the Amplification Factor
4.5 Second-Order Second-Order Refined Refined Analytical Analytical Model Model
4.5 Second-Order Second-Order Refined Refined Analytical Analytical Model Model
The above analysis is based on five assumptions (Section 4.2.1). The fifth The above analysis is based on five assumptions (Section 4.2.1). The fifth assumption, which ignores the horizontal forces, could cause some discrepancy between assumption, which ignores the horizontal forces, could cause some discrepancy between the analytical results and those calculated from FEM methods. A refined second-order the analytical results and those calculated from FEM methods. A refined second-order analytical model is developed to take into account the horizontal displacement. The analytical model is developed to take into account the horizontal displacement. The free- body
body diagram odiagram of a f a 22nd nd -kind microleverage mechanism is shown in Fig. 4.5. The rotation-kind microleverage mechanism is shown in Fig. 4.5. The rotation angle at the mobile end of the pivot beam,
angle at the mobile end of the pivot beam, θ θ , is equal to the rotation angle of the rigid , is equal to the rotation angle of the rigid lever arm and can be separated into two components:
lever arm and can be separated into two components: θ θ mm pp (at pivot) caused by the(at pivot) caused by the bending moment and
bending moment and θ θ hh pp (at pivot) caused (at pivot) caused by the horizontal force. Similarly, the rotation by the horizontal force. Similarly, the rotation angle at the joint between the output connection beam and the lever arm,
angle at the joint between the output connection beam and the lever arm,θ θ
,,
which is alsowhich is also equal to the rotation angle of the rigid lever arm, can also be separated into two equal to the rotation angle of the rigid lever arm, can also be separated into two components: the rotation anglecomponents: the rotation angle θ θ mm oo(at output) caused by the bending moment and (at output) caused by the bending moment and θ θ hh oo (at output) caused
(at output) caused by the horizontal force. by the horizontal force. We haveWe have
ho ho mo
mo θ θ θ
θ θ
θ
== ++
(4.13)(4.13)hp hp mp
mp θ θ θ
θ θ
θ
== ++
(4.14)(4.14)Equilibrium of horizontal forces on the leve
Equilibrium of horizontal forces on the lever arm gives:r arm gives:
hp hp hp hp ho
ho ho
ho k k
k
k θ θ θ θ
==
-- θ θ θ θ (4.15)(4.15)where
where k k θ θ hh oo and and k k θ θ hh pp are the rotational spring constants at output and pivot, respectively,are the rotational spring constants at output and pivot, respectively, when loaded with a horizontal force.
when loaded with a horizontal force.
Since the lever arm remains rigid during loading and the vertical output force Since the lever arm remains rigid during loading and the vertical output force does not cause any horizontal displacement or rotation of the output system (the fourth does not cause any horizontal displacement or rotation of the output system (the fourth assumption given before), we can have the following equation for the geometrical assumption given before), we can have the following equation for the geometrical requirement of equal horizontal displacement at the two joints: (i) between the pivot and requirement of equal horizontal displacement at the two joints: (i) between the pivot and lever arm and (ii) between output connection beam and the lever arm:
lever arm and (ii) between output connection beam and the lever arm:
hhp respectively, when loaded with a bending moment;
respectively, when loaded with a bending moment; k k hh mm oo and and k k h mhm pp are the horizontalare the horizontal spring constants at output and pivot, respectively, when loaded with a bending moment;
spring constants at output and pivot, respectively, when loaded with a bending moment;
k
k hh hh oo and and k k hh hh pp are the horizontal spring constants at output and pivot, respectively, whenare the horizontal spring constants at output and pivot, respectively, when loaded with a horizontal force.
loaded with a horizontal force.
