A&AE 550 MULTIDISCIPLINARY DESIGN OPTIMIZATION
FALL 2011
HOMEWORK ASSIGNMENT #2
DUE NOVEMBER 4, 2011
I.
CONSTRAINED MINIMIZATION IN
N
VARIABLES – DIRECT METHODS
NOTE:
This problem is based upon problem 3.34 from Arora,
Introduction to Optimum Design, second edition.
You do not need the textbook to solve this problem, but I wanted to attribute its source
.
Design a minimum weight hollow torsion rod with a fixed length,
l
= 4.25 ft, as illustrated below.
The torsion bar is subject to the following constraints:
1.
The calculated shear stress,
τ
, shall not exceed the allowable shear stress,
τ
a
, under the normal
operating torque,
T
0
= 145
×
10
3
inlb.
2.
The calculated angle of twist,
θ
, shall not exceed the allowable twist,
θ
a
= 4.25º, under the normal
operating torque,
T
0
= 145
×
10
3
inlb.
3.
The member shall not buckle under a short duration torque of
T
max
= 275
×
10
3
inlb.
The torsion bar shall be fabricated using an aluminum alloy with the following material properties: density,
ρ
= 175 lb/ft
3
(note units here); allowable shear stress,
τ
a
= 24 ksi (1 ksi = 1000 lb/in
2
); elastic modulus,
E
= 10,600 ksi; shear modulus,
G
= 4,000 ksi; and Poisson’s ratio,
ν
= 0.32.
The manufacturing process limits the outer diameter of the bar and the ratio of the outer and inner diameters of
the bar as follows:
5
≤
d
≤
20in
0.01
≤
t
d
≤
0.45
Useful expressions for the rod are:
Mass:
M
=
π
4
ρ
l
d
2
−
d
−
2
t
( )
2
⎡
⎣
⎢
⎤
⎦
⎥
Shear stress under torque:
τ
=
c
J
T
0
where
c
= distance from center of rod to outside edge (this is
d
/2) and
J
= polar moment of inertia (for a
hollow cylinder,
J
=
π
[
d
4
– (
d
–2
t
)
4
] / 32.
T
0
or
T
max
l
t
d
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View Full DocumentAAE 550, FALL 2011
HOMEWORK #2, PAGE 2
Angle of twist under torque:
θ
=
l
GJ
T
0
radians
⎡
⎣
⎤
⎦
Critical buckling torque:
T
cr
=
π
d
3
E
12
2
1
− ν
2
( )
0.75
1
−
d
−
2
t
( )
d
⎛
⎝
⎜
⎞
⎠
⎟
2.5
1)
Provide your optimization problem statement.
Clearly present the design variables you used as functions of
d
and
t
.
The design variables could simply be
d
and
t
; however, do not forget that transformations and / or
scaling may be helpful to you, especially since these two design variables will have different magnitudes.
Formulate the constraints into
g
i
(
x
)
≤
0 formulations and clearly present them in your submittal.
Consider
any geometric constraints that may restrict the value of one variable based upon the value of another
variable (or variables).
Present how you wish to treat the variable bounds, including the upper and lower
limit values.
Consider using volume of the rod rather than mass as the objective function.
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 Spring '97
 RAVINDRAN
 Shear Stress, Optimization, objective function, allowable shear stress, sqp, generalized reduced gradient

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