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Theses
Thesis/Dissertation Collections
1985
Balancing of the Shaking Force, Shaking Moment,
Input Torque and Bearing Forces in Planar Four Bar
Linkages
Timothy C. Hewitt
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Recommended Citation
AND BEARING FORCES IN PLANAR FOUR BAR LINKAGES
Approved by
USING
AUTOMATED OPTIMAL DESIGN
by
Timothy C. Hewitt
A Thesis Submitted
in
Partial Fulfillment
of the
Requirements for the Degree of
MASTER OF SCIENCE
in
Mechanical Engineering
Prof.
~~(=T"""'hes'isA""Q.=v""i',s""or)
Prof.
Prof.
Prof.
T(EB=e~p~a~I;t;m~e~n~t~H~e~a~dT)DEPARTMENT OF MECHANICAL ENGINEERING
ROCHESTER INSTITUTE OF TECHNOLOGY
ROCHESTER, NEW YORK
To : Whom it may Concern
Batavia, Ohio
45103
January 31, 1984
Regarding : Reproduction of thesis entitled" Balancing of
the Shaking Force, Shaking Moment, Input Torque,
and Bearing Forces in Planar Four Bar Linkages "
I, Timothy C.
Hewitt, do hereby grant
permission
to
the
Wallace
Memorial
Library to reproduce my thesis in whole, or
in part.
It is my understanding that
any
reproduction
will
not be for commercial use or profit.
Signed,
The
author wouldlike
to
take
this
_{opportunity}
to
thank
all
those
whohave
assistedhim
throughout
the
development
ofthis
thesis
andin
particular:Dr.
W.
F.
Halbleib,
the
author'sthesis
advisor,
for
his
patience,
advice anddirection
throughout
the
course ofthis
thesis.
In
_{addition,}
for
_{instilling}
the
importance
offundamental
ideas
in
the
author.Dr.
R.
C.
Johnson,
member ofthe
author'sthesis
committee,
whose effortsin
the
field
of automated optimaldesign
have
made a substantial portion ofthis
thesis
possible.
Dr.
P
_{Venkataraman,}
member ofthe
author'sthesis
committee,
for
his
insight
and suggestions_{concerning}
this
thesis.
Mr.
L.
Dearstyne
ofthe
Eastman
Kodak
_{Company,}
Rochester,
N.Y.
for
his
insight
and suggestions_{concerning}
the
initial
development
ofthe
problem.The
Eastman
Kodak
Company,
Rochester,
N.Y.
for
Technology,
_{Rochester,}
N.Y.
andthe
Aircraft
Engine
Business
Group
ofthe
General
Electric
_{Company,}
Evendale,
O.H.
for
useof
their
VAX/VMS
11/782
digital
computers.To
the
author's
wife and_{family,}
for
their
continuedThe
_{theory,}
development
and application of a computerprogram
to
balance
the
combined effects ofthe
_{shaking}
force,
shaking
moment,
input
torque
andindividual
_{bearing}
forces
in
four
bar
linkages
is
presented.The
_{theory}
assumesthe
linkage
to
consist of rigid_{bodies,}
andis
limited
to
balancing
planarfour
bar
linkages
otherthan
sliders.Balancing
is
accomplished_{using}
circular counterweights whichare
_{tangentially}
attachedto
the
_{bearing}
joints.
Counterweight
sizes andlocations
aredetermined
_{using}
nonlinear
_{programming}
techniques
where an objective_{function,}
dependent
upon allthe
kinetic
parameters,
is
minimized.The
_{balancing}
programis
capable of_{performing}
diverse
functions.
The
number of added_{counterweights,}
type
anddegree
of numerical quadrature and regional constraints on allimportant
_{balancing}
parameters canbe
varied.In
_{addition,}
the
programis
capable of_{balancing}
linkages
with offline massdistributions,
andto
some_{extent,}
emphasis canbe
placed onindividual
terms
such asthe
input
torque.
A
majorlimitation
of
the
theory
is
the
assumption of rigidlinks.
This
_{may}
notalways
be
valid and makesthe
programinsensitive
to
naturalexcessiveand
application
ofthe
_{balancing}
program.The
first
exampleshows
the
effect
of_{varying}
the
degree
andtype
of numericalintegration
used.
The
Gaussian
quadrature methodis
shownto
be
most_{efficient,}
withthe
optimum number of_{sampling}
pointsdetermined
to
be
10.
In
example_{two,}
aninline
four
bar
linkage
_{operating}
at a constantinput
speed of5000
rpmis
balanced
sothat
allkinetic
quantities are reducedfrom
75%
to
92%
overthe
unbalanced case.Similar
results are shownin
example
three
for
_{balancing}
afour
bar
linkage
with an offlinecoupler mass
distribution.
The
effect of_{adding}
from
oneto
three
counterweightsis
also_{investigated,}
withthe
resultsindicating
that additional counterweightsdo
not alwaysimprove
the
_{balancing}
situation.With
just
one counterweightadded,
the
important
kinetic
terms
are reduced an average of87%,
whilethe
addition ofthree
counterweights_{only}
reducesthese
same quantities an additional5%.
