**3. Analysis of quasi-static behavior**

### **3.1. Force and moment equation derivation**

Rigid body model of the compliant micro mechanism is considered. Free body diagram of each beam is sketched and a typical beam model is schematically shown in Fig. 3. Forces acting on each beam is broken down into x and y components as follows;

**Figure 3.** Free body diagram of beam 2

achieving a high force amplifying.

Depth 25 µm

25 µm

Y

800µm

X

Thin elastic beam

**1.stage** 

**Figure 2.** Novel compliant MEMS Force Amplifier

**3. Analysis of quasi-static behavior** 

**3.1. Force and moment equation derivation** 

Rigid body model of the compliant micro mechanism is considered. Free body diagram of each beam is sketched and a typical beam model is schematically shown in Fig. 3. Forces

100 µm

A B

Foutput

F input

**2.stage** 

Rigid beam

acting on each beam is broken down into x and y components as follows;

is rocked, then it is called a crank-rocker mechanism. To determine the moving limit of the micro mechanism, the relation between the lengths of beams turns out to be an important issue. Therefore, selecting the length of a beam plays a crucial role for the micro mechanism. Due the fact that, x1, x2 are assumed as length of the shortest beam and length of the longest beam, respectively, as x3, x4 are the mean lengths of the beams. If x1+x2<= x3+x4, at least one of the beams can rotate and If x1+x2= x3+x4, the mechanism is activated and crank has limited rotation this feature enables beams to pass horizontal positions closely to each other

25µm

The static force and moment equations of beam 2 is typically shown and derived as; Equation derivation of forces acting on beam 2 along x axis;

$$
\sum F\_x = 0\tag{3}
$$

$$F\_{12\,x} + F\_{32\,x} + F\_{52\,x} = 0\tag{4}$$

Equation derivation of forces acting on beam 2 along y axis;

$$
\sum F\_y = 0\tag{5}
$$

$$F\_{12y} + F\_{32y} + F\_{52y} = 0\tag{6}$$

Equation derivation of moments acting on beam 2 along z axis;

$$
\sum M\_z = 0\tag{7}
$$

$$R\_{12x} \ \ \ \ \ F\_{12y} - R\_{12y} \ \ \ \ \ \ F\_{12x} + R\_{32x} \ \ \ \ \ \ \ F\_{32y} - R\_{32y} \ \ \ \ \ \ \ F\_{32x} + R\_{52x} \ \ \ \ \ \ \ F\_{52y} - R\_{52y} \ \ \ \ \ \ \ \ F\_{52x} = 0 \tag{8}$$

Free body diagram of beam 3 is shown in Fig. 4 and equation derivation of forces acting on beam 3 along x axis;

$$\sum F\_x = 0\tag{9}$$

$$F\_{23\times} + F\_{43\times} = 0\tag{10}$$

Equation derivation of forces acting on beam 3 along y axis;

$$
\sum F\_y = 0\tag{11}
$$

$$F\_{23y} + F\_{43y} = 0\tag{12}$$

Equation derivation of moments acting on beam 3 along z axis;

$$
\sum M\_z = 0\tag{13}
$$

$$R\_{23x} \ \ \ \ \ \ F\_{23y} \ \ -R\_{23y} \ \ \ \ \ \ F\_{23x} + R\_{43x} \ \ \ \ \ \ \ F\_{43y} - R\_{43y} \ \ \ \ \ \ \ \ F\_{43x} = 0 \tag{14}$$

Free body diagram of beam 5 is shown in Fig. 5 and equation derivation of forces acting on beam 5 along x axis;

$$\sum F\_x = 0\tag{15}$$

$$F\_{65x} + F\_{25x} = 0\tag{16}$$

Equation derivation of forces acting on beam 5 along y axis;

$$
\sum F\_y = 0\tag{17}
$$

Dynamic and Quasi-Static Simulation of a Novel Compliant MEMS Force Amplifier by Matlab/Simulink 95

$$F\_{65y} + F\_{25y} = 0\tag{18}$$

Equation derivation of moments acting on beam 5 along z axis;

94 MATLAB – A Fundamental Tool for Scientific Computing and Engineering Applications – Volume 1

Equation derivation of forces acting on beam 3 along y axis;

Equation derivation of moments acting on beam 3 along z axis;

