5.4 Different Configurations of Two-stage Leverage
Mechanisms Mechanisms
As classified in section 5.2, there are sixteen different configurations of two-stage As classified in section 5.2, there are sixteen different configurations of two-stage leverage mechanisms for force amplification and sixteen different configurations for leverage mechanisms for force amplification and sixteen different configurations for displacement amplification. They are distinguished by different kinds of lever displacement amplification. They are distinguished by different kinds of lever combinations (the first- and second-kind for force amplification, the first- and third-kind combinations (the first- and second-kind for force amplification, the first- and third-kind for displacement amplification) and different sides of the pivot and output locations (“
for displacement amplification) and different sides of the pivot and output locations (“S S ””
for same and “
for same and “ D D” for different).” for different).
For the first- and second- kind of levers there is a sign difference (“+” or “-”) in For the first- and second- kind of levers there is a sign difference (“+” or “-”) in the
the analytical expressions of the analytical expressions of the amplification factor. However, amplification factor. However, for the subgroup of for the subgroup of pivotpivot and output being on same and different sides of the lever arm, the first-order analytical and output being on same and different sides of the lever arm, the first-order analytical amplification factor can not be distinguished. That is because the analysis is based on the amplification factor can not be distinguished. That is because the analysis is based on the assumptions given before without taking into account the strain energy consumed on the assumptions given before without taking into account the strain energy consumed on the lever arm and the internal horizontal forces acting on the pivot and connection beams.
lever arm and the internal horizontal forces acting on the pivot and connection beams.
Table 1 lists all sixteen configurations for a two-stage leverage mechanism for Table 1 lists all sixteen configurations for a two-stage leverage mechanism for force amplification and their deflections after loading. Also listed are the rotation angles force amplification and their deflections after loading. Also listed are the rotation angles of the two lever arms and the amplification factor of the mechanism. The starting (i.e., of the two lever arms and the amplification factor of the mechanism. The starting (i.e.,
before
before loading) loading) positions positions of of the the lever lever arm arm and and flexure flexure beams beams are are shaded shaded and and the the after- after-loading positions shown with solid lines (exaggerated for illustration purposes). The third loading positions shown with solid lines (exaggerated for illustration purposes). The third and fourth columns of Table 1 list the rotation angles of lever arms and the and fourth columns of Table 1 list the rotation angles of lever arms and the force-amplification factors of first- and second-stage, respectively. The pivot and connection amplification factors of first- and second-stage, respectively. The pivot and connection beam
beam are are sometimes deflected sometimes deflected into a into a twisted curve twisted curve when when the the output output and and the pthe pivot are ivot are onon different sides of the lever arm, contributing to the lower amplification factor than the “ different sides of the lever arm, contributing to the lower amplification factor than the “S S
--” group. The dimensions used for SUGAR simulation are as follows, with a double-ended
” group. The dimensions used for SUGAR simulation are as follows, with a double-ended tuning fork as the output:
tuning fork as the output:
lever arm distance between input and pivot
lever arm distance between input and pivot L L11 == L L22 = 200= 200
μμ
m, between pivot and m, between pivot and tuning forktuning fork ll11 ==ll22 ==
±±
1010μμ
m (negative sign for first-kind microlevers),m (negative sign for first-kind microlevers), width of lever 1 and lever 2 pivot beam, and connection beam between two width of lever 1 and lever 2 pivot beam, and connection beam between twomicrolever stages
microlever stagesww p p22 ==ww p p22 ==wwc2c2= 2= 2
μμ
m,m, length of the tuning fork beamlength of the tuning fork beam ll f f = 100= 100
μμ
m, microlever 1 pivotm, microlever 1 pivot ll p p11 = 6 and T-F= 6 and T-F connection beamsconnection beamsllcc11 = 6= 6
μμ
