Chapter 7 - Islamic University of...
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Chapter 7
Shafts and Shaft Components
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Chapter Outline
Dr. Mohammad Suliman Abuhaiba, PE
Introduction
Shaft Materials
Shaft Layout
Shaft Design for Stress
Deflection Considerations
Critical Speeds for Shafts
Miscellaneous Shaft Components
Limits and Fits
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Considerations of Shaft Design
Material Selection
Geometric Layout
Stress & strength: Static strength, Fatigue strength
Deflection and rigidity
Bending deflection
Torsional deflection
Slope at bearings & shaft-supported elements
Shear deflection due to transverse loading of
short shafts
Vibration due to natural frequency
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Shaft Materials
Deflection primarily controlled by
geometry, and material
Stress controlled by geometry, not
material
Strength controlled by material
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Shaft Materials
Shafts: commonly made from low
carbon, CD or HR steel, such as ANSI
1020–1050 steels.
Fatigue properties don’t usually benefit
much from high alloy content and heat
treatment.
Surface hardening usually used when the
shaft is being used as a bearing surface. Dr. Mohammad Suliman Abuhaiba, PE
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Shaft Materials
CD steel: typical for d < 3 in
HR steel common for larger sizes
Should be machined all over
Low production quantities
Lathe machining
High production quantities
Forming or casting
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Shaft Layout
Issues to consider for shaft layout:
Axial layout of components
Supporting axial loads
Torque transmission
Assembly & Disassembly
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Axial Layout of Components
Dr. Mohammad Suliman Abuhaiba, PE
Fig. 7-2
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Support load-carrying components between
bearings
Pulleys & sprockets often need to be mounted
outboard for ease of installation of belt or chain.
Some axial space between components is
desirable to allow for lubricant flow and to
provide access space for disassembly.
Load bearing components should be placed
near the bearings.
Primary means of locating components is to
position them against a shoulder of the shaft.
Axial Layout of Components
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A shoulder also provides a solid support to
minimize deflection and vibration of the
component.
When magnitudes of forces are reasonably
low, shoulders can be constructed with
retaining rings in grooves, sleeves between
components, or clamp-on collars.
Where axial loads are very small, it may be
feasible to do without shoulders entirely, and
rely on press fits, pins, or collars with setscrews
to maintain an axial location.
Axial Layout of Components
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Supporting Axial Loads
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Figure 7–3
Tapered roller bearings
used in a mowing
machine spindle.
This design represents
good practice for the
situation in which one
or more torque transfer
elements must be
mounted outboard.
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Supporting Axial Loads
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Figure 7–4
A bevel-gear drive
in which both
pinion and gear
are straddle-
mounted.
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Providing for Torque Transmission
Common means of transferring
torque to shaft
Keys
Splines
Setscrews
Pins
Press or shrink fits
Tapered fits
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Providing for Torque Transmission
Keys:
Moderate to high levels of torque
Slip fit of component onto shaft for easy
assembly
Designed to fail if torque exceeds
acceptable operating limits, protecting
more expensive components.
Positive angular orientation of
component, useful in cases where phase
angle timing is important. Dr. Mohammad Suliman Abuhaiba, PE
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Providing for Torque Transmission
Splines
Gear teeth formed on outside of shaft and
on inside of the hub of the load-
transmitting component.
Transfer high torques
Can be made with a reasonably loose slip
fit to allow for large axial motion between
the shaft and component while still
transmitting torque.
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Providing for Torque Transmission
Pins, setscrews in hubs, tapered fits, and
press fits
Low torque transmission
Press and shrink fits: used both for torque transfer and for preserving axial location.
A split hub with screws to clamp the hub
to the shaft. This method allows for
disassembly and lateral adjustments.
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Providing for Torque Transmission
Pins, setscrews in hubs, tapered fits, and
press fits
A two-part hub consisting of a split inner
member that fits into a tapered hole.
The assembly is then tightened to the shaft
with screws, which forces the inner part
into the wheel and clamps the whole
assembly against the shaft.
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Providing for Torque Transmission
Pins, setscrews in hubs, tapered fits, and
press fits
Tapered fits between the shaft and the
shaft-mounted device, such as a wheel
Used on the overhanging end of a shaft
Screw threads at shaft end then permit use
of a nut to lock the wheel tightly to shaft.
Easy disassembled
Does not provide good axial location of
the wheel on the shaft. Dr. Mohammad Suliman Abuhaiba, PE
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Assembly and Disassembly
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Figure 7–5
Bearing inner rings press-fitted to shaft while
outer rings float in the housing.
