Piezoelectric effect finite element modeling method considering electric field and rate related hysteresis characteristics

Abstract

The invention discloses a piezoelectric effect finite element modeling method considering electric field and rate related hysteresis characteristics, belongs to the technical field of piezoelectric driving theories, and aims to solve the problem that a finite element model with a fixed piezoelectric coefficient is large in error during displacement analysis. A piezoelectric effect finite element modeling method which adopts a time-varying piezoelectric coefficient and considers electric field and rate-dependent hysteresis characteristics is provided, and accurate description of piezoelectric displacement under excitation of different electric signals is realized. Comprising the following steps: S1, piezoelectric displacement measurement; s2, identifying a hysteresis model and parameters; s3, regression fitting of the hysteresis model; and S4, finite element modeling. According to the modeling method of the piezoelectric ceramic finite element model, the hysteresis model is combined with the piezoelectric constitutive equation in the finite element model, the modeling method of the piezoelectric ceramic finite element model considering the hysteresis characteristic is provided, and the established finite element model can accurately predict output displacement tracks under different electric signal excitation and force-electricity coupling excitation in the simulation process; and the method has a certain promotion significance on the research of shearing piezoelectric ceramic analog simulation.

Description

TECHNICAL FIELD

[0001] The application relates to a piezoelectric effect finite element modeling method considering electric field and rate-dependent hysteresis characteristics, and belongs to the technical field of piezoelectric driving. BACKGROUND

[0002] A piezoelectric actuator taking piezoelectric ceramic material as a core has extremely high resolution (in the nanometer range), extremely short response time (below 1 ms), high actuating force, long service life, low power consumption and good usability, and the superior performance of the piezoelectric ceramic material enables the piezoelectric driving technology to be widely applied in many fields such as precise positioning, micro-fluid driving, biomedicine, aerospace and the like. In this case, based on the piezoelectric constitutive equation of the piezoelectric material, a finite element model is established to analyze the piezoelectric effect and the inverse piezoelectric effect under different conditions. In the piezoelectric constitutive equation, the relationship between the electric quantity and the mechanical quantity is connected through the piezoelectric coefficient. In the finite element simulation of the piezoelectric material, the piezoelectric coefficient is mostly constant. However, in the actual use process of the piezoelectric ceramic, due to the characteristics of the material itself, there is hysteresis nonlinearity between the response and the input electric signal. When the frequency of the input electric signal changes, the response of the piezoelectric ceramic shows a frequency-dependent phenomenon. This is because the piezoelectric coefficient of the piezoelectric ceramic changes with the change of the amplitude and the frequency of the applied electric excitation signal in the use process. In this case, the finite element modeling using the constant piezoelectric coefficient will result in a large error. SUMMARY

[0003] The application aims to solve the problem of large error of the finite element model with fixed piezoelectric coefficient in displacement analysis, and proposes a piezoelectric effect finite element modeling method considering electric field and rate-dependent hysteresis characteristics by using time-varying piezoelectric coefficient, so as to realize accurate description of piezoelectric displacement under different electric signal excitation.

[0004] The above application purpose is realized through the following technical scheme:

[0005] I. Piezoelectric displacement measurement

[0006] A suitable piezoelectric ceramic material is selected, different electric excitation signals with different voltages and frequencies are applied to the piezoelectric ceramic material, and the piezoelectric ceramic material moves under the excitation signals. A certain measurement method is used to measure the displacement of the piezoelectric ceramic material under different voltage and frequency excitation. The voltage-displacement hysteresis loop curve is drawn according to the measured piezoelectric displacement under different excitation signals.

[0007] II. Hysteresis model and parameter identification

[0008] Based on the hysteresis characteristics of the voltage-displacement curve, a hysteresis model is selected to represent the hysteresis characteristics of the piezoelectric displacement. For the piezoelectric displacement under each voltage and frequency, a suitable intelligent algorithm is used to identify the unknown parameters in the selected hysteresis model, and finally the hysteresis model parameter values corresponding to the excitation signal of the voltage and frequency used in technical scheme one are obtained.

[0009] III. Hysteresis model regression fitting

[0010] The hysteresis model parameters corresponding to different voltages and frequencies obtained above are fitted with respect to the electric field and frequency to obtain the mathematical expression of the electric field and frequency dependent hysteresis model parameters. The electric field and frequency dependent hysteresis model parameters are combined with the hysteresis model to obtain a hysteresis model considering the electric field and frequency dependent hysteresis characteristics. The time-varying piezoelectric coefficients under different voltages and frequencies are calculated by the obtained hysteresis model.

