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DESIGN OF MACHINERY
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SOLUTION MANUAL 5-8-1
PROBLEM 5-8
Statement:
Design a linkage to carry the body in Figure P5-1 through the two positions P1 and P2 at the angles
shown in the figure. Use analytical synthesis without regard for the fixed pivots shown. Use the
free choices given below.
Given:
Coordinates of the points P1 and P2 with respect to P1:
P 1x
0.0
P 1y
0.0
P 2x
1.236
P 2y
2.138
Angles made by the body in positions 1 and 2:
210 .deg
θ P1
θ P2
147.5 .deg
β2
27.0 .deg
φ
204.4 .deg
γ2
40.0 .deg
ψ
74.0 .deg
Free choices for the WZ dyad :
z
1.075
Free choices for the US dyad :
s
1.240
Two argument inverse tangent
return 0.5 .π if x 0
atan2( x , y )
return atan
y
x
atan
y
x
Solution:
if x > 0
π otherwise
See Figure P5-1 and Mathcad file P0508.
1.
Note that this is a two-position motion generation (MG) problem because the output is specified as a complex
motion of the coupler, link 3. Because of the data given in the hint, the second method of Section 5.3 will be
used here.
2.
Define the position vectors R1 and R2 and the vector P21 using Figure 5-1 and equation 5.1.
P 1x
R1
R2
P 1y
P 2x
P 21x
P 2y
P 21y
R2
R1
P 21x = 1.236
P 21y = 2.138
p 21
3.
4.
P 21x
2
P 21y
2
p 21 = 2.470
From the trigonometric relationships given in Figure 5-1, determine α2 and δ2.
α2
θ P2
θ P1
α 2 = 62.500 deg
δ2
atan2 P 21x , P 21y
δ 2 = 120.033 deg
Solve for the WZ dyad using equations 5.8.
Z 1x
2nd Edition, 1999
z .cos( φ )
Z 1x = 0.979
Z 1y
z .sin( φ )
Z 1y = 0.444
DESIGN OF MACHINERY
A
cos β 2
B
sin β 2
C
cos α 2
W 1x
W 1y
θ
1
A = 0.109
D
sin α 2
D = 0.887
B = 0.454
E
p 21 .cos δ 2
E = 1.236
C = 0.538
F
p 21 .sin δ 2
F = 2.138
D .Z 1y
B . C .Z 1y
E
D .Z 1x
F
2 .A
A . C .Z 1y
D .Z 1x
B . C .Z 1x
F
D .Z 1y
E
W 1x = 1.462
W 1y = 3.367
2 .A
W 1x
2
w = 3.670
W 1y
atan2 W 1x , W 1y
θ = 246.528 deg
Solve for the US dyad using equations 5.12.
s .cos( ψ )
S 1x
A
cos γ 2
B
sin γ 2
C
cos α 2
U 1x
U 1y
σ
S 1x = 0.342
1
1
A . C .S 1x
s .sin( ψ )
S 1y
D
sin α 2
D = 0.887
B = 0.643
E
p 21 .cos δ 2
E = 1.236
C = 0.538
F
p 21 .sin δ 2
F = 2.138
D .S 1y
A . C .S 1y
E
B . C .S 1y
D .S 1x
F
D .S 1x
F
B . C .S 1x
D .S 1y
E
2 .A
U 1x
S 1y = 1.192
A = 0.234
2 .A
2
u
6.
1
A . C .Z 1x
2
w
5.
SOLUTION MANUAL 5-8-2
2
U 1x = 3.180
U 1y = 4.439
u = 5.461
U 1y
atan2 U 1x , U 1y
σ = 234.381 deg
Solve for links 3 and 1 using the vector definitions of V and G.
Link 3:
V 1x
z .cos( φ )
s .cos( ψ )
V 1x = 1.321
V 1y
z .sin( φ )
s .sin( ψ )
V 1y = 1.636
θ3
Link 1:
2nd Edition, 1999
atan2 V 1x , V 1y
2
v
V 1x
G 1x
w .cos( θ )
θ 3 = 231.086 deg
2
v = 2.103
V 1y
v .cos θ 3
u .cos( σ )
G 1x = 0.398
DESIGN OF MACHINERY
SOLUTION MANUAL 5-8-3
w .sin( θ )
G 1y
θ1
8.
u .sin( σ )
atan2 G 1x , G 1y
2
g
7.
v .sin θ 3
θ 1 = 54.796 deg
2
G 1x
g = 0.690
G 1y
Determine the initial and final values of the input crank with respect to the vector G.
θ 2i
θ
θ 2f
θ 2i
θ 2i = 301.323 deg
θ1
θ 2f = 274.323 deg
β2
Define the coupler point with respect to point A and the vector V.
rp
δp
z
φ
θ3
δ p = 26.686 deg
r p = 1.075
9.
G 1y = 0.564
Locate the fixed pivots in the global frame using the vector definitions in Figure 5-2.
ρ1
atan2 P 1x , P 1y
2
R1
P 1x
ρ 1 = 90.000 deg
2
R 1 = 0.000
P 1y
O 2x
R 1 .cos ρ 1
z .cos( φ )
O 2y
R 1 .sin ρ 1
z .sin( φ )
O 4x
R 1 .cos ρ 1
s .cos( ψ )
O 4y
R 1 .sin ρ 1
s .sin( ψ )
w .cos( θ )
w .sin( θ )
u .cos( σ )
u .sin( σ )
O 2x = 2.441
O 2y = 3.811
O 4x = 2.838
O 4y = 3.247
10. Determine the rotation angle of the fourbar frame with respect to the global frame (angle from the global X axis to
the line O2O4.
θ rot
O 2x , O 4y
atan2 O 4x
O 2y
θ rot = 54.796 deg
11. Determine the Grashof condition.
Condition( S , L , P , Q )
SL
S
PQ
P
L
Q
return "Grashof" if SL PQ
return "non-Grashof" otherwise
Condition( g , w , u , v ) = "Grashof"
12. DESIGN SUMMARY
2nd Edition, 1999
Link 2:
w = 3.670
θ = 246.528 deg
Link 3:
v = 2.103
Link 4:
u = 5.461
θ 3 = 231.086 deg
σ = 234.381 deg
DESIGN OF MACHINERY
SOLUTION MANUAL 5-8-4
Link 1:
g = 0.690
Coupler:
r p = 1.075
θ 1 = 54.796 deg
δ p = 26.686 deg
Crank angles:
θ 2i = 301.323 deg
θ 2f = 274.323 deg
13. Draw the linkage, using the link lengths, fixed pivot positions, and angles above, to verify the design.
O2
1.236
G1
Y
O4
U2
P2
S2
B2
V2
2.138
W2
A2
A1
X
P1
V1
S1
B1
2nd Edition, 1999
U1
Z2
Z1
62.5°
W1