Description
1. The shear stress induced along the z−axis when two cylinders are in contact with each
other is given by
τzy
pmax
= −
1
2
−
1
q
1 +
z
b
2
+
(
2 −
1
1 +
z
b
2
)
×
r
1 + z
b
2
− 2
z
b
(1)
where 2b is the width of the contact area and pmax is the maximum pressure developed
at the center of the contact area (see, Figure 1) giveny by
b =
”
2F
πl
1−v
2
1
E1
+
1−v
2
2
E2
1
d1
+ 1
d2
#1/2
(2)
pmax =
2F
πbl (3)
F is the contact force; E1, ν1 and E2, ν2 are the Young’s modulus and Poisson’s ratio
of the cylinders 1 and 2, respectively. d1 and d2 are the diameters of the two cylinders,
and l the axial length of contact.
In several applications such as roller bearings, a crack
originates at the point of maximum shear stress and propagates to the surface leading to
a fatigue failure. Hence, to locate the origin of a crack, it is necessary to find the point
at which the shear stress attains its maximum value.
Figure 1: Contact stress between two cylinders
Show that the problem of finding the location of the maximum shear stress for ν1 = ν2 = 0.3
reduces to maximizing the function
f(λ) = 0.5
√
1 + λ
2
−
√
1 + λ
2
1 −
0.5
1 + λ
2
+ λ (4)
where f = τzy/pmax and λ = z/b.
(a) Plot the graph of the function f(λ) given by Equation (4) in the range (0,3) and 1
identify its maximum.
(b) Find the maximum of the function using the following methods by writing a computer program and also using pen and paper.
i. Unrestricted search with a fixed step size of 0.1 from the starting point 0.0 1
ii. Unrestricted search with an accelerated step size using an initial step size of 1
0.1 from the starting point 0.0
iii. Exhaustive search method in the interval (0,3) to achieve an accuracy within 2
5% of the exact value.
iv. Dichotomous search method in the interval (0,3) to achieve an accuracy within 2
5% of the exact value.
v. Interval halving method in the interval (0,3) to achieve an accuracy within 5% 2
of the exact value.
vi. Fibonacci method with n = 10. 2
vii. Golden section method with n = 10. 2
(c) Compare the relative efficiencies of all the methods and comment on their behaviour. 2