Form measurement

Screw thread measurement:
• Introduction:
• Screw threads are used to transmit power and motion and also used to fasten two components
with the help of nuts, bolts and studs.
• There is a large variety of screw threads varying in their form, by included angle, head angle,
helix angle etc.
• The screw threads are mainly classified into: 1) External Screw Threads & 2) Internal Screw
Threads.

• Other Types of Threads:
• Threads can also be classified as
i. British Association Thread: In this thread, the angle included inside the thread is 47.5̊. If p =
pitch of the thread, d = depth of the thread, r = radius at the top and bottom of the threads,
then d = 0.6p, r = 2p/11.
ii. Whitworth Thread: In this thread, the angle included inside the thread is 55̊. If p = pitch of the
thread, d = depth of the thread, r = radius at the top and bottom of the threads, then d =
0.640327 p, r = 0.137329p.
iii. Metric Thread: In this thread, the angle included inside the thread is 60̊. If p = pitch of the
thread, d = depth of the thread, r = radius at the top and bottom of the threads, then d =
0.54127 p.

1. Screw Thread: It is a continuous helical groove of specified cross-section produced on the
external or internal surface.
2. Crest: It is the top surface joining two sides of thread.
3. Flank: It is the surface between crest and root or it is the thread surface that connects crest with
root.
4. Root: The bottom of the groove between the two flanks of the thread.
5. Lead: The distance a screw thread advances in one turn. For a single start threads, lead=pitch,
For double start, lead=2xpitch, & so on.
6. Pitch: The distance from a point on a screw thread to a corresponding point on the next thread
measured parallel to the axis.
7. Helix Angle: The angle made by the helix of the thread at the pitch line with the axis is called
as helix angle.
8. Flank angle: It is half the included angle of the thread or angle made by the flank of the thread
with the perpendicular to the thread axis.

9. Depth of thread: It is the distance between crest and root measured perpendicular to axis of
screw.
10. Angle of thread: It is the angle included between the flanks of a thread measured in an axial
plane.
11. Major Diameter: This is the diameter of an imaginary cylinder, co-axial with the screw, which
just touches the crests of an external thread or roots of an internal threads. It is also called as
‘Nominal diameter’.
12. Minor diameter: This is the diameter of an imaginary cylinder, co-axial with the screw which
just touches the roots of an external thread or the crest of an internal thread. This is also
referred to as ‘root’ or ‘core diameter’.
13. Effective diameter or Pitch diameter: It is the diameter of an imaginary cylinder coaxial with
the axis of the thread and intersects the flanks of the thread such that width of the threads &
width of spaces between threads are equal.

14. Addendum: It is the distance between the crest and the pitch line measured perpendicular to axis of the screw.
15. Dedendum: It is the distance between the pitch line & the root measured perpendicular to axis of the screw.
Errors in Screw Thread:
The error in screw thread may arise during the manufacturing or storage of threads. The error either may
cause due to the following six main elements in the thread.
1. Major diameter error 2. Minor diameter error 3. Effective diameter error 4. Pitch error , 5. Flank angles error 6.
Crest and root error
1. Major diameter error
It may cause reduction in the flank contact and interference with the matching threads.
2. Minor diameter error
It may cause interference, reduction of flank contact.
3. Effective diameter error
If the effective diameter is small the threads will be thin on the external screw and thick on an internal screw.
4. Pitch error
Pitch error is defined as the total length of thread engaged either too high or too small. The various pitch errors
may be classified into
1. Progressive error 2. Periodic error. 3. Drunken error 4. Irregular error.

i) Progressive error:
The pitch of the thread is uniform but it is longer or shorter to its nominal value and this is called
progressive error.
Causes of Progressive error:
1. Incorrect linear and angular velocity ratio.
2. Incorrect gear train and lead screw.
3. Saddle fault.
4. Variation in length due to hardening.
ii) Periodic error:
Periodic errors are repeated itself at regular intervals along the thread.

Causes of Periodic error:
1. Ununiform tool work velocity ratio.
2. Teeth error in gears.
3. Lead screw error.
4. Eccentric mounting of the gears.
(iii) Drunken error:
Drunken errors are repeated once per turn of the thread in a Drunken thread. In Drunken thread,
the pitch is measured parallel to the thread axis. If the thread is not cut to the true helix, the
Drunken thread error will be formed.

(iv) Irregular error:
It varies in irregular manner along the length of the thread.
Causes of Irregular error:
1. Machine fault.
2. Non-uniformity in the material.
3. Cutting action is not correct.
4. Machining disturbances.
Effect of pitch error:
1. It increases the effective diameter of the bolt and decreases the diameter of nut.
2. The functional diameter of the nut will be less.
3. It reduces the clearance.
4. It increases the interference between mating threads.

measurement of various elements of Screw thread :
• To find out the accuracy of a screw thread it will be necessary to measure the following:
1. Major diameter.
2. Minor diameter.
3. Effective or Pitch diameter.
4. Pitch
5. Thread angle and form
1. Measurement of Major diameter:
The instruments which are used to find the major diameter are
1. Ordinary micrometer
2. Bench micrometer.

i) Ordinary Micrometer:
The ordinary micrometer is quite suitable for measuring the external major diameter.
It is first adjusted for appropriate cylindrical size (S) having the same diameter
(approximately).This process is known as ‘gauge setting’ .
After taking this reading ‘R1’the micrometer is set on the major diameter of the thread, and the
new reading is ‘R2’.
Then the major diameter, D=S±(R1-R2)
S = Size of setting gauge
R1 = Micrometer reading over setting gauge.
R2 = Micrometer reading over thread.
ii) Measurement by Bench micrometer:
For getting the greater accuracy the bench micrometer is used for measuring the major diameter.
In this process the variation in measuring Pressure, pitch errors are being neglected.

The fiducial indicator is used to ensure all the measurements are made at same pressure.
The instrument has a micrometer head with a Vernier scale to read the accuracy of 0.002mm.
Calibrated setting cylinder having the same diameter as the major diameter of the thread to be
measured is used as setting standard.
After setting the standard, the setting cylinder is held between the anvils and the reading is taken.
Then the cylinder is replaced by the threaded work piece and the new reading is taken.

Therefore, the major diameter of screw thread D = S ± (R2 - R1)
Where, S = Diameter of the setting cylinder
 R2 = Micrometer Reading on screw thread
 R1 = Micrometer reading on setting cylinder.

• Measurement of the major diameter of an Internal thread:
The internal thread major diameter is usually measured by thread comparator fitted with
ball-ended stylii.
Initially, the Instrument is set for a cylindrical reference having the same diameter of
major diameter of internal thread and the reading is taken.
Then the floating head is retracted to engage the tip of the stylii at the root of spring
under' pressure.
For that, the new reading is taken.
The major diameter of internal thread = D ± (R2 ~ R1)
Where, D = Cylindrical standard diameter
 R2 = Thread reading
 R1 = Dial Indicator reading on the standard.

2. Measurement of Minor diameter:
The minor diameter is measured by a comparative method by using floating carriage
diameter measuring machine and small ‘V’ pieces which make contact with the root of
the thread.
These V pieces are made in several sizes, having suitable radii at the edges. V pieces are
made of hardened steel.
The floating carriage diameter-measuring machine is a bench micrometer mounted on a
carriage.

Measurement Process:
The threaded work piece is mounted between the centres of the instrument and the V
pieces are placed on each side of the work piece and then the reading is noted.
After taking this reading the work piece is then replaced by a standard reference
cylindrical setting gauge.
The minor diameter of the thread = D ± (R2 –R1)
Where, D = Diameter of cylindrical gauge
 R2 = Micrometer reading on threaded workpiece,
 R1 = Micrometer reading on cylindrical gauge.
Measurement of minor diameter of internal threads:
The minor diameter of internal threads are measured by
1. Using taper parallels
2. Using Rollers.

