Page 82 - C:\Users\trainee\AppData\Local\Temp\msoEAA3.tmp

Document Title
Fundamentals of Stress and Vibration Chapter Title
[A Practical guide for aspiring Designers / Analysts] 2. Engineering Mechanics

From the equations x = a cosθ and y = b sinθ , we have:

dx dy
= −a sinθ and = b cos θ - - - - (2.63)
dθ dθ

dy b cos θ b
Therefore, we have: = − = − cot θ
dx a sin θ a

2
d y b dθ b 1 1 b
2
And = − cosec θ ∗ = − − = - - - - (2.64)
3
2
2
dx 2 a dx a sin θ a sinθ a sin θ
Substituting equations (2.63) and (2.64) in equation (2.62) we get:
3
2 2
b
1 + − cot θ 3 3
2
3
2
3
2
2
2
2
2
2
a a sin θ b cos θ 2 a sin θ a sin θ + b cos θ 2
R = b = b 1 + a sin θ = b ∗ 3

2
2
2
2
a sin θ a cos θ 2
2
3

Simplifying the above expression further, we have:
3 3
2
2
3
2
2
2
2
2
2
2
a sin θ a sin θ + b cos θ 2 a sin θ + b cos θ 2
R = ∗ = - - - - (2.65)
3
3
b a cos θ ab
Equation (2.65) gives us the radius of curvature at any point on the ellipse and the radius of
curvature at points (A, B, C and D) is tabulated as follows:

Points on the Ellipse Angle Radius of Curvature (R)
a 3 a 2
A 90 =
0
ab b
b 3 b 2
0
B 0 =
ab a
a 3 a 2
C 270 =
0
ab b
b 3 b 2
D 180 =
0
ab a

Page 82 QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries,