LECTURE #27 CONVEYER
SYSTEMS
Course Name : DESIGN OF MACHINE ELEMENTS
Course Number: MET 214
Package handling requirements established from supply chain management strategies dictate transfer
rates of packages.
Package specifications (weight, volume, etc) and transfer rate requirements influence the conveyer
system configuration selected for a particular application.
Transfer rates, package specifications and conveyor system configuration influences drive requirements
for the conveyer system.
Conveyer configuration influences how drive requirements for the conveyor system are related to design
requirements for sub system components.
Various types of conveyor systems are available
To determine the drive requirements for a conveyor system, torque requirements
associated with conveyor motion must be reflected back to the drive shaft of the
motor.
Information provided in previous lectures identified how to reflect mass moment of
inertias between shafts of pulley systems and will be utilized to assist in the
development of the drive requirements for a conveyor system. In addition to reflecting
mass moment of inertia and/or torque loads as illustrated in the previous lecture,
conveyor systems transport packages having specified weights. The torque
requirements for a conveyor system are influenced by the movement of the packages.
To identify how to relate motion of the packages to torque requirements for the drive
of a conveyor system, consider the system shown below.
The total amount of mass being transported by the conveyor system involves the mass of the
individual packages in addition to the mass of the belt.
mT = m1 +m2 + m3 + m4 + m belt
Using Newton’s 2nd law we can relate mT to the force necessary to accelerate mT.
F=mT a
where
F =Force applied to conveyor belt in order to accelerate
belt and packages on belt
a = acceleration of conveyor belt
The linear acceleration a may be belt expressed in terms of angular measures as follows.
F  mT r11
where
r1 = radius of drive pulley of conveyor system
α1 = angular acceleration of drive pulley rads/sec2
The force F necessary to accelerate mT can be related to torque about the drive shaft as follows
TMT  r1F  r1mT r11  mT r121
Recall Newton’s 2nd law for angular measures
TJ
Accordingly, the total mass mT being accelerated by an acceleration of a may be expressed in terms
of an equivalent mass moment of inertia as shown below
T  mT r121
Tm  J
 J mT  mT r12
2
The total mass mT multiply by r1 establishes an equivalent mass moment of inertia which is to be
included in the analysis of the drive requirements for a conveyor system.
Prior to identifying how all of the torque requirement of a conveyor system are reflected to the shaft
of the motor, let us consider how friction effects due to linear motion of the belt impact the torque
requirements for a conveyor system.
To relate how friction accompanying linear motion of the belt impacts the torque requirements of a
conveyor system, consider the conveyor system illustrated below.
Belt motion creates a friction force between the conveyor belt and the support
platform which supports the conveyor belt. The magnitude of the friction force is
related to the normal force and the coefficient of friction existing between the belt and
the platform. The relationship is presented below.
where Ff = Force due to friction between conveyor belt and support
platform
u = coefficient of friction between belt and platform
N = Normal force acting to press belt against support platform
To relate the friction force Ff to a torque requirement for the motor of the conveyor
system, note that Ff can be related to a torque acting about shaft #1.
Ff  uN
Tf  r1Ff  r1uN  r1umT g
After accounting for rotational acceleration effects, the total torque required from a
motor for a conveyor system can be determined.
To identify the torque required for a motor, all motion effects relating to torque must
be transformed to the motor shaft.
To systematically transform all motion effects relating to torque to motor shaft, proceed
as follows:
1) Determine mass moment of inertia of all components undergoing rotational motion
and reflect to drive shaft of conveyor system using scale factors consistent with
reflected impedances.
2) Determine resisting torques accompanying rotational motion of rotating
components and reflect the resisting torques to drive shaft of conveyor system using
scaling factors consistent with reflected torques.
3) Transform all masses experiencing linear motion to equivalent mass moment of
inertias and reflect equivalent mass moment of inertias to drive shaft of conveyor
system.
4) Transform all forces associated with linear motion of conveyor system to equivalent
torques and reflect to drive shaft of conveyor system.
5) Combine the torques associated with steps 1-4.
6) Reflect torque in steps 1-5 to shaft of motor if motor is not directly connected to
conveyor system.
As an example of how to determine the drive requirements for a conveyor system, consider the
system shown below
A)
For angular motion:
1) Determine mass moment of inertia of rotating components and reflect to drive shaft of
conveyor system using scaling factors consisted with reflected impedances.
Jm2 = mass moment of inertia about shaft #2 includes mass moment of inertia of shaft #2, and
pulley #2 and any component rotating about shaft #2 . (from lecture #14) Shaft #1 is destination.
reflect Jm2 to shaft #1
2
r 
J 2 R1  J m 2  1 
 r2 
2
 r1 
r1  r2      1
 r2 
J 2 R1  J m 2
B)
C)
Jm1 = mass moment of inertia of about shaft #1: includes mass moment of inertia of shaft #1, and
pulley #1, and any component rotating about shaft #1.
J mT1  J m1  J 2 R1  J m1  J m 2
2) Determine resisting torques accompanying rotation of rotating components and reflect back to
drive shaft of conveyor system using scaling factors consisted with torques.
A)
Assume rotation of shaft 2 is accompanied by a viscous damping torque due to bearing behavior
that depends on speed of conveyor.
T2   2 n2
T2 R1  T2
n2
n
D
  2 n2  2   2 n2 1   2 n2
n1
n1
D2
T2 R1  T2   2 n2   2 n1
B)
Assume rotation of shaft #1 is accompanied by a viscous damping torque due to bearing behavior
that depends on speed of conveyor.
T1  1n1
C)
Combine the torque loads acting on shaft #1 with all torque loads reflected to shaft #1
TT1  T1  T2 R1
TT1  1n1   2 n1
TT1  n1 1   2 
D)
Alternatively, the torque loading on shaft #1 and/or shaft #2 could be independent of speed, and
modeled simply as a drag torque.
T1  Td 1
T2  Td 2
E)
where Td 1 =drag torque existing on shaft #1
Td 2 =drag torque existing on shaft #2
Reflecting T2 to shaft #1 and combining the two torques results in the following expression:
TT 1d  Td 1  Td 2
F)
Combination of torque types may exist depending on circumstances
3) Transform all masses experiencing linear motion to equivalent mass moment of inertias and
reflect equivalent mass moment of inertias to drive shaft of conveyor system:
Utilizing relationships presented earlier:
J eq1  mT r12
mT  m1  m2  m3  m4  mbelt
4) Transform all forces associated with linear motion of conveyor system to equivalent torques and
reflect to drive shaft of conveyor system.
Utilizing relationship presented earlier:
Tf  r1Ff  r1umT g
5) Combine the torques associated with steps 1-4