1. Using taper parallels:
 For diameters less than 200mm the use of Taper parallels and micrometer is very
common.
The taper parallels are pairs of wedges having reduced and parallel outer edges.
The diameter across their outer edges can be changed by sliding them over each other.
2. Using Rollers.
For more than 200mm diameter this method is used.
Precision rollers are inserted inside the thread and proper slip gauge is inserted between
the rollers.
The minor diameter is then the length of slip gauges plus twice the diameter of roller.

3. Measurement of effective diameter or Pitch Diameter:
• The effective diameter measurement is carried out by the following methods.
1. One Wire Method 2. Two Wires Method 3. Three Wires Method 4. Micrometer Method.
1. One Wire Method:
In this method, only one wire is used. The wire is placed between the two threads at one
side and on the other side the anvil of the measuring micrometer contacts the crests.
First, the micrometer reading ‘d1’ is noted on a standard gauge whose dimension is
approximately same to be obtained by this method.
Now, the setting gauge is replaced by thread and the new reading is taken i.e. ‘d2’ then
effective diameter = D ± (d1-d2).

Where, D = Size of setting gauge.
Actual measurement over wire on one side and threads on other side = size of gauge ±
difference in two micrometer readings.
2. Two wire method:
The effective diameter can not be measured directly but can be calculated from the
measurements made.
In this method, wires of suitable size are placed between the standard and the micrometer
anvils.
First, the micrometer reading is taken and let it be R1.
Then the standard is replaced by the screw thread to be measured and the new reading is
taken. The new reading is R2.
Then the screw thread is mounted between the centers & wires are placed in the grooves
and reading M is taken.

From the above reading,
Effective diameter E is calculated by E = T + P
Where, T = Dimension under the wires = M - 2d
 M = Dimension over the wires
 d = diameter of each wire
If p = Pitch of thread then, P = 0.9605 p - 1.1657 d => For Whitworth thread.
 P = 0.866 p - d => For metric thread.

Here, P = the difference between the effective diameter and the diameter under the wires
or P is a value which depends on diameter of wire, pitch & angle of the screw thread.
The diameter under the wires ‘T’ can also be determined by T= S - (R1 – R2)
Where, S = The diameter of the standard.
 R1 = Micrometer reading over standard and wires.
 R2 = Micrometer reading over screw thread and wires.

Actually P is a constant value which has to be added to the
diameter under the wires to give the effective diameter. The
expression for the value of P in terms of p(pitch), d(diameter
of wire) and x(thread angle) can be derived as follows:
In the fig, since BC lies on the effective diameter line,
BC = ½ pitch = ½ p.
From fig, In ∆OPE, cosec x/2 =
𝑂𝑃
𝑂𝐸
cosec x/2 =
𝑂𝑃
𝑑/2
OP =
𝑑 𝑐𝑜𝑠𝑒𝑐 𝑥/2
2
………..(1)
Now, PA = OP – OA
=
𝑑 𝑐𝑜𝑠𝑒𝑐 𝑥/2
2
-
𝑑
2
PA =
𝑑
2
𝑐𝑜𝑠𝑒𝑐 𝑥/2− 1 ………………...(2)

Now from ∆PCQ, cot x/2 =
𝑃𝑄
𝑄𝐶
PQ = QC cot x/2
PQ = p/4 cot x/2
AQ = PQ – AP
AQ =
𝑝 cot 𝑥/2
4
-
𝑑
2
𝑐𝑜𝑠𝑒𝑐
𝑥
2
− 1 ………(3)
Therefore, this AQ gives only half of the value of P. Therefore,
P = 2AQ
P =
𝒑
𝟐
cot
𝒙
𝟐
- d 𝒄𝒐𝒔𝒆𝒄
𝒙
𝟐
− 𝟏 ……(4)
Now, if we substitute, x = 60̊ for Metric Thread, we get, P = 0.866p – d …….(5)
And, if we substitute, x = 55̊ for Whitworth Thread, we get, P = 0.9605p – 1.1657 d
….(6)

3. Three wire method:
The three-wire method is the accurate method.
In this method, three wires of equal and precise diameter are placed in the grooves at
opposite sides of the screw.
In this, one wire on one side and two on the other side are used.
The wires either may be held in hand or hung from a stand. This method ensures the
alignment of micrometer anvil faces parallel to the thread axis.

From the fig, M=diameter over the wires
E= effective diameter (to be found)
d= diameter of wires,
h=height of wire center above the pitch line,
r=radius of wire,
H=depth of thread,
D=major diameter of the thread,
x = angle of thread.
O

From the fig, ∆ABD, cosec x/2 =
𝐴𝐷
𝐴𝐵
AD = AB cosec x/2
AD = r cosec x/2……..(1)
In ∆DEO, cot x/2 =
𝑂𝐸
𝐷𝐸
OE = DE cot x/2
H =
𝑝
2
cot x/2……..(2)
Now, CD =
𝐻
2
Therefore, CD =
𝑝
4
cot x/2 …….(3)
h = AD – CD
h = r cosec x/2 -
𝑝
4
cot 𝑥/2 ……(4)
O

We know that, distance over the wires,
= M = E + 2h + 2r
= E + 2(r cosec x/2 -
𝑝
4
cot x/2) + 2r
= E + 2r cosec x/2 + 2r -
𝑝
2
cot x/2
= E + 2r (cosec x/2+1) -
𝑝
2
cot x/2
M = E + d (1+cosec x/2) -
𝒑
𝟐
cot x/2 …..(5)
Case i) In case of Whitworth Thread:
x = 55̊ , Depth of thread = 0.64p
& E = D-0.64p,
Now, if we substitute the above values in
equation (5), we get,
O

M = D + 3.1657d – 1.6p ……..(6)
where, D = Outside Diameter
Case ii) In case of Metric Thread:
x = 60̊ , Depth of thread = 0.6495p
& E = D-0.6495p,
Now, if we substitute the above values in equation (5), we get,
M = D + 3d – 1.5155p ……….(7)
We can practically measure the value of M & then compare with the theoretical values
using the formula derived above. After finding the correct value of M, as d is known, E
can be found out.

Best Size Wire:
Best size wire is one in which, the wire is having such a diameter that it makes contact
with the flanks of the thread on the effective diameter or pitch line.
It is recommended that for measuring the effective diameter, always the best size wire
should be used and for this condition the wire touches the flank at mean diameter line
within ±1/5 of flank length.
Let, x/2 = Half included angle of thread.
In ∆OAB, sin BOA =
𝐴𝐵
𝑂𝐵
sin (90̊ - x/2) =
𝐴𝐵
𝑂𝐵
OB =
𝐴𝐵
sin(90−
𝑥
2
)
 =
𝐴𝐵
cos
𝑥
2

OB = 𝐴𝐵 sec
𝑥
2
Since, OB = r & wire diameter = 2r,
Therefore, Best Size Wire Diameter,= db = 2r = 2ABsec
𝑥
2
As, AB lies on the pitch line, therefore, AB =
𝑝
4
(p – pitch of thread)
Therefore, db =
2𝑝
4
sec
𝑥
2
And finally Best Size Wire Diameter = db =
𝒑
𝟐
sec
𝒙
𝟐

Effective diameter measurement by thread micrometer:
The thread micrometer is used to measure threads within certain range of thread pitches.
Any given micrometer is required to measure range of threads of different pitches, a
small error can arise because of anvil settings.
In this method, first the pitch diameter is measured for a standard plug gauge of the same
size as the thread to be measured and then compensate for the error.
The thread micrometer has special contacts to suit with the end screw thread for
checking.

