Micro-beam large deformation analysis and MEMS sensor control method and system

Abstract

The present application relates to the technical field of sensor control, and particularly to a micro-beam large deformation analysis and MEMS sensor control method and system. The method comprises: obtaining structural parameters, material parameters and end load parameters of a functionally graded micro-beam; establishing a micro-beam large deformation control equation containing an intrinsic material scale parameter and a material gradient index according to the obtained parameters; converting the control equation into an elliptic integral form to obtain an accurate analytical relationship between stress, end displacement and deformation configuration of the micro-beam; and performing real-time control on the micro-beam or a MEMS sensor containing the micro-beam based on the accurate analytical relationship. By introducing the intrinsic material scale parameter, the size effect under microscale is accurately quantified, and significant prediction deviation of the classical theory under microscale is avoided.

Description

Technical Field

[0001] This invention relates to the field of MEMS sensor control technology, and in particular to a method and system for large deformation analysis of microbeams and control of MEMS sensors. Background Technology

[0002] Microelectromechanical systems (MEMS) occupy a fundamental position in modern engineering. Microcantilever beams are core components of these architectures, serving as primary sensing and actuation elements in diverse technologies such as atomic force microscopy, biochemical detectors, inertial navigation units, and pressure sensors. As device dimensions continue to shrink to the sub-micrometer scale, experimental data consistently demonstrate significant differences between microscale mechanical behavior and traditional macroscopic predictions. For example, bending and torsion tests show that when the thickness of an epoxy resin beam decreases from 115 μm to 20 μm, its normalized bending stiffness increases by a factor of 2.4. This size effect exceeds the prediction range of classical elasticity theory; therefore, intrinsic scale-related energy mechanisms such as rotational gradients must be incorporated into theoretical models.

[0003] To accurately capture the size-dependent behavior of microstructures, the academic community has developed various non-classical continuum theories, mainly including nonlocal elasticity theory, strain gradient theory, and couple stress theory. Among them, the modified couple stress theory proposed by Yang et al., which only requires the introduction of one intrinsic material scale parameter, ensures both physical accuracy and mathematical operability, and has been widely used in the free vibration, static bending, and buckling analysis of microscale plates and beams.

[0004] Meanwhile, functionally graded materials (FGMs) offer an efficient structural strategy for MEMS fabrication. Unlike traditional laminated composites, which suffer from severe interfacial stress concentrations, FGMs achieve a smooth transition of internal properties by continuously changing the material composition (typically following a power-law distribution), thereby optimizing structural performance.

[0005] However, existing technologies for large deformation analysis of functionally graded microbeams still have the following shortcomings: Geometric nonlinearity is difficult to handle: When a microbeam undergoes large deformation (e.g., end displacement exceeds 20% of the beam length), geometric nonlinear effects become significant. Traditional linear or weakly nonlinear analysis methods (such as perturbation methods) will fail. Although numerical methods such as the finite element method can be used, the introduction of higher-order constitutive relations leads to a dramatic increase in computational complexity and convergence difficulties, making them unsuitable for real-time control of MEMS devices and on-chip algorithm deployment.

[0006] Size effects cannot be quantified: Classical continuum mechanics cannot capture the "smaller, stiffer" phenomenon at the microscale. Ignoring size effects will lead to significant deviations in the prediction of microbeam deformation behavior, thereby affecting the calibration accuracy and control performance of MEMS sensors using these as sensing elements. Existing patents (such as CN116910942A) only address the large deformation problem of macroscopic cantilever beams and cannot quantify size effects at the micrometer scale.

[0007] Lack of efficient and accurate analytical tools: Currently, there is a lack of analytical methods that can precisely couple the theory of modified couple stress, the properties of functionally graded materials, and large geometric deformations. This poses a bottleneck for applications requiring low power consumption and high real-time computation (such as on-chip MEMS sensor calibration).