Replacing the total rotation angle
Replacing the total rotation angle θ θ in equation (4.2) with the rotation angle duein equation (4.2) with the rotation angle due to the bending moment,
to the bending moment, θ θ mm oo and and θ θ mm pp, we have the following equation for the moment, we have the following equation for the moment equilibrium with respect to the joint between the pivot beam and the lever arm:
equilibrium with respect to the joint between the pivot beam and the lever arm:
( ( ))
in
in L L k k vvovvoll ll k k mmoo momo k k mm p p mpmp
F
F
==
δ δ++
θ θ++
θ θ θ θ++
θ θ θ θ (4.17)(4.17)Solving equations (4.13)-(4.16) for
Solving equations (4.13)-(4.16) for θ θ mm ooand and θ θ mm pp, we have:, we have:
where
where f f oo and and f f p p are functions of the eight spring constants in equations (4.13)-(4.16) and are functions of the eight spring constants in equations (4.13)-(4.16) and their expressions are cumbersomely long and not given here. However, for the special their expressions are cumbersomely long and not given here. However, for the special case of a resonant accelerometer, their expressions are much simpler and given later in case of a resonant accelerometer, their expressions are much simpler and given later in equations. (4.39) and (4.40).
equations. (4.39) and (4.40).
Substituting equation (4.18) and (4.19) into equation (4.17), we can obtain:
Substituting equation (4.18) and (4.19) into equation (4.17), we can obtain:
( ( ))
Comparing equation (4.20) with (4.2), it is clear that
Comparing equation (4.20) with (4.2), it is clear thatk k θ θ mm oo and and k k θ θ mm ppin equation (4.2) arein equation (4.2) are now replaced by
now replaced by k k θ θ mm oo f f oo and and k k θ θ mm pp f f p p in equation (4.20). The latter are now referred to asin equation (4.20). The latter are now referred to as the apparent rotational spring constants, and
the apparent rotational spring constants, and f f oo and and f f p p are the correction factors for theare the correction factors for the apparent rotational spring constants. In other words, the consideration of the horizontal apparent rotational spring constants. In other words, the consideration of the horizontal forces at pivot and output is equivalent to replacing the actual rotational spring constants forces at pivot and output is equivalent to replacing the actual rotational spring constants (with respect to a bending moment) with apparent ones. Furthermore, in this refined (with respect to a bending moment) with apparent ones. Furthermore, in this refined analysis, equation (4.1) from the first-order analysis still holds provided that the vertical analysis, equation (4.1) from the first-order analysis still holds provided that the vertical displacement of the output system caused by the bending moment and horizontal force displacement of the output system caused by the bending moment and horizontal force are negligible. The amplification factor and amplification coefficient can be obtained by are negligible. The amplification factor and amplification coefficient can be obtained by following the same procedure in the first-order analysis. After solving
following the same procedure in the first-order analysis. After solving δ δ and and θ θ fromfrom equations (4.1) and (4.20), one can obtain the following equations:
equations (4.1) and (4.20), one can obtain the following equations:
( ( ))
Fig. 4.5
Fig. 4.5 Free-body diagram Free-body diagram of a of a second-kind microlever second-kind microlever under loading.under loading.
((a)a)
((bb))
L L
δδ
Output Output
Input Input
θθ
Pivot Pivot
l l θθ++δδ
θθ
l l
Output Output P
Pivivotot
F
Fhphp FFhoho
Lever Arm Lever Arm F
Fvovo ==FFououtt
F Fvpvp
M
Mpp MMoo
F Finin F
Fhphp FFhoho F Fvovo F
Fvpvp
M Moo M
Mpp
Input Input
( ( ))
It is noted that the above analysis is for a second-kind microlever. For a first-kind It is noted that the above analysis is for a second-kind microlever. For a first-kind microleverage mechanism, For the single-stage microleverage mechanism design in a microleverage mechanism, For the single-stage microleverage mechanism design in a resonant accelerometer, results from both the first-order model and the refined resonant accelerometer, results from both the first-order model and the refined second-order model are plotted together with the SUGAR results, which are presented later in order model are plotted together with the SUGAR results, which are presented later in this Chapter. We will see that the refined model results are almost the same as the this Chapter. We will see that the refined model results are almost the same as the SUGAR results as shown in Section 4.9. This is a strong validation of both the SUGAR SUGAR results as shown in Section 4.9. This is a strong validation of both the SUGAR simulator and the analytical model.
simulator and the analytical model.