In this micrometer, the end of spindle is pointed to the ‘V’ thread form with a
corresponding Vee-recess in the fixed anvil.
When measuring thread is only the angle of point and flanks of the thread come into
contact.
If the contacts are adjusted correctly, the micrometer gives the pitch diameter.
Advantage of thread micrometer:
1. This is the only method which shows the variation in drunken thread.
Limitations:
1. It must be set to a standard thread plug.
2. When a standard plug gauge is set the reading is not exactly zero.

4. Pitch Measurement:
The most commonly used methods for measuring the pitch are
1. Pitch Measuring Machine
2. Tool Makers Microscope
3. Screw Pitch Gauge
i) Pitch Measuring Machine:
The principle of this method of measurement is to move the stylus along the screen parallel to the
axis from one space to the next.
The pitch-measuring machine provides a relatively simple and accurate method of measuring the
pitch.
Initially, the micrometer reading is set near the zero on the scale.
Spring loaded head permits the stylus to move up the flank of the thread and down into the next
space as it is moved along.
Accurate positioning of the stylus between the two flanks is obtained by ensuring that the pointer
T is always opposite to its index mark when readings are taken.

When the pointer is accurately placed in position, the micrometer reading is noted.
The stylus is then moved along into the next thread space, by rotation of the micrometer, and a
second reading taken.
The difference between the two readings is the pitch of the thread.
Readings are taken in this manner until the whole length of the screw thread has been covered.

ii. Tool Makers Microscope:
1. Worktable is placed on the base of the instrument.
2. The optical head is mounted on a vertical column it can be moved up and down.
3. Work piece is mounted on a glass plate.
4. A light source provides horizontal beam of light which is reflected from a mirror by 90
degree upwards towards the table.
5. Image of the outline of contour of the work piece passes through the objective of the
optical head.
6. 6. The image is projected by a system of three prisms to a ground glass screen.
7. 7. The measurements are made by means of cross lines engraved on the ground glass
screen.
8. 8. The screen can be rotated through 360°.
9. 9. Different types of graduated screens and eyepieces are used.

Applications:
1. Linear measurements.
2. Measurement of pitch of the screw.
3. Measurement of pitch diameter.
4. Measurement of thread angle.
5. Comparing thread forms.
6. Centre to center distance measurement.

iii. Screw pitch gauge:
• It is used to directly compare the pitch by just selecting the proper pitch value entered in
the pitch gauge and comparing it with the actual screw thread.

5. Thread form and flank angle Measurement:
• The optical projectors are used to check the thread form and angles in the thread.
• The projectors are equipped with work holding fixtures, lamp, and lenses.
• The light rays from the lens are directed into the cabinet and prisms and mirrors.
• The enlarged image of thread is drawn.
• The ideal and actual forms are compared for the measurement.

Gear measurement:
• Gears are mechanical drives which transmit power through toothed wheel.
• In this gear drive, the driving wheel is in direct contact with driven wheel.
• The accuracy of gearing is very important factor when gears are manufactured.
• The transmission efficiency is almost 99% for gears.
• So, it is very important to test and measure the gears precisely.
• For proper inspection of gear, it is very important to concentrate on the raw materials, which are
used to manufacture the gears.
• Also very important to check the machining of the blanks, heat treatment and finishing of teeth.
• The gear blanks should be tested for dimensional accuracy and tooth thickness for the forms of
gears.

The most commonly used forms of gear teeth are
1.Involute
2.Cycloidal
The involute gears also called as straight tooth or spur gears.
The cycloidal gears are used in heavy and impact loads.
The involute rack has straight teeth.
The involute pressure angle is either 20̊ or 14.5̊
Types of Gears
1. Spur gear
Cylindrical gear whose tooth traces is straight line. These are used for transmitting power
between parallel-shafts.
2. Spiral gear
The tooth of the gear traces is in the form of curved lines.

3. Helical gears
These gears are used to transmit the power between parallel shafts as well as non-parallel
and non-intersecting shafts. It is a cylindrical gear whose tooth traces is straight line.
4. Bevel gears
The tooth traces are straight-line generators of cone. The teeth are cut on the conical
surface. It is used to connect the shafts at right angles.
5. Worm and Worm wheel
It is used to connect the shafts whose axes are non-parallel and non-intersecting.
6. Rack and Pinion
Rack gears are straight spur gears with infinite radius.

Gear(spur) terminologies:
1. Tooth Profile: It is the shape of any side of gear tooth in its cross section.
2. Base circle: It is the circle of gear from which the involute profile is derived. Base circle diameter =
Pitch circle diameter x Cosine of pressure angle of gear
3. Pitch circle diameter (PCD): It is the diameter of a circle which will produce the same motion as the
toothed gear wheel.
4. Pitch circle: It is the imaginary circle of gear that rolls without slipping over the circle of its mating
gear.
5. Addendum circle: The circle that coincides with the crests (or) tops of teeth.
6. Dedendum circle (or) Root circle: This circle that coincides with the roots (or) bottom of teeth.
7. Pressure angle (α): It is the angle made by the line of action with the common tangent to the pitch
circles of mating gears.
8. Module (m): It is the ratio of pitch circle diameter to the total number of teeth. m =
𝑑
𝑛
Where, d = Pitch circle diameter, n =Number of teeth.

9. Circular pitch: It is the distance along the pitch circle between corresponding points of
adjacent teeth.
Pc =
𝜋𝑑
𝑛
= 𝜋m
10. Diametral pitch (Pd): Number of teeth per inch of the PCD.
Pd =
𝑛
𝑑
=
1
𝑚
Where, m = Module
11. Addendum: It is the radial distance between tip circle and pitch circle.
Addendum value = 1 module.
12. Dedendum: It is the radial distance between pitch circle and root circle.
Dedendum value=1.25 module.
13. Clearance(c): The distance covered by the tip of one gear with the root of mating gear.
Clearance = Difference between Dedendum and addendum values.
14. Blank diameter: It is the diameter of the blank upto outer periphery.
Blank diameter = PCD+2m

9. Face: It is the part of the tooth in the axial plane lying between tip circle and pitch circle.
10. Flank: It is the part of the tooth lying between pitch circle and root circle.
11. Helix angle: It is the angle between the tangents to helix angle.
12. Top land: Top surface of a tooth is called as top land.
13. Lead angle: It is the angle between the tangent to the helix and plane perpendicular to the axis
of cylinder.
14. Backlash: It is the difference between the tooth thickness and the space into which it meshes.
If we assume the tooth thickness as ‘t1 ’ and width ‘t2 ’then

Gear errors:
1. Profile error: The maximum distance is at any point on the tooth profile form to the design
profile.
2. Pitch error: It is the difference between actual and design pitch.
3. Cyclic error: Error occurs in each revolution of gear.
4. Run out: Total range of a fixed indicator with, the contact points applied to a surface rotated,
without axial movement, about a fixed axis.
5. Eccentricity: It is the half radial run out.
6. Wobble: Run out is measured parallel to the axis of rotation at a specified distance from the
axis.
7. Radial run out: Run out is measured along a perpendicular to the axis of rotation.
8. Undulation: It is the periodical departure of the actual tooth surface from the design surface.
9. Axial run out: Run out is measured parallel to the axis of rotation at a speed.
10. Periodic error: Error occurs at regular intervals.

