About Frequency Shift Principle

Transverse sensitivity

Cross-Axis Sensitivity

A measurement of what percentage of a force applied perpendicular to the sensitive axis of the device.

% of on-axis output

-3dB frequency Bandwidth Hz

Courtesy of NeoStone MicroFabrication Corporation.

The general performance requirement for an accelerometer is listed in Table 1.

1.3. About Frequency Shift Principle

The natural frequency of a mass-spring vibration system is determined by ω = (k/m)^1/2. Thus, if an external or environmental application such as force, acceleration can change the system parameters k or m [3], the natural frequency ω of the mass-spring system can be shifted responding to this external application, then if a relation between the external application and the natural frequency can be established, the external application can be measured by evaluating the shifted natural frequency. In addition, the frequency shift principle is attractive in performing precision measurement, due to its characteristics on linear response, intrinsic sensitivity and quasi-digital signal output.

Chapter 2

Design Consideration

The previous researches [3~8] show that, the important factors considered for designing structure of the microaccelerometer include response behavior, structure geometry, structural rigidity and dimension arrangement. The response behaviors mean that a linear dynamic response is preferred for its easiness to be processed, and a nature of significant variation tendency responding to the detected acceleration is the other requirement. In addition, a symmetrical geometry configuration is necessary to decouple a detected 2D acceleration into two independent acceleration components.

The structure having the rigidity and stiffness is necessary to reduce the deformation or dynamic influence due to other non-interested acceleration or gravitation. More, the dimensions of the structural components dominate the mechanical and electrical sensitivity of the device.

In this research, these factors are considered to meet the general performance requirement listed in Table 1. That is, the requirements about offset and its hysteresis, linearity error, cross-axis sensitivity and sensitivity are met by the design consideration on rigidity, linear response, decoupling capability and dimension optimization, respectively.

According to the factors, a 2D frequency shift microaccelerometer used for attitude and tilt sensing in a constant acceleration or gravitation filed is proposed, as shown in Figure 5. This accelerometer is configured with a central proof mass connected via four identical and symmetrical microbeams with high-aspect-ratio. This accelerometer is intended to operate by using electrostatic force and sense the acceleration by using natural frequency shift of the microbeam. Thus, on the two sidewalls of each

capacitors for driving and sensing the transverse vibration of the microbeam. When the microaccelerometer shown in Figure 5 is accelerated to right, the inertial effect of the proof mass M applies compression and tension respectively on the microbeams C and the D, and thereby shifts their natural frequencies. The electrode pad Gd generates a variable electrostatic force to actuate the microbeam D to vibrate at its shifted natural frequency and the vibrating frequency of the excited microbeam D is detected by the electrode pad Gs.

2.1. Consideration on Structure Rigidity

To meet the offset and offset hysteresis requirements, structural rigidity is necessary to against deformation due to gravitation, which is fulfilled by structure with high-aspect-ratio in the proposed design. The high-aspect-ratio characterization makes the microbeam itself is capable of reducing the deflection due to the connected proof mass under gravitation or acceleration perpendicular to the planar acceleration. The high-aspect-ratio characterization also provides the degree of freedom for the capacitor disposed on the sidewall of the microbeam to be expanded by employing UV-LIGA process to build up the microstructure [9, 10].

2.2. Consideration on Linear Response

The principle of the natural frequency shift of the microbeam is adopted by the microaccelerometer to measure acceleration to meet the linearity error requirement.

The merits of using natural frequency as an output signal are, on theoretically it has high sensitivity and linearly frequency shift in response to a measured physical or chemical signal, and on practically it posses a quasi-digital nature as being adopted to be an output signal, that making the resonant type sensor itself easy to be integrated into

2.3. Consideration on Structure Symmetry

To reduce the cross-axis sensitivity, the symmetric structure is chosen to decouple a planar acceleration into two independent accelerations, which induces little crosstalk.

In addition, due to its compact and simple configuration, the response of 2D

microaccelerometer in sensing acceleration can be described easily and precisely by the concise analytical model without complicated calculation.

