by construc-
tion.) The forces under which a is in equilibrium are re-
presented by the four sides of the closed polygon MNP^QM.
If our only object in the construction is to find NP^ or the
efficiency of the combination, we draw only the triangle
MNP^ but it must not be forgotten that (just as in the last
section) P^M is the sum of two distinct forces, and in
particular that it does not represent, either in magnitude
or in direction, the pressure between the surfaces of the pin
and eye, or shaft and bearing.
>5.88 THE MECHANICS OF MACHINERY. [CHAP. xn.
We have now to see how the point R^ can be practically
found. Let OR, the radius of the shaft, = r, and let there
be a small circle drawn about O with radius OS= r sin fa 1
Then R l can be found at once as the point in which a line
drawn through N touching this small circle, cuts the pin
circle. Obviously the angle OR^S is equal to <, for its
OS
sine is , and this is already made equal to sin by con-
struction. The small circle used in this construction has
received the name of friction circle, and as such we shall
refer to it.
If the sense of the effort were reversed, so as to be NM
instead of MN^ the construction would have been that of
Fig, 335, in which the same lettering is used as in the last
figure. Here the sense of rotation is reversed, but the sense
of the normal pressure is reversed also (PM changed to
MP\ so that the pressure comes on the lower side of the
bearing instead of the upper, and the friction, still opposing
the motion of a, again acts from left to right.
If, however, instead of changing the sense of the effort in
Fig. 334, we had reversed the sense of rotation by making
NP the effort and MN the direction of the resistance, the
line NR^S would have changed its position to the left instead
of the right of NR, the sense of the friction being reversed,
while it acts still on the same side of the shaft. There need
never be any hesitation as to whether NS lies to right or
left of NO. The points N, M and P are always given, and
NP
the point P^ must always lie between N and P, for \
which is equal to the efficiency, must always be less than
1 Under ordinary circumstances $ itself is not given only tan <>, the
friction-factor. For any such small angle tan , and the radius OS made = r tan #, which will save some trouble.
73-] FRICTION IN TURNIN
unity. It being thus known how MP^ lies
to MP, the relative slopes of their parallels NS and NO are
also known. In more complex cases, where this cannot so
readily be done, find the sense of the pressure, as a force
external to the body or element which we treat as the
moving one, and draw it so as to oppose, with this sense,
the rotation of the friction circle considered as a part of
the moving body. This rule, which must be thoroughly
mastered, finds many applications in 77.
It will be noticed at once that in the figure, in order to
make the construction as clear as possible, the angle has
been taken absurdly large. In ordinary circumstances, as
we have seen in 71, it is exceedingly small, and the
efficiency is, therefore, exceedingly near to unity. If the
point TV" were upon the periphery of the shaft or at an equal
radius, that is, if it coincided with R, the construction
would become identical with that given for a sliding pair.
^?! would coincide with R, the angle ONS would be equal
to (f>, and the friction circle would be superfluous. In that
case the efficiency of the turning pair would be exactly equal
to that of a sliding pair working with the same friction-factor.
As the point IV, however, is moved further away, the angle
at N becomes smaller than < and continually diminishes,
although the angle OR^S remains constant. Thus as the
radius of N increases the angle PMP\ diminishes, and the
efficiency becomes greater than that of a sliding pair, be-
coming more and more nearly unity as N goes further and
further off. This corresponds, of course, to the fact that
the radius of the friction is constant (and equal to the
peripheral radius of the shaft), while the radius of the effort
is constantly enlarged, so that a diminishing fraction of it is
required at the constant radius of the friction. If in any
combination, and the case is quite a possible one, the
592 THE MECHANICS OF MACHINERY. [CHAP. xii.
74. FRICTION IN SCREWS.
THE necessary treatment for a screw pair as regards
friction is in the first instance precisely that which we have
used for a sliding pair. The surface of the screw thread of
a slides upon that of the thread in the nut b (Figs. 336 and
337). The direction of sliding is taken to be the direction
of the tangent to the thread at its mean radius. The
resistance PN is taken as axial, the effort NM as at right
angles to the axis, and the latter is assumed to be applied
FIG. 336.
FIG. 337.
at a radius equal to the mean radius of the screw thread.
MP is the direction of pressure between the surfaces,
normal to the assumed tangent. Without friction NM
balances PN, the ratio between them being (as on p. 479),
NM
= tan a
pitch
of screw thread. With
PN circumference
friction there is precisely the same change as in the sliding
pair. In Fig. 336 the given effort NM \v\\\ balance only the
P IV
axial resistance P^N t the efficiency being as before.
MP\ is the sum of the friction QP 1 and the normal pressure
74-1 FRICTION IN SCREWS. 593
MQ, and is inclined at the angle to the direction of the
latter. If the resistance jP/Vis fixed and we have to find
the increased effort to balance it, we have it in NM^ Fig. 337,
the ratio - being the efficiency, and equal to the ratio
J. \ J.VJ. -.
P N
of the last figure. QP is now the frictional resistance,
M-J? the total reaction, and M\Q\ the normal pressure
between the surfaces.
The great loss by friction in screws, and the small
efficiency of a screw pair, is well known and often remarked
on. It must not be supposed, however, that this is
specifically due to the screw surface or anything connected
with it. It is due solely to the particular values of the
angles between effort and resistance and direction of motion
which happen to be common in screws. A sliding pair
working with the same angles would have precisely the same
efficiency. It may save misapprehension if this is always
clearly borne in mind.
The ratio of effort to resistance without friction is
equal to tan a. The ratio - \ with friction, (or the equal
ratio "W is ec l ual to tan ( a + <) The value of the effi "
ciency may therefore be stated algebraically as
NM PiN tan a
PN tan (a + ),
the counter-efficiency being, of course, the reciprocal of this.
The small efficiency of screws arises from the fact that in
them the angle a is always a small angle, so that although
< is no larger than in a sliding pair, (a + <) may be propor-
tionately very much larger than a. Enlargement of a would
tend to increase the efficiency, but at the same time might
QQ
594 THE MECHANICS OF MACHINERY. [CHAP. xn.
still faster diminish the "mechanical advantage" of the
screw, which is simply the ratio , numerically equal to
tan a.
