q 1
DESCRIPTION
Position Analysis of a 4 bar RRRR Linkage Using Complex Numbers. Q. P. Im. q 3. q 4. q 2. q 1. Re. O. Viewing the point P from two directions in the complex plane. Position Analysis of a 4 bar RRRR Linkage Using Complex Numbers. Q. P. Im. q 3. q 4. q 2. q 1. Re. O. - PowerPoint PPT PresentationTRANSCRIPT
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22
33
44
32 1 42 3 1 4
ii i il e l e l e l e 32 1 42 3 1 4
ii i il e l e l e l e
Viewing the point P from two directions in the Viewing the point P from two directions in the complex planecomplex plane
Viewing the point P from two directions in the Viewing the point P from two directions in the complex planecomplex plane
PP
OOReRe
ImIm
Position Analysis of a 4 bar RRRR Linkage Position Analysis of a 4 bar RRRR Linkage Using Complex NumbersUsing Complex Numbers
Position Analysis of a 4 bar RRRR Linkage Position Analysis of a 4 bar RRRR Linkage Using Complex NumbersUsing Complex Numbers
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Separating the realSeparating the realSeparating the realSeparating the real
PP
OOReRe
ImIm
2 2 3 3 1 1 4 4cos cos cos cosl l l l 2 2 3 3 1 1 4 4cos cos cos cosl l l l
Position Analysis of a 4 bar RRRR Linkage Position Analysis of a 4 bar RRRR Linkage Using Complex NumbersUsing Complex Numbers
Position Analysis of a 4 bar RRRR Linkage Position Analysis of a 4 bar RRRR Linkage Using Complex NumbersUsing Complex Numbers
2 2 3 3 1 1 4 4sin sin sin sinl l l l 2 2 3 3 1 1 4 4sin sin sin sinl l l l
And the imaginary partsAnd the imaginary partsAnd the imaginary partsAnd the imaginary parts
We have two unknownsWe have two unknownsWe have two unknownsWe have two unknowns3 4 and 3 4 and
And two scalar equations to solve for themAnd two scalar equations to solve for themAnd two scalar equations to solve for themAnd two scalar equations to solve for them
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Rearranging termsRearranging termsRearranging termsRearranging terms
PP
OOReRe
ImIm
3 3 1 1 4 4 2 2
3 3 1 1 4 4 2 2
cos cos cos cos
sin sin sin sin
l l l l
l l l l
3 3 1 1 4 4 2 2
3 3 1 1 4 4 2 2
cos cos cos cos
sin sin sin sin
l l l l
l l l l
Position Analysis of a 4 bar RRRR Linkage Position Analysis of a 4 bar RRRR Linkage Using Complex NumbersUsing Complex Numbers
Position Analysis of a 4 bar RRRR Linkage Position Analysis of a 4 bar RRRR Linkage Using Complex NumbersUsing Complex Numbers
2
1 1 4 4 2 2
2 21 1 4 4 2 2 3
cos cos cos
sin sin sin
l l l
l l l l
2
1 1 4 4 2 2
2 21 1 4 4 2 2 3
cos cos cos
sin sin sin
l l l
l l l l
Squaring and addingSquaring and addingSquaring and addingSquaring and adding
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Rearranging termsRearranging termsRearranging termsRearranging terms
PP
OOReRe
ImIm
Position Analysis of a 4 bar RRRR Linkage Position Analysis of a 4 bar RRRR Linkage Using Complex NumbersUsing Complex Numbers
Position Analysis of a 4 bar RRRR Linkage Position Analysis of a 4 bar RRRR Linkage Using Complex NumbersUsing Complex Numbers
2 2 2 2 2 21 1 4 4 2 2
2 2 2 2 21 1 4 4 2 2 3
1 4 1 4 1 2 1 2 4 2 4 2
1 4 1 4 1 2 1 2 4 2 4 2
cos cos cos
sin sin sin
2 cos cos 2 cos cos 2 cos cos
2 sin sin 2 sin sin 2 sin sin 0
l l l
l l l l
l l l l l l
l l l l l l
2 2 2 2 2 21 1 4 4 2 2
2 2 2 2 21 1 4 4 2 2 3
1 4 1 4 1 2 1 2 4 2 4 2
1 4 1 4 1 2 1 2 4 2 4 2
cos cos cos
sin sin sin
2 cos cos 2 cos cos 2 cos cos
2 sin sin 2 sin sin 2 sin sin 0
l l l
l l l l
l l l l l l
l l l l l l
4 4 1 1 2 2
4 4 1 1 2 2
2 2 2 21 2 4 3 1 2 1 2
2 cos cos cos
2 sin sin sin
2 cos 0
l l l
l l l
l l l l l l
4 4 1 1 2 2
4 4 1 1 2 2
2 2 2 21 2 4 3 1 2 1 2
2 cos cos cos
2 sin sin sin
2 cos 0
l l l
l l l
l l l l l l
Rearranging Rearranging furtherfurther
Rearranging Rearranging furtherfurther
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Substitute Substitute Substitute Substitute
PP
OOReRe
ImIm
Position Analysis of a 4 bar RRRR Linkage Position Analysis of a 4 bar RRRR Linkage Using Complex NumbersUsing Complex Numbers
Position Analysis of a 4 bar RRRR Linkage Position Analysis of a 4 bar RRRR Linkage Using Complex NumbersUsing Complex Numbers
4tan2
x
4tan2
x
And And And And
4 1 1 2 2 4 1 1 2 2
2 2 2 21 2 4 3 1 2 1 2
2 cos cos , 2 sin sin ,
2 cos
a l l l b l l l
c l l l l l l
4 1 1 2 2 4 1 1 2 2
2 2 2 21 2 4 3 1 2 1 2
2 cos cos , 2 sin sin ,
2 cos
a l l l b l l l
c l l l l l l
To getTo getTo getTo get
Or Or Or Or
2
2 2
1 20
1 1
x xa b c
x x
2
2 2
1 20
1 1
x xa b c
x x
2 2 0c a x bx c a 2 2 0c a x bx c a
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PP
OOReRe
ImIm
Position Analysis of a 4 bar RRRR Linkage Position Analysis of a 4 bar RRRR Linkage Using Complex NumbersUsing Complex Numbers
Position Analysis of a 4 bar RRRR Linkage Position Analysis of a 4 bar RRRR Linkage Using Complex NumbersUsing Complex Numbers
2 2 2
1 14 2 tan 2 tan
b a b cx
c a
2 2 2
1 14 2 tan 2 tan
b a b cx
c a
SolvingSolving
we get we get two roots! two roots!
SolvingSolving
we get we get two roots! two roots!
1 1 1 4 4 2 23
3
1 1 1 4 4 2 2
3
cos cos coscos
sin sin sin =sin
l l l
l
l l l
l
1 1 1 4 4 2 23
3
1 1 1 4 4 2 2
3
cos cos coscos
sin sin sin =sin
l l l
l
l l l
l
And And And And
The signs of sinThe signs of sin33 and cos and cos33 will let us will let us
determine the quadrant in which determine the quadrant in which 33 lies lies
The signs of sinThe signs of sin33 and cos and cos33 will let us will let us
determine the quadrant in which determine the quadrant in which 33 lies lies