Energy Method For Mode Identification In The Analysis Of High Frequency Vibration Of Quartz Crystal Plate

Vibration analysis of a quartz crystal plate is widely known as the core requirement in the quartz crystal resonator design.The vibration of a real quartz crystal plate is very complicated due to coupling of many vibration modes.The vibration of the quartz crystal plate can be treated as couplings of simple harmonic vibration modes and the dominant mode is usually the strongest at the specific frequency.For the calculation accuracy of high frequency vibration,it is necessary to consider enough vibration modes to ensure the accuracy of the calculations.It is still very difficult to solve the vibration of quartz crystal plate with the three-dimensional equations even under the ideal conditions,due to the coupled modes and the complicated boundary conditions.The Mindlin plate theory provides an effective method for the vibration analysis of quartz crystal plates.The core advantage of this theory is that the three-dimensional displacement is converted into the two-dimensional displacement series and the equations are simplified.However,at the same time,the Mindlin plate theory has to use the displacement truncation and correction factors to ensure the accuracy.In recent years,with the development of computer technology,the finite element method based on Mindlin high-order plate theory has made great progress,and the calculation accuracy has been effectively improved.As so many modes participate in the calculation,vibration analysis and mode identification become very difficult.For instance,when calculating with the finite element method based on Mindlin third-order plate theory,there are 16 modes to be considered in each node,and the eigenfrequency results usually very large in the foused intense.Nowadays,the process of mode identification for quartz crystal plates is still complicated.The mode shape of each frequency is plotted one by one for identification.The plottings of mode shapes intuitively represent the characteristics of each vibration mode of the quartz crystal plate at a specific frequency.When the number of modes involved in the calculation is relatively large,the mode shapes will be very complicated,and the identification of each mode will become difficult.On the other hand,since the mode shape identification method relies mainly on the graphic recognition,the limitation of this method is on the manual intervention.Identifying the modes in finite element calculations with mode shapes requires huge manual intervention by people and comsums a lot of time.Generally,it is important to find an efficient way to automatically identify the modes.Through oberving the characteristics of mode shapes and distributions,it is not hard to found that,as the dominant mode,the working mode vibrates as the strongest among all the modes.So we reasonably suppose that the mode vibration energy should also be dominant one,and the energy method for mode identification is thus proposed.Begin with the Mindlin first-order plate equations,the strain and kinetic energy of the plate are calculated and classified with each mode type to obtain the energy distributions.By calculating and comparing the distributions of the mode energy,the dominant mode can be correctly located,that is,each vibration mode can be identified througth the energy concerntrations.The energy predictions are also proved to be consistent with the mode shape plots.After the energy method for mode identification has been successfully used in the vibration of the quartz crystal plate,we introduce it is used in the vibration analysis with the thermal field.We obtain the novel energy-temperature relations wich reflect the frequency-temperature relations well and have a clear description at the conversions of vibration modes with the temperature variations.The energy method for mode identification can be applied to the finite element method based on Mindlin high-order plate theory,and the program is able to calculate and output the dominant modes information automatically.Our research proves that the energy method for mode identification can take full advantages of numerical calculation,and help the finite element program efficiently completing the classification and identification of vibration modes during the analytical process.The energy method for mode identification provides a very effective tool to the vibration analysis of quartz crystal plates,and the identificaiton became more intuitive and simple.Meanwhile the energy identification method has a novel description at the mode changes by the influence of the environment.This mode identification method can be extended to the vibration analysis of structure with coupled modes,providing numerical analysis solutions and improving current calculation software tools.