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Kinematics of Machines Friction Drives Unit-V
Yatin Kumar Singh Page 1
Introduction Belt Drives
Usually, power is transmitted from one shaft to another by means of belts, ropes, chains and gears, the salient
features of which are as follows:
1. Belts, ropes and chains are used where the distance between the shafts is large. For small distances, gears
are preferred.
2. Belts, ropes and chains are flexible type of connectors, i.e., they are bent easily.
3. The flexibility of belts and ropes is due to the property of their materials whereas chains have a number of
small rigid elements having relative motion between the two elements.
4 Belts and ropes transmit power due to friction between them and the pulleys. If the power transmitted
exceeds the force of friction, the belt or rope slips over the pulley.
5 Belts and ropes are strained during motion as tensions are developed in them.
6. Owing to slipping and straining action, belts and ropes are not positive type of drives, i.e. their velocity
ratios are not constant. On the other hand, chains and gears have constant velocity ratios.
Belt Drives
To transmit power from one shaft to another, pulleys are mounted on the two shafts. The pulleys are then
connected by an endless belt passing over the pulleys. The connecting belt is kept in tension so that motion of
one pulley is transferred to the other without slip. The speed of the driven shaft can be varied by varying the
diameters of the two pulleys.
For an un-stretched belt mounted on the pulleys, the outer and the inner faces become in tension and
compression respectively. In between there is a neutral section which has no tension or compression. Usually,
this is considered at half the thickness of the belt. The effective radius of rotation of a pulley is obtained by
adding half the belt thickness to the radius of the pulley.
A belt may be of rectangular section, known as Flat Belt [Fig. a] or of trapezoidal section, known as V- Belt
[Fig. b]. In case of a flat belt, the rim of the pulley is slightly crowned which helps to keep the belt running
centrally on the pulley rim. The groove on the rim of the pulley of a V-belt drive is made deeper to take the
advantage of the wedge action. The belt does not touch the bottom of the groove. Owing to wedging action, V-
Kinematics of Machines Friction Drives Unit-V
Yatin Kumar Singh Page 2
belts need little adjustment and transmit more power, without slip, as compared to flat belts. Also, a multiple
V-belt system, using more than one belt in the two pulleys, can be used to increase the power transmitting
capacity. Generally, these are more suitable for shorter centre distances.
For power transmission by ropes, grooved pulleys are used [Fig. c]. Rope is gripped on its sides as it bends
down in the groove reducing the chances of slipping. Pulleys with several grooves can also be employed to
increase the capacity of power transmission [Fig. d]. They may be connected in either of the two ways:
1. Using a continuous rope passing from one pulley to the other and back again to the same pulley in the next
groove and so on.
2. Using one rope for each pair of grooves.
The advantage of using continuous rope is that the tension in it is uniformly distributed. However, in case of
belt failure, the whole drive is put out of action. Using one rope for each groove poses difficulty in tightening
the ropes to the same extent but with the advantage that the system can continue its operation even if a rope
fails. The repair can be undertaken when it is convenient.
Rope Drives are usually, preferred for long centre distances between the shafts, ropes being cheaper as
compared to belts. These days, however, long distances are avoided and thus the use of ropes has been
limited.
Open and Crossed Belt Drives
1. Open-Belt Drive
An open belt drive is used when the driven pulley is desired to be rotated in the same direction as the driving
pulley as shown in Fig. Generally, the centre distance for an open-belt drive is 14 to 16 m. If the centre
distance is too large, the belt whips, i.e. vibrates in a direction perpendicular to the direction or motion. For
very shorter centre distances, the bell slip increases. Both these phenomena limit the use of belts for power
transmission.
While transmitting power, one side of the belt is more tightened (known as Tight Side) than the other
(known as Slack Side). In case of horizontal drives, it is always desired that the tight side is at the lower side
of two pulleys. This is because the sag of the belt will be more on the upper side than the lower side. This
slightly increases the angles of wrap of the belt on the two pulleys than if the belt had been perfectly straight
between the pulleys. In case the tight side is at the upper side, the sag will be greater at the lower side,
reducing the angle of wrap and slip could occur earlier. This ultimately affects the power to be transmitted.
Kinematics of Machines Friction Drives Unit-V
Yatin Kumar Singh Page 3
2. Crossed-Belt Drive
A crossed-belt drive is adopted when the driven pulley is to be rotated in the opposite direction to that of the
driving pulley.
A crossed-belt drive can transmit more power than an open-belt drive as the angle of wrap is more. However,
the belt has to bend in two different planes and it wears out more.
Velocity Ratio
Velocity ratio is the ratio of speed of the driven pulley to that of the driving pulley.
Let N1 = rotational speed of the driving pulley; N2 = rotational speed of the driven pulley; D1 =diameter of the
driving pulley; D2 =diameter of the driven pulley; t = thickness of the belt
Neglecting any slip between the belt and the pulleys and also considering the belt to be inelastic,
Speed of belt on driving pulley = Speed of belt on driven pulley
( + ) = ( + )
= =
+
+
Slip
The effect of slip is to decrease the speed of belt on the driving shaft and to decrease the speed of the driven
shaft.
Let ω1 = Angular velocity of the driving pulley; ω2 = Angular velocity of the driven pulley; S1 = percentage slip
between the driving pulley and the belt; S2 = percentage slip between the driven pulley and the belt; S = total
percentage slip
���������� ����d �� d����n� �u���y = � (
+
)
����d �� b��� �n ��� d����n� �u���y = [� (
+
)] (
−
)
This is also the speed of the belt on the driven pulley.
���������� ����d �� d����n �u���y = {[� (
+
)] (
−
) (
−
)}
As S is the total percentage slip,
Kinematics of Machines Friction Drives Unit-V
Yatin Kumar Singh Page 4
���������� ����d �� d����n ����� = [� (
+
)] (
−
)
{[� (
+
)] (
−
) (
−
)} = [� (
+
)] (
−
)
− � − �
×
=
− �
− � − � = − �
= + − .
Effect of slip is to reduce the velocity ratio,
= = (
+
+
) (
−
)
Also it is to be remembered that slip will first occur on the pulley with smaller angle of lap i.e. on the smaller
pulley.
