Dynamic modeling and performance analysis method for quartz double-end tuning fork resonator

Abstract

The invention discloses a dynamic modeling and performance analysis method for a quartz double-end tuning fork resonator, and the method comprises the steps: constructing a high-order displacement field which meets the condition that the free boundary stress is zero in prior to describe the deformation of a mass block, and employing a coupling scheme based on a virtual work principle; the tuning fork arms and the mass block are integrated into an energy conservation dynamic self-consistent system. According to the method, the linear characteristic frequency and modal of the system are solved, then geometric nonlinearity is introduced, and the amplitude-dependent nonlinear frequency response is accurately calculated through an iterative algorithm. According to the dynamic modeling and performance analysis method for the quartz double-end tuning fork resonator, the fundamental frequency prediction error is reduced to 0.34% from 1.6% of a traditional model, and the relative precision is improved by 79%; meanwhile, the non-physical residual stress at the free boundary is restrained to the magnitude of the maximum stress in the domain, the boundary stress distortion is fundamentally eliminated, and a reliable theoretical tool is provided for accurate design and optimization of the high-performance MEMS resonator.

Description

Technical Field

[0001] This invention belongs to the field of microelectromechanical systems (MEMS) modeling and design, specifically relating to a boundary self-consistent dynamics modeling and performance analysis method for a quartz double-ended tuning fork resonator, which is particularly suitable for the design, frequency prediction and optimization of high-performance resonators. Background Technology

[0002] Tuning fork resonators are a common type of low-frequency MEMS resonator, widely used in time and frequency references, accelerometers, gyroscopes, pressure sensors, atomic force microscopes, and other fields. Accurate prediction of their core performance parameter, the resonant frequency, is crucial for device sensitivity, signal-to-noise ratio, and reliability.

[0003] Traditional tuning fork resonator analysis treats the tuning fork arm as an equivalent Euler-Bernoulli beam and the mass block as a concentrated mass. Therefore, the tuning fork can be viewed as a coupled splice of multiple Euler beams. This simplified model offers concise analysis and clearly reveals its force-frequency characteristics. However, this model assumes the cross-section remains planar after deformation, neglects shear and warping, and cannot describe the non-uniform strain caused by the beam, resulting in errors compared to the real structure. The following problems exist: 1. Boundary stress distortion: The displacement field constructed by existing models (such as the Euler beam concentrated mass model and the quadratic warping model) cannot meet the physical condition that the free boundary stress is zero, resulting in non-physical residual stress at the boundary, which leads to model prediction distortion. 2. Inadequate coupling conditions: Traditional models use simple force-moment equilibrium conditions to handle the coupling between beams and mass blocks. Under the geometric mismatch of "line-surface" connections, energy conservation cannot be guaranteed, introducing systematic errors. 3. Neglecting nonlinear effects: The quasi-linear macroscopic behavior of tuning fork resonators has led most existing studies to neglect the geometric nonlinearity of tuning fork systems; 4. The nonlinear mechanism is unclear: Existing research has failed to reveal the nonlinear competition mechanism (hardening effect and softening effect) that exists inside the system, and cannot explain and predict the complex dynamic behavior under high amplitude.

[0004] Therefore, there is an urgent need for a more accurate, energy-conserving coupled model and solution method. Summary of the Invention

[0005] This invention provides a boundary-consistent and energy-coordinated modeling and analysis method for quartz double-ended tuning fork resonators to solve the aforementioned technical problem of inaccuracy in traditional models. Specifically, the technical solution is as follows: A method for dynamic modeling and performance analysis of a quartz double-ended tuning fork resonator includes the following steps: Step 1: Establish the displacement field of the Euler-Bernoulli beam model of the tuning fork arm and the self-consistent displacement field of the mass block, wherein the displacement field of the mass block is constructed by a preset warping function and simultaneously satisfies the condition that its free boundary stress is zero, as well as the displacement continuity condition with the tuning fork arm at the connection interface. Step 2: Based on the principle of virtual work, derive the dynamic coupling conditions between the tuning fork arm and the mass block at the connection interface; Step 3: Based on the displacement field from Step 1 and the coupling conditions from Step 2, assemble the global system and solve its linear eigenvalue problem to obtain the system's linear resonant frequency and modes; Step 4: Based on the linear modes obtained in Step 3, introduce geometric nonlinear strain and establish the nonlinear control equations of the system; Step 5: Solve the nonlinear control equations using an iterative algorithm to calculate the nonlinear frequency response of the resonator under finite amplitude.

