BELT AND CHAIN DRIVES I
Lecture #7
Course Name : DESIGN OF MACHINE ELEMENTS
Course Number: MET 214
As mentioned in a previous lecture, the power required for a motor to drive a load can
be determined by applying a service factor to the steady state power required by the
load. The steady state power required by a load maybe determined by forming the
product of the steady state torque and steady state speed requirements required by
the load.
Although a motor may be found that has an acceptable power rating, the steady state
torque and speed combination that the motor operates with to produce the power
may not match the torque and speed requirements required by the load even though
the horsepower requirement for the motor was established from the horsepower
requirements of the load. Belt and chain drives, in addition to other components such
as gears, may be used to transmit the required amount of power from the motor to
the load while permitting adjustments in the relative amounts of torque and speed
associated with the power being transferred. In general, belt drives are utilized when
rotational speeds are relatively high and must be reduced. Due to the manner of how
belt drives operate, if the speed is reduced when transferring power from one pulley
to another, the torque being transmitted from one pulley to another is increased by an
inverse amount which enables the total amount of power being transmitted from one
pulley to another to be constant.
To understand this feature of belt drives, recall the basic power equation.
P  T11
T1  Torque  N  m
1  AngularVelocity  rads / sec
or
PHP 
P1  Power  watt
Tn
63,000
PHP  Power  HorsePower
T  Torque  in  lbs
n  AngularVelocity  rpm
Assuming the amount of power transferred between a pair of pulleys is constant, the
torque speed combination at which each pulley operates may be related to the power
being transmitted between the pulleys by the following equation.
Fill in proper relationship
P  T11  T22
if T1  T2 then 1 ___ 2
if T2  T1 then 1 ___ 2
To understand how a belt drive system scales speed and torque in an inverse manner
while power is being transmitted from one pulley to another pulley of a pulley pair at a
constant amount, consider the figures shown below.
Although the figures shown below applies to a belt drive system, (i.e a system consisting
of a pair of pulleys connected by a belt) in order to understand how the speed and torque
combination may be varied between a pair of pulleys, it may be more convenient to view
the system provided on the previous slide as an item that is similar to a belt drive that
you may have more familiarity with, such as a spool of wire or a thimble with some
thread wrapped around the thimble. In this sense, the figures on the previous slide
depict how the length of the wire that is pulled from the spool is related to the rotation
of the spool.
Using the previous slide as reference, provide an equation for ∆l in terms of the variables
associated with the pulley
∆l =
Assume the amount of time it takes to remove a length of wire of ∆l is determined to be
∆t. Provide an expression for the linear velocity ν of the free end of the wire in terms of
∆l and ∆t.
ν=
Express ν in terms of the rotational variables associated with the pulley.
ν=
What does the combination of ∆θ/ ∆t represent?
Designate ∆θ/ ∆t =

Accordingly, when the above analysis is applied to a belt and pulley, assuming the belt
does not slip on the pulley, the belt linear velocity is related to the angular velocity of the
pulley by the equation presented below. The belt is assumed to be in contact with the
pulley at a distance r from the center of the pulley. The circle of radius r, which was used
in the figures presented in the previous slide to represent a pulley, is referred to as the
pitch circle of the pulley. In order to relate the angular velocity of a pulley to the linear
velocity of the belt, the belt is assumed to be at the radius r of the pitch circle of the
pulley.
v  r
where   linear velocity of belt in/sec, ft/sec, m/sec
r  radius of pitch circle of the pulley in, ft, m
  angular velocity of pulley rads/sec
Now consider a more general situation which involves a pair of pulleys where each pulley
has a pitch circle of a different diameter.
Assume the belt does not slip on either pulley and the belt does not stretch or change
length in any manner as pulley A and pulley B rotate, i.e. the length of the belt wrapped
around both pulleys is constant as pulley A and pulley B rotate.
Q: Assuming the belt contacts pulley B on the pitch circle of pulley B and pulley B rotates
by an amount of ∆θB , provide an expression for the arc length ∆lB that the belt traverses
in response to the rotation of pulley B.
∆lB =
Assuming pulley A rotates by an amount ∆θA, provide an expression that relates the arc
length ∆lA the belt travels while the pulley rotates an amount ∆θA.
∆lA =
Assuming the belt does not change length while the pulleys rotate, what is the nature of
the relationship existing between ∆lB and ∆lA?
∆lB =
Equating expressions for the arc lengths leads to the following relationship.
rA A  rB  B
or, more generally
rA A  rB B
Note if rB>rA
then θB< θA
Taking the time derivative of both sides of the equation above leads to the following result
d rA A  d rB B 

