Study on the influence of pendulum convex plate on the partial value of quartz flexible accelerometer

The partial value is an essential performance index for assessing the stability of quartz flexible accelerometers. By analyzing the structure of the quartz flexural accelerometer and the impact of the pendulum convex plate, the influence of the convex plate on the partial value of the accelerometer is studied by theoretical derivation and modeling simulation methods. The results show that the barycentric coordinate shifts by approximately 10.7678 × 10-12 µm with the change of the coplanarity of the convex plate, which causes mechanical zero position errors, increases the partial value, and affects the stability of the quartz flexural accelerometer. Therefore, the coplanarity of the convex plate should be maintained at ≤0.3 µm and the partial value should be within the error range of ≤ |±2| mg and provide improvement measures to establish a basis for further improving the stability of the quartz flexible accelerometer.


Introduction
The quartz flexible accelerometers are widely used in missiles, aircraft, and other fields due to their high sensitivity, stability, and accuracy [1] .It is affected by the influence of temperature, vibration, and other factors, and the key performance parameters (partial value and scale factor) may drift over time.The accelerometer exhibits issues such as poor long-term repeatability and complex operation during its use.Therefore, a significant amount of research has been conducted both domestically and internationally [2- 6] , and the international mainstream quartz flexible accelerometer products include Honeywell's QA-3000-030 accelerometer and European Inna Labs' AI-Q-2030 accelerometer [7] .China started late in the development of quartz flexible accelerometers and lagged behind foreign countries in terms of structural design, materials, accuracy, range, and other aspects.Therefore, further investigation into quartz flexible accelerometers is crucial.
In this paper, two methods of theoretical derivation and modeling simulation are used to study the influence of the pendulum convex plate on the partial value of the accelerometer and provide improvement measures to establish a basis for further improving the stability of the quartz flexible accelerometer.

Working principle of quartz flexible accelerometer
The pendulum is the essential component of the quartz flexible accelerometer.When there is an external input acceleration, the pendulum quality may generate a tiny deflection and swing, causing it to deviate 2 from its equilibrium position [8] .The capacitance value of the differential capacitance sensor between the upper and lower yoke irons and the coated pendulum changes, which is fed back to the servo circuit.After undergoing integral amplification, the balance current is output.The torque converter coil flowing through the current is subjected to electromagnetic force in the magnetic field, resulting in inertial force balance, so that the pendulum is restored to the balance position [9] .When the accelerometer system reaches the force balance, the current value of the loop flowing through the torque is the value of the external acceleration, and the direction of the acceleration is also the current direction.The swing range of the pendulum is determined by the height of the convex plate, and the stability of its working state directly affects the performance of the accelerometer [10] .

Pendulum convex plate
Each side of the pendulum has three convex plates, formed by the final etching stage.The thickness of the three convex plates is approximately 20 , and the upper and lower yoke irons fix the pendulum by clamping the convex plates.The three convex plates exert a significant influence on the direction, distance, and sensitivity of the swing:  The height of the convex plates determines the extent of the vibration range exhibited by the central mass block.
 The convex plate creates a differential capacitance between the mass pendulum and the magnetic yoke.
 It is the kinetic damping of the central mass block. The convex plate is coated with a layer of gold film, which serves as the lead wire for the capacitance sensor plate, the capacitance plate, and the torque converter moving coil.
The partial value refers to the non-zero value of the accelerometer output under the condition of zero acceleration.It is an important parameter for determining the performance of the quartz flexible accelerometer and is also a key factor in achieving stable operation of the inertial system.In the quartz flexible accelerometer, changes in the mechanical zero have a significant impact on the partial value.In the process of assembling the pendulum and the upper and lower yoke irons, if the flatness and coplanarity of the pendulum convex plate are not good, the uniformity and symmetry of the gap will also be affected, resulting in the center of gravity of the assembled pendulum component offset and a mechanical zero error, and affecting the partial value and the bias stability.To meet the practical application demands of high-precision quartz flexure accelerometers, it is imperative that the surface of the convex plate exhibits a smooth texture without any defects, and that the three convex plates are aligned in a coplanar manner.

