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Metrology
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SHEET 2
FLATNESS
1. A table of upright drilling machines 550x 550 mm was tested for flatness error. A
mesh consists of 4 similar squares of 250 mm side length is drawn on the table.
A sensitive level of constant 0.04 mm/m is used to measure the level of each
point relative to the preceding one as listed on the following table. Using SEMI-
ANALYTICAL method, determine the maximum out of flatness.
Position ba cb ed fe
Reading (div) -3 2 4 1
Position hg ih da gd Reading (div) l 2 2 4
𝑐 = 0.04 mm/m = 0.01 mm/250mm
1- Reading
2- Accumulative
3-make the lower left corner =0
4-make the upper right corner =0
5-rotate about the diagonal
Maximum out of flatness = (3.5+1.5)× 0.01=0.05 mm = 50 microns
2. A Rotchdal arm tester having 200 mm leg distance was calibrated on three
points x, y, z on a line. Their reading at point Z was 0.1 mm. The same line is
divided into 4 equal distances to be checked using a sensitive level. The
readings were 1 and 3 μm respectively on the line xy and were -1 and -3 μm
respectively on the line yz
The calibrated Rotchdal arm is used to check the flatness of a surface shown in Fig.
The readings
Line abc Afe edc Bdh ehk bcl hjl cjk
Readings(μm) 15 -6 5 -10 -3 106 116 -83
Find the height readings relative to the plane abf, then find the out of flatness.
position x x1 y y1 z
reading relative to the preceding one 0 1 3 -1 -3
Accumulative (mm) 0 1 4 3 0
𝑅 = 2𝑞 − 𝑝 + 𝑦
Where:
R: level of the point under the dial indicator plunger
q: level of the point under the middle leg
p: level of the point under the end leg
y: dial indicator reading
get the dial reading
0 = 2 × 4 − 0 + 𝑦
𝑦 = −8 𝜇𝑚
Correction
𝑐 = 𝑅 − 𝑅𝑟𝑒𝑓 = 100 − (−8) = 108𝜇𝑚
Line abc Afe edc Bdh ehk bcl hjl cjk
Readings(μm) 15 -6 5 -10 -3 106 116 -83
Rcorrected=R-C -93 -114 -103 -118 -111 -2 8 -191
abe c = 2b -a + Rabe c -93
afe e = 2f - a + Rafe e -114
edc c = 2d - e + Redc d -52
bdh h = 2d - b + Rbdh h -222
ehk k = 2h - e + Rehk k -441
bcl l = 2c - b + Rbcl l -188
hjl l = 2j - h + Rhjl j -209
cjk k = 2j - c + Rcjk k -514
�̅� =3400
10= 340
𝑦 =1558.8
10= 155.88
𝑧̅ =−1319
10= −131.9
𝐸𝑟𝑟𝑜𝑟 𝑖𝑛 𝑟𝑒𝑎𝑑𝑖𝑛𝑔𝑠 =−514 − −441
−514= 0.142
Error within ±0.142 from the reading
𝑎 =∑ 𝑦′2 ∑ 𝑥′𝑧′ − ∑ 𝑥′𝑦′ ∑ 𝑦′𝑧′
∑ 𝑥′2 ∑ 𝑦′2 − (∑ 𝑥′𝑦′)2
𝑎 = −0.4854
𝑏 =∑ 𝑥′2 ∑ 𝑦′𝑧′ − ∑ 𝑥′𝑦′ ∑ 𝑥′𝑧′
∑ 𝑥′2 ∑ 𝑦′2 − (∑ 𝑥′𝑦′)2
𝑏 = −0.34669
𝑧𝑖′′ = 𝑎𝑥𝑖
′ + 𝑏𝑦𝑖′ = −0.4854𝑥′ − 0.34669𝑦′
𝛿𝑖 = 𝑧𝑖′ − 𝑧𝑖
′′
Maximum out of flatness = |126.4887| + |−27.178| = 153.667 microns
Calculations:
x Y z x' =x-�̅�
y' z' x'2 y'2 x'y' y'z' x'z' z'' delta
