Ch-3: Measurement of screw thread and gear

MECHANICAL MEASUREMENT & METROLOGY
(3141901)
CH-3: MEASUREMENT OF SCREW THREAD AND GEAR
PREPARED BY:
PROF. SURAJ A. SHUKLA

INTRODUCTION TO SCREW THREAD MEASUREMENT
• Screw thread geometry has evolved since the early 19th century, thanks to the importance of threaded
fasteners in machine assemblies. The property of interchangeability is associated more strongly with screw
threads than with any other machine part.
• Perhaps, the Whitworth thread system, proposed as early as the 1840s, was the first documented screw thread
profile that came into use. A couple of decades later, the Sellers system of screw threads came into use in the
United States.
• Both these systems were in practice for a long time and laid the foundation for a more comprehensive unified
screw thread system. Screw thread gauging plays a vital role in industrial metrology.
• In contrast to measurements of geometric features such as length and diameter, screw thread measurement is
more complex. We need to measure inter-related geometric aspects such as pitch diameter, lead, helix, and
flank angle, among others.
• The following sections introduce screw thread terminology and discuss the measurements of screw thread
elements and thread gauging, which speeds up the inspection process.

 Screw Thread Terminology:
 Screw Thread: The American Society of Tool and Manufacturing Engineers (ASTME) defines a screw
thread as follows: screw thread is the helical ridge produced by forming a continuous helical groove of a
uniform section on the external or internal surface of a cylinder or cone.
 Form of thread: This is the shape of the contour of one complete thread, as seen in an axial section. Some of
the popular thread forms are British Standard Whitworth, American Standard, British Association, Knuckle,
Buttress, Unified, Acme, etc.
 External thread: The screw thread formed on the external surface of a workpiece is called an external
thread. Examples of this include bolts and studs.

 Internal thread: The screw thread formed on the internal surface of a workpiece is called an internal thread.
The best example of this is the thread on a nut.
 Axis of thread (pitch line): This is the imaginary line running longitudinally through the center of the screw.
 Fundamental triangle: It is the imaginary triangle that is formed when the flanks are extended until they
meet each other to form an apex or a vertex.
 The angle of the thread: This is the angle between the flanks of a thread measured in the axial plane. It is
also called an included angle.
 Flank angle: It is the angle formed between a flank of the thread and the perpendicular to the axis of the
thread that passes through the vertex of the fundamental triangle.
 Pitch: It is the distance between two corresponding points on adjacent threads, measured parallel to the axis
of the thread.
 Lead: It is the axial distance moved by the screw when the crew is given one complete revolution about its
axis.

 Lead angle: It is the angle made by the helix of the thread at the pitch line with the plane perpendicular to
the axis.
 Helix angle: It is the angle made by the helix of the thread at the pitch line with the axis. This angle is
measured in an axial plane.
 Major diameter: In the case of external threads, the major diameter is the diameter of the major cylinder
(imaginary), which is coaxial with the screw and touches the crests of an external thread. For internal threads,
it is the diameter of the cylinder that touches the root of the threads.
 Minor diameter: In the case of external threads, the minor diameter is the diameter of the minor cylinder
(imaginary), which is coaxial with the screw and touches the roots of an external thread. For internal threads,
it is the diameter of the cylinder that touches the crests of the threads. It is also called the root diameter.
 Addendum: It is the radial distance between the major diameter and pitch line for external threads. On the
other hand, it is the radial distance between the minor diameter and pitch line for internal threads.
 Dedendum: It is the radial distance between the minor diameter and pitch line for external threads. On the
other hand, it is the radial distance between the major diameter and pitch line for internal threads.

 Effective diameter or pitch diameter: It is the diameter of the pitch cylinder, which is coaxial with the axis
of the screw and intersects the flanks of the threads in such a way as to make the widths of threads and the
widths of spaces between them equal. In general, each of the screw threads is specified by an effective
diameter as it decides the quality of fit between the screw and a nut.
 Single-start thread: In the case of a single-start thread, the lead is equal to the pitch. Therefore, the axial
distance moved by the screw equals the pitch of the thread.
 Multiple-start thread: In a multiple-start thread, the lead is an integral multiple of the pitch. Accordingly, a
double start will move by an amount equal to two pitch lengths for one complete revolution of the screw.

MEASUREMENT OF EXTERNAL SCREW THREAD
ELEMENTS
Measurement of screw thread elements is necessary not only for manufactured components but also for
threading tools, taps, threading hobs, etc. The following sections discuss the methods for measuring major
diameter, minor diameter, effective diameter, pitch, angle, and form of threads.
 Measurement of major diameter:
 Using Ordinary Micrometer:
• For the measurement of the major diameter of an external thread, a good micrometer is quite suitable. The
micrometer is used as a comparator.
• It is first adjusted on a cylindrical standard of appropriate size having approximately the same diameter. This
standard is called setting gauge (of size ‘s’ approximately same as that of the major diameter ‘D’ of the
thread).
• After taking this reading, the micrometer is set on the major diameter of the thread and the reading is taken.
Then the major diameter, D = S ± (d1 – d2)
where, S = size of setting gauge
d1 = micrometer reading over setting gauge
d2 = micrometer reading over the thread

ELEMENTS
 Using Ordinary Micrometer:
• As there is point contact between the crests of the thread and measuring anvils, a light pressure must be
applied (do not handle the micrometer ratchet), otherwise, errors due to deformation of crest may be
introduced.
 Using Bench Micrometer:

ELEMENTS
• The ordinary micrometer method has deficiencies like the variation in measuring pressure, pitch errors in the
threads, zero error setting, etc. Therefore, for greater accuracy and convenience, a bench micrometer is
preferred. This instrument was designed by NPL to remove the deficiencies inherent in the normal hand
micrometer.
• In order to ensure that all the measurements are made at the same pressure, a fiducial indicator is used. This
ensures constant pressure for all the measurements. The instrument has a micrometer head with a vernier
scale to read to the accuracy of 0.002 mm.
• It is used as a comparator in order to avoid pitch errors of micrometer threads, zero error settings, etc. A
calibrated setting cylinder having nearly the same diameter as the major diameter of the thread to be
measured is used as setting the standard.
• The setting cylinder is held between the anvils and reading is taken. In this machine, there is no provision for
mounting the workpiece between the centers and it is to be held in hand.

