Error modeling method for asymmetric flexible beam structure of quartz flexible accelerometer

Abstract

The application belongs to the technical field of accelerometer, and particularly relates to a quartz flexible accelerometer asymmetric flexible beam structure error modeling method. The method calculates the torsion angle size of the additional torque generated in the swing shaft direction through the section characteristics of the flexible swing beam resistance to bending and torsion, and based on the non-parallel polar plate differential capacitance structure model, the influence of the equivalent torsion angle on the accelerometer output is calculated by using the virtual series method, thereby providing a theoretical reference for the structural symmetry and coaxiality design of the swing assembly of the quartz flexible accelerometer. The specific steps of the application are as follows: S1, determining the actual action positions of the inertial force and the electromagnetic force according to the defects of the swing assembly; determining the size and direction of the additional torque shear force caused by the actual action positions of the inertial force and the electromagnetic force; S2, modeling the torsion angle of the flexible swing beam in the swing shaft direction under the action of the additional torque; S3, obtaining the asymmetric flexible beam structure error of the quartz flexible accelerometer caused by the swing shaft torsion angle based on the non-parallel polar plate differential capacitance model.

Description

Technical Field

[0001] The present invention belongs to the technical field of accelerometers, and in particular relates to an error modeling method for an asymmetric flexible beam structure of a quartz flexible accelerometer. Background Art

[0002] A quartz flexure accelerometer primarily consists of a mechanical meter head and a servo circuit, one for sensing and detecting signals, respectively. When the accelerometer is operating, the pendulum assembly within the meter head senses input shaft acceleration and causes the pendulum beam to vibrate. This displacement signal is then output as a current signal via a differential capacitance detection circuit, thereby detecting the input acceleration. As the core component within the meter head, the ultra-thin flexure beam's high flexibility along the input axis ensures the accelerometer's high resolution, while its high strength along the output axis limits the pendulum assembly's freedom of movement. The pendulum assembly's structural characteristics significantly influence the accelerometer's output performance, particularly its dynamic performance.

[0003] Currently, due to the large discretization errors in the production process of quartz flexible accelerometer heads, there is significant variation between quartz flexible components produced in the same batch, resulting in significant performance variations in the assembled accelerometers. Due to manufacturing limitations, it is difficult to produce a completely symmetrical dual beam through processes such as chemical etching and laser cutting. Furthermore, the axis of the coil assembly is rarely completely symmetrical with the axis of the quartz pendulum. When the flexible pendulum beam is asymmetrical, the dual beams provide different support for the pendulum mass. When the pendulum and coil assembly are not coaxial, the lengths of the inertia arm and the feedback arm differ. Both situations result in an additional torque acting on the pendulum assembly. Ideally, the differential capacitance detection circuit senses the rotation angle of the pendulum assembly about the output axis. Under the influence of additional torque, the pendulum assembly experiences a torsion angle along the pendulum axis. This torsion angle, when detected by the differential capacitance circuit, causes an error in the accelerometer output.

[0004] In summary, during the processing of the pendulum piece, there is a lack of inspection steps for the symmetry of the pendulum beam, the coaxiality of the pendulum assembly and other performance; there is no means to inspect the structural symmetry quality of the pendulum assembly from the completion of pendulum assembly production to the packaging of the entire meter; there is no effective equivalent method to study the influence of the additional torque of the pendulum shaft on the output performance of the quartz flexible accelerometer. Summary of the Invention

[0005] In view of the above problems, the present invention provides a method for modeling the error of the asymmetric flexible beam structure of a quartz flexible accelerometer. This method calculates the torsion angle generated by the additional torque in the direction of the pendulum axis through the bending and torsional cross-sectional characteristics of the flexible pendulum beam, and based on the non-parallel plate differential capacitor structure model, adopts the virtual string method to calculate the influence of the equivalent torsion angle on the accelerometer output, providing a theoretical reference for the structural symmetry and coaxiality design of the quartz flexible accelerometer pendulum assembly.

[0006] The present invention provides a method for modeling the error of an asymmetric flexible beam structure of a quartz flexible accelerometer, and the specific steps are as follows:

[0007] S1. Determine the actual action positions of the inertial force and the electromagnetic force based on the defects of the pendulum assembly; determine the magnitude and direction of the additional torque shear force based on the actual action positions of the inertial force and the electromagnetic force;

[0008] S2. Simulate the flexible pendulum beam of the quartz flexible accelerometer as two parallel rectangular beams; obtain the cross-sectional characteristics of the flexible pendulum beam; and model the torsion angle along the pendulum axis generated by the flexible pendulum beam under the action of the additional torque.

