Microelectromechanical Resonant Accelerometer Designed with a High Sensitivity

This paper describes the design and experimental evaluation of a silicon micro-machined resonant accelerometer (SMRA). This type of accelerometer works on the principle that a proof mass under acceleration applies force to two double-ended tuning fork (DETF) resonators, and the frequency output of two DETFs exhibits a differential shift. The dies of an SMRA are fabricated using silicon-on-insulator (SOI) processing and wafer-level vacuum packaging. This research aims to design a high-sensitivity SMRA because a high sensitivity allows for the acceleration signal to be easily demodulated by frequency counting techniques and decreases the noise level. This study applies the energy-consumed concept and the Nelder-Mead algorithm in the SMRA to address the design issues and further increase its sensitivity. Using this novel method, the sensitivity of the SMRA has been increased by 66.1%, which attributes to both the re-designed DETF and the reduced energy loss on the micro-lever. The results of both the closed-form and finite-element analyses are described and are in agreement with one another. A resonant frequency of approximately 22 kHz, a frequency sensitivity of over 250 Hz per g, a one-hour bias stability of 55 μg, a bias repeatability (1σ) of 48 μg and the bias-instability of 4.8 μg have been achieved.


Introduction
Microelectromechanical accelerometers can be found in numerous applications such as inertial navigation systems, gaming, smartphones and mobile devices [1]. These devices are very attractive for high-precision measurement applications due to their high sensitivity, frequency output and large dynamic range [2][3][4]. In a silicon micro-machined resonant accelerometer (SMRA), the acceleration is measured through the differential frequency shift originated by axial loading between the two pull and push double-ended tuning fork (DETF) resonators. This type of resonant accelerometer benefits from a direct frequency shift between the resonators when sensing the input acceleration and this feature draws the extensive attention of researchers.
The sensitivity of the SMRA is defined as the differential output frequency of the resonators produced by an acceleration of 1 g. It is an important characteristic and deserves to be researched. A high sensitivity allows for the acceleration signal to be easily demodulated by frequency counting techniques [2] and decreases the overall noise on the readout electronics [5]. Recently, the design and fabrication of various mechanical resonant accelerometers have been studied [1,6,7]. Pinto et al., have presented the design of a very small and sensitive resonant accelerometer [4]. By using thin silicon-on-insulator (SOI)-based technologies compatible with "In-IC" integration, the accelerometer size has been reduced drastically (0.05 mm 2ˆ4 .2 µm) with a sensitivity of 22 Hz/g. Sandia

Operation Principle
The SMRA structure is shown as a schematic in Figure 1. This structure can be divided into four major components: a proof mass, micro levers, flexure suspensions and DETFs. All these components have coplanar faces. Two DETFs are joined via micro-levers to the proof mass. The proof mass is constrained to move along the y-axis by four flexures, which are linked to the frame mounted to the silicon substrate by four anchors. When acceleration along the input axis is applied to the device, the force from the proof mass is magnified by micro-lever and then transferred to the DETFs. This input applies axial loads, either tension or compression, to the DETFs, which produces a measurable natural frequency shift between the two resonators. The output of the SMRA is the differential frequency Sensors 2015, 15,[30293][30294][30295][30296][30297][30298][30299][30300][30301][30302][30303][30304][30305][30306][30307][30308][30309][30310] variation of the two resonators, and this differential arrangement enables a first-order cancellation of common parasitic sensitivities such as temperature.  Figure 1. Schematic view of the SMRA with a frame structure.

Dies Based on the SOI-MEMS Process
The SMRA has been fabricated with SOI processing and wafer-level vacuum packaging [18]. The main characteristic of the SOI process is the use of silicon-to-silicon direct bonding (SSDB) and high-aspect ratio inductively coupled plasma (ICP) etching technology [17,19]. This process offers an 80 μm-thick MEMS structure with a high aspect ratio up to 1:30, which will then reduce cross-axis sensitivity and increase the robustness of the sensors.
The process cross-section is schematically represented in Figure 2. The die is realized with three wafers, the substrate, the SOI device layer and the cover. The sensor surface is around 110 mm 2 and its thickness for the three layers is 700 μm. The SOI device layer is 80 μm thick and is manufactured using deep reactive ion etching on the SOI wafer with a high aspect ratio of up to 1:30. The residual stress is much less than that of a silicon-on-glass (SOG) process [20]. The cover and the active SOI layer are joined by an Au/Si eutectic bonding, forming a hermetic cavity that maintains the vacuum needed for a high-Q operation of the SMRA. To maintain the vacuum level over the long term, a getter is adhered to the inner surface of the cover, and once it is activated, the getter progressively absorbs and traps gaseous species. The packaged SMRA die was placed in a ceramic cartridge to protect the silicon structure and to facilitate the welding of the whole device. Figure 3 shows the wafer-level vacuum packaged SMRA dies, and Figure 4 shows the SEM (Scanning Electron Microscope) photo of the SMRA structure. The vacuum level of the sensor is measured at 10 Pa.

