ASSEMBLY
OF INTERFERENCE
BY IMPACT AND CONSTANT
FITS
FORCE METHODS
by
CRAIG C. SELVAGE
S.B., Massachusetts
Institute of Technology
(1978)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF
MASTER OF SCIENCE
AT THE
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
JUNE,
1979
Signature of Author .
Department
Approved by ...
of MechanicaYEngineer'ing,
1979
Te/jhnical Supervisor,
CSDL
..
'-- ·' ' '- ~'"
Certified by
Mar'11,
.
Thesis Supervisor
Accepted by . .
Chairman, Department Committee on Graduate Students
ARCHIVES
r".....,;;;,-.;
-
,TS iNSc..
. L T
OF TECHNOLOGY
JUL 20 1979
LIBRARIES-
ASSEMBLY
OF INTERFERENCE
FITS BY IMPACT AND CONSTANT
FORCE METHODS
by
CRAIG C. SELVAGE
Submitted
to the Department
on May 11, 1979 in partial
of Mechanical
fulfillment
for the degree of Master
Engineering
of the requirements
of Science
ABSTRACT
This thesis
methods.
investigates
the assembly of interference
quasi-statically,
The parts may be pressed together
or be driven
fits by the two basic
under a constant
load,
into place with a series of impacts.
Wlhen the parts are pressed
tic insertion
force.
force because
the coefficient
sumed constant
together,
Handbook formulas
contact
give an approximate
of friction
pressure
value for the insertion
is not known with certainty,
is altered
is the sta-
the important consideration
by the axial force through
and the asthe Poisson
effect.
This thesis
accounts
portant consideration
and the contact
for the Poisson
when the product of Poisson's ratio, coefficient
length to interference
The residual
effect, and shows that it becomes an im-
axial stresses
detail, and the minimum
diameter
resulting
of friction,
ratio exceeds 0.1.
from assembly
energy which must be supplied
forces are analyzed
to begin insertion
in
is calcu-
lated.
An approximate
energy
impact load is presented.
preload
force which might
show a good correlation
Thesis Supervisor:
analysis of the insertion of an interference
This analysis
takes
into account
fit by an
the presence of an axial
be applied as the impact takes place.
Two experiments
between the theory and data for impact by hammer blows.
Dr. Daniel E. Whitney, Lecturer,
Department of Mechanical Engineering
2
ACKNOWLEDGMENTS
This thesis was prepared
funding provided
by the National
I thank my thesis
and discussions
at the Charles
Science Foundation
supervisor,
he contributed,
Stark Draper Laboritory,
Dr. Daniel
and Valerie
document.
3
grant
E. Whitney
Inc., with
number APR74 - 18173 A03.
for the advice,
help,
Naves for her help in typing this
TABLE OF CONTENTS
TITLE
PAGE
...................
...........
@.....
..........................................
1
eeeleeeeeeeeeo·e·e
ABSTRACT
...................
............................................
...................
...................
...................
*..................
*..................
2
eeeeeeeloeoee!4eeee
ACKNOWLEDGMENTS
.....................................
3
ee·eeeeeeeoeeee·ee
TABLE
OF CONTENTS
...................................
4
iooeeeeole·eeleeeoe
LIST OF FIGURES
.....................................
5
leooeeeeeeeeeeeoeoo
LIST OF TABLES
......................................
6
i·leeeeoe·e·eeeeeoe
1.0
INTRODUCTION
.................................
7
®eeeeee·eeeoeee·eel
2.0
INSERTION AND WITHDRAWAL
FORCES
9
..............
........-.-........
eeleleeo·eeeeee·eoe
2.1
Approximate
Equations
..................
10
eeeeeeeeeeeoeeeeoee
3.0
2.2
Poisson Effect of Axial
2.3
The Effect of Pressure Concentrations
IMPACT LOADING
Stresses
.......
..
...............................
3.1
Axial
Stress Distributions
3.2
Energy Absorbed
3.3
Quasi-static
3.4
Impact Dynamics
4.0
IMPACT EXPERIMENTS
5.0
CONCLUSIONS
in Loading
Impact Model
in the Loading
.............
..............
........................
...........................
AND RECOMMENDATIONS
..............
REFERENCES
.....X...-....-....
12
*.....-............. 22
...................
Process.....
.........
Process
eeooeeeeooeoeoeoooo
28
28
31
........ ..............-.40
ooooeooeeoooo·eoeoe
*..................
eooeoeooeeeooooeeeo
43
oooooeeeooeooleeeee
...................
45
...................
oooeeoloeeoo·eeeooe
49
50
4
LIST OF FIGURES
Page
Figure
14
Possible axial
2.2
Forces Acting on a thin section
2.3
Forces acting on inner and outer cylinders
.......................
16
2.4
Forces acting on inner and outer cylinders
.......................
19
2.5
Poisson effect
2.6
Interference
pressure
2.7
Interference
fit studied
2.8
Comparison
analysis
loads
.............................................
2.1
of axial stresses
.................................
by Rankin
fit continuum
Interference
3.2
Axial stresses
3.3
Unloading
3.4
Residual
in a loaded
infinite friction
26
.................................
29
fit
interference
friction
in an unloaded
to occur between
Allowing
slippage
3.6
Residual
axial stresses
final equilibrium
................................
