## 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