Due
to
the
apparent success ofthe
developed
program,
the
author recommends
that
it
be
extendedto
sliders,
sixbars
andother practical
linkages.
In
addition,
the
_{validity}
ofthe
rigid
_{body}
assumption shouldbe
_{experimentally}
andPage
i
LIST
OF
FIGURES
i
i i
LIST
OF
TABLES
iii
i
vNOMENCLATURE
v1
.0
INTRODUCTION
1
2
.0
LITERATURE
SURVEY
4
3
.0
OVERVIEW
OF
THE
THEORY
10
4.0
DEVELOPMENT
OF
THE
KINEMATIC
AND
KINETIC
EQUATIONS
13
4.1
Two
Point
Mass
Model
13
4.2
Kinematic
Analysis
22
4.3
Newtonian
Approach
28
4.4
Lagrangian
Approach
43
4.5
Combined
Approach
52
5.0
DEVELOPMENT
OF
THE
OPTIMIZATION
PROBLEM
AND
GENERAL
BALANCING
PROGRAM
63
5.1
Objective
Function
Proposal
63
5.2
Statement
ofthe
Optimization
Problem
65
5.3
Summary
ofthe
AOD
Program
69
5.4
_{Summary}
ofthe
Analysis
andFinal
Subroutines
75
6
.0
PRESENTATION
OF
THE
EXAMPLE
PROBLEMS
80
6.1
Capabilities
ofthe
OPTBAL
Computer
Program
82
6.2
_{Balancing}
of anInline
Four
Bar
Linkage
93
6.3
_{Balancing}
of aFour
Bar
with anOffline
Coupler
Link
97
9.0
_{RECOMMENDATIONS}
_{117}
10.0
_{REFERENCES}
118
11.0
APPENDIX
A
PRINCIPLES
OF
ANGULAR
MOMENTUM
Al
12.0
APPENDIX
B
SOLUTION
OF
THE
TWO
POINT
MASS
MODEL
CONVERSION
BI
13
.0
APPENDIX
C
SUMMARY
OF
NUMERICAL
INTEGRATION
CI
14
.0
APPENDIX
D
Figure
Title
Page
1
General
Four
Bar
Linkage
withCircular
Counterweights
14
2
Two
Point
Mass
Model
14
3
Combined C.
G.
Location
of aGeneral
Link
with a
Circular
Counterweight
17
4
Two
Point
Mass
Model
of aCombined
C.G.
Location
17
5
Stick
Figure
of aFour
Bar
Linkage
23
6
Free
Body
Diagram
of aGeneral
Two
Point
Mass
Model
^97
Free
_{Body}
Diagram
of aTwo
Point
Mass
Model
Separated
from
its
Frame
53
8
Flowchart
ofthe
MODSER
Program
72
9
_{Relationship}
between
the
AOD
program andAnalysis,
Final,
andPgauss
Subroutines
76
10
Example
1
Inline
Four
Bar
Linkage
83
11
Example
1
Normalized
Rms
Input
Torque
Versus
the
Number
of_{Sampling}
Points
85
12
Example
1
Normalized
Rms
_{Shaking}
Force
Versus
the
Number
of_{Sampling}
Points
86
13
Example
1
Normalized
Rms
_{Shaking}
Moment
Versus
the
Number
of_{Sampling}
Points
87
14
Example
3
Four
Bar
Linkage
with anOffline
Coupler
98
15
Example
3
Normalized
Force
in
Joint
01
Versus
Crank
Angle
101
16
Example
3
Normalized
Force
in
Joint
02
Versus
Crank
Angle
102
17
Example
3
Normalized
Force
in
Joint
03
Versus
Crank
Angle
103
18
Example
3
Normalized
Force
in
Joint
04
Versus
Crank
Angle
104
19
Example
3
Normalized
_{Shaking}
Force
Versus
Crank
Angle
105
20
Example
3
Normalized
Input
Torque
Versus
Crank
Angle
106
21
Example
3
Normalized
_{Shaking}
Moment
(01)
Versus
Crank
Angle
107
22
Example
3
Normalized
_{Shaking}
Moment
(03)
Versus
Crank
Angle
108
Al
General
System
of nPoint
Masses
_{Rigidly}
Fixed
Together
A2
Table
_{Title}
_{Page}
1
Example
1
Parameters
ofthe
Unbalanced
Inline
Four
Bar
Linkage
83
2
Example
1
Influence
of_{Weighting}
Factors
onCounterweight
Parameters
89
3
Example
1
Influence
of_{Weighting}
Factors
onthe
_{Shaking}
_{Force,}
_{Shaking}
Moment
andInput
Torque
89
4
Example
1
Influence
ofthe
Torque
Regional
Constraint
onCounterweight
Parameters
91
5
Example
1
Influence
ofthe
Torque
Regional
Constraint
onthe
_{Shaking Force,}
_{Shaking}
Moment
andInput
Torque
91
6
Example
2
Counterweight
Parameters
94
7
Example
2
