圆形压电单晶执行器的挠度特性的模拟、分析与优化
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摘要
圆形压电单晶执行器(Circular piezoelectric unimorph actuator, CPUA)由圆形压电陶瓷片通过环氧树脂粘结在金属基层上构成,由于结构简单、输出位移相对较大,在采用压电执行器作为力或位移发生器的领域得到广泛应用。为对圆形压电单晶执行器以及基于圆形压电单晶执行器的系统进行性能预测和/或结构优化,建立其分析模型是关键。由于粘结层相对压电层和基层很薄,很多圆形压电单晶执行器模型只考虑了压电陶瓷晶片和基层,忽略了粘结层。如果在建模的过程中考虑粘结层应该能更精确模拟圆形压电单晶执行器的挠度变形。
本文利用圆形薄板小挠度弯曲理论建立了固支边部分覆盖圆形压电单晶执行器在作用电压控制下的挠度数学模型,在建模过程中将粘结层作为单独一层考虑。根据建立的数学模型,分析了电压、压强及圆形压电单晶执行器的几何参数和材料特性对其挠度的影响。建立了实验装置对圆形压电单晶执行器静态和动态挠度进行了测试。根据本文建立的模型以圆形压电单晶执行器变形体积最大为目标对圆形压电单晶执行器的几何结构进行了优化。
本文主要的研究工作和成果包括:
1.利用圆形薄板小挠度弯曲理论、压电本构方程和轴对称的平衡方程建立了固支边部分覆盖圆形压电单晶执行器在作用电压控制下的挠度数学模型,在建模过程中将粘结层作为单独一层考虑。
2.根据建立的数学模型,分析了作用于圆形压电单晶执行器的电压、压强及圆形压电单晶执行器的几何参数和材料特性对其挠度的影响。当圆形压电单晶执行器结构一定时,改变作用电压或压强可得到圆形压电单晶执行器不同的挠度,且圆形压电单晶执行器的中心挠度与作用电压和压强成正比;当压电层和基层的半径比在0.85-0.9左右时圆形压电单晶执行器中心挠度达到最大值,圆形压电单晶执行器变形最大;粘结层与基层厚度比越小,圆形压电单晶执行器挠度越大,实际设计圆形压电单晶执行器时,为了得到较大的挠度,可以在不影响压电层和基层的粘结的情况下减小粘结层的厚度;粘结层的柔性系数对圆形压电单晶执行器挠度影响很小,在优化设计圆形压电单晶执行器时可以不考虑粘结层的柔性系数。
3.建立了实验装置并对两种半径比的圆形压电单晶执行器的静态和动态挠度进行了测试。实验结果表明由建立的模型预测的圆形压电单晶执行器的静态挠度与实际测得值吻合,比忽略粘结层的模型更精确,相对最大误差最多可减小8.45%;忽略粘结层影响的情况下部分覆盖圆形压电单晶执行器挠度模拟误差明显大于半覆盖圆形压电单晶执行器挠度模拟误差;由于未考虑压电陶瓷的迟滞特性,本文建立的模型在预测圆形压电单晶执行器的动态特性时存在较大误差,需进一步研究。
4.根据本文建立的圆形压电单晶执行器的挠度模型,以圆形压电单晶执行器变形体积最大为目标对圆形压电单晶执行器的几何结构运用遗传算法进行了优化。优化结果表明,当基层半径一定时,压电层和基层的半径比为0.865,压电层、粘结层和基层的厚度为允许值的最小值时,圆形压电单晶执行器变形体积最大。
本文的研究工作对圆形压电单晶执行器以及基于圆形压电单晶执行器的系统进行性能预测和/或结构优化奠定了理论基础。
A circular piezoelectric unimorph actuator (CPUA), by bonding a circular polarized piezoelectric layer and a substrate layer to each other using the conducting epoxy as the bonding layer, is widely used for a variety of force and/or displacement generators because of the compact structure and relative large deflection. In order to predict and optimize the behavior of CPUAs and CPUAs based systems, the key problem lies in establishing the analytical model for CPUAs. The bonding layer of the CPUA is relatively very thin compared with the piezoelectric layer and the substrate layer, most models for the CPUAs took both the piezoelectric layer and the substrate layer into account while always neglected the bonding layer. Theoretically speaking, it is reasonable that the static deflection model for CPUAs considering the influence of the bonding layer can accurately model the deflections of CPUAs.