[0008] The invention patent with application number 202311161774 describes a method for controlling the deformation of a cantilever beam under concentrated force at its end. However, it only describes the large deformation under macroscopic conditions at a macroscopic scale and does not further quantify the two major deformations at the micro- and nano-scales, nor does it apply to MEMS sensors. In view of the problem that the microscopic scale cannot be quantified, there is an urgent need for a method that can overcome the above-mentioned defects and accurately and efficiently analyze and control the large deformation behavior of functionally graded microbeams. Summary of the Invention

[0009] To address the problem that existing technologies cannot accurately and efficiently handle the large geometric deformation and size effects of functionally graded beams at the microscale, this invention provides a method and system for large deformation analysis of microbeams and control by MEMS sensors.

[0010] Firstly, the present invention provides a method for large deformation analysis of microbeams and control using MEMS sensors, which employs the following technical solution: A method for large deformation analysis of microbeams and control using MEMS sensors includes: S1. Obtain the structural parameters, material parameters, and end load parameters of the functionally graded microbeam; S2. Based on the obtained parameters, establish the large deformation control equations for the microbeam, which include intrinsic material scale parameters and material gradient exponents; S3. Transform the governing equations into elliptic integral form and solve for the precise analytical relationship between the microbeam's stress, end displacement, and deformation configuration; S4. Real-time control of the microbeam or MEMS sensor containing the microbeam based on the precise analytical relationship.

[0011] Furthermore, the structural parameters include the beam length. ,width and thickness The material parameters include the elastic modulus of the ceramic and metallic phases constituting the microbeams. shear modulus Poisson's ratio and material gradient index The load parameters include the magnitude of the concentrated force at the end. and direction angle of action .

[0012] Furthermore, the step of establishing the large deformation control equations for the microbeam, which include intrinsic material scale parameters and material gradient exponents, based on the acquired parameters, includes establishing the large deformation control equations for the functionally graded microbeam under concentrated forces at its ends in any direction, based on the modified couple stress theory; the control equations include intrinsic material scale parameters used to characterize the size effect at the microscale. The material properties of the functionally graded microbeams are along the thickness direction. Its effective elastic modulus varies with a continuous gradient. and shear modulus It follows a power-law distribution, which can be expressed as: Among them, subscript and These represent the top and bottom materials of the microbeam, respectively. The range of values ​​is , It is a non-negative material gradient exponent.

[0013] Furthermore, the step of establishing the large deformation control equation of the microbeam, which includes intrinsic material scale parameters and material gradient exponents, based on the obtained parameters, also includes defining the total effective flexural stiffness of the functionally graded microbeam based on the modified couple stress theory. The total effective bending stiffness From macroscopic bending stiffness and micro-couple stiffness It is formed by stacking, that is ;in, ; In the formula, z0 is the position of the physical neutral axis, determined by the following formula: Then, based on the moment-curvature relationship of the beam, a system considering the total effective bending stiffness is established. The differential equation for the large deformation of a microbeam is expressed as: ,in, Let be the deformation angle at any point on the beam, and s be the arc length coordinate starting from the fixed end; finally, variable substitution is introduced. The differential equation is then integrated, and the deformation angle is obtained by combining the boundary conditions at the fixed and free ends. With arc length The integral relation.

[0014] Furthermore, the process of transforming the governing equations into elliptic integral form and solving them to obtain the precise analytical relationship between the microbeam's stress, end displacement, and deformation configuration includes transforming the large deformation governing equations into elliptic integral form. By solving the elliptic integral equations, the correspondence between the microbeam's stress state and the free end displacement and microbeam deformation configuration is obtained, thereby obtaining the precise solution for the large deformation of the functionally graded microbeam based on the modified couple stress theory. Furthermore, obtaining the exact solution for the large deformation of a functionally graded microbeam based on the modified couple stress theory includes the following specific steps: [The text abruptly ends here, likely due to an incomplete sentence or a formatting error.] With arc length The integral relationship can be obtained by introducing variable substitution. This is transformed into a combination of the first and second types of incomplete elliptic integrals; then, based on the transformed elliptic integral form, the load factor β and the free end rotation angle are obtained. The analytical relationship between them is: ,in, , For a complete elliptic integral of the first kind, This is a first-kind incomplete elliptic integral. , Finally, under known loads and direction angle In this case, the free end rotation angle can be obtained by solving the analytical relation. Then, the displacements of the free end in the horizontal and vertical directions are calculated using elliptic integral expressions. and The coordinates of any point on the beam are used to obtain the complete deformable configuration of the microbeam.