2.4. Consideration on Sensitivity and Optimization

To obtain a most sensitive microaccelerometer structure, the relationship between the detecting sensitivity and the dimension for the proof mass and the microbeam has to be realized [11]. To investigate the variation of sensitivity related to the variation of geometry dimension, the microaccelerometers with different dimensions confined within the constraint of W=L+2l are considered. In Figure 5, L is the width of the proof mass and l represents the length of the microbeam.

A

B

C _Gs D

Gd

Anchor Frame

Sensing Electrode Pad

Driving Electrode Pad

Proof Mass

Microbeam

M

Figure 5 (a). Top view.

L h l

Figure 5 (b). Elevation view

Figure 5. The structure of the 2D HAR resonant microbeam microaccelerometer

Chapter 3 Design

An accelerometer with concise structure having resonant microbeam to measure 2D acceleration is proposed. This structure is configured with a central proof mass suspended by four symmetrical and orthogonal high-aspect-ratio (HAR) microbeams.

This dual-axis design is able to decouple a two-axis signal from a 2D acceleration. An analytical model relating the linear relationship between the acceleration and the associated resonant frequency shift of microbeam is derived, and a finite element analysis (FEA) is also performed to confirm this model. The FEA result also shows that there is little cross talk between X and Y directions measurement, meaning that this structure is able to decouple a planar 2D acceleration into two independent acceleration components, and therefore the 1D analytical model can be used to evaluate the 2D acceleration on the X-Y plane. In addition, the model is verified by the testing results of one conventional dual-axis natural frequency shifted microaccelerometer (DFSM) [5].

The simulation result also shows that the sensitivity of the proposed HAR accelerometer is triple over that of the conventional DFSM.

3.1. Literatures Review

Due to the remarkable sensitivity and linear response, many micro sensors based on frequency shift principle have been developed, including micro accelerometers [4~7, 12~16]. In which, the resonant frequency of the accelerometer structure or its substructure e.g. vibrating beam will be shifted by the variations of structure strain, stress [4~7, 12~14], or rigidity [15, 16]. However, most of the previous micro accelerometers using frequency shift principle were only capable of measuring 1D

microaccelerometer (DFSM), which could successfully decouple a two-axis signal and detect a 2D acceleration. However, the rigidity of its microbridge fabricated by the surface micromachining was too weak to resist deflection caused by the connected proof mass, and thus additional constraint bridges were required to support the proof mass, leading to a complex structure, which not only complicated the model derivation and fabrication process, but also reduced the acceleration measurement sensitivity. Besides, due to the inherent limitation resulting from the surface micromachining, the available area underneath the microbridge for forming capacitors to respectively drive and detect vibration was difficult to enlarge.

Here a HAR (high-aspect-ratio) resonant microbeam accelerometer to detect a planar 2D acceleration is proposed, which can be fabricated by various novel micromachining techniques, such as DRIE [17], LIGA [18] and LIGA-like process [9, 11, 19]. In contrast to the conventional DFSM design [5], the proposed accelerometer is configured with a suspending proof mass supported by just four identical, orthogonal, and HAR microbeams, requiring no constraint bridges, since the HAR feature can enhance rigidity along the gravitational direction to support the proof mass without sacrificing the structural compliance in plane. In addition, comparing to the limited capacitor area formed underneath the supporting microbridges in conventional DFSM, the capacitors in the proposed design are placed at two sides of microbeams, then the area of the capacitors can be enlarged by HAR structure. Also, a thicker and heavier proof mass can be obtained without using the complex leverage mechanism [4, 12, 13]

to increase sensitivity.

An analytical model is first derived here to relate the natural frequency shift of microbeam and the applied acceleration by following the classic mechanics and vibration theory [20, 21]. Finite Element Analysis (FEA) and experimental data from

sensitivity improvement of the proposed design.