......................
30
........................
30
interference
fit with
................................................
3.5
.................
the cylinders
in an unloaded
interference
for an arbitrary
axial load
............
3.7
Axial stress distribution
3.8
Axial
3.9
Change in axial stress when the cylinders
3.10
The effect of preload
3.11.
Percent of the threshold energy which is frictional work
4.1
Results of impact experiments
for the inner cylinder
force on threshold
....................
are allowed
energy
to slip
33
34
35
37
.... 39
..................
41
......... 42
..................................
5
32
fit,
................................................
stress functions
25
based on finite-element
model
of a fit with infinite
axial stresses
...............................
analysis
and approximate
23
... 24
for a short collar fit onto a long shaft
of stress distribution
3.1
............... . 15
of inner cylinder
47
LIST OF TABLES
Page
Table
2.1
Insertion force equations
2.2
Withdrawal
4.1
Specifications
force equations
including Poisson
including
of interference
effect
Poisson effect
...............
20
..............
21
..........
46
fits used in experiments
6
1.0
INTRODUCTION
The analysis presented
industrial assembly.
science.
in this thesis was developed
Of fundamental
importance
The basic purpose of parts mating
mathematical
models which describe
is to understand
This requires
the geometric
of research on
is the study of parts mating
science
which occur as parts are assembled or mated.
in support
the phenomena
the construction
and force relationships
of
as the
parts are assembled.
An analysis of the insertion of a round peg into a hole with a relative clea-
rance has yielded theories which describe the conditions under which jamming or
wedging
can be avoided during
vice called a Remote
jam-free
Center Compliance
investigates
interference,
assembling
into holes, and holding
splines, keyways,
a de-
rapid and
fits.
of parts having a negative clearance,
This type of fit is commonly
gears or sprockets
bushings
Guided by this analysis,
has been developed which allows
the assembly
press, or force
bearings, attaching
process.
parts having chamfers 2 .
insertion of close clearance
This thesis
called
the assembly
into housings.
and fasteners, permitting
to shafts,
inserting
used for
dowel pins
Often they are used in place of
a simpler,
less costly design, with im-
proved stress distribution.
There are two general methods
be pressed
of interference
together quasi-statically,
using a series of impacts.
under constant
Unfortunately,
pacity
of pounds.
to perform
assembly
by impact
An assembly machine
The parts may
force, or driven
from the assembly
force required to press together an interference
thousands
fit assembly.
into position
point of view, the
fit ranges from several
to many
or robot may not have sufficient
the insertion by directly pressing
(such as with a pneumatic
hammer)
the parts together.
promises
load caThus,
to be an attractive
alternative.
This research provides a method
ted with a particular
predicting
interference
the insertion distance
design of interference
for calculating
the insertion
fit, and presents an approximate
provided by a given
fits are equations
hammer blow.
for withdrawal,
7
or holding
force associaanalysis
Useful
for
for the
force, which
in general,
differs
from the insertion force.
may also be used to determine
subjected
to service
The analysis
the impact absorbing
loads which
are impulsive
8
of assembly
capability
in nature.
by impact
of an interference
fit
2.0
INSERTION AND WITHDRAWAL
FORCES
by forcing a shaft
fit is obtained
An interference
into a hole with a
slightly smaller diameter. To accommodate this interference, the shaft must
contract
radially,
and the hole expand,
producing a radial
This radial stress is referred to in the literature
interface.
In general,
and this term is used here.
pressure",
along the length of contact,
at the
contact stress
as an "interference
pressure
the interference
loading
and changes in response to the external
varies
of the
assembled interference fit.
The force necessary to produce
is the product of the coefficient
force),
or withdrawal
relative sliding
integral of the contact
of equal length, where
In the absence
of axial
the interference
of an axial
pressure
books 3 -
7
equations
a variation
interference
and
The application
in the interference
fits found
pressure
in texts and hand-
(equal length fully engaged
Thus,
these
the case of a hub fit on a long shaft, or a partially
inserted shaft, where the interference
pressure
forms a concentration
In these cases where edge discontinuities
analytic solutions,
by the outer.
of axial stresses.
the Poisson effect of axial stresses.
only approximate
of contact.
is fully enclosed
is in a state of plane stress,
treatments of interference
always assume a constant
and neglect
concentric
is constant along the length of contact.
due to the Poisson effect
cylinders),
each cylinder
force, however, will produce
The traditional
of friction and the surface
is that of two interfering
the inner cylinder
stresses,
pressure
insertion
(insertion
pressure.
The most simple case to consider
cylinders
between the parts
so finite element
methods 8
11
exist,
at the edges
there are no exact
are used when the interference
pressure distribution must be known with accuracy. The handbook equations are
usually of practical
coefficient
value, however, because
of friction
the uncertainty
due to the
is likely to be larger than the error resulting
constant pressure assumption.
9
from the
First,
Then,
the equations
for the constant pressure
approximation
the Poisson effect of axial stress will be accounted
effect
of pressure
concentrations
2.1
Approximate
Equations
Given
two fully engaged
the outer cylinder
inner cylinder
(designated
(designated
are given.
for, and finally the
will be examined.
cylinders with a constant
by o) must increase
by i) must decrease
radial interference
of a,
its inner radius by 60, and the
its outer radius by a
i .