Axial clearance should be sufficient only to
allow for machinery vibrations. Note the
labyrinth seal on the right.
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Assembly and Disassembly
Figure 7–6
Similar to the arrangement of Fig. 7–5 except
that the outer bearing rings are preloaded.
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Assembly and Disassembly
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Figure 7–7
Inner ring of LH bearing is locked to the shaft between a
nut and a shaft shoulder.
The snap ring in the outer race is used to positively
locate the shaft assembly in the axial direction. Note the
floating RH bearing and the grinding runout grooves in
the shaft.
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Assembly and Disassembly
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Figure 7–8
Similar to Fig. 7–7: LH bearing positions the entire shaft
assembly. Inner ring is secured to the shaft using a snap
ring. Note the use of a shield to prevent dirt generated
from within the machine from entering the bearing.
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Shaft Design for Stress
Stresses are only evaluated at critical
locations
Critical locations are usually
On the outer surface
Where the bending moment is large
Where the torque is present
Where stress concentrations exist
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Shaft Stresses
Axial loads are generally small and
constant, so will be ignored in this section
Standard alternating and midrange
stresses
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Shaft Stresses
Customized for round shafts
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Shaft Stresses
Combine stresses into von Mises stresses
Substitute von Mises stresses into failure
criteria equation.
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Using modified Goodman line,
Shaft Stresses
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Shaft Stresses
DE-Gerber
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Shaft Stresses
DE-ASME Elliptic
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Shaft Stresses DE-Soderberg
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Shaft Stresses for Rotating Shaft
For rotating shaft with steady
bending and torsion
Bending stress is completely reversed
Torsional stress is steady
Previous equations simplify with Mm and
Ta = 0
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Checking for Yielding in Shafts
Soderberg criteria inherently guards
against yielding
ASME-Elliptic criteria takes yielding into
account, but is not entirely conservative
Gerber and modified Goodman criteria
require specific check for yielding
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Checking for Yielding in Shafts
Use von Mises max stress to check for
yielding,
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Example 7-1
At a machined shaft shoulder the small diameter d
is 1.100 in, the large diameter D is 1.65 in, and the
fillet radius is 0.11 in. The bending moment is 1260
lbf·in and the steady torsion moment is 1100 lbf·in.
The heat-treated steel shaft has an ultimate strength
of Sut = 105 kpsi and a yield strength of Sy = 82 kpsi.
The reliability goal is 0.99.
a) Determine the fatigue factor of safety of the
design using each of the fatigue failure criteria
described in this section.
b) Determine the yielding factor of safety. Dr. Mohammad Suliman Abuhaiba, PE
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Estimating Stress Concentrations
Stress concentrations depend on size
specifications, which are not known the
first time through a design process.
Standard shaft elements such as
shoulders and keys have standard
proportions, making it possible to
estimate stress concentrations factors
before determining actual sizes.
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Estimating Stress Concentrations
Shoulders for bearing & gear support
should match the catalog
recommendation for the specific bearing
or gear.
Fillet radius at the shoulder needs to be
sized to avoid interference with the fillet
radius of the mating component.
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Reducing Stress Concentration
at Shoulder Fillet
Dr. Mohammad Suliman Abuhaiba, PE
Fig. 7-9
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Reducing Stress Concentration
at Shoulder Fillet
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Fig. 7-9
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Reducing Stress Concentration
at Shoulder Fillet
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Fig. 7-9
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Example 7-2 This example problem is part of a larger case study.
See Chap. 18 for the full context. A double reduction
gearbox design has developed to the point that the
general layout and axial dimensions of the
countershaft carrying two spur gears has been
proposed, as shown in Fig. 7–10. The gears and
bearings are located and supported by shoulders,
and held in place by retaining rings. The gears
transmit torque through keys. Gears have been
specified as shown, allowing the tangential and
radial forces transmitted through the gears to the
shaft to be determined as follows.
Dr. Mohammad Suliman Abuhaiba, PE
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Example 7-2 Proceed with the next phase of the design, in which
a suitable material is selected, and appropriate
diameters for each section of the shaft are
estimated, based on providing sufficient fatigue and
static stress capacity for infinite life of the shaft, with
minimum safety factors of 1.5.
Dr. Mohammad Suliman Abuhaiba, PE
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Example 7-2
Dr. Mohammad Suliman Abuhaiba, PE
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Deflection Considerations
Deflection analysis at a single point of
interest requires complete geometry
information for the entire shaft.
