# Balancing of the Shaking Force, Shaking Moment, Input Torque and Bearing Forces in Planar Four Bar Linkages

228

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## Full text

(1)

(2)

(3)

(4)

author would

all

who

assisted

of

and

particular:

author's

advice and

course of

of

author.

member of

author's

whose efforts

### field

of automated optimal

### have

made a substantial portion of

possible.

member of

author's

and suggestions

of

and suggestions

of

problem.

(5)

and

of

use

of

computers.

wife and

continued

(6)

### development

and application of a computer

program

### the

combined effects of

and

presented.

assumes

consist of rigid

and

planar

other

sliders.

accomplished

### using

circular counterweights which

are

attached

sizes and

are

nonlinear

### techniques

where an objective

upon all

minimized.

program

capable of

number of added

and

### degree

of numerical quadrature and regional constraints on all

parameters can

varied.

program

capable of

### linkages

with offline mass

and

some

emphasis can

placed on

such as

major

of

### the

assumption of rigid

not

always

valid and makes

program

natural

(7)

excessive-and

of

program.

example

shows

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and

of numerical

### Gaussian

quadrature method

shown

most

with

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optimum number of

points

example

an

at a constant

speed of

rpm

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all

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quantities are reduced

over

unbalanced case.

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results are shown

example

a

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counterweights

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results

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not always

situation.

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one counterweight

### terms

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while

addition of

counterweights

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### these

same quantities an additional

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apparent success of

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author recommends

extended

six

and

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IT

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(18)

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(19)

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are

methods of

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methods

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over

years.

and

### [4]

published a simple

method

other

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moment and

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eliminated

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mass

required

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counterweights

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Reduction of

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use of a

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(20)

gave

consideration

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next

equations

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computer

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constraints could

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curves

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schemes

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problem as

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and

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closed

solution

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presented

and

and

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a general computer program which

analysis will

(21)

circular

computer program

an

program written

and

general

theory.

of

program will

### three

numerical examples.

parameters of

program will

an

efficient

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optimum number of

points

numerical

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will

of

example will

used

second example.

### This

example presents a practical

method of

an

effect of

### placing

additional counterweights on

and coupler

also

a

### linkage

with an offline coupler mass

most

practical

scheme of

example will

compared

unbalanced

### the

method of complete

and

### the

method of complete

and moment

(22)

of

a problem

and

researcher

years.

of

problem since

must

account

experimental and

methods exist

some of

most

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other

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summarized until

year

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published a

of

of

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sliders.

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complete work categorized major

and

and

papers

and

supplemented

a

of methods

and

### 71

references were cited and

major

techniques were

such

comprehensive

surveys

efforts of

will not

either of

above papers.

major

and

Berkoffs'

papers will

presented.

### important

theoretical methods

rigid

(23)

of

, or

a popular method

years.

and

where

was

or

eliminated.

attempts

make

center of

of

and can

categorized

static

### methods,

method of principal

cam

methods and

### duplicate

mechanism methods.

other

partial

methods

reduce

and

accomplished

methods or

springs

methods use

a

series and

squares approach

eliminate

of

while

springs

alters

path

which

ground.

and

### [4]

published a new method

a

method of

vectors.

method was

and

required

addition

of

counterweights

and output

and

showed

### the

quantitative effects of complete

a general

on other

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reactions such as

moment and

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paper

also provided a complete

on

analysis

### final

equations and popular methods of

(24)

a method which

chose

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number of counterweights

of

equations.

extended

work

practical

six

method of

reduces

net

on

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and

and

moment.

of complete

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and

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mechanism with

mirror symmetry.

### harmonic

methods were popular

moment.

of

and sometimes

second

of

### shaking

moment

was accomplished with

### rotary

masses which were

synchronized

out of phase.

and

### [23]

published a graphical method

### shaking

moment was minimized while

a

and moment

of

was

given

mass of

mechanism, and

all other

reactions.

### inertial

properties of a general

so

### the

amount of angular

(25)

### 1980

extended complete

and some six

effects of

coupler

motion

a ground

where

eliminated

moment.

and

some

methods

concentrated on

can never

eliminated

magnitude of

can

reduced.

### [7]

summarized available methods of

equivalents

### flywheels,

and cam subsystems are all capable

of

out

curve.

### Each

method stores and

reduce

### particular,

cam subsystems can

out

perfection.