Ratio of Friction Tensions
1. Flat Belt
Let T1 = tension on tight side; T2 = tension on slack side; θ = angle of lap of the belt over the pulley; μ=
coefficient of friction between the belt and the pulley
Consider a short length of belt subtending an angle δθ at the center of the pulley
Let R = normal (radial) reaction between the element length of belt and the pulley
T= Tension on slack side of the element; δT= increase in tension on tight side than that on slack side
T + δT = tension on tight side of the element
Tensions T and (T + δT) act in directions perpendicular to the radii drawn at the ends of the element. The
friction force μR will act tangentially to the pulley rim resisting the slipping of the elementary belt on the
pulley. Resolving the forces in the tangential direction,
� +
��
− + �
��
=
A� �� �� �����
��
≈
� + − − � = �� � = �
Resolving the forces in the radial direction,
−
��
− + �
��
=
A� �� �� �����
��
≈
��
Kinematics of Machines Friction Drives Unit-V
Yatin Kumar Singh Page 5
−
��
−
��
−
� ��
=
Neglecting product of two small quantities,
= ��
From (i) and (ii) ,
� = � �� ��
�
= ���
Integrating between proper limits,
∫ = ∫ � �
�
= �� �� = ��
It is to be noted that the above relation is valid only when the belt is on the point of slipping on the pulleys.
Power Transmitted
Let T1 = tension on the tight side; T2 = tension on the slack side; v = linear velocity of the belt; p= power
transmitted
Then, P = net force × distance moved/second = (T1 - T2) × v
This relation gives the power transmitted irrespective of the fact whether the belt is on the point of slipping
or not. If it is, the relationship between T1 and T2 for a flat belt is given by
= ��
If it is not, no particular relation is available to calculate T1 and T2.
Centrifugal Effect on Belts
While in motion, as a belt passes over a pulley, the centrifugal effect due to its own weight tends to lift the belt
from the pulley. Owing to symmetry, the centrifugal force produces equal tensions on the two sides of the
belt, i.e. on the tight sides and the slack sides. Consider a short element of belt
Let m = mass per unit length of belt; Tc = centrifugal tension on tight and slack sides of element; Fc =
centrifugal force on the element; r = radius of the pulley; v = velocity of the belt; δθ = angle of lap of the
element over the pulley
� = × = × ×
� = �� × ×
�
= � ��
A���, � =
��
Kinematics of Machines Friction Drives Unit-V
Yatin Kumar Singh Page 6
A� �� �� �����
��
≈
��
� =
��
= ��
From (i) and (ii)
�� = � �� �� = �
Thus centrifugal tension is independent of the tight and slack side tensions and depends only on the velocity
of the belt over the pulley.
� = =
� = + . = +
� = +
It can be shown that the power transmitted is reduced if centrifugal effect is considered for a given value
of the total tight side tension T.
(a) Centrifugal Tension considered
Friction tension on tight side = T - Tc = T1
Let T2 be the friction tension on the slack side.
���n = ��
= , � c�n���n� �� =
And �����, = − � = ( − ) � = ( − ) �
(b) Centrifugal Tension Neglected
Friction tension on tight side = T; Let T’2 be the friction tension on slack side.
′
= ��
= �� ′ =
�����, = − ′ � = ( − ) � = ( − ) �
As T1 is lesser than T, power transmitted is less when centrifugal force is taken into account.
Maximum Power Transmitted by Belt
If it is desired that a belt transmits maximum possible power, two conditions must be fulfilled simultaneously.
They are
1. Larger tension must reach the maximum permissible value for the belt
2. The belt should be on the point of slipping, i.e. maximum frictional force is developed in the belt.
���, = − � = ( − ) � = ( − ��
) � = �
Kinematics of Machines Friction Drives Unit-V
Yatin Kumar Singh Page 7
= − ��
=
�� = − � = � − � � = −
The maximum tension T in the belt should not exceed the permissible limit. Hence treating T as constant and
differentiating the power with respect to v and equating the same equal to zero.
�
= − =
�� = = �� =
Therefore, for maximum power, to be transmitted centrifugal tension in the belt must be equal to one-third of
the maximum allowable belt tension and the belt should be on the point of slipping.
� = − = − = = √
�
Friction in Bearings
When a shaft is rotating in a bearing, a pivot or a collar is provided on the shaft to take up the axial thrust.
Common examples are steam turbines, hydraulic turbines, propeller shaft of a ship, etc.
1. Flat Pivot Bearing
When the axial force is taken by the end of a shaft that is inserted in a recess to take up the thrust, it is
called a Pivot Bearing or Foot Step Bearing. A flat pivot bearing like the foot step bearing is shown in Fig.5.9
(a). Let W = load on the bearing; r = radius of bearing surface; p = intensity of pressure between rubbing
surfaces
(a) Uniform Pressure
When the pressure is uniformly distributed over the bearing surface area, then
=
�
.
Consider a ring of radius x and thickness dx of the bearing area, as shown in the figure.
A��� �� b����n� �u���c�, � = ��. �
���d ���n������d �� ��� ��n�, � = . �
���c���n�� �������nc� �� ���d�n� �� ��d�u� , � = �. � = ��. � �
���c���n�� ����u� �n ��� ��n�, = � � = �� � �
����� ���c���n�� ����u�, = ∫ �� � � = �� ∫ � � = �� |
�
| =
��
=
�
= � .
, = .
Kinematics of Machines Friction Drives Unit-V
Yatin Kumar Singh Page 8
� , = . � .
If the shaft of radius r2 is resting on a disc of radius r1 as shown in Fig.5.9(b), then
=
� −
.
= �� ∫ � � = �� |
�
| = ��
( − )
= �
−
−
= � .
����� =
−
−
= .
(b) Uniform Wear
The rate of wear depends upon the intensity of pressure and the rubbing velocity. The rubbing velocity
increases with the increase in radius. Therefore, the wear rate is proportional to the pressure p and radius x.
��� un����� ����, � = �� =
�
where C is a constant.