Silicon
Silicon dioxide Au Cover Substrate SOI device Figure 2. SOI-MEMS (silicon-on-insulator microelectromechanical system) process cross-section of the die.

Dies Based on the SOI-MEMS Process
The SMRA has been fabricated with SOI processing and wafer-level vacuum packaging [18]. The main characteristic of the SOI process is the use of silicon-to-silicon direct bonding (SSDB) and high-aspect ratio inductively coupled plasma (ICP) etching technology [17,19]. This process offers an 80 µm-thick MEMS structure with a high aspect ratio up to 1:30, which will then reduce cross-axis sensitivity and increase the robustness of the sensors.
The process cross-section is schematically represented in Figure 2. The die is realized with three wafers, the substrate, the SOI device layer and the cover. The sensor surface is around 110 mm 2 and its thickness for the three layers is 700 µm. The SOI device layer is 80 µm thick and is manufactured using deep reactive ion etching on the SOI wafer with a high aspect ratio of up to 1:30. The residual stress is much less than that of a silicon-on-glass (SOG) process [20]. The cover and the active SOI layer are joined by an Au/Si eutectic bonding, forming a hermetic cavity that maintains the vacuum needed for a high-Q operation of the SMRA. To maintain the vacuum level over the long term, a getter is adhered to the inner surface of the cover, and once it is activated, the getter progressively absorbs and traps gaseous species. The packaged SMRA die was placed in a ceramic cartridge to protect the silicon structure and to facilitate the welding of the whole device. Figure 3 shows the wafer-level vacuum packaged SMRA dies, and Figure 4 shows the SEM (Scanning Electron Microscope) photo of the SMRA structure. The vacuum level of the sensor is measured at 10 Pa.
3 getter progressively absorbs and traps gaseous species. The packaged SMRA die was placed in a ceramic cartridge to protect the silicon structure and to facilitate the welding of the whole device. Figure 3 shows the wafer-level vacuum packaged SMRA dies, and Figure 4 shows the SEM (Scanning Electron Microscope) photo of the SMRA structure. The vacuum level of the sensor is measured at 10 Pa.

Silicon
Silicon dioxide Au Cover Substrate SOI device Figure 2. SOI-MEMS (silicon-on-insulator microelectromechanical system) process cross-section of the die.

Theoretical Analysis
For each DETF, the natural frequency of the basic lateral vibration mode is expressed as [21] 3 3

Theoretical Analysis
For each DETF, the natural frequency of the basic lateral vibration mode is expressed as [21] 3 3

Theoretical Analysis
For each DETF, the natural frequency of the basic lateral vibration mode is expressed as [21] f 0 " 1 2π where E is the Young's modulus, ρ is the density of single-crystal silicon and l, w and t are the length, width and thickness of the resonant beam, respectively. The comb-drive structure and the resonant beam have the same thickness, and q s is the x-y plane area of the comb-drive structure. If q s is constant, it is clear that the natural frequency of one resonant beam depends on the length and the width but is independent of the thickness of the beam. When the acceleration a along the sensitive axis is applied (see Figure 1), the proof mass is subjected to the force F 1 = m 1 a, and each of the connecting masses is subjected to F 2 = m 2 a. The axial force on each resonant beam has been magnified by the micro-lever to be where A˚is the amplification factor of the sensitive structure (i.e., the effective amplification factor) and m 1 and m 2 are the mass of the proof mass and the connecting mass. Therefore, the resonant beam frequency f under acceleration can be found by energy analysis [22] to be f " 1 2π d where M e f f is the effective mass and K e f f is the axial effective stiffness of the DETF.
Setting α " 0.073 pA˚m 1`2 m 2 q al 2 Ew 3 t , the frequency shift between two DETFs is By substituting a = ng (where n is the applied acceleration in terms of g) into Equation (4) and by taking the derivative of ∆f with respect to n, the sensitivity can be expressed in terms of frequency (with units of Hz/g) where S res is the sensitivity of the DETF sensing element.

