Optimization in the Resonant Acceleromete Accelerometerr

The previous chapter presents the design and optimization of the single-stage The previous chapter presents the design and optimization of the single-stage microleverage mechanism in the resonant accelerometer. This section will present the microleverage mechanism in the resonant accelerometer. This section will present the

design and optimization of two-stage microleverage mechanism in the resonant design and optimization of two-stage microleverage mechanism in the resonant accelerometer [Su and Yang, 2000 (2)].

accelerometer [Su and Yang, 2000 (2)].

Among the sixteen configurations of the two-stage leverage mechanism for force Among the sixteen configurations of the two-stage leverage mechanism for force amplification, four of them (1D-1S, 1S-1S, 2D-1S and 2S-1S) have inside second-stage amplification, four of them (1D-1S, 1S-1S, 2D-1S and 2S-1S) have inside second-stage pivots which limit the amplification factor.

pivots which limit the amplification factor. Among the other twelve configurations of theAmong the other twelve configurations of the two-stage leverage mechanism, six configurations are feasible for the resonant two-stage leverage mechanism, six configurations are feasible for the resonant accelerometer application as shown schematically in Figs. 5.5 (a) to (f). The area between accelerometer application as shown schematically in Figs. 5.5 (a) to (f). The area between the two lever stages is

the two lever stages is used as a portion of used as a portion of the proof-mass. the proof-mass. Three configurations (1S-2S,Three configurations (1S-2S, 2S-2D and 2S-2S) have inside anchors.

2S-2D and 2S-2S) have inside anchors.

For the optimization of the two-stage microleverage mechanism, the amplification For the optimization of the two-stage microleverage mechanism, the amplification factors of a 1S-1S type are calculated with prescribed dimensions of pivot, tuning fork factors of a 1S-1S type are calculated with prescribed dimensions of pivot, tuning fork beams, connec

beams, connection beams, tion beams, and lever and lever ratio. ratio. For a For a symmetric two-stage lever symmetric two-stage lever design and design and aa constant beam thickness,

constant beam thickness, t t , of 50, of 50

μμ

m, substituting the following values into equationsm, substituting the following values into equations (5.2) – (5.7) for calculating spring constants and finally into equations (5.1) for (5.2) – (5.7) for calculating spring constants and finally into equations (5.1) for calculating the amplification factor:

calculating the amplification factor:

modulus of elasticity E = 1.65 x 10

modulus of elasticity E = 1.65 x 10¹¹¹¹N/mN/m²²,, input force,

input force,F F _in_in, = 1.5 x 10, = 1.5 x 10^-9^-9 N,N,

lever arm distance between input and pivot

lever arm distance between input and pivot L L₁₁== L L₂₂= 210= 210

μμ

m,m, lever arm distance between pivot and tuning fork

lever arm distance between pivot and tuning fork ll₁₁ ==ll₂₂ = 10= 10

μμ

m,m, tuning fork beam length

tuning fork beam lengthll_f_f= 100= 100

μμ

m,m,

Fig. 5.5 5.5 (a) (a) A A 1S-1D 1S-1D type type two–stage two–stage microleverage microleverage mechanism.mechanism.

Fig. 5.5 5.5 (b) (b) A A 1S-2D 1S-2D type type two-stage two-stage microleverage microleverage mechanism.mechanism.

Fig. 5.5 5.5 (c) (c) A A 1S-2S 1S-2S type type two-stage two-stage microleverage microleverage mechanism.mechanism.

Fig 5.5 5.5 (d) (d) A A 2S-1D 2S-1D type type two-stage two-stage microleverage microleverage mechanismmechanism

Fig 5.5 5.5 (e) (e) A A 2S-2D 2S-2D type type two-stage two-stage microleverage microleverage mechanism.mechanism.

Fig. 5.5 5.5 (f) (f) A A 2S-2S 2S-2S type type two-stage two-stage microleverage microleverage mechanism.mechanism.

width of lever 1 pivot, T-F beams, and T-F connection beams

width of lever 1 pivot, T-F beams, and T-F connection beamsww_p_p1₁ == ww_f_f== ww_cc1₁ = 4= 4

μμ

m, or as variables,m, or as variables,

width of lever 2 pivot, and connection beam between lever 1 and 2

width of lever 2 pivot, and connection beam between lever 1 and 2ww_p_p2₂ ==ww_cc2₂ = 2= 2

μμ

m, or as variables,m, or as variables,

length of lever 1 pivot and T-F connection beams

length of lever 1 pivot and T-F connection beams ll_p_p1₁ = 6 and = 6 and ll_cc1₁ = 2= 244

μμ

m, or asm, or as variables,

variables,

length of lever 2 pivot, and connection beam between lever 1 and 2

length of lever 2 pivot, and connection beam between lever 1 and 2ll_p_p2₂ ==ll_cc2₂ = 60= 60

μμ

m, or as variables,m, or as variables,

the amplification factor of individual lever,

the amplification factor of individual lever, A A₁₁ and and A A₂₂, and total amplification factor,, and total amplification factor, A A,, can be calculated and the results are shown in Figs. 5.6 – 5.11. The same dimensions are can be calculated and the results are shown in Figs. 5.6 – 5.11. The same dimensions are used in SUGAR simulation for comparison with the analytical results in Fig. 5.9.

used in SUGAR simulation for comparison with the analytical results in Fig. 5.9.

The amplification factors of the individual levers,

The amplification factors of the individual levers, A A₁₁ and and A A₂₂, and the total, and the total amplification factor,

amplification factor, A A, as a function of beam width of tuning fork, connection beam and , as a function of beam width of tuning fork, connection beam and pivot of

pivot of lever 1 lever 1 (the width of (the width of all beams are all beams are assumed to be assumed to be same) are shown same) are shown in Fig. 5in Fig. 5.6(a).6(a) and 5.6(b). The width of the first-stage lever pivot, T-F beam and connection beam are and 5.6(b). The width of the first-stage lever pivot, T-F beam and connection beam are chosen to be 4

chosen to be 4

μμ

m to increase the stiffness of the first stage lever to ensure that lever 2m to increase the stiffness of the first stage lever to ensure that lever 2 will have a good amplification factor. For the same reason, the lengths of lever 1 pivot will have a good amplification factor. For the same reason, the lengths of lever 1 pivot and T-F connection beams are 6

and T-F connection beams are 6

μμ

m, which is relatively short compared to those of them, which is relatively short compared to those of the second stage lever.

second stage lever. As the beam width As the beam width in lever 1 increases, in lever 1 increases, the amplification factor the amplification factor of of lever 1,

lever 1, AA₁₁, decreases while the amplification factor of lever 2,, decreases while the amplification factor of lever 2, A A₂₂, increases. An increase, increases. An increase

in the amplification factor of lever 1 results from an increase in the axial output spring in the amplification factor of lever 1 results from an increase in the axial output spring constant of lever 1 when the tuning fork, connection beam and pivot beam width in lever constant of lever 1 when the tuning fork, connection beam and pivot beam width in lever 1 increases, as shown in Fig. 5.6(b). A maximum amplification factor is obtained at an 1 increases, as shown in Fig. 5.6(b). A maximum amplification factor is obtained at an intermediate beam width of ~4

intermediate beam width of ~4

μμ

m.m.

The separate effects of the width of three flexure beams in lever 1: (i) tuning fork The separate effects of the width of three flexure beams in lever 1: (i) tuning fork beam, (ii)

