1
Displacement and Velocity Analysis
• 4 bar configurations – range of motion
• Grashof criteria
• Kinematic inversions
• Geometric inversions
• Dead center
• Circuits
• Graphical and Analytical Methods
2
Grashof, F. (1883). Theoretische Maschinenlehre. Vol. 2. Voss: Hamburg.
Kinematics Introduction
Grashof condition addresses mobility
s=shortest link
l=longest link
p=one remaining link
q=other remaining ling
If s+l ≤ p+q one link capable of full rotation (Class I kinematic chain)
If s+l> p+q no link capable of complete revolution (Class II kinematic chain)
3
Crank-rocker
Double Rocker Rocker-Crank
Double Crank
4 variations
Position of s
Displacement and Velocity Analysis
5
Displacement and Velocity Analysis
Geometric Inversions -
α α
Dead Center position Rocker-Crank Rocker-Crank
6
Displacement and Velocity Analysis
Crank-rocker – driving with short crank
- no dead center positions
Inversion 1,2 on 2 circuits
Circuit 1 Circuit 2
7
Displacement and Velocity Analysis
Crank-rocker
2 circuits
Circuit 1
Circuit 2
8
Kinematics Introduction
Rocker- crank – driving with longest crank
- 2 dead center positions
2 Inversions on 2 circuits
9
Displacement and Velocity Analysis
Displacement Analysis – Graphical Method
Input (driver) and
Output links
Arcs
10
Displacement and Velocity Analysis
• Displacement Analysis
• E&S 3.3
• Complex Polar Vector Notation
• Displacement Analysis
• S&R
• Velocity Analysis
• E&S 3.5
• S&R
• Acceleration
• S&R
• Homework
11
Displacement and Velocity Analysis
Displacement Analysis Analytical – E&S 3.3 (ex B)
52
355
4123
744
74
2
3
2
7
2
4
74
2
4
2
7
2
3
127
)arg()arg(
)arg()arg(
2arccos
cos2
cos
rrrr
rr
rrrr
rr
rrrrr
rrrrr
ineslaw
rrrdia
oac
c
4
5
21
11
inversionccw
inversioncw
knownrrr
knownrrr
c
oa
,,,
,,
543
21
12
Displacement and Velocity Analysis
Complex Polar Vector Notation
)(
)2(
)(
)(
)sincos()(
)sincos()sin(cos
)(
)sin(cos
ii
ii
ii
i
iii
yx
i
reirei
reirirei
reire
iriirire
reree
irrirrer
x
iy
r
x
iy
r
ri
)( rii
13
Displacement and Velocity Analysis
x
iy
r
r
iii
iii
iererredt
d
iererredt
d
)(
)()(
ii
iiii
iii
ierrerr
iierierrierer
iererdt
dredt
d
)2()(
)()()(
)()(
2
2
2
velocity
acceleration
14
Displacement and Velocity Analysis
Link Velocity
ground
0A
x
iy A
Ar
AV
2
2
Ai
AAA
i
AA
iA
AA
i
AA
AA
riorierV
eirerdt
rdV
err
dtd
A
AA
A
0
)cossin(2 AAAA irV
Cartesian form
15
Displacement and Velocity Analysis
AV
B
A BV
2
222
2
2
iririr
VVV
irV
irV
BAAB
ABBA
BB
AA
AV
BVBAV
Relative Velocity
BA
BA
BA
BA
rV
riV
2
2
16
Displacement and Velocity Analysis
2
3
y
x0A
A
B
2r
1r
3r
3322321
3
223
32
323211
321
coscos
)sin
arcsin(
)()(
rrrrr
rr
rr
rrrrrr
rrr
xx
yy
232 ,,: rrknow
Crank-slider
)(2 Arr )(3 BArr
1rposition
17
Displacement and Velocity Analysis
Velocity of B
BA
BAx
AxBABA
BAyBAxAxAyBx
BAx
Ax
By
BAAB
BAAi
B
BAAB
rr
ririV