Size critical locations for stress, then fill in
reasonable size estimates for other
locations, then perform deflection analysis.
Deflection of the shaft, both linear and
angular, should be checked at gears and
bearings. Dr. Mohammad Suliman Abuhaiba, PE
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Deflection Considerations Table 7–2: Typical Maximum Ranges for Slopes
and Transverse Deflections
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Deflection Considerations
Shaft deflection analysis is done with the
assistance of software.
Options include specialized shaft software,
general beam deflection software, and
FEA software.
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Example 7-3
In Ex. 7–2, a preliminary shaft geometry was
obtained on the basis of design for stress. The
resulting shaft is shown in Fig. 7–10, with proposed
diameters of
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Example 7-3
Check that the deflections and slopes at the gears
and bearings are acceptable. If necessary,
propose changes in the geometry to resolve any
problems.
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Adjusting Diameters for
Allowable Deflections
If any deflection is larger than allowed, since I is
proportional to d4, a new diameter can be found
from
Similarly, for slopes,
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Adjusting Diameters for
Allowable Deflections
Determine the largest dnew/dold ratio, then
multiply all diameters by this ratio.
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Example 7-4
For the shaft in Ex. 7–3, it was noted
that the slope at the right bearing is
near the limit for a cylindrical roller
bearing. Determine an appropriate
increase in diameters to bring this
slope down to 0.0005 rad.
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Angular Deflection of Shafts For stepped shaft with individual cylinder length li
and torque Ti, the angular deflection can be
estimated from
For constant torque throughout homogeneous
material
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Angular Deflection of Shafts Experimental evidence shows that these
equations slightly underestimate the
angular deflection.
Torsional stiffness of a stepped shaft is
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Critical Speeds for Shafts
A shaft with mass has a critical speed at
which its deflections become unstable.
Components attached to the shaft have
an even lower critical speed than the
shaft.
Lowest critical speed ≥ twice the operating speed.
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Critical Speeds for Shafts
For a simply supported shaft of uniform
diameter, the first critical speed is
For a group of attachments, Rayleigh’s
method for lumped masses gives
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Critical Speeds for Shafts
Eq. (7–23) can be applied to the shaft itself
by partitioning the shaft into segments.
Dr. Mohammad Suliman Abuhaiba, PE
Fig. 7–12
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Critical Speeds for Shafts
Influence coefficient: transverse deflection
at location i due to a unit load at location j
From Table A–9–6 for a simply supported
beam with a single unit load
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Critical Speeds for Shafts
Dr. Mohammad Suliman Abuhaiba, PE
Fig. 7–13
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Critical Speeds for Shafts
Taking a simply supported shaft with three
loads, the deflections corresponding to the
location of each load is
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Critical Speeds for Shafts
If the forces are due only to centrifugal
force due to the shaft mass,
Rearranging,
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Critical Speeds for Shafts
Non-trivial solutions to this set of simultaneous
equations will exist when its determinant equals
zero.
Expanding the determinant,
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Critical Speeds for Shafts
Eq. (7–27) can be written in terms of its
three roots as
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Critical Speeds for Shafts
Comparing Eqs. (7–27) and (7–28),
Define wii as the critical speed if mi is acting alone.
From Eq. (7–29),
Thus, Eq. (7–29) can be rewritten as
Dr. Mohammad Suliman Abuhaiba, PE
2
1i ii
ii
mw
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Critical Speeds for Shafts
Note that
The first critical speed can be
approximated from Eq. (7–30) as
Extending this idea to an n-body shaft, we
obtain Dunkerley’s equation,
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Critical Speeds for Shafts
Since Dunkerley’s equation has no loads
appearing in the equation, it follows that if each
load could be placed at some convenient
location transformed into an equivalent load, then
the critical speed of an array of loads could be
found by summing the equivalent loads, all
placed at a single convenient location.
For the load at station 1, placed at the center of
the span, the equivalent load is found from
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Example 7-5 Consider a simply supported steel shaft as
depicted in Fig. 7–14, with 1 in diameter and
a 31-in span between bearings, carrying two
gears weighing 35 and 55 lbf.
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Example 7-5 a) Find the influence coefficients.
b) Find % wy and % wy2 and the first critical speed
using Rayleigh’s equation, Eq. (7–23).
c) From the influence coefficients, find ω11 and ω22.
d) Using Dunkerley’s equation, Eq. (7–32), estimate
the first critical speed.
e) Use superposition to estimate the first critical
speed.
f) Estimate the shaft’s intrinsic critical speed. Suggest
a modification to Dunkerley’s equation to include
the effect of the shaft’s mass on the first critical
speed of the attachments.