ELEMENTS
• The cylinder is then replaced by the threaded workpiece and again the micrometer reading is noted.
If, D = Diameter of the setting cylinder,
R1 = Reading of micrometer on setting cylinder,
R2 = Reading of micrometer on the screw thread
Then, the major diameter of screw thread = D ± (R2 - R1)

ELEMENTS
 Measurement of minor diameter:
 Two V-piece method:
• This is also measured by a comparative process using small V-pieces which make contact with the root of the
thread. The V-pieces are available in several sizes having suitable radii at the edges.
• The included angle of V-pieces is less than the angle of the thread to be checked so that it can easily probe to
the root of the thread. To measure the minor diameter by V-pieces is suitable for only Whitworth and B.A.
threads which have a definite radius at the root of the thread. For other threads, the minor diameter is
measured by the projector or microscope.

ELEMENTS
 Two V-piece method:
• The measurement is carried out on a floating carriage diameter measuring machine in which the threaded work-
piece is mounted between centers and a bench micrometer is constrained to move at right angles to the axis of the
center by a V-ball slide.
• The dimensions of V-pieces play no important function as they are interposed between the micrometer faces and
the cylindrical standard when standard reading is taken. It is important while taking readings, to ensure that the
micrometer is located at right angles to the axis of the screw being measured.
• The selected V are placed on each side of the screw with their bases against the micrometer faces. The micrometer
head is then advanced until the pointer of the indicator is opposite the zero marks, and a note is made of the
reading.
• The screw is then replaced by standard reference disc or a plain cylindrical standard plug gauge of approximately
the core diameter of the screw to be measured and the second reading of the micrometer is taken.
If R1 = micrometer reading on standard cylinder gauge
R2 = Micrometer reading on the threaded workpiece
D = Diameter of the cylinder gauge
The minor diameter of the thread = D ± (R2 - R1)

ELEMENTS
 Measurement of Minor Diameter of Internal Threads:
Using taper parallels: For diameters, less than 200 mm the use of taper parallels and a micrometer is very
common. The taper parallels are pairs of wedges having parallel outer edges. The diameter across their outer
edges can be changed by sliding them over each other as shown in the figure below.
By using Rollers and slip gauges: This method is used for diameters more than 20 mm -diameter. In this
method, precision rollers are inserted inside the thread and proper slip gauges inserted between the rollers as
shown in the figure, so that firm contact is established. The minor diameter is then the length of slip gauges plus
twice the diameter of the roller.

ELEMENTS
 Measurement of Effective diameter (or Pitch diameter):
 The micrometer method:
• The thread micrometer resembles the ordinary micrometer, but it has special contacts to suit the end screw
thread form that is to be checked. In this micrometer, the end of the spindle is pointed to the Vee- thread form
with a corresponding Vee-recess in the fixed anvil.
• When measuring threads only, the angle of the point and the sides of Vee-anvil, i.e. the flanks of the threads
should come into contact with the screw thread.

ELEMENTS
 The micrometer method:
• If correctly adjusted, this micrometer gives the pitch diameter. This value should agree with that obtained by
measurement by outside diameter and pitch from the following relation:
Pitch diameter = D – 0.6403p (in case of Whitworth thread)
where depth of thread = 0.6403p,
D = outside dia.,
p = pitch.
• The limitation of the micrometer is that the micrometer must be set to a standard thread plug. If not done so
in the first instance, there will be an error due to the helix angle of the thread being measured.
• When setting the instrument to a standard plug gauge it will be observed that the reading is not exactly zero,
as previously inferred, when the spindle and anvil are brought together.
• A big advantage of thread micrometer is that this is the only method that shows the variation for the drunken
thread.

ELEMENTS
 The one-wire, two-wire or three-wire or rod method:
 One wire method:
• This method is used if a standard gauge of the same dimension as the theoretical value of the dimension over
the wire is available.
• First of all, the micrometer anvils are set over the standard gauge and the dimension is noted down.
Thereafter, the screw to be inspected is held either in hand or in a fixture, and the micrometer anvils are set
over the wire as shown in the figure.

ELEMENTS
 One wire method:
• Micrometer readings are taken at two or three different locations and the average value is calculated. This
value is compared with the value obtained with the standard gauge. The resulting difference is a reflection of
error in the effective diameter of the screw.
• An important point to be kept in mind is that the diameter of the wire selected should be such that it makes
contact with the screw along the pitch cylinder. The significance of this condition will become obvious in the
two-wire method explained in the next section.
If D = Size of setting gauge
d1 = Micrometer reading over standard gauge
d2 = Micrometer reading over thread under test
Then effective diameter = D ± (d1 – d2)
If d2 > d1, use +ve sign above and if d2 < d1, use –ve sign.

ELEMENTS
 Two wire method:
• In this method, one wire is placed between two threads at one side and on the other side, the anvil of the
measuring micrometer contacts the crests as shown in the figure.
• First, the micrometer reading is noted on a standard gauge whose dimension is nearly the same as to be
obtained by this method. Actual measurement over the wire on one side and threads on another side = size of
gauge ± difference in two-micrometer readings.

