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Fundamentals of Stress and Vibration
[A Practical guide for aspiring Designers / Analysts] 1. Mathematics for Structural mechanics
Let us assume the initial radius to be (r ) and the radius at any instant be (r). Integrating (r)
0
between the limits (r and r) and (θ) between the limits (0 to θ), we get:
0
r θ
dr r
cθ
= c dθ = log = cθ = r = r e - - - - (1.78)
0
r r 0
r 0 0
The constant (c) in equation (1.78) can assume a positive value if the motion is inward to
outward or a negative value if the motion is outward to inward.

ͳǤͳʹ ‘ ‡’– ‘ˆ ”ƒ†‹‡–

”ƒ†‹‡– ’Š›•‹ ƒŽŽ› ‹• –Š‡ †‹”‡ –‹‘ ‘ˆ ‰”‡ƒ–‡•– ˜ƒ”‹ƒ–‹‘ ‘ˆ ƒ ˆ— –‹‘Ǥ Š‡ ‡‰ƒ–‹˜‡ ‘ˆ ‰”ƒ†‹‡–
‰‹˜‡• –Š‡ †‹”‡ –‹‘ ‘ˆ •–‡‡’‡•– †‡ ”‡‡–Ǥ

‘” ƒ ‹” Ž‡ǡ ‹– ‹• ‹–—‹–‹˜‡ –Šƒ–ǡ –Š‡ ‰”ƒ†‹‡– ‹• ƒŽ‘‰ –Š‡ ”ƒ†‹ƒŽ †‹”‡ –‹‘ ‘” ‘”ƒŽǡ –Šƒ– ‹• ™Š‡”‡
–Š‡ ”ƒ†‹—• ƒ ˜ƒ”› ƒ– –Š‡ ƒš‹— ”ƒ–‡Ǥ

‡– —• ‘•‹†‡” ƒ ‹” Ž‡ ‘ˆ ”ƒ†‹—• ȋ” α ͵Ȍ ƒ† ‘”‹‰‹
ȋͲǡͲȌ ƒ• ‡–‡” ƒ• •Š‘™ ‹ ȏ ‹‰ ͳǤ͸ͶȐǤ

[Fig 1.64: Gradient of a circle]

2
2
2
Equation of the circle is given by: x + y − 3 = 0 - - - - (1.79)
∂f ∂f
∂x i + ∂y j
The direction of the gradient of a function f =
∂f 2 ∂f 2
∂x + ∂y

QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries,
Page 62