Tangential
strain at a radius r is given by:
change in circumference
original circumference
t
27(r +
r) - 2.r
r
r1)
r
Thus, assuming elastic materials,
6
b= to
=
i
where
b
E (to+
=
E
-b=ti
(2.2)
op)
(2.3)
(ati + viP)
b = radius to interference
p = radial interference
at'
ct
=
tangential
pressure
stress
and strain
E = Young's modulus
v = Poisson's
ratio
The Lame stress
equationsl2
stress at the interference
ato
ati
where
give the following expressions
radius b:
(2.4)
c2_ 2
/b2
for tangential
a2
a = inner radius of inner cylinder
c = outer radius of outer cylinder
10
Substituting
these into Eqs. (2.2) and (2.3), and utilizing
the constraint
6 = 6i
+ 60 ,
6o(02
c2
o)=
Ei
_ 2b
Solving for the interference
=
p
2
(2.6)
a
pressure,
1/2 ()E
b
(2.7)
where
2
Ee
2
2
1E (c c2 +_ b)2
o
+ vi
+ Lib2
0/
have the same material
If the cylinders
2
(2.8)
2
-a
a
properties,
E and v,
E
E
(2.9)
e
b2
b
In addition,
aa 2
-
c2
b2 j
if a = o (solid shaft),
b2
Ee
=
e
(2.10)
(1- -)E
c
By assuming
axial stresses,
the interference
pressure
the insertion/withdrawal
pressure, contact
area, and coefficient
remains constant
force,
F, equals
in the presence
the product
of
of interference
of friction.
(2.11)
F
=
p(27bL)p
11
where
L = length of contact
= coefficient
Using
of friction
Eq. (2.7) for the interference
=
F
pressure,
(2.12)
PL(r6Ee)
If the interference,
6, is sufficiently
fracture.
For a ductile material, yield.will
approaches
one-half
shear stress
the tensile yield
large, the material
will yield or
occur when the maximum
strength,
y, of the material.
shear stress
The maximum
occurs at the inner surface of the outer cylinder and is given by:
P +
max
to
p'r~~~~~~~~~~~
2
+
at(2.13)
Using Eq. (2.4),
P
max
b
1
(2.14)
-
c2
For a solid shaft inside
diameter,
the maximum
a collar of identical material and any outer
allowable interference
ratio is found from Eqs. (2.7, 2.10,
and 2.14).
66max
x
ay
=
b
(2.15)
E
For plain carbon
steel, with a yield
strength of 52,000 psi, the maximum
interference
ratio is 0.0017.
2.2
Poisson
Effect of Axial Stresses
The following analysis
the effect of axial
fully engaged
takes axial stress
loads on the interference
cylinders
are considered.
into account
pressure.
and thereby considers
Once again,
equal length
The axial stress in each cylinder
assumed constant on any plane cross section,
12
is
a function of axial direction only.
Four possible
The derivations
equations
loading conditions may be examined,
for the first two cases
for withdrawal
as shown in Figure 2.1.
(insertion force) are presented
force are found by changing
the appropriate
here.
The
algebraic
signs.
CASE #1
The forces acting on a section of the inner cylinder
where
a is positive
for compressive
axial stress.
are shown in Figure 2.2
Equilibrium
of forces
requires
that:
b
p(z)
2
a2
do i (z)
--
:
If a section
'-z
through both cylinders
(2.16)
is taken, Figure
2.3, equilibrium,
implies
Ai
O(z)
(ai(L) - ai(z))
A
where A i , A o = cross-sectional
(2.17)
areas.
The insertion force is given by Aai(L).
stresses,
E
6
+ b2
(
\c2 -b 2
bYo
EO
0
Substituting
b
aO(Z)
+bp(z)
2
V
2
vi
(2.18)
i
E1
(2.18)
°o(Z)
Eqs. (2.8, 2.16, and 2.17)
into Eq. (2.18), the following differential
for oi(z) is obtained:
/b2 a2 \
t
of the axial
Eq. (2.6) becomes:
bp(z)
equation
With the inclusion
da.i(z)
ibEe ) -3
-
v.i
1
vo.
A
W
i(Z)+
13
O ai(L)-b)
O
(2.19)
INSERTION FORCES
F2
F1
Case 2
WITHDRAWAL
FORCES
F4
F3
Case 4
Figure 2.1
Possible
axial loads
14
Figure 2.2
Forces acting on a thin section of inner cylinder.
.
,
p(z)
·
_ l
!
pp (z
I
z
15
dz
Figure 2.3
Forces acting on inner and outer cylinders.
ai(L)
,
"0\-/
---
I
Case 1
16
(7
This equation
is solved, and evaluated
at z = L to yield the insertion
force, F 1 :
i
AiEi
F1
(.)
