Chapter 82, Problem 12A

To determine

(a)

The coordinates in table form for the holes by using absolute programming.

Explanation of Solution

Given:

The length A is $170.00\text{ mm}$.

The length B is $130.00\text{ mm}$.

The length C is $122.40\text{ mm}$.

The length D is $30.00\text{ mm}$.

The length E is $30.00\text{ mm}$.

The length F is $300.40\text{ mm}$.

The length G is $450.00\text{ mm}$.

The length H is $249.30\text{ mm}$.

The length J is $269.30\text{ mm}$.

The length K is $218.70\text{mm}$.

The length L is $\text{104}\text{.00}\text{mm}$.

The length M is $228.00\text{ mm}$.

The length N is $95.10\text{ mm}$.

Pitch diameter of circular pattern with holes is $196.00\text{ mm}$.

Angle 1 is $41.75°$.

Concept Used:

Draw the diagram for the location of holes as shown below:

Write the expression for the location of first hole in absolute programming.

  $\left(x_1,y_1\right)=\left(A-C,H-B\right)$ ...... (1)

Here, $x_1$ is the $x$ coordinate of first hole from origin and $y_1$ is the $y$ coordinate of first hole from origin.

Write the expression for the location of second hole in absolute programming.

  $\left(x_2,y_2\right)=\left(x_1-L\cos \angle 1,y_1+L\sin \angle 1\right)$ ...... (2)

Here, $x_2$ is the $x$ coordinate of second hole from origin and $y_2$ is the $y$ coordinate of second hole from origin.

Write the expression for the location of third hole in absolute programming.

  $\left(x_3,y_3\right)=\left(M\cos \angle 1-x_1,M\sin \angle 1+y_1\right)$ ...... (3)

Here, $x_3$ is the $x$ coordinate of third hole from origin and $y_3$ is the $y$ coordinate of third hole from origin.

Write the expression for the location of fourth hole in absolute programming.

  $\left(x_4,y_4\right)=\left(F-x_1-\frac{P}{2}\cos 72°,y_1+\frac{P}{2}\sin 72°\right)$ ...... (4)

Here, $x_4$ is the $x$ coordinate of fourth hole from origin and $y_4$ is the $y$ coordinate of fourth hole from origin.

Write the expression for the location of fifth hole in absolute programming.

  $\left(x_5,y_5\right)=\left(F-x_1+\frac{P}{2}\cos 36°,y_1+\frac{P}{2}\sin 36°\right)$ ...... (5)

Here, $x_5$ is the $x$ coordinate of fifth hole from origin and $y_5$ is the $y$ coordinate of fifth hole from origin.

Write the expression for the location of sixth hole in absolute programming.

  $\left(x_6,y_6\right)=\left(G-x_1,N-y_1\right)$ ...... (6)

Here, $x_6$ is the $x$ coordinate of sixth hole from origin and $y_6$ is the $y$ coordinate of sixth hole from origin.

Write the expression for the location of seventh hole in absolute programming.

  $\left(x_7,y_7\right)=\left(F-x_1+\frac{P}{2}\cos 36°,y_1-\frac{P}{2}\sin 36°\right)$ ...... (7)

Here, $x_7$ is the $x$ coordinate of seventh hole from origin and $y_7$ is the $y$ coordinate of seventh hole from origin.

Write the expression for the location of eighth hole in absolute programming.

  $\left(x_8,y_8\right)=\left(F-x_1+\frac{P}{2}\cos 72°,\frac{P}{2}\sin 72°-y_1\right)$ ...... (8)

Here, $x_8$ is the $x$ coordinate of eighth hole from origin and $y_8$ is the $y$ coordinate of eighth hole from origin.

Write the expression for the location of ninth hole in absolute programming.

  $\left(x_9,y_9\right)=\left(F-x_1-\frac{P}{2},y_1\right)$ ...... (9)

Here, $x_9$ is the $x$ coordinate of ninth hole from origin and $y_9$ is the $y$ coordinate of ninth hole from origin.

Write the expression for the location of tenth hole in absolute programming.