beam, (ii) connection connection beam, and beam, and (iii) pivot (iii) pivot beam obeam on amplification n amplification factors of factors of both leveboth lever 1r 1 and 2 and the total amplification factor are shown in Fig. 5.7(a)-(c), respectively. The and 2 and the total amplification factor are shown in Fig. 5.7(a)-(c), respectively. The continuous increase in the amplification factors of both lever 1 and 2 and the total continuous increase in the amplification factors of both lever 1 and 2 and the total amplification factor with increasing width of the lever 1 T-F beam, shown in Fig. 5.7(a), amplification factor with increasing width of the lever 1 T-F beam, shown in Fig. 5.7(a), is expected because (i) the T-F beam consumes no bending energy; (ii) wider T-F beam is expected because (i) the T-F beam consumes no bending energy; (ii) wider T-F beam means greater vertical output spring constant for lever 1; (iii) wider T-F beam also results means greater vertical output spring constant for lever 1; (iii) wider T-F beam also results in a greater vertical output spring constant for lever 2. However, the greater width of the in a greater vertical output spring constant for lever 2. However, the greater width of the tuning fork may adversely affect the

tuning fork may adversely affect the measurement sensitivity.measurement sensitivity.

The amplification factor of lever 1 and the total amplification factor are almost The amplification factor of lever 1 and the total amplification factor are almost constant for a T-F connection beam width of less than 4

constant for a T-F connection beam width of less than 4

μμ

m, but then decrease rapidlym, but then decrease rapidly with wider T-F connection beams, as shown by Fig. 5.7(b). Similar effects on the with wider T-F connection beams, as shown by Fig. 5.7(b). Similar effects on the amplification factor are seen in Fig. 5.7(c) for pivot beam width. The decrease in amplification factor are seen in Fig. 5.7(c) for pivot beam width. The decrease in amplification factor of lever 1 at wider widths of the T-F connection beam or the pivot amplification factor of lever 1 at wider widths of the T-F connection beam or the pivot beam

beam is dis due ue to to increased increased moment bemoment bending nding energy energy consumed consumed at at these these two betwo beams as ams as their their width increases. Although an increase in the width of the T-F connection beam and the width increases. Although an increase in the width of the T-F connection beam and the pivot

pivot beam of beam of lever lever 1 ca1 causes an uses an increase in increase in the vertical the vertical output spring output spring constant of constant of lever 2lever 2 and, consequently, an increase in the amplification factor of lever 2, the decrease in the and, consequently, an increase in the amplification factor of lever 2, the decrease in the

amplification factor of lever 1 overrides the increase in the amplification factor of lever amplification factor of lever 1 overrides the increase in the amplification factor of lever 2. This leads to a decrease in the total amplification factor of the compound lever as the 2. This leads to a decrease in the total amplification factor of the compound lever as the width of the T-F connection beam and the pivot beam of lever 1 increases.

width of the T-F connection beam and the pivot beam of lever 1 increases.

The effects of the lengths of T-F, connection beam and lever 1 pivot, on the The effects of the lengths of T-F, connection beam and lever 1 pivot, on the amplification factor are shown in Fig. 5.8 (a), (b) and (c) respectively. Comparing Fig.

amplification factor are shown in Fig. 5.8 (a), (b) and (c) respectively. Comparing Fig.

5.8 with Fig. 5.7, it is seen that the effect of flexure beam length on amplification is less 5.8 with Fig. 5.7, it is seen that the effect of flexure beam length on amplification is less significant than the effect of beam width. As the length of T-F beam increases, the significant than the effect of beam width. As the length of T-F beam increases, the amplification factors of both lever 1 and 2 decrease because of decreasing vertical output amplification factors of both lever 1 and 2 decrease because of decreasing vertical output spring constants,

spring constants, k k _vvo_vvo_{,, 1}₁ and and k k _vvo_vvo_{,, 2}₂, of both levers. Increasing the length of the T-F, of both levers. Increasing the length of the T-F connection beam causes less drop in the vertical output spring constant than increasing connection beam causes less drop in the vertical output spring constant than increasing the width. The amplification factor decreases only slightly as the T-F connection beam the width. The amplification factor decreases only slightly as the T-F connection beam length increases after an initial increase to a maximum at relatively short beam length.

length increases after an initial increase to a maximum at relatively short beam length.

Comparing Fig. 5.8(b) with 5.8(c), it is seen that the length of the lever 1 pivot beam has Comparing Fig. 5.8(b) with 5.8(c), it is seen that the length of the lever 1 pivot beam has almost the same effect on amplification factor as the length of the T-F connection beam.

almost the same effect on amplification factor as the length of the T-F connection beam.