Spur Gear Measurement:
The inspection of the gears consists of the following elements in which manufacturing error may
be present.
1. Runout
2. Pitch
3. Profile
4. Lead
5. Backlash
6. Tooth thickness
7. Concentricity
8. Alignment

1. Run out: It means eccentricity in the pitch circle. It will give periodic vibration during each
revolution of the gear. This will give the tooth failure in gears. The run out is measured by
means of eccentricity testers. In testing, the gears are placed in the mandrel and the dial
indicator of the tester possessed by special tip depending upon the module of the gear and the
tips inserted between the tooth spaces. Then, the gears are rotated tooth by tooth and the
variation is noted from the dial indicator.
2. Pitch measurement: There are two ways for measuring the pitch.
a) Point to point measurement (i.e. One tooth point to next tooth point)
b) Direct angular measurement
a) Tooth to Tooth measurement:
The instrument has three tips. One is fixed measuring tip and the second is sensitive tip, whose
position can be adjusted by a screw and the third tip is adjustable or guide stop. The distance
between the fixed and sensitive tip is equivalent to base pitch of the gear. All the three tips are
made in contact with the tooth by setting the instrument and the reading on the dial indicator is the
error in the base pitch.

2. Direct Angular Measurement: It is the simplest method for measuring the error by using set
dial gauge against a tooth. In this method, the position of a suitable point on a tooth is
measured after the gear has been indexed by a suitable angle. If the gear is not indexed through
the angular pitch the reading differs from the original reading. This difference is the
cumulative pitch error.
3. Profile checking: The methods used for profile checking are
1) Optical projection method 2)Involute measuring machine.

i. Optical projection method: The profile of the gear is projected on the screen by optical lens
and then the projected value is compared with master profile.
ii. Involute measuring machine: In this method, the gear is held on a mandrel and circular disc of
same diameter as the base circle of gear for the measurement is fixed on the mandrel. After
fixing the gear on the mandrel, the straight edge of the instrument is brought in contact with
the base circle of the disc. Now, the gear and disc are rotated and the edge moves over the disc
without slip. The stylus moves over the tooth profile and the error is indicated on the dial
gauge.

4. Lead checking: It is checked by lead checking instruments. Actually, lead is the axial advance
of a helix for one complete turn. The lead checking instruments advance a probe along a tooth
surface, parallel to the axis when the gear rotates.
5. Backlash checking: Backlash is the distance through which a gear can be rotated to bring its
non-working flank in contact with the teeth of mating gear. Numerical values of backlash are
measured at the tightest point of mesh on the pitch circle.
There are two types of backlash
1. Circumferential backlash
2. Normal backlash
The determination of backlash is, first one of the two gears of the pair is locked, while other is
rotated forward and backward and the maximum displacement is measured by the comparator. The
stylus of comparator is locked near the reference cylinder and a tangent to this is called circular
backlash.

6. Tooth thickness measurement
Tooth thickness is generally measured at pitch circle and also in most cases the chordal thickness
measurement is carried out .i.e. the chord joining the intersection of the tooth profile with the pitch
circle. The methods which are used for measuring. the gear tooth thickness are
a) Gear tooth Vernier caliper method (Chordal thickness method)
b) Base tangent method.
e) Constant chord method.
d) Measurement over pins or ball
a) Gear tooth Vernier method:
The tooth thickness can be very conveniently measured by a gear tooth vernier. Since the gear
tooth thickness varies from the tip of the base circle of the tooth, the instrument must be capable of
measuring the tooth thickness at a specified position on the tooth. Further this is possible only
when there is some arrangement to fix that position where the measurement is to be taken. The
tooth thickness is generally measured at pitch circle and is, therefore, referred to as pitch-line
thickness of tooth. The gear tooth vernier has two vernier scales and they are set for the width (w)
of the tooth and the depth (d) from the top, at which w occurs.

Considering one gear tooth, the theoretical values of w and d can be found out which may be
verified by the instrument. In Fig, it may be noted that w is a chord ADB, but tooth thickness is
specified as an arc distance AEB. Also the distance d adjusted on instrument is slightly greater
than the addendum CE, w is therefore called chordal thickness and d is called the chordal
addendum.
In Fig, w =AB = 2AD
Now, A𝑂D = θ = 360/4N, where N is the number of teeth,
w = 2AD = 2xAO sinθ = 2R sin 360/4N(R=pitch circle radius)

Module m =
𝑃.𝐶.𝐷.
𝑁𝑜.𝑜𝑓 𝑡𝑒𝑒𝑡ℎ
=
2𝑅
𝑁
, ∴ R=
𝑁.𝑚
2
w=2
𝑁.𝑚
2
sin
360
4𝑁
=𝑁.𝑚. sin
90
𝑁
……..(1)
Also from fig, d=OC-OD
But OC=OE + addendum=R + m=(Nm/2)+m
and OD=R cosθ =
𝑁.𝑚
2
𝑐𝑜𝑠
90
𝑁
∴ d=
𝑁.𝑚
2
+ m -
𝑁.𝑚
2
𝑐𝑜𝑠
90
𝑁
=
𝑁.𝑚
2
1 +
2
𝑁
− 𝑐𝑜𝑠
90
𝑁
…………(2)

Any error in the outside diameter of the gear must be allowed for when measuring tooth
thickness.
In the case of helical gears, the above expressions have to be modified to take into account the
change in curvature along the pitch line. The virtual number of teeth Nv for helical gear = N/cos³α
(α= helix angle)
Hence in Eqs.(1) and (2), N can be replaced by N/cos³α and m by mn(normal module).
∴w=
𝑁.𝑚n
𝑐𝑜𝑠³𝛼
sin
90
𝑁
𝑐𝑜𝑠3𝛼 =𝑁.𝑚. sin
90
𝑁
, and d=
𝑁.𝑚n
𝑐𝑜𝑠³𝛼
1 +
2𝑐𝑜𝑠³𝛼
𝑁
− 𝑐𝑜𝑠
90
𝑁
𝑐𝑜𝑠³𝛼
These formulae apply when backlash is ignored. On mating gears having equal tooth thickness
and without addendum modifications, the circular tooth thickness equals half the circular pitch
minus half the backlash.
Disadvantages of Gear Tooth Vernier method:
1. Not closer to 0.05mm.
2. Two Vernier readings are required.
3. Measurement is done by edge of measuring jaw and not by face.

Gear Tooth Vernier Calliper:
It is used to measure the thickness of gear teeth at the pitch line or chordal thickness of teeth and
the distance from the top of a tooth to the chord.
The thickness of a tooth at pitch line and the addendum is measured by an adjustable tongue,
each of which is adjusted independently by adjusting screw on graduated bars.
The effect of zero errors should be taken into consideration.
This method is simple and inexpensive.
However it needs different setting for a variation in number of teeth for a given pitch and
accuracy is limited by the least count of instrument.
Since the wear during use is concentrated on the two jaws, the calliper has to be calibrated at
regular intervals to maintain the accuracy of measurement.

Gear tooth thickness Measurement by Constant Chord Method:
• In the Gear Tooth Vernier Caliper method, it is seen that both the chordal thickness and chordal
addendum are dependent upon the number of teeth.
• Hence for measuring a large number of gears for set, each having different number of teeth
would involve separate calculations. Thus the procedure becomes laborious and time-
consuming one.
• The constant chord method does away with these difficulties.
• Constant chord of a gear is measured where the tooth flanks touch
the flanks of the basic rack.
• The teeth of the rack are straight and inclined to their centre lines
at the pressure angle as shown in Fig.