  $\left(x_{10},y_{10}\right)=\left(x_1-E,K-y_1\right)$ ...... (10)

Here, $x_{10}$ is the $x$ coordinate of tenth hole from origin and $y_{10}$ is the $y$ coordinate of tenth hole from origin.

Write the expression for the location of eleventh hole in absolute programming.

  $\left(x_{11},y_{11}\right)=\left(A-C+D,J-y_1\right)$ ...... (11)

Here, $x_{11}$ is the $x$ coordinate of eleventh hole from origin and $y_{11}$ is the $y$ coordinate of eleventh hole from origin.

Calculation:

Substitute $170.00\text{mm}$ for $A$, $122.40\text{mm}$ for $C$, $\text{249}\text{.30}\text{mm}$ for $H$ and $\text{130}\text{.00}\text{mm}$ for $B$ in equation (1).

  $\begin{array}{c}\left(x_1,y_1\right)=\left(170.00-122.40,249.30-130.00\right)\\ =\left(-47.60\text{ mm},119.30\text{mm}\right)\end{array}$

The coordinates of first hole in absolute programming is $\left(-47.60\text{ mm},119.30\text{mm}\right)$.

Substitute $47.60\text{mm}$ for $x_1$, $\text{104}\text{.00}\text{mm}$ for $L$, $\text{119}\text{.30}\text{mm}$ for $y_1$ and $41.75°$ for $\angle 1$ in equation (2).

  $\begin{array}{c}\left(x_2,y_2\right)=\left(104.00\cos \angle 41.75°-47.60,119.30+104.00\sin \angle 41.75°\right)\\ =\left(-29.98\text{mm, 188}\text{.55}\text{mm}\right)\end{array}$

The coordinates of second hole in absolute programming is $\left(-29.98\text{mm, 188}\text{.55}\text{mm}\right)$.

Substitute $47.60\text{mm}$ for $x_1$, $\text{228}\text{.00 mm}$ for $M$, $\text{119}\text{.30}\text{mm}$ for $y_1$ and $41.75°$ for $\angle 1$ in equation (3).

  $\begin{array}{c}\left(x_3,y_3\right)=\left(228.00\cos \angle 41.75°-47.60,228.00\sin \angle 41.75°+119.30\right)\\ =\left(122.50\text{mm},271.12\text{mm}\right)\end{array}$

The coordinates of third hole in absolute programming is $\left(122.50\text{mm},271.12\text{mm}\right)$.

Substitute $47.60\text{mm}$ for $x_1$, $\text{300}\text{.40}\text{mm}$ for $F$, $\text{119}\text{.30}\text{mm}$ for $y_1$ and $196.00\text{ mm}$ for $P$ in equation (4).

  $\begin{array}{c}\left(x_4,y_4\right)=\left(300.40-47.60-\frac{196.00}{2}\cos 72°,119.30+\frac{196.00}{2}\sin 72°\right)\\ =\left(222.51\text{mm},212.50\text{ mm}\right)\end{array}$

The coordinates of fourth hole in absolute programming is $\left(222.51\text{mm},212.50\text{ mm}\right)$.

Substitute $47.60\text{mm}$ for $x_1$, $\text{300}\text{.40}\text{mm}$ for $F$, $\text{119}\text{.30}\text{mm}$ for $y_1$ and $196.00\text{ mm}$ for $P$ in equation (5).

  $\begin{array}{c}\left(x_5,y_5\right)=\left(300.40-47.60+\frac{196.00}{2}\cos 36°,119.30+\frac{196.00}{2}\sin 36°\right)\\ =\left(\text{332}\text{.08}\text{mm,176}\text{.90}\text{mm}\right)\end{array}$

The coordinates of fifth hole in absolute programming is $\left(\text{332}\text{.08}\text{mm,176}\text{.90}\text{mm}\right)$.

Substitute $47.60\text{mm}$ for $x_1$, $450.00\text{mm}$ for $G$, $\text{119}\text{.30}\text{mm}$ for $y_1$ and $\text{95}\text{.10}\text{mm}$ for $N$ in equation (6).