In this paper, based on the classical laminated plate theory, a new static deflection model for CPUAs subjected to applied voltage is established and the bonding layer is taken into account as an individual layer. According to the established analytical model, the influences of the voltage, pressure, structural parameters, and material properties of the CPUA on the transverse deflection are numerically simulated and the static and dynamic characteristics of the CPUA are experimentally tested. Based on the established analytical model, the structural parameters are optimized aiming at getting the maximum volume caused by deflection.
The major research works completed in this dissertation include:
1. A new static deflection model for CPUAs subjected to applied voltage is established based on the classical laminated plate theory, the constitutive equations for the piezoelectric layer, and the equilibrium equations for the axisymmetric plate. When modeling, the bonding layer is taken into account as one of single layers.
2. According to the established analytical model, the influences of the voltage and pressure applied to the CPUA and the structural parameters and material properties of the CPUA on the transverse deflection are numerically simulated. When the structure of the partially covered CPUA is determined, the different deflections can be available by changing the applied voltage and the pressure. The central deflection is directly proportional to the applied voltage and pressure and reaches a peak value when the radius ratio of the piezoelectric layer and the substrate layer is around 0.85-0.9. The smaller the thickness ratio of the bonding layer and the substrate layer is, the larger the central deflection is. When designing the CPUA, the large central deflection of the partially covered CPUA can be realized by reducing the thickness of the bonding layer on the promise that the cementation between the piezoelectric layer and the substrate layer can be ensured. There is a little influence of the elastic compliance constant of the bonding layer on the deflection of the partially covered CPUA and the influence of the elastic compliance constant of the bonding layer can be neglected when optimizing the performance of the partially covered CPUA.
3. The static and dynamic characteristics of the CPUAs with different radius ratio are experimentally tested by the established experimental setup The experimental results show that the predicted static deflections of the CPUA by the established deflection model in this paper agree well with the experimentally measured results, the established static deflection model considering the bonding layer is more accurate than the existed model neglecting the bonding layer, and the maximum relative error is reduced by 8.45%. The predicted deflection errors for the partially covered CPUA by the deflection model neglecting the bonding layer are obviously larger than those for the half covered CPUA. Because the hysteresis of the piezoelectric material is not considered when establishing the static deflection model, the error apparently exists when utilizing the static deflection model to predict the dynamic characteristic of the CPUA. If the deflection model for CPUAs is used to predict the dynamic deflections, the further research is needed.
4. According to the established analytical model for the CPUA, the structural parameters are optimized aiming at getting the maximum volume caused by deflection based on genetic algorithm. The results show that the volume caused by deflection reaches its maximum value when the radius of the substrate layer is constant and the radius radio of the piezoelectric layer and the substrate layer is 0.865 with the minimum value of the thickness of the piezoelectric layer, bonding layer, and substrate layer.
The research work in this paper lays the theoretic foundation for predicting and optimizing the behavior of CPUAs and CPUAs based systems.