Two important limiting cases occur. If a = o, tan a^o
and the efficiency = o. The screw becomes simply a series
of parallel rings, which we know to represent a screw of
zero pitch. If a = (90-^), tan (a + )= tan 90 == oc,
and again the efficiency is zero, although for a different
reason. In practice tan a varies commonly from 0*05 to
0*15, going occasionally as high as o'2. Tan , the
friction -factor, varies more widely. With ordinary screw
threads of square section, with surface contact and with
no special lubrication it may be 0*1 and even greater. 1 By
proper lubrication this will be greatly reduced, and where
along with complete lubrication there is point contact (as
with well-made worm gearing), the value of tan < may
fall as low as 0*01 and possibly less. The following table
EFFIC
tan a
IENCY -tan ( + ,,)
TAN a
Tan = o'oi
Tan = 0*02
Tan = o'os
Tan $ = o'io
Tan $ o'2i
O'OOO
O
O
O
O
O
O*O25
0-713
Q'555
0-458
0-203
0-II2
O*O5O
0-829
0-706
0*622
0-33I
0-196
0-075
0-883
0-741
0*699
0-429
0-270
O'lOO
0-906
0-828
0-766
0-495
0*325
0-125
0-924
0-858
0-805
0-552
0*376
0*150
o-935
0-877
0*829
0*592
0*414
0-175
o-943
0-893
0*849
0*627
0*450
0*200
0-950
0-904
0-865
0*656
0-480
O'225
o-954
0-913
0*876
0*678
0-505
0-250
0-958
0-920
0-886
0-698
0-527
1 See footnote at end of 71.
74-] FRICTION IN SCREWS. 595
gives values of the efficiency of a screw pair, calculated from
the formula already given, for a number of probable values of
tan a (o to 0-25) and for five different values of tan c/>. The
excessively low values of the efficiency for ordinary values of
a, when the friction factor- becomes large, is well shown in
the table, and the vital importance of thoroughly good
lubrication in any screw which is transmitting work needs
no further emphasis.
The line of axial pressure, PN, does not always lie along
one side of the screw parallel to its axis, but maybe distributed
over the whole thread equally, its resultant being coincident
with the axis of the screw. But as in that case the effort
(whether applied to the screw by a single lever or not), dis-
tributes itself round the thread in exactly the same way, no
error is caused by summing each one up, as we have done,
and treating it as if acting at one point only.
By the moment of the friction in a screw is generally
meant the moment of the additional effort taken up in
overcoming the friction, that is A/Mi X r in Fig. 337. The
work done against friction in a given time is equal (as with
a turning pair) to this moment multiplied by 2 TT and by
the number of revolutions of the screw in the given time, or
2 ic r n. MM\.
It will be noticed that MM^ is not itself the real frictional
resistance, which is represented by Q*P, but is greater than
it in the ratio * . In each revolution, however, the
cos a
frictional resistance is overcome through a distance greater
than 2 TT r in the same ratio (a distance equal to the length
of one turn of the helix), and therefore the work done
against friction is the same whether it be calculated from
MM, or from Q^P.
We have supposed the screw thread to be of square
Q Q 2
596 THE MECHANICS OF MACHINERY. [CHAP. xn.
section, so that the profile of the thread (as seen in the nut
section of the figures) is at right angles to the direction of
axial pressure. In a screw so made, other things being
equal, the friction is a minimum. With a screw thread of
triangular section, as Fig. 338, there is exactly the same
addition to the frictional resistance as we have already
noticed with a triangular guide block (p. 584). The normal
pressure producing friction is increased from NP to NR
(Fig. 338), i.e. in the ratio __, as before. In con
cos a
sequence of this there is an inward pressure on the screw
(and consequently an outward, bursting, pressure from the
screw on the nut) of the magnitude RP, which may under
some circumstances be very inconvenient.
FIG. 339-
In practice the force PN does not act directly upon the
screw thread, but generally upon the end, or upon some
other portions of the worm spindle. At the surface where
this pressure is transmitted there is therefore pivot friction,
which in practice often causes a further very serious loss of
effort, and therefore diminution of efficiency. This will be
considered in 75.
It is in many cases of great practical importance that a
screw should not be able to run back, that is that no axial
pressure, however great, should be able to turn it round. If
the block in Fig. 339 be part of a screw thread, find /Wthe
75-J FRICTION IN PIVOTS. 597
axial pressure, then the ratio of driving effort to normal
pressure must always be equal to ^ = tan a. But the
ratio of friction to normal pressure is ^ l ^ = tan <. So
Jong therefore as tan a is less than tan <, that is so long as
a is less than <, the screw cannot run back, no increase
whatever of pressure in the direction PN can in any way
move it.
75. FRICTION IN PIVOTS.
IT has been pointed out in the last two sections that
where there is any axial component of pressure in a turning
or a screw pair, that pressure will cause friction separate
from, and additional to, the friction proper to the pair. The
rubbing surface may in this case be the faces of one or more
collars on the shaft of the spindle, or may be formed by the
end of the shaft itself. Generically all such surfaces, rotat-
ing in contact under axial pressure, may be included under
the head of pivots.
\P \ p
s///////////////'
FIG. 340. FIG. 341. FIG. 342.
If P be the total pressure upon pivots such as those of
Figs. 340 to 342, we may say at once that in the case of the flat-
faced pivots the total frictional resistance will be /P, and in
p
the case l of the coned pivot/ . if / be the friction-factor
sin a
i See 72, p. 584.
598 THE MECHANICS OF MACHINERY. [CHAP. XH.
suitable for the particular case. This information is, how-
ever, of no use to us unless we know also the mean radius at
which we may assume the friction to act, which we have
therefore to find. If we suppose that at first, as is
intrinsically probable, the total pressure is uniformly dis-
tributed over the whole surface, then the mean radius of the
friction must be two-thirds of the radius of the pivot (in the
cases of Figs. 340 or 341, where the centre is not cut away).
But with such a distribution of pressure wear will at once
commence where the velocity of rubbing is greatest, that
is at the outer diameter of the pivot, and will gradually
extend itself inwards until the surfaces have so adjusted
themselves (if this be possible) that the rate of wear is
uniform over the whole. If the assumed friction-factor
remained still uniform over the whole area, the wear at any
point would be in direct proportion to the product of the
velocity and the intensity of pressure at that point. If the
wear is to be the same at every point, therefore, this product
must have a constant value, so that the intensity of pressure
at any point must vary inversely as its velocity, or simply as
its radius. If, therefore, we suppose the whole surface to be
divided into narrow concentric rings of equal breadth, the
area of each ring will be proportional to its radius, and the
intensity of pressure on each ring will be inversely propor-
tional to its radius. The amount of pressure on each ring
(intensity of pressure x area of ring) must, therefore, be the
same. Under these conditions the mean radius of friction will
be the half radius of the pivot, or (in the case of Fig. 342) the
arithmetical mean between its inner and its outer radius. The
moment of the frictional resistance would, therefore, be/^- P
2
r P
or f : , and the work done per minute against friction
2 sin a
75-1 FRICTION IN PIVOTS. 599
would be obtained by multiplying these quantities by 2 TT and
by the number of revolutions of the pivot per minute, as
p
fir r P n or fir r n respectively.
sin a
This result is generally taken as correct, and probably
enough it forms a reasonable working approximation, con-
sidering the very wide limits within which the factor f may
vary. It involves, however, in the first place, the apparently
impossible result that on some small, but not indefinitely
small, area at the centre of the pivot, the intensity of
pressure must be enormously great, so great as quite to
destroy the metal locally. Of this we have no physical
evidence in the condition of the surface of pivots after wear.
