，XIE Longtao，WANG Ji

Piezoelectric Device Laboratory，School of Mechanical Engineering & Mechanics，Ningbo University，Ningbo 315211，P.R.China

Abstract: The explosive demands for facial masks as vital personal protection equipment（PPE）in the wake of Covid-19 have challenged many industries and enterprises in technology and capacity，and the piezoelectric ceramic（PZT）transducers for the production of facial masks in the welding process are in heavy demand. In the earlier days of the epidemic，the supply of ceramic transducers cannot meet its increasing demands，and efforts in materials，development，and production are mobilized to provide the transducers to mask producers for quick production. The simplest solution is presented with the employment of Rayleigh-Ritz method for the vibration analysis，then different materials can be selected to achieve the required frequency and energy standards. The fully tailored method and results can be utilized by the engineers for quick development of the PZT transducers to perform precise function in welding.

Key words：piezoelectric ceramic（PZT）；transducer；facial mask welding；personal protection equipment（PPE）

0 Introduction

As one of the important personal protection equipment（PPE），facial masks of various specifica?tions are widely demanded in a sudden due to explo?sive and simultaneous outbreaks of Covid-19 world?wide. Since facial mask is one of the major regula?tions widely accepted and practiced around the world，people are suggested or even required man?datorily to wear facial masks in highly risky environ?ments. Therefore， facial mask demands surge worldwide. Many mask production lines are built and operating in a hurry yet beyond the capacity. All components needed in the production lines are in short supply. Among them，piezoelectric ceramic（PZT）ring for the ultrasonic welding process is one of the highly sought components related to the mask production with limited supply. When the epidemic spreads，many efforts have been made to produce the transducers quickly or to develop replacements.In the design process before production，it is critical to analyze the essential properties of ceramic ring in assistance with proper tools and procedure，such as vibration frequency and mode. As the obligation to China and global community，authors have released a guide on specifications，materials，and design con?siderations of PZT rings for facial mask production lines. PZT ring is used to generate ultrasonic waves of specific frequency，thus enabling the welding of mask fibers at a frequency of 15 kHz and 20 kHz with a piezoelectric cylinder dimension of approxi?mately 50 mm×17 mm×7 mm［1］. The subsequent development focuses on the efficient and simple pro?cedures for accurate analysis of vibrations of ceramic rings presented in this paper.

The ceramic ring is usually made of PZT ce?ramic with large elastic constants，or the hard PZT，for the generation of appropriate ultrasonic waves needed in the welding process. Most transducers for this objective are made of PZT-4，but as a replace?ment of PZT-4，PZT-8 can also have equivalent properties. For the high frequency vibrations of the PZT ring，the analysis is not a straightforward task with readily available methods and results，due to the complicated nature of finite elastic solids with material anisotropy［2-3］. The analysis should be per?formed by either approximate or numerical meth?ods，which are widely available but not specific to the particular transducer，and the process can be a challenge to many engineers. Therefore，recapitulat?ed the common procedure of the analysis with the Rayleigh-Ritz method，a detailed formulation is pro?vided so that engineers can analyze the parameters of PZT rings with a reliable guide in a fast man?ner［4］. In addition，results of both PZT-4 and -8 are demonstrated to give examples for material choices.All procedure and codes of the program are avail?able on the website of Piezoelectric Device Labora?tory（piezo.nbu.edu.cn）. We promise to provide full support on the analysis with the method outlined in this paper.

The formulation here is straightforward with the theory of piezoelectricity and wave propagation，which can be found in Refs.［5-8］. Although the dif?ferential equations of the vibration of a piezoelectric ring are well-known and the general procedure of an?alytical solutions have been discussed［5，9-10］，the closed form solutions are not available. The most ef?fective method for the solution is the Rayleigh-Ritz method［11-12］. In this paper，the Rayleigh-Ritz meth?od is tailored for this transducer with improved accu?racy and simplicity.

1 Mathematical Model

A typical PZT ring is shown in Fig. 1，where R1，R0are radii and h is the height. The PZT is al?ways polarized in the z direction in practice. The pa?rameters shown in Fig.1 and the material constants will be used in the following analysis.

Fig.1 Typical PZT ring

Considering deformation and electric field，we define the generalized strain energy in cylindrical co?ordinate as

where C，ε，e are the elastic，the piezoelectric，and the dielectric material constants. The components of the strain and the electric field are

For a typical PZT material，the material matri?ces of the piezoelectric，the elastic，and the dielec?tric constants are

The element values of the above matrices will be given later for specific materials.

Naturally，the kinetic energy of the PZT ring is

where ρ is the density of ceramic materials；u，v，w are the displacements；and t is the time.

Given the displacements and the electric poten?tial of the ceramic ring，components of strain and electric field as generalized strains in cylindrical co?ordinates are

where φ is the electric potential.

For simplicity，the normalized coordinates are

For free vibrations，the displacements and elec?tric potential of the ceramic ring in normalized coor?dinates are

where ω is the frequency of vibration and i=

In cylindrical coordinates，functions of the am?plitudes of displacements and electric potential of the ceramic ring can be further written as

where n is an integer representing the circumferen?tial wavenumber. If n=0，the vibration is axisym?metric.

Now the generalized strain energy in Eq.（1）with normalized coordinates is

The kinetic energy is

where the components of the generalized strain and parameters are

The amplitudes of the mechanical displace?ments and electric potential are expressed as the product of Chebyshev polynomials and boundary functions.

where the boundary functions are products of the two functions satisfying the boundary conditions at the two endsIn Eq.（12），all the dis?placements satisfy the cylindrical conditions of the ring，and the electric potential satisfies the boundary conditions of electric field of free vibrations by setting the boundary functions to 1. The integers I，J，K，L，M，N are the order of the Chebyshev poly?nomials for the displacements and potential evalua?tion.They take the same value in the actual computa?tion. The coefficients Aij，Bkl，Cmn，Dopare the ampli?tudes to be determined. The sth-order Chebyshev polynomials Ps(χ) (s=i，j，k，l，m，n，o，p；used here are

The energy functional of the PZT ring is

With the Rayleigh-Ritz method，the deriva?tives are

Hence，the eigenvalue for the free vibrations of the ceramic ring is

For the composition of stiffness and mass sub?matrices，the subscripts u，v，w，φ represent the dis?placements and electric potential；i，k，m，o the or?ders of the Chebyshev polynomialand s，j，l，n，p the orders of Chebyshev polynomialConsequently，the elements of stiffness and mass matrices are

where the elements of K and M are [ Kαβ]qrand[ Mαβ]qrwith

By eliminating the electric potential from Eq.（16），the piezoelectrically stiffened eigenfre?quency equation of the vibration is

where the piezoelectrically stiffened stiffness matrix is

By solving Eq.（19）for eigenvalues，the analy?sis of free vibrations of the PZT ring can be per?formed to assist in the design of vital components.Specifically，the appropriate and optimal choices of key parameters will help in the selection of material and in other considerations such as ultrasonic power.

A complete analytical procedure for the vibra?tions of a PZT ring is presented with the Rayleigh-Ritz method. The electric field in the eigenvalue equation is eliminated for efficiency. The procedure and formulation are almost standard by adopting the sophisticated techniques and selection of displace?ment functions. The detailed procedure specifically developed for the PZT ring with targeted applica?tions in ultrasonic welding will provide right tools in the design and improve the performance of transduc?ers for the facial mark production lines in full capaci?ty worldwide.

2 Simulation and Discussion

With the detailed process mentioned above，the procedure has to be tested and validated for the analysis. There are some pevious studies on similar structures with the Rayleigh-Ritz method and other numerical methods，and there are adequate referenc?es for the needed validation. The validation process is important in the analysis.

First，a completely free ring with inner-outer radius ratio R1/R0=2，h/R1=0.4，and ν=0.3 is analyzed. The material constants are C11=C33=E(1-ν)/[(1+ν)(1-2ν)]，C12=C13=νE/[(1+ν)(1-2ν)]，C44=C66=E/(2(1+ν) )，where E is the Young’s modulus. For comparison，the first non-dimensional frequencies of free vibrations，Ω=for wavenumber n=1，2，3 are calcu?lated and compared in Table 1.

Table 1 The first non-dimensional frequency of circular annular plate

Second，a ceramic PZT-4 ring with outer radi?us R1=25 mm， inner radius R0=8.5 mm， and thickness h=7 mm is analyzed for the frequency convergence in Table 2. For the calculation with the Rayleigh-Ritz method，orders of the Chebyshev polynomials are set as same as the circumferential wavenumber n. From the simulation results in Ta?ble 2，it is clear that with the 8th-order polynomials in each variable，the results are convergent.

Furthermore，the material properties of PZT-4 from the COMSOL material library are C11=139 GPa， C12=77.84 GPa， C13=74.28 GPa，C33=115.4 GPa， C44=25.64 GPa， C66=30.58 GPa， e15=12.7 C/m2， e31=-5.2 C/m2，e33=15.1 C/m2，and ε11=762.5，ε33=663.2 with ε0=8.854×10-12F/m. The vibration analysis is performed with COMSOL for comparison in Ta?ble 3.

Table 2 Convergence check of resonant frequencies of a PZT?4 ring

Table 3 Comparison of vibration frequencies of a PZT?4 ring with finite element method (FEM)

In another example，vibration frequencies of a PZT ring are given in Table 4 with an outer radius of R1=25 mm， inner radius of R0=8.5 mm，height of h=7 mm. The material constants from COMSOL data library of PZT-8 are： C11=146.9 GPa， C12=81.09 GPa， C13=81.05 GPa，C33=131.7 GPa， C44=31.35 GPa， C66=32.89 GPa；e15=10.34 C/m2，e31=-3.875 C/m2，e15=13.91 C/m2；ε11=904.4，ε33=561.6，ε0=8.854×10-12F/m.

Table 4 Vibration frequencies of a PZT?8 ring in this paper and FEM

3 Conclusions

The Rayleigh-Ritz method is a highly efficient and reliable method for the calculation of resonant vibration frequencies of structures. Combined the equations of a PZT ring and the generalized displace?ment representation by Chebyshev polynomials，a simple and efficient procedure based on the Ray?leigh-Ritz method for the calculation of vibration fre?quencies and mode shapes are presented and validat?ed. The method has also been verified from analysis of FEM by using both PZT-4 and PZT-8 materials.The accurate and efficient calculation of vibration frequency is the first step in the selection of structur?al parameters for accurate frequency needed by the transducer. Further analysis and design parameters related to the ultrasonic power can also be calculated with the modification and extension of the Raleigh-Ritz method presented in this paper. Moveover，the utilization of newer piezoelectric materials and differ?ent frequencies for welding and other processes can be accurately chosen with results from the method shown in this paper. We hope the proposed method，along with the codes in MATLAB can be freely available to engineers and students in the design for improvement of transducers needed in combating Covid-19 and related causes.