Effective Amplification Factor A *
When acceleration a along the input axis is applied to the device, the force from the proof mass is magnified by the micro-lever and then transferred to the DETFs. The effective amplification factor A˚is therefore defined as the ratio of the axial force of the DETF beam to the input inertial force of the proof mass. Because the structure is symmetrical with respect to both the xand y-axes, only one-quarter of the structure has been directly analyzed. Figure 5a shows the model and deformation of each part of the half structure under an inertial load a. All the deformations have been exaggerated for clarity. Because the connecting mass is symmetrical with respect to the y-axis, the bending moment and the horizontal force transferred from output beams will be counteracted. As a result, only the axial force can be transferred to the DETFs and will therefore cause vertical displacements. Figure 5b shows the equivalent model of a quarter of the structure. By supposing the flexure can be regarded as a vertical spring K 1 , half of the connecting mass and one DETF can be regarded as vertical springs of which stiffness are k f and k b .
By solving the boundary conditions for F xi , F yi , M i , F yo , F xo and M o (see Appendix A), the effective amplification factor can be obtained to be Because this is a fairly large output, the expression for A * is shown in Appendix A. Based on the analysis above, the sensitive structure should be designed to produce a high effective amplification factor A * , which proportionally contributes to the sensitivity.
where Alever = Fyo/Fyi represents the amplification factor of the micro-lever, and Klever = Fyi/dyi represents the spring constant of the micro-lever. Both Alever and Klever are decided by the geometry of the micro-lever, and K1 has no influence on them; therefore, the effective amplification factor A * increases as K1 decreases. As for the connecting mass, it should be rigid to ensure energy conservation in an ideal situation. In reality, if kb is 10 times greater than kf, the connecting mass can be regarded as rigid. The design for the connecting mass relies on this principle.

DETF Design Analysis
The corresponding enhancement to the sensitivity owes not only to micro-lever mechanisms' reasonable design, but also to the reasonable design for the resonators. As shown in Equation (6), where A lever = F yo /F yi represents the amplification factor of the micro-lever, and K lever = F yi /d yi represents the spring constant of the micro-lever. Both A lever and K lever are decided by the geometry of the micro-lever, and K 1 has no influence on them; therefore, the effective amplification factor Ai ncreases as K 1 decreases. As for the connecting mass, it should be rigid to ensure energy conservation in an ideal situation. In reality, if k b is 10 times greater than k f , the connecting mass can be regarded as rigid. The design for the connecting mass relies on this principle.

DETF Design Analysis
The corresponding enhancement to the sensitivity owes not only to micro-lever mechanisms' reasonable design, but also to the reasonable design for the resonators. As shown in Equation (6), the sensitivity of DETF S res increases with an increase to the beam length and decreases with an increase to the beam width. Figure 6 shows the variation trend of S res : the sensitivity of DETF increases steadily when the beam length changes from 300 µm to 1100 µm, while it decreases rapidly when the beam width changes from 1 µm to 10 µm, especially when less than 3 µm. Thinner, longer beams provide more sensitivity, but may also cause resonator mismatch due to process, with detrimental effects to the temperature stability of the sensor. This is a trade-off in the mechanical structure design, and the optimum geometry of DETF will be point out in next section.
where Alever = Fyo/Fyi represents the amplification factor of the micro-lever, and Klever = Fyi/dyi represents the spring constant of the micro-lever. Both Alever and Klever are decided by the geometry of the micro-lever, and K1 has no influence on them; therefore, the effective amplification factor A * increases as K1 decreases. As for the connecting mass, it should be rigid to ensure energy conservation in an ideal situation. In reality, if kb is 10 times greater than kf, the connecting mass can be regarded as rigid. The design for the connecting mass relies on this principle.