[0009] S3. Based on the non-parallel plate differential capacitance model, the asymmetric flexible beam structure error of the quartz flexible accelerometer caused by the pendulum shaft torsion angle is obtained.

[0010] Optionally, the specific steps of S2 are:

[0011] S21. Simulate the flexible pendulum beam of the quartz flexible accelerometer as two parallel rectangular cross-section beams, wherein the two parallel rectangular cross-section beams are twisted around the torsion center point C; set end boundary conditions of the two parallel rectangular cross-section beams;

[0012] S22, determine the equilibrium condition;

[0013] S23, determining physical conditions;

[0014] S24, determining the corner condition;

[0015] S25, determining the return boundary conditions;

[0016] S26, combining the boundary conditions in step 25 and the distance equations of the two rectangular cross-section beams;

[0017] S27, bringing the boundary conditions of the bending corner into the equilibrium conditions in step 22;

[0018] S28. Based on step 27, obtain the pendulum shaft torsion angle.

[0019] Optionally, the specific steps of step S3 are: when the pendulum assembly is subjected to additional torque, the plates on both sides of the sensitive capacitor are in a non-parallel state; the non-parallel plate differential capacitance is obtained by using a virtual string method; based on the differential capacitance, the asymmetric flexible beam structure error between the pendulum shaft torsion angle and the quartz flexible accelerometer is obtained.

[0020] Optionally, the two side plates of the sensitive capacitor generated by the pendulum plate under the influence of the additional torque are equivalent to the secondary displacement of the two side plates of the sensitive capacitor to obtain the differential capacitance.

[0021] Alternatively, the expression for the differential capacitance is:

[0022]

[0023] Among them, C1 and C2 are two sensitive capacitors in the differential capacitance detection circuit; ε0 represents the vacuum dielectric constant; ε r represents the relative dielectric constant; s represents the plate area; d0 is the plate distance when the two plates of the sensitive capacitor are in equilibrium; Δd is the displacement of the two plates of the sensitive capacitor due to the input acceleration; Δd′ represents the additional sensitive displacement caused by the additional torque; C0 represents the capacitance of the two plates of the sensitive capacitor when they are in equilibrium; ΔC represents the ideal capacitor output; ΔC′ represents the additional capacitor output caused by the additional displacement; ΔN is the distance the shear center deviates from the ideal symmetry axis.

[0024] Optionally, based on the differential capacitance, the asymmetric flexible beam structure error Y of the quartz flexible acceleration is obtained, and the expression is:

[0025] Y=K C (ΔC+ΔC′)

[0026] Among them, K C It is the amplification factor of the proportional link of the detection circuit.

[0027] Optionally, the end boundary conditions of the two parallel rectangular cross-section beams include boundary conditions of torsion angle, bending deflection and bending rotation angle.

[0028] Compared with the prior art, the present invention has at least the following beneficial effects:

[0029] 1. The method of the present invention uses the equilibrium equation to derive the dual-beam torsion problem and uses the virtual string method to calculate the output of the differential capacitance of non-parallel plates, which can clearly obtain the mechanical analysis and the obtained results are accurate:

[0030] This method uses equilibrium equations as its basis, with the displacement, deflection, and rotation angle of the pendulum beam ends as boundary conditions, to clearly define the force relationship between the two beams. It also employs the virtual string method to calculate the output of the differential capacitor with non-parallel plates, equating the torsion problem to the additional displacement sensitive to the differential capacitor and thus establishing the relationship between the torsion problem and the accelerometer output. This two-step equivalence method provides clear mechanical analysis, accurate calculation results, reduced computation time, and improved experimental efficiency.

[0031] 2. The method of the present invention proposes an equivalent method between the torque of the quartz flexible pendulum assembly along the pendulum axis and the output of the accelerometer, providing an evaluation method for the structural error of the quartz pendulum assembly:

[0032] The flexible pendulum is a critical structure in quartz accelerometers that provides sensitive acceleration input. Currently, during the production and assembly of quartz flexible accelerometers, there is no quality inspection of the pendulum assembly's structural symmetry and coaxiality, nor is there a method for evaluating these properties. This invention provides a method for calculating the equivalent output of an accelerometer when the input error source is an asymmetric or non-coaxial quartz pendulum assembly. This method provides a method for evaluating the structural error of the quartz pendulum assembly and offers theoretical insights for optimizing the structural design of quartz accelerometer pendulum assemblies. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] Figure 1 Schematic diagram of the cross section of the quartz double pendulum beam of the present invention;

