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8/8/2019 Acceleration Analysis
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Chapter 7Acceleration Analysis
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Definition
Rate of change of velocity with respect totime Angular
Linear
Position Vector
Velocity
dt
d
dt
dVA
j
PA peR
jjPApa ep
dt
dpje
dt
RV
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Definition
dt
jepd
dt
d jPAPA
A
A
sincoscossin 2 jpjpPA A
"Absolute"PPA AA
dt
dje
dt
dejp
jj
PA
A
jj
PA epjep2A
PAn
PAt
PA AAA
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Definition
Acceleration (difference)
Relative Acceleration
"bodysameon the"
PAAP AAA
APPA AAA
nPAtPAnAtAnPtP AAAAAA
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Graphical Analysis
Graphical Acceleration Analysis
Solve for
PAAP AAA
rAtt
A
CBA ,,on;acceleratilinear
,;onacceleratiangular 43
2rA
nn A
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Graphical Analysis
Example 7-1 Given 2, 3, 4, 2 ,3, 4,2
find 3 , 4 , AA, AB and AC
Velocity analysis alreadyperformed
1. Start at the end of thelinkage about which you havethe most information.Calculate the magnitude ofthe acceleration of point A,
222 AOAn
A 22 AOAt
A
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Graphical Analysis
Example 7-1 2. Draw the acceleration AA 3. Move next to a point which
you have some information,point B. Draw theconstruction line pp throughB perpendicular to BO4
4. Write the acceleration
difference equation for pointB vs. A
BAAB AAA
n
BA
t
BA
n
A
t
A
n
B
t
B AAAAAA 2
44 BOAn
B
n
BA
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Graphical Analysis
5. Draw construction line qqthrough point B andperpendicular to BA torepresent the direction ofABA
6. The vector equation can besolve graphically by drawing
the following vector diagram
23BAAn
BA
nBAtBAnAtAnBtB AAAAAA BAAB AAA
n
B
n
BA
n
A
t
A AAAA ?& t
B
t
BA
AA
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Graphical Analysis
7. The angular velocities of link3 and 4 can be calculated,
8. Solve for VC
4
4BO
At
B
BA
At
BA3
CAAC AAA
3cAtCA
2
3cAn
CA
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Graphical Analysis
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Analytical Analysis
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Fourbar Pin-Joint Linkage Vector Loop
Position
Velocity
Acceleration
Analytical Solution
01432 RRRR
01432
jjjj
decebeae
0432 432
jjj
ejcejbeja
0443322 244233222 jjjjjj ecjecebjebeajea
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Fourbar Pin-Joint Linkage
Analytical Solution
04433
22
2
44
2
33
2
22
jjn
B
t
BB
jjn
BA
t
BABA
jjn
A
t
AA
ecjec
ebjeb
eajea
AAA
AAA
AAA
0 BBAA AAA
identityEuler
partreal
partimaginary
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Fourbar Pin-Joint Linkage
Analytical Solution
4
2
43
2
32
2
222
3
4
coscoscossin
sin
sin
cbaaC
bB
cA
BDAE
AFCD
3
BDAE
BFCE
4
4
2
43
2
32
2
222
3
4
sinsinsincos
cos
cos
cbaaF
bE
cD
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Fourbar Pin-Joint Linkage
Analytical Solution
442
4444
33
2
3333
22
2
2222
sincoscossin
sincoscossin
sincoscossin
jcjc
jbjb
jaja
B
BA
A
A
A
A
0 BBAA AAA
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Analytical Solution
Slider-Crank
d
ebjeb
eajea
t
BB
jjn
BA
t
BABA
jjn
A
t
AA
AA
AAA
AAA
33
22
2
33
2
22
0 BBAA AAA
BABA
AA
BAAB AAA
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Analytical Solution
Slider-Crank
3
2
3332
2
222
3
3
2
32
2
2223
cossincossincos
sinsincos
bbaadb
baa
identityEuler
partreal
partimaginary
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Coriolis Acceleration Is an additional component
of acceleration present whena sliding joint is on a rotating
link Thy position vector is both
rotating and changing lengthas the system moves
Analytical Solution
2j
P peR
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Coriolis Acceleration The velocity expression is
The acceleration,
Analytical Solution
22
2
jj
P epep V
sliptrans PPP VVV
2222
2
22
22
2
jjjj
P
epjepejpjep A
slipcoriolisnormaltangetial PPPPPAAAAA
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Coriolis Acceleration
The acceleration,
Analytical Solution
2222
222
22 2 jjjj
P epjepejpjep A
slipcoriolisnormaltangetial PPPPPAAAAA
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Acceleration of Any Point
Once the angular acceleration of all the links arefound it is easy to define and calculate theacceleration of any point on any link for any input
position of the linkage
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Acceleration of Any Point
To find the acceleration of points S
22220
sincos222
jssej
SSRR
222222 cossin22
jsjsej
SV
22222
2
22222
sincos
cossin
js
jsS
A
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Acceleration of Any Point
To find the acceleration of points U
44440
sincos444
juuej
UR
444444
cossin44
jujsej
UV
444424
44444
sincos
cossin
ju
juUA
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Acceleration of Any Point
To find the acceleration of point P
PAAP
j
PA jppe
RRR
R
3333sincos33
PAAP
jPA jpjpe
VVV
V
333333cossin33
3333
2
3
33333
sincos
cossin
jp
jpPA
PAAP
A
AAA
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Jerk page 367-369 Human Tolerance to Acceleration page 364-366 Fourbar Inverted Slider-Crank page 354-360
Acceleration Analysis of the Geared Fivebar- page361 -362
Important topics to review