Flatness

Using Dial Indicator

Flatness error is the departure of the surface from a true flat plane

Objectives:

To determine the maximum out of flatness of a steel sheet using dial

indicator

Materials:

Three block gauges (Provide a rest to workpiece on three points on the bottom surface.

This will stop the movement along z-axis, rotation with respect to x-axis and y-axis)), dial

indicator, dial indicator holder and the measured sheet

Apparatus:

Experimental work:

A network is marked on the tested surface.

Determine the z coordinate of the measured point

Use round contact tip

m n o p

i j k l

e f g h

a b c d

Reading:

position a b c d e f g h i j k l m n o p

Reading 8.54 8.53 8.52 8.61 8.51 8.52 8.52 8.58 8.54 8.50 8.55 8.57 8.50 8.48 8.45 8.54

Profile:

0

30

60

908.3

8.4

8.5

8.6

8.7

0

30

60

90

Chart Title

8.6-8.7

8.5-8.6

8.4-8.5

8.3-8.4

Calculation:

x y z x'=x-xbar y' z' x'^2 y'^2 x'y' y'z' x'z' z'' delta

0 0 8.54 -45 -45 0.011875 2025 2025 2025 -0.53437 -0.53437 0.000675 0.0112

0 30 8.51 -45 -15 -0.01812 2025 225 675 0.271875 0.815625 -0.01553 -0.0026

0 60 8.54 -45 15 0.011875 2025 225 -675 0.178125 -0.53437 -0.03173 0.0436

0 90 8.5 -45 45 -0.02812 2025 2025 -2025 -1.26562 1.265625 -0.04793 0.0198

30 0 8.53 -15 -45 0.001875 225 2025 675 -0.08438 -0.02813 0.016425 -0.01455

30 30 8.52 -15 -15 -0.00812 225 225 225 0.121875 0.121875 0.000225 -0.00835

30 60 8.5 -15 15 -0.02812 225 225 -225 -0.42187 0.421875 -0.01598 -0.01215

30 90 8.48 -15 45 -0.04812 225 2025 -675 -2.16562 0.721875 -0.03218 -0.01595

60 0 8.52 15 -45 -0.00812 225 2025 -675 0.365625 -0.12187 0.032175 -0.0403

60 30 8.51 15 -15 -0.01812 225 225 -225 0.271875 -0.27187 0.015975 -0.0341

60 60 8.55 15 15 0.021875 225 225 225 0.328125 0.328125 -0.00023 0.0221

60 90 8.45 15 45 -0.07813 225 2025 675 -3.51563 -1.17188 -0.01643 -0.0617

90 0 8.61 45 -45 0.081875 2025 2025 -2025 -3.68438 3.684375 0.047925 0.03395

90 30 8.58 45 -15 0.051875 2025 225 -675 -0.77813 2.334375 0.031725 0.02015

90 60 8.57 45 15 0.041875 2025 225 675 0.628125 1.884375 0.015525 0.02635

90 90 8.54 45 45 0.011875 2025 2025 2025 0.534375 0.534375 -0.00067 0.01255

sum 720 720 136.45 0 0 7.11E-15 18000 18000 0 -9.75 9.45

aver 45 45 8.528125

Errors including in this method are:

- block error

- Tilting error

�̅� =720

16= 45

𝑦 =720

16= 45

𝑧̅ =136.45

16= 8.528125

𝑎 =∑ 𝑦′2 ∑ 𝑥′𝑧′ − ∑ 𝑥′𝑦′ ∑ 𝑦′𝑧′

∑ 𝑥′2 ∑ 𝑦′2 − (∑ 𝑥′𝑦′)2

𝑎 =18000 × 9.45 − 0 × −9.75

18000 × 18000 − 02 = −0.4854

𝑏 =∑ 𝑥′2 ∑ 𝑦′𝑧′ − ∑ 𝑥′𝑦′ ∑ 𝑥′𝑧′

∑ 𝑥′2 ∑ 𝑦′2 − (∑ 𝑥′𝑦′)2

𝑏 =444000 × −9.75 − 76208 × 9.45

18000 × 18000 − 02= −0.34669

𝑧𝑖′′ = 𝑎𝑥𝑖

′ + 𝑏𝑦𝑖′ = 0.000525𝑥′ − 0.00054𝑦′

𝛿𝑖 = 𝑧𝑖′ − 𝑧𝑖

′′

Maximum out of flatness = |0.0436| + |−0.0403| = 0.0839𝑚𝑚 = 83.9 𝑚𝑖𝑐𝑟𝑜𝑛𝑠

0.08

Flatness

Using Sensitive Level

Objectives:

To determine the maximum out of flatness of a drilling machine worktable

using sensitive level

Materials:

Two identical gauge blocks, Optical clinometer

Micrometer clinometers: clinometers used to measure small angle of inclination

with reference to the horizontal plane by placing on the surface and raising the spirit

level by micrometer until the level is accurately horizontal

Apparatus:

Experimental work:

A network is marked on the tested surface.