But apart from this difficulty, the method of investigation
involves two assumptions of which one is obviously wrong
and the other doubtful. The first is the constancy of the
friction-factor, and the second the possibility of uniform
wear, or wear without alteration of the shape of the surface.
As to the first, if we may apply here Mr. Tower's results
( 71), we know that the friction-factor must vary at each
point according to its velocity and pressure. If we suppose
the whole surface of the pivot divided into equal small areas,
the total frictional resistance on each would be about the
same 1 for all areas having the same velocity, and otherwise
would vary more or less as the square root of the velocity.
The wear on each small area will be proportional to its
velocity and its total frictional resistance, and therefore to
v *Jv or zA The velocity of each small area being propor-
tional to its radius, we may, therefore, say that on these
assumptions the wear at any point will be proportional to
1 This supposes perfect lubrication, which is not unreasonable, but it
applies results of journal friction to a pivot, which it is quite possible
\ve are not entitled to do.
6oo THE MECHANICS OF MACHINERY. [CHAP. xn.
the square root of the cube of its radius, or r\. But it is
probable that at the extremely slow velocities existing close
to the centre of a pivot the friction-factor must be propor-
tionately higher than elsewhere, and the wear, therefore,
greater than we have assumed. Further, the rapid wear at
the periphery of the pivot must speedily reduce dispropor-
tionately the intensity of pressure there to such an extent
that the constancy of frictional resistance on equal areas is
no longer even approximately true. Starting from some
simple assumptions, such as those first mentioned, as to the
distribution of pressure and the value of the friction-factor,
it is of course easy to calculate mathematically the conditions
of wear, the best form of " anti-friction " pivot, and so forth.
In fact, however, the actual physical conditions under which
a pivot wears are only sufficiently known to show that the
usual assumptions about them are entirely misleading.
Until the physical side of the matter is more completely
studied, it does not seem as if further purely mathematical
investigation could of itself lead to any useful result.
So long as our knowledge of the probable mean friction-
factor for a pivot is as uncertain as it is at present, the
formulae given on p. 598 for frictional moment, and work
done against friction, no doubt give results sufficiently
accurate for the rough approximation which is all we can at
present hope to obtain. It seems almost certain, however,
that the actual mean radius of the frictional resistances is
greater than the half radius of the pivot, lying between it
and the two-thirds radius. 1
The total axial load on a shaft is often extremely great in
proportion to its area, so great that without increase of area
it would cause so great an intensity of pressure as to bring
1 The efficiencies given in the table in footnote to 71 include the
frictional loss in the pivot as well as in the screw friction.
;6.] FRICTION IN TOOTHED GEARING.
60 1
about inconvenient and probably irregular wear. To obviate
this, "thrust collars " (Fig. 343) are often used (as in marine
engines to take the thrust of the screw propeller), by
multiplying which any required amount of surface can be
obtained. In actual pivots the pressure is often distributed
through several disks (Fig. 344) placed one below the other.
If these be properly lubricated, each will move relatively to
the one next it, and the sum of all the relative rotations will
FIG. 343.
FIG. 344.
be equal to the whole rotation of the shaft relatively to its
bearing, which is thus (more or less) uniformly distributed
among the disks. By this means, although the pressure is
not less than it would otherwise be, the velocity of rubbing,
and therefore the wear of each pair of surfaces, is much
reduced and rendered more uniform. Such bearings work
very well in practice under very heavy pressures.
76. FRICTION IN TOOTHED GEARING.
IN toothed gearing of all kinds a very considerable amount
of work is wasted in overcoming the resistance of the teeth
to sliding on one another, as we have seen that they must
do. 1 In 1 8 we saw how to obtain the amount of sliding,
1 18 and 71.
602 THE MECHANICS OF MACHINERY. [CHAP. xu.
or distance through which rubbing takes place, during the
whole contact of any pair of teeth. We saw further that the
mean velocity with which the teeth slid upon one another
might be expressed 1 as
= "L. v = T ^L ( + - \ v,
ra 4 \ r r^/
The first expression requires measurement of the tooth
profiles (for s\ the second does not. The value of ra, the
distance moved through by a point on the pitch circle while
a pair of teeth remain in contact, requires to be known in
both cases. In order that two pairs of teeth may always be
in contact ra must not be less than twice the pitch, although
in practice it is not uncommonly only i'6 to 1*8 times the
pitch. If we insert 2p in the equations instead of ra we get
2p 2 \ r
The work lost per second (the velocities being supposed to
be in feet per second) by friction between the teeth will be
found by multiplying either of these expressions by the
frictional resistance// 3 , where /is the friction-factor and P
the mean total normal pressure between the teeth.
Work lost in friction per second
l(l + L\ VfP.
The useful work done per second is (very approximately)
VP? so that the efficiency of the wheel gearing is
i i
2p
and the counter-efficiency
1 See p. 128. Note that r and r^ are here written for r^ and r .
2 This assumes the pressure P to be in the direction of motion of the
teeth instead of normal to their surfaces.
;6.] FRICTION IN TOOTHED GEARING. 603
2p
In either case may be substituted for/ when necessary.
Surfaces such as those of wheel-teeth are often enough rough,
and work with very imperfect lubrication, so that / is com-
paratively large ; but in spite of this the actual loss by tooth
friction is comparatively very small, much smaller than it is
often imagined to be. Thus, for example, with f taken as
much as 0*2, the efficiency of a pair of wheels of 2 and 6 feet
diameter, with teeth of 3 inches pitch, is still about 97 per
cent., so far, that is, as mere friction between the teeth is
concerned.
It must be remembered that instead of ( - + ) in the
\r rJ
above equations, ( - ) must be used if one of the
wheels (whose radius is r^) is an annular wheel.
We have calculated above the mean value of the efficiency
of transmission by toothed gearing. It is important now to
see how that efficiency can be found graphically for any
single position of the teeth. Let a and b (Fig. 345) be a
pair of spur wheels turning about A and B respectively, and
in contact at O. Let a^ and b l be the circles with which the
teeth have been described ; further, let O l and O 2 be the first
and last points of contact, respectively, of a pair of teeth.
Let a be the driving wheel and .MA 7 " the driving effort, while
the resistance on b is assumed to act at C in the direction
CF. Without friction we should find the resistance from
the effort, for position of contact at O^ as follows : Join O^O,
this gives us the direction normal to the surfaces of the
teeth, and therefore the direction of pressure between them.
The wheel a is balanced under a known force MN, a
6o2 THE MECHANICS OF MACHINERY. [CHAP. xn.
or distance through which rubbing takes place, during the
whole contact of any pair of teeth. We saw further that the
mean velocity with which the teeth slid upon one another
might be expressed l as
= s ^v= r ^( L + l\ v.
ra 4 \ r 1\J
The first expression requires measurement of the tooth
profiles (for s\ the second does not. The value of ra, the
distance moved through by a point on the pitch circle while
a pair of teeth remain in contact, requires to be known in
both cases. In order that two pairs of teeth may always be
in contact ra must not be less than twice the pitch, although
in practice it is not uncommonly only i'6 to i p 8 times the
pitch. If we insert 2p in the equations instead of ra we get
P . v = ( +
2p 2\ r r^
The work lost per second (the velocities being supposed to
be in feet per second) by friction between the teeth will be
found by multiplying either of these expressions by the
frictional resistance fP, where /is the friction-factor and P
the mean total normal pressure between the teeth.
Work lost in friction per second
= 1 fc/5P-(+ -} VfP.
2p 2 \r rj
The useful work done per second is (very approximately)
VP? so that the efficiency of the wheel gearing is
i i
2p
and the counter-efficiency
1 See p. 128. Note that r and r^ are here written for r\ and r 2 .
2 This assumes the pressure P to be in the direction of motion of the
teeth instead of normal to their surfaces.
76.] FRICTION IN TOOTHED GEARING. 603
z+l./^+^l+l
In either case may be substituted for/ when necessary.
Surfaces such as those of wheel-teeth are often enough rough,
and work with very imperfect lubrication, so that / is com-
paratively large ; but in spite of this the actual loss by tooth
friction is comparatively very small, much smaller than it is
often imagined to be. Thus, for example, with f taken as
much as 0*2, the efficiency of a pair of wheels of 2 and 6 feet
diameter, with teeth of 3 inches pitch, is still about 97 per
cent., so far, that is, as mere friction between the teeth is
concerned.
It must be remembered that instead of ( - 4- ) in the
1+1)
r rJ
above equations, ( - j must be used if one of the
Sr rj
wheels (whose radius is r^) is an annular wheel.
We have calculated above the mean value of the efficiency
of transmission by toothed gearing. It is important now to
see how that efficiency can be found graphically for any
single position of the teeth. Let a and b (Fig. 345) be a
pair of spur wheels turning about A and B respectively, and
in contact at O. Let a^ and b be the circles with which the
teeth have been described ; further, let O l and O 2 be the first
and last points of contact, respectively, of a pair of teeth.
Let a be the driving wheel and J/^Vthe driving effort, while
the resistance on b is assumed to act at C in the direction
CF. Without friction we should find the resistance from
the effort, for position of contact at O it as follows : Join O^O,
this gives us the direction normal to the surfaces of the
teeth, and therefore the direction of pressure between them.
The wheel a is balanced under a known force MN, a
6o:|. THE MECHANICS OF MACHINERY. [CHAP. xn.
resistance acting along OO^ and the sum of these two, which
must pass through A. We find (as on p. 273) the join E of
the directions J/TVand OO^ and resolve the force MN in
the directions EA and EO. This has been done in the
triangle MNR, where RN is the pressure on the tooth
surfaces. To find the resistance at C we have only to re-
peat a similar operation, resolving RN in the directions FC
and FB. This gives us NM^ for the resistance. In this
case, of course, there being no friction, and both effort and
resistance being assumed to act at the radius of the pitch
circles, MN = NM^ Taking now the same position of the
mechanism, but assuming a friction-factor = tan < for the
rubbing of the teeth, we can set off O^E^ making the
angle < with O^E, the normal to the surfaces, and resolve
MN parallel to E^A and E&. This gives us SNfor the
sum of the pressure and friction at O^. Carrying this on to
/;, and resolving ,57V 7 " in the directions F^B and F^C, we get
for the net resistance NP instead of NM^ the efficiency
NP NP
being , which in this case is equal to -- The small
.-*
arrows at Oi show the direction in which the teeth slide on
each other, which determines the position of O^E^. The
relative motion of the teeth continues the same in sense
(although its velocity diminishes) until O is reached. Here
the point of contact of the teeth is also the virtual centre, and
there is no sliding and therefore no friction, 1 so at this instant
the efficiency of transmission is unity. It is in fact equal to
the efficiency of transmission of two plain cylinders rolling
on one another without slipping. At any intermediate point
NP
between Oi and O the efficiency lies between - and
1 The resistance to rolling, which is sometimes called "rolling
friction," is here disregarded, as being practically negligible in comparison
with the friction proper.'
76.] FRICTION IN TOOTHED GEARING.
605
unity, its value becoming greater as O is approached. After
the centre is passed the efficiency again diminishes, but not
so rapidly as before, because the sense of sliding of the teeth
is now reversed (as shown by the arrows at O 2 ). The line
O f2 2 , whose direction is that of the sum of the pressure and
frictional resistance, is now more inclined to the line of centres
than the normal O. 2 O, whereas before O was reached the
FIG. 345.
corresponding line (O^-^) was less inclined to the line of
centres than the normal. This appears to be the real ex-
planation of the statement so often made that the frictional
resistance of the teeth as they approach the line of centres is
greater than their frictional resistance as they recede from it.
This is often expressed by saying that the friction during the
" arc of approach " is greater than during the " arc of recess."
606 THE MECHANICS OF MACHINERY. [CHAP. xn.
It will be noticed that during approach it is the roots 1 of the
driving teeth which act upon the points of the driven teeth,
while during recess the conditions are reversed, and the points
of the driving teeth act on the roots of the driven. In order
therefore to increase the efficiency as much as possible wheels
have sometimes been made with only point-teeth upon the
driver and root-teeth upon the follower. Such teeth have
no contact before reaching the line of centres, and if they
are to work well the arc of recess should therefore be made
much greater than usual.
If EI be the efficiency of a pair of wheels at the commence-
ment of contact of a pair of teeth, and 2 at the close of the
contact, then an approximation to the mean efficiency E, as
close as is generally obtained from the formulas on page 602
above, is given by
E = ^ + * + 0-5.
4
1
For most practical purposes E = 1 is quite sufficiently
2
accurate, and can be found of course by the very simplest
construction.
If it is required at the same time to take into account the
frictional resistance of the shafts, nothing more is necessary
than the construction of Fig. 334 in 73. Instead of draw-
ing the lines from E, E^ F v &c., through A and B, they
must be drawn to touch the friction circles which have these
points as centres, the side on which they touch being deter-
mined as on p. 589. In Fig. 345 they touch to the left of
the centres in both cases as dotted.
