MULTI-STAGE MICROLEVERAGE MECHANISMS
By stacking multiple single-stage levers together to make a multistage microlever, By stacking multiple single-stage levers together to make a multistage microlever, higher and higher amplification factors could be achieved as more and more single-stage higher and higher amplification factors could be achieved as more and more single-stage levers are added. As presented in Chapter 5, a two-stage microleverage mechanism can levers are added. As presented in Chapter 5, a two-stage microleverage mechanism can have a much higher amplification factor than a single-stage microleverage mechanism.
have a much higher amplification factor than a single-stage microleverage mechanism.
For instance, in the resonant accelerometer, the amplification factor of the two-stage can For instance, in the resonant accelerometer, the amplification factor of the two-stage can be
be higher higher than than 80 80 while while the the amplification amplification factor factor of of a a single-stage single-stage microleveragemicroleverage mechanism is much lower for a fixed lever ratio. Questions to be asked are: (i) can mechanism is much lower for a fixed lever ratio. Questions to be asked are: (i) can multiple stages of micro-levers be stacked together to achieve any specified or desired multiple stages of micro-levers be stacked together to achieve any specified or desired amplification factor? and (ii) is there any limitation for the maximum amplification amplification factor? and (ii) is there any limitation for the maximum amplification factor? This chapter presents some answers to these questions related to multistage factor? This chapter presents some answers to these questions related to multistage microlevers. For a specified output system, the achievable amplification factor is not microlevers. For a specified output system, the achievable amplification factor is not infinite but it is determined by the stiffness of the output system and the lever ratio. The infinite but it is determined by the stiffness of the output system and the lever ratio. The calculation of the maximum achievable amplification factor for a given output system is calculation of the maximum achievable amplification factor for a given output system is presented
presented in in this this chapter. chapter. Also Also discussed discussed is is how how to to obtain obtain the the maximum maximum amplificationamplification factor by an optimum number of lever stages. The theory for multiple-stage factor by an optimum number of lever stages. The theory for multiple-stage microleverage mechanism design is applied to the resonant accelerometer. Finally, the microleverage mechanism design is applied to the resonant accelerometer. Finally, the compliance relationship in the microleverage mechanisms is discussed.
compliance relationship in the microleverage mechanisms is discussed.
6.1.
6.1. Analysis Analysis of of the the Multi-stage Multi-stage Microleverage Microleverage Mechanism Mechanism
In this section, a general method for calculating the amplification factor of In this section, a general method for calculating the amplification factor of multiple stage microleverage mechanisms is presented. The amplification factor of the multiple stage microleverage mechanisms is presented. The amplification factor of theiithth microlever stage can be expressed as a generalized equation as follows:
microlever stage can be expressed as a generalized equation as follows:
( ( ))
where k k vvo, ivvo, i is the output vertical (axial) spring constant of theis the output vertical (axial) spring constant of the iiththmicrolever stage,microlever stage,k k vvp,ivvp,i the vertical (axial) spring constant of the pivot for the output system for the
output system for the iithth lever arm,lever arm, L Lii total length of thetotal length of the iithth lever arm. While the“-“ signlever arm. While the“-“ sign in the numerator is for the first-kind levers, the
in the numerator is for the first-kind levers, the “+” sign is for the second “+” sign is for the second kind.kind.
In order to determine the amplification factor of a lever stage in a compound In order to determine the amplification factor of a lever stage in a compound lever, four spring constants in equation (6.1) need to be calculated. The axial and bending lever, four spring constants in equation (6.1) need to be calculated. The axial and bending spring constants of the pivot in each lever stage are already given in equation (4.23). The spring constants of the pivot in each lever stage are already given in equation (4.23). The calculation of the axial and bending spring constants of the output system of each lever calculation of the axial and bending spring constants of the output system of each lever stage follows the same procedure as in the case of the two-stage microlever analyzed in stage follows the same procedure as in the case of the two-stage microlever analyzed in Chapter 5.
Chapter 5.
The
The iithth microlever stage and the connecting beam Cmicrolever stage and the connecting beam Cii+1+1 make a serial connectionmake a serial connection
to become the output system for the downstream lever stage, (
to become the output system for the downstream lever stage, (i+i+1)1)thth. Therefore, the. Therefore, the output axial and rotational spring constants of the (
output axial and rotational spring constants of the (i+i+1)1)thth stage are:stage are:
1
Among the unknown variables in equation (6.2), the axial and rotational spring constants, Among the unknown variables in equation (6.2), the axial and rotational spring constants, k
k vv vvcc,, ii and and k k θ θ mmcc,, ii, of the connection beam C, of the connection beam Cii, can be obtained with equations similar to, can be obtained with equations similar to (4.23). The input rotational spring constant of the
(4.23). The input rotational spring constant of the iithth microlever stage connected together microlever stage connected together with all the upstream lever stages and the external output system is
with all the upstream lever stages and the external output system is
,, ,,
,, k k k k k
k θ θ mm I I ii
==
θ θ momo ii++
θ θ mm p p ii (6.3)(6.3)The input axial spring constant of the
The input axial spring constant of the iithth microlever stage connected together with all themicrolever stage connected together with all the upstream lever stages and the output system is
upstream lever stages and the output system is
(
Equation (4.23) is used to calculate the axial and rotational spring constants of the pivot Equation (4.23) is used to calculate the axial and rotational spring constants of the pivot beam of lever
beam of lever #1. #1. Equations (6.3) and Equations (6.3) and (6.4) are then (6.4) are then used to caused to calculate the input lculate the input axial and axial and
rotational spring constants of lever #1, while equation (4.23) is used to calculate the axial rotational spring constants of lever #1, while equation (4.23) is used to calculate the axial and rotational spring constants of the connection beam between microlever stages 1 and and rotational spring constants of the connection beam between microlever stages 1 and 2. Equation (6.2) can now be used to calculate the output axial and bending spring factor of each microlever stage can be calculated. The total amplification factor is
factor of each microlever stage can be calculated. The total amplification factor is
( ( ))
constant of each lever stage in aconstant of each lever stage in ann-stage microlever obeys the following equation:-stage microlever obeys the following equation:
(
This equation means that, with each microlever stage (with an amplification factor of This equation means that, with each microlever stage (with an amplification factor of A Aii)) added, the overall input axial spring constant of the compound microlever decreases by a added, the overall input axial spring constant of the compound microlever decreases by a factor of
factor of A Aii22//eeii..