Amplification Factor of Two-Stage Microleverage Amplification Factor of Two-Stage Microleverage

5.2 Amplification Factor of Two-Stage Microleverage

Mechanisms

Mechanisms

The amplification factor of a two-stage microleverage mechanism is obtained by The amplification factor of a two-stage microleverage mechanism is obtained by multiplying the amplification factors of the two single-stage microlevers. If the same multiplying the amplification factors of the two single-stage microlevers. If the same assumptions are made for each stage of the two-stage microlever as in the case of a assumptions are made for each stage of the two-stage microlever as in the case of a single-stage microlever discussed in Chapter 4, the total amplification factor for a single-stage microlever discussed in Chapter 4, the total amplification factor for a two-stage microleverage mechanism can be expressed as following:

stage microleverage mechanism can be expressed as following:

( ( ))

constant of the pivot for the ii^thth microlever stage, respectively;microlever stage, respectively; k k _vvo,ivvo,i the output verticalthe output vertical (axial) spring constant of the

(axial) spring constant of the ii^thth microlever stage,microlever stage, k k _{θ θ mmo,o, ii} the output rotational springthe output rotational spring constant of the

constant of the ii^thth microlever stage,microlever stage, ll_ii the length between the pivot and output system for the length between the pivot and output system for  the

the ii^thth lever arm (positive sign for a second-kind microlever stage, and negative sign for alever arm (positive sign for a second-kind microlever stage, and negative sign for a first-kind lever stage),

first-kind lever stage), L L_ii the length between the pivot and input system of thethe length between the pivot and input system of the ii^thth lever lever  arm. The calculations of all the spring constants in equation (1) are given in the arm. The calculations of all the spring constants in equation (1) are given in the following.

following.

The axial and rotational spring constants of the pivot in each microlever stage are The axial and rotational spring constants of the pivot in each microlever stage are ((ii= 1, 2):= 1, 2):

where fiber of the beam).

fiber of the beam).

The axial/rotational spring constants at the output of the first-stage microlever can The axial/rotational spring constants at the output of the first-stage microlever can  be calcu

 be calculated by lated by treating the treating the following two following two components as components as serially-connected springs: serially-connected springs: (i)(i) the external output system and (ii) the beam connecting the external output system and  the external output system and (ii) the beam connecting the external output system and  the first-stage lever arm, i.e.,

the first-stage lever arm, i.e.,

(5.3)

where k k _vvovvo and and k k _θ θ mmooare the vertical (axial) and rotational spring constants of the externalare the vertical (axial) and rotational spring constants of the external output system for the two-stage microlever, respectively;

output system for the two-stage microlever, respectively; k k _{vv vvcc,, 11} and and k k _{θ θ mmcc,, 11} the axial and the axial and  rotational spring constants of beam C

rotational spring constants of beam C₁₁ connecting the external output system and theconnecting the external output system and the first-stage lever arm, respectively. The other connection beam C

first-stage lever arm, respectively. The other connection beam C₂₂ is between the first and is between the first and  second microlever. The spring constants of the connection beams are:

second microlever. The spring constants of the connection beams are:

(5.4)

where ll_{c,c, ii} is the pivot beam length,is the pivot beam length, t t _{c,c, ii} beam thickness,beam thickness, ww_{c,c, ii} beam width,beam width, I  I _{c,c, ii} moment of moment of  inertia about the neutral axis at mid-width.

inertia about the neutral axis at mid-width.

Similarly, the axial/rotational spring constants at the output of the second-stage Similarly, the axial/rotational spring constants at the output of the second-stage microlever can be calculated by treating its “output system” as the first-stage microlever  microlever can be calculated by treating its “output system” as the first-stage microlever  connected with the output system. It is noted that the rotation angle of the first-stage connected with the output system. It is noted that the rotation angle of the first-stage microlever caused by the vertical output force of the second-stage microlever is microlever caused by the vertical output force of the second-stage microlever is negligible as compared with the rotational angle of the second-stage lever arm. Then, an negligible as compared with the rotational angle of the second-stage lever arm. Then, an equation similar to equation (5.3) can be written for the second-stage microlever:

equation similar to equation (5.3) can be written for the second-stage microlever:

(5.5)

stage microlever, and k k _{vv vv I I ,, 11} can be calculated as following,can be calculated as following,

(

 pivot as two as two springs connesprings connected in cted in parallel aparallel as far s far as sustaining as sustaining a pure a pure bending bending moment atmoment at the input point of the first-stage microlever is concerned, therefore,

the input point of the first-stage microlever is concerned, therefore,

1

Substituting all the spring constants calculated from equations (5.2)-(5.7) into Substituting all the spring constants calculated from equations (5.2)-(5.7) into (5.1), the overall amplification factor can be obtained.

(5.1), the overall amplification factor can be obtained.