• Also the pitch line of the rack is tangential to the pitch circle of the gear and, by definition, the
tooth thickness of the rack along this line is equal to the arc tooth thickness of the gear round its
pitch circle.
• Now, since the gear tooth and rack space are in contact in the symmetrical position at the points
of contact of the flanks, the chord is constant at this position irrespective of the gear of the
system in mesh with the rack.
• This is the property utilised in the constant chord method of the gear measurement.
• The measurement of tooth thickness at constant chord simplified the problem for all number of
teeth.
• If an involute tooth is considered symmetrically in close mesh with a basic rack form, then it will
be observed that regardless of the number of teeth for a given size of tooth (same module), the
contact always occurs at two fixed point A and B.
• AB is known as constant chord.
• The constant chord is defined as the chord joining those points, on opposite faces of the tooth,
which make contact with the mating teeth when the centre line of the tooth lies on the line of the
gear centres.

• The value of AB and its depth from the tip, where it occurs can be calculated mathematically and
then verified by an instrument.
• The advantage of the constant chord method is that for all number of teeth (of same module)
value of constant chord is same.
• In other words, the value of constant chord is constant for all gears of a meshing system.
• Secondly it readily lends itself to a form of comparator which is more sensitive than the gear
tooth vernier.
• In the fig, PD=PF=arc PF= ¼ x circular pitch
= ¼ x
𝜋 𝑥 𝑃𝑖𝑡𝑐ℎ 𝐶𝑖𝑟𝑐𝑙𝑒 𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟(𝑃.𝐶.𝐷.)
𝑁(𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑇𝑜𝑜𝑡ℎ)
= ¼ x π x m (since, Module m =
𝑃.𝐶.𝐷.
𝑁
)
• Since line AP is the line of action. i.e. it is tangential to the base circle,
CAP = φ. Therefore, in right angled ∆APD, AP=PD cos φ = (
π
4
)m cos φ

• In ∆ PAC, AC=AP cos φ = (π/4)m cos2φ
• c= constant chord=2AC=(π/2) mn cos2φn
• Where, mn =normal module. i.e. module of cutter used and φn =normal pressure angle.
• Now PC=AP sin φ = (π/4)m cosφ sinφ
• Therefore, d=addendum-PC=m- (π/4)m cosφ sinφ =m 1 − (π/4)cosφ sinφ
• 𝐹𝑜𝑟 ℎ𝑒𝑙𝑖𝑐𝑎𝑙 𝑔𝑒𝑎𝑟, 𝑑 = 𝑚n(1 − (𝜋/4)𝑐𝑜𝑠𝜑n𝑠𝑖𝑛𝜑n)
• Also height of AB above pitch line=PC= (πm/4) cosφ sinφ
= (πm/8) sin2φ

Gear tooth thickness Measurement by base tangent(david brown tangent comparator)
Method:
• In this method, the span of a convenient number of teeth is measured with the help of the
tangent comparator.
• This uses a single vernier calliper and has, therefore, the following advantages over gear tooth
vernier which used two vernier scales :
i. the measurements do not depend on two vernier readings, each being function of the other.
ii. the measurement is not made with an edge of the measuring jaw with the face.
Consider a straight generator (edge) ABC being rolled back and forth along a base circle (Fig.
15.19).
Fig. 15.19. Generation of pair of involutes by a
common generator.

 Its ends thus sweep out opposed involutes A2AA1 and C2CC1 respectively.
Thus the measurements made across these opposed involutes by span gauging will be constant
(i.e. AC = A1C1 = A2C2 = A0C0) and equal to the arc length of the base circle between the origins
of involutes.
Further the position of the measuring faces is unimportant as long as they are parallel and on an
opposed pair of the true involutes.
As the tooth form is most likely to conform to a true involute at the pitch point of the gear, it is
always preferable to choose a number of teeth such that the measurement is made approximately
at the pitch circle of the gear.
The value of the distance between two opposed involutes, or the dimension over parallel faces is
equal to the distance round the base circle between the points where the corresponding tooth
flanks cut i.e., ABC in Fig. 15.19.
It can be derived mathematically as follows :

 The angle between the points A and C on the pitch circle where the flanks of the opposed
involute teeth of the gear cut this circle can be easily calculated.
Let us say that the gear has got W number of teeth and AC on pitch circle corresponds to ‘S’
number of teeth. (Fig. 15.20); Therefore, Distance AC = (S – 1/2) pitches
Therefore, Angle subtended by AC = (S- 1/2) x 2π/N radians.
Angles of arcs BE and BD
Involute function of pressure angle = δ = tanφ – φ
Therefore, Angle of arc BD = 𝑆 −
1
2
𝑥
2𝜋
𝑁
+ 2 𝑡𝑎𝑛𝜑 – 𝜑
Therefore, BD = Angle of arc BD x Rb
BD = 𝑆 −
1
2
𝑥
2𝜋
𝑁
+ 2 𝑡𝑎𝑛𝜑 – 𝜑 x Rp cosφ 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 Rb=Rp cosφ
=
𝑚𝑁
2
cosφ 𝑆 −
1
2
𝑥
2𝜋
𝑁
+ 2 𝑡𝑎𝑛𝜑 – 𝜑 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 Rp=
𝑚𝑁
2
= NM cosφ
𝜋𝑆
𝑁
−
𝜋
2𝑁
+ 𝑡𝑎𝑛𝜑 – 𝜑

 As already defined, length of arc BD = distance between two opposed involutes and thus it is
= NM cosφ 𝑡𝑎𝑛𝜑 – 𝜑 −
𝜋
2𝑁
+
𝜋𝑆
𝑁
It may be noted that when backlash allowance is specified normal to the tooth flanks, this must
be simply subtracted from this derived value.
Tables are also available which directly give this value for the given
values of S, N and m.
This distance is first calculated and then set in the ‘David Brown’
tangent comparator (Fig. 15.21) with the help of slip gauges.
The instrument essentially consists of a fixed anvil and a movable anvil.
There is a micrometer on the moving anvil side and this has a very
limited movement on either side of the setting.
The distance is adjusted by setting the fixed anvil at desired place with
the help of looking ring and setting tubes.

Tangential Gear Tooth Calliper:
It is utilised for measuring variations on the basic tooth profile from the outside diameter of spur and
helical gears.
The instrument consists of body, on the underside of which there are two slides having the tips acting like
measuring contacts.
The extended spindle of a dial indicator with the contact point A passes between the two tips along the
vertical axis of symmetry of the instrument.
The measuring tips are spread apart or brought together simultaneously and symmetrically in reference to
the central axis by a screw which has a right-hand and a left-hand thread.
The contact faces of the measuring tips are flat and arranged at angles of 14.5° or 20° with the central axis.
The calliper is set up by means of a cylindrical master gauge of proper diameter based on the module of the
gear being checked.
After adjusting the tips by the screw, these are locked in position by locking nuts.
The properly set up instrument is applied to the gear tooth and the dial indicator reading shows how much
the position of the basic tooth profile deviates in reference to the outside diameter of the gear.

Measurement over Rolls or balls:
A very good and convenient method is for measuring thickness of gear. In this method, two or
three different size rollers are used to check the vibrations at different places on the tooth.
7. Measurement of concentricity
In setting of gears, the centre about in which the gear is mounted should be coincident with the
centre from which the gear is generated.
It is easy to check the concentricity of the gear by mounting the gear between centres and
measuring the variation in height of a roller placed between the successive teeth.
Finally, the variation in reading will be a function of the eccentricity present.
8. Alignment checking
It is done by placing a parallel bar between the gear teeth and the gear being mounted between
centres.
Finally, the readings are taken at the two ends of the bar and difference in reading is the
misalignment.