### Other

methods summarized

### torque

regional constraints

synthesis

see

### example),

and rearrangement of

### internal

mass redistribution.

previous methods

on

elimination of

or at most

quantities.

as shown

### in

various published example

of

parameters

### have

an adverse effect

on other

such as

mid

researchers

combined

effects of

(26)

was

use of

computer.

and

similar

### interactive

computer programs

which

and permitted

user

### link

and counterweight properties

reduce other

and

### [9]

used computer aided optimal

### to

generate mechanisms

synthesis while

### 1975

also used nonlinear

counterweight

and angular

while

quantities

### through

a series of optimizations.

effects of

parameters were

curves

were used

### determine

which counterweight scheme resulted

overall

computer

and

equations

with

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constraints on ground

were of

### 16th

order and required extensive

computer work

solved.

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procedures

a

series of

### determine

which method provided

overall results.

were

and

semiiterative

nature.

problem of

### the

combined effects seems

suited

even with

aid of

all of

### previously

mentioned procedures required some manual

and at

(27)

problem was

solved

and

who used

optimal

minimize

maximum

maximum

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search process of

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problem was

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who

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method showed

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method

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apparent success of

and

author's

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(28)

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vector analysis.

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approaches.

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evaluation of all

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equations

analysis are

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general objective

will

and a

statement of

### the

optimization problem will

problem

### techniques

of automated optimal

(29)

and angular

of each

so

objective

minimized.

of

are :

are

as rigid

effects of

or natural

effects of

on

### linkage

and counterweights are assumed

same plane.

### to

misalignment are not considered.

of

### the

mechanism are assumed

a

analysis

not

sliders

account.

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circular counterweights of constant

and

are used.

### the

counterweights can

attached

some of

of

of

### the

counterweight attachment

also

of

mechanism are

pin

"linkage" as

and

mandates

assumption of

pair contact.

system

conservative.

effects of

(30)

of each of

### the

above assumptions

upon

application and

of all

rigid

not always

speed

or

where

of each

comparison

mass.

### these

assumptions provide a close approximation

real

(31)

general model of

shown

consists of

and one

or

considered

and

applied

one of

at

~, or

### Circular

counterweights of

constant

are shown as

a

general

and

counterweight

### thickness,

radius and angular

must all

variable.

### From

an analysis point of

a simpler yet

### dynamically

equivalent model of

### the

real counterweighted

### two

point mass model.

applied

mechanism

and

model reduces

number of

variables

### for

a general analysis.

shown

### the

model consists of

point masses

attached

same

massless

each

masses are

magnitude and

are restricted

upon

given

### body

coordinate system axes.

shown

mass must

on

" r "

and

mass

must

on

" " coordinate.

rules of vector

(32)

:

with

(33)

of

positive

of

masses.

properties of

rigid

can

### by

an equivalent system of point masses.

model must

all

of

and

equivalent

real

conditions

equivalence

real

and

equivalent model

require

:

mass of

must

same

models.

### The

center of mass of

must remain

same

models.

mass moment of

of

must

remain

same about

centroidal

or about a

on

above set of

statements

and

ensure

of motion will

satisfied.

### third

statement ensures

second

of motion will

met.

### The

additional statement

mass moment of

same about a

not

0' and

0'

,

also applies

and

a

shown

valid

appendix

conversion

a real

with a

a

point mass model

will

shown

a

process.

with

real

### link

and counterweight parameters

### known,

an

equivalent combined center of

will

(34)

general

about a

general case

r '

a null

analysis can

and

properties of

center of

values of

point

mass model

as shown

of

steps must

rules

### dynamic

conversions as stated

above.

:

of a

with a

counterweight

e.g.

and

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are several coordinate

systems

spatial

xyz

and

coordinates

as r9z' ' .

### The

spatial coordinates are considered

with

set of

not

### any

angular velocity.

(35)

with

point

.

r9z '

so

and

with

each of

are

and

:

of a

i

*

mass of

sum of

real

and

or :

m

m. +

x and

coordinates of

center of

given as :

=

cos

=

sin

(36)

stated

of

real

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and counterweight

=

+

+

+

=

+

+

+CX-

+

and

and

result

produces :

=

+

equations

and

equation

x^-i-and

and

:

liT=

+ +

__

obtain

value of

, equation

equation

and equations

and

### 5

are again substituted

x

and

respectively.

end result

:

=

+

+

m

+

+

mass moment of

of

real

### link

and

counterweight must

about

centroidal

point

.

parallel axis

### the

given mass moments of

c

=

+

"

XiT> +

"

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