1 See p. 126.
77-] FRICTION IN LINKS AND MECHANISMS. 607
77. FRICTION IN LINKS AND MECHANISMS.
THE determination of the whole frictional resistances, or
of the total efficiency of transmission in a link or in an entire
mechanism, involves no more than the right use and com-
bination of the constructions already given, which we shall
now illustrate by some examples. Let it be borne in mind,
before proceeding, that what we are finding here is only the
efficiency of a link or mechanism in one particular position
under the action of the given forces. Its efficiency varies as
its position changes, and the value of its mean efficiency
throughout one revolution, or other complete cycle of changes
of position, requires to be found separately. This matter
will be considered later on. 1
We have seen how to determine the resultant direction
of friction and surface pressure in a pin joint or turning
pair. The most important point now before us is the cor-
responding determination in the case of a link connected
with its neighbours by two such pairs, such for example as
an ordinary coupling or connecting rod. The direction of
the resultant just mentioned may in this case either cross the
axis of the rod or lie parallel to it, and this resultant has in
general four possible positions. Its direction line may con-
veniently be called the friction axis of the rod, and is
always different from its geometrical axis. The four cases
just mentioned are shown in Figs. 346 to 349, of which we
shall first look at Fig. 346 alone. The link b is the one of
which we require to find the friction axis. The mechanism
turns in the direction of the arrows on d and c, and c is the
driving link. The forces /j and/ 2 acting on b from c and a,
1 Keep in view always, in working out the efficiency of a machine,
the remarks at the end of 71.
6oS THE MECHANICS OF MACHINERY. [CHAP. xn.
without friction, would have the sense of the arrows shown,
and would coincide in direction with the axis of the link.
The angle be is (in the position shown) increasing, and the
angle ba is simultaneously decreasing. The rotation of b
relatively to c and a is, therefore, represented by the small
arrows on ^, contra-clock-wise or left-handed in both cases.
The direction line of the sum of the force f^ and the friction at
its joint must touch the friction circle of the joint, and touch
FIG. 347.
FIG. 348.
it on such a side as to oppose the rotation of b relatively to c,
i.e. to oppose the motion of the pin in the eye, or of the eye
over the pin. This direction line must, therefore, lie to the
left of/ 1} and by exactly similar reasoning we can see that it
must lie to the right of / 2 . But the directions of the reac-
tions at the two ends of the link must coincide ; if they did
not, there would be an unbalanced moment acting upon b,
and the mechanism could not be in equilibrium. Hence the
77-] FRICTION IN LINKS AND MECHANISMS. 609
friction axis f can be drawn at once as a line touching the
two friction circles, the one to the left and the other to the
right of its centre.
If the motion of the mechanism had been the same, but
with the link a the driving link instead of <:, we should have
had the case of Fig. 347. The sense of/[ andy^, as forces
acting upon b, would have been reversed. The rotation of
b relatively to c and a would, however, have remained un-
changed. The friction axis/ would, therefore, have crossed
the axis of the link in the reversed sense to that of the last
case. Its position is shown in the figure.
In the mechanism sketched the link b has the same sense
of rotation relatively to a and to c. But in such a mechanism
as that of Fig. 348, it has opposite senses of rotation rela-
tively to its adjacent links. The angle be is increasing, and
the angle ba decreasing. The sense of rotation of b relatively
to c is right-handed, and relatively to a left-handed. The
link c is the driving link, and the friction axis/ touches both
friction circles on the same side, and lies parallel to the
geometrical axis of the link. In Fig. 349, the link a is taken
as the driving link, everything else remaining unchanged.
The change in the friction axis exactly corresponds to that in
Fig. 347 above. It remains parallel to the axis of the link,
but lies on the opposite side of it.
In dealing with a link in this way it is essential to re-
member that the sense of the forces must always be taken as
that corresponding to their action on the link, and not from
it. 1 Similarly the sense of rotation opposed by the friction
is that of the link itself relatively to its neighbour in each
case, and not of its neighbour relatively to it. If these things
are clearly kept in mind in working from link to link through
1 See p. 589, 73.
R R
610 THE MECHANICS OF MACHINERY. [CHAP. xn.
a mechanism, the necessary constructions will not give any
trouble.
Before going on to any more complex cases we shall work
out completely two examples from those we have just looked
at, taking first the mechanism of Fig. 350. A force/ ( = jRS)
acts at C '; we require to find its balance at A, taking into
account friction at all the four pins. The circles at the joints
represent the friction circles, not the pins, which are omitted
for the sake of clearness. 1 The given force/ intersects the
FIG. 350.
friction axis of b in M. We first resolve it in the direction
of that axis, and along a direction through M touching the
friction circle. of cd at P. The sense of the component in
the last-named direction is from P to M, which determines
the side on which it shall touch the friction circle. This
resolution gives us ST(m the figure separately drawn) as the
component of f c acting along the friction axis of b. The
required force/ cuts the friction axis in IV, and to find it we
As to size of the friction circles see p. 591.
77-1 FRICTION IN LINKS AND MECHANISMS. 611
have only further to resolve ST (reversed in sense, as acting
on a and not on c) in the directions NA and 1VQ, the last
being determined in the same way as MP. This gives us
the triangle SUT, of which the side SU represents the
required value of/ a . The dotted lines give the correspond-
ing construction disregarding friction, so that
is the
total efficiency of the mechanism for the position sketched.
Exactly the same problem is solved in Fig. 351 for the
mechanism of Fig. 348. The same lettering is used, so that
the force lines do not need to be traced out in detail. In
both cases - is the total efficiency of the chain. In both
i
FIG. 351.
cases, it will be seen, the construction is practically identical
with that of Fig. 127, 40, with the substitution of P and Q
for the two virtual centres, and of the friction axis of the
middle link for its geometrical axis.' The more general con-
struction of Fig. 128, 40, cannot be applied here with any
approach to accuracy.
The position of the friction axis of the connecting rod of
an ordinary steam-engine undergoes all the four changes of
Figs. 352 to 355, during each revolution. From the com-
mencement of the forward stroke the angle ft increases and
R R 2
612 THE MECHANICS OF MACHINERY. [CHAP. xn.
the angle y diminishes, and the friction axis has the position
shown in Fig. 352, where the small arrows on b show its sense
of rotation relatively to the crosshead and the crank, its two
adjacent links. When a becomes 90, i.e. when the crank is
in its mid-position, the angle (3 has obtained its maximum
value, and while the crank is in its next quadrant it con-
tinually diminishes, the angle y still diminishing also. The
position of the friction axis is shown in Fig. 353. During
the next quadrant of the crank's motion, the angles /? and y
both increase, the forces acting on the rod change sign (the
engine now making a backward stroke), and the friction
352'
FIG. 353.
FIG. 354.
FIG. 355.
axis simply changes sides, remaining parallel to the axis of
the rod (Fig. 354). In the last quadrant, y still increases,
but y3 diminishes. The friction axis takes the position
shown in Fig. 355, which is just reversed (corresponding
to the reversal of the force signs) from that of Fig. 352.