rrrrV
rr
Vimaginaryreal
iririV
ririeV
VVV
)(
)(
)(
)0(
)sincos()sincos())0sin()0(cos(
23
22
23
333222
32
BAV
BV
AV
22 ,
18
Displacement and Velocity Analysis
22 ,
sec/1
"13
"6
22
3
2
rad
rr
rr
BA
A
19
Displacement and Velocity Analysis
Homework 4 : Problem 3.10 E&S
Find
Given geometry:
iAB
mmiBA
mmiDE
mmiCE
mmiCB
mmiBB
mmiDA
mmiAA
5.981
1080104
395.2232
5.251322
645631
32285.15
505.4619
205.185.7
00
0
0
0
0
65 ,, EV
20
Displacement and Velocity Analysis
r4
r3
r2
r1
Bo Ao
A
B
2
2
2 4
d
d
d3
x
iy S&R ref 2
))(())((
1
22
2
4132
2121
21
4132
iiii
ii
ii
iiiiB
errerrdede
ee
errde
ererererr
dd
d
Complex conjugate
Loop 1
2 unknowns
Complex polar vector notation
21
Displacement and Velocity Analysis
221
2
2
2
1
221
2
2
2
121
2
2
2
1
2
cos2
cos2)( 22
rrrrd
rrrreerrrrdii
)cos(sintan
sin0sinsin:
coscoscos:
)sin(cos)sin(cos
22122
22
122
1222
212
rrr
dri
rdrR
ridir
errde
d
dd
dd
dd
ii d
Loop 1
identity
22
Displacement and Velocity Analysis
r4
r3
r2
r1
Bo Ao
A
B
2
2
2 4
d
d
d3
x
iy
3
2
4
22
33
2
4
22
333
)()(
3
)(
3
)(
3
22
3
2
4
3344
2)cos(
)cos(2)(
))(())((
33
33
3344
drrdr
rdrdreedr
edredrdrr
deerdeererer
d
d
ii
ii
iiiiii
dd
dd
dd
Loop 2
diiideerer
34
34
23
Displacement and Velocity Analysis
r4
r3
r2
r1
Bo Ao
A
B
2
2
2 4
d
d
d3
x
iy
)(:2
2)cos(
3
3
2
4
22
33
d
d
solutions
drrdr
r4 r3
r2
r1
Bo Ao
A
B
2
2
2
4
d
d
d
3
x
iy
Inversion 1
Inversion 2
24
Displacement and Velocity Analysis
4
334
4433
4433
43
34
)sinsin(sin
sinsinsin:
coscoscos:
43
34
rdr
rdri
rdrR
erdeer
deerer
d
d
d
iii
iii
d
d
),,(
),,(
),,(
),,(
434
433
221
221
drrf
drrf
rrf
rrfd
d
Loop 2
25
Displacement and Velocity Analysis
)sincos(
)sincos()sincos(
)(
4444
33332222
443322
4132
432
4132
ir
irir
ierierier
ererererdt
d
iii
iiii
velocity
26
Displacement and Velocity Analysis
Real and imaginary parts
4433
4433
44222
44223
222
222
4
3
4433
4433
coscos
sinsin
coscos
sin2sin
cos
sin
coscos
sinsin
rr
rr
rr
rr
r
r
rr
rr
Cramer’s Rule
27
Displacement and Velocity Analysis
),,,,,,(,
)sin(
)sin()(
)sin(
)sin()(
432432243
43
23
4
224
43
24
3
223
rrrf
rr
rr
Again using Cramer’s rule
Using identities
28
Kinematic Analysis using Cartesian Vector Notation
Scalar Product – Dot Product
...0
1
cos
etcji
kkjjii
bababaabba zzyyxx
Vector Product – Cross Product
...,
sin
etcikxjkjxi
bbb
aaa
kji
bxa
abbxa
zyx
zyx
29
Kinematic Analysis using Cartesian Vector Notation
Differentiation of Vectors
First Derivative- velocity
rrr
rrr
tolarperpendicubemustrdt
dr
ttdtd
dtdrrr
rr
tt
)(
limlim00
r
y
x
30
Kinematic Analysis using Cartesian Vector Notation
Differentiation of Vectors
Second Derivative -acceleration
)(2)(
)()()(
)(
)(
)(
rrrrr
rrrrrdt
d
rrdt
dr
rdt
drdt
drrr
rrdt
dr
rrr
31
Kinematic Analysis using Cartesian Vector Notation
Find B,4
Vector Loop Equation
21
12
rrd
rdr
r4
r3
r2
r1
Bo Ao
A
B
2
2
2 4
4 Bar Linkage