3.2. Structure Design and Operation Principle

The structure of the proposed high-aspect-ratio dual-axis frequency shift microaccelerometer is shown in Figure 5 Fig. 1. This microaccelerometer is operated using electrostatic force, and two respective electrode pads Gd and Gs adjacent to the two side walls of each microbeam are placed to form two capacitors to drive and sense the transverse vibration of the microbeams A, B, C and D. To distinguish the term of

“microbridge” in conventional DFSM design [5], “microbeam” is adopted here to describe the HAR beam in current design with larger thickness.

Figure 6 shows the microaccelerometer under acceleration along Y direction, the inertial effect of the proof mass M applies a tensile and a compression on the microbeams A and the B, respectively, which will shift the natural frequencies of this two microbeams, as shown in Figure 7. To detect such Y-directional acceleration, the driving electrode pads Gd shown in Figure 5 generate the variable electrostatic forces to respectively actuate the microbeams A and B to vibrate at their natural frequencies. On the other side, the sensing electrode pads Gs respectively sense the natural frequency variations of the excited microbeams A and B. Thus, if a relationship between the acceleration and the shifted natural frequency of the microbeam subject to the axial load is realized, the Y-directional acceleration can be evaluated by the natural frequency shift.

Similarly, the X-directional acceleration can also be evaluated by the same way using microbeams C and D.

Figure 5. The proposed HAR 2D resonant microbeam accelerometer. (Repeat)

Figure 6. The exggeratedly deformation tendency of the microbeams due to a Y-directional acceleration.

C Figure 8. The boundary of microbeam C comprising a fixed

clamped end and a movable clamped end [21].

δ y

y

x axial load

axial load

Figure 7. The primary vibration mode of the clamped-clamped microbeam B subject to an axial load.

3.3. Analytical Model Derivation

Here a model based on Euler-Bernoulli beam theory is derived to estimate the natural frequency shift of the microbeam. Comparing to the proof mass, the inertial effect of the microbeam is negligible, and thus is ignored in this derivation.

3.3.1. Analytical Model of the Microbeam

To estimate the frequency shift of the microbeams A, B, C, and D, as shown in Figures 5 and 7, the continuous system theory in [20] depicting the transverse vibration of elastic beam subject to an axial force T, with a clamped-clamped boundary condition, are used.

The governing equation is

₂ 0

The clamped-clamped boundary condition is expressed as

( ) ( )

To evaluate the fundamental natural frequency of the microbeam, the assumed mode method is used to solve Eq. (1). The 1^st mode shape function of the microbeam is derived and shown as

φ₁ =C₁cosh

( )

a_nx +C₂sinh

( )

a_nx +C₃cos

( )

b_nx +C₄sin

(

b_nx

)

(2)

By substituting the clamped-clamped boundary condition into Eq. (2) to solve the undetermined constant coefficients C1, C2, C3, and C4, the characteristic equation can be obtained as

(

an^{2 −}bn²

)

^sinh

^{( ) ( )}

an ^sin bn ⁺²anbn^[1−cosh

^{( ) ( )}

an ^cosbn ^]=0 (5)

From Eq. (5), the shifted natural frequency of microbeam subject to an axial load T can be calculated. However, when the axial load on the microbeam is zero, the fundamental natural frequency of the microbeam becomes

₁ 22.4 ₄ ml

= EI

ω (6)

3.3.2. Load Analysis of Microbeams

The boundary constraints of each microbeam A, B, C or D, can be considered as one end fixed on the anchor frame, and the other one is clamped on the movable proof mass accelerated along the Y-direction, as shown in Figure 8 Fig. 4, by taking microbeam C as an example. It can be seen that the deflection on the movable clamped end of microbeam C is caused by a concentrated force P and a moment Mo at the end due to the proof mass under acceleration. Thus, the resultant deflection of the microbeam C due to the assumed concentrated force P together with the moment

EI Pl 12

= 3

δ (7)

Based on the structural continuity, the bending deflections of the microbeams C and D, as shown in Figure 8, must equal to the elongation of the microbeam A and the shrinkage of the microbeam B, respectively, due to the axial tension TA and the compression TB. Therefore, ^B mass M under acceleration a, must satisfy following relation

B microbeam in terms of acceleration can be obtained.