ELEMENTS
Let, E = Effective diameter,
H = Diameter over wire (maximum),
Y = Diameter under wire (minimum), d = wire diameter
From the figure, we can write Y = H – 2d.
Let the pitch value P’, which depends on the pitch of thread ‘P’, and wire diameter d, then we can write
Effective diameter, E = Y + P’
For metric thread the value of P’ = 0.866P - d

ELEMENTS
Derivation for the equation of the pitch value:
Referring to the above figure,
E = Y + 2AB
From the figure consider ΔODG,
sin (θ / 2) = OD / OG
⸫
cosec (θ / 2) = OG / OD
⸫
OG = OD cosec (θ / 2)
= (d / 2) cosec (θ / 2) (⸫
OD = OC = d / 2 = radius of wire)

ELEMENTS
Consider ΔJAG,
tan (θ / 2) = AJ / AG
⸫cot (θ / 2) = AG / AJ
⸫AG = AJ cot (θ / 2)
Since IJ lies on effective diameter,
⸫IJ = (1/2) × pitch = (1/2) × P
IJ = IA + AJ
⸫AJ = (1/2) × IJ = (1/2) × (1/2) × P
⸫AJ = (1/4) × P
⸫AG = (P / 4) × cot (θ / 2)

ELEMENTS
From the figure, OG = OB + BG
⸫BG = OG – OB
Substituting the value of OG,
⸫BG = (d / 2) × cosec (θ / 2) – (d / 2) (⸫OB = OC = OD = d / 2)
= (d / 2) [cosec (θ / 2) – 1]
From the figure, AG = AB + BG
⸫AB = AG – BG
Substituting the values of AG and BG, we get
AB = (P / 4) cot (θ / 2) – (d / 2) [cosec (θ / 2) – 1]

ELEMENTS
From the above equations, we can write
P’ = 2AB
⸫
P’ = 2 [(P / 4) cot (θ / 2) – (d / 2) (cosec (θ / 2) – 1)]
⸫
P’ = [(P / 2) cot (θ / 2) – d (cosec (θ / 2) – 1)]
This equation is for the pitch value P’ in terms of pitch P, the diameter of wire ‘d’ and thread angle ‘θ’.
The value of Y can be calculated using a floating carriage micrometer as shown in the figure.
Let, D1 = micrometer reading over standard and wire (Diameter)
D2 = micrometer reading over screw thread and wire (diameter)
D = diameter of the standard
Then, Y = D ± (D1 – D2)

ELEMENTS
Limitation: Measurement by the two-wire method using a micrometer is not satisfactory where high accuracy is
required. Using a micrometer, alignment is not possible by two-wire, therefore a floating carriage machine is
required for satisfactory results.

ELEMENTS
 Three wire method:
• This method of measuring the effective diameter is an accurate method. In this, three wires or rods of known
diameter are used: one on one side and two on the other side.
• This method ensures the alignment of micrometer anvil faced parallel to the thread axis. The wires may be
either held in hand or hung from a stand so as to ensure freedom to the wires to adjust themselves under
micrometer pressure.

ELEMENTS
Let,
E = Effective diameter of screw thread
H = Diameter over wire (maximum)
Y = Diameter over wire (minimum)
d = Diameter of wire
Referring the above figure, consider ΔOLG,
sin (θ / 2) = OL / OG
cosec (θ / 2) = OG / OL
OG = OL × cosec (θ / 2)
OG = (d / 2) × cosec (θ / 2) (⸫OL = d / 2)

ELEMENTS
Considering ΔNIK,
tan (θ / 2) = IK / NI
⸫ cot (θ / 2) = NI / IK
As distance NI = M and IK = GK / 2 = P / 2
cot (θ / 2) = M / (P / 2)
⸫ M = (P / 2) × cot (θ / 2)
From the figure, we have
RG = M / 2
⸫ RG = (P / 4) × cot (θ / 2)

ELEMENTS
From the figure, we have OR + RG = OG.
⸫ OR = OG – RG
Substituting value of OG and RG,
⸫ OR = (d / 2) × cosec (θ / 2) - (P / 4) × cot (θ / 2)
From the figure, we can write,
H = E + 2OR + 2OQ
= E + 2 [(d / 2) × cosec (θ / 2) - (P / 4) × cot (θ / 2)] + d
⸫H = E + d [cosec (θ / 2) + 1] – (P / 2) × cot (θ / 2)
For metric thread:
Depth of thread = 0.6495P, θ = 60˚ and E = D – 0.6495 P
For Whitworth thread:
Depth of thread = 0.64P, θ = 55˚ and E = D – 0.64 P

ELEMENTS
Derivation of best wire size:
• If there is a possibility of the thread angle being incorrect, the wire used should be such as to touch the thread
exactly on the pitch line and the inspection of the effective diameter from reading over such wires will be
independent of any error in the thread angle. Such wires are called the “best” wire.
• In the case of best wire size, at point C and D the wire touching the flank of thread lies on the pitch line or the
effective diameter line as shown in the figure. The line OD is perpendicular to the flank position of the thread.

ELEMENTS
Derivation of best wire size:
Let the half the included angle of thread be (θ / 2). Then in ΔOAD,
∠ AOD = 90 - (θ / 2)
⸫ sin (90 – (θ / 2)) = AD / OD
⸫ cos (θ / 2) = AD / OD
⸫ OD = AD sec (θ / 2)
As OD = r = db / 2, where db = best wire diameter and AD = P / 4 , where P = pitch, substituting value of OD
and AD,
⸫ db = P / 2 × sec (θ / 2)
This is the equation for best wire size in terms of P and θ.

ELEMENTS
 Measurement of Thread Angle:
• Thread angle and form are usually checked by optical means, using an optical projector which may be of the
horizontal or vertical form. Cabinet projectors are available which are compact and take up relatively little
space.
• Also measuring microscopes may be used, one form of which is a tool maker’s microscope which can also be
used to carry out a complete check of the thread elements.
• The four essential elements of a projection system are:
1. Source of light - a lamp.
2. Collimating lens - gives a parallel beam of light.
3. Projection lens - combination lens forms a real image on the screen.
4. Screen-opaque or translucent screen.
• The enlarged image of the thread form appears on the ground-glass screen on which is mounted the template
or drawing of the form made to scale equal to the magnification of the two forms, ideal and projected (actual)
are then compared.

ELEMENTS
 Measurement of Thread Angle:
• The simplest and probably the most effective protractor for this class of work is shown as the N.P.L.
protractor as it was developed at the National Physical Laboratory.
• A vernier protractor, of the type shown in the figure, usually has a circular scale calibrated in half-degrees,
minutes of the arc being read off a tangent screw dial.
• Each flank angle is measured separately as shown, the protractor arm being rotated in turn to each flank of
the projected thread image. Hence, the thread included angle θ is the sum of the two flank angles (θ / 2).