Comparison
ofNormalized
Rms
Quantities
95
8
Example
2
Comparison
ofNormalized
Maximum
Quantities
95
9
Example
3
Parameters
ofthe
Unbalanced
Four
Bar
Linkage
with anOffline
Coupler
Link
98
10
Example
3
Counterweight
Parameters
_{99}
11
Example
3
Comparison
ofNormalized
Rms
Quantities
99
Al
_{Definitions}
ofAngular
Momentum
A5
CI
Abscissas
and_{Weighting}
Factors
for
Gaussian
Integration
C6
Scalars
C0R
Cp
Ex
Fij
FSH
FSHRMS
(F
)max
SHRMS
Fx,Fy
R
BRMS
(F
BRMS
i
)max
Angular
momentum about point0
Resultant
couple about point0
Penalty
function
coefficient
Kinetic
_{energy}
Bearing
joint
force
between
joints
i
and_{j}
Shaking
force
Rms
F
sh
Maximum
allowableF
SHRMS
X
andY
components ofthe
externalforce
appliedto
the
follower
link
Resultant
force

_{Maximum}
bearing
force
Rms
value ofF

_{Maximum}
allowable
F
BRMS
Mass
moment ofinertia
ofthe
ith
link
about point_{j}

_{Length}
of
link
i

Length
from
joint
0'to
the
point ofexternal
load
_{application,}
point "a"
Length
_{to}
the
_{centroid} ofthe
ith
real
link
from
joint
_{0.}
(lb
in
M
SH/O
1
counterweight
from
joint
_{Oi}
(in)
liT
Length
to
the
total,
or combinedcentroid
for
the
ith
link
from
joint
_{Oi}
(in)
lil
'li2
~
Length
to
the
masses ofthe
two
point mass model
1
,1
Body
coordinates_{r,Q,z'',}
for
the
ilT
i2T
combined centroid
location
(in)
Shaking
moment aboutjoint
0
_{i}
(lb
in)
H;h/o
rms
"
Rms
value fMSH/0i
(lb
in)
i
(M
,)max
Maximum
allowableM
,(lb
in)
SH/OlRMS
SH/OiRMS
m .
Mass
ofthe
ith
reallink
(lb
s2/in)
m
_{ic}
~Mass
ofthe
ith
counterweight(lb
s
/in)
m
_{ix}
~Mass
ofthe
ith
combined centroid(lb
s2/in)
2
m
_{ll'mi2}
~Masses
ofthe
two
point mass model(lb
s
/in)
Ne
Number
of_{equality}
constraintsfor
optimizationNr
Number
of regional constraintsfor
optimizationNv
Number
ofdesign
variablesfor
optimizationp
Penalty
function
Q
Constrained
objectivefunction
Qu
Unconstrained
penalty
function
=Q
+_{P}
r ,W

_{Gaussian}
quadrature abscissas and
weighting
factors
T
Generalized
externalforce
(lb
in)
91
DRMS
(TDRMS
)max
TL
XiT
,YiTx ,y
IT
JiT
wlfw2
OL:
e
e
e
iC
IT
6W
60
Rms
value ofT
D
(lb
in)
Maximum
allowable_{T^rms}
(lb
~*n)
Loading
torque
(lb
in)
Distance
from
the
fixed
coordinatesystem origin
to
the
combined centroidfor
the
ith
link
(in)
Distance
from
the
_{translating}
coordinatesystem origin
to
the
combined centroidfor
the
ith
link
(in)
Weighting
factors
for
the
objectivefunction
Angle
between
the
r_{body}
coordinateand
the
ith
reallink
centroid(rad)
Angle
between
the
x_{translating}
coordinate and
the
r_{body}
coordinatefor
the
ith
link
(rad)
Angle
between
the
r_{body}
coordinateand
the
ith
counterweight centroid_{(rad)}
Angle
between
the
x_{translating}
coordinate and
the
combined centroidlocation
for
the
ith
link
_{(rad)}
virtual work as a result of
<?9,
(lb
in
)
virtual
displacement
ofthe
crank_{(rad)}
*
*
*
*
"o
_
n0f JJ1C'1^U11T'
"'12
2*2
Ao
=_{A}
_{/}
_{L^e, (mn+}
m19
)
CoR
=C0r/
^(mll
+m12
>
Dl
=Li/Ll
=
1/L1
da
=VL1
dil'di2
='"il^
Ll'
anci^.2
^
L
1
respectively
*
2*2
ET
=_{ET/}
_{L1e1(m11+}m12)
2,
F..
=_{F../}
_{L1e1(m11+}m12)
2
FSH
=_{FSE/}
_{Ll9l(mH+}m12>
*
_{i}
2
I.
=_{Ii/L1(m11+m12)}
* * ....
i
,i_
=
_{2/}
81
and3/
_{6]_}
respectively
MSH/0.
=MSH/0./
Lll(mll+ m12>* *
M
..,M
.2
= _{m11/(m11+m12)} _{and}
m2
/
(m
11
+ m12)P
,,P.
