3. Limits, Fits & Tolerances
• It is impossible to manufacture any part precisely to a given dimension.
• Even if by chance the part is exact, it is impossible to measure it accurately to prove it.
• If attempts are made to achieve perfect size the cost of production will increase tremendously.
Therefore, some permissible variation has to be allowed.
Limits
The dimension of a manufactured part can only be made to lie between maximum and minimum limits.
Tolerance
The permissible variation in size or dimension is called tolerance. The difference between the maximum and minimum
limit is known as tolerance zone.
System of writing tolerance
Unilateral: - Dimension of a part is allowed only on one side of basic size.
Example − 25+0.00
+0.02
, 25−0.02
−0.00
, 25+0.02
+0.05
Bilateral: - Dimension of a part is allowed both sides of basic size.
Example − 25−0.02
+0.02
, 25−0.03
+0.05
Terminologies
Shaft ― Not only refers to diameter of circular shaft but also to any external dimension of a component.
Hole ― Not only refers to the diameter of hole but also to any internal diameter of a component.
Basic Size ― It is the standard size of a part with reference to which the limits of variation of a size are determined.
Actual Size ― Dimension as measured on manufactured part.
Zero-line ― A straight line drawn to represent the basic size.
Deviation ― The algebraic difference between the actual size and basic size.
Upper deviation ― The algebraic difference between maximum limit of size and corresponding basic size.
Lower deviation ― The algebraic difference between minimum limit of size and corresponding basic size.
4. Fundamental Deviation ― It is one of that two deviations either upper or lower which is nearest tot the zero line for
either hole or shaft.
For hole it is denoted by Capital letters – A, B, C, …… ZA, ZB, ZC.
For shaft it is denoted by small letters – a, b, c, …… za, zb, zc.
Tolerance Grade ― It is an indication of degree of accuracy of manufacture & is designated by “IT” followed by a
number.
There are IT01, IT0, IT1, IT2, IT3, IT4, IT5, IT6, IT7, IT8, IT9, IT10, IT11, IT12, IT13, IT14, IT15.
Only tolerances from IT5 are used.
i = 0.45D(1 3⁄ )
+ 0.001D
D = √D1 × D2
𝐃 is geometric mean dimension of dimension steps between which is given basic size is lying.
D1 & D2 are given, D is in mm & 𝐢 is in microns(μ)
IT5 7i IT11 100i
IT6 10i IT12 160i
IT7 16i IT13 250i
IT8 25i IT14 400i
IT9 40i IT15 640i
IT10 64i IT16 1000i
Fits
It is the degree of tightness or looseness between 2 mating parts.
Clearance Fit ― Shaft is always smaller than the hole.
Interference Fit ― The minimum permissible diameter of the shaft is bigger than the maximum allowable diameter.
Transition Fit ― It lies midway between interference fit and clearance fit.
5. Allowance
It is defined as the minimum clearance between the mating parts, and is equal to difference between Lower limit of hole
and Upper limit of shaft (or) maximum interference between the mating parts.
Hole basis system
A hole is created first, shaft is designed based on hole.
Shaft basis system
A shaft is created first, hole is designed based on shaft.
Designation
Example
Find out the limits & tolerances for 50H7d8 and Dimension steps are 30 & 50?
Basic Size = 50mm
Fundamental Deviation for Hole is 𝐇 and tolerance is 𝐈𝐓𝟕.
Fundamental Deviation for Shaft is 𝐝 and tolerance is 𝐈𝐓𝟖.
D = √D1 × D2 = √30 × 50 = 38.7 mm
i = 0.45D(1 3⁄ )
+ 0.001D = 0.45 × 38.7(1 3⁄ )
+ 0.001 × 38.7 = 1.6μm
Tolerance for Hole = IT7 = 16i = 16 × 1.6μm = 25.6μm = 0.0256mm
Tolerance for Shaft = IT8 = 25i = 25 × 1.6μm = 40μm = 0.04mm
Fundamental deviation for hole is denoted as H which is equal to zero (0) i.e., Lower limit for hole is zero.