ELEMENTS
 Pitch Measurement:
Errors in the pitch of a nut or bolt have a serious effect on the accuracy of the fit product so it is to be measured
more accurately.
 Pitch Measuring Machine:
• Measurement of the pitch of a screw thread by pitch measuring machine is a relatively simple and accurate
method.
• The threaded component whose pitch is to be measured is mounted between the centers on the pitch
measuring machine. Then a spring-loaded stylus, of a size such that it contacts the thread flank at points near
the pitch line as shown in the figure.

ELEMENTS
 Pitch Measuring Machine:
• A range of stylus is supplied with the machine, together with tables giving correct stylus to be used according
to the type and pitch of the thread under test.
• When the stylus is correctly positioned in the Vee, the carriage of the machine, which carries the stylus and
indicator, can be displaced parallel to the work axis by means of a micrometer screw to which a large dial is
attached.
• When the pointer is accurately in position, the micrometer reading is noted. The stylus is then moved along
into the next thread space, by rotation of the micrometer and second reading is taken.
• Successive measurements P1, P2, P3, etc. are read from the micrometer as shown in the figure. For the exact
positioning of the stylus, the stylus should touchpoint A and B on flanks along the pitch line.
• The difference between the two readings is the pitch of the thread. Readings are taken in this manner until the
whole length of the job has been covered.

ELEMENTS
 Optical Projection Method:
• A profile projector or tool maker’s microscope may be used for projecting a screw thread. This technique is
suitable for external screw thread only because the internal thread cannot be projected.
• A screw thread projected by means of a profile projector is shown in the figure. The pitch of the screw thread
AB will be given by
AB = (BC / cos θ) × (1 / magnification)
• The length BC, perpendicular to the thread flank can be measured experimentally and hence the pitch can be
calculated from the above equation.

MEASUREMENT OF INTERNAL SCREW THREAD
ELEMENTS
 Measurement of Major diameter:
• The major diameter of an internal thread is normally measured by some form of horizontal comparator fitted
with ball ended styli of radius less than the root radius of the thread to be measured.
• The stylus is attached to a floating head which is kept in contact with the spindle of the dial indicator. The
floating head is constrained towards the indicator by a spring. A comparator measures the distance x as
shown in the figure.
D = √𝑥2 – (P / 2)2

ELEMENTS
 Measurement of Minor diameter:
• When the value of minor diameter is less than 20 mm, taper parallels and micrometers are used for
measurement. For diameter above 20 mm, calibrated rollers and slip gauges are used.
 Using Taper Parallels:
• The taper parallels are a pair of wedges having reduced and parallel outer edges. The diameter across their
outer edges can be changed by sliding them over each other.
• Inside the thread, a pair of taper parallel is inserted and adjusted until firm contact is established with the
minor diameter as shown in the figure. The diameter over the outer edges is measured with a micrometer.

ELEMENTS
 Measurement of Minor diameter:
• When the value of minor diameter is less than 20 mm, taper parallels and micrometers are used for measurement.
For diameter above 20 mm, calibrated rollers and slip gauges are used.
 Using slip gauges and rollers:
• In this method, precision rollers are inserted inside the thread and proper slip gauges inserted between the rollers as
shown in the figure, so that firm contact is established.
• The minor diameter is then the length of slip gauges plus twice the diameter of the roller.
Dm = x + 2d
where, Dm = minor diameter
x = slip gauge distance
d = roller diameter

ELEMENTS
 Measurement of Effective diameter:
• The effective diameter of the internal screw thread is measured using ball-ended styli of best wire size in a
thread measuring machine. These are so designed to contact the thread on or near the effective diameter line.

ELEMENTS
 Measurement of Effective diameter:
• For different threads, the different stylus is needed. The machine is set to a master consisting of a pair of Vee
notched arms analogizing the thread as shown in the figure.
• The Vee notches in the jaw blades are chosen to correspond to the included angle of the thread to be
measured and the size of the gauge block combination is determined from the knowledge of the thread pitch
and a constant for the jaw blades.
• The Vee notches are separated by slip gauges of length x which gives the effective diameter as follows:
Let P = pitch and ɸ = flank angle.
Consider ΔABC in the given figure,
tan ɸ = AC / BC = (P / 4) / Z
⸫ Z = (P / 4) / tan ɸ
Now, Y / 2 = L1 – Z = L2 – Z
⸫ Y = (L1 + L2) – 2Z
⸫ E = slip gauge distance + Y
⸫ E = x + Y

ELEMENTS
 Measurement of Pitch:
• Using an adaptor, the pitch of an internal thread can be measured by any pitch measuring machine. This
adaptor carries a fine bar that can be inserted into the ring, the stylus being fined to the bar end as shown in
the figure, engaging with the thread in the usual manner.
• The ring gauge is mounted on a faceplate or the headstock of the machines, which will accommodate a range
of various sizes. A special set up on the faceplate becomes essential for very large rings.

ELEMENTS
 Screw Thread Gauges:
• When a large number of components with screw thread are to be produced then the measurement of various
elements of screw thread will be very expensive and time-consuming.
• In mass production, these procedures can not be adapted. Gauges are scaleless inspection tools at a rigid
design, which are used to check the dimensions of manufactured parts.
• They also check the form and relative position of the surfaces of parts. They do not measure the actual size or
dimension of the part. They are only used to determine whether the inspected part has been made within the
specified limits or not.
• Taylor's principle can be applied to thread. According to Taylor's principle, a GO gauge should check both
geometric features and size and thus be of full form, whereas NO GO gauge should check only one
dimension major diameter and the effective diameter.
Classification of thread gauges:
1. Working gauges: Working gauges arc used to check the product as it is being manufactured. These are also
used to determine the acceptance or rejection of the product.
2. Setting or checking gauges: Checking gauges are generally plugged type gauges with the help of which
adjustable thread ring gauges, thread snap gauges, and other thread comparators are set for checking size of
master or basic gauges.