Had the machine been a pump instead of an engine, so
that the crank was the driving link instead of the piston, we
should have had each position of the axis reversed. Thus
Fig. 356 corresponds to Fig. 352. The forces have the same
sense, but the sense of rotation of the crank, now the driving
77-] FRICTION IN LINKS AND MECHANISMS. 613
link, is reversed. 1 The angle /? is therefore diminishing
and y increasing, and the position of the friction axis is the
same as formerly in Fig. 355, that is, reversed from its former
position.
FIG. 356.
As an illustration of the determination of the efficiency of
a mechanism containing both sliding and turning pairs, we
cannot do better than take the ordinary steam-engine
mechanism, such as is sketched in Fig. 357. The direction
FIG. 357-
of the effort in an engine is always fixed, but the direction of
the resistance varies very much, and upon its position in any
given case the actual efficiency must depend. In the case
1 If the sense of the crank's rotation were left unchanged, the sense
of the forces would have to be reversed, and the result would be the
same.
614 THE MECHANICS OF MACHINERY. [CHAP. xn.
sketched the direction of the resistance is made such as it
would be if the engine were driving machinery by means of
a spur wheel or pinion of a radius equal to that of the point
A. The circles represent, as before, the friction circles, and
not the pins. We draw first the line MP, making the angle
(f> with the normal MR, and resolve SM t the piston pressure,
in the direction of J/^Pand of MN, the friction axis of the
connecting rod. This gives us TM as the force acting in
the latter direction. Changing its sign, and resolving it in
the directions f a and JVQ, we find at once MR as the
required value of/,, the resistance at A. Without friction
the resistance balanced would be MR^ ; the efficiency is
therefore . If the friction circles have been enlarged
in any ratio, on Professor Smith's plan (see p. 591), it must
not be forgotten that the value of tan < (the friction angle
for the sliding block) must be enlarged in the same ratio. If
this is not convenient, the efficiencies of the sliding pair and
of the connecting rod and crank shaft must be determined
separately, and afterwards multiplied together.
The loss of efficiency found by such a construction as this
refers only, of course, to the particular forces which have
been taken into account. There is no difficulty in finding
similarly the losses caused by friction due to the weight of
the moving parts. If in such a case as that of the last
figure, for example, there be some very large weight, as that
of a fly-wheel, upon the bearing, the frictional resistance
caused by it may be estimated and allowed for separately.
This is probably the most convenient plan, because any such
resistance is the same for every position of the mechanism,
so that one calculation serves for all. But it can also be
found graphically with the greatest ease. Thus let MU (in
the last figure) be any such weight, acting vertically down-
77-] FRICTION IN LINKS AND MECHANISMS. 615
wards through the centre of the shaft. Adding MUto TM
we get TU as the sum of the weight and the connecting rod
pressure on the crank, and this sum must pass through the
point W, the join of the lines of action of the weight with
the friction axis. Through JFdraw WA\ parallel to UT, and
resolve UT in the direction of f a and of N^ Q v which gives
URc as the new value of the force balanced at A, allowing
for the journal friction caused by the weight MU.
The donkey-pump mechanism of Fig. 184 affords us a
very instructive, but somewhat more complex, example. Let
FIG. 358.
it be assumed that this mechanism is to be used for a steam-
engine (where the main resistance is, as usual, to the rotation
of the shaft) and not for a pump, and find its efficiency in the
position shown in Fig. 358. Here we start with the given
piston pressure f c = RS, acting upon the sliding frame c.
This is balanced by a pressure or resistance from the block
to the frame, and also by side pressures in the two guides.
We have in the first instance to assume the points A and B
at which the resultant pressures in the guides act ; these
we have no means of determining. The pressure from the
616 THE MECHANICS OF MACHINERY. [CHAP xn.
guide to the link c will be upwards at A and downwards at
B, and as we know the sense of motion of c relatively to the
guides (as shown by small arrows) we can draw the lines AE
and DB, making the friction angle with the normals AE^
and D-J3, at A and B respectively. Without friction the
direction of pressure from b to c would be simply the normal
D\E-b which line is the real geometrical axis of the link b,
which we know to represent (see p. 399) an infinitely long
connecting rod. With friction the direction of pressure
must be along the friction axis of b. Of this line we know,
firstly, that it must touch the friction circle of the crank pin on
the under side in the figure, as at P, for exactly the same
reason as in Fig. 353 above, which represented the similar
position of the slider crank mechanism. Secondly, we know
that it must be inclined at an angle = < to the normal to the
surfaces of the block and frame. We can therefore at once
draw it as PN or DE. We have now the condition that the
link c is in equilibrium under four forces, of which one, /,
is given completely, and the other three are given in di-
rection only, as AE, ED, and BD. We can employ for
resolution the construction used formerly in p. 312, 41.
Calling the pressures at A, B, and P, f a , / and />, re-
spectively, we know that
/+/- = -(/* +/,)
But the sum of f c and f a must pass through C, the join of
their directions, and similarly the sum oif b andj^ must pass
through D. To find^ then we have first to resolve^ along
AC and DC, which gives us TR (see separate figure) for
the component along DC, which is the sum of f b and f p .
Next we resolve this component in the directions DN and
DB, which gives us TU as the required reaction between b
and c. Had there been no friction we should have had to
resolve f c along the direction AE^, E^D^, and D^B^ and the
77-] FRICTION IN LINKS AND MECHANISMS. 617
force polygon would have been the simple rectangle 7?,SZJ U-^
(see also p. 311). To complete our problem, let f a be the
given direction of the resistance upon the crank shaft, whose
magnitude we require to find. We find N, its point of
intersection with the friction axis of b, just as in Fig. 357, and
resolve TU along the direction off a and of NQ, the point Q
being found as before. This gives us TV for the resistance
which we had to determine. Without friction we should
have had to resolve T^U^ in the directions of f a and of
N-i O, which gives us T^ V r The efficiency of the mechan-
ism as a whole is, therefore, for this particular position,
TV
FIG. 359.
It will be sufficient to take one example with a screw ; for
instance the right- and left-handed screw coupling of Fig. 359.
The coupling is pulled with a constant tension f a =f bt it is
required to find the moment necessary to turn the screw in
either direction under this tension. Suppose first that the
screw has to be turned so as to tighten up the coupling.
Through A and B draw normals to the screw threads, meeting
at C r Set off MO=f a =f b . Through O draw OP l \\ BC^
and OQ l \\ AC^ making the direction PiQ 1 normal to the
axis of the screw, that is in the direction of the intended
6iS THE MECHANICS OF MACHINERY. [CHAP. xn.
turning effort. Then without friction the effort P^ <2. applied
at a radius equal to the mean radius of the screw thread, will
be just sufficient to turn it. The moment of the effort will
be PiQi x r, if r be the mean radius of the screw. The
sense of motion of the screw relatively to the nuts is shown
by the small arrows at A and B. With friction therefore
the directions C^A and C\B will be changed to CA and CB,
the angles CACi and CBC\ being each = . Drawing
parallels to these lines in the force polygon we get PQ as the
effort required instead of P\Q\, and the moment necessary
to turn the screw, including frictional resistance, is PQ x r,
if r be, as before, the mean radius of the screw thread.