Determine the angle of each line

Turn the turret and turn the adjustment screw until the air bubble is

centered

Take the reading

Network:

Readings:

Level of each point relative to the preceding one

Position ba cb dc fe gf hg ji kj

Reading (angle in minute) forward

-12 -17 -15 -18 -17 -15 -18 -10

Reading (angle in minute) backward

+1 -2 -3 +1 -1 -3 -1 -2

Position lk nm on po ea ie mi

Reading (angle in minute) forward

-23 -18 -17 -16 -3 -4 -3

Reading (angle in minute) backward

+2 -1 -2 -3 -15 -13 -13

Distance between the two measured point = 100 mm

Calculations:

forward forward mm backward backward mm average

ba -12 -0.34907 1 0.029089 -0.18908

cb -17 -0.49451 -2 -0.58178 0.043631

dc -15 -0.43634 -3 -0.02873 -0.2038

ea -3 -0.08727 -15 -0.43634 0.174534

fe -18 -0.5236 1 0.029089 -0.27635

gf -17 -0.49451 -1 -0.02909 -0.23271

hg -15 -0.43634 -3 -0.02873 -0.2038

ie -4 -0.11636 -13 -0.37816 0.130901

ji -18 -0.5236 -1 -0.02909 -0.24726

kj -10 -0.29089 -2 -0.58178 0.145444

Lk -23 -0.66905 2 0.581776 -0.62541

Mi -3 -0.02873 -13 -0.37816 0.174715

Nm -18 -0.5236 -1 -0.02909 -0.24726

On -17 -0.49451 -2 -0.58178 0.043631

Po -16 -0.46542 -3 -0.02873 -0.21835

m n o p

i j k l

e f g h

a b c d

0.481 0.234 0.278 0.06

0.306 0.059 0.204 -0.421

0.175 -0.101 -0.334 -0.538

0 -0.189 -0.145 -0.349

accumulative

0.481 0.234 0.278 .06

0.306 0.059 0.204 -0.421

0.175 -0.101 -0.334 -0.538

0 -0.189 -0.145 -0.349

Make the right down corner = 0

0.481 0.350 0.510 0.409

0.306 0.175 0.46 -0.072

0.175 0.015 -0.102 -0.189

0 -0.073 0.087 0

Make the left upper corner = 0

0 -0.131 0.029 -0.072

-0.015 -0.146 0.115 -0.393

0.015 -0.145 -0.262 -0.349

0 -0.073 0.087 0

Rotate about the diagonal

0 -0.119 0.053 -0.036

-0.039 -0.146 0.127 -0.369

0.003 -0.157 -0.262 -0.325

-0.036 -0.085 -0.063 0

Maximum out of flatness = |0.127| + |−0.325| = 0.452𝑚𝑚 = 452 𝑚𝑖𝑐𝑟𝑜𝑛𝑠

0.45

0

100

200

300

0

30

60

90

Chart Title

0-0.2

-0.2-0

-0.4--0.2

Flatness

Using Rochdale Arm

Objectives:

To determine the maximum out of flatness of a planner machine waorktable

using Rochdale Arm

Materials:

Rochdale Arm, dial indicator

Apparatus:

Experimental work:

A network is marked on the tested surface (use the rochdale arm)

The arm is placed on a reference plate and takes the

dial reading.

The arm is made to rest on the tested surface on its feet

such that the feet are on two successive marked

points and the dial plunger is on contact with the

next point.

The reading of the dial is observed and recorded

The arm is then moved to another position till the

entire area is scanned.

The readings are then analyzed to determine the

surface profile, (X, Y, Z coordinates of every

point).

Network:

Readings:

