**3.3. Position analysis**

The micro mechanism is a single degree of freedom mechanism and position analysis provides to inform the positions of other links and points as one of the links moves or rotates.

To find out position problem of the micro mechanism, nonlinear and transcendental vector loop equations that are derived and solved.

The vector loops are schematically shown in Fig. 8. There are two vector loop equations such as;

First vector loop equation:

$$R\_2 + R\_3 = R\_1 \tag{39}$$

**Figure 8.** Vector loops for the force amplifier

Deriving equations according to coordinates of x and y:

$$r\_2 \, "\, \cos\theta\_2 + r\_3 \, "\, \cos\theta\_3 = r\_1\tag{40}$$

$$-r\_2 \, "\, \sin \theta\_2 - r\_3 \, "\, \sin \theta\_3 = 0\tag{41}$$

Second vector loop equation:

$$R\_2 + R\_5 + R\_6 = R\_{cs} \tag{42}$$

Vector loop equations along x--axis

$$r\_2 \, ^\ast \cos \theta\_2 + r\_5 \, ^\ast \cos \theta\_5 + r\_6 \, ^\ast \cos \theta\_6 = r\_{cs} \, ^\ast \cos \theta\_{cs} \tag{43}$$

Vector loop equations along y--axis

$$r\_2 \, ^\ast \sin \theta\_2 + r\_5 \, ^\ast \sin \theta\_5 + r\_6 \, ^\ast \sin \theta\_6 = r\_{cs} \, ^\ast \sin \theta\_{cs} \tag{44}$$

By quasi-static analysis, it is claimed that (360°-Θ3) and Θ2 decreases linearly and are equal to each other during both quasi-static and dynamic simulations run by Matlab/Simulink. As seen in Fig. 9, it is calculated that as Θ5 goes from 70° to 74.0248°, Θ6 reduces from 110° to 105.9756°. Thus, as Θ2 rotates 20°, both Θ5 and Θ2 rotates approximately 4.02° and slightly different from each other. The relation both between Θ5 and Θ2, Θ6 and Θ2 are linear.

**Figure 9.** Plot of Θ5 and Θ6 according to first stage crank angle, Θ2

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

**Figure 8.** Vector loops for the force amplifier

Second vector loop equation:

Vector loop equations along x--axis

Vector loop equations along y--axis

Deriving equations according to coordinates of x and y:

2 23 31

2 23 3 *r r* \* sin \* sin 0

 2 25 56 6 \* cos \* cos \* cos \* cos *cs cs rrrr* 

 2 25 56 6 \* sin \* sin \* sin \* sin *cs cs rrrr* 

*rrr* \* cos \* cos 

*RRRR* <sup>256</sup> *cs* (42)

 

 

By quasi-static analysis, it is claimed that (360°-Θ3) and Θ2 decreases linearly and are equal to each other during both quasi-static and dynamic simulations run by Matlab/Simulink. As seen in Fig. 9, it is calculated that as Θ5 goes from 70° to 74.0248°, Θ6 reduces from 110° to

 

> 

(40)

(41)

 

  (43)

(44)

Displacement ratio is defined as Uoutput/Uinput. As the micro mechanism operates under an input force along – x direction, the first stage crank angle starts decreasing and pass from 0° and again starts increasing in an opposite direction and the ratio of output displacement to input displacement decreases as shown in Fig. 10. Beams 5 and 6 moves along –x and –y directions and the length of beams 5 and 6 are 8 times of beams 2 and 3. So, the input displacement increases rapidly than output displacement at close to zero degree crank angles. At negative crank angle values defining opposite directions, the slider gets close to initial position on contrast, beams 5 and 6 continue to get close to their vertical positions meaning that input displacement goes on to increase whereas output displacement begin to decrease. Therefore, after zero-crank angle, the displacement ratio continues to decrease according to Θ2.

As the micro mechanism displays, both the second stage crank angle, (Θ6-90°) and the first stage crank angle, Θ2 get close to zero degree, the force amplification defined as Foutput/Finput starts increasing and when Θ2 is 0° and (Θ6-90°) is at about small values, the micro mechanism provides high output force and force amplifying sharply increases as seen in Fig. 11 under 1.7\*10-7 in [N]. Also, there are two peaks in force amplification by quasi-static run. As, the first crank angle is close to zero but at still positive value, the force amplifying reaches 5093 and after that step first crank angle gets negative value but it is still close to zero, the force amplification ratio is 4830 at negative direction due to the fact that the slider motion begin to move in opposite direction and also, output force is in opposite direction. It is claimed that the toggle position of the micro mechanism is a very crucial issue meaning that if the initial conditions such as crank angles are adjusted properly to enable both crank angle pass 0° at the same time, the ratio of the output to the input force applied to the mechanism goes to infinity at zero degree crank angles.

**Figure 10.** Plot of displacement ratio according to first stage crank angle, Θ2

**Figure 11.** Plot of force amplifying according to first stage crank angle, Θ2