=
_{M.,d}
., and
M.9
d.9
respectively
il
i2
ii ii i^1*2
2TD
=TD
/
L^dn^
m12).2 i2
Te,
=
Te/
Liei(mn+ mi2)2
i ' ~
i
i
=e,
/
e
e ,e ,e
unit vectors
for
the
_{body}
coordinates[E]
Equality
constraints matrix[F]
Matrix
offinal
items
to
be
printed outi',j',k'
unit vectors
for
the
fixed
coordinatesi,2,k
unit vectors
for
the
_{translating}
coordinates[R]
Regional
constraint matrix[V]
Scaled
design
variablesfor
optimization1.0
INTRODUCTION
Unbalanced
linkages
that
operate athigh
speed,
containmassive
links
or experiencelarge
externalloads
_{may}
be
subjected
to
excessive
_{vibration,}
noise and wear.Prediction
of
the
amount ofimbalance
is
based
uponthe
magnitude ofthe
shaking
force,
shaking
moment,
input
torque
andindividual
bearing
forces.
Reduction
of some or all ofthese
kinetic
terms
is
desirable
in
_{balancing}
the
linkage.
There
are_{many}
different
methods of_{balancing}
_{linkages,}
but
mosthave
_{only}
been
appliedto
the
slider crank .Effective
methodsfor
balancing
linkages
otherthan
_{sliders,}
have
_{only}
been
developed
overthe
last
fifteen
years.In
_{1969,}
Berkoff
andLowen
_{[4]}
published a simpletechnique
for
_{completely}
eliminating
the
_{shaking}
force.
But
the
methodignored
the
otherkinetic
effects,
and oftenincreased
the
_{shaking}
moment andinput
torque.
Berkoff,
in
1973
_{[5],}
alsodeveloped
a methodthat
_{completely}
eliminatedboth
the
_{shaking}
force
and_{shaking}
momentin
four
bar
linkages
with
inline
massdistributions.
The
technique,
however,
required
the
addition oflarge
gearedinertia
counterweightswhich ended
_{up}
doubling
the
mass ofthe
linkage,
and most ofthe
kinetic
terms
too.
Reduction ofthe
input
torque
has
alsobeen
accomplishedby
a number of methods.Most
notably,
the
use of a
flywheel,
which absorbs andtransmits
_{energy}
to
methods
gavelittle
considerationto
the
_{remaining}
kinetic
quantities.
_{Balancing}
ofthe
combined effectsis
the
nextlogical
_{step,}
but
the
equationsinvolved
are of sufficientcomplexity
that
all practical considerations arelimited
to
computer
_{applications.}
_{In}
_{1975,}
_{Sadler}
_{[33]}
and others usednonlinear
_{programming}
to
balance
these
combined effects.Their
results werein
general_{better,}
andfar
more practicalthan
_{existing}
closedform
techniques
since regionalconstraints could
be
usedto
limit
_{many}
kinetic
terms.
None
the
_{less,}
the
formulation
required separate optimizationtrials
for
each_{quantity}
to
be
minimized,
and a manualdevelopment
oftradeoff
curvesto
selectthe
best
overallcounterweight configuration.
Additional
schemeshave
been
presented since
_{then,}
but
mosthave
nothandled
the
problem asefficiently
and_{effectively}
asLee
and_{Cheng}
[21].
_{They}
choseto
minimize a single objectivefunction
expressedin
terms
ofthe
input
torque
andthe
ground_{bearing}
forces.
Reduction
ofthe
_{shaking}
force
and_{shaking}
momentis
also accomplishedsince
_{they}
aredependent
uponthe
_{bearing}
forces.
Lee
andCheng
alsodeveloped
anefficient,
closedform
solutionfor
the
kinetic
analysis,
and showedtheir
methodto
be
superiorto
severalexisting
techniques.
The
objective ofthis
thesis
is
to
balance
the
combinedkinetic
effects,
using
the
theory
presented_{by}
Lee
and_{Cheng,}
and
to
_{develop}
a general computer program whichincorporates
their
ideas.
The
analysis willbe
limited
to
_{balancing}
circular
_{counterweights.}
_{The}
computer programis
developed
from
an_{existing}
_{optimization}
_{program} _{written}by
R.C.
Johnson
[18],
andis
modified
to
handle
the
general_{balancing}
theory.Capabilities
ofthe
_{balancing}
program willbe
demonstrated
_{using}
three
numerical examples._{First,}
_{key}
parameters of
the
program willbe
determined
for
an_{accurate,}
but
efficientanalysis.
The
optimum number of_{sampling}
pointsfor
numerical_{quadrature,}
andthe
degree
to
which emphasis_{may}
be
placed onindividual
_{terms,}
such asthe
input
torque,
willbe
investigated.
Results
ofthe
first
example willthen
be
used
in
the
second example.This
example presents a practicalmethod of
_{balancing}
aninline
four
bar
linkage.
The
effect ofplacing
additional counterweights onthe
follower
and couplerlinks
is
alsoinvestigated.
In
the
third
example,
afour
bar
linkage
with an offline coupler massis
balanced.