DETF Design Analysis
The corresponding enhancement to the sensitivity owes not only to micro-lever mechanisms' reasonable design, but also to the reasonable design for the resonators. As shown in Equation (6), the sensitivity of DETF Sres increases with an increase to the beam length and decreases with an increase to the beam width. Figure 6 shows the variation trend of Sres: the sensitivity of DETF increases steadily when the beam length changes from 300 μm to 1100 μm, while it decreases rapidly when the beam width changes from 1 μm to 10 μm, especially when less than 3 μm. Thinner, longer beams provide more sensitivity, but may also cause resonator mismatch due to process, with detrimental effects to the temperature stability of the sensor. This is a trade-off in the mechanical structure design, and the optimum geometry of DETF will be point out in next section.

Reasonable Design of the Structure
By substituting Equation (7) into Equation (5), there is a tremendous dimension system about the sensitivity S g . To obtain a high S g , the Nelder-Mead method under constraint conditions is used as the optimization algorithm for each part of the SMRA. The Nelder-Mead method is a technique for minimizing an objective function in multidimensional space. It uses the concept of a simplex, which is a special polyhedron with N + 1~2N vertexes in N dimensions [16]. In this paper, the energy-consumed concept (energy-consumed concept: based on the conservation of energy law, micro-lever mechanisms with boundary conditions are optimized to consume a low amount of energy and show high-force transmission efficiency from the proof mass to the resonators) is applied to this method and used as the structure optimization algorithm. The flow chart for this algorithm under constraint conditions is shown in Figure 7. The sensitivity is regarded as the negative objective function to obtain a maximum result. Limited by the layout size, the proof mass area is assumed to be lower than 8 mm 2 . The first vibrating mode (the first mode: when SMRA is under acceleration, the proof mass will generate an inertial force, amplified by the micro-lever mechanism; then, the amplified inertial force will cause axial push and pull loading on the DETF resonators) frequency can be expressed as 7 mechanism; then, the amplified inertial force will cause axial push and pull loading on the DETF resonators) frequency can be expressed as Create function constraints and r is a pseudorandom number from 0 to 1.  In the testing environment, the first mode frequency should be larger than 2.1 kHz to stay away from low-frequency external vibrations. Substituting m 1 into Equation (9), the spring constant K 1 + K lever is assumed to be more than 71.4 N/m. The width of each beam is assumed to be equal to or larger than 4.5 µm which is limited by the processing level. Therefore, the range of each beam's geometry is set as shown in Table 1 for our design requirements. Because k b is assumed to be 10 times greater than k f , the connecting mass can be regarded as rigid, which helps to reduce the energy consumed. All the above requirements are used as constraint conditions for the Nelder-Mead method.
The SMRA geometric dimensions after optimization are shown in Table 1. Its sensitivity is determined to be 216.35 Hz/g. Limiting by the layout size and processing level and based on the energy-consumed concept, several dimensions have been corrected slightly. The final sensitivity is 211.5 Hz/g, 66.1% higher than the previous structure's sensitivity of 127.33 Hz/g [14]. The FEM (finite-element analyses) result of the first vibrating mode is 1994.59 Hz, 4.7% lower than the theoretical value of 2.1 kHz. Meanwhile, the energy consumed in each component of this sensitive structure is shown in Table 2. The DETF in this work consumes 59.6% of the total energy, while the DETF of the earlier structure consumes only 6.58 ppm (parts per million). Table 3 shows the ratio of the sensitivity improvement contributed by the DETF, the new micro-lever and the proof mass. This means that the sensitivity improvement is mainly attributed to both the re-designed DETF and the reduced energy loss on the lever. The micro-lever mechanisms (given the boundary conditions) consume lower energy and show high force transmission efficiency from the proof mass to the DETF resonators. If another optimum sensitivity is required, users can simply change the constraint conditions of the Nelder-Mead algorithm and repeat the steps above.