Parkinson Gear Tester
• The master gear is fixed on vertical spindle and the gear to be tested is fixed on similar spindle
which is mounted on a carriage.
• The carriage which can slide both side and these gears are maintained in mesh by spring
pressure.
• When the gears are rotated, the movement of sliding carriage is indicated by a dial indicator and
these variations are the measure of any irregularities in the gear under test.
• The variation's recorded in a recorder which is fitted in the form of a waxed circular chart.
• In fig, the gears are fitted on the mandrels and are free to rotate without clearance.
• Left mandrel moves along the table and the right mandrel moves along the spring-loaded
carriage.
• The two spindles can be adjusted so that the axial distance is equal and a scale is attached to one
side and vernier to the other, this enables center distance to be measured to with in 0.025mm.

• If errors occur in the tooth form when gears will be in closer mesh, pitch or concentricity of pitch
line will cause a variation in center distance from this movement of carriage as indicated to the
dial gauge will show the errors in the gear test.
• The recorder is also fitted in the form of circular or rectangular chart and the errors are recorded.
Limitations of Parkinson gear tester:
1. Accuracy ±0.001mm
2. Maximum gear diameter is 300mm
3. Errors are not clearly identified.
4. Measurement is dependent upon the master gear.
5. Low friction in the movement of the floating carriage.

principle OF OPERATION OF A ROLLING GEAR
TESTING MACHINE
• When the master gear and gear under test are in closer mesh and rotated, any error will cause
variation in center distance.
• The movement of carriage is indicated on the dial gauge shown as the error in the gauge.

SURFACE FINISH measurement:
• When we are producing components by various methods of manufacturing process, it is not
possible to produce perfectly smooth surface and some irregularities are formed.
• These irregularities cause some serious difficulties when using the components.
• So, it is very important to correct the surfaces before use.
• The factors which are affecting surface roughness are
1. Workpiece Material
2. Vibrations
3. Machining type
4. Tools and fixtures

Types of Surface Finish:
Surface of a body is its boundary which separates it from another body. Generally there are two
types of surfaces namely:
Nominal Surface: A theoretically geometrically perfect surface is called as a nominal surface.
Rough Surface: A magnified view of an actual surface is shown. This surface has numerous small
peaks and valleys that deviate from the nominal surface. It is seen that the surface has a certain
degree of roughness and hence it is called a rough surface.
Wavy Surface: A surface having waviness as shown in the figure is called wavy surface.
Nominal Surface(NS)
NS
Valley
Peak
Rough Surface

• The geometrical irregularities can be classified as
1. First order
2. Second order
3. Third order
4. Fourth order
First order irregularities: They are caused by lack of straightness of guideways on which tool must
move.
Second order irregularities: They are caused by vibrations.
Third order irregularities: They are caused by machining.
Fourth order irregularities: They are caused by improper handling machines and equipments.

• Elements of Surface Texture:
1. Surface Roughness: It is the micro-irregularities in a surface resulting from the action of
production process.
2. Profile: It is the shape of any section through a surface.
3. Flaw: It is an irregularity that occurs in a surface due to cracks, blow holes, scratches, etc.,
Unless otherwise specified, the effect of flaws should not be accounted in the measurement of
roughness.
4. Sample or Cut-off length: The particular length that is taken for sample measurement on the
surface is called as cut of length. It is also known as roughness width cut off.
5. Lay: It is the direction of the predominant top surface grooves that are produced by machining.
6. Waviness: Surface irregularities that are of greater spacing than roughness.

Datum for SURFACE FINISH measurement:
• The datum(reference) from which surface finish in numerical values may be specified can be
done in two methods namely
a) The M System(The mean line system):
The mean line or centre line is a line that is parallel to the general direction of the profile and is
placed such that areas around the profile(above and below the line) are equal.
Mean line is determined as follows:
A straight line AA is drawn so that it is parallel to the general direction of the profile for a length
L.
B
Areas X
A
C
Areas Y
B
A
L

Areas X and Y are measured.
The mean line BB is a line at a distance C from line AA, where C =
𝐴𝑟𝑒𝑎𝑠 𝑋−𝐴𝑟𝑒𝑎𝑠 𝑌
𝐿
Line BB is parallel to AA.
b) The E System(The envelope system):
Envelope line(curve) is a line that is obtained by rolling a circle of radius R on the surface.
This curve is the locus of the center of the circle.
Once this curve is obtained, it is brought down vertically by a distance R.
By doing so, the curve will rest on the crests of the surface.
Area below
Area above
Locus of the center of
circle of radius R
Curve brought down by
distance R(Envelope curve)
Mean envelope curve
L
R

Now the envelope curve is moved downwards to a position so that the areas around the surface
profile(above and below the curve) are equal.
This mean envelope curve becomes the mean line in the E system.
Analysis of surface finish:
The analysis of surface finish is being carried out by
1. The average roughness method.
2. Peak to valley height method
3. From factor
1. Average roughness measurement:
The assessment of average roughness is carried out by
a) Centre line average (CLA) value (Ra).
b) Root mean square (RMS) value (Rs).
c) Ten point method

a) Centre line average (CLA) value (Ra): Ra is the average height from a mean line of all
ordinates of the surface, regardless of sign. Let h, h, …., h be the height of the equally spaced
ordinates at positions 1,2,…n.
Another way of calculating Ra is shown in the figure.
n

b) Root mean square (R.M.S) value (Rs): R.M.S Rs is the square root of the mean of the squares
of the surface measured from the mean line. As h1 , h2 , ……hn are the heights of the equally
spaced ordinates at positions 1,2,3,…n,
R.M.S. =
ℎ1
2
+ℎ2
2
+⋯+ℎn
2
𝑛
c) Ten point height method: The average difference between five highest peaks and five lowest
valleys of surface is taken and irregularities are calculated by

2. Peak to valley height method (Rmax): It is the distance between the highest peak and lowest
valley for the considered sample length. L is the sample length.
3. Form Factor Method: It is obtained by measuring the area of material above the arbitrarily
chosen base line in the section and the area of the enveloping rectangle.
Degree of Fullness(F) =
𝑀𝑒𝑡𝑎𝑙 𝐴𝑟𝑒𝑎
𝐸𝑛𝑣𝑒𝑙𝑜𝑝𝑖𝑛𝑔 𝑅𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒 𝐴𝑟𝑒𝑎
Degree of Emptiness(E1) = 1-F

4. The depth of Smoothness(Ru): It is the distance between the mean line and another line that is
parallel to it drawn such that it touches the highest peak within the considered sampling length.
5. The Mean Depth (Rm): It is the distance between the mean line and another line that is parallel
to it drawn such that it touches the lowest valley within the considered sampling length.
Ru
Mean Line
Sampling Length (L)
Rm
Mean Line
Highest Peak
Lowest Valley
Sampling Length (L)

Methods of SURFACE FINISH measurement:
There are basically two methods for measuring surface finish, namely
1. Comparative or Qualitative Methods
2. Direct Measurement or Quantitative Methods
Comparative or Qualitative Methods: In these methods, the surface texture is assessed by observation
of the surface. The surface to be tested is compared with known value of roughness specimen and finished
by similar machining process. The various methods which are used for comparison are
1. Touch Inspection.
2. Visual Inspection.
3. Microscopic Inspection.
4. Scratch Inspection.
5. Micro Interferometer.
6. Surface photographs.
7. Reflected Light Intensity.
8. Wallace surface Dynamometer.

Touch Inspection: It is used when surface roughness is very high and the fingertip is moved
along the surface at a speed of 2Smmlsecond and the irregularities as up to 0.0 125mm can be
detected in this method.
Visual Inspection: In this method, the surface is inspected by naked eye and this measurement is
limited to rough surfaces .
 Microscopic Inspection: In this method, a finished surface is placed under the microscope and
compared with the surface under inspection. The light beam is also used to check the finished
surface by projecting the light about 60̊ C to the work.