本文利用圆形薄板小挠度弯曲理论建立了固支边部分覆盖圆形压电单晶执行器在作用电压控制下的挠度数学模型,在建模过程中将粘结层作为单独一层考虑。根据建立的数学模型,分析了电压、压强及圆形压电单晶执行器的几何参数和材料特性对其挠度的影响。建立了实验装置对圆形压电单晶执行器静态和动态挠度进行了测试。根据本文建立的模型以圆形压电单晶执行器变形体积最大为目标对圆形压电单晶执行器的几何结构进行了优化。
本文主要的研究工作和成果包括:
1.利用圆形薄板小挠度弯曲理论、压电本构方程和轴对称的平衡方程建立了固支边部分覆盖圆形压电单晶执行器在作用电压控制下的挠度数学模型,在建模过程中将粘结层作为单独一层考虑。
2.根据建立的数学模型,分析了作用于圆形压电单晶执行器的电压、压强及圆形压电单晶执行器的几何参数和材料特性对其挠度的影响。当圆形压电单晶执行器结构一定时,改变作用电压或压强可得到圆形压电单晶执行器不同的挠度,且圆形压电单晶执行器的中心挠度与作用电压和压强成正比;当压电层和基层的半径比在0.85-0.9左右时圆形压电单晶执行器中心挠度达到最大值,圆形压电单晶执行器变形最大;粘结层与基层厚度比越小,圆形压电单晶执行器挠度越大,实际设计圆形压电单晶执行器时,为了得到较大的挠度,可以在不影响压电层和基层的粘结的情况下减小粘结层的厚度;粘结层的柔性系数对圆形压电单晶执行器挠度影响很小,在优化设计圆形压电单晶执行器时可以不考虑粘结层的柔性系数。
3.建立了实验装置并对两种半径比的圆形压电单晶执行器的静态和动态挠度进行了测试。实验结果表明由建立的模型预测的圆形压电单晶执行器的静态挠度与实际测得值吻合,比忽略粘结层的模型更精确,相对最大误差最多可减小8.45%;忽略粘结层影响的情况下部分覆盖圆形压电单晶执行器挠度模拟误差明显大于半覆盖圆形压电单晶执行器挠度模拟误差;由于未考虑压电陶瓷的迟滞特性,本文建立的模型在预测圆形压电单晶执行器的动态特性时存在较大误差,需进一步研究。
4.根据本文建立的圆形压电单晶执行器的挠度模型,以圆形压电单晶执行器变形体积最大为目标对圆形压电单晶执行器的几何结构运用遗传算法进行了优化。优化结果表明,当基层半径一定时,压电层和基层的半径比为0.865,压电层、粘结层和基层的厚度为允许值的最小值时,圆形压电单晶执行器变形体积最大。
本文的研究工作对圆形压电单晶执行器以及基于圆形压电单晶执行器的系统进行性能预测和/或结构优化奠定了理论基础。
A circular piezoelectric unimorph actuator (CPUA), by bonding a circular polarized piezoelectric layer and a substrate layer to each other using the conducting epoxy as the bonding layer, is widely used for a variety of force and/or displacement generators because of the compact structure and relative large deflection. In order to predict and optimize the behavior of CPUAs and CPUAs based systems, the key problem lies in establishing the analytical model for CPUAs. The bonding layer of the CPUA is relatively very thin compared with the piezoelectric layer and the substrate layer, most models for the CPUAs took both the piezoelectric layer and the substrate layer into account while always neglected the bonding layer. Theoretically speaking, it is reasonable that the static deflection model for CPUAs considering the influence of the bonding layer can accurately model the deflections of CPUAs.
In this paper, based on the classical laminated plate theory, a new static deflection model for CPUAs subjected to applied voltage is established and the bonding layer is taken into account as an individual layer. According to the established analytical model, the influences of the voltage, pressure, structural parameters, and material properties of the CPUA on the transverse deflection are numerically simulated and the static and dynamic characteristics of the CPUA are experimentally tested. Based on the established analytical model, the structural parameters are optimized aiming at getting the maximum volume caused by deflection.
The major research works completed in this dissertation include:
1. A new static deflection model for CPUAs subjected to applied voltage is established based on the classical laminated plate theory, the constitutive equations for the piezoelectric layer, and the equilibrium equations for the axisymmetric plate. When modeling, the bonding layer is taken into account as one of single layers.
2. According to the established analytical model, the influences of the voltage and pressure applied to the CPUA and the structural parameters and material properties of the CPUA on the transverse deflection are numerically simulated. When the structure of the partially covered CPUA is determined, the different deflections can be available by changing the applied voltage and the pressure. The central deflection is directly proportional to the applied voltage and pressure and reaches a peak value when the radius ratio of the piezoelectric layer and the substrate layer is around 0.85-0.9. The smaller the thickness ratio of the bonding layer and the substrate layer is, the larger the central deflection is. When designing the CPUA, the large central deflection of the partially covered CPUA can be realized by reducing the thickness of the bonding layer on the promise that the cementation between the piezoelectric layer and the substrate layer can be ensured. There is a little influence of the elastic compliance constant of the bonding layer on the deflection of the partially covered CPUA and the influence of the elastic compliance constant of the bonding layer can be neglected when optimizing the performance of the partially covered CPUA.