Simulation and Experiments for the SMRA
The micro-lever mechanism can be evaluated by the output frequency by supposing one resonator's natural frequency is f 10 . When it is subjected to an inertial force, the frequency becomes f 1 . By substituting f 10 into Equation (1) and f 1 into Equation (3), we obtain 0.073 f 2 10 pA˚q 1`2 q 2 q ρal 2 Ew 3 By setting c " 0.073ρal 2 Ew 3 , the effective amplification factor A˚for one resonator can be determine by Equation (10) as follows: Equation (11) can be used to judge whether the design of the micro-lever mechanism is reasonable.
The SMRA has been simulated by FEA software, from which the sensitive structure was subjected to an input acceleration in the range of˘40 g (see Figure 8). By fitting 13 datasets, the output frequency, simulated frequency and sensitivity agree with the designed values, as shown in Table 4. Moreover, the effective amplification factor is calculated through Equation (11). The nonlinearity of S g within˘40 g is 49.66 ppm (parts per million). The SMRA has been simulated by FEA software, from which the sensitive structure was subjected to an input acceleration in the range of ±40 g (see Figure 8). By fitting 13 datasets, the output frequency, simulated frequency and sensitivity agree with the designed values, as shown in Table 4. Moreover, the effective amplification factor is calculated through Equation (11). The nonlinearity of Sg within ±40 g is 49.66 ppm (parts per million).    The initial testing was performed in open air at the Sci & Tech Micro Inertial Technology Lab of the Nanjing University. Three SMRA prototypes (A1-5, A1-7, and A1-8) had been chosen for testing these prototypes adopted a self-excited oscillation loop with automatic gain control (AGC) as the drive circuit, and the packaged SMRA dies were finally placed in a ceramic cartridge. The ceramic cartridge package was put on a socket that was wire-connected to an off-chip circuit on a PC board. During our testing, the output was connected to an oscilloscope. Without any input acceleration in A1-5, the resonant frequency of one DETF was 22,447.45 Hz and that of the other DETF was 22,179.4 Hz. The gaps between the normalized frequencies are attributed to thermal and residual stress during the process. Substituting the measured frequency into Equation (5), the theoretical sensitivity is determined to be 249.46 Hz/g.
The PC board of A1-5 was then placed vertically on a rotating platform with a constant temperature control. When this prototype was subjected to 1 g, the resonant frequency for the pull resonator was 22,574.23 Hz, while the push resonator was 22,047.41 Hz. The increased frequency of  The initial testing was performed in open air at the Sci & Tech Micro Inertial Technology Lab of the Nanjing University. Three SMRA prototypes (A1-5, A1-7, and A1-8) had been chosen for testing these prototypes adopted a self-excited oscillation loop with automatic gain control (AGC) as the drive circuit, and the packaged SMRA dies were finally placed in a ceramic cartridge. The ceramic cartridge package was put on a socket that was wire-connected to an off-chip circuit on a PC board. During our testing, the output was connected to an oscilloscope. Without any input acceleration in A1-5, the resonant frequency of one DETF was 22,447.45 Hz and that of the other DETF was 22,179.4 Hz. The gaps between the normalized frequencies are attributed to thermal and residual stress during the process. Substituting the measured frequency into Equation (5), the theoretical sensitivity is determined to be 249.46 Hz/g.
The PC board of A1-5 was then placed vertically on a rotating platform with a constant temperature control. When this prototype was subjected to 1 g, the resonant frequency for the pull resonator was 22,574.23 Hz, while the push resonator was 22,047.41 Hz. The increased frequency of the pull resonator was 126.78 Hz, and the decreased frequency of the push resonator was 131.99 Hz. The total frequency shift was therefore translated to a sensitivity of 258.77 Hz/g, only 3.6% higher than the calculation of 249.46 Hz/g. Figure 9 shows experimental points and a linear fitting of the measured differential frequency for the acceleration of sin(θ) g on the three SMRA prototypes: A1-5, A1-7, A1-8. The rotating angle θ was adjusted to be 0˝, 5˝, 15˝, 25˝, 45˝, 65˝, 75˝, 85˝, 90˝, 95˝, 105˝, 115˝, 135˝, 155˝, 165˝, 175˝, 180˝, 185˝, 195˝, 205˝, 225˝, 245˝, 255˝, 265˝, 270˝, 275˝, 285˝, 295˝, 315˝, 335˝, 345˝, and 355˝, respectively [23]. Good linearity of these prototypes is observed in this range of operation. By fitting the 32 sets of the differential frequency, the average sensitivity within 1 g turns out to be 254.3 Hz/g. As shown in Figure 10, when the SMRA prototypes were subjected to an input acceleration in the range of˘40 g with a constant temperature control, the testing nonlinearity of the sensitivity is within 100 ppm. All of the above results have helped to confirm our theory.
180°, 185°, 195°, 205°, 225°, 245°, 255°, 265°, 270°, 275°, 285°, 295°, 315°, 335°, 345°, and 355°, respectively [23]. Good linearity of these prototypes is observed in this range of operation. By fitting the 32 sets of the differential frequency, the average sensitivity within 1 g turns out to be 254.3 Hz/g. As shown in Figure 10, when the SMRA prototypes were subjected to an input acceleration in the range of ±40 g with a constant temperature control, the testing nonlinearity of the sensitivity is within 100 ppm. All of the above results have helped to confirm our theory.   Figure 9. Variation of the differential resonant frequency Δf for three SMRA prototypes between the resonators in the range of ±1 g. To study the bias stability, the A1-5's input axis was kept horizontal to insure that the input accelerometer was 0 g, and then the whole accelerometer was kept powered for 20 min. In this working state, the output data of this prototype was recorded at a 1 Hz sampling rate for 60 min. To avoid a temperature influence, the sample had been put on a rotating platform under a constant 20 °C . Then, the above steps were repeated for seven times. All the tested data have been presented in Figure 11 with a one-hour bias stability of 55 μg and a bias repeatability of 48 μg. The random bias variance was then characterized using Allan variance, a method proposed for clock  Figure 9. Variation of the differential resonant frequency ∆f for three SMRA prototypes between the resonators in the range of˘1 g.
To study the bias stability, the A1-5's input axis was kept horizontal to insure that the input accelerometer was 0 g, and then the whole accelerometer was kept powered for 20 min. In this working state, the output data of this prototype was recorded at a 1 Hz sampling rate for 60 min. To avoid a temperature influence, the sample had been put on a rotating platform under a constant 20 °C . Then, the above steps were repeated for seven times. All the tested data have been presented in Figure 11 with a one-hour bias stability of 55 μg and a bias repeatability of 48 μg. The random bias variance was then characterized using Allan variance, a method proposed for clock To study the bias stability, the A1-5's input axis was kept horizontal to insure that the input accelerometer was 0 g, and then the whole accelerometer was kept powered for 20 min. In this working state, the output data of this prototype was recorded at a 1 Hz sampling rate for 60 min. To avoid a temperature influence, the sample had been put on a rotating platform under a constant 20˝C. Then, the above steps were repeated for seven times. All the tested data have been presented in Figure 11 with a one-hour bias stability of 55 µg and a bias repeatability of 48 µg. The random bias variance was then characterized using Allan variance, a method proposed for clock systems [24]. Allan variance calculation is applied to the frequency reading and plotted in Figure 12. The Allan variance flattens around 3 s and then shows an increase trend as the averaging time increases. The flatten floor is known as the Allan deviation, which indicates the random parts of the bias-instability is 4.8 µg. The increase trend part is believed to be caused by the temperature drift.  [24]. Allan variance calculation is applied to the frequency reading and plotted in Figure 12. The Allan variance flattens around 3 s and then shows an increase trend as the averaging time increases. The flatten floor is known as the Allan deviation, which indicates the random parts of the bias-instability is 4.8 μg. The increase trend part is believed to be caused by the temperature drift.  As a result, compared to the studies of [2,11,25,26], after reasonable geometrical design, the SMRA reported in this paper stands out for its high-sensitivity of over 210 Hz/g, the input range of ±40 g, one-hour bias stability of 55 μg and the bias repeatability of 48 μg.