If the screw has to be slackened instead of being tightened
up, the friction angle has to be set off in the opposite sense,
as C 2 A and C^B. The corresponding effort, greatly less than
before, and of course reversed in sense, is shown at P 2 Q.>,
the turning moment being P 2 Q 2 x ?' We have already ( 74)
noticed the necessary relations between the magnitude of the
angles < and in order that the screw may not " run down.' ;
Here it will be seen at once that if < = $, C 2 would coincide
with S, and the points P 2 and Q 2 in the force polygon would
come together, so that the effort required to slacken the screw
would be zero. If 6 were greater than <, C 2 would fall
above S, and some effort would be required to prevent the
screw slackening itself. As this would entirely destroy the
usefulness of the couplingj the case is one in which a finely
pitched thread (i.e. a small angle 6) is essential, and a
too small friction-factor ( = tan <) detrimental instead of
desirable.
We have already pointed out that the constructions of
this section have for their object the determination of the
efficiency of a mechanism in one particular position only.
In general what is practically required is the average
77-] FRICTION IN LINKS AND MECHANISMS. 619
efficiency of the mechanism during one complete cycle of
changes of position, as, for instance, the average efficiency of
the mechanism of a steam-engine during one complete
revolution of the crank shaft. To obtain the efficiency of
the mechanism in a number of different positions, and then
find the average value of the efficiencies so obtained, would
be a long process, because each determination of efficiency
requires two complete constructions, one to determine the
balanced resistance with and the other without friction, the
efficiency being the ratio between these two quantities. But
this is unnecessary. In every case we do, or easily can,
start with a diagram of work, that is a curve (as Fig. 14.6,
p. 321) whose ordinates represent pressures (here efforts),
and whose abscissae represent the distances through which
these pressures are exerted. All that is necessary to do is to
determine by construction the resistances with friction, and
plot these out into a diagram whose base represents the
distance travelled by the point at which the resistance
acts (as A in Fig. 357 above). Apart from work done
against friction this diagram would have an area equal
to that of the effort diagram, as we have seen in 43.
Having, however, taken friction into account, its area will re-
present the net work done against useful resistance, and will
be less than the area of the effort diagram by an amount
corresponding exactly to the work expended in overcoming
frictional resistances. The ratio between the areas of the two
diagrams will be the mean efficiency of the whole mechanism.
It is not necessary that we should give here any detailed
example of this determination for a whole mechanism ; all
the necessary constructions for it have been given very fully.
It will be sufficient to give the simple case of the determina-
tion of the work lost in the friction of the guide block of
a steam-engine, and the average efficiency of the guides.
620 THE MECHANICS OF MACHINERY. [CHAP. xn.
Fig. 360 represents this case (which we have already looked
at) for one position, Fig. 361 shows the determination of the
average efficiency. AB (Fig. 360) is the known piston
effort, balanced at B by a resistance in the direction of the
friction axis of the connecting rod, BM, and by a normal
pressure and frictional resistance whose sum lies in the
direction B B^, making an angle to the vertical. Drawing
the force polygon we get BM for the pressure transmitted
through the connecting rod, and BA^ for the effort which
would be required to balance this pressure if there were no
friction. The efficiency is therefore, in this position, ~~ l
Jj A
*
FIG. 360.
FIG. 361.
If an effort diagram has been drawn, BA will be equal to its
ordinate at the point B, which must have been turned down
into its present position, and we must now turn up BA^
above B as an ordinate for our new (net effort) curve.
But this turning down and up of lines is somewhat incon-
venient, and it is much more handy to turn the whole
construction through a right angle, and work it as in Fig. 361.
Here CADE is the effort diagram (an indicator card drawn
to a straight base), BA the effort at B, and NB the direction
(as BM in Fig. 360) of the friction axis of the connecting
77-1 FRICTION IN LINKS AND MECHANISMS. 621
rod. Through B draw BM at right angles to BN, and
through A draw AM making an angle equal to with the
horizontal. Then project M to A l on the line BA^ and
the required point on the net effort curve is at once
obtained. It will be seen at once that the figure BA^AM
in Fig. 361 is identically equal to the similarly lettered figure
in Fig. 360. Each of its sides is, however, turned through
90 to suit the direction in which the effort has been
originally set out. A similar construction for FD gives us
FD for the net effort at D. The shaded area HADED^
represents the work expended in overcoming the friction of
the guide block, and the ratio of areas - gives the
Cx^Tl I. J I~*4
mean efficiency for the motion from C to E.
The plan just used of turning the construction through a
right angle is one which the student will find useful in a
number of cases, especially where work or energy diagrams
are concerned, but it is not necessary to give further
examples of it.
In finding the mean efficiency of any mechanical com-
bination other than the very simplest, it is very desirable
that for at least one position the student should make the
complete determination of resistance both with and without
friction, and should see that at each separate joint there is a
loss of efficiency. Without this double determination there
exist no ready means of checking possible mistakes in
the position of friction axes, &c., which may notably affect
the resultant efficiency, without, however, making it so
conspicuously wrong as to be otherwise evident.
In a very large number of machines, more or less complex
in appearance, the friction (so far as it is caused by known
and measurable forces) can be quite easily estimated by the
methods given by treating them as a sequence of separate
622 THE MECHANICS OF MACHINERY. [CHAP. xn.
mechanisms, and graphically or otherwise combining the
whole. In really complex mechanisms, however, such as
those of Figs. 128 and 240, the determination of the frictional
efficiency is much more difficult. Our knowledge, however,
of the real value of the friction-factor in any particular case
is so very vague that the error of an approximation which
is mathematically exceedingly rough may still be much less
than the probable error of our estimation of this most
essential element in our data.
An excellent collection of examples of graphic frictional
estimations will be found in the Zttr graphischen Statik
der Masckinengetriebe 1 of Professor Gustav Hermann, of
Aachen. These include a stone-breaking machine, pulley
tackle, screw-jack with worm gearing, geared crane, and
various forms of steam-engine.
78. FRICTION IN BELT-GEARING.
IT was mentioned in 7 1 that there is a large and im-
portant class of cases in which it is desired that the surfaces
between which friction occurs shall not move relatively to
each other ; and in which the frictional resistance alone is
relied upon to prevent this motion. In these cases no work
is expended in overcoming friction, because no motion takes
place under it; the frictional contact between the surfaces
does not affect the efficiency of the apparatus ; and instead
of wishing to diminish the frictional resistance as much as
possible, it is essential to the working of the machine that
it should have at least some definite, and generally very
large, amount.