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Setscrews
Setscrews resist axial & rotational motion
They apply a compressive force to create
friction
Dr. Mohammad Suliman Abuhaiba, PE Fig. 7–15
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Setscrews
Holding power: Resistance
to axial motion of collar or
hub relative to shaft
Typical factors of safety
=1.5 to 2.0 for static, and 4
to 8 for dynamic loads
Length ≈ 0.5 shaft
diameter
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Keys and Pins Used to secure
rotating elements
and to transmit
torque
Dr. Mohammad Suliman Abuhaiba, PE
Fig. 7–16
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Tapered Pins
Taper pins are sized by diameter at large
end
Small end diameter is
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Tapered Pins
Table 7–5: Some standard sizes in inches
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Keys Keys come in
standard
square and
rectangular
sizes
Shaft diameter
determines key
size
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Keys
Failure of keys
direct shear
bearing stress
Key length is designed to provide desired
factor of safety
Factor of safety should not be excessive,
so the inexpensive key is the weak link
Key length is limited to hub length
Key length ≤ 1.5 times shaft diameter to
avoid problems from twisting
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Keys
Multiple keys may be used to carry
greater torque, typically oriented 90º
from one another
Stock key material is typically low carbon
cold-rolled steel, with dimensions slightly
under the nominal dimensions to easily fit
end-milled keyway
A setscrew is sometimes used with a key
for axial positioning, and to minimize
rotational backlash Dr. Mohammad Suliman Abuhaiba, PE
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Gib-head Key
Gib-head key is tapered so that when firmly
driven it prevents axial motion
Head makes removal easy
Projection of head may be hazardous
Dr. Mohammad Suliman Abuhaiba, PE
Fig. 7–17
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Woodruff Key
Woodruff keys have deeper penetration
Useful for smaller shafts to prevent key from
rolling
Dr. Mohammad Suliman Abuhaiba, PE
Fig. 7–17
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Woodruff
Key
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Woodruff Key
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Stress Concentration Factors
for Keys For keyseats cut by standard end-mill
cutters, with a ratio of
r/d = 0.02, Peterson’s charts give
Kt = 2.14 for bending
Kt = 2.62 for torsion without the key in place
Kt = 3.0 for torsion with the key in place
Keeping the end of the keyseat at least a
distance of d/10 from the shoulder fillet will
prevent the two stress concentrations from
combining.
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Example 7-6
A UNS G10350 steel shaft,
heat-treated to a minimum
yield strength of 75 kpsi, has
a diameter of 1 7/16 in. The
shaft rotates at 600 rpm and
transmits 40 hp through a
gear. Select an appropriate
key for the gear.
Dr. Mohammad Suliman Abuhaiba, PE
Fig. 7–19
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Retaining Rings
Retaining rings are often used instead of a
shoulder to provide axial positioning
Dr. Mohammad Suliman Abuhaiba, PE
Fig. 7–18
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Retaining Rings
Retaining ring must seat well in bottom of
groove to support axial loads against the
sides of the groove.
This requires sharp radius in bottom of
groove.