[0011] IV. Finite element modeling

[0012] A piezoelectric ceramic finite element model is established in the finite element analysis software, and the piezoelectric coefficients in the piezoelectric constitutive equation of the finite element model are fixed values. By writing a program, the fixed piezoelectric coefficients in the finite element model are replaced by the electric field and frequency dependent time-varying piezoelectric coefficients calculated in technical scheme three, and the piezoelectric ceramic finite element model considering the electric field and frequency dependent hysteresis characteristics is completed.

[0013] The effectiveness of the proposed model is verified by comparing the output displacement of the established piezoelectric ceramic finite element model considering the electric field and frequency dependent hysteresis characteristics, the output displacement of the traditional finite element model and the experimental data. On this basis, the piezoelectric displacement under the action of force-electric coupling is analyzed by simultaneously applying electric signals and mechanical loads to the model.

[0014] Compared with the prior art, the present application has the following beneficial effects:

[0015] The present application combines the hysteresis model with the piezoelectric constitutive equation in the finite element model, and proposes a modeling method for a piezoelectric ceramic finite element model considering hysteresis characteristics. The established finite element model can accurately predict the output displacement trajectory under different electric signal excitations and force-electric coupling excitations during simulation, which has certain promoting significance for the simulation and research of shear piezoelectric ceramics. BRIEF DESCRIPTION OF DRAWINGS

[0016] Figure 1 The overall implementation flowchart of the shear piezoelectric ceramic finite element model considering hysteresis characteristics is shown in the figure.

[0017] Figure 2 The modification flowchart of the inp file is shown in the figure.

[0018] Figure 3Displacement contour of the shear piezoelectric ceramic finite element model considering hysteresis at different time (a) 0 s (b) 0.001 s (c) 0.002 s (d) 0.003 s (e) 0.004 s (f) 0.005 s under sinusoidal electric excitation in one period;

[0019] Figure 4 Displacement comparison chart of the shear piezoelectric ceramic finite element model considering hysteresis and traditional model and experimental data;

[0020] Figure 5 Hysteresis loop curve comparison chart of the shear piezoelectric ceramic finite element model considering hysteresis and traditional model and experimental data; DETAILED DESCRIPTION

[0021] In order to better understand the purpose, structure and function of the present application, the present application will be described in further detail below with reference to the accompanying drawings.

[0022] Step one: piezoelectric displacement measurement experiment

[0023] Take the shear piezoelectric ceramic with the model number NAC2902-H2.8 from Noliac, Denmark as an example to carry out the piezoelectric displacement measurement experiment. Build a piezoelectric displacement measurement device, connect the signal generator, power amplifier and shear piezoelectric ceramic with wires, the laser emitted by the one-dimensional laser Doppler vibration meter is irradiated on a point in the movement direction of the shear piezoelectric ceramic, and the excitation signal generated by the signal generator is synchronized with the displacement signal measured by the one-dimensional laser Doppler vibration meter to display in the oscilloscope. By changing the voltage and frequency values of the excitation signal, the displacement of the shear piezoelectric ceramic under different voltages and frequencies is recorded. According to the measured experimental data, the voltage-displacement hysteresis loop curves under different conditions are drawn.

[0024] Step two: hysteresis model establishment and parameter identification

[0025] Take the numerical analysis software MATLAB as an example to carry out hysteresis modeling and parameter identification. In this patent, the Bouc-Wen model is selected to characterize the piezoelectric displacement hysteresis characteristics. By changing the parameter values in the Bouc-Wen model, the Bouc-Wen model can reflect different hysteresis loop curves. Based on the experimental piezoelectric displacement data, the particle swarm optimization is used to identify the parameters in the Bouc-Wen model. The objective function of the particle swarm optimization algorithm is selected as the mean square error between the experimental data and the model data. Finally, the corresponding Bouc-Wen model parameter values under different voltages and frequencies are obtained.

[0026] Step three: hysteresis model parameter fitting

[0027] The parameters in the Bouc-Wen model are fitted with the electric field and frequency correlation. The electric field strength can be obtained by dividing the voltage by the thickness of the shear piezoelectric ceramic. The mathematical expression of the electric field and frequency correlation of each parameter in the model is obtained F c (E,f). The mathematical expression of the electric field and frequency correlation of the model parameters is combined with the Bouc-Wen model to obtain the electric field and frequency correlation hysteresis model F BW (E,f). The electric field and frequency correlation hysteresis model can calculate the piezoelectric displacement under the electric field and frequency within the identification range. Taking the excitation signal of E=200V / mm and f=200Hz as an example, the piezoelectric displacement calculated under this condition is used to draw the voltage-displacement hysteresis loop curve. The displacement of each point of the hysteresis loop curve is removed from the corresponding voltage to calculate the time-varying piezoelectric coefficient d 15 (t).