ELEMENTS
 Plug screw gauges for internal threads:
• In order to gauge nuts or internal threads, it is obvious that a full form plug screw gauge, made accurately to
the minimum dimensions of the internal thread, will ensure that all dimensions are not less than the minimum
permissible if it will assemble.
• Major, minor, and effective diameters will be checked, and the gauge will ensure that pitch, angle or form
errors arc not reducing the virtual effective diameter below the minimum.
• Plug gauges are made of different patterns, most commonly used patterns are discussed in the following:
Full Form GO gauge: This is an external screw-threaded plug gauge of the type shown in the figure, of a
gauging length equal to the working length of engagement, and made to the low limits of size.

ELEMENTS
 Plug screw gauges for internal threads:
Effective Diameter NO GO gauge: The screw plug gauge is made to the effective diameter high limit, having
restricted contact with the working threads, and with a length sufficient to accommodate two or three thread
forms. This type of gauge is shown in the figure.
Minor Diameter NO GO gauge: This is a plain cylindrical plug gauge manufacture to the high limit of the
minor diameter. Such a gauge is shown in the figure.

ELEMENTS
 Ring screw gauges for external screw threads:
• For the production gauging of bolts, the equivalent mating surface of the bolt threads is known as a ring
gauge. The GO ring gauge has a full form of thread, the NO-GO gauge being truncated on the minor
diameter and cleared on the major diameter at the root of the thread. A GO thread ring gauge is shown in the
figure.

ERRORS IN THREADS
During the manufacturing or storage of screw threads, errors can arise. There are following important elements
of a thread, and errors in any one of these may cause rejection.
 Errors in Major diameter: Error in the major diameter may cause interference between mating threads or a
reduction in the flank contact. In the case of internal threads, these errors cause weakening of screw by
reduction of the wall thickness.
 Errors in Minor diameter: These errors may similarly cause interference, flank contact reduction, or
weakness by reduction of the cross-section of the root.
 Errors in Effective diameter: The effective diameter is an important element of the screw thread, so errors
in this element will cause either interference between the thread flanks or general slackness of fit between
mating parts.
 Errors in Thread angle:
• These errors of thread cause interference between the bolt and nut, and to accommodate it, the effective
diameter of nut has to be increased. These errors increase the virtual effective diameter of a bolt and decrease
that of the nut.
• The effective diameter of an incorrect bolt must be decreased to permit a correct mating thread to mate and
similarly, the effective diameter of an incorrect nut must be increased.

ERRORS IN THREADS
 Errors in Pitch:
• Threads are generated by a point cutting tool. The ratio of the linear velocity of tool and angular velocity of
work must be correct and this ratio must be maintained constant otherwise pitch error will occur in this case.
Cumulative pitch error is the total pitch error in the overall length of thread.
• The pitch errors are further classified as:
 Progressive Error:
• If the pitch of the thread is uniform but is longer or shorter than its nominal value, the error is called
progressive error. This error occurs when the tool work velocity ratio is incorrect though it may be constant.
It occurs due to the use of incorrect gear on an approximate gear train between work and lead screw.

ERRORS IN THREADS
 Periodic Error:
• This type of error occurs when the tool-work velocity ratio is not constant.
• The periodic error repeats itself at equal intervals along the thread. Errors due to these causes will be cyclic,
i.e. the pitch will increase to a maximum, reduce through normal to a minimum and so on.
 Drunken Error:
• A drunken error is a particular case of a periodic pitch error repeating at intervals of one pitch. For the
drunken thread, the pitch measured parallel to the thread axis will always be correct.

ERRORS IN THREADS
 Periodic Error:
 Drunken Error:
• The development of the thread helix will be a curve and not a straight line as shown in the figure. Such errors
are extremely difficult to determine and except on large threads will not have any great effect.
 Irregular Errors:
• Irregular errors are those which vary in an irregular manner along the length of the thread.
• Fault in the machine, irregular cutting action resulting from non-uniformity in the material of the screw,
disturbances in machining set up, etc. are the possible causes of these errors.

TYPES OF GEARS
 Spur gears: These gears are the simplest of all gears. The gear teeth are cut on the periphery and are parallel
to the axis of the gear. They are used to transmit power and motion between parallel shafts.
 Helical gears: The gear teeth are cut along the periphery, but at an angle to the axis of the gear. Each tooth
has a helical or spiral form. These gears can deliver higher torque since there is a number of teeth in a mesh
at any given point of time. They can transmit motion between parallel or non-parallel shafts.

TYPES OF GEARS
 Herringbone gears: These gears have two sets of helical teeth, one right hand and the other left-hand,
machined side by side.
 Worm and worm gears: A worm is similar to a screw having single or multiple start threads, which form the
teeth of the worm. The worm drives the worm gear or worm wheel to enable transmission of motion. The
axes of worm and worm gear are at right angles to each other.

TYPES OF GEARS
 Bevel gears: These gears are used to connect shafts at any desired angle to each other. The shafts may lie in
the same plane or in different planes.
 Hypoid gears: These gears are similar to bevel gears, but the axes of the two connecting shafts do not
intersect. They carry curved teeth, are stronger than the common types of bevel gears, and are quiet- running.
These gears are mainly used in automobile rear axle drives.

TYPES OF GEARS
 Gear tooth terminology:
• Pitch surface: The surface of the imaginary rolling cylinder (cone, etc.) that replaces the toothed gear.
• Pitch circle: A normal section of the pitch surface.
• Addendum circle: A circle bounding the ends of the teeth, in a normal section of the gear.
• Dedendum circle or Root circle: The circle bounding the spaces between the teeth, in a normal section of
the gear.
• Addendum: The radial distance between the pitch circle and the addendum circle.

TYPES OF GEARS
• Dedendum: The radial distance between the pitch circle and the root circle.
• Clearance: The difference between the Dedendum of one gear and the addendum of the mating gear.
• The face of a tooth: That part of the tooth surface lying outside the pitch surface.
• The flank of a tooth: The part of the tooth surface lying inside the pitch surface.
• Top land: The top surface of a gear tooth.
• Bottom land: The bottom surface of the tooth space.