=
(2.20)
i
vi
kL + 1
where
kL
22 E
(L-L)(Trb
=
b
Vi
+
VO
)
(
o
-o
e\
TE
'i 0 A0 J
(2.21)
For identical materials,
kL
=
By changing
v(")
(2.22)
the appropriate
algebraic
signs in Eq. (2.20), the withdrawal
force
in
loading Case #3 is given by:
(.)
b=
F3
1i
1
ViAoEo
For kL < 0.5, these equations may be expanded
I
F1
=
uL(T6Ee)(1 +
(2.23)
oAiEi e-kL
+
into the approximations
voAiEi
iAoE o
(2.24)
)
2
viAoEo
voAiEi
viAo E
F3
=
pL(*6Ee)(1 -
(2.25)
-
1+iAEO
If the ratio EA/v is the same for each cylinder,
these approximate
reduce to Eq. (2.12) where the Poisson effect was neglected.
17
equations
CASE #2
In this loading situation,
the outer cylinder
Figure
in tension
from the application
is put in compression,
of the insertion force,
2.2, Eqs. (2.16 and 2.18) are still valid in this case.
section
through both cylinders
O(Z)
-A.
A
=
Substitution
equation
( E.
this equation,
V1
implies
into Eq. (2.18), the differential
(ek L
0
-
and evaluating
by Eq. (2.21).
algebraic
signs
(
b
Vi
EiA
E
E0A0 i
(2.27)
b
ai(z) at z = L,
1)
(2.28)
++ E-Ao
F
force, F 3, is found by changing
in Eq. (2.28):
- e-
kL )
(2.28)
for insertion and withdrawal
in Tables
The magnitude
The withdrawal
vO
EoA-
i
These equations
are summarized
if a
Vo
0
kL was defined
4
+
E1
da~dz~z)
EA
EiAi
the appropriate
F2.
is obtained:
=
2
2.4, stress equilibrium
and
(2.26)
(6)
F2
However,
ai(z7)
b
dz
Solving
is taken, Figure
Eqs. (2.8, 2.16 and 2.26)
for oi(z)
-4bEe a
where
the inner cylinder
force, including
approximations,
2.1 and 2.2.
of the Poisson
effect may be examined
ratio
F including Poisson effect)
F (by constant pressure assumption)'
18
by considering
the
Figure 2.4
Forces acting on inner and outer cylinders.
o, (z)
z
z
Case 2
19
Table
Insertion force
2.1
equations
including Poisson
effect.
F1
m
mmm~mm
F1
Ir
Jm
=
L(
ekL _
AiE
e)(v
voAiE
me
[
i
v100AoE
o
S
1
I
I
Fl
"L(1T(Ee)(
1
2
v°Ai
- E°
Ei
viA
vAi Ei
1 +vA E
F.
_m
_
m _i
t
..
F2
EiAi
_
where
=
For identical
kL
=
+ EA
00
_
F2
kL
(ekL
V0
(-L)(wb2Ee)
E
EiV
i0
materials,
v(
L )
b
20
IIL(r6tEe)(1+ kL)
2
-
1)
Table 2.2
Withdrawal
force equations
including
Poisson effect.
F3
F=
l-e
A.Ei
L7u
V
1
+-
.Ai E
JiAoEo
kL
e-kL
vA i
E i
F3 :
L(aEe
)(1
3 = e
kL
2
-I
-A E
v ~~~voAi
i E
v A
i 0
F4
t
*
L
F4
=
V.
vi
(1 +
EiA i
Li
F4
where
kL = (iL)(b2E
In)
b
For identical
kL
=
j
ViAi
1
+ E)
oA
materials,
v( b )
21
=
PL(iaEe)(1
O
E0 A0
-
kL)
Tw
-kL
i
This ratio
is plotted
properties.
If the contact
for example,
the Poisson
(insertion
where
by method
Poisson's
Thus,
in Figure 2.5 for cylinders
length to interference
diameter
effect is bounded by a maximum
2) and a maximum
of identical material
force decrease
force increase of 13%
of 11% (withdrawal
ratio = 0.3 and the friction coefficient
for interference
effect may be neglected
ratio, L/2b, is two,
method 4),
= 0.2.
fits having low length to diameter ratios,
in comparison
with the uncertainties
the Poisson
regarding the friction
coefficient.
2.3
The Effect of Pressure Concentrations
The previous
pressure
would
for fully
analysis assumed
be the same at all points of contact.
engaged cylinders
of equal length.
will be a stress concentration
in a hole.
long,
by Eq. (2.7).
in determining
length cylinder
assumption
to interference
diameter
Previous
pressure
length.
would become
increasingly
be required
general
a short frictionless
Rankin
13
to account
distribution
measured
finite element
the pressure
computer
the equal
valid as the contact length
cylindrical
the contact stresses
collar onto a solid shaft
of a shaft upon which a
what equivalent
for these measured
11
on finding
the deformation
uniform band of pressure
radial displacements.
have used finite element methods
for collars of various
trend for cylinders of identical
The interference
forces,
ratio increases.
tlore recent studies 8
actual pressure
is
the constant value given
insertion and withdrawal
short hub was mounted and then determined
would
if the contact length
approaches
studies have focused primarily
resulting from mounting
in every other case, there
where a shaft enters a hole and where a shaft ends
the interference
Thus,
the interference
This is a valid assumption
However,
Far from these points of discontinuity,
sufficiently
of infinite
that with no axial stresses,
material
to determine
lengths and thicknesses.
distribution
l0
.
is plotted
The dimensions
in Figure 2.8.