[0034] Figure 2 Schematic diagram of differential capacitor torsion of the present invention;

[0035] Figure 3 A schematic diagram of the offset of an example coil assembly in the present invention;

[0036] Figure 4 Schematic diagram of the quartz double pendulum beam structure of the present invention;

[0037] Figure 5 The simulation results of the torsion angle calculation of the pendulum assembly in the present invention are as follows;

[0038] Figure 6 This is the error value simulation result under the action of additional torque of the differential capacitor in the present invention. DETAILED DESCRIPTION

[0039] In order to more clearly understand the above-mentioned objects, features and advantages of the present invention, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments. It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments can be combined with each other. In addition, the present invention can also be implemented in other ways different from those described herein. Therefore, the scope of protection of the present invention is not limited by the specific embodiments disclosed below.

[0040] A specific embodiment of the present invention, as Figure 1-6 , discloses a method for modeling the error of asymmetric flexible beam structure of quartz flexible accelerometer, the specific steps are as follows:

[0041] S1. Determine the actual action positions of the inertial force and the electromagnetic force based on the defects of the pendulum assembly; determine the magnitude and direction of the additional torque shear force based on the actual action positions of the inertial force and the electromagnetic force.

[0042] For example, the asymmetry of the flexible pendulum beam can cause the shear center of the cross-section of the flexible pendulum beam to shift; the non-coaxiality between the quartz pendulum piece and the coil assembly in the pendulum assembly can cause the center of mass of the pendulum assembly to not coincide with the center of force of the coil. When the shear center of the flexible pendulum beam cross-section, the center of mass of the pendulum assembly, and the center of mass of the coil are not coaxial, the pendulum assembly will be subjected to additional torque along the pendulum axis.

[0043] S2. Model the flexible pendulum beam of the quartz flexible accelerometer as two parallel rectangular beams; obtain the cross-sectional properties of the flexible pendulum beam (section moment of inertia and section polar moment of inertia); and model the pendulum axis torsion angle generated by the flexible pendulum beam along the pendulum axis under the magnitude and direction of the additional torque shear force. The steps are as follows:

[0044] S21, two parallel rectangular cross-section beams such as Figure 1 As shown, N is the distance between the centers of the two rectangular cross-section beams, and the pendulum beam is assumed to twist around the torsion center point C; the end boundary conditions of the two parallel rectangular cross-section beams are set, including the boundary conditions of the pendulum axis torsion angle, bending deflection and bending rotation angle.

[0045] (1) The boundary conditions of the torsion angle are:

[0046]

[0047] in, is the torsion angle at the end of the first rectangular cross-section beam 1, is the torsion angle at the end of the second rectangular cross-section beam 2, is the pendulum shaft torsion angle.

[0048] (2) The boundary conditions for bending deflection are:

[0049]

[0050] Among them, ω1 is the deflection of the end of the first rectangular cross-section beam 1, ω2 is the deflection of the end of the second rectangular cross-section beam 2, N1 is the distance from the centroid of the cross-section of the first rectangular cross-section beam 1 to point C, and N2 is the distance from the centroid of the cross-section of the second rectangular cross-section beam 2 to point C.

[0051] (3) The boundary conditions of the bending corner are:

[0052] θ1=θ2=θ

[0053] Among them, θ1 is the bending angle of the end of the first rectangular cross-section beam 1, θ2 is the bending angle of the end of the second rectangular cross-section beam 2; θ represents the bending angle of the swing tongue.

[0054] S22. Determine the equilibrium condition, the expression is:

[0055]

[0056] Among them, ∑F y represents the resultant force of the first rectangular cross-section beam 1 and the second rectangular cross-section beam 2 in the y-axis direction; F1 and F2 are the forces of the first rectangular cross-section beam 1 and the second rectangular cross-section beam 2 relative to the linear displacement, respectively; Represents the sum of the bending moments of the first rectangular section beam 1 and the second rectangular section beam 2. M1 and M2 are the bending moments of the first rectangular section beam 1 and the second rectangular section beam 2 relative to the bending angle respectively; M0 represents the magnitude and direction of the additional torque shear force on the pendulum assembly; ∑M x represents the sum of the torques of the first rectangular cross-section beam 1 and the second rectangular cross-section beam 2; is the torque of the first rectangular cross-section beam 1 and the second rectangular cross-section beam 2 relative to the torsion angle.

[0057] S23. Determine the physical conditions, the expression is:

[0058]

[0059] Among them, l represents the length of the first rectangular cross-section beam 1 and the second rectangular cross-section beam 2; I1 and I2 are the section moments of inertia of the first rectangular cross-section beam 1 and the second rectangular cross-section beam 2 respectively; E is the elastic modulus of the material; It1 and It2 are the section polar moments of inertia of the first rectangular cross-section beam 1 and the second rectangular cross-section beam 2 respectively; G is the shear modulus of the material.