A SUGAR simulation of the results of two different types of two-stage A SUGAR simulation of the results of two different types of two-stage microlevers, 1S-2S and 1S-2D, are presented in Fig. 5.9 for comparison with the 1

microlevers, 1S-2S and 1S-2D, are presented in Fig. 5.9 for comparison with the 1^st^st-order -order analytical results as shown in Fig. 5.7(c). The amplification factors calculated from the analytical results as shown in Fig. 5.7(c). The amplification factors calculated from the first-order analytical model are slightly higher in the case of the 1S-2S lever and much first-order analytical model are slightly higher in the case of the 1S-2S lever and much higher in the case of 1S-2D lever than the SUGAR results. The first-order analytical higher in the case of 1S-2D lever than the SUGAR results. The first-order analytical model gives reasonable predictions of the amplification factor when the pivot and output model gives reasonable predictions of the amplification factor when the pivot and output system are on the same side of the lever arm in both the first- and second-stage levers.

system are on the same side of the lever arm in both the first- and second-stage levers.

When the pivot and the output system are on different sides, different horizontal force When the pivot and the output system are on different sides, different horizontal force

generates different amplification factor. The 1

generates different amplification factor. The 1^st^st-order analytical model can not show the-order analytical model can not show the difference. However, for a two-stage leverage mechanism, the 2

difference. However, for a two-stage leverage mechanism, the 2^nd^nd-order analytical model-order analytical model is too complicated to pursue. To distinguish the subgroup of “S” (pivot and output are at is too complicated to pursue. To distinguish the subgroup of “S” (pivot and output are at the same side) and “D” (pivot and output are at different side) in the two-stage leverage the same side) and “D” (pivot and output are at different side) in the two-stage leverage mechanism, the SUGAR simulation is the main approach here since all of the beam mechanism, the SUGAR simulation is the main approach here since all of the beam deformation is in elastic regime.

deformation is in elastic regime.

The effect of the width of lever 2 flexure beams on amplification factor is shown The effect of the width of lever 2 flexure beams on amplification factor is shown in Figs. 5.10(a) and (b) for the lever 2 connection beam and pivot beam, respectively. The in Figs. 5.10(a) and (b) for the lever 2 connection beam and pivot beam, respectively. The width of the lever 2 connection beam has the same effect on the amplification factor as width of the lever 2 connection beam has the same effect on the amplification factor as the pivot beam. A comparison between Fig. 5.11 and Fig. 5.10 shows that the width of the pivot beam. A comparison between Fig. 5.11 and Fig. 5.10 shows that the width of the lever 2 flexure beams has more significant effect on total amplification factor than the lever 2 flexure beams has more significant effect on total amplification factor than that of the lever 1 flexure beams. This is a result of sharp decreases in the amplification that of the lever 1 flexure beams. This is a result of sharp decreases in the amplification factor of lever 2 at increasing width of lever 2 flexure beams.

factor of lever 2 at increasing width of lever 2 flexure beams.

Figure 5.11 shows the effect of the length of the lever 2 connection beam on the Figure 5.11 shows the effect of the length of the lever 2 connection beam on the amplification factors, which is opposite to the effect of the width of lever 2 flexure beams amplification factors, which is opposite to the effect of the width of lever 2 flexure beams shown in Fig. 5.10. While increasing the length of the lever 2 connection beam has no shown in Fig. 5.10. While increasing the length of the lever 2 connection beam has no effect on the amplification factor of lever 1, it has a pronounced effect on the effect on the amplification factor of lever 1, it has a pronounced effect on the amplification factor of lever 2 and the total amplification factor. The effect of the length amplification factor of lever 2 and the total amplification factor. The effect of the length of lever 2 pivot beam on the amplification factors has been found to be the same as that of lever 2 pivot beam on the amplification factors has been found to be the same as that of the length of the lever 2 connection beam.

of the length of the lever 2 connection beam.

