10. Comparison of different sections at the level of the tooth

#### 10.1. Section A-A

After applying the fuzzy logic and plotting the tooth curve, we notice that the green curve is far from the ideal curve; this is the normal vector method. But blue tracing is closer to this one because of having dots that define the involute curve, the red curve is closer to the curve of fuzzy logic (see Figure 9).

Δef ¼ f <sup>1</sup> � f <sup>2</sup> ¼ 25, 77–25, 72 Δef ¼ 0:05 mm Δef % ¼ 8:33 Δeg ¼ g1 � g2 ¼ 25, 82–25, 72 Δeg ¼ 0:15 mm Δeg% ¼ 25

Fuzzy Logic Applications in Metrology Processes http://dx.doi.org/10.5772/intechopen.79381 57

So, it is intolerable to accept miscalculations of more than Δeg % = 25 by the normal vector method, so the piece was rejected. On the other hand, if we use the same database, we find errors of fuzzy logic for this piece: Δef % = 8.33 mm, while the piece was accepted. It was

In this case, there is no difference between the two methods (fuzzy logic and the normal vector), the error is zero. However, the method FL does not influence the measurement of spur

concluded that the fuzzy logic method is closer to the ideal measurement.

Figure 12. Combination of the x and y coordinates by forming the tooth.

10.3. Section C-C

gears (see Figures 11 and 12).

#### 10.2. Section B-B

In that case, we find that the intersection between the involute curve and the outer circle gives a large deviation as expected. In addition, this leads to an increase in errors, that is, the increase in the gap (see Figure 10).

For example, if we determine the height of the tooth by the formula h = ha + hf + Δe (see Figure 10), then the percentage of the error according to the definition can be calculated as follows:

Figure 11. Section C-C.

$$\Delta e\_f = f\_1 - f\_2 = 25,77 - 25,72$$

$$\Delta e\_f = 0.05 \text{ mm}$$

$$\Delta e\_f \%= 8.33$$

$$\Delta e\_{\%} = \mathbf{g}\_1 - \mathbf{g}\_2 = 25,82 - 25,72$$

$$\Delta e\_{\%} = 0.15 \text{ mm}$$

$$\Delta e\_{\%} \%= 25$$

So, it is intolerable to accept miscalculations of more than Δeg % = 25 by the normal vector method, so the piece was rejected. On the other hand, if we use the same database, we find errors of fuzzy logic for this piece: Δef % = 8.33 mm, while the piece was accepted. It was concluded that the fuzzy logic method is closer to the ideal measurement.

#### 10.3. Section C-C

10. Comparison of different sections at the level of the tooth

56 Fuzzy Logic Based in Optimization Methods and Control Systems and Its Applications

After applying the fuzzy logic and plotting the tooth curve, we notice that the green curve is far from the ideal curve; this is the normal vector method. But blue tracing is closer to this one because of having dots that define the involute curve, the red curve is closer to the curve of fuzzy

In that case, we find that the intersection between the involute curve and the outer circle gives a large deviation as expected. In addition, this leads to an increase in errors, that is, the increase

For example, if we determine the height of the tooth by the formula h = ha + hf + Δe (see Figure 10), then the percentage of the error according to the definition can be calculated as

10.1. Section A-A

logic (see Figure 9).

10.2. Section B-B

Figure 11. Section C-C.

follows:

in the gap (see Figure 10).

In this case, there is no difference between the two methods (fuzzy logic and the normal vector), the error is zero. However, the method FL does not influence the measurement of spur gears (see Figures 11 and 12).

Figure 12. Combination of the x and y coordinates by forming the tooth.