TYPES OF GEARS
• Circular thickness (tooth thickness): The thickness of the tooth measured on the pitch circle. It is the length
of an arc and not the length of a straight line.
• Tooth space: The space between successive teeth.
• Width of space: The distance between adjacent teeth measured on the pitch circle.
• Backlash: The difference between the tooth thickness of one gear and the tooth space of the mating gear.

TYPES OF GEARS
• Circular pitch (p): The width of a tooth and space, measured on the pitch circle. It is equal to the pitch
circumference divided by the number of teeth.
p = πd / Z
• Diametral pitch (P): The number of teeth of a gear per unit pitch diameter. The diametral pitch is hence the
number of teeth divided by the pitch diameter.
p = Z / d

TYPES OF GEARS
• Module: Pitch diameter divided by the number of teeth. The pitch diameter is usually specified in
millimeters.
m = πd / Z
• Fillet radius: The small radius that connects the profile of a tooth to the root circle.
• Base circle: An imaginary circle used in involute gearing to generate the involutes that form the tooth
profiles.

TYPES OF GEARS
• Pitch point: The point of tangency of the pitch circles of a pair of mating gears.
• Common tangent: The line tangent to the pitch circle at the pitch point.
• Line of action: Line normal to a pair of mating tooth profiles at their point of contact.
• Path of contact: The path traced by the contact point of a pair of tooth profiles.
• Pressure angle (ɸ): The angle between the common normal at the point of tooth contact and the common
tangent to the pitch circles. The pressure angle is also the angle between the line of action and the common
tangent.

ERRORS IN SPUR GEARS
• A basic understanding of the errors in spur gears during manufacturing is important before we consider the
possible ways of measuring the different elements of gears.
• A spur gear is a rotating member that constantly meshes with its mating gear. It should have the perfect
geometry to maximize the transmission of power and speed without any loss.
• From a metrological point of view, the major types of errors are as follows:
 Gear blank runout errors:
• Gear machining is done on the gear blank, which may be a cast or a forged part. The blank would have
undergone preliminary machining on its outside diameter (OD) and the two faces.
• The blank may have radial runout on its OD surface due to errors in the preliminary machining. In addition, it
may have excessive face runout.
• Unless these two runouts are within prescribed limits, it is not possible to meet the tolerance requirements at
later stages of gear manufacture.

 Gear tooth profile errors:
• These errors are caused by the deviation of the actual tooth profile from the ideal tooth profile.
• Excessive profile error will result in either friction between the mating teeth or backlash, depending on
whether it is on the positive or negative side.
 Gear tooth errors:
• This type of error can take the form of either tooth thickness error or tooth alignment error. The tooth
thickness measured along the pitch circle may have a large amount of error.
• On the other hand, the locus of a point on the machined gear teeth may not follow an ideal trace or path. This
results in a loss in the alignment of the gear.
 Pitch errors:
• Errors in pitch cannot be tolerated, especially when the gear transmission system is expected to provide a
high degree of positional accuracy for a machine slide or axis. Pitch error can be either a single pitch error or
accumulated pitch error.
• Single pitch error is the error in actual measured pitch value between adjacent teeth. Accumulated pitch error
is the difference between theoretical summation over any number of teeth intervals and summation of actual
pitch measurement over the same interval.

 Runout errors:
• This type of error refers to the runout of the pitch circle. Runout causes vibrations and noise and reduces the
life of the gears and bearings. This error creeps in due to inaccuracies in the cutting arbor and tooling system.
 Lead errors:
• This type of error is caused by the deviation of the actual advance of the gear tooth profile from the ideal
value or position. This error results in poor contact between the mating teeth, resulting in loss of power.
 Assembly errors:
• Errors in an assembly may be due to either the center distance error or the axes alignment error. An error in
center distance between the two engaging gears results in either backlash error or jamming of gears if the
distance is too little.
• In addition, the axes of the two gears must be parallel to each other, failing which misalignment will be a
major problem.

MEASUREMENT OF TOOTH THICKNESS
• Various methods are recommended for the measurement of the gear tooth thickness. There is a choice of
instruments such as the gear tooth caliper, and span gauging or tooth span micrometer.
• Constant chord measurement and measurement over rolls or balls are additional options. Two such methods,
namely measurement with gear tooth caliper and tooth span micrometer are discussed in detail here.
 Measurement of Gear Tooth Calipers:
• This is one of the most commonly used methods and perhaps the most accurate one. The figure above
illustrates the construction details of the gear caliper. It has two vernier scales, one horizontal and the other
vertical.

• The vertical vernier gives the position of a blade, which can slide up and down. When the surface of the
blade is flush with the tips of the measuring anvils, the vertical scale will read zero.
• The blade position can be set to any required value by referring to the vernier scale. From the figure, it is
clear that tooth thickness should be measured at the pitch circle (chord thickness C1C2 in the figure).
• Now, the blade position is set to a value equal to the addendum of the gear tooth and locked into position
with a locking screw. The caliper is set on the gear in such a manner that the blade surface snugly fits with
the top surface of a gear tooth.

• The two anvils are brought into close contact with the gear, and the chordal thickness is noted down on the
horizontal vernier scale.

• The value of width (w) and addendum height (h) can be obtained as follows.
From the figure, AB = w and CD = h.
So, AD + BD = w and AD = BD.
⸫ w = 2 × AD
From ΔAOD,
sin θ = AD / AO
⸫ AD = AO sin θ = R sin θ
Angle AOD = θ = 360 / 4N
⸫
θ = 90 / N
⸫θ = R sin (90 / N)
Substituting value of AD,
⸫ w = d sin (90 / N), where d = pitch circle diameter

• Now the height h can be calculated as follows:
From the figure,
OC = OE + EC
But OE = r = Nm / 2 and EC = addendum = module = m
⸫ OC = (Nm / 2) + m
From the figure,
OC = OD + DC
From ΔADO, cos θ = OD / r
⸫ OD = r cos θ
⸫ OD = (Nm / 2) cos (90 / N)
Substituting value of OD and OC,
(Nm / 2) + m = (Nm / 2) cos (90 / N) + DC

But DC = h
⸫ h = (Nm / 2) + m - (Nm / 2) cos (90 / N)
⸫ h = (Nm / 2)[1 + (2 / N) - cos (90 / N)]

 Constant Chord Method:
• In the above 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 a set, each having a different number of teeth would involve
separate calculations. Thus the procedure becomes laborious and time-consuming.
• The constant chord method does away with these difficulties. Constant chord of gear is measured where the
tooth flanks touch the flanks of the basic rack. The teeth of the rack are straight inclined to their center lines
at the pressure angle.