The
mostpractical
_{balancing}
scheme ofthe
third
example willthen
be
compared
to
the
the
unbalanced_{linkage,}
the
method of completeforce
_{balancing}
andthe
method of completeforce
and moment2.0
LITERATURE
SURVEY
The
_{balancing}
offour
bar
linkages
has
been
a problemfor
both
the
designer
andthe
researcherfor
_{many}
years.This
_{may}
be
_{partially}
attributed
to
the
_{complexity}
ofthe
problem sincemany
different
quantities
mustbe
taken
into
account(ie.
shaking
force,
shaking
moment,
input
torque,
bearing
forces).
Various
experimental andtheoretical
methods existto
balance
some of
these
_{parameters,}
but
mosthave
_{only}
been
appliedto
the
slider crank_{family.}
Techniques
for
_{balancing}
linkages
other
than
sliders were not_{thoroughly}
summarized until1968.
In
that
yearLowen
andBerkoff
_{[22]}
published a_{survey}
ofinvestigations
_{concerning}
the
_{balancing}
offour
bar
linkages
with special emphasis on mechanisms other
than
sliders.The
complete work categorized major
techniques
for
_{balancing}
the
shaking
force
and_{shaking}
_{moment,}
listed
119
_{references,}
andtranslated
11
important
papersfrom
Russian
andGerman
into
English.
In
_{1977,}
Berkoff
supplementedthe
initial
_{survey}
_{by}
publishing
a_{summary}
of methodsfor
_{determining}
and_{balancing}
the
input
torque.
In
that
_{paper,}
71
references were cited andmajor
_{balancing}
techniques weredescribed.
With
suchcomprehensive
literature
surveysalready
published,
the
efforts of
this
thesis
will not_{try}
to
duplicate
either ofthe
above papers.
Rather,
the
majortechniques
discussed
in
Lowen
and
Berkoffs'
papers will
be
presented.In
_{addition,}
important
theoretical methodsfor
balancing
planar,
rigid_{body}
be
discussed.
The
method
of_{balancing}
_{the}
_{Shaking}
force
, orforce
balancing,
has
been
a popular methodfor
_{many}
years.Lowen
and
Berkoff
_{[22]}
summarized
_{many}
investigations
wherethe
shaking
force
was_{completely}
or_{partially}
eliminated.Complete
force
_{balancing}
attemptsto
makethe
center ofgravity
ofthe
linkage
stationary,
and canbe
categorizedinto
static
_{balancing}
_{methods,}
method of principal_{vectors,}
cammethods and
duplicate
mechanism methods.On
the
other_{hand,}
partial
_{balancing}
methods_{only}
reducethe
_{shaking}
_{force,}
andmay
be
accomplished_{using}
harmonic
_{balancing}
methods or_{by}
adding
springsto
the
linkage.
Harmonic
_{balancing}
methods usea
Fourier
series andGaussian
least
squares approachto
eliminate
lower
harmonics
ofthe
_{shaking}
_{force,}
while_{adding}
springs
to
the
linkage
altersthe
paththrough
whichthe
shaking
force
travels
to
ground.In
1969,
Berkoff
andLowen
[4]
published a new methodfor
_{completely}
force
_{balancing}
alinkage
termed
the
method oflinear
independent
vectors.This
method was
_{very}
easy
to
apply,
and_{only}
requiredthe
additionof
two
_{rotating}
counterweightsfixed
to
the
input
and outputlinks.
Lowen,
Trepper,
andBerkoff
[24]
in
1973
showedthe
quantitative effects of complete
force
balancing
a generalinline
four
bar
linkage
on otherdynamic
reactions such asthe
input
torque,
shaking
moment and_{bearing}
forces.
The
paperalso provided a complete
tutorial
onkinetic
analysissummarizing
final
equations and popular methods ofanalysis.
independent
vectors
_{by}
_{developing}
a method which_{automatically}
chose
the
minimum
number of counterweightsfor
full
force
balancing,
without
the
derivation
ofkinematic
equations.Many
others
have
extendedBerkoffs
initial
workto
practicalsix
bar
_{linkages;}
Balasubramanian
[3]
1978,
Berkoff
_{[6]}
_{1979,}
Dearstyne
_{[10]}
1983.
Although
the
method offorce
_{balancing}
significantly
reduces
the
netforce
onthe
_{frame,}
it
may
actually
increase
the
_{shaking}
moment.Lowen
and_{Berkoff,}
in
their
initial
survey,
described
techniques
for
_{completely}
and_{partially}
eliminating
the
shaking
moment.Methods
of complete_{shaking}
moment
_{balancing}
include
cam actuated_{oscillating}
counterweights,
andthe
addition of aduplicate
mechanism withmirror symmetry.
_{Also,}
harmonic
methods were popularfor
partially
eliminating
the
shaking
moment.Elimination
ofthe
first,
and sometimesthe
secondharmonic
ofthe
_{shaking}
momentwas accomplished with
two
_{rotary}
masses which weresynchronized
180
degrees
out of phase.In
_{1971,}
Lowen
andBerkoff
[23]
published a graphical method_{whereby}
the
_{shaking}
moment was minimized while
_{maintaining}
afull
force
balance.
Complete
force
and moment_{balancing}
ofinline
four
bars
wasgiven
_{by}
Berkoff
[4]
in
1973.