m,m,length of microlever 2 pivot, and connection beam between two microlever stages length of microlever 2 pivot, and connection beam between two microlever stages
ll p2 p2 ==llcc22 = 60= 60
μμ
m.m.Figure 5.4 shows the comparison of the amplification factor of two-stage Figure 5.4 shows the comparison of the amplification factor of two-stage leverage mechanisms of different configurations. The two-stage levers with the pivot and leverage mechanisms of different configurations. The two-stage levers with the pivot and output on the same side of the lever arm (e.g., 2S-1S, 2S-2S, 1S-1S and 1S-2S output on the same side of the lever arm (e.g., 2S-1S, 2S-2S, 1S-1S and 1S-2S configurations) have much higher amplification factor than those with the pivot and configurations) have much higher amplification factor than those with the pivot and output
output
2 2
L
LEVER ARMEVER ARM ROTATION, ROTATION,
1S-1D 1S-1D
T
TWO-STAGE LEVER WO-STAGE LEVER CONFIGURATION
CONFIGURATION EACH STAGEEACH STAGE
Table 5.1. Beam Beam Deflections and Amplification Factor Deflections and Amplification Factor of Two-stage of Two-stage MicroleversMicrolevers
1S-2D
AMPLIFICATION FACTOR MPLIFICATION FACTOR TOTAL
10 10
L
LEVER ARMEVER ARM ROTATION, ROTATION,
2S-1D 2S-1D
T
TWO-STAGE LEVER WO-STAGE LEVER CONFIGURATION
Table 5.1. Beam Beam Deflections and ADeflections and Amplificatiomplification Fn Factor of actor of Two-stage Microlevers Two-stage Microlevers (continu(continued)ed)
2S-2D
AMPLIFICATION MPLIFICATION FACTOR FACTOR TOTAL TOTAL
0 0 20 20 40 40 60 60 80 80
A A m m p p l l i i f f i i c c a a t t i i o o n n F F a a c c t t o o r r
2
2DD--11DD 22DD--22DD 22DD--11SS 22DD--22SS 22SS--11DD 22SS--22DD 22SS--11SS 22SS--22SS Ty
Typpes es of of TwTwo-so-sttagage e MiMicrcrololeveverer A
A11 AA22 TToottaallAA
0 0 20 20 40 40 60 60 80 80
A A m m p p l l i i f f i i c c a a t t i i o o n n F F a a c c t t o o r r
1
1DD--11DD 11DD--22DD 11DD--11SS 11DD--22SS 11SS--11DD 11SS--22DD 11SS--11SS 11SS--22SS Ty
Typepes os of f TwTwo-o-ststagage e MiMicrcrololeveverer A
A11 AA22 TToottaallAA
Fig. 5.4
Fig. 5.4 Comparison of Comparison of the amplithe amplification ffication factors of actors of two-stage mtwo-stage microlevers icrolevers of diffof differenterent configurations.
configurations.
on the different sides of the lever arm. The increased resistance of a microlever to on the different sides of the lever arm. The increased resistance of a microlever to rotational and axial displacement when the output and pivot are on the different sides of a rotational and axial displacement when the output and pivot are on the different sides of a lever arm leads to lower force-amplification factor. Among the sixteen configurations, lever arm leads to lower force-amplification factor. Among the sixteen configurations, the second-stage pivots in 1D-1S, 1S-1S, 2D-1S and 2S-1S are inside the enclosure the second-stage pivots in 1D-1S, 1S-1S, 2D-1S and 2S-1S are inside the enclosure formed by the connection beam, first-stage and second-stage lever arm. Therefore, the formed by the connection beam, first-stage and second-stage lever arm. Therefore, the pivot
pivot can can not not be be longer longer than than the the connection connection beam. beam. Such Such a a constraint constraint limits limits thethe amplification factor and so those four co
amplification factor and so those four configurations are not commonly used.nfigurations are not commonly used.
In summary, the optimum design of two-stage microlever should be of In summary, the optimum design of two-stage microlever should be of 1S/2S-1S/2S types of combinations, i.e., the pivot and connection beam of both the first- and 1S/2S types of combinations, i.e., the pivot and connection beam of both the first- and second-stage microlever are
second-stage microlever are on the same on the same side of tside of the lever arm. he lever arm. In addition, stifIn addition, stiffnessfness (axial spring constant) of the first-stage microlever (connected to output system) should (axial spring constant) of the first-stage microlever (connected to output system) should be significantly
be significantly greater than greater than that of that of the second-stage. the second-stage. A wider A wider gap bgap between the etween the first- and first- and second-stage microlever would allow the use of very long pivot and connection beams second-stage microlever would allow the use of very long pivot and connection beams for the second-stage microlever, which can reduce its axial spring constant and increase for the second-stage microlever, which can reduce its axial spring constant and increase the force-amplification factor.
the force-amplification factor.