• 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 utilized 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 the tooth (same module), the contact always occurs
at two fixed points A and B. AB is known as a constant chord.

• The Constant chord is defined as the chord joining those points, on opposite faces of the tooth, which makes
contact with the mating teeth when the centerline of the tooth lies on the line of the gear centers.
• The true value of AB and its depth from the tip, where it calculated mathematically and then verified by an
instrument.
• The advantage of the constant chord method is that for all number of teeth (of the same module) value of
constant chord is the 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 a comparator which is more sensitive than the gear tooth Vernier.

• The chord length AB can be calculated as follows:
From figure,
Length DP = PE = (1 / 4) × circular pitch = (1 / 4) × πm
Consider ΔPAD,
Angle APD = ɸ
cos ɸ = AP / DP
⸫AP = DP cos ɸ
⸫AP = (1 / 4) × πm cos ɸ
Consider ΔPCA,
Angle CAP = ɸ
cos ɸ = CA / AP
⸫ CA = AP cos ɸ = (1 / 4) × πm cos2 ɸ

• The depth h can be calculated as follows:
Consider ΔPAC,
sin ɸ = CP / AP
CP = AP sin ɸ
⸫CP = (1 / 4) × πm cos ɸ sin ɸ
GP = GC + CP
Where, GP = addendum = module = m
GC = h
⸫ m = h + (1 / 4) × πm cos ɸ sin ɸ
⸫ h = m - (1 / 4) × πm cos ɸ sin ɸ

PARKINSON’S GEAR TESTER
 Principle:
• A master gear (standard gear) is mounted on a fixed vertical spindle and the gear to be tested on another
similar spindle mounted on a sliding carriage. These gears are maintained in the mesh by spring pressure.
• If spring-loaded pair of gears in the close mesh is rotated, any errors in tooth form, pitch or concentricity of
pitch line, will cause a variation of center distance. Thus, movements of the carriage, as indicated by the dial
gauge, indicate errors in the gear under test.

 Construction:
• A gear tester for testing spur gears is shown in figure This types of testers are also available for bevel, helical
and worm gears. The gears are mounted on two spindles so that they are free to rotate without measurable
clearance.
• The master gear is mounted as an adjustable carriage whose position can be adjusted to enable a wide range
of gears diameters to be accommodated and it can be clamped in any desired position.
• The gear under test is mounted on a floating spring-loaded carriage so that the master gear and the gear under
test may have meshed together under controlled spring pressures.
 Working:
• Using gauge blocks between the spindles set the dial gauge to read zero at the correct center distance, and
adjust the spring loading. Set limit marks on the dial gauge.
• Mount the master gear and the gear to be tested, and note the variation in the dial indicator reading when the
gears are rotated. If it falls outside the limit marks, the gear is not acceptable.

 Working:
• The variations in dial gauge indicator leadings are a measure of any irregularities in the gear under test,
alternatively, a recorder can be fitted, in the form of a waved circular chart and records made of the gear
variation in accuracy of mesh.
• The below figure shows a reproduction of a few typical charts with a reduced scale and magnified radial errors.
 Limitations:
1. Friction in the movement of the floating carriage reduces the sensitivity.
2. It is not suitable for gear with more than 300 mm in diameter.
3. Measurements are directly dependent upon master gear or reference gear.
4. Errors are not clearly identified for type profile, pitch, helix, and tooth thickness and are indistinguishably
mixed.
5. The accuracy is of the order of ± 0.001 mm

MEASUREMENT OF TOOTH PROFILE
• The profile is the portion of the tooth flank between the specified form circle and the outside circle or start of
tip chamfer. Profile tolerance is the allowable deviation of the actual tooth form from the theoretical profile in
the designated reference plane of rotation.
• As the most commonly used profile for spur and helical gears is the involute profile, our discussions are limited
to the measurement of involute profile and errors in this profile. We will now discuss two of the preferred
methods of measuring a tooth profile.

 Tool Makers’ Microscope:
• Tool Makers Microscope is an optical device used to view and measure very fine details, shapes, and dimensions on
small and medium-sized tools, dies, and workpieces.
• It is equipped with a glass table that is movable in two principal directions and can be read to 0.01 mm. The
microscope is also equipped with a protractor to measure the angular dimensions.
• It is also equipped with surface illumination. Provision is available to adjust the height of the viewing head to get a
sharp image of the object.
• The probe is brought into contact with the tooth profile. To obtain the most accurate readings, it is essential that the
feeler (probe) is sharp, positioned accurately, and centered correctly on the origin of the involute at 0° of the roll.

• The machine is provided with multiple axes movement to enable measurement of the various types of gears.
The measuring head comprising the feeler, electronic unit, and chart recorder can be moved up and down by
operating a handwheel.
• The arbor assembly holding the gear can be moved in two perpendicular directions in the horizontal plane by
the movement of a carriage and a cross-slide. Additionally, the base circle disk on which the gear is mounted
can be rotated by 360°, thereby providing the necessary rotary motion for the gear being inspected.
• The feeler is kept in such a way that it is in a spring-loaded contact with the tooth flank of the gear under
inspection. As the feeler is mounted exactly above the straight edge, there is no movement of the feeler if the
involute is a true involute.

• If there is an error, it is sensed due to the deflection of the feeler and is amplified by the electronics unit and
recorded by the chart recorder.
• The movement of the feeler can be amplified 250, 500, or 1000 times, the amplification ratio being selected by
a selector switch. When there is no error in the involute profile, the trace on the recording chart will be a
straight line.
• Gleason gear inspection machine, a product of Gleason Metrology Systems Corporation, USA, follows the
fundamental design aspect of any testing machine with the capability to handle up to 350 mm diameter gears. It
also integrates certain object-oriented tools to achieve faster cycle times and better human- machine interaction.