Table A–15–16 and A–15–17: Stress
concentrations for flat-bottomed grooves
Typical stress concentration factors are
high, around 5 for bending and axial, and
3 for torsion Dr. Mohammad Suliman Abuhaiba, PE
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Nomenclature
for Cylindrical
Fit
Dr. Mohammad Suliman Abuhaiba, PE
Fig. 7–20
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Nomenclature for Cylindrical Fit
Upper case letters refer to hole
Lower case letters refer to shaft
Basic size: nominal diameter and is same for both parts, D = d
Tolerance: difference between max &
min size
Deviation: difference between a size and the basic size
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Nomenclature for Cylindrical Fit
Upper deviation = max limit - basic size
Lower deviation = min limit - basic size
Fundamental deviation: either the upper or the lower deviation, depending on
which is closer to the basic size
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Nomenclature for Cylindrical Fit
Hole basis: a system of fits corresponding to a basic hole size. The fundamental
deviation is H
Shaft basis: a system of fits corresponding to a basic shaft size. The fundamental
deviation is h
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Tolerance Grade Number
International tolerance grade numbers designate groups of
tolerances such that the tolerances
for a particular IT number have the
same relative level of accuracy but
vary depending on the basic size
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Tolerance Grade Number
IT grades range from IT0 to IT16, but
only IT6 to IT11 are generally needed
Specifications for IT grades are listed
in Table A–11 for metric series and A–
13 for inch series
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Tolerance Grades – Metric Series
Dr. Mohammad Suliman Abuhaiba, PE
Table A–11
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Tolerance Grades – Inch Series
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Table A–13
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Fundamental Deviation Letter Codes
Shafts with clearance fits
◦ Letter codes c, d, f, g, and h
◦ Upper deviation = fundamental deviation
◦ Lower deviation = upper deviation –
tolerance grade
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Fundamental Deviation Letter Codes
Shafts with transition or interference
fits
◦ Letter codes k, n, p, s, and u
◦ Lower deviation = fundamental deviation
◦ Upper deviation = lower deviation +
tolerance grade
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Fundamental Deviation Letter Codes
Hole
◦ The standard is a hole based standard, so
letter code H is always used for the hole
◦ Lower deviation = 0 (Dmin = D)
◦ Upper deviation = tolerance grade
Fundamental deviations for letter
codes are shown in Table A–12 for
metric series and A–14 for inch series
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Fundamental Deviations – Metric series
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Table A–12
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Fundamental Deviations – Inch series
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Table A–14
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Specification of Fit
A particular fit is specified by giving the
basic size followed by letter code and IT
grades for hole and shaft.
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Specification of Fit
Ex: a sliding fit of a nominally 32 mm
diameter shaft and hub would be
specified as 32H7/g6
32 mm basic size
hole with IT grade of 7 (DD in Table A–11)
shaft with fundamental deviation
specified by letter code g (fundamental
deviation F in Table A–12)
shaft with IT grade of 6 (tolerance Dd in Table A–11) Dr. Mohammad Suliman Abuhaiba, PE
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Specification of Fit
Appropriate letter codes and IT grades for
common fits are given in Table 7–9
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Preferred Fits (Clearance)
Dr. Mohammad Suliman Abuhaiba, PE
Table 7–9
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Preferred Fits (Transition & Interference)
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Table 7–9
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Procedure to Size for Specified Fit
Select description of desired fit from Table
7–9.
Obtain letter codes & IT grades from
symbol for desired fit from Table 7–9
Use Table A–11 (metric) or A–13 (inch) with
IT grade numbers to obtain DD for hole and
Dd for shaft
Use Table A–12 (metric) or A–14 (inch) with
shaft letter code to obtain F for shaft
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Procedure to Size for Specified Fit
For hole
For shafts with clearance fits c, d, f, g, and
h
For shafts with interference fits k, n, p, s,
and u
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Example 7-7
Find the shaft and hole dimensions for a
loose running fit with a 34-mm basic size.
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Example 7-8
Find the hole and shaft limits for a medium
drive fit using a basic hole size of 2 in.
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Stress in Interference Fits
Interference fit generates pressure at
interface
Treat shaft as cylinder with uniform
external pressure
Treat hub as hollow cylinder with uniform
internal pressure
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Stress in Interference Fits
The pressure at the interface, from Eq.
(3–56) converted into terms of
diameters,
If both members are of the same
material,
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Stress in Interference Fits
is diametral interference
Taking into account the tolerances,
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Stress in Interference Fits
From Eqs. (3–58) and (3–59), with radii
converted to diameters, the tangential
stresses at the interface are
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Stress in Interference Fits
The radial stresses at the interface are
simply
The tangential and radial stresses are
orthogonal and can be combined using a
failure theory
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Torque Transmission from
Interference Fit Estimate the torque that can be
transmitted through interference fit by
friction analysis at interface
Use the min interference to determine the
min pressure to find the max torque that
the joint should be expected to transmit.
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Project
Crushing and steel cutting machine
Design considerations: Due Monday
13/10/2014
Design specifications: Due Monday
13/10/2014
action plan: Due Monday 13/10/2014
10/20/2014
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Design considerations
Cut steel with different shape and size into
small pieces (with different mechanical
properties)
Expected capacity: 20 to 50 ton per day
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Design specifications
Cut steel with different shape
size
small pieces: 5*5*5cm max
different mechanical properties: specify
most critical mech properties: hardness,
Ultmate, yield, …
Expected capacity: 30+-5
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action plan
Literature review: books, papers, videos,
manufactures
Concept design:
Detailed design
Machine layout
Design for strength
For deflection
For resonance
Design for ease of manufacturing and
assembly
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