Theoretical research
The pendulum is simplified as shown in Figure 1.The bending stiffness of the flexible beam is: Where  is the width of the pendulum (unit: mm);  is the length of the pendulum (unit: mm); h is the thickness of the pendulum (unit: mm); s is the elastic compliance constant of quartz (unit:   ⁄ ); L is the distance from the centroid of the pendulum component to the end of the flexible beam (unit: mm); m is the quality of pendulum component (unit: g).
As a result of an erroneous installation, the input axis of the quartz flexible accelerometer is inconsistent with the input direction of the measured acceleration, resulting in an angular error, as shown in Figure 2. (5) The input signal is: By substituting Equation (1) into Equation ( 6), we can obtain: The simplified model equation of a quartz flexible accelerometer is: where E is the output of the accelerometer (unit: V);  is a scaling factor (unit: V/g);  is a partial value (unit: mg);  is the n−order nonlinear coefficient;  is the installation error of input axis relative to IA (input reference shaft) around OA(output reference shaft) (unit: rad);  is the installation error of the input shaft relative to IA (input reference shaft) around PA (swing reference shaft) (unit: rad);  is the accelerations along the PA axis (unit:   ⁄ );  is the accelerations along the OA axis (unit:   ⁄ ).It can be seen from the simplified equation that the parameters that have a significant influence on the performance of the accelerometer are scale factor and partial value.Due to the "rectification effect" of the second-order nonlinear term, the partial value will change.When   sin  , from the second-order nonlinear term in Equation ( 7), it can be concluded that the installation error affects the partial value.Solid works modeling analysis The SolidWorks 3D software is used to model the upper and lower yoke iron, compensation ring, magnetic steel, skeleton, and pendulum of the quartz flexible accelerometer.The structure is shown in Figure 3.The pendulum modeling is shown in Figure 4.  .6323647Under the assumption that all other conditions remain unchanged, a convex plate of the pendulum piece is elevated, disregarding the quality of the elevated section of the plate.After the reassembling process, as shown in Figure 5, there is an observed alteration in the height of the convex plate from 20 μm to 20.1 μm.Due to the non-coplanarity of the convex plate, it causes an inclination in the pendulum assembly and the change of the center of gravity coordinates, as shown in Figure 6. Figure 6.Coordinate chart.The findings indicate that in comparison to the ideal assembly, the center of gravity undergoes a shift and deviates from its geometric center position, resulting in mechanical zero error and influencing the partial value.Further experiments were conducted by incrementally adding a convex platform of 0.01  each time.The changes in the XYZ coordinates of the barycenter were observed and recorded, as shown in Table 1.Using Origin drawing software, the change in barycenter coordinates is depicted in a 3D trajectory diagram, as shown in Figure 7.According to the coordinate equation between two points in threedimensional space, we can obtain: The MATLAB software is used to solve and draw the changing curve, as shown in Figure 8.The results show that each time the convex plate increases by 0.01 , the barycenter coordinate shifts by about 10.7678 10 .The centroid position of the pendulum deviates from the geometric center, and the partial value increases.This is a mechanical zero phenomenon caused by only changing the height of one convex plate.In practical production scenarios, discrepancies in the heights of the three convex plates can lead to more significant errors.

