LECTURE 26- MASS MOMENT OF INERTIA OF PULLEY SYSTEMS

Course Name : DESIGN OF MACHINE ELEMENTSCourse Number: MET 214

Moment of Inertia combinations

Power transmission systems often include a variety of power transmission components mounted to multiple drive shafts. Example system is shown below

To assist in the determination of the acceleration, torque, power, time to stop, time to start and/or speed performance associated with the operation of a power transmission system including conveyor systems, the mass moment of inertia of the entire system needs to be identified.The mass moment of inertia of a pulley can be determine from adjustments to the mass moment of a short cylinder. Envision the formation of a pulley by removing the center portion from a solid cylinder.

where Mass moment of inertia of pulley

Mass moment of inertia of cylinder #1 (or just mass moment)

Mass moment of inertia of cylinder #2

Where t is thickness of cylinders Area moment of inertia of cylinder #1 (or just mass moment)

Area moment of inertia of cylinder #2

12 mCmCmp JJJ

)(12 CCmP JJtJ

mPJ

1mCJ

2mCJ

1C

J

2C

J

Alternatively, the mass moment of inertia Jm can be expressed in terms of the radius of gyration of mass km .

Recall Torque is related to angular acceleration by the following relationship

For constant α

For customized units:

wherefinal speed after accelerating, rpminitial speed prior to accelerating ,rpmchange in time during which α is experienced

22

2

mmmm

mm

mm

Kg

wJmKJ

m

JK

m

JK

mJT

tJT m

t

nJT m

105.

if nnn

t

n

n

i

f

Alternatively,

It will be assumed that the Jm values for pulleys, including the effects of thru hole and/or a V groove are available from the suppliers of pulleys.Mass moment if inertia for solid shafts have already been identified.

where mass moment of inertia of solid shaft mass of solid shaft weight of solid shaft radius of shaft

22 )(2

1

2

1R

g

WRMJ TMS

MSJ

TMWR

t

nwKT

t

ngwK

T

t

nK

g

wT

m

m

m

308

105.

105.

2

2

2

To determine the mass moment of inertia of a combination involving a pulley mounted on a shaft, simply add the mass moment of inertias of the components.

where

mass moment of inertia of shaft mass moment of inertia of pulley

total mass moment of inertia of shaft and pulley

mSmPmT JJJ

mPJ

mSJ

mTJ

Combinations involving two or more shafts with power transfer between shafts using pulleys.When two shafts are interconnected by a pulley arrangement, the mass moment of inertia of the two pulleys and the two shafts can be combined to form a single equivalent mass moment of inertia to simplify calculations involving the torque required from a motor to achieve a desired acceleration, time to stop, time to start, etc.

Reflected Impedances:A technique for combining mass moments of inertias and other mechanical aspects effecting angular acceleration to form a single, equivalent alternative representation useful in simplifying calculations involving angular accelerations, torque, etc.Consider the single pulley labeled as pulley #1 connected to a shaft labeled as shaft #1. The motor supplies a torque to shaft #1

The amount of work associated with rotating pulley #1 and shaft #1 due to an external torque T1 applied to the shaft by the motor can be determined as follows.

where Torque applied to shaft #1 by motor to rotate pulley #1 and shaft #1 Angular rotation of pulley #1 due to T1

Total mass moment of inertia of shaft #1 and pulley #1

Angular acceleration associated with motion of pulley #1

11111 ][1

mTJTW

1T1

1mTJ1

Assume a pulley system is formed by using a belt to link pulley #1 to a second pulley labeled as pulley #2 mounted to a second shaft labeled as shaft #2.

Note:

where Total work performed by motor to rotate shaft #1 and pulley #1 and shaft #2 and pulley #2 work performed to rotate pulley #1 and shaft #1

work performed by pulley #1 to rotate shaft #2 and pulley #2

1/211WWWT

1T

W

1W

1/2W

Where Torque required to rotate pulley #2 and shaft #2

Angular rotation of pulley #2 due to T2

Angular acceleration associated with motion of pulley #2

Mass moment of inertia of pulley #2 and shaft #2

Recall the following relationships exist for belt drives

Substituting for θ2 and α2 in terms of θ1 and α1, the second term can be rewritten as follows:

Using the above enables the expression for WT1 to be reconfigured in terms of α1θ1 as shown below

22111 TTWT

2211 ][][211

mTmTT JJW

2T2

2

2mTJ

12

122211 r

rrr

12

122211 r

rrr

12

122211 r

rrr

11

2

2

1 ][22111

r

rJJW TmTmT

11

2

2

12

12

11

2

12222

r

rJ

r

r

r

rJJ

mT

mTmT

The reconfigured expression for represents the total amount of work to be performed by the motor to rotate pulley #1 and shaft #1 and pulley #2 and shaft #2 with the requirements for pulley #2 and shaft #2 expressed in terms associated with pulley #1. can be expressed as the product of the total amount of torque that must be applied to pulley #1 to rotate both pulleys.

total amount of torque applied to shaft #1 so pulley #1 and shaft #1 and pulley #2 and shaft #2can rotate with their respective requirements.