a 0 0 0 -340 -155.88 131.9 115600 24298.57 52999.2 -20560.6 -44846 219.078 -87.178
b 200 0 0 -140 -155.88 131.9 19600 24298.57 21823.2 -20560.6 -18466 121.998 9.901963
f 100 173.2 0 -240 17.32 131.9 57600 299.9824 -4156.8 2284.508 -31656 110.4913 21.40867
c 400 0 -93 60 -155.88 38.9 3600 24298.57 -9352.8 -6063.73 2334 24.91804 13.98196
e 200 346.4 -114 -140 190.52 17.9 19600 36297.87 -26672.8 3410.308 -2506 1.904621 15.99538
d 300 173.2 -52 -40 17.32 79.9 1600 299.9824 -692.8 1383.868 -3196 13.41133 66.48867
h 400 346.4 -222 60 190.52 -90.1 3600 36297.87 11431.2 -17165.9 -5406 -95.1754 5.075379
k 700 346.4 -441 360 190.52 -309.1 129600 36297.87 68587.2 -58889.7 -111276 -240.795 -68.3046
l 600 0 -188 260 -155.88 -56.1 67600 24298.57 -40528.8 8744.868 -14586 -72.162 16.06196
j 500 173.2 -209 160 17.32 -77.1 25600 299.9824 2771.2 -1335.37 -12336 -83.6687 6.568671
sum 3400 1558.8 -1319 0 -5.7E-14 0 444000 206987.9 76208 -108752 -241940
3. The flatness of a surface shown in Fig. was tested using a Rotchdal arm tester having 200 mm leg distance. The readings were:
Line Abc Afe edc Bdh ehk bcl hjl cjk
Readings(μm) 15 -6 5 -10 -3 106 116 -83
The Rotchdal arm tester was calibrated on three points x, y, z on a line. Its readings
at point Z was 0.1 mm while the reading of a sensitive level was 8 μm on line xy and
was -10 μm on line yz. Find the height readings relative to the plane ace.
position x y z
reading relative to the preceding one 0 8 -10
Accumulative (mm) 0 8 -2
𝑅 = 2𝑞 − 𝑝 + 𝑦
get the dial reading
−2 = 2 × 8 − 0 + 𝑦
𝑦 = −10 𝜇𝑚
𝑐 = 𝑅 − 𝑅𝑟𝑒𝑓 = 100 − (−10) = 110𝜇𝑚
Line abe Afe edc Bdh ehk bcl hjl cjk
Readings 15 -6 5 -10 -3 106 116 -83
Rcorrected=R-C -95 -116 -105 -120 -113 -4 6 -193
abe c = 2b -a + R abe b 47.5
afe e = 2f - a + R afe f 58
edc c = 2d - e + Redc d 52.5
bdh h = 2d - b + R bdh h -73
ehk k = 2h - e + R ehk k -259
bcl l = 2c - b + R bcl l -51.5
hjl l = 2j - h + R hjl j -65.25
cjk k = 2j - c + R cjk k -323.5
𝐸𝑟𝑟𝑜𝑟 𝑖𝑛 𝑟𝑒𝑎𝑑𝑖𝑛𝑔𝑠 =−323.5 − −259
−323.5= 0.142
Error within ±0.2 from the reading
�̅� =3400
10= 340
𝑦 =1558.8
10= 155.88
𝑧̅ =−290.75
10= −29.075
𝑎 =∑ 𝑦′2 ∑ 𝑥′𝑧′ − ∑ 𝑥′𝑦′ ∑ 𝑦′𝑧′
∑ 𝑥′2 ∑ 𝑦′2 − (∑ 𝑥′𝑦′)2
𝑎 = −0.29175
𝑏 =∑ 𝑥′2 ∑ 𝑦′𝑧′ − ∑ 𝑥′𝑦′ ∑ 𝑥′𝑧′
∑ 𝑥′2 ∑ 𝑦′2 − (∑ 𝑥′𝑦′)2
𝑏 = −0.19137
𝑧𝑖′′ = 𝑎𝑥𝑖
′ + 𝑏𝑦𝑖′ = −0.29175𝑥𝑖
′ − 0.19137𝑦𝑖′
𝛿𝑖 = 𝑧𝑖′ − 𝑧𝑖
′′
Maximum out of flatness = |73.21953| + |−99.9508| = 173.1703microns
Calculations:
x Y z x' =x-�̅�
y' z' x'2 y'2 x'y' y'z' x'z' z'' delta
a 0 0 0 -340 -155.88 29.075 115600 24298.57 52999.2 -4532.21 -9885.5 129.0258 -99.9508
b 200 0 47.5 -140 -155.88 76.575 19600 24298.57 21823.2 -11936.5 -10720.5 70.67576 5.899244