ANSYS static analysis
The finite element simulation software ANSYS is utilized for conducting static analysis.The material properties of quartz are specified as follows: a density of 2.203 g/cm 3 , a Poisson ratio of 0.17, and a Young's modulus of 7.25×10 4 MPa.The mesh is divided and fixed constraints are applied.In practical applications, the motion of a pendulum is influenced by the upper and lower limits of the yoke iron.The maximum displacement of the pendulum occurs at the lower end of its surface, and the value is 20 , which corresponds to the height of a single-sided convex plate.
In an ideal scenario, the partial value is zero.When an external acceleration of 1 g is applied along the Z-axis, the pendulum surface swings back and forth along the Z-axis.Disregarding the limitations imposed by the yoke iron, the pendulum reaches a maximum displacement of 1.8511867 mm, as shown in Figure 9.The maximum stress on the top of the two flexible beams is 48.966789MPa, as shown in Figure 10.When the external input acceleration value is increased to 1.35 g, the maximum stress of the flexible beam is 66.105154 MPa, which is less than the allowable bending stress of the quartz material of 67 MPa. Figure 10.Stress diagram.Other conditions remain unchanged, and the displacement constraint is applied to the elevated part of the convex plate so that the pendulum produces a deflection of 0.1  without external acceleration input.This is equivalent to the pendulum component in the assembly process.Since the convex plate coplanarity error becomes 0.1 , a shift in the center of gravity of the swing component and a tilt of approximately 0.00257° will occur, generating a partial value.At this time, when the pendulum is subjected to the acceleration of 1 g along the Z-axis direction, the partial value is approximately 0.16 mg, and the maximum displacement of the pendulum surface is 1.8511867 mm, as shown in Figure 11.The maximum stress on the top of the flexible beam is 48.966793MPa, as shown in Figure 12. Figure 12.Stress diagram.As shown in Figure 13, as the coplanarity error of the convex plate increases, when the coplanarity error is ≤ 0.3 , the maximum stress value of the flexible beam is about 48.96683 MPa, and the maximum displacement generated by the pendulum surface is 1.8511867 mm and the partial value is ≤ |±2| mg, which will not affect the stability of the quartz flexible accelerometer.
When the coplanarity error surpasses 0.3 , the stress on the flexible beam has a significant upward trend, and the partial value is greater than 2 mg.The coplanarity error starts to impact the stability of the accelerometer.Taking the error of coplanarity as 0.4 , the maximum displacement generated by the pendulum surface is 2.5916613 mm when the external input acceleration value is 1.35g, as shown in Figure 14.The maximum stress of the flexible beam is 68.553507MPa, which exceeds the allowable bending stress of quartz material of 67 MPa, and the flexible beam cannot be used due to fracture, as shown in Figure 15.The simulation results indicate that to improve the stability of the quartz flexible accelerometer, it is necessary to limit the coplanarity error of the convex plate to ≤ 0.3 , and the partial value to ≤ |±2| mg.

Improvement measures
Currently, there is an increasing demand for high-performance quartz flexible accelerometers, which necessitates higher precision in the processing and detection of the pendulum.The height, flatness, and coplanarity of the pendulum convex plates are crucial geometric parameters that can impact the stability of quartz flexural accelerometers.To enhance the flatness of the convex plates and minimize clamping deformation, the non-contact ion beam figuring method can be employed during the processing stage.Traditional measurement techniques, which are predominantly contact-based and possess low precision, are no longer sufficient to meet the current accuracy requirements.To further improve the measurement accuracy of the convex plates, a fast, high-precision, and non-contact laser displacement sensor or a spectral confocal sensor can be used.

Conclusion
In this paper, an analysis is conducted on the structure of the quartz flexible accelerometer and the role of pendulum convex plates to investigate their impact on the accelerometer's bias.Theoretical research and simulation outcomes demonstrate that reducing the coplanarity and flatness error of the convex plates leads to a decrease in the influence on the partial value, thereby enhancing the stability of the quartz flexible acceleration.Hence, to minimize machining errors on convex plates, non-contact processing and measurement methods are chosen.This ensures that the coplanarity error of the convex plate is ≤ 0.3 , and the partial value is ≤ |±2| mg.This study lays a foundation for further improving the stability of quartz flexible accelerometers.

