ME 481 - H06 Name ________________________
Use generalized coordinates {q} and joint constraints {

} for the slider crank shown below.




 x y
 x
2
2
2
3











 y
 x y

3
3
4
4
4







 
2
 r
3
2
 
 r
4
3
 

4



























    r
4
2
   
3
C r
3
1
2
A
B
C

2
 y

4
4

2 t







 y
2
A
2
A x
2
A
1
2 y
1
B
2 x
1
B
B
3
G
3 y
3
G
3 x
3
3 C
C
3
C
4
4 y
4 x
4
AB = R = 0.985 inch
BC = L = 4.33 inch
BG
3
= 1.1 inch
G
2
is at A
2
(balanced crank)
G
3
is on centerline of link 3
G
4
is at C
4
(simple piston model) x
2
axis along centerline of link 2 x
3
axis along centerline of link 3

2
= 1000 rpm CCW constant
2
4
A 
0
0
C 
0

0
 
'
2
 
1
A
B 

0
0
R
0
BLUEPRINT INFORMATION
 
'
3
B 
BG
3
0
 
3
'
C 
L BG
3
0 example for B3
   
3 3

  
3 3
'
B
 
3


 cos sin


3
3
 sin

3 cos

3



ME 481 - H06 Name ________________________
1) Evaluate residuals {

} for rough estimates of generalized coordinates {q} at t = 0.005 sec shown below. Comment on the relative precision of {

} versus {q}.





0
0
0 .
4363
2 .
0 inch inch rad inch
( 25

)










__________ _____
__________ _____
__________ _____
__________ _____













0 .
5
0 .
1745
0
5 .
0
0 rad inch rad inch inch
( 0

(

10

)
)













__________ _____
__________ _____
__________ _____
__________ _____
__________ _____






2) Use geometric equations to determine better estimates for {q} at time t = 0.005 sec. Then evaluate new residuals. Comment on precision of {q} and {

} between parts 1) and 2).



0
0 inch inch



2 t

__________
__________ _____










__________ _____
__________ _____
__________ _____
__________ _____






 







__________ _____
__________ _____
__________ _____
0
0 inch rad ( 0

)












__________ _____
__________ _____
__________ _____
__________ _____
__________ _____






3) Evaluate the Jacobian
  q
for your better estimate of {q} at t = 0.005 sec.














______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______
______














ME 481 - H06 Name ________________________
4) Use your code to perform a Newton-Raphson position solution at t = 0.010 sec. Calculate piston position x
4
and determinant of the Jacobian. Validate with geometric equations. x
4
(Newton-Raphson) ____________ det
  q
____________ x
4
(geometric) ____________
5) Compute piston velocity x
4
and acceleration right-hand-side (RHS) vectors
 
and
 
 x 
4
at t = 0.010 sec using a matrix solution with

. Validate with geometric equations. x
4
(matrix) ____________
 x 
4
(matrix) ____________ x
4
(geometric) ____________  x 
4
(geometric) ____________
EXTRA CREDIT
Place a loop around your solution for part 5) using 0 ≤ t ≤ 0.06 sec and provide MATLAB graphs for piston position x
4
and acceleration  x 
4
as functions of crank angle

2
. Validate using results from geometric equations on the same MATLAB graphs.
EXTRA EXTRA CREDIT
Modify your slider crank code for part 5) to analyze the four bar in Notes_04_05. This should only require modifying the last three rows in your constraint vector, your Jacobian matrix and your acceleration RHS vector. use

2
= 65° validation

= 13.151°
3
2
3
= 10 rad/sec CW
= _______________
  
  
3
2
= 2 rad/sec
2
CCW
= +7.0627 rad/sec
2

4
= -65.173°
4
= -5.3533 rad/sec
  
4
= _______________