[0028] Step four: finite element model establishment

[0029] The overall process of establishing a finite element model of shear piezoelectric ceramic considering hysteresis characteristics is shown in Figure 1 Taking the finite element analysis software Abaqus as an example, a geometric model with a size of 5mm×5mm×0.5mm is established. According to the selected piezoelectric ceramic material parameters, the material properties are assigned to the geometric model, and the piezoelectric coefficient is set to the fixed value of the material 6.69×10 -10 C / N. The material direction is specified according to the default coordinate system direction, the piezoelectric unit (C3D8E) is created, and the mesh is divided. The boundary condition is set to apply E=200V / mm and f=200Hz sinusoidal electric excitation to the model, which is completed by inputting the corresponding values of the excitation signal into the table of boundary condition application in the software. When creating the analysis step, the analysis step time should correspond to the excitation signal time. When the excitation signal frequency is 200Hz, the analysis step time is set to 0.005s. After completing the finite element model creation, the inp file for piezoelectric analysis is exported, and the job is submitted for finite element analysis to obtain the traditional fixed piezoelectric coefficient d 15 The output displacement of the finite element model.

[0030] In MATLAB, the d 15 (t) calculated in step two is written into the txt text through fopen, fprintf, and fclose commands. Then the Python program is called through the system command to modify the inp file. The modification process is as follows Figure 2As shown, the Python program reads the contents of the inp and txt files using the open, f.readline, and f.close commands based on a predetermined path and filename. It then locates the fixed value d to be replaced using the keyword "*Piezoelectric,dependencies=1,type=E\n" in the inp file. 15 The line in question can be manipulated using commands such as `readlines` to display the data. 15 Replace the value with d 15 The value of (t) is used to modify the inp file. A USDFLD subroutine is written to make the defined field variable, d... 15 (t) changes over time during finite element analysis.

[0031] Displacement contour plots of a shear piezoelectric ceramic finite element model considering hysteresis characteristics at different times within one cycle of sinusoidal electric excitation are shown below. Figure 3 As shown, the shear piezoelectric ceramic undergoes reciprocating shear motion within one cycle. To verify the effectiveness of the model and its superiority over traditional models, a single node in the finite element model's field output was selected to obtain the XY displacement curves. The results were then compared with experimental data and a fixed d... 15 The results are compared with those of the traditional finite element model, such as Figure 4 As shown in the figure. It can be observed that by using a fixed d... 15 The traditional finite element model has relatively large errors, while the d-model... 15 The output displacement of the finite element model considering the electric field and rate-dependent hysteresis (t) shows good agreement with the experimental data. A comparison of the hysteresis loop curves for the three cases is shown in the figure below. Figure 5 As shown, the results obtained from the traditional finite element model cannot describe the hysteresis characteristics of piezoelectric displacement, and the error is relatively large. Compared with the traditional finite element model with a constant piezoelectric coefficient, the finite element model considering electric field and rate-dependent hysteresis can more accurately describe the piezoelectric displacement under electric field excitation. Based on the above, the piezoelectric displacement under electromechanical coupling can also be analyzed by applying mechanical loads to the finite element model.

[0032] This invention, based on a series of embodiments, clearly demonstrates its actual operation and potential application scenarios. Those skilled in the art can improve, adjust, or replace these embodiments without departing from the basic spirit and core innovation of this invention, to adapt to different practical needs and usage scenarios. Furthermore, the core principles of this invention can be referenced, combined with specific circumstances and material properties, to make flexible innovations and improvements. Therefore, this invention is not limited to the specific embodiments disclosed herein, and all embodiments falling within the scope of the claims of this application are protected by this invention.

Claims

1. A finite element modeling method for the piezoelectric effect considering electric field and rate-dependent hysteresis characteristics, characterized in that: Includes the following steps: S1. Piezoelectric displacement measurement; S2. Hysteresis model and parameter identification; S3. Lag model regression fitting; S4. Finite element modeling.

2. The finite element modeling method for piezoelectric effect considering electric field and rate-dependent hysteresis characteristics according to claim 1, characterized in that: The specific steps of S1. piezoelectric displacement measurement are as follows: S11. Select a suitable piezoelectric ceramic material and apply electrical excitation signals of different voltages and frequencies to it. The piezoelectric ceramic moves under the excitation signal. Taking the NAC2902-H2.8 shear piezoelectric ceramic from Noliac, Denmark as an example, a piezoelectric ceramic displacement measurement experiment is conducted. A piezoelectric displacement measurement device is built, and the signal generator, power amplifier, and shear piezoelectric ceramic are connected by wires. The laser emitted by the one-dimensional laser Doppler vibrometer is irradiated at a point in the direction of motion of the shear piezoelectric ceramic. The excitation signal generated by the signal generator and the displacement signal measured by the one-dimensional laser Doppler vibrometer are synchronously displayed on the oscilloscope. S12. By changing the voltage and frequency values ​​of the excitation signal, record the displacement of the shear piezoelectric ceramic under different voltages and frequencies. Plot the voltage-displacement hysteresis loop curves under different conditions based on the measured experimental data.