[0060] S24. Determine the corner condition, the expression is:

[0061]

[0062] Furthermore, F1=F2, so M1=M2.

[0063] S25. Determine the boundary conditions, the expression is:

[0064]

[0065] S26. Combine the boundary conditions and equations N1+N2=N in step 25:

[0066]

[0067] Wherein, N represents the distance between the centroid of the cross section of the first rectangular cross-section beam 1 and the centroid of the cross section of the second rectangular cross-section beam 2;

[0068] S27. Substitute the boundary conditions of the bending corner into the equilibrium conditions in step 22, and we have:

[0069]

[0070] S28. Based on step 27, obtain the pendulum shaft torsion angle, which is expressed as:

[0071]

[0072] Wherein, b1 represents the width of the first rectangular cross-section beam 1; b2 represents the width of the second rectangular cross-section beam 2.

[0073] S3. Based on the non-parallel plate differential capacitance model, the asymmetric flexible beam structure error of the quartz flexible accelerometer caused by the pendulum shaft torsion angle is obtained.

[0074] Furthermore, when the pendulum assembly is subjected to additional torque, the two plates of the sensitive capacitor become non-parallel. The virtual string method is used to calculate the differential capacitance of the two non-parallel plates, obtaining the equivalent relationship between the pendulum shaft torsion angle and the accelerometer output.

[0075] Specifically, see Figure 2 , the pendulum shaft torsion angle generated by the two side plates of the sensitive capacitor under the influence of the additional torque The differential capacitance is obtained by equivalently shifting the plates on both sides of the sensitive capacitor. The expression is:

[0076]

[0077] Among them, C1 and C2 are two sensitive capacitors in the differential capacitance detection circuit; ε0 represents the vacuum dielectric constant; ε r Represents the relative dielectric constant; s represents the plate area; d0 is the plate spacing when the plates on both sides of the sensitive capacitor are in equilibrium; Δd is the displacement of the plates on both sides of the sensitive capacitor due to the input acceleration; Δd′ represents the additional sensitive displacement caused by the additional torque; C0 represents the capacitance of the plates on both sides of the sensitive capacitor when they are in equilibrium; ΔC represents the ideal capacitor output; ΔC′ represents the additional capacitor output caused by the additional displacement of the plates on both sides of the sensitive capacitor; ΔN is the distance between the shear center of the plates on both sides of the sensitive capacitor and the ideal symmetry axis.

[0078] Based on the differential capacitance, the asymmetric flexible beam structure error Y of the quartz flexible acceleration is obtained, and the expression is:

[0079] Y=K C ·(ΔC+ΔC′) (9)

[0080] Among them, K C It is the amplification factor of the proportional link of the detection circuit.

[0081] S4. Example verification:

[0082] This embodiment is a double flexible pendulum beam with asymmetric thickness. The thickness distribution is h1 = 22 μm, h2 = 20 μm. The quartz pendulum piece and the coil assembly in the pendulum assembly are not coaxial. The coil assembly is offset along the output axis by Δx = 0.01 mm. The structure is shown in the figure. Figure 3 The point of action of the feedback force F is offset from the original symmetry axis Δx, and the offset of the mass center of the swing part Δx' is:

[0083]

[0084] Wherein, m 线圈组件 is the total mass of the coil assembly, m 摆组件 is the total mass of the swing assembly, and m 摆组件 is the mass of the coil assembly and the quartz swing piece.

[0085] The structure of the flexible double-beam is shown in Figure 4 , wherein, l is the length of the swing beam, b is the width of the single swing beam, and h1 and h2 are the thicknesses of the swing beam.

[0086] The additional torque on the swing beam is M0 = m -5 a(Δx-Δx') = 3.4 x 10 2 N·mm, wherein a = 9.8 m / s -6 is the input acceleration.

[0087] The torsion angle θ can be calculated according to formula (7) Through ANSYS finite element simulation, the torsion angle of the asymmetric structure model under 1g acceleration is calculated to be 7.9 x 10 -6 rad, which shows that the calculation result is basically accurate, and the simulation result is shown in Figure 5 .

[0088] Further, ΔN = 0.46 mm and Δd' = 4.14 x 10 -6 mm can be obtained.