B

When the axial loads in Eq. (11) due to 1D acceleration is substituted into Eqs. (3), (4), and (5), the corresponding natural frequency shift for the stressed microbeams can be calculated. Thereby, the relationship between acceleration and shifted natural

frequency is established.

Ma P

P

T

^B

T

^A

Figure 9. Fig. 5. Free body diagram of the proof mass M under acceleration a, subject to axial forces TA and TB from the microbeams A and B, and two

reactions P, respectively, from the beams C and D.

3.3.3. Buckling Analysis for Proof Mass

When a microbeam is buckled due to the axial compression, its vibration frequency is zero, from Eqs. (3)-(5). Thus, a buckling analysis is necessary to determine the maximum measurable acceleration for the compressed clamped-clamped microbeam.

The allowable maximum axial compression occurred on microbeam B, as shown in Figure 10, has to be less than the ultimate buckling load [22], i.e.

max

To determine the proof mass dimension, the bulking load limitation of Eqs. (11) and (12) are used. It is found that

_{max max} 2 a

T_B ≅ M (13)

Thus, from Eqs. (12) and (13), the width L of proof mass M can be determined by specifying the length l of the microbeam and the maximum detectable acceleration amax.

T

B

B

T

B

Figure 10. Buckling of the clamped-clamped microbeam B subject to the axial load TB under the maximum acceleration. ^B

3.3.4. Calculation of 2D Acceleration

If the proposed structure design can decouple the signals from two axes coupled 2D acceleration, the 1D analytical model can be used to estimate a planar 2D acceleration shown in Figure 11. Use Eqs. (11), (3), (4), and (5) to calculate the shifted natural frequencies for the microbeams A, B, C, and D under axial loads, the X and Y acceleration components ax and ay can be evaluated respectively, and then the vector sum of these two components represents the planar acceleration a.

M A

B

C D

X Y

a (acceleration)

Figure 11. The exaggeratedly deformation tendency of the

3.4. Analytical Simulation

Figure 12. The characteristic dimension of the accelerometer; the thickness of the proof mass and that of the microbeam are identical, indicated by h as

shown in Figure 5(b).

w l l

L

The material properties of nickel and the geometry parameters of the microaccelerometer listed in Table 2 are used for analytical and FEA simulation.

Figure 12 shows the characteristic dimension L is the width of the square proof mass, while l and w are the length and the width of the microbeam, respectively. The analytical simulation results are shown in Figure 13, and the symbols Ta, Ca, Da, Tf, Cf and Df are explained in Table 3, in which, the capital letter T and C respectively represent the frequency shift due to tension and compression, and D is the frequency difference between these two frequency shifts, while the index a and f represent the result from analytical simulation and finite element analysis, respectively. The conditions for analytical or FEA simulations are listed Table 4.

Table 2: Table 1: Geometry parameters and mechanical properties in simulations Material Ni Dimension and Property

Length of the microbeam l (µm) 1000 Width of the microBeam w (µm) 5 Thickness of the Beam and the Proof Mass h (µm) 20

1^st mode natural frequency (KHz) of the clamped-clamped microbeam

fo 25.107

The assigned Maximum Detectable Acceleration (G) 10

Young’s Modules E (Gpa) 210

Mass Density ρ(Kg/m³) 8800

Mass of the Proof Mass M (Kg) 3.521×10⁻⁵

Width of the Proof Mass L (µm) 14144.9

Figure 13. The frequency shift of the microbeam of the proposed accelerometer at

Table 3: Symbol used in the simulation results

Figure 14. The frequency shift difference between the analytical model and FEA at F1-8 and F2-8 conditions.

