各向异性压电板非线性高频振动方程及其求解研究
摘要
石英晶体板的厚度剪切振动作为压电谐振器的主要工作模态在频率控制和传感器等领域都有着极为广泛的应用。传统的高阶板理论例如Mindlin板理论和Lee板理论都可以有效地分析线性压电板的高频振动,其结果对压电谐振器设计有着重要意义。随着谐振器工作频率和精度要求的不断提高以及谐振器的微型化,各种非线性现象日益突出并最终会对谐振器的频率稳定性带来显著影响。本文建立了考虑几何和材料非线性的石英晶体板高频振动的二维方程,接着利用解析法对获得的非线性方程进行了变换和求解,最后分析了各种非线性效应对石英晶体谐振器工作频率以及相关电路参数的影响。
作为石英晶体板非线性高频振动分析的基础,本文利用自然修正的线性Mindlin三阶板方程,获得了基频和三阶泛音厚度剪切振动的精确截止频率。为了研究厚度剪切模态和寄生模态的耦合情况,我们计算了基频和三阶泛音厚度剪切振动的频谱关系。计算结果表明在一定长厚比情况下厚度剪切模态和弯曲模态存在强烈耦合,而频谱关系可以用于确定石英晶片的最佳尺寸并以此来减弱和避免这种强烈耦合。
我们建立了无限大各向同性板单一厚度剪切振动的非线性方程,并利用伽辽金法将该非线性方程转化为关于时间变量的非线性常微分方程。接着分别用摄动法和同伦分析法对这个方程进行了解析求解,获得了非线性厚度剪切振动的频幅关系。我们发现非线性振动的频率不仅依赖于板的厚度而且与振幅存在密切关系。数值计算结果表明相比于摄动解,同伦分析解能更快地收敛到精确解,同时该方法提供了一个辅助参数来调节收敛区域和收敛速度。
本文建立了考虑几何和材料非线性的石英晶体板高频振动的非线性二维方程,特别包括了强烈耦合的厚度剪切模态和弯曲模态。由于所建立的非线性二维方程十分复杂,直接求解非常困难。基于长厚度剪切波假设和厚度剪切近似理论,本文利用伽辽金法将厚度剪切振动的非线性控制方程转化为关于时间变量的常微分方程。通过摄动法和同伦分析法的解析求解,获得了石英晶体板非线性厚度剪切振动的频幅关系。计算结果表明由几何和材料非线性引起的频率漂移并不明显,这促使我们考察电场效应的影响。
我们建立了考虑强电场作用下石英晶体板高频振动的非线性二维方程。同样基于长厚度剪切波假设和厚度剪切近似理论,本文用伽辽金法将控制厚度剪切振动的非线性方程转化为关于时间变量的常微分方程。接着用逐次近似法获得了非线性厚度剪切振动的频率响应关系,并绘制了不同振幅比和不同电压下的频率响应曲线图。数值计算结果表明由电场引起的频率漂移十分明显,需要引起足够的重视。然后计算了AT切石英晶体谐振器的部分电路参数,讨论了非线性效应对电路参数的影响。最后利用伽辽金法将强烈耦合的厚度剪切模态和弯曲模态的控制方程组都转化为关于时间变量的非线性常微分方程组,并用逐次近似法对该非线性方程组进行了解析求解,获得了两种模态强烈耦合情况下的频率响应曲线。
考虑几何和材料非线性的石英晶体板的非线性高频振动方程的建立和求解是石英晶体谐振器非线性分析的基础,也可以用于有限元分析和即将开展的偏场效应的解析求解。本文得到的结果为石英晶体谐振器的精确设计和分析提供了有效指导。
Thickness-shear vibration of a quartz crystal plate is one of the most widely used functioning modes of piezoelectric resonators with broad applications for frequency control and detection. The classical plate theories, including Mindlin and Lee plate theories, are linear systems and have been successfully used for linear analysis of high frequency vibrations of piezoelectric plates and yielded important results for the design of piezoelectric resonators. With the growing demands for higher and more precise frequency and miniaturization of piezoelectric structures, the nonlinear phenomena have emerged and eventually caused frequency instability of resonators. In order to study the frequency shifts due to the nonlinear effects, we have established nonlinear two-dimensional equations of high frequency vibrations of quartz crystal plate with the consideration of kinematic and material nonlinearities. We further employed analytical methods to transform and solve these complicated nonlinear equations and investigated the different effects of nonlinearities of quartz crystal resonators on its frequency and corresponding circuit parameters.