f 100 173.2 58 -240 17.32 87.075 57600 299.9824 -4156.8 1508.139 -20898 66.70547 20.36953
c 400 0 0 60 -155.88 29.075 3600 24298.57 -9352.8 -4532.21 1744.5 12.32576 16.74924
e 200 346.4 0 -140 190.52 29.075 19600 36297.87 -26672.8 5539.369 -4070.5 4.385188 24.68981
d 300 173.2 52.5 -40 17.32 81.575 1600 299.9824 -692.8 1412.879 -3263 8.355472 73.21953
h 400 346.4 -73 60 190.52 -43.925 3600 36297.87 11431.2 -8368.59 -2635.5 -53.9648 10.03981
k 700 346.4 -259 360 190.52 -229.925 129600 36297.87 68587.2 -43805.3 -82773 -141.49 -88.4352
l 600 0 -51.5 260 -155.88 -22.425 67600 24298.57 -40528.8 3495.609 -5830.5 -46.0242 23.59924
j 500 173.2 -65.25 160 17.32 -36.175 25600 299.9824 2771.2 -626.551 -5788 -49.9945 13.81953
sum 3400 1558.8 -290.75 0 -5.7E-14 0 444000 206987.9 76208 -61845.4 -144120
4. A Rotchdal arm tester having 200 mm leg distance was calibrated on three
points x, y, z on a line. Its readings at point Z was - 0.1 mm. The same line is
divided into 4 equal distance to be checked using a sensitive level. The readings
were 1 and 3 u.m respectively on the line xy and were -2 and -4 u.m respectively
on the line yz The calibrated Rotchdal arm is used to check the flatness of a surface shown in
Find the height readings relative to the plane ace.
position x x1 y y1 z
reading relative to the preceding one 0 1 3 -2 -4
y accumulative 0 1 4 2 -2
𝑅 = 2𝑞 − 𝑝 + 𝑦
get the dial reading
−2 = 2 × 4 − 0 + 𝑦
𝑦 = −10 𝜇𝑚
Correction
𝑐 = 𝑅 − 𝑅𝑟𝑒𝑓 = −100 − (−8) = −92 𝜇𝑚
line abc afe edc bdh ehk bcl hjl cjk
Readings(μm) 15 -6 5 -10 -3 106 116 -83
Rcorrected=R-C 107 86 97 82 89 118 128 49
abe c = 2b -a + R abe b -53.5
afe e = 2f - a + R afe f -43
edc c = 2d - e + Redc d -48.5
bdh h = 2d - b + R bdh h 38.5
ehk k = 2h - e + R ehk k 166
bcl l = 2c - b + R bcl l 171.5
hjl l = 2j - h + R hjl j 41
cjk k = 2j - c + R cjk k 131
Line abc afe edc Bdh ehk bcl Hjl cjk
Readings( μm) 15 -6 5 -10 -3 26 36 -43
𝐸𝑟𝑟𝑜𝑟 𝑖𝑛 𝑟𝑒𝑎𝑑𝑖𝑛𝑔𝑠 =166 − 131
166= 0.2108
Error within ±0.2 from the reading
�̅� =3400
10= 340
𝑦 =1558.8
10= 155.88
𝑧̅ =272
10= 27.2
𝑎 =∑ 𝑦′2 ∑ 𝑥′𝑧′ − ∑ 𝑥′𝑦′ ∑ 𝑦′𝑧′
∑ 𝑥′2 ∑ 𝑦′2 − (∑ 𝑥′𝑦′)2
𝑎 = −0.4854
𝑏 =∑ 𝑥′2 ∑ 𝑦′𝑧′ − ∑ 𝑥′𝑦′ ∑ 𝑥′𝑧′
∑ 𝑥′2 ∑ 𝑦′2 − (∑ 𝑥′𝑦′)2
𝑏 = −0.34669
𝑧𝑖′′ = 𝑎𝑥𝑖
′ + 𝑏𝑦𝑖′ = −0.4854𝑥𝑖
′ − 0.34669𝑦𝑖′
𝛿𝑖 = 𝑧𝑖′ − 𝑧𝑖
′′
Maximum out of flatness = |379.5954| + |−246.278| = 625.8734microns
6. A table of upright drilling machines 500x500 mm was tested for flatness error by
using A Rotchdale arm having 200 mm leg distance. The mesh shown in the
accompanying figure was drawn on the m/c table. The arm was first placed on a
reference straight edge and its reading was observed as 2.15 mm. the readings
on table were as listed on the following table.