dt
dt
d rA A  drA
d

 A  rA A
dt
dt
dt
drA
Assuming the radius of pulley A does not change with time, i.e., rA = constant, then
0
dt
d rA A 
d A
 rA
 rA A
dt
dt
Similar results hold for pulley B. Accordingly,
rAA  rBB
To relate the angular accelerations of pulley A to pulley B requires an additional derivative
to be applied to both sides of the equation.
d  rA A  d  rBB 

dt
dt
rA A  rB B
where

d
dt
To understand the nature of the relationships existing between the pulleys, let us collect
the following equations
r A A  rB B
rA A  rBB
rA A  rB B
To demonstrate how the equations can be applied to effect changes in the speed of
rotation of shafts in power transmission systems, reorganize the equations as follows:
B   A
rA
rB
B   A
rA
rB
B   A
rA
rB
1
If it is desired to have pulley B rotate one-third the rate of pulley A, wB  wA , develop a
3
relationship between the radii of the pulleys.
rA = ______
Conversely, express the radius of rB in terms of rA to achieve the speed reduction ratio
where w  1 w
B
3
A
rB = ______
Consider the following change in units:
r
B   A A
rB
 60sec s   1rev 
nA  rpm    A  rads / sec  
 min   2  rads 
 60sec s   1rev 
nB  rpm   B  rads / sec  
 min   2  rads 
Accordingly,
nB rpm  nA rpm
rA
rB
where nA  speed of rotation of shaft A, rpm
nB  speed of rotation of shaft B, rpm
To relate the linear speed of the belt to the rotational speed  of a pulley, recall the
formulation of ν in terms of 
νb = r 
where vb  belt speed
r  radius of pulley
  angular speed of pulley, rads/sec
The expression for belt speed in terms of the rotation rate of the pulley may be
customized for a particular set of units.
Dn
vb 
12
where vb  belt speed in ft/min
D  diameter of pulley, inches
n  speed of pulley in rpm
To identify the conversion factors involved in the customized expression for νb note the
following.
D
vb    
2
D  rev   2 radians   1 ft 
vb  n 

2  min  
rev
 12in 
D n  2  nD
vb 

2 12 
12
where D  pulley diameter in inches
vb  belt speed in ft/min
Combing constants yields the following result.
vb  .262Dn
A belt runs over a 30 in pulley which rotates at 150 rpm. What is the speed of the belt
in fpm and fps.
vb 
v ft / sec 
Belt drives are designed so that belt speeds are typically around 4000 fpm.
In a belt drive system, the pulley diameters are 8 in and 20 in. The small pulley turns at
800 rpm. How fast does the large pulley rotate?
n20 =
What is the belt speed in the pulley system above?
νb =
Assuming the smaller pulley is the driving pulley, what is the speed ratio?
mw =
To investigate how a belt drive system effects torque, the assumption will be made that
the belt drive transmits power between the pulleys of a belt drive system without a loss.
Consider the system shown below.
PA  PB
PA  TAA  TBB  PB
Recall
 A rA  B rB  B   A
rA
rB
Substituting for 𝜔B yields the following
expression for relating TA to TB
TA A  TB A
TA
rA
rB
rB
 TB
rA
Compare the expression for TB with the expression for
TB  TA
B
rB
rA
B   A
rA
rB
Notice whatever change is affixed to B by the ratio of rA/rB, the opposite relationship is
imparted to TB. In this manner PB can be made equal to PA but PB can have a different
combination of torque and speed than what is associated is PA.
Recall P=FV
 N  m   ft  lbs 
P  Power
 s   s 
F  ForceN , lb
 m ft 
V  LinearVelocity ,

 s sec 
The units on power may be customized by noting the following
HP 
P
FV

550 550
where 550 is conversion factor from (ft-lbs)/sec to HP
If V is given in ft/min, another conversion factor is needed to obtain Horse Power.
 ft  1 min 
F lb vb 


min  60sec 

where
HP 
550
Fvb
Fvb
HP 

55060 33,000
F=tension force existing in belt
νb=belt velocity ft/min