Figure 1 .
Figure 1.Simplified diagram of the pendulum.Given an acceleration  of input axis, the pendulum component will produce an angle : 180 i L mL a 2K     (1)

Figure 2 .
Figure 2. Angle due to installation error.The output shaft has interference acceleration, which will add an error signal to the pendulum.0 0 sin a a   (3) The input signal is: 0 0 cos sin i i A a a a a        (4) It can be concluded from Figure 2 that: 0 co in ; s s

Figure 3 .
Figure 3. Modeling diagrams.Figure 4. Pendulum structure diagram.The built model is assembled, and the barycenter coordinate at this time is:  11029.09426196 4596.30270074 232.6323647Under the assumption that all other conditions remain unchanged, a convex plate of the pendulum piece is elevated, disregarding the quality of the elevated section of the plate.After the reassembling process, as shown in Figure5, there is an observed alteration in the height of the convex plate from 20 μm to 20.1 μm.Due to the non-coplanarity of the convex plate, it causes an inclination in the pendulum assembly and the change of the center of gravity coordinates, as shown in Figure6.

Figure 4 .
Figure 3. Modeling diagrams.Figure 4. Pendulum structure diagram.The built model is assembled, and the barycenter coordinate at this time is:  11029.09426196 4596.30270074 232.6323647Under the assumption that all other conditions remain unchanged, a convex plate of the pendulum piece is elevated, disregarding the quality of the elevated section of the plate.After the reassembling process, as shown in Figure5, there is an observed alteration in the height of the convex plate from 20 μm to 20.1 μm.Due to the non-coplanarity of the convex plate, it causes an inclination in the pendulum assembly and the change of the center of gravity coordinates, as shown in Figure6.

Figure 5 .
Figure 5. Assembly drawing.Figure6.Coordinate chart.The findings indicate that in comparison to the ideal assembly, the center of gravity undergoes a shift and deviates from its geometric center position, resulting in mechanical zero error and influencing the partial value.Further experiments were conducted by incrementally adding a convex platform of 0.01  each time.The changes in the XYZ coordinates of the barycenter were observed and recorded, as shown in Table1.Table1.Center of gravity coordinate statistics.

Figure 9 .
Figure 9. Displacement distribution.Figure10.Stress diagram.Other conditions remain unchanged, and the displacement constraint is applied to the elevated part of the convex plate so that the pendulum produces a deflection of 0.1  without external acceleration input.This is equivalent to the pendulum component in the assembly process.Since the convex plate coplanarity error becomes 0.1 , a shift in the center of gravity of the swing component and a tilt of approximately 0.00257° will occur, generating a partial value.At this time, when the pendulum is subjected to the acceleration of 1 g along the Z-axis direction, the partial value is approximately 0.16 mg, and the maximum displacement of the pendulum surface is 1.8511867 mm, as shown in Figure11.The maximum stress on the top of the flexible beam is 48.966793MPa, as shown in Figure12.

Figure 11 .
Figure 11.Displacement distribution.Figure12.Stress diagram.As shown in Figure13, as the coplanarity error of the convex plate increases, when the coplanarity error is ≤ 0.3 , the maximum stress value of the flexible beam is about 48.96683 MPa, and the maximum displacement generated by the pendulum surface is 1.8511867 mm and the partial value is ≤ |±2| mg, which will not affect the stability of the quartz flexible accelerometer.When the coplanarity error surpasses 0.3 , the stress on the flexible beam has a significant upward trend, and the partial value is greater than 2 mg.The coplanarity error starts to impact the stability of the accelerometer.Taking the error of coplanarity as 0.4 , the maximum displacement generated by the pendulum surface is 2.5916613 mm when the external input acceleration value is 1.35g, as shown in Figure14.The maximum stress of the flexible beam is 68.553507MPa, which exceeds the allowable bending stress of quartz material of 67 MPa, and the flexible beam cannot be used due to fracture, as

Figure 13 .
Figure 13.Variation curve of maximum stress of the flexible beam.

Table 1 .
Center of gravity coordinate statistics.