Simulation Object: Symbol

for Analytical Simulation

Symbols for FEA Simulation Natural Frequency Shift

Due to Tension Ta Tf

Natural Frequency Shift

Due to Compression Ca Cf

Difference of Two above Natural Frequencies Shifts

Da = Ta-Ca Df = Tf-Cf

Table 4: Analytical and FEA simulation conditions

Y direction (G) Simulation Direction

Acc. Dim.

X Direction (G)

Conditions Start End Inc

F1-8 1D Acceleration 0 0 8 1

F2-8 2D Acceleration 8 0 8 1

Figure 13 shows that the natural frequency shifts of microbeams under tension and compression are all quite linear with respect to the applied accelerations below 8 G, and the detecting sensitivity of Da is about 2,400 Hz/G. By buckling analysis, the microbeam subject to axial compression buckles at 10 G, and the corresponding resonant frequency becomes zero, thereby the difference of the frequency shift increases abruptly at 10 G.

3.5. Finite Element Analysis

To realize the dynamic effect of the suspended proof mass, the inertia effect of the microbeam, and the structural cross talk, which are ignored in the analytical model, the FEA software ANSYS® is employed to verify the analytical results. FEA elements shell 63 and beam 4 are respectively adopted to depict the behaviors of the proof mass in acceleration detecting and the microbeams in natural frequency shifting, respectively.

The imposed boundary condition on the position of the microbeam connected to the anchor frame is clamped and stationary. Whereas, the boundary condition on the other end of the microbeam connected to the proof mass is a movable clamped end, and only translations along X, Y and Z directions are allowed. Through static and prestressed modal analysis of FEA, the natural frequency shift of the microbeam responding to the acceleration is obtained.

In Table 4, the condition F1-8 means an external 1D acceleration from 0 to 8 G in Y direction with an increment of 1 G in simulations, and the case F2-8 means an external 2D acceleration comprised by a Y-directional acceleration changing from 0 G to 8 G by an increment of 1 G in Y-direction and a constant acceleration of 8 G in X direction. To verify the capability of the structure to decouple a planar 2D acceleration into two independent components ax and ay, and validate the 1D analytical model to

performed for cases F1-8 and F2-8.

The deviation of simulation results between analytical model and FEA are shown in Figure 14. The solid line Ta-Tf (F1-8) in Figure 14 is less than 1.1 Hz indicating that the mass of microbeam ignored in analytical model causes little deviation to FEA result, on the other side, the dash-dot line Ta-Tf (F2-8) is less than 1 Hz, means that there is little deviation when using the 1D derived model to do a 2D simulation, lines Ca-Cf (F1-8) and Ca-Cf (F2-8) also have the same results. In addition, the maximum shrinkage of microbeam with length of 1000 μm before buckling is less than 0.007 μm, as shown in Figure 14. It indicates that not only the cross talk of the proposed microaccelerometer is very small and negligible, but also the assumption of clamped-clamped boundary condition in the analytical model has very little influences on the simulation result. The result verifies that the derived 1D analytical model is able to estimate two independent acceleration components of a 2D acceleration, decoupled by the proposed structure.

3.6. Performance Evaluation

The considered design factors i.e. linearity, rigidity, decoupling capability and sensitivity, which are achieved by frequency shift principle, high-aspect ratio structure, structure symmetry and dimension optimization. The results below show the coincidence between of the considered factors and the performance of the proposed 2D mcroaccelerometer structure.

3.6.1. Linearity Evaluation

The linearity error of the frequency output shown in Figure 15 is less than 4% for the natural frequency shifted due to tension or compression only, and is less than 7% for

Linearity Error

0 1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7

Acceleration (G)

Linearity Error (%)

Figure 15. Linearity error of the shifted natural frequency.

3.6.2. Deviation Evaluation of 1D analytical model to FEA

The simulation results in Figure 16 shows that, the absolute deviation between the derived model and FEA is less than 0.6 Hz, and the corresponding relative one is less

The simulation results in Figure 16 shows that, the absolute deviation between the derived model and FEA is less than 0.6 Hz, and the corresponding relative one is less

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