���d ���n������d �� ��� ��n�, � = . ��. � = � . �
����� ���d ���n������d �� ��� b����n�, = � ∫ � = � |�| = �
��, =
�
Fig.5.9 Flat Pivot bearing
���c���n�� ����u� �n ��� b����n�, = �� � � = ��. �. �
����� ���c���n�� ����u� �n ��� b����n�, = �� ∫ � � = ��. |
�
| = ��. =
�
.
Kinematics of Machines Friction Drives Unit-V
Yatin Kumar Singh Page 9
If the shaft of radius r2 is resting on a disc of inner radius r1, then
= �� ∫ � � = ��. |
�
| = ��. ( − ) =
� +
= .
2. Conical Pivot Bearing
Consider a conical pivot bearing as shown in Fig.5.10.
Let pn = intensity of normal pressure on the cone; α = semi cone angle; μ = coefficient of friction between the
shaft and the bearing; r = radius of the shaft
Consider a small ring of radius x and thickness dx. Let dl be the length of the ring along the cone, so that
= �. �
A��� �� ��� ��n�, = ��. = ��. � �
(a) Uniform Pressure
������ ���d �c��n� �n ��� ��n�, � = �� × × � �
V����c�� ���d �c��n� �n ��� ��n�, � = � �
� , = ∫ � = � ∫ � � = � |� | = �
��, =
�
���c���n�� ���c� �c��n� �n ��� ��n� ��n��n�����y �� ��d�u� �, � = � × �
���c���n�� ����u� �c��n� �n ��� ��n�, = � . � = ��. . � . � . �
����� ���c���n�� ����u�, = ��. . � ∫ � � = ��. � |� | = ( ) . � �. �
= ( ) . � � .
(b) Uniform Wear
In the case of uniform wear, the intensity of normal pressure varies inversely with the distance. Therefore,
pn · x = C, where C is a constant
���d ���n������d �� ��� ��n�, � = . ��. � = � . �
����� ���d ���n������d �� ��� b����n�, = ∫ � = � ∫ � = � .
��, =
�
���c���n�� ����u� �c��n� �n ��� ��n�, = ��. . � . � . � = ��. . � . �. �
Kinematics of Machines Friction Drives Unit-V
Yatin Kumar Singh Page 10
����� ���c���n�� ����u� �c��n� �n ��� b����n�, = ��. . � ∫ �. � = ��. . . �
= ( ) � � .
Fig.5.10 Conical pivot bearing
3. Truncated Conical Pivot Bearing
Consider a truncated conical pivot bearing as shown in Fig.5.11. Then
�n��n���y �� un����� �����u�� �� ����n by, =
� −
(a) Uniform Pressure
����� ���c���n�� ����u� �c��n� �n ��� b����n�, = ��. . � ∫ � . � = ��. . �
( − )
= �� [
� −
] .
� ( − )
= ( ) . � . �
−
−
= � . . � .
�����, =
−
−
.
(b) Uniform Wear
����� ���c���n�� ����u� �c��n� �n ��� b����n�, = ��. . � ∫ �. � = ��. . � ( − )
= ��. [
� −
] � ( − ) = � �
+
= � � .
Kinematics of Machines Friction Drives Unit-V
Yatin Kumar Singh Page 11
�����, = + .
Fig.5.11 Truncated pivot bearing
4. Flat Collar Bearing
A single collar bearing is shown in Fig.5.12(a) and the multiple-collar bearing in Fig.5.12(b).
Let r1 and r2 = inner and outer radii of the bearing, respectively.
A��� �� ��� b����n� �u���c�, = �( − )
(a) Uniform Pressure
�n��n���y �� �����u��, = =
� −
���c���n�� ����u� �n ��� ��n� �� ��d�u� � �nd ���ckn��� ��, = ��. . � . �
����� ���c���n�� ����u�, = ��. ∫ � . � = ( ) �. ( − ) = ( ) � .
−
−
.
= � .
(b) Uniform Wear
��� un����� ���� ���d ���n������d �� ��� ��n�, � = . ��. � = � . �
����� ���d ���n������d �� ��� c�����, = � ∫ � = � . −
=
� −
Kinematics of Machines Friction Drives Unit-V
Yatin Kumar Singh Page 12
���c���n�� ����u� �c��n� �n ��� ��n�, = �. � . = ��. . �. �
����� ���c���n�� ����u� �n ��� b����n�, = ��. ∫ �. � = ��. ( − ) =
� +
.
= � .
For a multi-collared bearing having n collars, multiply by n.
Fig.5.12 Flat collar bearings
Rolling Friction
Consider a cylinder or sphere rolling on a flat surface, as shown in Fig.5.13 (a). When there is no deformation
of the surface on which rolling is taking place, then the point of contact will be a line in the case of a cylinder
and a point in the case of a sphere, as shown in Fig.5.13(a). If the surface deforms, then the shape of the
surface will be as shown in Fig.5.13 (b). Let the distance between the point of contact B and the point A
through which the load W passes be b and F be the force required for rolling. Then rolling moment is equal to
F · h and the resisting moment is W · b. For the equilibrium of forces, we have
F · h = W · b
b is known as the coefficient of rolling friction, and has linear dimension.
Let Fr = force applied to the body for rolling; Fs = force applied to the body for sliding
���n, � = ( ) .
� = �
The body rolls without sliding, if Fr < Fs or μ > b/h. The body will slide if Fs < Fr or μ < b/h. The body will
either roll or slide if μ = b/h.
Kinematics of Machines Friction Drives Unit-V
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Fig.5.13 Rolling of a cylinder or sphere
Anti-Friction Bearings
In the case of anti-friction bearings, the point of contact between the journal and the bearing elements is
either a point (as in the case of ball bearings) or a line (as in the case of roller bearings). The ball bearing
consists of a number of hardened balls mounted between two hardened races. The inner race is fitted on the
shaft and the outer race is a tight fit into the bearing housing. The balls are kept at a fixed distance from one
another in a brass cage. The distortion of balls is very little and the rolling friction is very low. They are
mainly used to carry radial loads. In roller bearings, either right cylindrical or tapered rollers are present.
They are used to carry heavy loads, both radial and thrust.