3. The finite element modeling method for piezoelectric effect considering electric field and rate-dependent hysteresis characteristics according to claim 2, characterized in that: S2. Hysteresis model and parameter identification, the specific steps are as follows: S21. By changing the parameter values ​​in the Bouc-Wen model, different hysteresis loop curves are reflected in the Bouc-Wen model. S22. Based on the experimentally measured piezoelectric displacement data, particle swarm optimization was used to identify the parameters in the Bouc-Wen model. S23. The objective function of the particle swarm optimization algorithm is chosen as the mean square error between the experimental data and the model data. Finally, the corresponding Bouc-Wen model parameter values ​​for different voltages and frequencies are obtained.

4. The finite element modeling method for piezoelectric effect considering electric field and rate-dependent hysteresis characteristics according to claim 3, characterized in that: S3. Lag model regression fitting, the specific steps of which are as follows: S31. Perform electric field and frequency correlation fitting on the parameters in the Bouc-Wen model. The electric field strength can be obtained by dividing the voltage by the shear thickness of the piezoelectric ceramic, and obtain the mathematical expression for the electric field and frequency correlation of each parameter in the model. S32. Combine the mathematical expressions related to the electric field and frequency of the model parameters with the Bouc-Wen model to obtain a hysteresis model related to the electric field and frequency. S33. The electric field and rate-dependent hysteresis model can calculate the piezoelectric displacement under the electric field and frequency within the identified range. Taking an excitation signal of E = 200 V / mm and f = 200 Hz as an example, the voltage-displacement hysteresis loop curve is plotted using the piezoelectric displacement calculated under this condition. The voltage corresponding to the displacement at each point of the hysteresis loop curve is removed to calculate the time-varying piezoelectric coefficient.

5. The finite element modeling method for piezoelectric effect considering electric field and rate-dependent hysteresis characteristics according to claim 4, characterized in that: S4. Finite element modeling, the specific steps are as follows: S41. Using the finite element analysis software Abaqus as an example, establish a geometric model with dimensions of 5mm × 5mm × 0.5mm. Based on the selected piezoelectric ceramic material parameters, assign material properties to the geometric model, where the piezoelectric coefficient uses a fixed value of 6.69 × 10⁻¹⁰ C / N. Specify the material orientation according to the default coordinate system, create piezoelectric elements (C3D8E), and complete the mesh generation. Set the boundary conditions to apply a sinusoidal electric excitation of E = 200V / mm and f = 200Hz to the model. This is done by inputting the corresponding values ​​of the excitation signal into the boundary condition table in the software. When creating the analysis step, the analysis step time must correspond to the excitation signal time; when the excitation signal frequency is 200Hz, the analysis step time is set to 0.005s. After completing the finite element model creation, export the inp file for piezoelectric analysis and submit the job for finite element analysis to obtain the output displacement of the traditional fixed piezoelectric coefficient finite element model under sinusoidal electric excitation of E = 200V / mm and f = 200Hz. S42. In MATLAB, use the commands fopen, fprintf, and fclose to write the result calculated in step two into a txt file, and then use the system command to call the written Python program to modify the inp file. A Python program reads the contents of .inp and .txt files based on predetermined paths and filenames using the commands `open`, `f.readline`, and `f.close`. It locates the line containing the fixed value to be replaced using the keyword "*Piezoelectric,dependencies=1,type=E\n" in the .inp file, and then uses commands like `readlines` to replace the value with the correct one, thus modifying the .inp file. A USDFLD subroutine is written to make the defined field variables change over time during finite element analysis. S43. To verify the effectiveness of the model and its superiority over traditional models, a single node in the finite element model's field output was selected to obtain the XY displacement curves. The results were then compared with experimental data and results obtained using a fixed traditional finite element model. It was found that the fixed traditional finite element model had relatively large errors, while the output displacement of the finite element model considering the electric field and rate-dependent hysteresis showed better agreement with the experimental data. It was also found that the results obtained from the traditional finite element model could not describe the hysteresis characteristics of piezoelectric displacement, resulting in significant errors. Compared to the traditional constant piezoelectric coefficient finite element model, the finite element model considering the electric field and rate-dependent hysteresis can more accurately describe the piezoelectric displacement under electric field excitation.

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