Position a be bfg afe bdh geh
Reading (mm) 2.39 2.05 1.95 2.44 2.15
R corrected 0.24 -0.1 -0.2 0.29 0
abc c = 2b -a + R abe b -0.12
afe e = 2f - a + R afe f 0.1
bfg g = 2f - b + R bfg g 0.22
geh h = 2e - g + R geh h -0.22
bdh h = 2d - b + R bdh d -0.315
𝑎 = −0.4854
𝑏 =∑ 𝑥′2 ∑ 𝑦′𝑧′ − ∑ 𝑥′𝑦′ ∑ 𝑥′𝑧′
∑ 𝑥′2 ∑ 𝑦′2 − (∑ 𝑥′𝑦′)2
𝑏 = −0.34669
𝑧𝑖′′ = 𝑎𝑥𝑖
′ + 𝑏𝑦𝑖′ = −0.00072𝑥𝑖
′ + 0.000115𝑦𝑖′
𝛿𝑖 = 𝑧𝑖′ − 𝑧𝑖
′′
Maximum out of flatness = |0.385962| + |−0.34512| = 0.73108𝑚𝑚
7. A Rotchdal arm tester having 200 mm leg distance was calibrated on three points x, y, z on a line. Its readings at point Z was 0.1 mm. The reading of a sensitive level on the same line needs to divide the line into parts as shown. The readings were 1 and 3 μm respectively on the line xy and were -1 and -3 μm respectively on the line yz
x____ xl _____ y ____ yl _____ z
The calibrated Rotchdal arm is used to check the flatness of a surface shown in Fig. The readings were:
Line a be afe edc Bdh ehk bcl hjl cjk
Readings(μm) 15 -6 5 -10 -3 106 116 -83
Find the height readings relative to the plane abf, then find the out of flatness.
8. A Rotchdal arm tester having 200 mm leg distance was calibrated on a reference
straight edge and its reading was 45 μm .The calibrated Rotchdal arm is used to
check the flatness of a surface shown in Fig. The readings were:
Line abc Afe edc Bdh ehk bcl hjl cjk
Readings(μm) 15 -6 5 -10 -3 106 116 -83
Find the height readings relative to the plane abf.
R corrected -30 -51 -40 -55 -48 61 71 -128
abe c = 2b -a + R abe c -30
afe e = 2f - a + R afe e -51
edc c = 2d - e + Redc d -20.5
bdh h = 2d - b + R bdh h -96
ehk k = 2h - e + R ehk k -189
bcl l = 2c - b + R bcl l 1
hjl l = 2j - h + R hjl j -83
cjk k = 2j - c + R cjk k -264
𝐸𝑟𝑟𝑜𝑟 𝑖𝑛 𝑟𝑒𝑎𝑑𝑖𝑛𝑔𝑠 =−264 − −189
−264= 0.284
Error within ±0.284 from the reading
𝑎 = −0.13967
𝑏 = 0.24466
𝑧𝑖′′ = 𝑎𝑥𝑖
′ + 𝑏𝑦𝑖′ = −0.13967𝑥𝑖
′ + 0.24466𝑦𝑖′
𝛿𝑖 = 𝑧𝑖′ − 𝑧𝑖
′′
Maximum out of flatness = |126.4887| + |−45.2562| = 171.7449μm
9. The following figure shows the sensitive level readings in divisions which
were taken on a square grid drawn on a planer machine. The grid overall
dimensions were 800x600 mm. The level constant was 0.04 mm/m.
It required to determine the maximum out of flatness of the plane table
surface analytically and semi analytic.
semi analytic
𝑐 = 0.04 𝑚𝑚/𝑚 = 0.008𝑚𝑚/200𝑚𝑚
6 3 -2 -6 7
-3 9 2 -4 -1
-6 -2 -7 -3 1
0 -1 -2 1 -2
Accumulative:
-3 0 -2 -8 -1
-9 0 2 -2 -3
-6 -8 -1 -4 -3
0 -1 -3 -2 -4
Make the left down corner = 0
-3 1 0 -5 3
-9 -9 1 1 1
-6 -7 -7 -1 1
0 0 1 1 0
Make the right upper corner = 0
0 4 3 -2 6
-7 3 6 3 3
-5 -6 2 0 2
0 0 -1 1 0
Rotate about the diagonal
0 3.25 1.5 -4.25 3
-6 3.2 5.5 1.75 1
-3 -4.75 2.5 -0.2 1
3 2.25 0.5 1.75 0
∑ 12
𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛 =12
20= 0.6
-0.6 2.65 0.9 -4.85 2.4
-6.6 2.6 4.19 1.115 0.4
-3.6 -5.35 1.9 -0.8 0.2
2.4 1.65 -0.1 1.15 -0.6
Maximum out of flatness = |4.19| + |−5.4|=10.3
10.3 × 0.008 = 0.0824 𝑚𝑚