LECTURE #27 CONVEYER SYSTEMS

Course Name : DESIGN OF MACHINE ELEMENTSCourse 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.

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

Recall Newton’s 2nd law for angular measures

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

11rmF T

1211111 rmrmrFrT TTMT

JT

The total mass mT multiply by 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.

21

121

rmJJTrmT

Tm

m

T

T

21r

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.

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:

uNFf

gumruNrFrT Tff 111

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

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.

A) 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

22

2

2

121

2

2

122

1

1

1

mR

mR

JJr

rrr

r

rJJ

B) 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.

C)

2) Determine resisting torques accompanying rotation of rotating components and reflect back to drive shaft of conveyor system using scaling factors consisted with torques.

C) Assume rotation of shaft 2 is accompanied by a viscous damping torque due to bearing behavior that depends on speed of conveyor.

B) Assume rotation of shaft #1 is accompanied by a viscous damping torque due to bearing behavior that depends on speed of conveyor.

2121 11 mmRmmT JJJJJ

122222

222

122

1

222

1

222

222

1

1

nnTT

nD

Dn

n

nn

n

nTT

nT

R

R

111 nT

C) Combine the torque loads acting on shaft #1 with all torque loads reflected to shaft #1

D) Alternatively, the torque loading on shaft #1 and/or shaft #2 could be independent of speed, and modeled simply as a drag torque.

where =drag torque existing on shaft #1=drag torque existing on shaft #2

E) Reflecting T2 to shaft #1 and combining the two torques results in the following expression:

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:

211

1211

21

1

1

11

nTnnT

TTT

T

T

RT

22

11

d

d

TT

TT

1dT

2dT

211 ddT TTTd

beltT

Teq

mmmmmmrmJ

4321

211

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:

5) Combine the torques associated with steps 1-4

or if viscous dragging effects

6) Transform torque in step 5 to shaft of motor. Recall basic relation governing transformation of torques in pulley/ chain drive systems.

where Torque supplied by motorPitch diameter of sprocket attached to motor

Note:

gumrFrT Tff 11

gumrTTrmJJT

TTJJT

TddTmmS

fTmS eqT

121121211

11 111

gumrnrmJJT TTmmS 1121121211

1

111

11

11

D

D

n

nDnDn

n

nTT

nTnT

m

mmm

mm

mm

m

m

D

T

Accordingly:

Applying the scalar factor above to the equations for T1 results in the following torque requirements for the motor

or if viscous effects are considered

Substituting

Or if viscous effects are considered

11 D

DTT mSm

1

11

2111

2121 D

Dgumr

D

DTT

D

DrmJJT m

Tm

ddm

TmmM

1

11

12111

2121 D

Dgumr

D

Dn

D

DrmJJT m

Tmm

TmmM

11

11

D

Dnn

D

D

mm

mm

21

21

2

1

2121

mT

mddm

mTmmM

Dgum

D

DTT

D

DrmJJT

2

2

121

2

1

2121

mTm

mm

mTmmM

Dgumn

D

D

D

DrmJJT

CBSASTM 2

Note:

If αm = 0 then

Or if viscous effects are considered

2

1

2121 D

DrmJJJ m

Tmmeqs

22 1

211

21m

Tm

ddm

Tm

ddm

Dgum

D

DTT

Dgum

D

DTTT

2

2

121

mTm

mm

Dgumn

D

DT