This
method,
however,
doubled
the
total
mass ofthe
mechanism, andsignificantly
increased
all other
dynamic
reactions.Wiederich
Roth
[43]
in
1976
determined
link
inertial
properties of a generalforce
balanced
four
bar
linkage
sothat
the
amount of angularminimized.
Bagci
[1]
in
1980
extended completeforce
balancing
to
offline
four
bars
and some sixbars
_{using}
balancing
idler
loops.
These
idler
loops
transferred
the
effects of
the
couplerlink
motionto
a ground_{bearing}
joint,
where
Lancaster
type
balancers
eliminatedthe
_{shaking}
moment.In
contrast
to
_{balancing}
the
_{shaking}
force
andmoment,
some
_{balancing}
methodshave
concentrated on_{reducing}
the
input
torque.
Although
it
can neverbe
eliminated_{totally,}
the
magnitude of
torque
fluctuations
canbe
_{significantly}
reduced.In
_{1977,}
Berkoff
_{[7]}
summarized available methods oftorque
balancing.
As
described
_{by}
_{Berkoff,}
_{flywheels,}
_{spring}
equivalents
to
_{flywheels,}
and cam subsystems are all capableof
_{smoothing}
outthe
torque
curve.Each
method stores anddischarges
_{energy}
to
reducetorque
fluctuations.
In
particular,
cam subsystems canbe
designed
to
flatten
outthe
torque
to
perfection.Other
methods summarized_{by}
Berkoff
include
using
torque
regional constraints_{during}
kinematic
synthesis
(
see[9]
for
example),
and rearrangement oflink
geometry
by
internal
mass redistribution.All
the
previous methodsdiscussed
have
focused
onthe
elimination of
one,
or at mosttwo
_{balancing}
quantities.However,
as shownin
various published example_{problems,}
balancing
ofindividual
parametersmay
have
an adverse effecton other
dynamic
quantities,
such as_{bearing}
forces.
_{During}
the
mid1970's
researchersbegan
_{attempting}
to
balance
the
combined
dynamic
effects ofthe
shaking
force,
shaking
moment,
effects
is
_{very}
difficult
to
achieve,
but
wasfacilitated
_{by}
the
use ofthe
digital
computer.In
1975
Oldham
_{[30],}
andSmith
_{[36],}
_{developed}
_{similar}_{interactive}
computer programswhich
provided
full
force
balancing,
and permittedthe
userto
manipulate
link
and counterweight propertiesto
reduce otherdynamic
effects.
_{Conte,}
_{George,}
_{Mayne,}
andSadler
in
1975
[9]
used computer aided optimal
design
to
generate mechanismsthrough
kinematic
synthesis while_{minimizing}
individual
balancing
quantities.
Sadler
_{[33]}
in
1975
also used nonlinearprogramming
to
determine
counterweightsizes
and angularlocationSj
while_{minimizing}
individual
_{balancing}
quantitiesthrough
a series of optimizations.Once
the
effects ofminimizing
individual
parameters were_{known,}
tradeoff
curveswere used
to
determine
which counterweight scheme resultedin
the
best
overallbalance.
Aside
from
digital
computerapplications,
Trepper
andLowen
[38]
in
1975
developed
equations
for
_{balancing}
the
shaking
force
withflexible
constraints on ground
_{bearing}
forces.
These
_{equations,}
however,
were ofthe
16th
order and required extensivecomputer work
to
be
solved.Elliot
andTesar
_{[11]}
in
1977
developed
proceduresfor
_{applying}
_{existing}
techniques
in
aseries of
trials
to
determine
which method providedthe
best
overall results.
These
trials
weretime
consuming
andsemiiterative
in
nature.Thus
the
problem of_{balancing}
the
combined effects seems
best
suitedfor
the
computer.
_{However,}
even with
the
aid ofthe
_{computer,}
all ofthe
_{previously}
mentioned procedures required some manual
_{iteration,}
and atThis
problem was_{partially}
solved_{by}
Tricamo
andLowen
_{[40]}
in
1983,
who usedautomated
optimaldesign
to
minimizethe
maximum
_{bearing}
_{force.}
_{The}
maximum_{bearing}
force
was notalways
limited
to
_{any}
specific_{bearing}
location,
but
variedwith
counterweight
arrangement._{So,}
asthe
search process ofdetermining
acounterweight
configuration was carried_{out,}
the
actual
location
ofthe
maximum_{bearing}
force
was changing.In
addition,
Tricamo
and Lowen's method placed regional andequality
constraints on otherimportant
_{balancing}
_{quantities,}
such as
the
_{shaking}
force.
The
problem wasfurther
refined_{by}
Lee
and_{Cheng}
[21]
whodefined
a general objectivefunction
in
terms
ofthe
ground_{bearing}
forces
andthe
input
torque.
The
main
idea
behind
_{reducing}
the
ground_{bearing}
forces
is
to
balance
the
_{shaking}
force,
shaking
moment andtwo
ofthe
bearing
forces
in
oneterm.
The
input
torque
term
was addedto
incorporate
_{sensitivity}
to
that
quantity.Parameters
ofthe
counterweights werethen
determined
_{using}
Heurisitic
computer optimization
to
minimizethe
objectivefunction.