d
d
d3
Closed vector loop
Head to tail chain
Or
Two branch paths
+
32
Kinematic Analysis using Cartesian Vector Notation
21 rrd
xxyy
d rrrr
rrrrd
rrrrd
rrrrdd
2121
221
2
2
2
1
221
2
2
2
1
2
2121
tan
cos2
cos2
)()(
33
Kinematic Analysis using Cartesian Vector Notation
drr
rdr
34
43
r4
r3
r2
r1
Bo Ao
A
B
2
2
2 4
d
d
d3
2nd loop equation
d
drrdr
drdrr
3
3
2
4
22
3
3
22
3
2
4
2cos
cos2
xxyy
d rrrr
2121tan
3r
4r
+ inversion 1
- inversion 2
3
34
Kinematic Analysis using Cartesian Vector Notation
)(
2cos
cos2
54
4
2
3
22
45
54
22
4
2
3
d
drrdr
drdrr
r4
r3
r2
r1
Bo Ao
A
B
2
2
2 4
4 Bar Linkage
d
d
d3
5
35
Kinematic Analysis using Cartesian Vector Notation
Velocity Analysis
r4
r3
r2
r1
Bo Ao
A
B
2
2
2 4
4 Bar Linkage
d
d
d3
Two paths to common point B
443322
1
444111333222
4132
0,0
)(
rrr
axisfixedrbut
rrrrrrrrdt
d
rr
rrrr
36
Kinematic Analysis using Cartesian Vector Notation
443322
443322
444433332222
444333222
443322
)(
)(
)()()(
xxx
yyy
yxyxyx
yxyxyx
rrrj
rrri
irjrirjrirjr
jrirkjrirkjrirk
vectorsunit
rrr
yxyx
yxyx
yxyx
xyyx
rrrr
rrrr
rrrr
rrrr
4334
233224
4334
424223
Same as complex polar
r2y=r2sinθ2 , etc….
37
Kinematic Analysis using Cartesian Vector Notation
Acceleration
r4
r3
r2
r1
Bo Ao
A
B
2
2
2 4
4 Bar Linkage
d
d
d3
......
..........)))((()(
))(())(())((
)(
2
2
22
2
22222
2222222
444443333322222
443322
jririrjr
jrirkkjrirk
rrrrrr
rdt
rd
rrrdt
d
yxyx
yxyx
Expand in vector form
Take cross products
38
Kinematic Analysis using Cartesian Vector Notation
Rearranging
xx
yy
x
y
xx
yy
x
y
xx
yy
yyyxxx
xxxyyy
rr
rr
Br
Ar
rr
rr
rB
rA
B
A
rr
rr
Brrrrrrj
Arrrrrri
43
43
3
3
4
43
43
4
4
3
4
3
43
43
4
2
43
2
32
2
2224433
2
2
43
2
32
2
2224433
)
)
39
Kinematic Analysis using Cartesian Vector Notation
)( 54 d
yxyx
yxyx
yxyx
xyyx
rrrr
rrrr
rrrr
rrrr
4334
233224
4334
424223
xx
yy
x
y
xx
yy
x
y
rr
rr
Br
Ar
rr
rr
rB
rA
43
43
3
3
4
43
43
4
4
3
d3
5
r4
r3
r2
r1
Bo Ao
A
B
2
2
2
4
d
d
d3
xxyy
d rrrr
rrrrd
2121
221
2
2
2
1
tan
cos2
drrdr
3
2
4
22
3
2cos
drrdr
4
2
3
22
45 2
cos
All the equations
known
knownrrrr
222
4321
,,
,,,
40
Kinematic Analysis using Cartesian Vector Notation
Oscillating Slider
)(
1
4
4
1342
1
oo abrrfr
DOF
f
n
Position equation
33
20012
,
),(,,
runknowns
abrfrknowns
2r
3r
f
2
d
4
3
24
3
d
0a
a
0b
d
41
Kinematic Analysis using Cartesian Vector Notation
222
03
2
3
22
22
0
2
00
000000
21
)()(
)()(
)()(
)()()()(
fababr
rfdalso
ababddd
jabiabd
jaaiaajabiabd
rrddiagonal
yyoxx
yyoxx
yyxx
yyxxyyxx
d
xox
yoyd
xox
yoyd
rf
rf
ab
ab
ab
ab
3
3
1
3
1
)(tantan
tantan
b
ad
42
0
1
2
3
4
5
6
7
0 90 180 270 360
R3
R3
2
2
3
Oscillating Slider
r2=a=2.20”
f=2.15”
r1=4.51”
0
10
20
30
40
50
60
70
80
90
100
0 90 180 270 360
Th 3 vs Th2
Th 3 vs Th2
43
Kinematic Analysis using Cartesian Vector Notation
Velocity
Homework #2
Derive expressions for Velocity and acceleration