screws
TRANSCRIPT
Screws 1
1 SCREWS
Threaded fasteners as screws, nuts and bolts are important components of
mechanical structures, and machines. Screws may be used as removable
fasteners or as devices for moving loads.
1.1 Screw thread
The basic arrangement of a helical thread wound around a cylinder is illus-
trated in Fig 1.1. The terminology of an external screw threads is, Fig. 1.1:
• pitch denoted by p is the distance, parallel to the screw axis, between
corresponding points on adjacent thread forms having uniform spacing;
• major diameter denoted by d is the largest (outside) diameter of a screw
thread.
• minor diameter denoted by dr or d1, is the smallest diameter of a screw
thread.
• pitch diameter denoted by dm or d2 is the imaginary diameter for which
the width of the threads and the grooves are equal.
Screws 2
The standard geometry of a basic profile of an external threads is shown in
Fig. 1.2, and it is basically the same for both Unified (inch series) and ISO
(International Standards Organization, metric) threads.
The lead denoted by l is the distance the nut moves parallel to the screw
axis when the nut is given one turn. A screw with two or more threads
cut beside each other is called multiple-threaded screw. The lead is equal to
twice the pitch for a double-threaded screw, and to 3 times the pitch for a
triple-threaded screw. The pitch p, lead l, and lead angle λ are represented in
Fig. 1.3. Figure 1.3(a) shows a single thread right hand screw and Fig. 1.3(b)
shows a double-threaded left hand screw. All threads are assumed to be
right-hand, unless otherwise specified.
A standard geometry of an ISO profile, M (metric) profile, with 60◦ sym-
metric threads is shown in Fig. 1.4. In Fig. 1.4 D(d) is the basic major
diameter of internal (external) thread, D1(d1) is the basic minor diame-
ter of internal (external) thread, D2(d2) is the basic pitch diameter, and
H = 0.5(3)1/2p.
Metric threads are specified by the letter M preceding the nominal major
diameter in millimeters and the pitch in millimeters per thread. For example:
M 14× 2
Screws 3
M is the SI thread designation, 10 mm is the outside (major) diameter, and
the pitch is 2 mm per thread.
Screw size in the Unified system is designated by the size number for
major diameter, the number of treads per inch, and the thread series, like
this:
5”8
− 18 UNF
5”8
is the the outside (major) diameter where the double tick marks mean
inches, and 18 threads per inch. Some Unified thread series are:
UNC Unified National Coarse
UNEF Unified National Extra Fine
UNF Unified National Fine
UNS Unified National Special
UNR Unified National Round (round root)
The UNR series threads have improved fatigue strengths.
1.2 Power screws
For application which require power transmission, the Acme, Fig. 1.5, and
square threads, Fig. 1.6, are used.
Power screws are used to convert rotary motion to linear motion of the
Screws 4
meting member along the screw axis. These screws are used to lift weights
(screw-type jacks) or exert large forces (presses, tensile testing machines).
The power screws can also be used to obtain precise positioning of the axial
movement.
A square-threaded power screw with a single thread having the pitch
diameter dm, the pitch p, and the helix angle λ is considered in Fig. 1.7.
Consider that a single thread of the screw is unrolled for exactly one turn.
The edge of the thread is the hypotenuse of a right triangle and the height
is the lead. The hypotenuse is the circumference of the pitch diameter circle
(Fig. 1.8). The angle λ is the helix angle of the thread.
The screw is loaded by an axial compressive force F , Figs. 1.7 and 1.8.
The force diagram for lifting the load is shown in Fig. 1.8(a), (the force
Pr acts to the right). The force diagram for lowering the load is shown in
Fig. 1.8(b), (the force Pl acts to the left). The friction force is
Ff = µN,
where µ is the coefficient of dry friction and N is the normal force. The
friction force is acting opposite to the motion.
Screws 5
The equilibrium of forces for raising the load gives
∑Fx = Pr − N sin λ − µN cos λ = 0, (1.1)
∑Fy = F + µN sinλ − N cosλ = 0. (1.2)
Similarly, for lowering the load one may write the equations
∑Fx = −Pl − N sin λ + µN cosλ = 0, (1.3)
∑Fy = F − µN sin λ − N cos λ = 0. (1.4)
Eliminating N and solving for Pr
Pr =F (sinλ + µ cosλ)
cos λ − µ sinλ, (1.5)
and for lowering the load
Pl =F (µ cos λ − sin λ)
cosλ + µ sin λ. (1.6)
Using the relation
tanλ = l/πdm,
and dividing the equations by cosλ one may obtain
Pr =F [(l/πdm) + µ]1 − (µl/πdm)
, (1.7)
Pl =F [µ − (l/πdm)]1 + (µl/πdm)
. (1.8)
Screws 6
The torque required to overcome the thread friction and to raise the load is
Tr = Prdm
2=
Fdm
2
(l + πµdm
πdm − µl
). (1.9)
The torque required to lower the load (and to overcome a part of the friction)
is
Tl =Fdm
2
(πµdm − l
πdm + µl
). (1.10)
When the lead, l, is large or the friction, µ, is low the load will lower itself. In
this case the screw will spin without any external effort, and the torque Tl in
Eq. (1.10) will be negative or zero. When the torque is positive, Tl > 0 Tl in
Eq. (1.10), the screw is said to be self-locking. The condition for self-locking
is
πµdm > l.