[0015] Furthermore, the real-time control of the microbeam or MEMS sensor containing the microbeam based on the precise analytical relationship includes designing structural parameters, calibrating mechanical properties, or controlling spatial motion of the functionally graded microbeam or MEMS sensor containing the microbeam according to the correspondence. Specifically, based on the precise solution of the large deformation of the functionally graded microbeam, a signal output error model of the MEMS piezoresistive sensor is constructed; the error model is used to quantify the sensor output voltage predicted by the classical continuous medium theory, which ignores size effects. The output voltage predicted based on the modified couple stress theory Deviation between , is represented as: Then, based on the signal output error model, the sensitivity and linearity of the MEMS sensor are precisely calibrated, and the material gradient exponent of the microbeam is designed in reverse based on the target performance. and geometric dimensions .

[0016] Furthermore, the real-time control of the microbeam or MEMS sensor containing a microbeam based on the precise analytical relationship also includes the output voltage of the MEMS piezoresistive sensor. Maximum strain at the root of the fixed end of the microbeam Proportional to the maximum strain at the root of the fixed end of the microbeam The obtained free end displacement and The solution yields the force. Direction angle By changing the direction of the applied force, the deformation configuration of the functionally graded microbeam under the same applied force magnitude but different directions can be calculated, thereby realizing accurate deformation prediction and control when a specific directional force is applied to the microbeam or MEMS sensor.

[0017] Secondly, a microbeam large deformation analysis and MEMS sensor control system includes: The data acquisition module is configured to acquire the structural parameters, material parameters, and end load parameters of the functionally graded microbeam. The equation construction module is configured to establish, based on the acquired parameters, the large deformation control equations of the microbeam, which include intrinsic material scale parameters and material gradient exponents. The analytical solution module is configured to transform the governing equations into elliptic integral form and solve for the precise analytical relationship between the microbeam's stress, end displacement, and deformation configuration. The control execution module is configured to perform real-time control of the microbeam or MEMS sensor containing the microbeam based on the precise analytical relationship.

[0018] Thirdly, the present invention provides a computer-readable storage medium storing a plurality of instructions adapted for loading and execution by a processor of a terminal device of the microbeam large deformation analysis and MEMS sensor control method.

[0019] Fourthly, the present invention provides a terminal device, including a processor and a computer-readable storage medium, wherein the processor is used to implement various instructions; the computer-readable storage medium is used to store multiple instructions, the instructions being adapted to be loaded and executed by the processor to provide a microbeam large deformation analysis and MEMS sensor control method.

[0020] In summary, the present invention has the following beneficial technical effects: This invention is the first to combine the modified couple stress theory with geometrically accurate large deformation elliptic integral solutions for the analysis of functionally graded microbeams. By introducing intrinsic material scale parameters, the size effect at the microscale is accurately quantified, avoiding the significant prediction bias of classical theory at the microscale (numerical examples show that for pure metallic microbeams, the signal output error of classical theory can reach 382.4%).

[0021] This invention provides an analytical or semi-analytical framework for solving elliptic integrals. Compared to iterative numerical methods such as the finite element method, this method is computationally highly efficient, requires no iteration, and can be easily embedded into low-power microprocessors (MCUs), providing a highly efficient algorithm engine for real-time calibration and reverse design of MEMS sensors.

[0022] Comprehensive functionality: This method is not only applicable to concentrated loads at the end of any direction, but can also handle functionally graded material properties, calculate critical buckling loads and post-buckling equilibrium paths, and can be coupled with the electrical model of MEMS sensors to perform signal error analysis, providing comprehensive theoretical guidance for the design, calibration and control of high-precision MEMS devices.

[0023] Compared with the nonlinear finite element method, whose convergence deteriorates with increasing load, the elliptic integral solution of the present invention can maintain high accuracy and stability under any load level. Attached Figure Description

[0024] Figure 1 This is a schematic diagram of a microbeam large deformation analysis and MEMS sensor control method according to Embodiment 1 of the present invention; Figure 2 This is a schematic diagram of the geometric relationship of the functionally graded microbeam of Embodiment 1 of the present invention undergoing large deformation under the action of concentrated force at the end; Figure 3 This is a schematic diagram illustrating the influence of different material gradient indices and size effect parameters on the effective bending stiffness of the microbeam in Embodiment 1 of the present invention. Figure 4 This is a schematic diagram of the displacement change of the free end of the microbeam when the load direction angle is 45° in Embodiment 1 of the present invention; Figure 5 This is a schematic diagram of the displacement change of the free end of the microbeam when the load direction angle is 60° in Embodiment 1 of the present invention; Figure 6 This is a schematic diagram of the displacement change of the free end of the microbeam when the load direction angle is 90° in Embodiment 1 of the present invention; Figure 7 This is a schematic diagram of the displacement change of the free end of the microbeam when the load direction angle is 120° in Embodiment 1 of the present invention; Figure 8 This is a surface plot of the MEMS sensor output voltage error versus the material gradient index in Embodiment 1 of the present invention. Detailed Implementation

[0025] The present invention will be further described in detail below with reference to the accompanying drawings.

[0026] Example 1 Reference Figure 1 The microbeam large deformation analysis and MEMS sensor control method of this embodiment specifically includes the following steps: S1. Parameter Acquisition Obtain the structural parameters, material parameters, and end load parameters of the functionally graded microbeam.

[0027] like Figure 2 As shown, the microbeam is a uniform cross-section Euler-Bernoulli beam with a length of [missing information]. Width is Thickness is Microbeams are made of ceramics (such as SiC, It is a composite of two phases of materials, namely metals (such as Al and Ni), and the material properties are along the thickness direction. ( The gradient exhibits a power-law variation. The obtained macroscopic material parameters include: the elastic modulus of the top surface material (such as ceramics). shear modulus The elastic modulus of the base material (such as metal) shear modulus Poisson's ratio (Assumed to be constant); and the gradient exponent describing the rate of change of components. .

[0028] Load parameters include the magnitude of the concentrated force acting on the free end. and its angle with the horizontal direction ( ).

[0029] S2. Establishment of governing equations Based on the modified couple stress theory, the governing equations for large deformation are established.

[0030] S2.1. Calculate the total effective bending stiffness .

[0031] First, determine the position of the physical neutral axis. Due to material asymmetry, the neutral axis deviates from the geometric mid-plane, and its position is determined by the axial force equilibrium condition: in, Solving this equation yields .

[0032] Then, calculate the macroscopic bending stiffness. and micro-couple stiffness : The total effective bending stiffness is .in, It is an intrinsic material scale parameter that characterizes the microscale size effect.

[0033] S2.2. Establish the differential equation.

[0034] Based on the Euler-Bernoulli beam theory, considering large geometric deformations, the bending moment at any point on the beam... With curvature The relationship is From the static equilibrium of the infinitesimal segment, the governing equations can be obtained: (1) Boundary conditions are: fixed end Place Free end The bending moment is zero at that point, i.e. .

[0035] S2.3. Solve by integration.

[0036] make Substitute into equation (1) and integrate once, combining with the free end boundary conditions. We can obtain: (2) in The unknown angle is at the free end.

[0037] S3. Solving Elliptic Integrals By substituting the variables in equation (2), we can transform it into an elliptic integral.

[0038] S3.1. Introduce a transformation. Let... and introduce new variables , making Through differential transformation, the integral with respect to s can be transformed into the integral with respect to s. The points.

[0039] S3.2. Obtain the load-deformation relationship. From equation (2) at the fixed end... (correspond ) to the free end (correspond Integrating, we get: Right now: (3) in, It is a complete elliptic integral of the first kind. It is a first-kind incomplete elliptic integral. .

[0040] S3.3. Solve for the deformable configuration. For a given load... and direction Equation (3) contains only one unknown. (or Solving equation (3) using numerical methods yields the following results. and Then, the coordinates of the free end can be calculated. By finding the coordinates of any point on the beam, the complete large deformation configuration can be obtained.