F23y

**θ<sup>3</sup>**

**Figure 4.** Free body diagram of beam 3

F23x

Equation derivation of forces acting on beam 5 along y axis;

beam 5 along x axis;

23 43 0 *x x F F* (10)

<sup>0</sup> *<sup>y</sup> <sup>F</sup>* (11)

23 43 0 *y y F F* (12)

<sup>0</sup> *Mz* (13)

23 23 23 23 43 43 43 43 \* \* \* \*0 *RFRFRFRF xy yx xy yx* (14)

Free body diagram of beam 5 is shown in Fig. 5 and equation derivation of forces acting on

**R43**

**3**

**R23**

F43y

65 25 0 *x x F F* (16)

<sup>0</sup> *<sup>y</sup> <sup>F</sup>* (17)

<sup>0</sup> *<sup>x</sup> <sup>F</sup>* (15)

F43x

$$
\sum M\_z = 0\tag{19}
$$

$$R\_{65x} \ \ ^\ast F\_{65y} - R\_{65y} \ ^\ast F\_{65x} + R\_{25x} \ ^\ast F\_{25y} - R\_{25y} \ ^\ast F\_{25x} = 0 \tag{20}$$

Free body diagram of beam 6 is shown in Fig. 6 and equation derivation of forces acting on beam 6 along x axis;

$$
\sum F\_x = 0\tag{21}
$$

$$F\_{g\,6x} + F\_{5\,6x} = 0\tag{22}$$

Equation derivation of forces acting on beam 6 along y axis;

$$
\sum F\_y = 0\tag{23}
$$

$$F\_{y\%y} + F\_{5\%y} = 0\tag{24}$$

Equation derivation of moments acting on beam 6 along z axis;

$$
\sum M\_z = 0\tag{25}
$$

$$R\_{g\delta x} \, \prescript{\*}{}{F}\_{g\delta y} - R\_{g\delta y} \, \prescript{\*}{}{F}\_{g\delta x} + R\_{5\delta x} \, \prescript{\*}{}{F}\_{5\delta y} - R\_{5\delta y} \, \prescript{\*}{}{F}\_{5\delta x} = 0 \tag{26}$$

**Figure 5.** Free body diagram of beam 5

**Figure 6.** Free body diagram of beam 6

Free body diagram of slider is shown in Fig. 7 and equation derivation of forces acting on slider along x- and y- axes;

$$\sum F\_x = 0\tag{27}$$

$$F\_{output} + F\_{34 \, x} = 0\tag{28}$$

$$
\sum F\_y = 0\tag{29}
$$

$$F\_{s4y} + F\_{34y} = 0\tag{30}$$

Thus, 14 force and moment equations are derived. Equations of relation between internal forces of beams;

$$F\_{32x} = F\_{23x} \tag{31}$$

$$F\_{32y} = F\_{23y} \tag{32}$$

$$F\_{43x} = F\_{34x} \tag{33}$$

$$F\_{43y} = F\_{34y} \tag{34}$$

$$F\_{52x} = F\_{25x} \tag{35}$$

Dynamic and Quasi-Static Simulation of a Novel Compliant MEMS Force Amplifier by Matlab/Simulink 97

$$F\_{52y} = F\_{25y} \tag{36}$$

$$F\_{65x} = F\_{56x} \tag{37}$$

$$F\_{65y} = F\_{56y} \tag{38}$$

8 equations are derived from the relations between internal forces of beams.

#### **Figure 7.** Free body diagram of slider

96 MATLAB – A Fundamental Tool for Scientific Computing and Engineering Applications – Volume 1

**Rg6**

Fg6y

Free body diagram of slider is shown in Fig. 7 and equation derivation of forces acting on

**6**

**R56**

34 0 *output x F F* (28)

<sup>0</sup> *<sup>y</sup> <sup>F</sup>* (29)

4 34 0 *sy y F F* (30)

Thus, 14 force and moment equations are derived. Equations of relation between internal

32 23 *x x F F* (31)

32 23 *y y F F* (32)

43 34 *x x F F* (33)

43 34 *y y F F* (34)

52 25 *x x F F* (35)

<sup>0</sup> *<sup>x</sup> <sup>F</sup>* (27)

F56x **<sup>θ</sup><sup>6</sup>**

F56y

**Figure 6.** Free body diagram of beam 6

slider along x- and y- axes;

Fg6x

forces of beams;