Friction Circle
Consider a journal bearing, as shown in Fig.5.14. When the journal is at stand still, then the point of contact is
at A. The load W is balanced by the reaction R. When the journal starts rotating in the clockwise direction,
then the point of contact shifts from A to B. The resultant of normal reaction R and force of friction F = μR is S,
as shown in Fig.5.14 (b). For the equilibrium of the journal, we have
W = S
= . = . � = . � �n��� � b��n� ����� = . � .
A circle drawn with centre O and radius OC = r sin ϕ = rμ is called the friction circle.
Fig.5.14 Friction circle
Kinematics of Machines Friction Drives Unit-V
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Film Friction
Thin-Film Lubrication: Thin-film lubrication is defined as a condition of lubrication in which the lubricant
film is relatively thin and there is partial metal-to-metal contact. This mode of lubrication is seen in door
hinges and machine tool slides. It is also called boundary lubrication. The conditions resulting in thin-film
lubrication are excessive load, insufficient surface area or oil supply, low speed, and misalignment.
Thick-Film Lubrication: When the bearing and the journal are completely separated from each other by the
lubricant film, then it is called thick-film lubrication. Since there is no contact between the surfaces,
properties of surface, like surface finish, have little or no influence on the performance of the bearing. The
resistance to relative motion between the journal and the bearing arises from the viscous resistance of the
lubricant. Therefore, the performance of the bearing is affected by the viscosity of the lubricant. Thick-film
lubrication could be hydrodynamic or hydrostatic.
In a journal bearing working under thick-film lubrication regime, the frictional resistance depends upon the
following parameters:
 Dynamic Viscosity of the Lubricant
 Speed of the Journal
 Unit Pressure of the Lubricant, and
 Radial Clearance Ratio
The coefficient of friction depends upon the factor (μn/p), where μ = dynamic viscosity of lubricant, n =
speed of journal, and p = unit pressure of lubricant. The value of this factor at which the coefficient of friction
is minimum is called the bearing modulus, as shown in Fig.5.15. Below this value, there exists boundary (or
thin-film) lubrication and above this value there shall be full-film (or thick-film) lubrication.
Fig.5.15 Variation of coefficient of friction with μn/p
Mitchell (or Tilting Pad) Thrust Bearing
To maintain thick-film lubrication in the case of rotating members, a wedge-shaped converging space should
be available for the lubricant between the journal and the bearing. For flat surfaces, the same purpose can be
served if one surface is made slightly inclined. A Mitchell thrust bearing consists of a series of metallic pads
Kinematics of Machines Friction Drives Unit-V
Yatin Kumar Singh Page 15
arranged around a rotating collar fixed to the shaft, as shown in Fig.5.16. Each pad is held by the housing of
the bearing to prevent rotation but free to tilt about its stepped edge. The oil carried by the moving collar is
dragged around the pad. Thus, an oil film of wedge shape is formed and a considerable pressure is built up to
carry the axial load.
Friction Clutch
A clutch is a device used to transmit the rotary motion of one shaft to another when desired. The axes of the
two shafts are coincident. In friction clutches the connection of the engine shaft to the gear box shaft is
affected by friction between two or more rotating concentric surfaces. The surfaces can be pressed firmly
against one another when engaged and the clutch tends to rotate as a single unit.
1. Disc Clutch (Single – Plate Clutch)
A disc clutch consists of a clutch plate attached to a splined hub which is free to slide axially on splines cut on
the driven shaft. The clutch plate is made of steel and has a ring friction lining on each side. The engine shaft
supports a rigidly fixed fly wheel. A spring loaded pressure plate presses the clutch plate firmly against the
flywheel when clutch is engaged. When disengaged, the spring press against a cover attached to the fly wheel.
Thus both the fly wheel and pressure plate rotate with the input shaft. The movement of the clutch plate is
transferred to the pressure plate through a throat bearing.
Kinematics of Machines Friction Drives Unit-V
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Figure shows the pressure plate pulled back by the release levers and the friction linings on the clutch plate
are no longer in contact with the pressure plate or the fly wheel. The fly wheel rotates without driving the
clutch plate and thus the driven shaft.
When the foot is taken off from the clutch pedal, the pressure on the throat bearing is released. As a result the
spring becomes free to move the pressure plate to bring it in contact with the clutch plate. The clutch plate
slide on the splined hub and is tightly gripped between the pressure plate and the fly wheel. The friction
between the lining on the clutch plate and the fly wheel on one slide and the pressure plate, on the other,
cause the clutch plate and hence the driven shaft to rotate. In case the resisting torque on the driven shaft
exceeds the torque at the clutch, clutch slip will occur.
2. Multiple Plate Clutch:
In multiple clutch, the number of frictional linings and the metal plates are increased which increases the
capacity of the clutch to transmit torque. Fig shows simplification diagram. The friction rings are splined on
their outer circumference and engage with corresponding splines on the flywheel. They are free to slide
axially. The friction material thus, rotates with the fly wheel and the engine shaft. The number of friction rings
depends upon the torque to be transmitted.
The driven shaft also supports disc on the splines which rotate with the driven shaft and can slide axially. If
the actuating force on the pedal is removed, a spring presses the discs into contact with the friction rings and
torque is transmitted between the engine shaft and the driven shaft. If n1 is the number of plates on diving
and n2 is the number of plates on driven shaft then the number of active surfaces will be n = n1 + n2 ‒ 1
3. Cone Clutch
In a cone clutch, the contact surfaces are in the form of cones. In the engaged position, the friction surfaces of
the two cones A and B are in complete contact due to spring pressure that keeps one cones pressure against
Kinematics of Machines Friction Drives Unit-V
Yatin Kumar Singh Page 17
the other all the time. When the clutch is engaged the torque is transmitted from the diving shaft to the driven
shaft through the flywheel and the friction cones. For disengaged the clutch, the cone B is pulled back though
a lever system against the force of the spring.
The advantage of a cone clutch is that the normal forces on the contact surfaces are increased. If F is the axial
forces, Fn is the normal force and α is the semi cone angle of the clutch, then for a conical collar with uniform
wear theory.
� =
�
�
=
� −
�
= � . ; � =
−
Where b is the width of the cone face. Remember as pr is constant in case of uniform wear theory which is
applicable to clutches be on the safer side, P is to be normal pressure at the radius considered i.e. at the inner
radius ri it is Pi and at the mean radius Rm it is pm .