[0041] S4. Control Applications The forces obtained from step S3 With displacement The precise correspondence between functionally graded microbeams and their deformable configurations allows for precise control of functionally graded microbeams or MEMS devices composed of them. For example, in MEMS switches, the required contact displacement can be determined based on the desired contact displacement. This method allows for the precise inverse calculation of the required driving voltage (corresponding to the load). ).exist Figure 1 In the process, this step is reflected in the final stage of "spatial motion control of the microbeam".

[0042] Based on Example 1, this embodiment further applies the method to signal error analysis and calibration of MEMS piezoresistive sensors.

[0043] Step S5. Construct a signal output error model.

[0044] For MEMS cantilever beam sensors with piezoresistive elements integrated at the root, the output voltage Proportional to the maximum surface strain at the root . The deformed configuration obtained in step S3 can be calculated. The method of this invention (considering...) is used respectively. and ) and classical continuum theory methods (let ,have to ) Calculate the same load Below This leads to the corresponding predicted output voltage. and Define relative error , Figure 5MEMS sensor output voltage error and material gradient exponent and size effect parameters A three-dimensional surface plot of the relationship.

[0045] Step S6. Sensor calibration and reverse design.

[0046] This error model can be used to accurately calibrate MEMS sensors. For example... Figure 5 As shown, by drawing and , The relationship diagram allows us to visually see that, Larger (e.g.) When the accuracy is low, the prediction error of classical theory increases dramatically. For applications requiring high precision, the model of this invention must be used for calibration. Furthermore, designers can use this model to reverse-engineer the optimal gradient exponent based on the target range and sensitivity requirements. and geometric dimensions To achieve performance optimization.

[0047] Experimental verification The accuracy, efficiency, and practicality of the method of this invention in solving large deformations of functionally graded microbeams, quantifying microscale size effects, and analyzing and calibrating MEMS sensors are verified. By comparing it with classical continuous medium theory and traditional finite element method, the technical advantages of this invention are clarified.

[0048] The submicron-scale functionally graded microbeam (the core sensing element of a MEMS sensor) was selected, and its structural and material parameters are as follows: Structural parameters: Beam length L = 100 μm, width b = 10 μm, thickness h = 2~20 μm; Material parameters: The top surface is SiC ceramic (elastic modulus E1=410GPa, shear modulus G1=188GPa), the bottom surface is Al metal (elastic modulus E2=70GPa, shear modulus G2=26GPa), Poisson's ratio ν=0.3; Material gradient exponent n: 0, 0.5, 1, 2, 5, 10, ∞; Intrinsic material size parameter l: 0~1μm (characterizing size effect); Load parameters: end concentrated force P = 1~20μN, direction angle α = 45°, 60°, 90°, 120°.

[0049] Comparison Methods Method 1: Classical continuum theory (ignoring size effects, no couple stress term); Method 2: Traditional nonlinear finite element method (ANSYS modeling, mesh refinement, iterative solution); Method 3: The method of this invention (modified couple stress theory + exact solution of elliptic integral).

[0050] Evaluation indicators Deformation prediction error: The relative deviation between the predicted and measured values ​​of the displacement at the free end of the microbeam; Computational efficiency: Solving time for a single working condition (ms); Sensor output error: Predicted output voltage error (%) of MEMS piezoresistive sensor; Robustness: Convergence and stability of solutions under large loads.

[0051] Experimental steps (1) Parameter input: Enter the microbeam structure, material, and load parameters, and set the material gradient exponent n, intrinsic scale parameter l, and load direction angle α; (2) Equation solution: Three comparative methods were used to calculate the large deformation configuration, free end displacement, and root maximum strain of the microbeam; (3) Error calculation: Based on the accurate solution of this invention, the deformation prediction error and sensor output error of classical theory and finite element method are calculated; (4) Efficiency test: Record the solution time of the three methods under a single working condition and compare the calculation efficiency; (5) Robustness verification: Gradually increase the end load and test the convergence of each method under large deformation; (6) Application verification: Based on the accurate solution of this invention, the sensitivity calibration and structural inverse design of MEMS sensor are completed.

[0052] Experimental Results and Analysis 1. Deformation prediction accuracy Classical theory: Ignoring size effects, the smaller the microbeam thickness, the larger the prediction deviation; when h=2μm, the free end displacement error reaches 382.4%. Traditional finite element method: Iterative convergence is difficult under large deformations, and divergence occurs when the load P>15μN, with an average error of 12.3%; The method of this invention has a full-condition deformation prediction error of ≤0.8%, accurately matches the measured deformation configuration, and accurately quantifies the size effect.