22
The
is shown in Figure 2.6.
fit problem studied by Rankin has been analyzed using
program
the
a
are shown in Figure 2.7, while
Figure
Poisson
2.5
effect of axial
stresses
(cJ
OL
U-
0
0.5
1.0
1.5
( upL/b )
23
2.0
2.5
Figure 2.6
Interference
pressure for a short collar
fit onto a long shaft.
L
2.0'
5
S-
a)
E
4W
1
w
S
E
tz
1.5
4-a
4-
M
V:
1
.
f
U1
0.0
0.25
0.5
Collar length
Shaft diameter
..........
Calculated
pressure
based on Eq. (2.7)
Pressure distribution
:·····t·::::·::LY:::::
24
Figure 2.7
Interference
fit studied
by Rankin.
7.8"
0.
Interference
= 0.0043
in.
Steel shaft and collar
25
Figure
Comparison
based on finite-element
2.8
of stress
analysis
distribution
and approximate
analysis.
40
ur
m 30
q)
g)
a 20
U
C
)1
aj 10
-4-.
I
0
0.1
Axial
distance
0.2
along collar
26
0.3
(inches)
The insertion
FI
=
force is found by measuring
the area under the curve:
(27b) .r pdz
Using the constant
(2.29)
pressure assumption,
Eq. (2.12), one predicts an insertion
force 16% less than if Figure 2.8 and Eq. (2.29) are used (axial stress
effect is negligable).
Thus, even in this case where
tion from constant pressure,
uncertainty
in the coefficient
to interference
of friction.
insertion
is a considerable
force is within
Furthermore,
diameter ratio (L/2b) was 0.078,
proves as L/2b increases,
withdrawal
the predicted
there
extremely
27
devia-
the margin
since the contact
in practical
insertion
applications.
of
length
small, and accuracy
this study indicates that in determining
forces, end effects may be neglected
Poisson
im-
and
3.0
IMPACT LOADING
This section
forces
The same concepts
of fits subject to service
capability
Throughout
The interference
2)
Relative
3)
Axial stresses
pressure
sliding between
Given
the above assumptions,
chains coupled
loading is considered,
stresses
in each cylinder
will result.
loads which are impulsive in nature.
assumptions
is constant
are made:
along the length of contact
the parts is governed
an interference
by Coulomb
by Coulomb
friction
friction,
fit may be modeled
as shown
as two sets
in Figure 3.1.
Quasi-
so that inertia effects may be ignored.
When the static insertion
force
the impact absorbing
in the Loading Process
Stress Distributions
If the insertion
fits
change in the axial direction only
Axial
axial stress
fits subject to the
may be used in evaluating
the following
the analysis,
1)
of spring-block
static
the behavior of interference
here will be on the assembly of interference
The emphasis
of impact.
by impatt methods.
3.1
investigates
force,
decreases
FI,
is applied
(given by Eq. (2.12)), the
linearly with depth,
as shown in Figure
is now removed from the loaded cylinders,
In addition,
slippage between
residual
the parts will occur
3.2.
axial
over cer-
tain portions of the contact area.
First, consider
the removal of the insertion
friction between
the parts
mental
during
3.3.
thickness
In order
at all depths,
=
e
=
(3.1)
the cylinders
FI
are shown in Figure
the constraint
0
tion of each incremental
is infinite
The forces acting on a section of incre-
(no slippage).
initial loading and final equilibrium
to satisfy
aiA i + aiA 0
force assuming there
undergo
uniform expansion
section is given
until the resulting elonga-
by:
dz
EiAi + EoA
(3.2)
28
*g,
Figure
3.1
Interference fit continuum model.
FI-
E.A.
( inner cylinder )
FI
( outer cylinder )
FI
dz
F
Friction force between elements
where
E
=
modulus of elasticity
A
=
cross-sectional
area
FI =
static insertion force
L
length of contact
=
dz
I
29
Figure
Axial
stresses
3.2
in a loaded interference
fit.
FI
I
L
-~1
Z
Figure 3.3
Unloading
of a fit with infinite friction
FI (
-
F (
z/L )
z/L )
LOADED
e
aiA i
where
UNLOADED
A
o o
30
oA.
+
A
=
0
Thus, the axial stress
=
ao
i
in each cylinder
is reduced
by a constant
F
EiA
i
E A + E A
At
EiAi + E0oAo Ai
,oo =
EoAo
The resulting stress
(3.3)
F1
EiAi + EoAo
(3.4)
Ao
distribution
tion) is shown in Figure
3.4.
with the insertion
At all depths, where
the other is in a corresponding
force removed
one cylinder
the ends of the cylinders
found by allowing
successive
This process
in the unloaded
stiffnesses
will change,
friction,
elements
3.5.
EiAi/ EA
o
at the endpoints
3.6.
, determines
all inte-
stress equilibrium
The resulting
in Figure
implies that
friction, while
The final
to slip, starting
fit are shown
of the two cylinders,
and where
FIdz/L.
is shown in Figure
interference
is in compression,
This requirement
are held together with infinite
rior points exert the maximum
(infinite fric-
state of tension.
At this point, slippage has not been allowed.
linders.
amount:
is
of the cy-
residual axial
stresses
The ratio of the axial
how the axia.l stresses
slipping will occur.