We have utilized the linear third-order Mindlin plate equations with natural correction and obtained exact cut-off frequencies of the fundamental and third-order thickness-shear modes as the foundation of nonlinear analysis. In order to study the couplings and interactions between thickness-shear and spurious modes, we have calculated frequency spectra in the vicinity of fundamental thickness-shear and third-order overtone modes. From frequency spectra, we have found that the coupling between thickness-shear and flexural modes depends strongly on the aspect ratios (length/thickness) and frequency spectra is the most basic tool to obtain optimal aspect ratios for decreasing and avoiding this coupling.
The nonlinear equation of simple thickness-shear vibrations of an infinite and isotropic plate has been established and transformed into a nonlinear ordinary differential equation by Galerkin approximation. Then we solved this nonlinear equation by the perturbation method and homotopy analysis method, respectively. The amplitude-frequency relation we obtained showed the nonlinear frequency not only depends on amplitude but also related to the thickness of plate. Compared with results from perturbation method, our analytical solutions converge faster while homotopy analysis method provides an auxiliary parameter to control convergence region and speed.
The nonlinear two-dimensional equations of high frequency vibrations of a quartz crystal plate for strongly coupled thickness-shear and flexural modes with the consideration of kinematic and material nonlinearities have been established. These equations are too complicated to solve directly. Based on the assumption of long thickness-shear waves and thickness-shear approximation, the nonlinear equation of thickness-shear modes has been transformed into an ordinary differential equation depends on time by Galerkin approximation. The amplitude-frequency relations of thickness-shear vibration have been obtained by perturbation and homotopy analysis methods. Numerical results showed neither kinematic nor material nonlinearities are the main factors of frequency shifts which suggest us to examine the effect of electrical field.
The two-dimensional equations of high frequency vibration of a quartz crystal plate under a strong electric field have been established. Also based on the assumption of long thickness-shear waves and thickness-shear approximation, we utilized Galerkin approximation to transform the nonlinear partial differential equation of thickness-shear vibrations into an ordinary differential equation depending only on time. By successive approximation method, we obtained electrical current frequency-response relation and plotted nonlinear frequency-response curves for different amplitude ratios and driving voltages. Numerical results showed that frequency shift caused by the electrical field is significant and should be treated as the primary cause of frequency instability. We further computed some circuit parameters of AT-cut quartz crystal plates as simple resonators, and investigated the effect of nonlinearities on circuit parameters. The strongly coupled equations of vibrations of the thickness-shear and flexural modes have all been transformed into a system of nonlinear ordinary equations. By successive approximation method, we obtained frequency-response relations with the consideration of strong coupling between vibrations modes.
The two-dimensional equations of high frequency vibrations of a quartz crystal plate with the consideration of material and kinematic nonlinearities we established and solved lay a firm theoretical foundation for nonlinear analysis of quartz crystal resonators, which also can be used for the finite element analysis and further analytical studies of nonlinear effects of bias fields on quartz crystal resonators. The results we obtained provide guidance for the precise analysis and design of quartz crystal resonators.
作为石英晶体板非线性高频振动分析的基础,本文利用自然修正的线性Mindlin三阶板方程,获得了基频和三阶泛音厚度剪切振动的精确截止频率。为了研究厚度剪切模态和寄生模态的耦合情况,我们计算了基频和三阶泛音厚度剪切振动的频谱关系。计算结果表明在一定长厚比情况下厚度剪切模态和弯曲模态存在强烈耦合,而频谱关系可以用于确定石英晶片的最佳尺寸并以此来减弱和避免这种强烈耦合。
我们建立了无限大各向同性板单一厚度剪切振动的非线性方程,并利用伽辽金法将该非线性方程转化为关于时间变量的非线性常微分方程。接着分别用摄动法和同伦分析法对这个方程进行了解析求解,获得了非线性厚度剪切振动的频幅关系。我们发现非线性振动的频率不仅依赖于板的厚度而且与振幅存在密切关系。数值计算结果表明相比于摄动解,同伦分析解能更快地收敛到精确解,同时该方法提供了一个辅助参数来调节收敛区域和收敛速度。
本文建立了考虑几何和材料非线性的石英晶体板高频振动的非线性二维方程,特别包括了强烈耦合的厚度剪切模态和弯曲模态。由于所建立的非线性二维方程十分复杂,直接求解非常困难。基于长厚度剪切波假设和厚度剪切近似理论,本文利用伽辽金法将厚度剪切振动的非线性控制方程转化为关于时间变量的常微分方程。通过摄动法和同伦分析法的解析求解,获得了石英晶体板非线性厚度剪切振动的频幅关系。计算结果表明由几何和材料非线性引起的频率漂移并不明显,这促使我们考察电场效应的影响。
我们建立了考虑强电场作用下石英晶体板高频振动的非线性二维方程。同样基于长厚度剪切波假设和厚度剪切近似理论,本文用伽辽金法将控制厚度剪切振动的非线性方程转化为关于时间变量的常微分方程。接着用逐次近似法获得了非线性厚度剪切振动的频率响应关系,并绘制了不同振幅比和不同电压下的频率响应曲线图。数值计算结果表明由电场引起的频率漂移十分明显,需要引起足够的重视。然后计算了AT切石英晶体谐振器的部分电路参数,讨论了非线性效应对电路参数的影响。最后利用伽辽金法将强烈耦合的厚度剪切模态和弯曲模态的控制方程组都转化为关于时间变量的非线性常微分方程组,并用逐次近似法对该非线性方程组进行了解析求解,获得了两种模态强烈耦合情况下的频率响应曲线。
考虑几何和材料非线性的石英晶体板的非线性高频振动方程的建立和求解是石英晶体谐振器非线性分析的基础,也可以用于有限元分析和即将开展的偏场效应的解析求解。本文得到的结果为石英晶体谐振器的精确设计和分析提供了有效指导。