  $\begin{array}{c}\left(x_6,y_6\right)=\left(450.00-47.60,95.10-119.30\right)\\ =\left(402.40\text{mm, }-\text{24}\text{.20}\text{mm}\right)\end{array}$

The coordinates of sixth hole in absolute programming is $\left(402.40\text{mm, }-\text{24}\text{.20}\text{mm}\right)$.

Substitute $47.60\text{mm}$ for $x_1$, $\text{300}\text{.40}\text{mm}$ for $F$, $\text{119}\text{.30}\text{mm}$ for $y_1$ and $196.00\text{ mm}$ for $P$ in equation (7).

  $\begin{array}{c}\left(x_7,y_7\right)=\left(300.40-47.60+\frac{196.00}{2}\cos 36°,119.30-\frac{196.00}{2}\sin 36°\right)\\ =\left(\text{332}\text{.08}\text{mm, 61}\text{.70}\text{mm}\right)\end{array}$

The coordinates of seventh hole in absolute programming is $\left(\text{332}\text{.08}\text{mm, 61}\text{.70}\text{mm}\right)$.

Substitute $47.60\text{mm}$ for $x_1$, $\text{300}\text{.40}\text{mm}$ for $F$, $\text{119}\text{.30}\text{mm}$ for $y_1$ and $196.00\text{ mm}$ for $P$ in equation (8).

  $\begin{array}{c}\left(x_8,y_8\right)=\left(300.40-47.60-\frac{196.00}{2}\cos 72°,\frac{196.00}{2}\sin 72°-119.30\right)\\ =\left(283.08\text{mm, }-\text{26}\text{.10}\text{mm}\right)\end{array}$

The coordinates of eighth hole in absolute programming is $\left(283.08\text{mm, }-\text{26}\text{.10}\text{mm}\right)$.

Substitute $47.60\text{mm}$ for $x_1$, $\text{300}\text{.40}\text{mm}$ for $F$, $\text{119}\text{.30}\text{mm}$ for $y_1$ and $196.00\text{ mm}$ for $P$ in equation (9).

  $\begin{array}{c}\left(x_9,y_9\right)=\left(300.40-47.60-\frac{196.00}{2},119.30\right)\\ =\left(350.08\text{mm, 119}\text{.30}\text{mm}\right)\end{array}$

The coordinates of ninth hole in absolute programming is $\left(350.08\text{mm, 119}\text{.30}\text{mm}\right)$.

Substitute $47.60\text{mm}$ for $x_1$, $30.00\text{ mm}$ for $E$, $\text{119}\text{.30}\text{mm}$ for $y_1$ and $\text{218}\text{.70}\text{mm}$ for $K$ in equation (10).

  $\begin{array}{c}\left(x_{10},y_{10}\right)=\left(47.60-30.00,218.70-119.30\right)\\ =\left(17.60\text{ mm, 99}\text{.40 mm}\right)\end{array}$

The coordinates of tenth hole in absolute programming is $\left(-17.60\text{ mm, }-\text{99}\text{.40 mm}\right)$.

Substitute $\text{170}\text{.00}\text{mm}$ for $A$, $122.40\text{mm}$ for $C$, $119.30\text{ mm}$ for $y_1$, $269.30\text{mm}$ for $J$ and $30.00\text{ mm}$ for $D$ in equation (10).

  $\begin{array}{c}\left(x_{11},y_{11}\right)=\left(170.00-122.40+30.00,269.30-119.30\right)\\ =\left(77.60\text{ mm, 150}\text{mm}\right)\end{array}$

The coordinates of eleventh hole in absolute programming is $\left(-77.60\text{ mm, }-\text{150}\text{mm}\right)$.