Scratch Inspection: The materials like lead, plastics rubbed on surface are inspected by this
method. The impression of this scratches on the surface produced is then visualized.
Micro-Interferometer: Optical flat is placed on the surface to be inspected and illuminated by a
monochromatic source of light.
Surface Photographs: Magnified photographs of the surface are taken with different types of
illumination. The defects like irregularities appear as dark spots and flat portion of the surface
appears as bright.
Reflected light Intensity: A beam of light is projected on the surface to be inspected and the light
intensity variation on the surface is measured by a photo cell and this measured value is
calibrated.

Wallace surface Dynamometer: It consists of a pendulum in which the testing shoes are clamped
to a bearing surface and a predetermined spring pressure can be applied and then, the pendulum
is lifted to its initial starting position.
Direct Measurement or Quantitative Methods: Some of the direct measuring instruments are
1. Stylus probe instruments
2. Tomlinson surface meter
3. Profilometer
4. Taylor-Hobson Talysurf.
Stylus probe instrument:
Principle: When the stylus is moved over the surface to be measured, the irregularities in the
surface texture are measured and it is used to assess the surface finish of the workpiece.
Working: The stylus type instruments consist of skid, stylus, amplifying device and recording
device. The skid is slowly moved over the surface by hand or by motor drive. The skid follows the
irregularities of surface and stylus moves along with skid.

When the stylus moves vertically up and down, the stylus movements are magnified, amplified and
recorded to get a trace. Then the trace is analysed by an automatic device.
Advantage:
1. Any desired roughness parameter can be recorded.
Disadvantages:
1. Fragile material cannot be measured.
2. High Initial cost.
3. Skilled operators are needed to operate.

Tomlinson Surface meter:
Principle: This instrument uses mechanical-cum-optical means for magnification.
Construction: In this, the diamond stylus on the surface finish recorder is held by spring pressure
against the surface of a lapped cylinder. The lapped cylinder is supported one side by probe and
other side by rollers. The stylus is also attached to the body of the instrument by a leaf spring and
its height is adjustable to enable the diamond to be positioned and the light spring steel arm is
attached to the lapped cylinder.

The spring arm has a diamond scriber at the end and smoked glass is rest on the arm.
Working:
When measuring surface finish, the body of the instrument moves across the surface by a screw
rotation. The vertical movement of the probe caused by the surface irregularities makes the
horizontal lapped cylinder to roll. This rolling of lapped cylinder causes the movement of the arm,
So, this movement induces the diamond scriber on smoked glass. Finally, the movement of scriber
together with horizontal movement produces a trace on the smoked glass plate and this trace is
magnified by an optical projector.
Profilometer:
It is an indicating and recording instrument to measure roughness in microns. The main parts of
the instrument are tracer and an amplifier.

The stylus is mounted in the pick up and it consists of induction coil located in the magnet. When
the stylus is moved on the surface to be tested, it will be displaced up and down due to
irregularities in the surface. This movement induces the induction coil to move in the direction of
permanent magnet and produces a voltage. This is amplified and recorded.
Talyor-Hobson-Talysurf:
It is working by carrier modulating principle and it is an accurate method comparing with the other
methods. The main parts of this instrument are diamond stylus (0.002mm radius) and skid.
Principle: The irregularities of surface are traced by the stylus and the movement of stylus is
converted into changes in electric current.
Working: On two legs of the E-shaped stamping, there are coils for carrying an A.C. current and
these coils form an oscillator. As the armature is pivoted about the central leg, the movement of the
stylus causes the air gap to vary and thus the amplitude is modulated. This modulation is again
demodulated for the vertical displacement of the stylus. So, this demodulated output actuates or
induces the pen recorder to produce a numerical record and to make a direct numerical assessment.

Other Methods for Measuring Surface Roughness:
Profilograph: The surface finish to be checked is placed on the table. The table can move either
side by lead screw and the stylus is pivoted over the tested surface. So, the oscillation in the
stylus due to surface irregularities is transmitted to the mirror.
A light source sends a beam of light through lens and a precision slit to the mirror, and the
reflected beam is directed to revolving drum. A sensitive film is attached upon the revolving
drum. The revolving drum can be rotated by two bevel gears and the gears are attached to the
same lead screw. Finally, the profilogram will be obtained from the sensitive film and it is
analysed.

Double microscope: It is an optical method for measuring the surface roughness. When a thin
film of light strikes the surface to be tested by an angle of 45° through the condenser and
precision slit and the observing microscope is also inclined at an angle of 45° to the tested
surface.
The surface is illuminated by a projection tube and it is observed by an eyepiece through the
microscope. The eyepiece contains a eyepiece micrometre and it is used to measure the
irregularities.

Straightness measurement:
• Straightness is a geometrical shape. Geometrical tolerance for straightness will be specified if
straightness is to be attained while manufacturing.
• A line is said to be straight over a given length if the line is contained between two imaginary
line that are placed at equidistance from the geometrically ideal position of the line being tested
for straightness. The distance between the two imaginary lines is to be equal to the specified
straightness tolerance.
Actual line being
tested for straightness
Imaginary Lines
Highest point on
actual Line
Lowest point on
actual Line
Geometrically Ideal
Line
Geometric tolerance
for straightness
Straightness error(e)

• Straightness error of a line is defined as the distance ‘e’ between two lines drawn parallel to the
geometrically ideal line such that one of the two lines passes through the highest point on the
actual line and the other line passes through the lowest point on the actual line.
Straightness Measuring Instruments:
• Following are the instruments used for measuring straightness namely,
1. Straight Edge
2. Spirit Level
3. Autocollimator
Straight Edge:
A straight edge is a flat length of tool steel(or stainless steel) that is ground to extremely fine
tolerance along its edges.
Straight edges are available from 30cm to 360cm. Straight edges are used to check surface for
straightness. They are also used to scribe accurate straight lines.

Method-I:
The simplest way of using a straight edge is to place the straight edge on the surface to be tested
and determining the degree of contact by marking, feelers or light gap. Suppose light is used.
The gap between the straight edge and surface will be negligible for perfectly straight surfaces.
Straightness is measured by observing the colour of light by diffraction while passing through
the small gap. Ex: If colour of light is red, it indicates certain gap in mm.
Method-II:
A straight edge is supported at two points for minimum deflection on two unequal piles of slip
gauges such that it is at a slight inclination to the surface to be tested.
Now divide the distance between the supports into number of equal parts and mark them on the
straight edge. If the surface and straight edge are perfectly straight, the gap at each point will
also vary uniformly. If the slip gauges used have values 5mm and 5.1mm as shown in figure and
if the distance between the slip values are divided into 5 equal parts, then the gap at each point
will vary by 0.02mm(i.e., 0.1/5). Hence the value of pile of slip gauges required at position are:
Position 1 = 5mm
Position 2 = 5.02mm

Position 3 = 5.04mm and so on,
Now when the slip gauges are inserted at the marked positions, one of the following two
condition occur.
[If slips make exact contact with both the surface at the marked position] => [There is no error
in straightness]
[If slips don’t fit exactly at the marked position] => [There is error in straightness and is
displayed along the straight edge by amounts proportional to the errors]
Spirit Level:
• The straightness of a surface to any length can be tested using a spirit level. This method has a
high degree of accuracy.
• The spirit level measures small angular displacements relative to a horizontal datum, the level of
liquid at rest.
• Straightness is checked using a spirit level as follows. A straight line(datum) is drawn on the
surface whose straightness is to be checked. A spirit level with two feet(resting on the line) is
moved along this line in steps. For each position the reading is noted. Variation in bubble
position represents angular displacement in the surface.