 Involute Measuring Machine:
• This machine is designed for checking the involute profiles of the spur and helical gears. The involute
measuring machine is shown in the figure below. if a straight edge is rolled around a base circle without
slipping, the stylus of the dial gauge attached to the straight edge would traverse a true involute.
• A ground circular disc having exactly the same diameter as the base circle of the gear under test and with the
gear to be tested is mounted on the mandrel.
• The straight edge of the instrument is brought in contact with the base circle of the disc. As the gear along with
disc is rotated, the straight edge moves over the disc without slip.
• The stylus of the dial gauge brought in contact with a tooth profile. When the gear and disc are rotated, the
stylus is slid along the involute curve.
• The deviation of the tooth from the correct involute is indicated by a dial indicator of accuracy 0.001mm. A
master involute template is also provided with the machine for setting and calibration of the machine.

MEASUREMENT OF PITCH
• Pitch is the distance between corresponding points on equally spaced and adjacent teeth. Pitch error is the
difference in distance between equally spaced adjacent teeth and the measured distance between any two
adjacent teeth.
 Pitch measuring instruments:
• These instruments enable the measurement of chordal pitch between successive pairs of teeth. The instrument
comprises a fixed finger and a movable finger, which can be set to two identical points on adjacent teeth along
the pitch circle.
• The pitch variation is displayed on a dial indicator attached to the instrument, as shown in the figure below. In
some cases, the pitch variation is recorded on a chart recorder, which can be used for further measurements.
• A major limitation of this method is that readings are influenced by profile variations as well as runout of the
gear.

MEASUREMENT OF PITCH
 Pitch checking instruments:
• A pitch-checking instrument is essentially a dividing head that can be used to measure pitch variations. The
instrument can be used for checking small as well as large gears due to its portability.
• The figure below explains the measuring principle for spur gear. It has two probes - one fixed, called the anvil,
and the other movable, called the measuring feeler. The latter is connected to a dial indicator through levers.
• The instrument is located by two adjacent supports resting on the crests of the teeth. A tooth flank is butted
against the fixed anvil and locating supports. The measuring feeler senses the corresponding next flank.
• The instrument is used as a comparator from which we can calculate the adjacent pitch error, actual pitch, and
accumulated pitch error.

MEASUREMENT OF RUNOUT
• Runout is caused when there is some deviation in the trajectories of the points on a section of a circular surface
in relation to the axis of rotation.
• In the case of gear, runout is the resultant of the radial throw of the axis of gear due to the out of roundness of
the gear profile. Runout tolerance is the total allowable runout.
• In the case of gear teeth, runout is measured by a specified probe such as a cylinder, ball, cone, rack, or gear
teeth. The measurement is made perpendicular to the surface of revolution.
• On bevel and hypoid gears, both axial and radial runouts are included in one measurement. A common method
of runout inspection, called a single-probe check and shown in the figure, uses an indicator with a single probe
whose diameter makes contact with the flanks of adjacent teeth in the area of the pitch circle.

MEASUREMENT OF RUNOUT
• On the other hand, in a two-probe check illustrated in the figure, one fixed and one free-moving probe, are
positioned on diametrically opposite sides of the gear and make contact with identically located elements of the
tooth profile.
• The range of indications obtained with the two-probe check during a complete revolution of the gear is twice
the amount resulting from the single-probe check.

MEASUREMENT OF LEAD
• Lead is the axial advance of a helix for one complete rotation about its axis. In the case of spur gears, lead
tolerance is defined as the allowable deviation across the face width of a tooth surface.
• Control of lead is necessary in order to ensure adequate contact across the face width when gear and pinion are
in mesh. The figure below illustrates the procedure adopted for checking the lead tolerance of spur gear.
• A measuring pointer traces the tooth surface at the pitch circle and parallels to the axis of the gear. The
measuring pointer is mounted on a slide, which travels parallel to the center on which the gear is held.
• The measuring pointer is connected to a dial gauge or any other suitable comparator, which continuously
indicates the deviation. The total deviation shown by the dial indicator over the distance measured indicates the
amount of displacement of the gear tooth in the face width traversed.

MEASUREMENT OF BACKLASH
• If the two mating gears are produced such that tooth spaces are equal to tooth thicknesses at the reference
diameter, then there will not be any clearance in between the teeth that are getting engaged with each other.
• This is not a practical proposition because the gears will get jammed even from the slightest mounting error or
eccentricity of bore to the pitch circle diameter. Therefore, the tooth profile is kept uniformly thinned, as shown
in the figure. This results in a small play between the mating tooth surfaces, which is called a backlash.
• We can define backlash as the amount by which a tooth space exceeds the thickness of an engaging tooth.
Backlash should be measured at the tightest point of mesh on the pitch circle, in a direction normal to the tooth
surface when the gears are mounted at their specified position.
• Backlash value can be described as the shortest or normal distance between the trailing flanks when the driving
flank and the driven flank are in contact. A dial gauge is usually employed to measure the backlash.
• Holding the driver gear firmly, the driven gear can be rocked back and forth. This movement is registered by a
dial indicator having its pointer positioned along the tangent to the pitch circle of the driven gear.

CONCENTRICITY OF TEETH
• It is an important item and should be checked to ensure that the setup and equipment are in good order. If teeth
are not concentric then fluctuating velocity will be noticed on the pitch line-while transmitting motion.
• This also leads to inaccuracy of parts, when being used for indexing purposes. Tooth concentricity can be
checked by:
1. mounting the gear between the bench centers, placing a standard roller in each tooth space and then using dial
indicator,
2. using a projector in which case the teeth are brought against a stop and each image of tooth on screen should
coincide with a line on the screen,
3. using a gear testing fixture fitted with a spring-loaded slide and dial indicator, in which case spring exerts a
constant pressure on the mating teeth and the movement of the dial indicator gives the measure of the
eccentricity of teeth.

Ch-3: Measurement of screw thread and gear

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