Conclusion:

The coordinate of each hole is shown in below table:

Hole	X-coordinate (in mm)	Y-coordinate (in mm)
1	$-47.60$	$\text{119}\text{.30}$
2	$-29.98$	$\text{188}\text{.55}$
3	$122.50$	$271.12$
4	$222.51$	$212.50$
5	$323.08$	$176.90$
6	$\text{402}\text{.40}$	$-24.20$
7	$323.08$	$\text{119}\text{.30}$
8	$283.08$	$-26.10$
9	$350.08$	$\text{119}\text{.30}$
10	$-17.60$	$-99.40$
11	$-77.60$	$-\text{150}\text{.00}$

To determine

(b)

The coordinates in table form for the holes by using incremental programming.

Explanation of Solution

Concept Used:

Write the expression for the location of first hole in incremental programming.

  $\left({x^{\prime }}_1,{y^{\prime }}_1\right)=\left(x_1,y_1\right)$ ...... (12)

Here, ${x^{\prime }}_1$ is the $x$ coordinate of first hole from origin and ${y^{\prime }}_1$ is the $y$ coordinate of first hole from origin.

Write the expression for the location of second hole in incremental programming.

  $\left({x^{\prime }}_2,{y^{\prime }}_2\right)=\left(x_2-x_1,y_2-y_1\right)$ ...... (13)

Here, ${x^{\prime }}_2$ is the $x$ coordinate of second hole from origin and ${y^{\prime }}_2$ is the $y$ coordinate of second hole from first hole.

Write the expression for the location of third hole in incremental programming.

  $\left({x^{\prime }}_3,{y^{\prime }}_3\right)=\left(x_3-x_2,y_3-y_2\right)$ ...... (14)

Here, ${x^{\prime }}_3$ is the $x$ coordinate of third hole from origin and ${y^{\prime }}_3$ is the $y$ coordinate of third hole from second hole.

Write the expression for the location of fourth hole in incremental programming.

  $\left({x^{\prime }}_4,{y^{\prime }}_4\right)=\left(x_4-x_3,y_4-y_3\right)$ ...... (15)

Here, ${x^{\prime }}_4$ is the $x$ coordinate of fourth hole from origin and ${y^{\prime }}_4$ is the $y$ coordinate of fourth hole from third hole.

Write the expression for the location of fifth hole in incremental programming.

  $\left({x^{\prime }}_5,{y^{\prime }}_5\right)=\left(x_5-x_4,y_5-y_4\right)$ ...... (16)

Here, ${x^{\prime }}_5$ is the $x$ coordinate of fifth hole from origin and ${y^{\prime }}_5$ is the $y$ coordinate of fifth hole from fourth hole.

Write the expression for the location of sixth hole in incremental programming.

  $\left({x^{\prime }}_6,{y^{\prime }}_6\right)=\left(x_6-x_5,y_6-y_5\right)$ ...... (17)

Here, ${x^{\prime }}_6$ is the $x$ coordinate of sixth hole from origin and ${y^{\prime }}_6$ is the $y$ coordinate of sixth hole from fifth hole.

Write the expression for the location of seventh hole in incremental programming.

  $\left({x^{\prime }}_7,{y^{\prime }}_7\right)=\left(x_7-x_6,y_7-y_6\right)$ ...... (18)

Here, ${x^{\prime }}_7$ is the $x$ coordinate of seventh hole from origin and ${y^{\prime }}_7$ is the $y$ coordinate of seventh hole from sixth hole.

Write the expression for the location of eighth hole in incremental programming.

  $\left({x^{\prime }}_8,{y^{\prime }}_8\right)=\left(x_8-x_7,y_8-y_7\right)$ ...... (19)

Here, ${x^{\prime }}_8$ is the $x$ coordinate of eighth hole from origin and ${y^{\prime }}_8$ is the $y$ coordinate of eighth hole from seventh hole.

Write the expression for the location of ninth hole in incremental programming.

  $\left({x^{\prime }}_9,{y^{\prime }}_9\right)=\left(x_9-x_8,y_9-y_8\right)$ ...... (20)

Here, ${x^{\prime }}_9$ is the $x$ coordinate of ninth hole from origin and ${y^{\prime }}_9$ is the $y$ coordinate of ninth hole from eighth hole.

Write the expression for the location of tenth hole in incremental programming.