𝐥𝐢𝐦𝐢𝐭𝐬 𝐟𝐨𝐫 𝐡𝐨𝐥𝐞 = 𝟓𝟎 𝟎.𝟎𝟎
𝟎.𝟎𝟐𝟓𝟔
Fundamental deviation for shaft is given as = −𝟏𝟔𝐃 𝟎.𝟒𝟒
𝛍𝐦 = −16 × (38.7)0.44
= −80 μm
So lower limit of shaft = Fund. Deviation + Tolerance = −0.08 − 0.04 = −0.12mm
So upper limit of shaft = Fundamental Deviation = −0.08
𝐥𝐢𝐦𝐢𝐭𝐬 𝐟𝐨𝐫 𝐒𝐡𝐚𝐟𝐭 = 𝟓𝟎−𝟎.𝟏𝟐
−𝟎.𝟎𝟖
6. Limit gauges
These are the gauges which are used to check the limits of a part. They are of two types GO & NOGO gauges.
• When the component inspected by using GO & NOGO gauges, if the GO gauge is entering and NOGO gauge is not
entering into component, the component is said to be acceptable. If both the gauges are entering (or) both gauges
are not entering, then the component is said to be rejectable.
• GO gauge is always made for the maximum material limit of the component (Upper limit or Lower limit which is
farther from zero line), such that it always enters in the component (shaft or hole).
• The length of GO gauge is at least equal to the length of depth of the part to be inspected.
• NOGO gauge is made for the minimum material limit of the component such that it will not enter into the
component (Hole or Shaft).
• Plug gauges are used for holes and ring gauges are used for shafts.
Max material limit of a component ⇒ {
Smallest hole (L − limit of hole)
Largest shaft (H − limit of shaft)
} Checked by 𝐆𝐎 gauge
Min material limit of a component ⇒ {
Largest hole (H − limit of hole)
Smallest shaft (L − limit of shaft)
} Checked by 𝐍𝐎𝐆𝐎 gauge
Limit gauges for Holes-
Limit gauges for Shafts-
7. Taper gauges-
• If the top surface of component is lying in between the limits of the recess, the component is said to be
acceptable.
• If the top surface is going above the upper limit of recess (or) below the lower limit of recess the component is
said to be rejectable
Design of limit gauges
Basic Principle
According to basic principle no rejectable component is allowed to accept, but there is no problem in rejecting the
acceptable component.
Taylor’s Principle
• GO-Gauge is made for the maximum material limit if the component and it has to incorporate as many
dimensions (length, breadth & height) as possible to inspect in one stroke at one stage.
• NOGO-Gauge is made for the minimum material limit of the component. Separate NOGO gauges must be used for
each and every dimension i.e., separate for length and breadth and height.
• We can’t use separate limit gauges for GO gauge and 1 gauge for NOGO.
If we use following GO & NOGO gauges for hole of l=25.03 & b=40.06,
This component satisfies both GO & NOGO condition but it should be rejected actually but if Taylor principle (one gauge
for GO & separate gauges for diff. dimensions for NOGO) is used the NOGO gauge passes through hole which rejects the
component.
→ If Taylor principle is violated, then it accepts rejectable components and basic principle in design of limit
gauges will be violated.
8. Principle of provision of limits and tolerances on limits
Whenever the limit gauges used for inspection of the components also has to be manufactured in industry. Therefore, it
is difficult to manufacture the limit gauges to the exact dimensions.
It is required to provide limits and tolerances on limit gauges such that basic principle in designing of limit gauges
should not be violated.
Tolerances for limit gauge should be 10% of work tolerance (tolerance for hole or shaft)
Gear Tolerance = work allowance = 10% of work tolerance
On the limit gauges it is recommended to provide only unilateral tolerances.
Example (Hole)
Hole → 20−0.04
+0.06
mm ⇒ work tolerance = 0.06 − (−0.04) = 0.1 mm
Gear tolerance = 10% of work tolerance =
10
100
× 0.1mm = 0.01mm
Work allowance = 10% of work tolerance =
10
100
× 0.1mm = 0.01mm
Limit gauges are designed such that it may reject accepted components but should not accept rejectable components.
For GO limit gauge, as it has surface contact with component it is given wear allowance. For NOGO limit gauge, it has only
gear tolerance.