Line abc aho bhn cio bcd bip djp cjq cde

Readings(mm) 1.98 1.96 1.89 1.91 1.93 2.06 1.96 1.99 2.02

R corrected 0.05 0.03 -0.04 -0.02 0 0.13 0.03 0.06 0.09

Line ekq qkr qrs els rlf rst cfg smg tmf

Readings(mm) 2.11 1.96 2.01 2.21 1.97 1.88 1.99 1.91 2.07

R corrected 0.18 0.03 0.08 0.28 0.04 -0.05 0.06 -0.02 0.14

The height readings relative to the plane aco

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

Line Rule Point Level Line Rule Point Level

abc c = 2b -a + R abe b 0.025 ekq q = 2kk -e + R ekq k -0.025

aho o = 2h - a + R aho h 0.015 qkr r = 2k - q + R qkr r -0.11

bhn n = 2h - b + R bhn n -0.035 qrs s = 2r - q + Rqrs s -0.23

cio o = 2i - c + R cio i -0.01 els s = 2l - e + R els l -0.235

bcd d = 2c - b + R bcd d -0.025 rlf f = 2l - r + R rlf f -0.32

bip p = 2i - b + R bip p 0.085 rst t = 2s - r + R rst t -0.4

djp p = 2j - d + R djp j 0.015 cfg g = 2f - c + R cfg g -0.58

cjq q = 2j - c + R cjq q 0.09 smg g = 2m - s + R smg m -0.395

cde e = 2d - c + R cde e 0.04 tmf f = 2m - t + R tmf f -0.25

𝐸𝑟𝑟𝑜𝑟 𝑖𝑛 𝑟𝑒𝑎𝑑𝑖𝑛𝑔𝑠 =−0.32 − −0.25

−0.32= 0.21875

Error within ±0.21875 from the reading

�̅� =1060

20= 70.66667

𝑦 =229.4967

20= 15.29978

𝑧̅ =−1.53

20= −0.102

𝑎 =∑ 𝑦′2 ∑ 𝑥′𝑧′ − ∑ 𝑥′𝑦′ ∑ 𝑦′𝑧′

∑ 𝑥′2 ∑ 𝑦′2 − (∑ 𝑥′𝑦′)2

𝑎 = −0.00289

𝑏 =∑ 𝑥′2 ∑ 𝑦′𝑧′ − ∑ 𝑥′𝑦′ ∑ 𝑥′𝑧′

∑ 𝑥′2 ∑ 𝑦′2 − (∑ 𝑥′𝑦′)2

𝑏 = −0.00116

𝑧𝑖′′ = 𝑎𝑥𝑖

′ + 𝑏𝑦𝑖′ = −0.00289𝑥′ − 0.00116𝑦′

𝛿𝑖 = 𝑧𝑖′ − 𝑧𝑖

′′

Maximum out of flatness = |0.253024| + |−0.24046| = 0.493484𝑚𝑚

0.49348

0

-0.6

-0.4

-0.2

0

0.2

0

13.2

5

26.5

39.7

5

53

66.

25

79.

5

92.7

5

106

119

.25

13

2.5

145.

75

159

0-0.2

-0.2-0

-0.4--0.2

-0.6--0.4

Calculation:

x y z x'=x-xbar y' z' x'^2 y'^2 x'y' y'z' x'z' z'' delta

a 0 0 0 -70.6667 -15.2998 0.102 4993.778 234.0833 1081.184 -1.56058 -7.208 0.221974 -0.11997

b 26.5 0 0.025 -44.1667 -15.2998 0.127 1950.694 234.0833 675.7403 -1.94307 -5.60917 0.145389 -0.01839

c 53 0 0 -17.6667 -15.2998 0.102 312.1111 234.0833 270.2961 -1.56058 -1.802 0.068804 0.033196

d 79.5 0 -0.025 8.833333 -15.2998 0.077 78.02778 234.0833 -135.148 -1.17808 0.680167 -0.00778 0.084781

e 106 0 0.04 35.33333 -15.2998 0.142 1248.444 234.0833 -540.592 -2.17257 5.017333 -0.08437 0.226366

f 132.5 0 -0.32 61.83333 -15.2998 -0.218 3823.361 234.0833 -946.036 3.335352 -13.4797 -0.16095 -0.05705

g 159 0 -0.58 88.33333 -15.2998 -0.478 7802.778 234.0833 -1351.48 7.313295 -42.2233 -0.23754 -0.24046

h 13.25 22.94967 0.015 -57.4167 7.649893 0.117 3296.674 58.52087 -439.231 0.895038 -6.71775 0.15706 -0.04006

i 39.75 22.94967 -0.01 -30.9167 7.649893 0.092 955.8403 58.52087 -236.509 0.70379 -2.84433 0.080475 0.011525

j 66.25 22.94967 0.015 -4.41667 7.649893 0.117 19.50694 58.52087 -33.787 0.895038 -0.51675 0.00389 0.11311

k 92.75 22.94967 -0.025 22.08333 7.649893 0.077 487.6736 58.52087 168.9351 0.589042 1.700417 -0.07269 0.149695

l 119.25 22.94967 -0.235 48.58333 7.649893 -0.133 2360.34 58.52087 371.6573 -1.01744 -6.46158 -0.14928 0.01628

m 145.75 22.94967 -0.395 75.08333 7.649893 -0.293 5637.507 58.52087 574.3795 -2.24142 -21.9994 -0.22586 -0.06714

n 0 45.89935 -0.035 -70.6667 30.59957 0.067 4993.778 936.3335 -2162.37 2.050171 -4.73467 0.168731 -0.10173

o 26.5 45.89935 0 -44.1667 30.59957 0.102 1950.694 936.3335 -1351.48 3.121156 -4.505 0.092146 0.009854

p 53 45.89935 0.085 -17.6667 30.59957 0.187 312.1111 936.3335 -540.592 5.722119 -3.30367 0.015561 0.171439

q 79.5 45.89935 0.09 8.833333 30.59957 0.192 78.02778 936.3335 270.2962 5.875117 1.696 -0.06102 0.253024

r 106 45.89935 -0.11 35.33333 30.59957 -0.008 1248.444 936.3335 1081.185 -0.2448 -0.28267 -0.13761 0.129609

s 132.5 45.89935 -0.23 61.83333 30.59957 -0.128 3823.361 936.3335 1892.073 -3.91674 -7.91467 -0.21419 0.086194

t 159 45.89935 -0.4 88.33333 30.59957 -0.298 7802.778 936.3335 2702.962 -9.11867 -26.3233 -0.29078 -0.00722

Sum 1060 229.4967 -1.53 -5E-08 0 -2.2E-16 39911.21 3862.375 -4054.44 7.229147 -110.704

Average 70.66667 15.29978 -0.102