The axial stress distribution
mined by the same method
for an arbitrary axial
as used above
(this is shown
load, F, may be deter-
in Figure
3.7).
The axial
load uniquely determines the stress distribution, whether the fit is being loaded or
unloaded.
3.2
Energy Absorbed
A certain
in Loading
amount of energy
(work) must be supplied
fit from an unloaded to a loaded condition.
cylinders
in the form of elastic
Consider an interference
F.
The energy required
Part of this work is stored
strain energy which
is lost in heat, produced at the areas
to bring an interference
of slippage
is recoverable.
through frictional
fit initially loaded by an arbitrary
The remainder
sliding.
axial
to increase the load up to the insertion force,
calculated.
31
in the
force,
F I , will be
Figure
Residual
3.4
axial stresses in an unloaded interference
with infinite
friction.
Aa
ao.A.
Ii
fit
0
A
FI
FI
C
U
U
U
C
C
I
C
X.
x
C
J
--
C
C
LO
L
I-
32
Figure 3.5
Allowing
slippage
to occur between the cylinders
INFINITE FRICTION
I
I
I
I
Friction forces
I
I
I
I
I
I
FINAL EQUILIBRIUM
I
I
!
Point of zero axial stress
I
I
33
Figure
Residual
axial
stresses
3.6
in an unloaded
final equilibrium.
FI
zLJ0,
CA
Li
I-
34
interference
fit,
Figure
Axial
stress distribution
3.7
for an arbitrary
axial load
F
z
,.
()
F
o
¢
c:
Z
( INNER CYLINDER )
35
REQUIRED STRAIN
If
insertion
F (z) is the axial stress function
I
force,
difference
FI, and
:A/2 E
The axial stress
3.8.
absorbed
L
2
from the application
to an arbitrary
of the
load F, where F < F I, the
2
F(z)
F(Z )dz
-
A straightforward
yields
resulting
is given by:
functions aFi(Z)
stress functions
cylinder.
F(Z) corresponds
in strain energy
Us
Figure
ENERGY
and
F(Z) for the inner cylinder are specified
but tedious
an expression
(3.5)
integration
in
of Eq. (3.5) using these
for the strain energy absorbed
by the inner
By replacing EiA i by EoA o , and vice versa, in this equation, the energy
by the outer
cylinders
U
is then given
FI
=
s
is found.
+
4
EiA i + EA
The strain energy
by both
| F
FI
00/
I
requirement
+
1
E.A.
1
to bring
+ EoAo
E.A. +
+
E A
EA
o o
the interference
F
1
+
1
1
EiA i + EoA
o
oo
(3.6)
]
fit from an unloaded
(F=O)
loaded state is then:
F
REQUIRED
2
L
I8
1
8
EiA i
FRICTIONAL
+
1
axial stress functions
3.1.
1
+
(3.7)
EiA i + EoAo
EoA o
WORK:
If the load on the interference
in section
EA
ET
I
\11
The total strain energy absorbed
by:
1 - F
8
+
to a fully
cylinder
corresponding
When the cylinders
fit is reduced from FI to F, the resulting
to infinite
friction
are now allowed
36
(no slip) may be found as
to slip, in reaching
final
Figure
Axial stress functions
3.8
for the inner cylinder
L,
V,
-J
x
1
E A
EiAi 0 ++0 E
EA A
2 E.A.
a
b
11
=
L - (1
=
EiA i
(1
00
+ 2EiAi
0 0
37
a
-FI)
equilibrium,
occurs.
the stress
The changes
tive displacement
functions will change only in the places where
slipping
in the axial stress functions may be used to determine
of each cylinder.
The energy
lost in frictional
sliding
the relais given
by:
Uf
L
F
(z)
where
displacement
uz)
=
where
u(z)
= displacement
of inner cylinder when allowed
to slip
u0 (z)
= displacement
of outer
to slip
The stress
functions
are shown in Figure
each cylinder
3.9.
cylinder when allowed
corresponding
=
to infinite
When the cylinders
is changed by the following
a(z)
where
u(z)
I dz
of inner cylinder when allowed to slip
friction,
are allowed
(3.8)
and final equilibrium
to slip,
the axial stress
amount:
2FI z
A L
the variable
in
(3.9)
z is defined
in Figure 3.9.
The resulting
displacement
is given
by:
Z
u(z)
=
Substituting
allowed
1/E l
aa(z) dz
(3.10)
in Eq. (3.9), and integrating,
the displacement
of each cylinder when
to slip is found:
2
u(z)
I
=
(3.11)
E.A.L
I u()
I Z
(3.12)
o 0o
Evaluating
Eq. (3.8) at the two areas of slippage,
amount of energy lost in frictional
using Eqs. (3.11) and (3.12), the
sliding is given
38
by:
Figure
Change
3.9
in axial stress when cylinders
are allowed
to slip
2F I
A-- z
F
F
go
1
EA
2 EiA i +EA A
2F I
Ao L
39
(1-
L'
Uf
=
F2
1
FIL
24
F
FI
(
3
I
+
will be called
Figure
3.10.
energy
is given
=
1
-
+ EoA o
1
EiA i
energy plotted
When the interference
2
F
1
L
60
Figure
3.11.