Example
problems presented_{by}
Lee
and_{Cheng}
showedtheir
method
to
be
superiorto
force
_{balancing,}
orforce
and momentbalancing.
Comparisons
to
Tricamo
and Lowen's nonlinearprogramming
method showedLee
andCheng's
methodto
be
marginally
better.
The
apparent success ofLee
andCheng
has
led
to
the
author'sinterest
in
developing
a similar_{program,}
3.0 OVERVIEW
OF
THE
THEORY
The
_{theory}
of_{balancing}
the
combined effects of a planarfour
bar
linkage
is
divided
into
two
parts:1.
The
derivation
of equationsfor
the
kinematic
andkinetic
analysis.
In
_{particular,}
general equationsfor
the
input
torque,
shaking
force,
shaking
moment and_{bearing}
forces
must
be
found.
2.
Development
ofthe
optimization_{problem,}
and application ofthe
_{existing}
automated optimization programto
the
specificproblem of
_{balancing}
four
bar
linkages.
Equations
_{describing}
the
motion ofthe
four
bar
mechanismare obtained
_{using}
vector analysis.This
methodis
analogousto
the
complex number method as presented_{by}
_{Shigley}
andUicker
[34],
Equations
for
the
forces
and couples arederived
by
combining
the
Newtonian
andLagrangian
approaches.This
method allows
direct
evaluation of allkinetic
quantitieswithout
the
use of matrix solutions.Once
the
equationsfor
the
kinetic
analysis are_{known,}
then
the
general optimization problem canbe
formulated.
A
general objective
function
willbe
_{proposed,}
and aformal
statement of
the
optimization problem willbe
given.
With
the
problem
_{defined,}
the
techniques
of automated optimaldesign
variables
(ie.
radius
and angularlocation
of eachcounterweight)
sothat
the
objectivefunction
is
minimized.The
_{key}
_{assumptions}
ofthe
_{theory}
are :1.
All
links
aretreated
as rigidbodies.
Therefore,
the
effects of
_{stiffness,}
_{damping}
or naturalfrequencies
have
been
_{ignored.}
2.
The
effects of_{gravity}
onthe
linkage
have
been
ignored.
3.
The
linkage
and counterweights are assumedto
lie
in
the
same plane.
Couples
due
to
misalignment are not considered.4.
All
links
ofthe
mechanism are assumedto
have
afinite
length.
In
otherwords,
the
analysisdoes
nottake
slidersinto
account.5.
_{Only}
circular counterweights of constantthickness
anddensity
are used._{Further,}
the
counterweights can_{only}
be
attached
tangentially
to
some ofthe
_{bearing}
joints
ofthe
linkage.
Effects
ofthe
counterweight attachmentbrackets
have
alsobeen
ignored.
6.
All
joints
ofthe
mechanism arelower
pair,
pinjoints.
The
term
"linkage" asdefined
by Shigley
andUicker
_{[34],}
mandates
this
assumption oflower
pair contact.7.
The
systemis
conservative.Thus
the
effects offriction
The
_{validity}
of each ofthe
above assumptionsdepends
upon
the
_{particular}
application andlinkage
to
be
balanced.
For
_{example,}
_{the}
_{assumption}
_{of} _{all}_{the}
_{links}
_{being}
_{rigid}_{may}
not always
be
true
for
_{extremely}
high
speed_{applications,}
orwhere
the
stiffness
of eachlink
is
_{relatively}
low
in
comparison
to
the
link
mass._{However,}
for
_{many}
_{applications,}
these
assumptions provide a close approximationto
the
real4.0
DEVELOPMENT
OF
THE
KINEMATIC
AND
KINETIC
EQUATIONS
4.1
Two
Point
Mass
Model
A
general model ofthe
four
bar
linkage
to
be
balanced
is
shown
in
Figure
1.
It
consists ofthree
_{moving}
links
and onefixed
link.
Link
1
is
the
input
or_{crank,}
link
2
is
consideredthe
_{coupler,}
link
3
is
the
follower
andlink
4
is
the
frame.
In
general,
counterweights
_{may}
be
appliedto
_{any}
one ofthe
_{moving}
links
atjoints
_{0',,}
0
~, orO3.
Circular
counterweights ofconstant
thickness
are shown asdashed
lines
in
Figure
1.
For
ageneral
_{analysis,}
the
link
length,
mass,
thickness
andcounterweight
_{thickness,}
radius and angularlocation
must allbe
variable.
From
an analysis point ofview,
a simpler yet_{dynamically}
equivalent model of
the
real counterweightedlinkage
is
the
two
point mass model.
First
appliedto
mechanism_{balancing}
_{by}
Wiederich
andRoth
_{[43],}
the
model reducesthe
number ofvariables
for
a general analysis.As
shownin
Figure
_{2,}
the
model consists of
two
point masses_{rigidly}
attachedto
the
samemassless
link.