Figure -47
We know,
=
��
�
+ =
�� �
�
(
+
) = ��
Engagement force is required
�′
= � + �� � = � � + �� � = � � + � �
� = � � + � �
� = � � � − �

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Friction Drives

  • 1. Kinematics of Machines Friction Drives Unit-V Yatin Kumar Singh Page 1 Introduction Belt Drives Usually, power is transmitted from one shaft to another by means of belts, ropes, chains and gears, the salient features of which are as follows: 1. Belts, ropes and chains are used where the distance between the shafts is large. For small distances, gears are preferred. 2. Belts, ropes and chains are flexible type of connectors, i.e., they are bent easily. 3. The flexibility of belts and ropes is due to the property of their materials whereas chains have a number of small rigid elements having relative motion between the two elements. 4 Belts and ropes transmit power due to friction between them and the pulleys. If the power transmitted exceeds the force of friction, the belt or rope slips over the pulley. 5 Belts and ropes are strained during motion as tensions are developed in them. 6. Owing to slipping and straining action, belts and ropes are not positive type of drives, i.e. their velocity ratios are not constant. On the other hand, chains and gears have constant velocity ratios. Belt Drives To transmit power from one shaft to another, pulleys are mounted on the two shafts. The pulleys are then connected by an endless belt passing over the pulleys. The connecting belt is kept in tension so that motion of one pulley is transferred to the other without slip. The speed of the driven shaft can be varied by varying the diameters of the two pulleys. For an un-stretched belt mounted on the pulleys, the outer and the inner faces become in tension and compression respectively. In between there is a neutral section which has no tension or compression. Usually, this is considered at half the thickness of the belt. The effective radius of rotation of a pulley is obtained by adding half the belt thickness to the radius of the pulley. A belt may be of rectangular section, known as Flat Belt [Fig. a] or of trapezoidal section, known as V- Belt [Fig. b]. In case of a flat belt, the rim of the pulley is slightly crowned which helps to keep the belt running centrally on the pulley rim. The groove on the rim of the pulley of a V-belt drive is made deeper to take the advantage of the wedge action. The belt does not touch the bottom of the groove. Owing to wedging action, V-
  • 2. Kinematics of Machines Friction Drives Unit-V Yatin Kumar Singh Page 2 belts need little adjustment and transmit more power, without slip, as compared to flat belts. Also, a multiple V-belt system, using more than one belt in the two pulleys, can be used to increase the power transmitting capacity. Generally, these are more suitable for shorter centre distances. For power transmission by ropes, grooved pulleys are used [Fig. c]. Rope is gripped on its sides as it bends down in the groove reducing the chances of slipping. Pulleys with several grooves can also be employed to increase the capacity of power transmission [Fig. d]. They may be connected in either of the two ways: 1. Using a continuous rope passing from one pulley to the other and back again to the same pulley in the next groove and so on. 2. Using one rope for each pair of grooves. The advantage of using continuous rope is that the tension in it is uniformly distributed. However, in case of belt failure, the whole drive is put out of action. Using one rope for each groove poses difficulty in tightening the ropes to the same extent but with the advantage that the system can continue its operation even if a rope fails. The repair can be undertaken when it is convenient. Rope Drives are usually, preferred for long centre distances between the shafts, ropes being cheaper as compared to belts. These days, however, long distances are avoided and thus the use of ropes has been limited. Open and Crossed Belt Drives 1. Open-Belt Drive An open belt drive is used when the driven pulley is desired to be rotated in the same direction as the driving pulley as shown in Fig. Generally, the centre distance for an open-belt drive is 14 to 16 m. If the centre distance is too large, the belt whips, i.e. vibrates in a direction perpendicular to the direction or motion. For very shorter centre distances, the bell slip increases. Both these phenomena limit the use of belts for power transmission. While transmitting power, one side of the belt is more tightened (known as Tight Side) than the other (known as Slack Side). In case of horizontal drives, it is always desired that the tight side is at the lower side of two pulleys. This is because the sag of the belt will be more on the upper side than the lower side. This slightly increases the angles of wrap of the belt on the two pulleys than if the belt had been perfectly straight between the pulleys. In case the tight side is at the upper side, the sag will be greater at the lower side, reducing the angle of wrap and slip could occur earlier. This ultimately affects the power to be transmitted.
