Download Homework 6

ME 481 - H06
Name ________________________
Use generalized coordinates {q} and joint constraints {} for the slider crank shown below.
x 2 
y 
 2  r 
 2   2 
   2 
 x 3  r 
q   y 3    3 
   3 
 3  r4 
x 4   
    4 
y 4 
 4 
r2 A  r1 A 
 B
B 
r3   r2  
C
C

  r4   r3  
4




y4


  2   2 t 
B
G3
3
2
AB = R = 0.985 inch
BC = L = 4.33 inch
BG3 = 1.1 inch
G2 is at A2 (balanced crank)
G3 is on centerline of link 3
G4 is at C4 (simple piston model)
x2 axis along centerline of link 2
x3 axis along centerline of link 3
2 = 1000 rpm CCW constant
C
4
A
B3
y3
x3
B2
y2 x2
C3
G3
y4
A2
y1
A1
s 2 ' A  
0

0
0
s 4 ' C   
0
C4
x4
x1
s 2 ' B  
r1 A
R

0
0
 
0
example for B3
BLUEPRINT INFORMATION
 BG 3 
L  BG 3 
s 3 ' B  
s 3 ' C  


 0 
 0 
r3 B  r3  A 3 s 3 ' B
cos 3
A 3   
 sin 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 inch


 _______________ 


 _______________ 
0 inch




 0.4363 rad (25) 
 _______________ 




2.0 inch


 _______________ 

q  
  _______________ 
0.5 inch

 0.1745 rad (10)
 _______________ 




5.0 inch


 _______________ 




0 inch


 _______________ 


 _______________ 
0 rad (0)
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 inch


 _______________ 


 _______________ 
0 inch




 2 t  __________ 
 _______________ 




 _______________ 
 _______________ 
q  _______________ 
  _______________ 
 _______________ 
 _______________ 




 _______________ 
 _______________ 




0 inch


 _______________ 


 _______________ 
0 rad (0)
3) Evaluate the Jacobian  q  for your better estimate of {q} at t = 0.005 sec.
 
q
 ______
 ______

 ______

 ______
  ______

 ______
 ______

 ______
 ______

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______ 
______ 
______ 

______ 
______ 

______ 
______ 

______ 
______ 
ME 481 - H06
Name ________________________
4) Use your code to perform a Newton-Raphson position solution at t = 0.010 sec. Calculate
piston position x4 and determinant of the Jacobian. Validate with geometric equations.
x4 (Newton-Raphson) ____________
 
det  q ____________
x4 (geometric) ____________
5) Compute piston velocity x 4 and acceleration x 4 at t = 0.010 sec using a matrix solution with
right-hand-side (RHS) vectors  and  . Validate with geometric equations.
x 4 (matrix) ____________
x 4 (geometric) ____________
x 4 (matrix) ____________
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 , velocity 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°
 2 = 10 rad/sec CW
 = 2 rad/sec2 CCW
2
validation
 3 = 13.151°
 3 = _______________
 = +7.0627 rad/sec2
3
 4 = -65.173°
 4 = -5.3533 rad/sec
 = _______________
4