[0006] Furthermore, the displacement field function of the mass block mentioned in step one is defined as: ; in, and Let be the warping function. and These are the unknown amplitude functions of the mass block along the x and y directions, respectively.

[0007] Furthermore, the warping function and Based on the aforementioned free boundary stress condition, it is determined that at the boundary... The following conditions must be met: ; Furthermore, the warping function and Let it be a higher-order polynomial function: ; ; Furthermore, in step two, the derivation of coupling conditions based on the principle of virtual work specifically involves: at the connection interface, setting the virtual work of the tuning fork arm equal to the virtual work of the mass block, thereby deriving the coupling boundary conditions.

[0008] Furthermore, in step five, the iterative method includes the following sub-steps: Step 5.1: Solve the linear eigenvalue problem to obtain the linear frequencies and modes, and use these as initial values; Step 5.2: Based on the linear mode shape, calculate the equivalent axial tension and boundary coupling terms caused by geometric nonlinearity; Step 5.3: Substitute the axial tension and boundary coupling terms as prestress and correction terms into the governing equations, and resolve the eigenvalue problem to obtain the updated frequencies and modes; Step 5.4: Repeat steps 5.2 and 5.3 until the change in frequency and equivalent axial tension is less than the preset tolerance, thereby obtaining a convergent nonlinear resonant frequency.

[0009] Furthermore, in step three, the global system is solved by combining the determinant scanning method and the Brent method to obtain the linear resonant frequency and modes of the system.

[0010] A medium storing a computer program, which, when executed by a processor, implements the aforementioned method for dynamic modeling and performance analysis of a quartz double-ended tuning fork resonator.

[0011] A MEMS resonator design system includes a memory and a processor. The memory stores a computer program, and when the processor executes the program, it implements the aforementioned method for dynamic modeling and performance analysis of a quartz double-ended tuning fork resonator. The system is used to predict and optimize the frequency characteristics of the quartz tuning fork resonator.

[0012] The dynamic modeling and performance analysis method for quartz double-ended tuning fork resonators provided by this invention reduces the fundamental frequency prediction error from 1.6% in the traditional classical model to within 0.34%, improving the relative accuracy by 79%. It also suppresses the boundary residual stress to the order of 10⁻⁸ of the maximum stress in the domain, strictly satisfying the free boundary stress condition, and providing a reliable benchmark for reliability assessment based on the stress field. Attached Figure Description

[0013] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0014] Figure 1 This is a schematic diagram of the quartz double-ended tuning fork resonator structure and its theoretical model provided in an embodiment of the present invention; Figure 2 This is a comparison diagram of shear stress distribution at the free boundary provided in the embodiments of the present invention; Figure 3 This is a schematic diagram of the linear solution process provided in an embodiment of the present invention; Figure 4 This is a schematic diagram of the convergence process of the nonlinear iterative algorithm provided in an embodiment of the present invention; Figure 5The nonlinear amplitude-frequency response curve provided in the embodiment of the present invention. Detailed Implementation

[0015] The embodiments of this application are described in detail below. Examples of the embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain this application, and should not be construed as limiting this application.

[0016] The technical solution adopted in this application is as follows: An Euler beam model is used to simulate the tuning fork arm, and a mass block model is used to simulate the base, simplifying calculations by utilizing system symmetry; a mass block displacement field is constructed, introducing rotation angle and Poisson deformation function, strictly satisfying the condition that the free boundary stress is zero, and ensuring continuous displacement at the beam-block interface; coupling conditions are derived based on the principle of virtual work, avoiding errors in traditional force-torque balance, and ensuring energy conservation and displacement coordination; geometric nonlinear effects are introduced to establish a Duffing-type frequency-amplitude relationship; a systematic solution process is provided, using eigenvalue problems to find linear frequencies and iterative methods to calculate nonlinear frequency deviations. This method reduces theoretical prediction errors to within 0.5%, reduces repeated fabrication and laser trimming, shortens the R&D cycle, and provides a reliable basis for predicting device fatigue life and thermal drift, applicable to the design and optimization of high-performance MEMS resonators. Specifically, the dynamic modeling and performance analysis method for the quartz double-ended tuning fork resonator in this application includes the following steps: Step 1: Establish the displacement field of the Euler-Bernoulli beam model of the tuning fork arm and the self-consistent displacement field of the mass block. The displacement field of the mass block is constructed by a preset warping function and simultaneously satisfies the condition that its free boundary stress is zero and the displacement continuity condition at the connection interface with the tuning fork arm.

[0017] This application takes a Z-cut, 2° double-ended quartz tuning fork resonator as an example, and its specific structural parameters are shown in the table below: Table 1 Structural Parameters of the Embodiment ; The elastic constant matrix C' (unit: GPa) of the quartz crystal used is: ; Quartz crystal density is .

[0018] Among them, such as Figure 1 As shown, the tuning fork arm adopts an Euler-Bernoulli beam model, and its displacement field function is: ; The mass block adopts a higher-order warping displacement field model, and its displacement field function is: ; in, and For warping function, and These are the unknown amplitude functions of the mass block along the x and y directions, respectively, and are functions of position x and time t. They need to be determined by solving the system control equations and boundary conditions.

[0019] Warp function and Determined based on the aforementioned free boundary stress condition. Based on the condition that the free boundary stress is zero. Its free boundary The following conditions must be met: ; Furthermore, the warping function and Let it be a higher-order polynomial function: ; .

[0020] like Figure 2 The comparison of free boundary stress distributions shown in the diagram demonstrates that the model in this application suppresses the boundary residual stress to 10% of the maximum stress within the domain. -8 The magnitude is such that it strictly satisfies the free boundary stress condition, providing a reliable benchmark for reliability assessment based on the stress field.

[0021] Based on the expressions for the kinetic and strain energy of the mass block, calculate the integration constants I1 to I6. Calculate the following terms using numerical integration (such as Gaussian integration): ; ; Based on Hamilton's principle, the dynamic equations and kinematic equations of the beam and the mass block are derived respectively, ensuring energy conservation and physical consistency.

[0022] Euler beam: ; ; in, The modulus of the tuning fork arm material is Young's modulus. For material density, Let be the moment of inertia of the tuning fork arm cross section about the neutral axis. Let be the cross-sectional area of ​​the tuning fork arm.

[0023] Euler beam governing equations: ; Mass block: ; ; Mass block governing equations: ; ; The governing equations for the mass block are a set of coupled second-order partial differential equations.

[0024] Simplify calculations by utilizing structural symmetry: The planar out-of-phase vibration mode of a tuning fork resonator has the widest application, and this mode exhibits high symmetry. Therefore, by placing the origin of the coordinate system of an Euler beam at the midpoint of the beam's length, the mode shape function can be simplified to an even function form using the symmetry.

[0025] ; in, The simplified model is as follows: Figure 1 As shown.

[0026] Step 2: Based on the principle of virtual work, derive the dynamic coupling conditions between the tuning fork arm and the mass block at the connection interface.

[0027] The displacement at the interface between the beam and the mass block is continuous: ; The corner at the interface between the beam and the mass block is continuous: ; The virtual work done at the interface between the beam and the mass block is the same: ; in, ; ; The consistency of virtual work avoids the errors introduced by the traditional force-torque balance condition at the interface between curved and flat surfaces.

[0028] Step 3: Based on the displacement field from Step 1 and the coupling conditions from Step 2, assemble the global system and solve its linear eigenvalue problem to obtain the system's linear resonant frequency and modes.

[0029] As mentioned earlier, by utilizing structural symmetry, the mode shape function of the tuning fork arm can be simplified to an even function form: ; in, .

[0030] Harmonic solution , Substituting into the governing equations of the mass block, we obtain the following... and The system of ordinary differential equations is transformed into a first-order state-space form. .

[0031] Combined with the boundary conditions of the fixed end of the mass block Given the conditions for continuity of displacement, rotation, and virtual work at the beam-mass block interface, construct a vector of unknown coefficients. homogeneous linear equation system .

[0032] The determinant scanning method and Brent's method are used together to find the determinant of the coefficient matrix. frequency value This value is the linear resonant frequency of the system. The solution process is as follows: Figure 3 As shown, Figure 3 (a) shows the coarse scan result. Figure 3 (b) shows the results of fine scanning.

[0033] The fundamental frequency results obtained are shown in Table 2 below. Compared with experimental values ​​and other models, the prediction error of the present invention is only 0.34%, which is significantly better than the traditional Euler beam model (+5.4%) and the quadratic warping model (-1.6%), verifying the high accuracy of the present invention in the linear domain.

[0034] Table 2 Comparison of fundamental frequency prediction results and errors of different models ; Step 4: Based on the linear modes obtained in Step 3, introduce geometric nonlinear strain and establish the nonlinear control equations of the system.

[0035] Introducing geometric nonlinearity, a Duffing-type frequency-amplitude relationship is established: .

[0036] Step 5: Solve the nonlinear control equations using an iterative algorithm and calculate the nonlinear frequency response of the resonator under finite amplitude.

[0037] Step five, the iterative method includes the following sub-steps: Step 5.1: Solve the linear eigenvalue problem to obtain the linear frequencies and modes, and use these as initial values.

[0038] Step 5.2: Based on the linear mode shape, calculate the equivalent axial tension and boundary coupling terms caused by geometric nonlinearity.

[0039] Step 5.3: Substitute the axial tension and boundary coupling terms as prestress and correction terms into the governing equations, and resolve the eigenvalue problem to obtain the updated frequencies and modes.

[0040] Step 5.4: Repeat steps 5.2 and 5.3 until the change in frequency and equivalent axial tension is less than the preset tolerance, thereby obtaining a convergent nonlinear resonant frequency.

[0041] Specifically, considering geometric nonlinearity, the complete Green-Lagrange strain tensor This explains finite deformation, and for an Euler-Bernoulli beam with predominantly lateral motion, the main nonlinear terms appear in the axial strain components: .

[0042] 2. Based on the linear mode shape obtained in step 3 Calculate the equivalent axial tension caused by the expansion and contraction of the neutral surface. : .

[0043] 3. Substitute the axial tension N as a prestress term into the nonlinear governing equation of the beam: .

[0044] 4. Resolve the eigenvalue problem to obtain the updated frequencies. and mode .

[0045] 5. Recalculate the axial tension based on the new mode shape. .

[0046] 6. Repeat steps 3-5 until the relative change in frequency calculated between two adjacent iterations is less than the preset tolerance (e.g., 1e-6). The iterative convergence process is as follows: Figure 4 As shown, the convergent nonlinear resonant frequency is finally obtained. This algorithm can achieve 10^10 resonant frequencies within 10 iterations. -6 The convergence tolerance ensures the efficiency and reliability of solving nonlinear problems.

[0047] The linear and nonlinear prediction results of the method of this invention are compared with the experimental values, as shown in Table 3 below. It can be seen that the nonlinear correction reduces the absolute deviation of the theoretical prediction from 121 Hz to 119 Hz. Although the correction magnitude is small, the direction is consistent, reflecting the coherence of the model from micro-vibration to finite amplitude dynamic description.

[0048] Table 3. Comparison of linear and nonlinear models in this invention .

[0049] at the same time, Figure 5The nonlinear amplitude-frequency response curves calculated using the iterative algorithm of this invention are shown, revealing that the system exhibits significant softening behavior. Theoretical analysis indicates that this behavior is a macroscopic manifestation of the competition between the hardening effect dominated by the tension of the neutral surface of the tuning fork arm and the softening effect dominated by the coupling modulation of the mass block boundary, revealing the intrinsic physical nature of the macroscopic quasi-linear behavior of the quartz DETF.

[0050] This application establishes a boundary-consistent dynamic model of a quartz double-ended tuning fork resonator. Through... Figure 1 Define the model, through Figure 2 Its boundary physical authenticity was verified by... Figure 3 Table 2 demonstrates the reliability and high accuracy of its linear solution process, and through... Figure 4 and Figure 5 The robustness of its nonlinear iterative algorithm and its ability to characterize nonlinear competition mechanisms are demonstrated. Table 3 further verifies the model's micro-correction effect in the nonlinear domain. This model provides a reliable theoretical tool for the design and optimization of high-performance MEMS resonators.

[0051] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the above embodiments do not limit the present invention in any way, and all technical solutions obtained by equivalent substitution or equivalent transformation fall within the protection scope of the present invention.

Claims

1. A method for dynamic modeling and performance analysis of a quartz double-ended tuning fork resonator, characterized in that, Includes the following steps: Step 1: Establish the displacement field of the Euler-Bernoulli beam model of the tuning fork arm and the self-consistent displacement field of the mass block, wherein the displacement field of the mass block is constructed by a preset warping function and simultaneously satisfies the condition that its free boundary stress is zero, as well as the displacement continuity condition with the tuning fork arm at the connection interface. Step 2: Based on the principle of virtual work, derive the dynamic coupling conditions between the tuning fork arm and the mass block at the connection interface; Step 3: Based on the displacement field from Step 1 and the coupling conditions from Step 2, assemble the global system and solve its linear eigenvalue problem to obtain the system's linear resonant frequency and modes; Step 4: Based on the linear modes obtained in Step 3, introduce geometric nonlinear strain and establish the nonlinear control equations of the system; Step 5: Solve the nonlinear control equations using an iterative algorithm to calculate the nonlinear frequency response of the resonator under finite amplitude.

2. The method for dynamic modeling and performance analysis of a quartz double-ended tuning fork resonator according to claim 1, characterized in that, The displacement field function of the mass block mentioned in step one is defined as follows: ; in, and Let be the warping function. and These are the unknown amplitude functions of the mass block along the x and y directions, respectively.

3. The method for dynamic modeling and performance analysis of a quartz double-ended tuning fork resonator according to claim 2, characterized in that, The warping function and Based on the aforementioned free boundary stress condition, it is determined that at the boundary... The following conditions must be met: 。 4. The method for dynamic modeling and performance analysis of a quartz double-ended tuning fork resonator according to claim 3, characterized in that, The warping function and Let it be a higher-order polynomial function: ； 。 5. The method for dynamic modeling and performance analysis of a quartz double-ended tuning fork resonator according to claim 1, characterized in that, In step two, the derivation of coupling conditions based on the principle of virtual work specifically involves: at the connection interface, setting the virtual work of the tuning fork arm equal to the virtual work of the mass block, and deriving the coupling boundary conditions accordingly.

6. The method for dynamic modeling and performance analysis of a quartz double-ended tuning fork resonator according to claim 1, characterized in that, Step five, the iterative method includes the following sub-steps: Step 5.1: Solve the linear eigenvalue problem to obtain the linear frequencies and modes, and use these as initial values; Step 5.2: Based on the linear mode shape, calculate the equivalent axial tension and boundary coupling terms caused by geometric nonlinearity; Step 5.3: Substitute the axial tension and boundary coupling terms as prestress and correction terms into the governing equations, and resolve the eigenvalue problem to obtain the updated frequencies and modes; Step 5.4: Repeat steps 5.2 and 5.3 until the change in frequency and equivalent axial tension is less than the preset tolerance, thereby obtaining a convergent nonlinear resonant frequency.

7. The method for dynamic modeling and performance analysis of a quartz double-ended tuning fork resonator according to claim 1, characterized in that, In step three, the global system is solved using a combination of the determinant scanning method and the Brent method to obtain the linear resonant frequencies and modes of the system. The linear resonant frequencies and modes of the coupled system are then solved using the determinant scanning method and the Brent method.

8. A medium storing a computer program, characterized in that, When the computer program is executed by the processor, it implements the dynamic modeling and performance analysis method for a quartz double-ended tuning fork resonator as described in any one of claims 1 to 7.

9. A MEMS resonator design system, characterized in that, The system includes a memory and a processor, the memory storing a computer program, and the processor executing the program to implement the dynamic modeling and performance analysis method for a quartz double-ended tuning fork resonator as described in any one of claims 1 to 7, the system being used to predict and optimize the frequency characteristics of the quartz tuning fork resonator.

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