[0089] When the swing assembly is in the equilibrium position, the static capacitance C0 = 30.9 pf, and the plate spacing d0 = 0.02 mm at equilibrium, so that Through MAXWELL electric field simulation, the error capacitance of the asymmetric structure model under the action of torsion is calculated to be 0.012 pf, which shows that the calculation result is basically accurate, and the simulation result is shown in Figure 6 .

[0090] Finally, the output error value of the accelerometer under the action of the additional torque can be equivalent to Y' = K C ·ΔC'.

[0091] It can be concluded that the output equivalent method of the asymmetric flexible beam structure of the quartz flexible accelerometer can effectively calculate the influence of the additional torque on the output of the accelerometer, the mechanical concept is clear, and the calculation time is greatly saved.

[0092] Therefore, the present invention adopts the above method to provide an equivalent output calculation method of the accelerometer when the input error source is the asymmetric or non-coaxial structure of the quartz pendulum assembly, provides an evaluation method for the structural error of the quartz pendulum assembly, and provides a theoretical reference for the structural optimization design of the quartz accelerometer pendulum assembly.

[0093] The above description is only a preferred specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or substitutions that can be easily thought of by any technician familiar with this technical field within the technical scope disclosed by the present invention should be covered by the scope of protection of the present invention.

Claims

1. A method for modeling the error of an asymmetric flexible beam structure of a quartz flexible accelerometer, characterized in that: The specific steps are as follows: S1. Determine the actual action positions of the inertial force and the electromagnetic force based on the defects of the pendulum assembly; determine the magnitude and direction of the additional torque shear force based on the actual action positions of the inertial force and the electromagnetic force; S2. Simulate the flexible pendulum beam of the quartz flexible accelerometer as two parallel rectangular beams; obtain the cross-sectional characteristics of the flexible pendulum beam; and model the torsion angle along the pendulum axis generated by the flexible pendulum beam under the action of the additional torque. The specific steps are as follows: S21. Simulate the flexible pendulum beam of the quartz flexible accelerometer as two parallel rectangular cross-section beams, wherein the two parallel rectangular cross-section beams are twisted around the torsion center point C; set end boundary conditions of the two parallel rectangular cross-section beams; S22, determine the equilibrium condition; S23, determining physical conditions; S24, determining the corner condition; S25, determining the return boundary conditions; S26, combining the boundary conditions in step 25 and the distance equations of the two rectangular cross-section beams; S27, bringing the boundary conditions of the bending corner into the equilibrium conditions in step 22; S28, based on step 27, obtaining the pendulum shaft torsion angle; S3. Based on the non-parallel plate differential capacitance model, the asymmetric flexible beam structure error of the quartz flexible accelerometer caused by the pendulum shaft torsion angle is obtained. The specific steps are: when the pendulum assembly is subjected to the additional torque, the two side plates of the sensitive capacitor are in a non-parallel state; the non-parallel plate differential capacitance is obtained by the virtual string method; based on the differential capacitance, the asymmetric flexible beam structure error between the pendulum shaft torsion angle and the quartz flexible accelerometer is obtained; the two side plates of the sensitive capacitor generated by the pendulum plate under the influence of the additional torque are equivalent to the secondary displacement of the two side plates of the sensitive capacitor to obtain the differential capacitance.

2. The error modeling method for the asymmetric flexible beam structure of a quartz flexible accelerometer according to claim 1, characterized in that: The expression for differential capacitance is: Among them, C1 and C2 are two sensitive capacitors in the differential capacitance detection circuit; ε0 represents the vacuum dielectric constant; ε r represents the relative dielectric constant; s represents the plate area; d0 is the plate distance when the two plates of the sensitive capacitor are in equilibrium; Δd is the displacement of the two plates of the sensitive capacitor due to the input acceleration; Δd′ represents the additional sensitive displacement caused by the additional torque; C0 represents the capacitance of the two plates of the sensitive capacitor when they are in equilibrium; ΔC represents the ideal capacitor output; ΔC′ represents the additional capacitor output caused by the additional displacement; ΔN is the distance the shear center deviates from the ideal symmetry axis.

3. The method for modeling the asymmetric flexible beam structure error of a quartz flexible accelerometer according to claim 2, characterized in that: Based on the differential capacitance, the asymmetric flexible beam structure error Y of the quartz flexible acceleration is obtained, and the expression is: Y=K C ·(ΔC+ΔC′) Among them, K C It is the amplification factor of the proportional link of the detection circuit.

4. The method for modeling the asymmetric flexible beam structure error of a quartz flexible accelerometer according to claim 1, characterized in that: The end boundary conditions of two parallel rectangular beams include boundary conditions of torsion angle, bending deflection and bending rotation.

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