1 Brunswick, Vieweg u. Sohn, 1879.
78.] FRICTION IN BELT-GEARING. 623
The most important case of this kind is that of the trans-
mission of work by means of belt-gearing. Let there be
given any pulley, as in Fig. 362, with a strap resting upon
it, and at each end of the strap a weight. For mere static
equilibrium, so long as the pulley works frictionless in its
bearings, W^ = W^ and the smallest addition to either
FIG. 362.
weight will cause it to descend, lifting the other. But sup-
pose the pulley to be fixed, so that its rotation is entirely
prevented, and that it be desired to make W^ large enough
to lift W^. In order that IV 2 may move, it has now not
only to balance W^ but also to overcome the whole friction
between the strap and the pulley caused by the tensions 7\
and T 2 in the strap. If we call this whole frictional resist-
ance F, the condition of possible motion is
W 2 = W^ + F, or T; = TI + F, and
W^~ W l = T 2 - 2\ = F.
The value of ^depends essentially upon (a) the friction-
factor for the belt and pulley surfaces, (b) the tension in the
belt, and (c) the angle of contact a. It can be shown by
integration 1 that for flat pulleys
r 2 = *;*>
f= TS- Z", = (**- i);
1 The proof is therefore not given. It will be found in all books on
the subject which utilise higher mathematics, e.g. Rankine, Machinery
6z.4 THE MECHANICS OF MACHINERY. [CHAP. xn.
while if the pulley be grooved with a V^aped groove of
angle 20, the power ~~ must be substituted for fa. In
sin tf
these expressions e is the number 272 (nearly), the base of
the natural system of logarithms,/ is the friction-factor, a
is the arc of contact / circular measure. In the case
sketched in the figure, where motion is to take place in the
direction of 7I>, T 2 is greater than T lt and the index in the
formula must be used with the positive sign. If it had
been required to find to what amount T 2 would have to be
reduced before W-^ could begin to move downwards against
it, the same formula would be used, but with a negative
instead of a positive index. Thus if W^ = 100 Ibs., a =
1 80 = TT, and/ = 0-4, the smallest value of T 2 which will
lift Vl\ will be 100 (2'72' 4X7r ) = 351 pounds, and the
value to which T 2 must be reduced in order to allow W\ to
fall must be 100 (2'72~' 4X ' r ) = 28*5 pounds. Or otherwise,
if 71 = 500 pounds, and T 2 = i pound, /remaining as before,
we can find the value of a, in order that the system may be
balanced, 1 as
i . 71
a = -rloge = 15-54.
/ ^2
This is in circular measure, and is therefore equivalent to
= 2 '47 complete turns. Thus with the assumed (very
large) friction-factor, two and a half turns of a cord round a
fixed pulley would give friction enough to enable i pound to
and Millwork, art. 310 A, or Cotterill, Applied Mechanics, art. 123.
Professor Cotterill also gives a graphic construction for finding the
variation of tension and frictional resistance along the belt.
1 It will be remembered that the natural logarithm, or logarithm to
the base e, of any number can be obtained by multiplying its common
logarithm by the number 2*30.
7S-]
FRICTION IN BELT-GEARING.
625
hold against 500, or in other words to prevent a weight of
500 pounds from running down. The hauling of a rope
round a capstan is of course a familiar example of this.
We have now to apply these results to movable pulleys,
such as those used in belt-gearing. Let us suppose we have
a belt pulley such as is shown in Fig. 363, where the resist-
ance to the motion of the pulley is a weight W, acting at an
arm r, the radius of the pulley being J?. In the first instance
suppose the weight to rest on the ground, and that T 2 ( =
7^e A ) is very small. At first the strap will slip round on the
pulley and the weight will remain unmoved. But if we con-
tinuously increase T 2 we not only increase T lt but increase
FIG. 363.
also the frictional resistance to slipping, T 2 -
At some
point the moment of the frictional resistance becomes equal
to the moment of the weight, i.e.,
(T 2 - T,}R - Wr,
and after this the pulley turns, lifting the weight, and the
strap ceases to slip on the pulley. This is the condition
under which all belt pulleys work. With any further increase
of tension in the strap we do not increase (T 2 7^), so long
as the motion is uniform, as the equality of moments just
s s
626 THE MECHANICS OF MACHINERY. [CHAP. xn.
stated must always exist. 1 By substitution in our former
equations we now have
or if we take r = R and write n for I>, the strap is in equilibrium under given conditions
80.] PULLEY TACKLE.
as to friction. To obtain this we must make - - 2
b Tl
If this were done, P o, so that the smallest possible
pressure at P would brake any load. We do not know /
accurately enough to carry out these conditions by any means
exactly, but even without this a very powerful and handy
brake can be made of this type. 1
80. PULLEY TACKLE. 2
THE efficiency of transmission by pulley tackle depends
not only on the pin friction of the sheaves, but still more
upon the work expended against the stiffness of the rope in
bending it round the sheaves and then straightening it again
(see p. 629). Physically this is equivalent to an increase of
the radius of the resistance (on the bending-on side) and
a decrease of the radius of the effort (on the bending-off
side), as shown in the sketch, Fig. 370. For a pin link chain
the value of the small displacement d?can be easily calculated,
and from it the loss of efficiency can be found. This is,
however, not possible with a rope ; the necessary data for the
calculation do not exist, and the loss of efficiency must be
estimated by use of an empirical formula based on experience.
According to Redtenbacher, if F be the pull in a rope of
diameter d running on a sheave of radius R, the resistance
/ V2\
to bending on or off the pulley is approximately ( -. )F.
3 ^
The formula of Eytelwein, used by Rankine, gives a larger
1 Descriptions of a number of important brakes will be found in a
paper by Mr. W. E. Rich in the Proc. Inst. Meek. Eng. July 1876.
See also Zeitschrift d. V. Deutsch. Ing. 1881, p. 321 ; Proc. Inst. Mech.
Eng. July 1858 and July 1877, the latter containing description of Mr.
Froude's turbine dynamometer.
2 The subjects dealt with in 78 to 80 have been perhaps nowhere
better handled than by Professor Ritter in his Technische Mechanik, to
which we are especially indebted in the present section.
636 THE MECHANICS OF MACHINERY. [CHAP. xn.
resistance
2'I
We have no means of knowing on
what experiments either is based ; the former may perhaps
be rightly preferred.
FIG. 370.
FIG. 371.
Let Fig. 371 represent a sheave of any pulley block, turning
in the direction of the arrow, f'^ and F 2 the effort and resist-
ance, that is the tensions in its two cords or "parts." R is
the radius of trie sheave, r of the pin, and fr that of the
friction circle. Without friction (and neglecting also the
resistance due to the stiffness of the rope) the force S
balancing F^ and F zt which is equal in magnitude to their
sum, but reversed in sense, would pass through the centre of
the sheave, and F[ = F 2 . Allowing for friction we know
that .Smust lie at a distance = fr from the centre. Assuming
F^ and F