Dividing both sides of this inequality by πdm, and using l/πdm = tanλ,
yields
µ > tan λ. (1.11)
The self-locking is obtained whenever the coefficient of friction is equal to or
greater than the tangent of the thread lead angle.
Screws 7
The torque, T0, required only to raise the load when the friction is zero,
µ = 0, is obtained from Eq. (1.9)
T0 =F l
2π. (1.12)
The screw efficiency e can be defined as
e =T0
Tr=
F l
2πTr. (1.13)
For square threads the normal thread load, F , is parallel to the axis of the
screw, Figs 1.6 and 1.7. The preceding equations can be applied for square
threads.
For Acme threads, Figs 1.5, or other threads, the normal thread load is
inclined to the axis due to the thread angle 2α and the lead angle λ. The lead
angle can be neglected (is small) and only the effect of the thread angle is
considered, Fig. 1.9. The angle α increases the frictional force by the wedging
action of the threads. The torque required for raising the load is obtained
from Eq. (1.9) where the frictional terms must be divided by cos α
Tr =Fdm
2
(l + πµdm sec α
πdm − µl sec α
). (1.14)
Equation (1.14) is an approximation because the effect of the lead angle has
been neglected. For power screws the square thread is more efficient than
Screws 8
the Acme thread. The Acme thread adds an additional friction due to the
wedging action. It is easier to machine an Acme thread than a square thread.
In general, when the screw is loaded axially, a thrust bearing or thrust
collar may be used between the rotating and stationary links to carry the
axial component, Fig. 1.10. The load is concentrated at the mean collar
diameter dc. The torque required is
Tc =Fµcdc
2, (1.15)
where µc is the coefficient of collar friction.
Example
A double square-thread power screw has the major diameter d = 64 mm
and the pitch p = 8 mm. The coefficients of friction is µ = 0.08 and the
coefficient of collar friction µc = 0.08. The mean collar diameter is dc = 80
mm. The external load on the screw is F = 10 kN.
Find:
1. the lead, the pitch (mean) diameter and the minor diameter;
2. the torque required to raise the load;
3. the torque required to lower the load;
4. the efficiency.
Screws 9
Solution
1. From Fig. 1.6(a):
the minor diameter is
dr = d − p = 64 − 8 = 56 mm,
the pitch (mean) diameter is
dm = d − p/2 = 64 − 4 = 60 mm.
The lead is
l = 2 p = 2 (8) = 16 mm.
2. The torque required to raise the load is
Tr =Fdm
2
(l + πµdm
πdm − µl
)+
Fµcdc
2
=104(60)(10−3)
2
[16 + π0.08(60)π60 − 0.08(16)
]+
104(0.08)(80)(10−3)2
= 49.8 + 32 = 81.8 N m.
3. The torque required to lower the load is
Tl =Fdm
2
(πµdm − l
πdm + µl
)+
Fµcdc
2
=104(60)(10−3)
2
[π0.08(60) − 16π60 + 0.08(16)
]+
104(0.08)(80)(10−3)2
= −1.54 + 32 = 30.45 N m.
Screws 10
The screw is not self-locking (the first term in the above expression is nega-
tive).
4. The overall efficiency is
e =F l
2πTr=
104(16)(10−3)2π(81.8)
= 0.31.
Major diameter
Pitch diameter
Minor diameter
Pitch p
45◦ chamfer
RootCrest
Thread angle 2α
Figure 1.1
dm
d
dr
, d2
, d1
p
4
60◦
30◦
Roo
t(o
rm
inor
)di
amet
erd
r
Pit
chdi
amet
erd
m
Maj
ordi
amet
erd
p
8p
Root
Axis of thread
Figure 1.2
Crest
l
p
λ
l
p
λ
(a)
Single thread-right hand
(b)
Double thread-left hand
Figure 1.3
H
8
3H8
p
8
30◦60◦
External threads
Internal threads
Figure 1.4
60◦H
5H8
H
4
p
2p
2
p
4
p
D1
d1
,
D2
d2
,
Dd ,
p
2
drdm
d
0.3p
(a) Acme
2α = 29◦
Figure 1.5
dr
p
2
2α = 29◦
p
p
2
p
ddm
(b) Acme stub
drd
Figure 1.6
p
2
p
2
p
(a)
5◦α = 5◦
dm dr d
p
p
2
p
2
(b) Modified square
Square
dm
dm
Nut
λ
F/2
p
Figure 1.7
F/2
F
µN
N
F
λ
πdm
l
Figure 1.8
πdm
N
F
µN
l
λ
(a) (b)
Pr Pl
yy
x x
αF
cos α
2α =Threadangle
Figure 1.9
F
dc
F
2
Nut
Collar
F
2
F
2F
2
Figure 1.10