Thickness-shear vibration of a quartz crystal plate is one of the most widely used functioning modes of piezoelectric resonators with broad applications for frequency control and detection. The classical plate theories, including Mindlin and Lee plate theories, are linear systems and have been successfully used for linear analysis of high frequency vibrations of piezoelectric plates and yielded important results for the design of piezoelectric resonators. With the growing demands for higher and more precise frequency and miniaturization of piezoelectric structures, the nonlinear phenomena have emerged and eventually caused frequency instability of resonators. In order to study the frequency shifts due to the nonlinear effects, we have established nonlinear two-dimensional equations of high frequency vibrations of quartz crystal plate with the consideration of kinematic and material nonlinearities. We further employed analytical methods to transform and solve these complicated nonlinear equations and investigated the different effects of nonlinearities of quartz crystal resonators on its frequency and corresponding circuit parameters.
We have utilized the linear third-order Mindlin plate equations with natural correction and obtained exact cut-off frequencies of the fundamental and third-order thickness-shear modes as the foundation of nonlinear analysis. In order to study the couplings and interactions between thickness-shear and spurious modes, we have calculated frequency spectra in the vicinity of fundamental thickness-shear and third-order overtone modes. From frequency spectra, we have found that the coupling between thickness-shear and flexural modes depends strongly on the aspect ratios (length/thickness) and frequency spectra is the most basic tool to obtain optimal aspect ratios for decreasing and avoiding this coupling.
The nonlinear equation of simple thickness-shear vibrations of an infinite and isotropic plate has been established and transformed into a nonlinear ordinary differential equation by Galerkin approximation. Then we solved this nonlinear equation by the perturbation method and homotopy analysis method, respectively. The amplitude-frequency relation we obtained showed the nonlinear frequency not only depends on amplitude but also related to the thickness of plate. Compared with results from perturbation method, our analytical solutions converge faster while homotopy analysis method provides an auxiliary parameter to control convergence region and speed.
The nonlinear two-dimensional equations of high frequency vibrations of a quartz crystal plate for strongly coupled thickness-shear and flexural modes with the consideration of kinematic and material nonlinearities have been established. These equations are too complicated to solve directly. Based on the assumption of long thickness-shear waves and thickness-shear approximation, the nonlinear equation of thickness-shear modes has been transformed into an ordinary differential equation depends on time by Galerkin approximation. The amplitude-frequency relations of thickness-shear vibration have been obtained by perturbation and homotopy analysis methods. Numerical results showed neither kinematic nor material nonlinearities are the main factors of frequency shifts which suggest us to examine the effect of electrical field.
The two-dimensional equations of high frequency vibration of a quartz crystal plate under a strong electric field have been established. Also based on the assumption of long thickness-shear waves and thickness-shear approximation, we utilized Galerkin approximation to transform the nonlinear partial differential equation of thickness-shear vibrations into an ordinary differential equation depending only on time. By successive approximation method, we obtained electrical current frequency-response relation and plotted nonlinear frequency-response curves for different amplitude ratios and driving voltages. Numerical results showed that frequency shift caused by the electrical field is significant and should be treated as the primary cause of frequency instability. We further computed some circuit parameters of AT-cut quartz crystal plates as simple resonators, and investigated the effect of nonlinearities on circuit parameters. The strongly coupled equations of vibrations of the thickness-shear and flexural modes have all been transformed into a system of nonlinear ordinary equations. By successive approximation method, we obtained frequency-response relations with the consideration of strong coupling between vibrations modes.
The two-dimensional equations of high frequency vibrations of a quartz crystal plate with the consideration of material and kinematic nonlinearities we established and solved lay a firm theoretical foundation for nonlinear analysis of quartz crystal resonators, which also can be used for the finite element analysis and further analytical studies of nonlinear effects of bias fields on quartz crystal resonators. The results we obtained provide guidance for the precise analysis and design of quartz crystal resonators.
引文
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