Example (Shaft)
9. Angular Measurement
Taper and Internal diameter measurement
tan
θ
2
=
O2A
O1A
=
r2 − r1
(h1 + r1) − (h2 + r2)
Two rollers of radius r1, r2 and height gauge
tan
θ
2
=
O2A
O1A
=
r2 − r1
(h2 − r2) − (h1 − r1)
tan θ =
y
h1 + r1
w = r1 + r1 sec θ + y
y = (h1 + r1) × tan θ
𝐰 = 𝐫𝟏 + 𝐫𝟏 𝐬𝐞𝐜 𝛉 + (𝐡 𝟏 + 𝐫𝟏) × 𝐭𝐚𝐧 𝛉
11. Single Dove tail
tan
θ
2
=
r2 − r1
(m2 − r2) − (m1 − r1)
Other provision one roller of radius ‘r’, slip gauges and micro meter.
tan θ =
S2 − S1
m1 − m2
Double dovetail
𝐭𝐚𝐧
𝛉
𝟐
=
𝐫𝟐 − 𝐫𝟏
(
𝐒 𝟏
𝟐 + 𝐫𝟏) − (
𝐒 𝟐
𝟐 + 𝐫𝟐)
12. Internal Diameter Measurements
Micro meter
Measurement of internal diameter using inside micro meter is similar to that of the outside diameter. But used for
measurement of plane inside diameter only.
Two balls of diameter ‘d’ and slip
D = 2d + s
Two balls of diameter d1, d2 and depth gauge
D = r2 + r1 + O1A
(O1A)2
= (O1O2)2
− (O2A)2
= (r1 + r2)2
− ((h1 + r1) − (h2 + r2))
2
Three balls of radius ‘r1’ and fourth ball of radius ‘r2’ and depth gauge
D
2
= r1 + O1A
D = 2(r1 + O1A)
O1A = √(O1O2)2 − (O2A)2
O1A = √(r1 + r2)2 + (H − (r1 + r2 + h))2
Pin method
OQ = OP + PQ
OP = √OP2 − AP2 = √L2 − (S 2⁄ )2
OP × PQ = AP × PB
PQ =
AP × PB
OP
=
S
2 ×
S
2
√L2 − (S
2⁄ )
2
13. Surface Finish
• Surface texture is a characteristic quality of an actual surface due to small departures from its original
geometric form which occurring at regular or irregular interval tend to form a pattern or texture on surface.
• Roughness is the short wavelength irregularities arising from the production process which comprise
individual scratch or tool marks.
• The Lay is tool marks or scratch marks taken collectively which characterizes the particular process.
• Waviness is the longer wavelength irregularities upon which roughness is super imposed. Waviness may be
induced by vibration, imperfect turning of a grinding wheel, chatter, heat treatment.
Quantitative parameters used in measurement of surface finish
Maximum Peak to valley height (Rt)
Rt = maximum peak to minimum valley = h4 − v5
As the value of Rt is increasing, the surface finish of given surface is
becoming poor.
If two surfaces have same Rt, it doesn’t mean they have same roughness.
Rt is not a good measure of roughness.
To overcome this, ten-point average (Rz) is used.
Rz =
(h1 + h2 + h3 + h4 + h5) − (v1 + v2 + v3 + v4 + v5)
5
Average Roughness (Ra)
This is also called ‘centre line average’ (CLA).
Ra =
A1 + A2 + A3 + A4 + ⋯ + An
L
=
∑ Ai
L
=
A
L
L = Sampling length
A = Total Area
Sampling length has some standard values like 0.025, 08, 0.25, 0.8, 2.5, 8.25mm. But 0.8mm, 2.5mm are commonly used
sampling lengths to find surface finish.
Ra =
|h1| + |h2| + |h3| + ⋯ + |hn|
n
×
1000
V. M
=
1
L
∫ h
L
0
dl
Ra =
∑ Ai
L
×
1
H. M
×
1000
V. M
14. Root mean square value (RMS)
RMS = √
h1
2
+ h2
2
+ h3
2
+ ⋯ + hn
2
n
= √
1
L
∫ h2
L
0
𝑑𝑙
RMS = 1.1Ra
⇒ Rz < Ra < Rt
⇒ Rz < Ra < RRMS < Rt
If two or more surfaces are having the same values of Ra, Rt, Rz indicates that both surfaces have same value of surface
roughness.