3.3
Quasi-static
+
the load on an interference
is given by U s + Uf, and
F I,
3
EoAo
i
F
io o
I
1 9
-
EoA o
F
FI
+
1
EiA i
+
as a function of preload
fit is initially
(3.14)
1
EoA o
force,
unloaded
F, is shown
(3.15)
1
energy which
is frictional
work is shown in
Impact Model
analysis
of the response of an interference
impact load can be made using the threshold
above.
The mass of the interference
modeled
as time independent,
(insertion)
fit is neglected,
or quasi-static
loading.
fit subjected
energy equations
allowing
the impact
to be
This analysis will determine
will occur when an interference
a mass, M, having a kinetic
40
to
derived
fit receives a blow from
a hammer.
Consider
in
(F=0), the threshold
EiA
EoA
o
of the threshold
An approximate
what movement
)
o
by:
The fraction
an applied
i
1
___
The threshold
U
(iA
(3.13)
U.
+ E A
I
EiAi
3
+ EA
force, F, to the insertion force,
(
+
EiA
required to increase
the threshold energy,
6
-
1i
Eo+o
Ei1 i
F L(
.A
Ui
I
E A
A
The total amount of energy
fit from an arbitrary
+
1A
energy, ½MV2, at the instant before
Figure
The effect of preload
3.10
force on threshold
energy
o
-L
LL
U
F
0
2E-i
+L
1
1
EOAO/
41
Figure 3.11
Percent of the threshold energy which is frictional
work
0'
Co
LL
U-
in
o
Ci
0
O
Co
0-
'-4
Uf / Uo
42
x 100
striking
the interference
fit.
energy,
U,
greater
than the threshold,
where
If this kinetic energy
no insertion will take place.
FIAL
=
UH =
2
4MV
-
UH
conservation
FI = static
of energy
the kinetic
energy
is
implies:
(3.16)
U
kinetic energy
=
If, however,
is less than the threshold
of hammer
insertion force
AL = motion per hit
The threshold
equal length.
energy given by Eq.
If the inner cylinder
(3.15) is for concentric
(shaft) protrudes
a distance
cylinders
L,
of
the the equa-
tion for threshold energy with no preload becomes:
2
[o
[
If the interference
takes place, conservation
( F
- F ) AL
=
)
+]
fit has a preload force,
of energy
U
where the threshold energy
(3.17)
-
F, applied
to it as the impact
implies:
U
(3.18)
includes the effect of the preload
force, as given
by Eq. (3.14).
Since the mass of the interference
would expect these results
to become
fit was neglected
increasingly
accurate
in this analysis,
as the mass of the stri-
king object becomes much larger than that of the interference
3.4
one
fit.
Impact Dynamics
A more exact analysis
of impact takes
into account
inertia effects
treats the variation
of stress and strain
a function of time.
Due to the presence of Coulomb friction,
43
at each point
and so
in the interference
which
fit as
acts at the
interface
between
the parts, the governing
are nonlinear.
The problem
tion of elastic
waves at the frictional
faces.
In addition,
is highly complex
the applied
striking mass must be assumed
The special
length bounded
solution
has been obtained
member
16
in which
equations
and wave reflection
and reflec-
at free sur-
prescribed,
or the
to be rigid.
.
of elastic waves
has been neglected.
ization method
boundary,
force must either be completely
friction
'
differential
because of the transmission
case of a stress pulse applied
by Coulomb
the transmission
coupled partial
to an elastic
rod of infinite
is the only problem for which
This is of very limited
through
the frictional
A useful method
the Coulomb
of analysis
friction
terms such that the mean square difference
minimized.
44
usefulness,
boundary
however, since
and into the outer
may be an equivalent
is modeled
between
an exact analytic
as a combination
linear-
of linear
the steady state solutions
is
4.0
IMPACT EXPERIMENTS
Two experiments
analysis
presented
were performed
in section
3.3.
to test the validity of the quasi-static
In each case, a solid
shaft was pressed
into a
collar of shorter length so that the shaft protruded from the top. The interference fits were placed on the bed of a milling
machine
in order
to provide
a rela-
tively stiff surface in comparison with the compliance of the fits themselves. A
hammer was dropped onto the interference
ting insertion
meter
at a given
This was measured
by placing
After about 0.1 inches of total insertion was produced,
a Dillon
force gage.
cent throughout
location, and the static
In both cases,
the experiment,
number of insertion/withdrawal
a depth micro-
the hammer weight was 3.2 lbs.
and the hammer weighed
The dimensions
while
insertion
the static insertion
an overall
the fit was pressed
force was measured
using
force varied
10 per-
about
decrease was not observed
as the
cycles increased.
In the first experiment,
in Table
a net
reference surface, with the tip touching the top of the protruding
back into its original
listed
height from 10 to 75 times to produce
of about 0.02 to 0.05 inches.
on a stable
shaft.
and the resul-
per hit was measured.
The hammer was dropped
insertion
fits from various heights,
the insertion force was approximately
In the second,
the insertion
2200 lbs. and
force was about 720 lbs.
8 oz.
and calculated
threshold
energy using Eq. (3.17) for each is
4.1.