Conclusions/Outlook
This paper presents the design and experimental evaluation of an SMRA. We apply energy-consumed concept and the Nelder-Mead algorithm on this sensor to address the design issues and to increase its sensitivity. This SOI-MEMS fabricated SMRA has a closed-form sensitivity of 211.5 Hz/g, its FEM value is 203 Hz/g, and the experimental value is 254.3 Hz/g. The nonlinearity of the Sg is below 100 ppm within the input range of ±40 g. All the results exhibit good agreement with the theoretical results. The sensitivity of the SMRA has increased 66% compared to the previous work by using a novel optimization algorithm. This improvement is mainly attributed to both the re-designed DETF and the reduced energy loss on the micro-lever. All the above work provides a reference for the geometrical design of other MEMS sensors. systems [24]. Allan variance calculation is applied to the frequency reading and plotted in Figure 12. The Allan variance flattens around 3 s and then shows an increase trend as the averaging time increases. The flatten floor is known as the Allan deviation, which indicates the random parts of the bias-instability is 4.8 μg. The increase trend part is believed to be caused by the temperature drift.  As a result, compared to the studies of [2,11,25,26], after reasonable geometrical design, the SMRA reported in this paper stands out for its high-sensitivity of over 210 Hz/g, the input range of ±40 g, one-hour bias stability of 55 μg and the bias repeatability of 48 μg.