Form factor (K)
𝐾 =
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑚𝑒𝑡𝑎𝑙
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒
=
𝐴 𝑚
𝐿 × 𝑅𝑡
𝐾 > 0.8 → 𝑔𝑜𝑜𝑑 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑓𝑖𝑛𝑖𝑠ℎ
Designation of Surface finish
ISO Method
Roughness values Ra
Micro meters (µm)
Roughness N
ISO Grade Numbers
𝐍−𝟐𝟓%
+𝟓𝟎%
50 N12 37.5-75
25 N11 18.75-37.5
12.5 N10 9.6-18.75
6.3 N9 4.8-9.6
3.2 N8 2.4-4.8
1.6 N7 1.2-2.4
0.8 N6 0.6-1.2
0.4 N5 0.3-0.6
0.2 N4 0.15-0.3
0.1 N3 0.08-0.15
0.05 N2 0.04-0.08
0.025 N1 0.02-0.04
Indian Method
Symbol Ra (μm) Roughness grade number
▽▽▽▽ <0.25 N1, N2, N3
▽▽▽ <0.252 to 1.6 N4, N5, N6
▽▽ 1.6 to 8 N7, N8, N9
▽ 8 to 25 N10, N11
∼ >25 N12
15. Representation of surface finish on drawings
Example
The meaning of symbol is on the given surface is the surface finish to be obtained is 1.2
to 1.4 microns with shaping operation, such that the direction of lay is ⏊lar to the length
of workpiece and after shaping (machining) the sample length taken to find surface
roughness is 2.5 mm.
Methods of finding Surface finish
Touch Inspection
This is a comparison method, we can only find which surface rough compared to other and cannot be accessed the
degree of surface finish and flaws can’t be detected.
Visual Inspection
Judging the surface finish by naked eye and is always likely to misleading when we go for smooth surface, so it is used
only for rough surface.
Surface Photograph
In this method magnified photographs of the surface are taken with different types of illumination.
Micro Interferometer
Here an optical flat is placed on the surface to be inspected and illuminated by a monochromatic source of light.
Wallace Surface Dynamometer
The surface finish can be measured through friction. A pendulum is allowed to swing over surface jus by touching, the
time of swing determines the roughness.
Reflected Light Intensity
It is based in the direction in which the light is reflected.
Direct Instruments
Tanlinson Surface meter
As the workpiece is moving the stylus is finding the peaks & valleys. The
arm is moving up and down which rotates the pointer due to friction. When
the pointer is rotating, it removes smoke from the glass producing
magnified version of surface structure. Now by measuring the heights of
irregularities the values of Ra, Rt & Rz can be calculated.
16. Surface Profilograph
As the workpiece is moving the stylus is finding peaks and valleys.
The L-shaped arm is rotating w.r.t pivot and it reflects the light
incident on it.
The reflected light incident on sensitized paper, because the drum is
connected to work table, it is also rotating and producing magnified
version of surface structure.
Talyserf
𝐡(peak/valley height) ∝ difference in 𝐄𝐌𝐅
As the workpiece is moving, the stylus is finding the peaks and valleys.
The arm is getting tilting and the gap between arm & AB and arm & AC
will be changing which produces different amounts of EMF in AB & AC.
This difference in EMF is directly proportional to the heights of the
irregularities. By connecting output of voltmeter to the amplifier it is
possible to get any amount of magnification.
Abbot’s Profilometer
It acts as a simple transformer. As workpiece moves the stylus finds peaks & valleys.
The Y type of stamp is moving up and down. So, that the length of the core in the
transformer is changing. EMF generated in the secondary winding of a transformer
is varying. By measuring EMF generated in secondary winding the heights of the
irregularities of a surface are estimated. i.e., height of irregularities re directly
proportional to EMF.
Piezo electric crystal
Height of irregularities ∝ EMF
As the work piece is moving the stylus is finding the peaks & valleys. The arm is
moving up & down. The movement of the arm produces deflection in the
piezoelectric crystal material.
Due to deflection EMF is induced in the piezoelectric crystal and by measuring
this EMF the heights of irregularities of the surface can be estimated.
Perthometer
Because the EMF generated by the piezoelectric crystal is in the order of milli or micro volts, the output of piezoelectric
crystal can be connected to computer. So that the computer can generate the surface irregularity diagram, measures the
heights of irregularities and calculate the values of Ra, Rt, Rz and giving the output in the form of surface irregularity
diagram along with the values of Ra, Rt, Rz. this is called Perthometer.
Plastic replica technique
Because of limited size of the platform the above equipment can’t be used for the measurement of surface finish of a very
large surfaces and also for finding surface finish of internal surfaces like hole. Because the arm and stylus are not
accessible to move inside the hole.
So, for above cases the surface finish can be measured by using plastic replica technique.
In plastic replica technique a heated thermos plastic piece is kept on the surface and apply a little amount of force so that
the soft plastic metal is flowing into the each and every surface irregularity. Continue to apply the force until the plastic
piece is brought down to room temperature. Now take out the plastic piece, the surface finish produced in the plastic
piece is same as that of the actual surface. Therefore, plastic piece is kept on the platform of any of the above surface
finish measuring equipment and measure surface finish, only thing is peaks becomes valleys and vice versa.