  • 3. Kinematics of Machines Friction Drives Unit-V Yatin Kumar Singh Page 3 2. Crossed-Belt Drive A crossed-belt drive is adopted when the driven pulley is to be rotated in the opposite direction to that of the driving pulley. A crossed-belt drive can transmit more power than an open-belt drive as the angle of wrap is more. However, the belt has to bend in two different planes and it wears out more. Velocity Ratio Velocity ratio is the ratio of speed of the driven pulley to that of the driving pulley. Let N1 = rotational speed of the driving pulley; N2 = rotational speed of the driven pulley; D1 =diameter of the driving pulley; D2 =diameter of the driven pulley; t = thickness of the belt Neglecting any slip between the belt and the pulleys and also considering the belt to be inelastic, Speed of belt on driving pulley = Speed of belt on driven pulley ( + ) = ( + ) = = + + Slip The effect of slip is to decrease the speed of belt on the driving shaft and to decrease the speed of the driven shaft. Let ω1 = Angular velocity of the driving pulley; ω2 = Angular velocity of the driven pulley; S1 = percentage slip between the driving pulley and the belt; S2 = percentage slip between the driven pulley and the belt; S = total percentage slip ���������� ����d �� d����n� �u���y = � ( + ) ����d �� b��� �n ��� d����n� �u���y = [� ( + )] ( − ) This is also the speed of the belt on the driven pulley. ���������� ����d �� d����n �u���y = {[� ( + )] ( − ) ( − )} As S is the total percentage slip,
  • 4. Kinematics of Machines Friction Drives Unit-V Yatin Kumar Singh Page 4 ���������� ����d �� d����n ����� = [� ( + )] ( − ) {[� ( + )] ( − ) ( − )} = [� ( + )] ( − ) − � − � × = − � − � − � = − � = + − . Effect of slip is to reduce the velocity ratio, = = ( + + ) ( − ) Also it is to be remembered that slip will first occur on the pulley with smaller angle of lap i.e. on the smaller pulley. Ratio of Friction Tensions 1. Flat Belt Let T1 = tension on tight side; T2 = tension on slack side; θ = angle of lap of the belt over the pulley; μ= coefficient of friction between the belt and the pulley Consider a short length of belt subtending an angle δθ at the center of the pulley Let R = normal (radial) reaction between the element length of belt and the pulley T= Tension on slack side of the element; δT= increase in tension on tight side than that on slack side T + δT = tension on tight side of the element Tensions T and (T + δT) act in directions perpendicular to the radii drawn at the ends of the element. The friction force μR will act tangentially to the pulley rim resisting the slipping of the elementary belt on the pulley. Resolving the forces in the tangential direction, � + �� − + � �� = A� �� �� ����� �� ≈ � + − − � = �� � = � Resolving the forces in the radial direction, − �� − + � �� = A� �� �� ����� �� ≈ ��
  • 5. Kinematics of Machines Friction Drives Unit-V Yatin Kumar Singh Page 5 − �� − �� − � �� = Neglecting product of two small quantities, = �� From (i) and (ii) , � = � �� �� � = ��� Integrating between proper limits, ∫ = ∫ � � � = �� �� = �� It is to be noted that the above relation is valid only when the belt is on the point of slipping on the pulleys. Power Transmitted Let T1 = tension on the tight side; T2 = tension on the slack side; v = linear velocity of the belt; p= power transmitted Then, P = net force × distance moved/second = (T1 - T2) × v This relation gives the power transmitted irrespective of the fact whether the belt is on the point of slipping or not. If it is, the relationship between T1 and T2 for a flat belt is given by = �� If it is not, no particular relation is available to calculate T1 and T2. Centrifugal Effect on Belts While in motion, as a belt passes over a pulley, the centrifugal effect due to its own weight tends to lift the belt from the pulley. Owing to symmetry, the centrifugal force produces equal tensions on the two sides of the belt, i.e. on the tight sides and the slack sides. Consider a short element of belt Let m = mass per unit length of belt; Tc = centrifugal tension on tight and slack sides of element; Fc = centrifugal force on the element; r = radius of the pulley; v = velocity of the belt; δθ = angle of lap of the element over the pulley � = × = × × � = �� × × � = � �� A���, � = ��
  • 6. Kinematics of Machines Friction Drives Unit-V Yatin Kumar Singh Page 6 A� �� �� ����� �� ≈ �� � = �� = �� From (i) and (ii) �� = � �� �� = � Thus centrifugal tension is independent of the tight and slack side tensions and depends only on the velocity of the belt over the pulley. � = = � = + . = + � = + It can be shown that the power transmitted is reduced if centrifugal effect is considered for a given value of the total tight side tension T. (a) Centrifugal Tension considered Friction tension on tight side = T - Tc = T1 Let T2 be the friction tension on the slack side. ���n = �� = , � c�n���n� �� = And �����, = − � = ( − ) � = ( − ) � (b) Centrifugal Tension Neglected Friction tension on tight side = T; Let T’2 be the friction tension on slack side. ′ = �� = �� ′ = �����, = − ′ � = ( − ) � = ( − ) � As T1 is lesser than T, power transmitted is less when centrifugal force is taken into account. Maximum Power Transmitted by Belt If it is desired that a belt transmits maximum possible power, two conditions must be fulfilled simultaneously. They are 1. Larger tension must reach the maximum permissible value for the belt 2. The belt should be on the point of slipping, i.e. maximum frictional force is developed in the belt. ���, = − � = ( − ) � = ( − �� ) � = �
  • 7. Kinematics of Machines Friction Drives Unit-V Yatin Kumar Singh Page 7 = − �� = �� = − � = � − � � = − The maximum tension T in the belt should not exceed the permissible limit. Hence treating T as constant and differentiating the power with respect to v and equating the same equal to zero. � = − = �� = = �� = Therefore, for maximum power, to be transmitted centrifugal tension in the belt must be equal to one-third of the maximum allowable belt tension and the belt should be on the point of slipping. � = − = − = = √ � Friction in Bearings When a shaft is rotating in a bearing, a pivot or a collar is provided on the shaft to take up the axial thrust. Common examples are steam turbines, hydraulic turbines, propeller shaft of a ship, etc. 1. Flat Pivot Bearing When the axial force is taken by the end of a shaft that is inserted in a recess to take up the thrust, it is called a Pivot Bearing or Foot Step Bearing. A flat pivot bearing like the foot step bearing is shown in Fig.5.9 (a). Let W = load on the bearing; r = radius of bearing surface; p = intensity of pressure between rubbing surfaces (a) Uniform Pressure When the pressure is uniformly distributed over the bearing surface area, then = � . Consider a ring of radius x and thickness dx of the bearing area, as shown in the figure. A��� �� b����n� �u���c�, � = ��. � ���d ���n������d �� ��� ��n�, � = . � ���c���n�� �������nc� �� ���d�n� �� ��d�u� , � = �. � = ��. � � ���c���n�� ����u� �n ��� ��n�, = � � = �� � � ����� ���c���n�� ����u�, = ∫ �� � � = �� ∫ � � = �� | � | = �� = � = � . , = .
  • 8. Kinematics of Machines Friction Drives Unit-V Yatin Kumar Singh Page 8 � , = . � . If the shaft of radius r2 is resting on a disc of radius r1 as shown in Fig.5.9(b), then = � − . = �� ∫ � � = �� | � | = �� ( − ) = � − − = � . ����� = − − = . (b) Uniform Wear The rate of wear depends upon the intensity of pressure and the rubbing velocity. The rubbing velocity increases with the increase in radius. Therefore, the wear rate is proportional to the pressure p and radius x. ��� un����� ����, � = �� = � where C is a constant. ���d ���n������d �� ��� ��n�, � = . ��. � = � . � ����� ���d ���n������d �� ��� b����n�, = � ∫ � = � |�| = � ��, = � Fig.5.9 Flat Pivot bearing ���c���n�� ����u� �n ��� b����n�, = �� � � = ��. �. � ����� ���c���n�� ����u� �n ��� b����n�, = �� ∫ � � = ��. | � | = ��. = � .