  $\left({x^{\prime }}_{10},{y^{\prime }}_{10}\right)=\left(x_{10}-x_9,y_{10}-y_9\right)$ ...... (21)

Here, ${x^{\prime }}_{10}$ is the $x$ coordinate of tenth hole from origin and ${y^{\prime }}_{10}$ is the $y$ coordinate of tenth hole from ninth hole.

Write the expression for the location of eleventh hole in incremental programming.

  $\left({x^{\prime }}_{11},{y^{\prime }}_{11}\right)=\left(x_{11}-x_{10},y_{11}-y_{10}\right)$ ...... (22)

Here, ${x^{\prime }}_{10}$ is the $x$ coordinate of eleventh hole from origin and ${y^{\prime }}_{10}$ is the $y$ coordinate of eleventh hole from tenth hole.

Calculation:

Substitute $-47.60\text{mm}$ for $x_1$ and $119.30\text{ mm}$ for $y_1$ in equation (12).

  $\left({x^{\prime }}_1,{y^{\prime }}_1\right)=\left(-47.60\text{mm, 119}\text{.30}\text{mm}\right)$

The coordinates of first hole in incremental programming is $\left(-47.60\text{mm, 119}\text{.30}\text{mm}\right)$.

Substitute $-47.60\text{mm}$ for $x_1$, $-29.98\text{ mm}$ for $x_2$, $188.55\text{mm}$ for $y_2$ and $119.30\text{ mm}$ for $y_1$ in equation (13).

  $\begin{array}{c}\left({x^{\prime }}_2,{y^{\prime }}_2\right)=\left(-29.98+47.60,188.55-119.30\right)\\ =\left(17.62\text{mm, 69}\text{.25}\text{mm}\right)\end{array}$

The coordinates of second hole in incremental programming is $\left(17.62\text{mm, 69}\text{.25}\text{mm}\right)$.

Substitute $-29.98\text{ mm}$ for $x_2$, $188.55\text{mm}$ for $y_2$, $122.50\text{ mm}$ for $x_3$ and $271.12\text{ mm}$ for $y_3$ in equation (14).

  $\begin{array}{c}\left({x^{\prime }}_3,{y^{\prime }}_3\right)=\left(122.50+29.98,271.12-188.55\right)\\ =\left(152.48\text{ mm,}82.57\text{ mm}\right)\end{array}$

The coordinates of third hole in incremental programming is $\left(152.48\text{ mm,}82.57\text{ mm}\right)$.

Substitute $\text{222}\text{.51 mm}$ for $x_4$, $212.50\text{mm}$ for $y_4$, $122.50\text{ mm}$ for $x_3$ and $271.12\text{ mm}$ for $y_3$ in equation (15).

  $\begin{array}{c}\left({x^{\prime }}_4,{y^{\prime }}_4\right)=\left(222.51-122.50,212.50-271.50\right)\\ =\left(\text{100}\text{.01 mm, }-\text{58}\text{.62}\text{mm}\right)\end{array}$

The coordinates of fourth hole in incremental programming is $\left(\text{100}\text{.01 mm, }-\text{58}\text{.62}\text{mm}\right)$.

Substitute $\text{222}\text{.51 mm}$ for $x_4$, $212.50\text{mm}$ for $y_4$, $332.08\text{mm}$ for $x_5$ and $176.90\text{mm}$ for $y_5$ in equation (16).

  $\begin{array}{c}\left({x^{\prime }}_5,{y^{\prime }}_5\right)=\left(332.08-222.51,176.90-221.50\right)\\ =\left(109.57\text{mm, }-35.60\text{mm}\right)\end{array}$

The coordinates of fifth hole in incremental programming is $\left(109.57\text{mm, }-35.60\text{mm}\right)$.

Substitute $402.40\text{ mm}$ for $x_6$, $-24.20\text{mm}$ for $y_6$, $332.08\text{mm}$ for $x_5$ and $176.90\text{mm}$ for $y_5$ in equation (17).