Equating the above expressions for WT1 and cancelling the θ1 terms enables the following expression to be derived for the total torque necessary to perform WT1.

where

Given a two pulley system with the pulley shaft combinations having mass moment of inertias and

respectively, the amount of torque TT1 that must be applied to shaft #1 so pulley #1 and shaft #1 will accelerate with α1 and pulley #2 and shaft #2 will accelerate with can be determined from the product of Jeq with α1.

The term involving is referred to as the reflected impedance. In this example, the reflected

impedance is given a special designation

The reflected impedance represents the resistance to the acceleration of pulley #1 due to pulley #2 with the effects of pulley #2 being described in terms of pulley #1.

111TT TW

1TT

1

2

2

1 ][211

r

rJJT mTmTT

11

TeqT JT ][2

2

121

r

rJJJ mTmTeq

1mTJ

2mTJ

2

2

12

r

rJmT

2

112 r

r

2

2

1212

r

rJJ mTR

The reflected impedance consists of two terms. The JmT2 represents the resistance (impedance) to motion due to mass moment of inertia effects. The term (r1/r2)2 represents how the impedance due to mass moment JMT2 must be adjusted so its overall effect can be consolidated with shaft #1 and pulley #1When determining reflected impedances in pulley systems, so the overall effects can be combined into the context of a single perspective, the following statement is helpful for generalizing the procedure for determining reflected impedances.

Reflected Impedances:Rotational mechanical impedances can be reflected (transferred) through a pulley system by multiplying the mechanical impedance by the following scale factor.

reflected impedance scale factor = [radius of destination pulley]2

[radius of source pulley]2

In the example analyzed previously, the term involving JmT2 has been modified by the impedance scaling factor (r1/r2)2

Where the reflected impedance of JMT2

radius of destination pulleyradius of source pulley

2

2

122 1

r

rJJ MTR

12RJ

1r2r

Pulley system example : Air compressor, wood working and metal working tools, rudiment perspective for conveyor systems.

Typical arrangement involving a belt drive.

Total torque required from motor drive can be resolved into components to facilitate establishing drive requirements for system

WhereTotal torque applied to shaft #1

Equivalent mass moment of inertia of system about shaft #1

Angular acceleration of shaft #1

Torque load on shaft #2 due to load force FN

Radius of pulley #1

Radius of pulley #2

2

11 21 r

rTJT LmeqS

1TT

meqJ

1

2L

T

1r

2r

Angular effects reflected through

system

Linear effect due to load forces propagated

through system

Total torque TT1 is determined by resolving TT1 into two components

Components due to angular effects propagated to drive shaft #1

Component due to load forces propagated through system.

As shown in example below each component is determined separately, and the effects combined to determine the total torque the motor must supply to the system.

1meqJ

2

12 r

rTL

Angular effects propagated through system

Note: Effects of FN are considered with linear effects and is not considered as part of angular effects.

Where Total mass moment of inertia about shaft #2

Mass moment of inertia of pulley #2

Mass moment of inertia for grinding wheel

Mass moment of inertia of shaft #2

MGmSmPmT JJJJ 222

2mTJ

2mPJ

2mSJ

mGJ

Where Total mass moment of inertia for shaft #1

Mass moment of inertia for pulley #1

Mass moment of inertia for shaft #1

Where

Torque to be applied to shaft #1 by motor to acceleration equivalent inertia

Equivalent inertia of system about shaft# 1

Angular acceleration of shaft #1

111mSmPmT JJJ

1mTJ

1mPJ

1mSJ

111eqS JT

2

2

1211

r

rJJJ mTmTeq

1

2

2

1 ][211

r

rJJT mTmTS

1S

T

1eq

J

1

Linear effects due to load forces propagated through systemConsider a load force applied to grinding wheel.

Where Normal force applied to part being ground

Tangential force due to grinding operation

Co-efficient of dynamic friction of grinding wheel

NNT FF

NF

TF

N

Where

Torque about axis #2 due to grinding operation

Radius of grinding wheel

Where

Torque about axis #1 due to grinding operation on axis #2

Radius of pulley #1

Radius of pulley #2

Where

Total torque required from motor to perform grinding operation.

TGL FrT 2

2L

T

Gr

2

121 r

rTT LL

1L

T

1r

2r

111 LST TTT

1TT

Note:

Comments about andIf speed of grinding wheel is constant during grinding operation, then

Where

Power required from motor to operate grinder

Speed of axis if during grinding operations presumed steady

Speed of axis #2 during grinding operations presumed steady

2

11

2

2

12211

][r

rT

r

rJJT LmTmTT

12

121

r

rrr

1mTJ 2mTJ

001

2212 r

r

2

1211 r

rTTT LLT

21 211 LTT TTP

1TP

1

2

Example: Consider the configuration shown below

Find time to accelerate shaft #1 from n1=0 to n2=550 rpm

is attached to shaft 2 and is not shown in the figure.

2

22

22

21

21

sec474.2sec218.sec00726.sec0038.sec0068.

24

lbftJlbftJlbftJlbftJlbftJ

lbsftT

mload

mP

mS

mP

mS

m

mloadJ