Conclusions/Outlook
This paper presents the design and experimental evaluation of an SMRA. We apply energy-consumed concept and the Nelder-Mead algorithm on this sensor to address the design issues and to increase its sensitivity. This SOI-MEMS fabricated SMRA has a closed-form sensitivity of 211.5 Hz/g, its FEM value is 203 Hz/g, and the experimental value is 254.3 Hz/g. The nonlinearity of the Sg is below 100 ppm within the input range of ±40 g. All the results exhibit good agreement with the theoretical results. The sensitivity of the SMRA has increased 66% compared to the previous work by using a novel optimization algorithm. This improvement is mainly attributed to both the re-designed DETF and the reduced energy loss on the micro-lever. All the above work provides a reference for the geometrical design of other MEMS sensors. As a result, compared to the studies of [2,11,25,26], after reasonable geometrical design, the SMRA reported in this paper stands out for its high-sensitivity of over 210 Hz/g, the input range of˘40 g, one-hour bias stability of 55 µg and the bias repeatability of 48 µg.

Conclusions/Outlook
This paper presents the design and experimental evaluation of an SMRA. We apply energy-consumed concept and the Nelder-Mead algorithm on this sensor to address the design issues and to increase its sensitivity. This SOI-MEMS fabricated SMRA has a closed-form sensitivity of 211.5 Hz/g, its FEM value is 203 Hz/g, and the experimental value is 254.3 Hz/g. The nonlinearity of the S g is below 100 ppm within the input range of˘40 g. All the results exhibit good agreement with the theoretical results. The sensitivity of the SMRA has increased 66% compared to the previous work by using a novel optimization algorithm. This improvement is mainly attributed to both the re-designed DETF and the reduced energy loss on the micro-lever. All the above work provides a reference for the geometrical design of other MEMS sensors.
Other key performances like bias stability, bias repeatability, and Allan variance are also shown in the paper. It should be noted that the testing results are prone to temperature shifts. Therefore, how temperature and residual stress influence the SMRA's performance remain to be elucidated, and this will be explored in the future work. A careful study on the model for the thermal stress of SMRAs is now under way.

Appendix A
By supposing an inertial load a is applied to the SMRA, the ends of the input and output beams are loaded with vertical forces F yi (for the input beam) and F yo (for the output beam), horizontal forces F xi (for the input beam) and F xo (for the output beam) and bending moments M i (for the input beam) and M o (for the output beam). The axial force and moment of each beam on the micro-lever in Figure 5b are shown in Table A1. Table A1. The axial force and moment of each beam on the micro-lever.

Beam Number
Axial Force F j Moment M j (j = 1, 2, 3, 4, 5) According to the theory of Castigliano's method [27], the displacements and rotation angles of the input and output beams can be expressed by the following equations: where I is the beam's bending moment in the x-y plane, l is the beam length and q is the beam cross-section with a subscript i, o, p, and lever representing the input beam, the output beam, the pivot beam, and the lever arm; l out and l in are the input and output arm length of the micro-lever. Applying force to the proof mass leads to the following equation: Moreover, the proof mass can be regarded as rigid, and therefore, the boundary conditions at the end of input beam can be expressed as d xi " d zi " 0 (A8) d yi "`m 1 a´4F yi˘{ 4K 1 (A9) Similarly, the boundary conditions at the end of output beam can be expressed as where K 2 is equal to the spring constant of one DETF beam k f connected in series with the spring constant of the half connecting mass k b . By solving these boundary conditions for F xi , F yi , M i , F yo , F xo and M o , the effective amplification factor can be obtained as