Full length articleFinite element model updating of piezoceramic thin-walled tube transducers
Introduction
There are broad applications of various types of piezoceramic transducers; one of the most common types is the piezoceramic thin-walled tube transducer poled in the radial direction [1]. Although the design of a piezoceramic transducer often calls for finite element (FE) analysis [2], [3], building a reliable FE model for a piezoceramic transducer is still an issue because the required piezoceramic material values provided by the manufacturers are often incomplete or inaccurate. The causes of inaccurate values stem from several factors, including the polycrystalline nature of piezoceramics, statistical variations in their composition, and production-related influences. For some material properties provided in a typical manufacturer’s specification sheet, an error of is quite possible [4]. To overcome this problem, it is desired to develop a FE model updating method for correcting the parameters of piezoceramic material.
Whereas various measurements, such as nodal displacement, nodal velocity, and eigen data [5], [6], [7] have been utilized for correcting the material parameters of piezoceramic material, the most commonly used measurement is the electric impedance function [8], which is a frequency response function (FRF) that could be easily measured in laboratory. Roughly, using impedance functions to update the piezoceramic material parameters can be classified into two categories: FE-based and non-FE-based methods. While non-FE-based methods evaluate material parameters by directly formulating analytical relationship between those parameters and the measured impedance functions [9], [10], [11], [12], [13], [14], [15], FE-based methods must build a FE model initially using nominal material values, then update them based on measurements.
The proposed method belongs to the category of FE-based model updating techniques using measured impedance functions; thus, the literature review will focus on this category as well. Despite considerable progress on the FE model updating techniques for mechanical and civil structures [16], only a few published articles are related to the updating of the material parameters of piezoceramic devices. Among which, De Boe et al. [17], assuming that dielectric parameters were given, developed a method that required a scaling method for the piezoelectric stiffness matrix to update the elastic and piezoelectric parameters of a piezoelectric bar. Piranda et al. [18], [19] provided a scheme for updating the elastic, piezoelectric and dielectric properties of a 2-dimensional multi-layer piezoelectric bar by matching the updated and measured admittance curves. Using a Newton-CG inversion optimization technique, Kaltenbacher et al. [20] proposed a method to reconstruct all material parameters of a piezoceramic thickness resonator. Joo et al. [21] adopted the steepest-descent optimization technique to determine the material parameters of a piezoelectric transformer. Perez et al. [22], [23] employed both sensitivity analysis and a Nelder–Mead optimization technique over a wide frequency band of impedance function to correct material constants of a piezoceramic disk. For tuning material parameters of thick piezoelectric disks, Jonsson et al. [24] used a feedback algorithm with the measured harmonic overtones. Kiyono et al. [25] employed a gradient-based optimization algorithm to determine the full piezoelectric complex parameters of piezoelectric disks. Relying on measured eigenfrequencies, which could have been extracted from a measured impedance function, of short/open-circuit models and making use of a genetic algorithm as a global optimization technique, Montemurro et al. [7] developed a strategy to identify electromechanical properties of an active plate. In summary, all mentioned articles had employed a global optimization technique. As the optimization was based on the impedance or admittance function, its outcome might match the analytical and measured impedance curves well, but the updated values of material parameters could be far away from their true values. In addition, the selection of measured frequency range and the initial parameter guess also had a large impact on the numerical stability. In their numerical implementations, as an iterative procedure was necessary, all those articles required to rebuild a new FE model at each iteration from scratch with the updated material parameters.
Never has the topic of updating material parameters of piezoceramic tube transducers been investigated before, even though many articles have been devoted to the electromechanical effects and dynamic characteristics of such transducers [26], [27], [28], [29], [30], [31]. The main objective of this study is to develop a novel FE model updating method for correcting the sensitive material parameters of a piezoceramic thin-walled tube transducer, using only a few characteristic quantities extracted from a measured impedance function of the transducer. The novelty lies in the following aspects: (1) We begin with the identification of the sensitive material parameters, recognizing that not all parameters are sensitive to the intended functions of a thin-walled piezoceramic tube transducer. (2) We utilize linear parameterization models for the mechanical, piezoelectric and dielectric stiffness matrices in terms of the sensitive parameters; consequently, these matrices can be updated at each iteration easily. Because there is no need to build a new FE model from scratch again, the proposed method significantly improve the computational efficiency. (3) We update the sensitive dielectric parameter directly from the impedance value at a very low frequency region, in accordance with the concept of capacitance. (4) We update the elastic parameters by using a short-circuit model, along with the measured resonant frequencies; and the piezoelectric parameter by using an open-circuit model, together with the measured anti-resonant frequencies. Note that both the measured resonant frequencies and anti-resonant frequencies could be easily extracted from a measured impedance function.
Two numerical examples are given in this paper. The purpose of the first example, which is a computer simulation study, is for illustrating the detailed procedure and demonstrating the correctness and effectiveness of the proposed method. The second example is a laboratory study; its goal is to evaluate the performance of the proposed method on a realistic piezoceramic tube transducer.
Section snippets
Preliminaries
We review background materials that are crucial to this article, including constitutive equations of piezoelectric materials, transducer impedance function, and short- and open-circuit finite element models of a piezoelectric transducer. Throughout the article, matrices and vectors are denoted by boldface letters; the notation convention for piezoelectric materials follows the IEEE standard [32].
Sensitivity-based model updating method
An iterative sensitivity-based (SB) method is developed to update material parameters of piezoelectric tube transducers. At the th iteration, the mathematical relationship between the measured output and the parameters to be updated is written as where is the error in the measured output and is the measured data vector; is the perturbation in the updating parameters; and subscript “” indicates the th iteration. The parameter vector represents the “actual”
Reduced constitutive equations and parameters
Not all ten parameters in Eq. (8) are sensitive to the intended functions of a thin-walled piezoceramic tube transducer. First, we show the reduced constitutive equations, derived from the original d-form equations, involve only 4 sensitive d-form material parameters. Note that the e-form constitutive equations cannot be easily reduced, but implementing the proposed model updating method needs to use e-form parameters for updating stiffness matrices. Thus, the conversions between the four
Updating sensitive parameters
When the only available measurement is the impedance of a piezoceramic thin-walled tube transducer, the 4 sensitive parameters of the transducer are updated through 3 steps. First, the dielectric parameter can be estimated directly by the value of at near zero. Second, the elastic parameters and are updated through a short-circuit model, whose modal frequencies are the resonant frequencies associated with . Lastly, after , , and having been
Numerical examples
Two numerical examples are presented in this paper. The first example is based on a simulated transducer, whose target values of the updating parameters are known; its purpose is to test the accuracy and efficiency of the proposed method, as well as to illustrate the detailed numerical procedure. The second example is to demonstrate the applicability of the proposed method to a real transducer. Throughout the numerical studies, the unit system is MKS (Meter-Kilogram-Second) system. For
Conclusions
This study developed a methodology for updating the sensitive parameters of a piezoceramic thin-walled tube transducer based on a few characteristic data extracted from the measured impedance function (MIF) of the transducer. Four d-form parameters , , , and were identified to be sensitive. First, the permittivity parameter could be updated directly by using the value of the MIF corresponding to a low frequency. Second, the elastic parameters and were updated
CRediT authorship contribution statement
Sau-Lon James Hu: Conceptualization, Methodology, Writing - review & editing, Supervision. Liang Su: Investigation, Data curation, Writing - original draft. Shuai Cong: Software, Validation.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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