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Bài giảng Kinematics and dynamics of machines - Chapter 2: Kinematics of mechanism and planar mechanisms with only lower pairs - Phạm Huy Hoàng

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KINEMATICS OF MECHANISM  
AND  
PLANAR MECHANISMS WITH ONLY  
LOWER PAIRS  
PHAM HUY HOANG  
I. Notes and Recall:  
1. Notes:  
- Size of link is “not limited”.  
- Always identify: point M belongs to link i: Mi  
2. Two points of the same link (rigid body)  
    
n
a a a a  
B A  
i i  
B A B A  
    
i i  
BA  
i i  
v v v  
B A B A  
i i i i  
  
AB  
AB  
2
i
AB  
i
AB  
AB  
i
Absolute velocity of Ai , Bi  
a
B A  
i i  
Relative velocity and  
acceleration between Ai  
and Bi  
i
n
a
Normal and tangential  
components of relative  
acceleration  
B A  
v
i i  
B A  
i i  
i
3. Two different points (belonging to two links sliding on  
each other) locating at the same position  
r
v
A A  
i j  
a
A A  
i j  
k
a
A A  
i j  
i
i
  
i j  
    
Absolute velocity of Ai , Aj  
v v v  
A A A A  
i j i j  
Relative sliding velocity  
two links I and j  
|| tt  
r
v
A A  
i j  
a
A A  
i j  
k
a
A A  
i j  
i
i
k
A A  
a
2 v  
i A A  
i j  
i j  
Absolute acceleration of Ai , Aj  
Coriolis and sliding components of  
relative acceleration between Ai , Aj  
  
i j  
Coriolis acceleration  
   
k
r
a a a a  
A A  
i j  
A A  
A A  
i j  
i j  
||tt  
c
Cross product  
(vector product) and  
right-hand rule or  
screw rule  
b
a
   
c
a
b  
II. Example 1:  
1  
3
3
l a,l a 3,l a,l a,CAB 60  
AB BC BD CD  
2
2
const  
1
?,v ?,v ?  
3 D  
2
2
?,a ?  
3
2
AB  
   
   
v v  
v v // AC  
B B  
2 1  
C C  
2 3  
a  
1
v v v  
C
B
C B  
2 2  
2
2
// AC AB BC  
?
a? BC  
1
2
1  
v
D
2
2
v
C B  
2 2  
1
v
3
Velocity polygon  
BC  
p
// AC  
c c  
2 3  
v v v  
C B  
2 2  
C
B
2
2
b b  
2 1  
// AC AB BC  
a? BC  
?
1
2
BC  
p
// AC  
c c  
2 3  
b b  
2 1  
// AC ()  
   
v v   
2
2
C C  
3 2  
v a  
B
1
2
3
3
()  
BC
v
  
a  
v
C B  
2 2  
2
1
2
1
C B  
1
2 2  
v   
C
2
3
BC 3  
// AC ( )  
v
v v  
D
C
D C  
2 2  
3
3
2
2
v a  
B
1
2
2
2
?
?
CD()  
2a  
a  
1
1
CD  
2
BC  
3
2 3  
d
p
2
// AC  
c c  
2 3  
D
v
2
2
b b  
2 1  
v
C B  
2 2  
1
v
3
 BA   
a a  
a a //AC  
B B  
C C  
2 1  
2
3
2
1
a  
n
a
a  
a  
a  
C B  
C2  
B
2
C B  
2 2  
2 2  
// AC  BA  
 CB  
BC  
? BC  
2
1
3a  
2
1
2
BC  
2
?
a  
2
9
1  
n
a a  
a  
C B  
a  
C B  
C
B
2
2
2 2  
2 2  
 CB  
// AC  BA  
BC  
2
1
3a  
2
2
BC  
2
?
a  
? BC  
2
1
9
2
D
B
BC  
1
// AC  
c c  
2 3  
1  
C
A
0
3
n
C2B2  
Acceleration polygon  
b b  
2 1  
a
C B  
2 2  
ε
2
BC  
// AC  
c c  
2 3  
1
a
3
n
C2B2  
// AC ()  
b b  
2 1  
   
a a   
C C  
2
2
n
2
1
3 2  
a
a  
C B  
3
9
2 2  
()  
BC  
a
  
a
a
2
C 8  
2
1
C B  
2 2  
2
8
C B  
2
1
a   a  
2 2  
B
  
2
2 9  
BC 9 3  
* Theorem: Any moving link has a corresponding velocity  
image with similar shape.  
* Theorem: Any moving link has a corresponding  
acceleration image with similar shape.  
d
2
* Theorem: Any moving link has a corresponding velocity  
image as well as acceleration image with similar shape.  
d
2
III. Example 2:  
l a,l a 3,  
AB AC  
1  
CAB 90  
const  
1
?,?,  
2
3
?,?  
2
3
AB  
   
v v  
v BC  
B B  
2 1  
B
3
a  
1
1  
  
2 3  
v
v v  
B
B
B B  
3 2  
3
2
BC  
AB // BC  
a  
? BC  
?
3
1
p c  
// BC  
v
3
B B  
3 2  
1
b
3
v
B
3
BC  
3
b b  
2 1  
v
v v  
B
B
B B  
3 2  
3
2
BC  
AB // BC  
a  
? BC  
?
3
1
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