  • 9. Kinematics of Machines Friction Drives Unit-V Yatin Kumar Singh Page 9 If the shaft of radius r2 is resting on a disc of inner radius r1, then = �� ∫ � � = ��. | � | = ��. ( − ) = � + = . 2. Conical Pivot Bearing Consider a conical pivot bearing as shown in Fig.5.10. Let pn = intensity of normal pressure on the cone; α = semi cone angle; μ = coefficient of friction between the shaft and the bearing; r = radius of the shaft Consider a small ring of radius x and thickness dx. Let dl be the length of the ring along the cone, so that = �. � A��� �� ��� ��n�, = ��. = ��. � � (a) Uniform Pressure ������ ���d �c��n� �n ��� ��n�, � = �� × × � � V����c�� ���d �c��n� �n ��� ��n�, � = � � � , = ∫ � = � ∫ � � = � |� | = � ��, = � ���c���n�� ���c� �c��n� �n ��� ��n� ��n��n�����y �� ��d�u� �, � = � × � ���c���n�� ����u� �c��n� �n ��� ��n�, = � . � = ��. . � . � . � ����� ���c���n�� ����u�, = ��. . � ∫ � � = ��. � |� | = ( ) . � �. � = ( ) . � � . (b) Uniform Wear In the case of uniform wear, the intensity of normal pressure varies inversely with the distance. Therefore, pn · x = C, where C is a constant ���d ���n������d �� ��� ��n�, � = . ��. � = � . � ����� ���d ���n������d �� ��� b����n�, = ∫ � = � ∫ � = � . ��, = � ���c���n�� ����u� �c��n� �n ��� ��n�, = ��. . � . � . � = ��. . � . �. �
  • 10. Kinematics of Machines Friction Drives Unit-V Yatin Kumar Singh Page 10 ����� ���c���n�� ����u� �c��n� �n ��� b����n�, = ��. . � ∫ �. � = ��. . . � = ( ) � � . Fig.5.10 Conical pivot bearing 3. Truncated Conical Pivot Bearing Consider a truncated conical pivot bearing as shown in Fig.5.11. Then �n��n���y �� un����� �����u�� �� ����n by, = � − (a) Uniform Pressure ����� ���c���n�� ����u� �c��n� �n ��� b����n�, = ��. . � ∫ � . � = ��. . � ( − ) = �� [ � − ] . � ( − ) = ( ) . � . � − − = � . . � . �����, = − − . (b) Uniform Wear ����� ���c���n�� ����u� �c��n� �n ��� b����n�, = ��. . � ∫ �. � = ��. . � ( − ) = ��. [ � − ] � ( − ) = � � + = � � .
  • 11. Kinematics of Machines Friction Drives Unit-V Yatin Kumar Singh Page 11 �����, = + . Fig.5.11 Truncated pivot bearing 4. Flat Collar Bearing A single collar bearing is shown in Fig.5.12(a) and the multiple-collar bearing in Fig.5.12(b). Let r1 and r2 = inner and outer radii of the bearing, respectively. A��� �� ��� b����n� �u���c�, = �( − ) (a) Uniform Pressure �n��n���y �� �����u��, = = � − ���c���n�� ����u� �n ��� ��n� �� ��d�u� � �nd ���ckn��� ��, = ��. . � . � ����� ���c���n�� ����u�, = ��. ∫ � . � = ( ) �. ( − ) = ( ) � . − − . = � . (b) Uniform Wear ��� un����� ���� ���d ���n������d �� ��� ��n�, � = . ��. � = � . � ����� ���d ���n������d �� ��� c�����, = � ∫ � = � . − = � −
  • 12. Kinematics of Machines Friction Drives Unit-V Yatin Kumar Singh Page 12 ���c���n�� ����u� �c��n� �n ��� ��n�, = �. � . = ��. . �. � ����� ���c���n�� ����u� �n ��� b����n�, = ��. ∫ �. � = ��. ( − ) = � + . = � . For a multi-collared bearing having n collars, multiply by n. Fig.5.12 Flat collar bearings Rolling Friction Consider a cylinder or sphere rolling on a flat surface, as shown in Fig.5.13 (a). When there is no deformation of the surface on which rolling is taking place, then the point of contact will be a line in the case of a cylinder and a point in the case of a sphere, as shown in Fig.5.13(a). If the surface deforms, then the shape of the surface will be as shown in Fig.5.13 (b). Let the distance between the point of contact B and the point A through which the load W passes be b and F be the force required for rolling. Then rolling moment is equal to F · h and the resisting moment is W · b. For the equilibrium of forces, we have F · h = W · b b is known as the coefficient of rolling friction, and has linear dimension. Let Fr = force applied to the body for rolling; Fs = force applied to the body for sliding ���n, � = ( ) . � = � The body rolls without sliding, if Fr < Fs or μ > b/h. The body will slide if Fs < Fr or μ < b/h. The body will either roll or slide if μ = b/h.
  • 13. Kinematics of Machines Friction Drives Unit-V Yatin Kumar Singh Page 13 Fig.5.13 Rolling of a cylinder or sphere Anti-Friction Bearings In the case of anti-friction bearings, the point of contact between the journal and the bearing elements is either a point (as in the case of ball bearings) or a line (as in the case of roller bearings). The ball bearing consists of a number of hardened balls mounted between two hardened races. The inner race is fitted on the shaft and the outer race is a tight fit into the bearing housing. The balls are kept at a fixed distance from one another in a brass cage. The distortion of balls is very little and the rolling friction is very low. They are mainly used to carry radial loads. In roller bearings, either right cylindrical or tapered rollers are present. They are used to carry heavy loads, both radial and thrust. Friction Circle Consider a journal bearing, as shown in Fig.5.14. When the journal is at stand still, then the point of contact is at A. The load W is balanced by the reaction R. When the journal starts rotating in the clockwise direction, then the point of contact shifts from A to B. The resultant of normal reaction R and force of friction F = μR is S, as shown in Fig.5.14 (b). For the equilibrium of the journal, we have W = S = . = . � = . � �n��� � b��n� ����� = . � . A circle drawn with centre O and radius OC = r sin ϕ = rμ is called the friction circle. Fig.5.14 Friction circle
  • 14. Kinematics of Machines Friction Drives Unit-V Yatin Kumar Singh Page 14 Film Friction Thin-Film Lubrication: Thin-film lubrication is defined as a condition of lubrication in which the lubricant film is relatively thin and there is partial metal-to-metal contact. This mode of lubrication is seen in door hinges and machine tool slides. It is also called boundary lubrication. The conditions resulting in thin-film lubrication are excessive load, insufficient surface area or oil supply, low speed, and misalignment. Thick-Film Lubrication: When the bearing and the journal are completely separated from each other by the lubricant film, then it is called thick-film lubrication. Since there is no contact between the surfaces, properties of surface, like surface finish, have little or no influence on the performance of the bearing. The resistance to relative motion between the journal and the bearing arises from the viscous resistance of the lubricant. Therefore, the performance of the bearing is affected by the viscosity of the lubricant. Thick-film lubrication could be hydrodynamic or hydrostatic. In a journal bearing working under thick-film lubrication regime, the frictional resistance depends upon the following parameters:  Dynamic Viscosity of the Lubricant  Speed of the Journal  Unit Pressure of the Lubricant, and  Radial Clearance Ratio The coefficient of friction depends upon the factor (μn/p), where μ = dynamic viscosity of lubricant, n = speed of journal, and p = unit pressure of lubricant. The value of this factor at which the coefficient of friction is minimum is called the bearing modulus, as shown in Fig.5.15. Below this value, there exists boundary (or thin-film) lubrication and above this value there shall be full-film (or thick-film) lubrication. Fig.5.15 Variation of coefficient of friction with μn/p Mitchell (or Tilting Pad) Thrust Bearing To maintain thick-film lubrication in the case of rotating members, a wedge-shaped converging space should be available for the lubricant between the journal and the bearing. For flat surfaces, the same purpose can be served if one surface is made slightly inclined. A Mitchell thrust bearing consists of a series of metallic pads
  • 15. Kinematics of Machines Friction Drives Unit-V Yatin Kumar Singh Page 15 arranged around a rotating collar fixed to the shaft, as shown in Fig.5.16. Each pad is held by the housing of the bearing to prevent rotation but free to tilt about its stepped edge. The oil carried by the moving collar is dragged around the pad. Thus, an oil film of wedge shape is formed and a considerable pressure is built up to carry the axial load. Friction Clutch A clutch is a device used to transmit the rotary motion of one shaft to another when desired. The axes of the two shafts are coincident. In friction clutches the connection of the engine shaft to the gear box shaft is affected by friction between two or more rotating concentric surfaces. The surfaces can be pressed firmly against one another when engaged and the clutch tends to rotate as a single unit. 1. Disc Clutch (Single – Plate Clutch) A disc clutch consists of a clutch plate attached to a splined hub which is free to slide axially on splines cut on the driven shaft. The clutch plate is made of steel and has a ring friction lining on each side. The engine shaft supports a rigidly fixed fly wheel. A spring loaded pressure plate presses the clutch plate firmly against the flywheel when clutch is engaged. When disengaged, the spring press against a cover attached to the fly wheel. Thus both the fly wheel and pressure plate rotate with the input shaft. The movement of the clutch plate is transferred to the pressure plate through a throat bearing.
  • 16. Kinematics of Machines Friction Drives Unit-V Yatin Kumar Singh Page 16 Figure shows the pressure plate pulled back by the release levers and the friction linings on the clutch plate are no longer in contact with the pressure plate or the fly wheel. The fly wheel rotates without driving the clutch plate and thus the driven shaft. When the foot is taken off from the clutch pedal, the pressure on the throat bearing is released. As a result the spring becomes free to move the pressure plate to bring it in contact with the clutch plate. The clutch plate slide on the splined hub and is tightly gripped between the pressure plate and the fly wheel. The friction between the lining on the clutch plate and the fly wheel on one slide and the pressure plate, on the other, cause the clutch plate and hence the driven shaft to rotate. In case the resisting torque on the driven shaft exceeds the torque at the clutch, clutch slip will occur. 2. Multiple Plate Clutch: In multiple clutch, the number of frictional linings and the metal plates are increased which increases the capacity of the clutch to transmit torque. Fig shows simplification diagram. The friction rings are splined on their outer circumference and engage with corresponding splines on the flywheel. They are free to slide axially. The friction material thus, rotates with the fly wheel and the engine shaft. The number of friction rings depends upon the torque to be transmitted. The driven shaft also supports disc on the splines which rotate with the driven shaft and can slide axially. If the actuating force on the pedal is removed, a spring presses the discs into contact with the friction rings and torque is transmitted between the engine shaft and the driven shaft. If n1 is the number of plates on diving and n2 is the number of plates on driven shaft then the number of active surfaces will be n = n1 + n2 ‒ 1 3. Cone Clutch In a cone clutch, the contact surfaces are in the form of cones. In the engaged position, the friction surfaces of the two cones A and B are in complete contact due to spring pressure that keeps one cones pressure against
  • 17. Kinematics of Machines Friction Drives Unit-V Yatin Kumar Singh Page 17 the other all the time. When the clutch is engaged the torque is transmitted from the diving shaft to the driven shaft through the flywheel and the friction cones. For disengaged the clutch, the cone B is pulled back though a lever system against the force of the spring. The advantage of a cone clutch is that the normal forces on the contact surfaces are increased. If F is the axial forces, Fn is the normal force and α is the semi cone angle of the clutch, then for a conical collar with uniform wear theory. � = � � = � − � = � . ; � = − Where b is the width of the cone face. Remember as pr is constant in case of uniform wear theory which is applicable to clutches be on the safer side, P is to be normal pressure at the radius considered i.e. at the inner radius ri it is Pi and at the mean radius Rm it is pm . Figure -47 We know, = �� � + = �� � � ( + ) = �� Engagement force is required �′ = � + �� � = � � + �� � = � � + � � � = � � + � � � = � � � − �