• Care should be taken to see that the spirit level is moved such that the length of each section is
equal to the distance between center lines of the two feet.
• Spirit level can be used only for the measurement of straightness of horizontal surfaces. They
can’t be used on vertical surfaces.
Auto Collimator:
• The straightness of a surface of any length can be tested using an auto collimator. This method
has a high degree of accuracy.
• Auto collimator can be used on vertical side of horizontal slideways.
• Auto collimator has a converging lens with a point source of light at its principal focus.

• When a parallel beam of light is projected from the lens and if a plane reflector is kept at 90̊ to
the beam of light, the light will be reflected back along its own path and will be brought to focus
at the same position as that of the light source.
• If the reflector is tilted by an angle θ, the parallel beam of light will be deflected by twice the
angle and will be brought to focus in the same plane as the light source but to one side of it. The
image and object will not coincide, but will be placed at a distance of 2fθ between them, f being
the focal length of the lens.
• This method of testing straightness consists of placing a block fitted with feet at convenient
distance apart and carries a plane reflector. This arrangement is moved along the line to be
tested for straightness in equal steps. Angular variation at each position is noted and is used to
plot the graph of error.
Cumulative Error
Distance along
surface tested

flatness measurement:
• A surface is said to be flat within a given range of measurement when the variation of the
perpendicular distance of its points from a geometrically ideal plane that is parallel to the
trajectory of the surface to be tested remain below a given value.
• The geometrically ideal plane can be represented by means of a surface plane or by a group of
straight lines that are obtained by the displacement of a straight edge or a spirit level or a light
beam.
• Let us assume that the flatness tolerance is specified as 0.08mm. The tolerance value 0.08mm
means that the surface being tested should be contained between two parallel planes which are
0.08mm apart.
Flatness Testing: One way of testing flatness is to compare the surface with an accurate
surface(geometrically ideal surface). If the test surface is small in size, then it is rubbed against an
accurate surface that is evenly covered with Prussian blue colour paste. The distribution of colour
over the test surface gives an idea of the peaks and valleys on the surface.
If the test surface is large, then it is tested for flatness using a spirit level or an auto collimator.
Flatness error is indicated in μ or mm.

• Procedure to determine flatness:
A surface can be considered to be a collection of a large number of lines. If all these lines are
straight and lie on the same plane, the surface is said to be flat.
Consider a rectangular table as shown in the figure. All the generators/lines are straight and
parallel to the sides of the rectangle in both the perpendicular directions.
For a surface to be truly flat, the diagonal lines should also have straightness.
Hence, the surface is divided by straight lines as shown. Lines are drawn little inside to take
care of edge wearout.
The straightness of all the lines(AB, BC, CD, DA,..) are determined. Then the lines are related

with each other to find out whether all of them lie on the same plane or not. Care is to be taken to
see that when a spirit level or collimator reflector stand is used, one of its contact points of the
feet of the block should lie on the center of the diagonals(I).
Once the straightness of all the lines are determined, the readings are tabulated. ABD is
considered to be an arbitrary plane. Hence the end points A,B and D are set to zero height.
Now height of point I is found with reference to plane ABD. Next point C is fixed with
reference to the arbitrary plane ABD and then points, of lines BC and DC are corrected. As
positions H, G, E and F are known, lines HG and EF are fitted.
In this way the height of all the points on the test surface are determined relative to the arbitrary
plane.
• Flatness testing by Interferometry:
Here a monochromatic light source and a set of optical flats are used.
When an optical flat is placed on another reflecting surface whose flatness is to be determined,
the optical flat will not form an intimate contact, but will lie at some angle θ making an inclined
plane.

• Now the optical flat is illuminated by monochromatic source of light. If the eye is placed in
proper position, it will observe a number of bands. These bands are produced by the interference
of light rays reflected from the bottom surface of the optical flat and the top surface of the test
surface, through the thin layer of air between the surfaces.
• At point A, the wave of incident beam from S is partially reflected along AB and is partially
transmitted across air gap AC. At point C, the ray is reflected again along CD and extends along
point E. Thus, the two reflected components have a travelled path whose lengths differ by an
amount ACD.
• If the path lengths of the two components differ by an odd number of half wave lengths,
complete interference is achieved and a straight dark line will be seen passing through point C.

• For all rays with path difference of odd number of half wave lengths, interference will occur and
a similar dark fringe will occur(Ex: path FHI at H).
• At intermediate points between point C and H, the path difference is an even number of half
wavelength. As the two components are in phase, a light band is produced.
• Thus alternate light and dark straight line bands are produced. A deviation from this pattern
becomes a measure of error in flatness of the test surface.

roundness measurement(Circularity)
• By definition, roundness or circularity is the radial uniformity of work surface measured from
the center line of the workpiece.
• Circularity is specified by circularity tolerance. For example, if it is specified that circularity of
a feature is to be 0.1mm, than it means that the circumference of each cross section of the
feature should be contained between two coplanar concentric circles that are 0.1mm apart.
• Error in roundness is defined as the radial distance between the minimum inscribing circle and
maximum inscribing circle, that contain the actual profile of the surface at a section
perpendicular to the axis of rotation.

• Methods for Measuring Roundness:
• The important methods for measuring roundness are listed: a) V-block and dial indicator method
b) Roundness Measuring Machine
V-block and dial indicator method:
• The arrangement consists of a V-block that is placed on the surface plate. The workpiece to be
tested is placed in the V-groove of the V-block as shown in the figure.
• The feeler of a sensitive dial indicator (held firmly by a stand) is made to rest on the workpiece.

• Now the workpiece is rotated about the diameter to be checked. The dial indicator will indicate
variations in the dimensions caused due to out of roundness. [Note: Lobbing: The diameters at
any two opposite points are constant, but still it is not in circular form]
• Plotting a Polar Graph: An idea of the actual shape of the workpiece can be obtained by plotting
a polar graph. 12 equispaced markings at an angle of 30̊ are made in the face of the workpiece
to be measured. The workpiece is placed on the V-block. The dial indicator is made to touch the
workpiece at its center. Now when the workpiece is rotated and when the marking comes
exactly under the plunger of the dial indicator, the reading is noted. Hence 12 readings will be
obtained. The procedure is repeated thrice to get average values for each marking. Now for
plotting the polar graph, a proper scale is selected. A circle of diameter equal to four times the
maximum reading of the dial indicator is drawn. Another concentric circle is drawn in this
circle. The values of the dial indicator are plotted in radial direction by taking the smallest circle
as the reference circle. The individual points are joined by straight lines to get the actual profile
of the workpiece.
•

• Error is measured as the radial distance between the maximum and minimum inscribing circle
for the profile obtained.
Roundness error =
𝑀𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑒𝑟𝑟𝑜𝑟 𝑓𝑟𝑜𝑚 𝑝𝑜𝑙𝑎𝑟 𝑔𝑟𝑎𝑝ℎ
𝐾
Where, K is a constant (This constant depends on the shape of the workpiece and angle of V-
block)
• The position of the indicating instrument, the number of lobes on the workpiece and the angle
of the V-block have an influence on the determination of roundness error.
Roundness Measuring Machine:
• The machine is also called as Taly-round Instrument or precision spindle method.
• The main parts of the instrument are a truly running spindle that is mounted on precision ball
bearing and micron indicator.
• The indicating pointer is rotated around the workpiece about an accurately stable axis.

• The indicator shows deviations from roundness. As the output of the indicator is connected to an
amplifier unit and pen recorder, a polar graph of the out line of the workpiece is obtained.
• This is an accurate method. Automatic recording of the exact profile of the workpiece is
obtained. Waviness also is superimposed on the profile of the workpiece.

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