[0053] 2. Computational efficiency Classic theory: Single-condition processing time ≈ 5ms; Traditional finite element method: After mesh refinement, the single-condition processing time is approximately 1200ms, and even longer under large deformations; The method of this invention: no iteration is required, the elliptic integral is solved directly, and the time taken for a single working condition is ≈8ms, which meets the real-time control requirements of MEMS.

[0054] 3. Sensor output error According to classical theory, the output voltage prediction error of MEMS sensors can reach up to 415.6% under small thickness and high gradient exponent. The method of this invention corrects the size effect, resulting in an output error of ≤1.2% and a significant improvement in calibration accuracy.

[0055] 4. Robustness The elliptic integral solution of this invention converges stably under any load level without divergence; the traditional finite element method suffers from poor convergence under large deformation and cannot be used for extreme working condition analysis.

[0056] 5. Engineering Application Results Based on the method of this invention, the reverse design of a MEMS sensor was completed. The optimal material gradient index n=2 and thickness h=5μm improved the linearity of the sensor by 28.5% and the sensitivity met the requirements of high-precision detection.

[0057] Experimental conclusions The method of this invention can accurately quantify microscale size effects, overcoming the shortcomings of large prediction bias in classical theories; The analytical solution of elliptic integrals is far more efficient than traditional finite element method, and it is suitable for low-power MCU embedded deployment; It exhibits strong robustness under large deformations and exhibits no convergence issues, making it suitable for full-condition analysis of MEMS. It can be directly used for precise calibration of MEMS sensors, structural reverse design and real-time control, and has outstanding practical engineering value.

[0058] The flowchart in Figure 1 illustrates the complete technical process of this invention, which is as follows: parameter acquisition → establishment of control equations → elliptic integral solution → deformation analysis → sensor error modeling → design / calibration / control. By clearly presenting the method logic, it reflects the closed-loop process from theoretical modeling to engineering application.

[0059] Figure 2 shows a schematic diagram of the geometric relationship of large deformation of a functionally graded microbeam. By labeling the fixed end, free end, concentrated force P at the end, load direction angle α, configuration before and after deformation, arc length coordinate s, and deformation angle θ of the microbeam, the geometric relationship of large deformation under stress of the microbeam is intuitively displayed, and the physical boundaries and variable definitions of the theoretical model are clarified.

[0060] Figure 3 illustrates the influence of different material gradient indices and size effect parameters on effective bending stiffness. The horizontal axis represents l / h (intrinsic dimensional parameter / beam thickness), and the vertical axis represents the effective bending stiffness of the microbeam. The curves correspond to different material gradient indices n (0, 0.5, 1, 2, 5, 10, ∞). The size effect-dominated region (larger l / h, significantly improved stiffness, exhibiting "the smaller the value, the stiffer") and the classical theoretical limit region (l / h→0, macroscopic mechanical properties) are marked. The coupled influence of material gradient and size effect on microbeam stiffness is quantified to verify the effectiveness of the modified couple stress theory.

[0061] Figure 4 Figure 5 , Figure 6 and Figure 7 The figure shows the displacement variation of the free end of the microbeam under different parameters, corresponding to the load direction angles α=45°, 60°, 90°, and 120° respectively; the horizontal axis is l / h, the vertical axis is the horizontal / vertical displacement of the free end (X / L, Y / L), and the curves are the gradient exponents n of different materials; it shows the variation law of displacement with size effect, load direction, and material gradient, and the displacement nonlinearity is obvious under large deformation; it verifies the accurate prediction ability of the method of the present invention for large deformation under load in arbitrary direction and multi-parameter coupling.

[0062] Figure 8 shows a three-dimensional surface plot of the relationship between the output voltage error of the MEMS sensor and the material gradient exponent and size effect parameter. The three-dimensional coordinate axes are the material gradient exponent n, the size effect parameter l / h, and the output voltage error ratio (V_classical / V_MCEST). The surface intuitively shows the trend of the classical theoretical error with n and l / h. The larger l / h and the larger n are, the more the classical theoretical prediction error increases sharply. The prediction deviation that ignores the size effect is quantified, which proves the necessity of the method of the present invention for high-precision calibration of MEMS sensors.

[0063] In summary, this invention creatively combines the modified couple stress theory with the elliptic integral method to provide an efficient and high-precision analytical tool for large deformation problems of functionally graded microbeams. This tool not only overcomes the shortcomings of existing technologies in quantifying size effects at the microscale, but also provides solid technical support and a practical algorithmic foundation for the design, calibration, and real-time control of high-precision MEMS sensors.

[0064] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention.

[0065] Example 2 This embodiment provides a microbeam large deformation analysis and MEMS sensor control system, including: The data acquisition module is configured to acquire the structural parameters, material parameters, and end load parameters of the functionally graded microbeam. The equation construction module is configured to establish, based on the acquired parameters, the large deformation control equations of the microbeam, which include intrinsic material scale parameters and material gradient exponents. The analytical solution module is configured to transform the governing equations into elliptic integral form and solve for the precise analytical relationship between the microbeam's stress, end displacement, and deformation configuration. The control execution module is configured to perform real-time control of the microbeam or MEMS sensor containing the microbeam based on the precise analytical relationship.

[0066] A computer-readable storage medium storing a plurality of instructions adapted for loading and execution by a processor of a terminal device of the microbeam large deformation analysis and MEMS sensor control method.

[0067] A terminal device includes a processor and a computer-readable storage medium, the processor being used to implement various instructions; the computer-readable storage medium being used to store multiple instructions, the instructions being adapted to be loaded and executed by the processor as described in the microbeam large deformation analysis and MEMS sensor control method.

[0068] The above are all preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Therefore, all equivalent changes made in accordance with the structure, shape and principle of the present invention should be covered within the scope of protection of the present invention.

Claims

1. A method for large deformation analysis of microbeams and control using MEMS sensors, characterized in that, include: Obtain the structural parameters, material parameters, and end load parameters of the functionally graded microbeam; Based on the obtained parameters, establish the large deformation control equations for the microbeam, which include intrinsic material scale parameters and material gradient exponents; The governing equations are transformed into elliptic integral form, and the precise analytical relationship between the microbeam's stress, end displacement, and deformation configuration is obtained by solving them. Real-time control of microbeams or MEMS sensors containing microbeams is performed based on the precise analytical relationship described above.

2. The method for large deformation analysis of microbeams and control by MEMS sensors according to claim 1, characterized in that, The structural parameters include the beam length. ,width and thickness The material parameters include the elastic modulus of the ceramic and metallic phases constituting the microbeams. shear modulus Poisson's ratio and material gradient index The load parameters include the magnitude of the concentrated force at the end. and the direction angle of action .

3. The method for large deformation analysis and MEMS sensor control of microbeams according to claim 2, characterized in that, The process of establishing a large deformation control equation for a microbeam, including intrinsic material scale parameters and a material gradient exponent, based on the acquired parameters, includes establishing a large deformation control equation for a functionally graded microbeam under concentrated force at its end in any direction, based on the modified couple stress theory. The control equation includes intrinsic material scale parameters characterizing the size effect at the microscale. The material properties of the functionally graded microbeams are along the thickness direction. Its effective elastic modulus varies with a continuous gradient. and shear modulus It follows a power-law distribution, which can be represented as: Among them, subscript and These represent the top and bottom materials of the microbeam, respectively. The range of values ​​is , It is a non-negative material gradient exponent.

4. The method for large deformation analysis and MEMS sensor control of microbeams according to claim 3, characterized in that, The process of establishing a large deformation control equation for the microbeam based on the acquired parameters, including intrinsic material scale parameters and material gradient exponents, also includes defining the total effective flexural stiffness of the functionally graded microbeam based on the modified couple stress theory. The total effective bending stiffness From macroscopic bending stiffness and micro-couple stiffness It is formed by stacking, that is ;in, ; In the formula, z0 is the position of the physical neutral axis, determined by the following formula: Then, based on the moment-curvature relationship of the beam, a system considering the total effective bending stiffness is established. The differential equation for the large deformation of a microbeam is expressed as: ,in, Let be the deformation angle at any point on the beam, and s be the arc length coordinate starting from the fixed end; finally, variable substitution is introduced. The differential equation is then integrated, and the deformation angle is obtained by combining the boundary conditions at the fixed and free ends. With arc length The integral relation.

5. The method for large deformation analysis and MEMS sensor control of microbeams according to claim 4, characterized in that, The process of transforming the governing equations into elliptic integral form and solving them to obtain the precise analytical relationship between the microbeam's stress, end displacement, and deformation configuration includes transforming the large deformation governing equations into elliptic integral form, solving the elliptic integral equations to obtain the correspondence between the microbeam's stress state and the free end displacement and microbeam deformation configuration, thereby obtaining the precise solution for the large deformation of the functionally graded microbeam based on the modified couple stress theory.

6. The method for large deformation analysis and MEMS sensor control of microbeams according to claim 5, characterized in that, Obtaining the exact solution for the large deformation of a functionally graded microbeam based on the modified couple stress theory includes the following specific steps: [The text abruptly ends here, likely due to an incomplete sentence or a formatting error.] With arc length The integral relationship can be obtained by introducing variable substitution. This is transformed into a combination of the first and second types of incomplete elliptic integrals; then, based on the transformed elliptic integral form, the load factor β and the free end rotation angle are obtained. The analytical relationship between them is: ,in, , For a complete elliptic integral of the first kind, This is a first-kind incomplete elliptic integral. , Finally, under known loads and direction angle In this case, the free end rotation angle can be obtained by solving the analytical relation. Then, the displacements of the free end in the horizontal and vertical directions are calculated using elliptic integral expressions. and The coordinates of any point on the beam are used to obtain the complete deformable configuration of the microbeam.

7. The method for large deformation analysis and MEMS sensor control of microbeams according to claim 6, characterized in that, The real-time control of the microbeam or MEMS sensor containing the microbeam based on the precise analytical relationship includes structural parameter design, mechanical performance calibration, or spatial motion control of the functionally graded microbeam or MEMS sensor containing the microbeam according to the correspondence. Specifically, based on the precise solution of the large deformation of the functionally graded microbeam, a signal output error model of the MEMS piezoresistive sensor is constructed. The error model is used to quantify the sensor output voltage predicted by the classical continuous medium theory, which ignores size effects. The output voltage predicted based on the modified couple stress theory Deviation between , represented as: Then, based on the signal output error model, the sensitivity and linearity of the MEMS sensor are precisely calibrated, and the material gradient exponent of the microbeam is designed in reverse based on the target performance. and geometric dimensions .

8. The method for large deformation analysis and MEMS sensor control of microbeams according to claim 7, characterized in that, The real-time control of the microbeam or MEMS sensor containing a microbeam based on the precise analytical relationship also includes the output voltage of the MEMS piezoresistive sensor. Maximum strain at the root of the fixed end of the microbeam Proportional to the maximum strain at the root of the fixed end of the microbeam The obtained free end displacement and The solution yields the force. Direction angle By changing the direction of the applied force, the deformation configuration of the functionally graded microbeam under the same applied force magnitude but different directions can be calculated, thereby realizing accurate deformation prediction and control when a specific directional force is applied to the microbeam or MEMS sensor.

9. The method for large deformation analysis and MEMS sensor control of microbeams according to claim 1, characterized in that, include: The data acquisition module is configured to acquire the structural parameters, material parameters, and end load parameters of the functionally graded microbeam. The equation construction module is configured to establish, based on the acquired parameters, the large deformation control equations of the microbeam, which include intrinsic material scale parameters and material gradient exponents. The analytical solution module is configured to transform the governing equations into elliptic integral form and solve for the precise analytical relationship between the microbeam's stress, end displacement, and deformation configuration. The control execution module is configured to perform real-time control of the microbeam or MEMS sensor containing the microbeam based on the precise analytical relationship.

10. A computer-readable storage medium storing a plurality of instructions adapted to be loaded and executed by a processor of a terminal device as described in any one of claims 1-8, a method for large deformation analysis of microbeams and control of MEMS sensors.

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