On
each_{link,}
these
masses are_{free}
to
_{vary}
in
magnitude and
distance,
but
are restrictedto
lie
uponthe
givenbody
coordinate system axes.As
shownin
Figure
_{2,}
the
first
mass must
lie
onthe
_{body}
" r "_{coordinate,}
andthe
second
massmust
lie
onthe
_{body}
" " coordinate.The
rules of vectorFigure
1
:_{General}
_{Four}
_{Bar}
_{Linkage}
with
Circular
Counterweights
LINK
1
LINK
3
X
X
of
Figure
2
defines
the
positivelocations
ofthe
masses.As
stated
in
_{[43],}
the
inertial
properties of_{any}
rigid_{body}
can
be
represented
_{by}
an equivalent system of point masses.However,
the
model must_{obey}
allthe
basic
laws
of_{dynamics,}
andbe
equivalentto
the
reallinkage.
The
conditionsfor
dynamic
equivalence
between
the
real_{mechanism,}
and_{any}
equivalent modelrequire
that
:1.
The
total
mass ofthe
ith
link
mustbe
the
samefor
both
models.2.
The
center of mass ofthe
ith
link
must remainin
the
same
location
for
both
models.3.
The
total
mass moment ofinertia
ofthe
ith
link
mustremain
the
same aboutthe
ith
links
centroidal_{location,}
or about a
_{bearing}
joint
onthat
link.
In
the
above set of_{remarks,}
statements_{(1.)}
and_{(2.)}
ensureEuler's
first
law
of motion willbe
satisfied.The
third
statement ensures
that
Euler's
secondlaw
of motion willbe
met.The
additional statementin
(3.)
_{requiring}
the
mass moment ofinertia
to
be
the
same about a_{bearing}
joint,
not_{only}
includes
the
frame
joints,
0' and0'
,
but
also appliesfor
joints
02
and0
aThis
peculiarity
is
shownto
be
validin
appendixA.
The
conversionfrom
a reallink
with acounterweight
to
atwo
point mass modellink
willbe
shownin
atwo
_{step}
process.First,
withthe
reallink
and counterweight parameters_{known,}
anequivalent combined center of
_{gravity}
location
willbe
general
link
_{rotating}
about a_{moving}
joint.
This
general casedescribes
link
2.
_{However,}
if
r 'is
a nullvector,
then
the
analysis can
be
applied
to
links
1
and3.
With
the
properties ofthe
combined
center of_{gravity}
_{known,}
the
values ofthe
two
pointmass model
_{may}
be
found
as shownin
Figure
4.
Each
ofthese
two
steps must
_{obey}
the
rulesfor
dynamic
conversions as statedabove.
Figure
3
:Combined
C.G.
Location
of aGeneral
Link
with a
Circular
Counterweight
J
counterweighti
_{combined}
_{e.g.}_{location}
\
In
both
Figures
3
and_{4,}
there
are several coordinatesystems
;
the
spatialXYZ
coordinates,
_{translating}
xyzcoordinates,
andthe
_{body}
coordinatesdefined
as r9z' ' .The
spatial coordinates are considered
to
be
_{stationary}
withthe
fixed
set ofstars,
hence
they
do
nothave
any
angular velocity.translate
withthe
general
point_{Oj}
.Finally,
the
r9z 'coordinates
are
fixed
to
the
link
so_{they}
translate
androtate
with
the
body.
Unit
vectors
for
each ofthe
coordinate
systems
are
defined
in
_{Figures}
3
and_{4.}
Figure
4
:Two
Point
Mass
Model
of aCombined
C.G.
Location
i
_{combined}
eg
location
*
%
Refering
back
to
Figure
3,
the
total
mass ofthe
ith
link
is
just
the
sum ofthe
reallink
andcounterweight
_{masses,}
or :m
iT
m. +
mic
(1)
In
addition,
the
x and_{y}
coordinates ofthe
_{combined}
_{center} _{of}gravity
may
be
given as :xiT
=iiT
coseiT
y.T
=liT
sin6iT
(2)
Or,
statedin
terms
ofthe
reallink
and counterweightparameters,
xiT
=_{micliccos(9i}
+_{Qic)}
+milicos(i
+i
)
(4)
mic
+rn,;
yiT
=micli
sin(i
+_{eic)}
+m1lisin(ei
+CX)
(5)
mic
+mi
Squaring
equations
2
and3
_{separately,}
and_{adding}
the
resultproduces :
2
2
2
lU
=xiT
+^TT
<6>
Substituting
equations4
and5
into
equation6
for
x^iand_{yiT}
respectively,
and_{solving}
for
1
T
:liT=
[mZicl.2c
+ +_{2miclicmilicos(eic0<i)]}
_{(7)}
__To
obtainthe
value ofj
, equation3
is
divided
by
equation2,
and equations
4
and5
are again substitutedfor
xT
and_{yT}
respectively.
The
end result_{being}
:Tan(iT)
=miclicsin(9i
+i(.)
+milisin(ei
+OCi
)
(8)
micliccos(1
+
_{.c)}
+m.l1cos(ei
+<Ki)
Last,
the
total
mass moment ofinertia
ofthe
reallink
andcounterweight must
be
determined
aboutthe
combined
centroidallocation,
pointT
.Using
the
parallel axistheorem
to
transfer
the
given mass moments ofinertia,
t
c2
2
X1T
={lic
+micC(xTc
"XiT> +