3. The static and dynamic characteristics of the CPUAs with different radius ratio are experimentally tested by the established experimental setup The experimental results show that the predicted static deflections of the CPUA by the established deflection model in this paper agree well with the experimentally measured results, the established static deflection model considering the bonding layer is more accurate than the existed model neglecting the bonding layer, and the maximum relative error is reduced by 8.45%. The predicted deflection errors for the partially covered CPUA by the deflection model neglecting the bonding layer are obviously larger than those for the half covered CPUA. Because the hysteresis of the piezoelectric material is not considered when establishing the static deflection model, the error apparently exists when utilizing the static deflection model to predict the dynamic characteristic of the CPUA. If the deflection model for CPUAs is used to predict the dynamic deflections, the further research is needed.
4. According to the established analytical model for the CPUA, the structural parameters are optimized aiming at getting the maximum volume caused by deflection based on genetic algorithm. The results show that the volume caused by deflection reaches its maximum value when the radius of the substrate layer is constant and the radius radio of the piezoelectric layer and the substrate layer is 0.865 with the minimum value of the thickness of the piezoelectric layer, bonding layer, and substrate layer.
The research work in this paper lays the theoretic foundation for predicting and optimizing the behavior of CPUAs and CPUAs based systems.
引文
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[64] Mo C, Wright R, Slaughter S W and Clerk W W. Behaviour of a unimorph piezoelectric actuator [J]. Smart Materials and Structures, 2006, 15(4): 1094-1102.
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[67]刘品宽,孙立宁.双压电复合薄圆板驱动器的理论分析[J]压电与声光, 2002 24(2): 111-115.
[68]黎毅力,张建辉,夏齐霄.泵用压电振子动态特性的研究[J].中国机械, 2007, 18(17): 2088-2093.
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[71] Andrzej B D and Piotr P. Theory of piezoelectric axisymmetric bimorph [J]. Sensor and Actuator A, 1997, 58: 203-212.
[72] Morris C J and Forster F K. Optimization of a circular piezoelectric bimorph for a micropump driver [J]. Journal of Micromechics and Microengineering, 2000, 10(3): 459-465.
[73] Muralt P, Baborowski J, Ledermann N. Piezoelectric MEMS with PZT thin films: integration and application [J]. Issues Piezoelectr MEMS Technol, 2002, 3(12): 231–260.
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[78] Hiroshi S and Shigemi S. An improved equivalent circuit of piezoelectric transducers including the effect of dielectric loss [J]. Journal of the Acoustical Society of Japan (E), 1985, 6(3): 225–233.
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[80] Robert B Meyer. Piezoelectric Effects in Liquid Crystals [J]. Physical Review Letters, 1969, 22(18): 918-921.
[81] Park S H, Chuang S L. Piezoelectric effects on electrical and optical properties of wurtzite GaN/AlGaN quantum well lasers [J]. Applied Physics Letters, 1998, 2(24): 3103-3109.
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[86] Liu C, Cui T and Zhou Z. Modal analysis of a unimorph piezoelectrical transducer [J]. Microsystem Technologies, 2003, 9(6-7): 474-479.
[87] Wang G Q, Sankar B V and Louis N C. Analysis of a composite piezoelectric circular plate with initial stresses for MEMS [A]. Proceedings of 2002 ASME International Mechanical Engineering Congress and Exposition, 2002, 11: 17-22.
[88] Zhang T and Wang Q M. Valveless piezoelectric micropump for fuel delivery in direct methanol fuel cell (DMFC) devices [J]. Journal of Power Sources, 2005, 140(1): 72-80.
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[92] [EB/OL] http://www.annon.com.cn/medical-transducer.htm.
[93] [EB/OL] http://www.yt178.com/Html/chaoshengbochanpinzhanshi/product/904366.html.
[94] [EB/OL] http://www.rishengele.com/cailiao.htm.
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