  $\begin{array}{c}\left({x^{\prime }}_6,{y^{\prime }}_6\right)=\left(402.40-332.08,-24.20-176.90\right)\\ =\left(70.32\text{mm},-201.10\text{mm}\right)\end{array}$

The coordinates of sixth hole in incremental programming is $\left(70.32\text{mm},-201.10\text{mm}\right)$.

Substitute $402.40\text{ mm}$ for $x_6$, $-24.20\text{mm}$ for $y_6$, $332.08\text{mm}$ for $x_7$ and $\text{61}\text{.70}\text{mm}$ for $y_7$ in equation (18).

  $\begin{array}{c}\left({x^{\prime }}_7,{y^{\prime }}_7\right)=\left(332.08-402.40,61.70+24.20\right)\\ =\left(-70.32\text{mm},85.90\text{ mm}\right)\end{array}$

The coordinates of seventh hole in incremental programming is $\left(-70.32\text{mm},85.90\text{ mm}\right)$.

Substitute $283.08\text{ mm}$ for $x_8$, $-26.10\text{mm}$ for $y_8$, $332.08\text{mm}$ for $x_7$ and $\text{61}\text{.70}\text{mm}$ for $y_7$ in equation (19).

  $\begin{array}{c}\left({x^{\prime }}_8,{y^{\prime }}_8\right)=\left(283.08-332.08,-26.10-61.70\right)\\ =\left(-49.00\text{mm,}-87.80\text{mm}\right)\end{array}$

The coordinates of eighth hole in incremental programming is $\left(-49.00\text{mm,}-87.80\text{mm}\right)$.

Substitute $283.08\text{ mm}$ for $x_8$, $-26.10\text{mm}$ for $y_8$, $350.08\text{ mm}$ for $x_9$ and $119.30\text{mm}$ for $y_9$ in equation (20).

  $\begin{array}{c}\left({x^{\prime }}_9,{y^{\prime }}_9\right)=\left(350.08-283.08,119.30+26.10\right)\\ =\left(67.00\text{mm, 145}\text{.40}\text{mm}\right)\end{array}$

The coordinates of ninth hole in incremental programming is $\left(67.00\text{mm, 145}\text{.40}\text{mm}\right)$.

Substitute $-17.60\text{ mm}$ for $x_{10}$, $-99.40\text{mm}$ for $y_{10}$, $350.08\text{ mm}$ for $x_9$ and $119.30\text{mm}$ for $y_9$ in equation (21).

  $\begin{array}{c}\left({x^{\prime }}_{10},{y^{\prime }}_{10}\right)=\left(-17.60-350.08,-99.40-119.30\right)\\ =\left(-367.68\text{mm},-218.70\text{ mm}\right)\end{array}$

The coordinates of tenth hole in incremental programming is $\left(-367.68\text{mm},-218.70\text{ mm}\right)$.

Substitute $-17.60\text{ mm}$ for $x_{10}$, $-99.40\text{mm}$ for $y_{10}$, $-77.60\text{ mm}$ for $x_{11}$ and $-150.00\text{ mm}$ for $y_{11}$ in equation (22).

  $\begin{array}{c}\left({x^{\prime }}_{11},{y^{\prime }}_{11}\right)=\left(-77.60+17.60,-150.00+99.40\right)\\ =\left(-60.00\text{mm},-50.60\text{mm}\right)\end{array}$

The coordinates of eleventh hole in incremental programming is $\left(-60.00\text{mm},-50.60\text{mm}\right)$.

Conclusion:

The coordinate of each hole is shown in below table:

Hole	X-coordinate (in mm)	Y-coordinate (in mm)
1	$-47.60$	$\text{119}\text{.30}$
2	$17.62$	$\text{69}\text{.25}$
3	$152.48$	$82.57$
4	$100.01$	$-58.62$
5	$109.57$	$-35.60$
6	$\text{70}\text{.32}$	$-201.10$
7	$-70.32$	$85.90$
8	$-49.00$	$-87.80$
9	$-67.00$	$145.40$
10	$-367.68$	$-218.70$
11	$-60.00$	$-\text{50}\text{.60}$