9. FRICTION
9.1 DRY FRICTION (Coulomb friction)
The forces resulting from the contact of two plane surfaces can be expressed in terms of
the normal force N and friction force f, or the magnitude R and angle of friction .
If slip is impending, the magnitude of the maximum friction force is
f  s N
s - coefficient of static friction (depends on the materials of the
contacting surfaces and the conditions of the surfaces – Table 9.1)
and its direction opposes the impending slip. The angle of friction equals the angle of
static friction tan s   s , or  s  arctan( s ) . The surfaces are not in relative motion.
If the surfaces are sliding, the magnitude of the friction force is
f  k N
k - coefficient of kinetic friction (generally, is smaller than that of s)
and its direction opposes the relative motion. The angle of friction equals the angle of
kinetic friction tan k   k , or  k  arctan( k ) .
9.2 APPLICATIONS
Effects of friction forces (wear, loss of energy, generation of heat) are often undesirable,
but many devices cannot function properly without friction forces and may actually be
designed to create them (e.g., car’s brakes and tires).
THREADS - wood screws, machine screws
- pitch – the axial distance p from one thread to the next
slope – angle 
tan 
p
2r
r – mean radius of the thread
- the couple required for impending rotation and axial motion opposite to the
direction of F is (Fig. 9.16b)
M  rF tan( s   )
- the couple required for impending rotation and axial motion in the
direction of F is (Fig. 9.17)
M  rF tan( s   )
- replacing s with k in these expressions gives the couple necessary to
rotate the shaft at a constant rate
- if s = , the couple M is zero
WEDGES - bifacial tools with the faces set at a small acute angle
- when pushed forward, the faces exert large lateral forces as a result of the small
angle between them; that large lateral force can be used to lift a load
JOURNAL BEARINGS - supports designed to allow the supported object to move
- the couple required for impending slip of the circular shaft is
M  rF sin  s
F – total load on the shaft
- replacing s with k in this expression gives the couple necessary to rotate the shaft
at a constant rate
- this type of journal bearing is too primitive (the contacting surfaces of the shaft
and bearing would quickly become worn), and designers usually incorporate “ball”
or “roller” bearings to minimize friction
BELT FRICTION
- example: a rope wrapped through an angle  around a fixed cylinder; a large force
T2 exerted on one end can be supported by a relatively small force T1 applied to the
other end (it is assumed that the tension T1 is known)
- the maximum force T2 that can be applied without causing the rope to slip when
the force on the other end is T1, is
T2  T1e s 
 is in radians
- replacing s by k gives the force required to cause the rope to slide at a constant rate
THRUST BEARINGS AND CLUTCHES
- a thrust bearing supports a rotating shaft that is subjected to an axial load
- example (Fig. 22): the conical end of the shaft is pressed against the mating conical
cavity by an axial load; the couple necessary to rotate the shaft at a constant rate is
2  k F  ro3  ri3 
 2

M
2 
3 cos  ro  ri 
- a simpler thrust bearing: the bracket supports the flat end of a shaft of radius r that
is subjected to an axial load F; the couple necessary to rotate the shaft at a constant
rate is obtained by setting   0, ri  0 and ro  r
2
M   k Fr
3
- these bearings are good examples of the analysis of friction forces, but they would
become worn too quickly, and designers usually minimize friction by
incorporating “roller” bearings
- a clutch is a device used to connect and disconnect two coaxial rotating shafts
- example (Fig. 9.25): two disks of radius r attached to the ends of the shafts
- when the clutch is engaged by pressing the disks together with axial forces F, the
shafts can support a couple M due to the friction forces between the disks;
the largest couple the clutch can support without slipping is
2
M   s Fr
3