All the results presented in Figs. 5.6 –5.11 agree well with the compliance All the results presented in Figs. 5.6 –5.11 agree well with the compliance analysis given in Section 5.3. In all, the stiffness (axial spring constant) of the first-stage analysis given in Section 5.3. In all, the stiffness (axial spring constant) of the first-stage

microlever (connected to output system) should be significantly greater than that of the microlever (connected to output system) should be significantly greater than that of the second-stage. A wider gap between the first- and second-stage microlever would allow second-stage. A wider gap between the first- and second-stage microlever would allow the use of very long pivot and connection beams for the second-stage microlever, which the use of very long pivot and connection beams for the second-stage microlever, which can reduce its axial spring constant and increase the force-amplification factor.

can reduce its axial spring constant and increase the force-amplification factor.

0 T-F, T-F connection beam, and pivot beam (SUGAR).

F, T-F connection beam, and pivot beam (SUGAR).

0 Lever 1 Beam Width, micron

Lever 1 Beam Width, micron

All dimension

Fig. 5.6 5.6 (b) (b) First-stage First-stage lever lever spring spring constantconstantk k and second-stage lever amplificationand second-stage lever amplification factor as a function of lever 1 flexure beam width (SUGAR).

factor as a function of lever 1 flexure beam width (SUGAR).

All dimdimensionension in microns

Width of T-F Connection Beam , micron Width of T-F Connection Beam , micron

All dimension connection beam width (SUGAR).

connection beam width (SUGAR).

Width of Pivot Beamivot Beam, micron, micron

All dimension

First-kind L eveeversrs

Fig. 5.7 (c) Amplification factors,

Fig. 5.7 (c) Amplification factors, A A₁₁,, A A₂₂ and and A A, as a function of the lever 1 pivot width., as a function of the lever 1 pivot width.

Pivot & Connectio Beam Width, micron Pivot & Connectio Beam Width, micron

All dimension

First-kind L eveeversrs

Fig.

Fig. 5.7 5.7 (d) (d) Amplification Amplification factors,factors, A A₁₁,, A A₂₂ and and A A, as a function of width of the lever 1, as a function of width of the lever 1 pivot and connection beam (SUGAR).

pivot and connection beam (SUGAR).

Lever 1 T-F Beam Length, micron Lever 1 T-F Beam Length, micron

All dimension

Lever 1 Connectonnection Bion Beam Lengtheam Length, micron, micron

All dimension

Lever 1 Pivot Beam ivot Beam Length, micronLength, micron

All dimension

Width of Pivot Beam, mivot Beam, micronicron

All dimension

Fig. 5.9 5.9 Comparison Comparison of of SUGAR SUGAR with with analytical analytical results results (SUGAR).(SUGAR).

0 Width of 2nd Connection Beam, micron

Width of 2nd Connection Beam, micron

All dimension

Fig. 5.10 5.10 (a) (a) Effect Effect of of second-stage second-stage lever lever connection widtconnection width h (SUGAR).(SUGAR).

Width of 2nd PPivot Beam, mivot Beam, micronicron

All dimension

Fig. 5.10 (b) Effect of second-stage l(b) Effect of second-stage lever pivot width on the amplification fever pivot width on the amplification factors,actors, A A₁₁,, A A₂₂ and

and A A(SUGAR).(SUGAR).

0 Lever 2 Connection Beam Length, micron

Lever 2 Connection Beam Length, micron

All dimension

Fig. 5.11 (a) (a) Effect Effect of of second-stage second-stage lever lever connection beam connection beam length length (SUGAR)(SUGAR)

Lever 2 Pver 2 Pivot Beam Lengtivot Beam Length, micronh, micron

All dimension

Fig. 5.11 (b) Effect (b) Effect of second-stage of second-stage lever pivot llever pivot length on the ength on the amplification factors,amplification factors, A A11,, A

A₂₂ and and A A(SUGAR).(SUGAR).

CHAPTER 6

In document Compliant Leverage Mechanism Design for MEMs Application (Page 119-136)