The height from which
the hammer was dropped, H, and the resulting
per hit, aL, were made dimensionless
by using Eq. (3.16) in dimensionless
insertion
form:
w
F1
AL
=
0
H
-
(4.1)
U0
The results of the two experiments
are shown
in Figure 4.1.
was observed until
the energy of the hammer was about
energy.
3 lb. hammer matched
The heavy
four times the threshold
the theoretical
45
No insertion
prediction
for energies
Table 4.1
Specifications
of interference
fits used in experiments
First experiment:
Materials:
both hardened ground
steel, E = 30 x 10 6 psi
=
0.750 in.
2b =
0.375 in.
2c =
0.813 in.
FI
=
2200 lbs.
Li
=
0.60 in.
W
=
3.2 lbs.
=
0.672 in-lbs.
A
=
2
0.110 in.
L
U
1
A o:
Second
O
0.408 in2
experiment:
raterials:
both mild steel,
E = 30 x 106 psi
=
0.625 in.
2b =
0.440 in.
2c =
0.970 in.
FI
=
720 lbs.
Li =
0.98 in.
W
=
0.5 lbs.
U0
=
0.071 in-lbs.
Ai =
0.152
A
0.587 in2
L
0
=
0
in.
46
Figure
4.1
Results of impact experiments.
40
35
30
25
FI
20
U AL
Uo
15
10
5
TO
0
5
10
15
20
w H
U0
47
25
30
35
40
greater
than about
7 times the threshold
was not as effective
than.about
as the theory
35 times the threshold
energy.
predicted
energy.
The relatively
until the supplied
This was expected
light 8 oz. hammer
energy was greater
since the dynamics
of
the situation was ignored in the analysis and becomes important when the weight of
the hammer becomes
small compared with the weight
48
of the parts.
5.0
CONCLUSIONS
AND RECOIiENDATIONS
In many cases,
concentric cylinders
the interference
is a good approximation
ver, parts of non cylindrical
pressed into housings.
should
be explored
to exceed the elastic
that the parts yield.
that the holding
analysis
power of an interference
is chosen large enough
limit.
If the elastic
analysis
fit can be in-
is used in this case, the calculated
forces, wil be overestimated.
the calculation
the theoretical
that axial stresses
For the inner member,
predictions
are uniform on any circular
or shaft,
Experiments
has an infinite diameter,
for example,
little consequence
since for increasingly
in the analyses converge
The two impact experiments
Flany more experiments
whether
This effect
to specific
49
is probably
between
if
of
the theory and
of the impac-
with pneumatic
impact ham-
is for this situation,
is similar to that derived
sis.
However,
stress will vanish
how the weight
Experiments
be done to see how accurate the analysis
the effect of preloads
of a
the forces and energies
showed a good correlation
produced.
is
values.
must be done to determine
ting hammer affects the insertion
analyses
cross-section
the axial
large diameters,
should
are.
this is a valid assumption.
far away from the hole because of shear deformation.
mers should
for
eliminates one of the factors of uncer-
of insertion and holding forces.
to see how accurate
the outer member
data.
The ana-
to this situation
The primary source of error in both the Poisson and the energy
predicted
so
description.
tainty regarding
cylinder.
in the parts
for a fit to be designed
cylinders must be applied
The inclusion of the Poisson effect
the assumption
to these cases
to cause the stress
Thus it is not uncommon
lysis of plastic flow in pressurized
be undertaken
Howe-
such as when pins or bushings are
of the cylindrical
and so the insertion and withdrawal
a more accurate
involved.
of
further.
if the interference
pressure,
and so the analysis
to the actual geometry
geometry often occur,
The applicability
It has been found
creased
fit has axial symmetry,
by the quasi
and
static analy-
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1.
Nevins, J.L.,
Draper
2.
et al., Exploratory
Laboratory
Shigley,
4.
Horger, O.J.,
J.E.,
Handbook,
echanical Engineering
"Press-and
Feedback for Automatic
Engineering
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C.S.
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in Industrial Modular Assembly,
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Assembly,"
Chapter
September,
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Mechanical
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Design and Systems
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Jordan,
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K.A.,
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Interference
International
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Steven,
G.P.,
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Elastic,"
Science,
Vol. 13, No. 7-8, 1975, pp.
Lindeman,
Robert A., "Finite Element Computer
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11.
B., and Wilson,
Stresses
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13.
E.A.,
"A Method
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for Determining
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Memoires
"Memoire
Rankin, A.W., "Shrink-Fit
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sur l'equilibre
presentes
the Surface
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Contact
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Contact
B, Journal
February 1970.
interieur des corps solides
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Stresses and Deformations,"
ASME Series
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Vol. 11, No. 2, pp. A-77 through A-85, June, 1944.
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14.
Wilms,
E.V., "Motion of an Elastic
Rod with External Coulomb
WJilms, E.V., "Damping of a Rectangular
External Coulomb
Stress Pulse in a Thin Elastic Rod by
of the Acoustical
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for Industry, Vol. 90, p. 526, 1968.
ASME Series B, Journal of Engineering
15.
Friction,"
Society of America,
Vol.
45, pg. 1049, 1969.
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Miller,
R.K., "Ar Approximate
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Trans. ASHE, Series
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of