Kinematic Analysis of Mechanisms

FOUNDATION of MECHANICS 1
Presentation03: Kinematics analysis of mechanisms
Outline
• Four-bar linkage: introduction; velocity and acceleration analyses (graphical
approach).
• Crank-slider mechanism: position, velocity, and acceleration analyses
(graphical and analytical approaches)
approaches).
• General analytical approach: the matrix formulation.
• Elements for the analytical study of Relative Motions.
FOUR-BAR LINKAGE
Four-bar
Four
bar linkage
GRASHOF’s rule
a: longest bar, b: shortest bar
c, d: intermediate length bars.
– a + b < c + d  Grashof mechanism
– a + b > c + d  non-Grashofian mechanism
– a + b = c + d  Change-point mechanism
FOUR-BAR LINKAGE
Grashof-type
yp four-bar
CRANK – ROCKER
Dead-point configurations
Grashof-type four-bar
TWO CRANKS
Grashof-type four-bar
TWO ROCKERS
FOUR-BAR LINKAGE
Change-point mechanism
Isosceles
linkage
Parallelogram linkage
Antiparallel
linkage
Locomotive
Table lamp
FOUR-BAR LINKAGE
Position analysis
Known: geometry, θ1
2
A
O1A
AB
BO3
B
3
1
B
2
A
θ1
O1
3
1
4
O1O3
O3
2’
θ1
O1
4
B’
B
3’
O3
FOUR-BAR LINKAGE
Position analysis
NO SOLUTION
A
θ1
1
O1
4
O3
A
B≡B’
SINGULARITY
θ1
1
O1
4
O3
FOUR-BAR LINKAGE
Velocity analysis
Known: geometry, position, 1
C24
2
vA
2
C12 ≡ A
vB
1 1
C14 ≡ O1
B ≡ C23
3
3
4
C34 ≡ O3
1
vA, 2
vB, 3
FOUR-BAR LINKAGE
Acceleration analysis
Known: geometry, position, 1 (assumed
as constant), 2, 3
2
A
B
3
1
aBn
1
aA
2
aBAn (aBAt?  aBAn)
3
aBn (aBt?  aBn)
aBt
a
aA BAn
4
O1
aBt
aBn
O3
aBAt
aA
aBAn
aB = aBn + aBt
aBn
aB
aBAt
aA
aBAn
aB
aB = aA + aBAt + aBAt
RRRP (or 3R-P) KINEMATIC CHAIN
Crank-Slider mechanism
Crank-Slotted mechanism
CRANK-SLIDER MECHANISM
Velocity analysis
1
C31
vA, 2
vA
vB
C24
2
C34
A ≡ C12
1
2
1
C23 ≡ B
vB
C14 ≡ O
4
3
CRANK-SLIDER MECHANISM
Kinematic analysis: analytical method
A

l
r

B
O
s
Position
sB  r cos( )  l cos( );

Velocity

sB    r (sin( ) 

Acceleration

sin( )   sin( )
sin(2 )
2 1   2 sin 2 ( )




  
);

 cos( )    cos( )    sin( )   2sin( );



2


sB    r sin( )  r  cos( )  l  sin( )  l  2 cos( )
2
cos( )
cos( )
KINEMATIC ANALYSIS: ANALYTICAL METHOD
Matrix formulation
Position:

q
s
f ( q, s )  0

d f ( q, s )
dt


q, s, q
Velocity:
Closure equations

s
f
q

q
 
Acceleration: q, s, q, s, q

f
s

s0



1
1

s

s   B h( q ) q  k ( q ) q
det( B )  0

  k ( q ) 

ds
' 2
2
s
 k (q) q 
q  k q k q
dt
q

1
det( B )  0 
B
h

1 DOF systems:
q := independent variable
s ::= dependent variables

SINGULARITY
KINEMATIC ANALYSIS: ANALYTICAL METHOD
Matrix formulation: example (Crank-slider)
Position:
q : 
f ( q, s )  0
Closure equations
 
s :  
 sB 
r cos( )  l cos( )  sB  0

 r sin( )  l sin( )  0
Velocity:
d f ( q, s )
dt

f
q

q
h
f
s

s0
det( B )  0


1
B

  r sin(( ) 
 l sin(( ) 1   
 r cos( )     l cos( ) 0      0



 s
 B
0


s   B h( q ) q  k ( q ) q

det( B )  l cos( )

 

2
KINEMATIC ANALYSIS: ANALYTICAL METHOD
Matrix formulation: example (Crank-slider)
d f ( q, s )
Velocity:
dt

f
q
h
 

2


q
f
s

s0
det( B )  0


1


s   B h( q ) q  k ( q ) q
B
1 
cos( )


  0



r
sin(

)





 
l
cos(

)
cos(

)




   r cos( ) 


 
  r sin( )  r tan( ) cos( ) 
 sB   1  tan( ) 


  k ( q ) 

ds
' 2
2
 k (q) q 
q  k q k q
Acceleration: s 
dt
q
cos(( )
sin(
i ( )
  
 
   




 
2
 


cos(
)
cos(
)



   
  r cos( )  r tan( ) sin( ) 
 sB    r sin( )  r tan( ) cos( ) 

RELATIVE MOTION
Position
( P - O0 )  ( P - O1 )  (O1 - O0 )  i1 x1  j1 y1  i0 x 0  j0 y 0
Velocity
1
P 2
vP 
y1
j1
y0
i1
O1
x1
d ( P - O0 )

dt




d i1
d j1




 i 0 x 0 j0 y 0 i1 x1 j1 y1
x1 
y1 
dt
dt
j0
O0
i0
x0
 v O1    ( P  O1 )  v r  vT  v r
RELATIVE MOTION
( P - O0 )  ( P - O1 )  (O1 - O0 )  i1 x1  j1 y1  i0 x 0  j0 y 0




v P  i0 x 0 j0 y 0 i1 x1 j1 y1   ( P  O1 )
Acceleration



  d i 
d j1  
d ( P  O1 )
1




a P  i 0 x 0 j0 y 0 i1 x1 j1 y1

x1 
y1   ( P  O1 )   
dt
dt
dt

 aO    ( P  O1 )   2 ( P  O1 )  a r  2   v r 
 aT  a r  a C
KINEMATICS: SUMMARY
T i /P bl
Topic/Problem
M h d
Methods
• Kinematics of a p
particle
 Cartesian p
planar vectors;
Complex Numbers
• Kinematics of a rigid body
 Cartesian planar vectors;
Complex Numbers
(Rivals theorem, Instant Centre of
Rotation, Kennedy-Aronhold theorem,
g motions))
Rotational/Translational/Rolling
• Kinematic analysis of mechanisms
(Position, Velocity, and Acceleration analyses,
Relative motion)
 Graphical approach;
Analytical approaches:
• explicit formulation
• matrix formulation
• Complex Numbers