TS 1  J mT 1  J eq1 1  TT1  T f
TS 1  J m1  J m 2  mT r12 1  Td 1  Td 2   r1umT g


or if viscous dragging effects


TS1  J m1  J m 2  mT r12 1  1   2 n1  r1umT g
6) Transform torque in step 5 to shaft of motor. Recall basic relation governing transformation of
torques in pulley/ chain drive systems.
Tm nm  T1n1
n
Tm  T1 1
nm
Note:
nm Dm  n1 D1  
where Tm  Torque supplied by motor
Dm  Pitch diameter of sprocket attached to motor
n1 Dm

nm D1
Accordingly:
Tm  TS 1
Dm
D1
Applying the scalar factor above to the equations for T1 results in the following torque requirements
for the motor

TM  J m1  J m 2  mT r12
 DD
m
1  Td 1  Td 2 
1
Dm
D
 r1umT g m
D1
D1
or if viscous effects are considered

TM  J m1  J m 2  mT r12
 DD
m
1  1   2 n1
1
Dm
D
 r1umT g m
D1
D1
Substituting
1   m
Dm
D1
n1  nm
Dm
D1

TM   J m1  J m 2  mT r12


2

D
D
 m  Td 1  Td 2  m  umT g m
D1
2

2

D
 m  1   2  m

 D1
 Dm 


D
 1

Or if viscous effects are considered
TM

  J m1  J m 2  mT r12


TM  AS 2  BS  C
 Dm

 D1




2

D
 nm  umT g m
2

Note:
J eqs

  J m1 J m 2 mT r12


 Dm 


 D1 

2



If αm = 0 then
D 
D
D
D
Tm  Td 1  Td 2  m   umT g m  Td 1  Td 2  m  umT g m
2
D1
2
 D1 
Or if viscous effects are considered
2
D 
D
Tm   1   2  m  nm  umT g m
2
 D1 