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
in-lb.
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
in-lb.
3.
The member shall not buckle under a short duration torque of
T
max
= 275
×
10
3
in-lb.
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