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Fundamentals of Stress and Vibration 1. Mathematics for Structural mechanics
[A Practical guide for aspiring Designers / Analysts]
n
d y n
Example: has order n and degree n.
dx n

šƒ’Ž‡ ͳǣ ‡– —• ‘•‹†‡” ƒ •‹–—ƒ–‹‘ ™Š‡”‡ ƒ ‘’‘‡– ‹• ‘‘Ž‹‰ǡ ƒ† –Š‡ ”ƒ–‡ ‘ˆ ‘‘Ž‹‰ ‹•
’”‘’‘”–‹‘ƒŽ –‘ –Š‡ ‹•–ƒ–ƒ‡‘—• –‡’‡”ƒ–—”‡ ‘ˆ –Š‡ ‘’‘‡– ‹–•‡ŽˆǤ ••—‹‰ ƒ
’”‘’‘”–‹‘ƒŽ‹–› ‘•–ƒ– ȋ Ȍǡ ‡•–ƒ„Ž‹•Š ƒ –‹‡ ȋ–Ȍ – –‡’‡”ƒ–—”‡ ȋ Ȍ ”‡Žƒ–‹‘•Š‹’ ƒ– ȋ–ͳ α ͳȌ ƒ†
ȋ–ʹ α ʹȌǤ

ƒ–Š‡ƒ–‹ ƒŽŽ›ǡ ™‡ Šƒ˜‡ǣ

Rate of change of temprature ∝ Instantaneous temperature

dT dT
= ∝ T = = −cT - - - - (1.17)
dt dt
Š‡ ‡‰ƒ–‹˜‡ •‹‰ ‹ ‡“—ƒ–‹‘ ȋͳǤͳ͹Ȍ ‹†‹ ƒ–‡• –Šƒ– –Š‡ –‡’‡”ƒ–—”‡ ‹• †‡ ”‡ƒ•‹‰ ™‹–Š –‹‡Ǥ

‡™”‹–‹‰ ‡“—ƒ–‹‘ ȋͳǤͳ͹Ȍ ƒ† ‹–‡‰”ƒ–‹‰ǡ ™‡ ‰‡–ǣ

T 2 t 2
dT T 2 T 2
= −c dt = log = −c t − t = = e −c t 2 −t 1 or T = T e −c t 2 −t 1 - - - - (1.18)
1
2
2
1
T T 1 T 1
T 1 t 1

– ƒ „‡ ‘„•‡”˜‡† –Šƒ–ǡ –‡’‡”ƒ–—”‡ ȋ ʹȌ ‹• ‰‘– „› —Ž–‹’Ž›‹‰ ȋ ͳȌ „› ƒ ˆƒ –‘” e c t 2 −t 1

šƒ’Ž‡ ʹǣ †‡”‹˜‡ ƒ† ’Ž‘– –Š‡ ”‡Žƒ–‹‘•Š‹’ „‡–™‡‡ Ǯ›ǯ ƒ† Ǯšǯǡ ’”‘˜‹†‡†ǡ ™Š‡ ȋš α ͳȌ ƒ– ȋ›α ͳȌǤ
Š‡ †‹ˆˆ‡”‡–‹ƒŽ ‡“—ƒ–‹‘ ‹• ‰‹˜‡ „›ǣ

dy y
= k - - - - (1.19)
dx x

dy dx
Equation . can be rewritten as: = k = log y = k log x + c
e
e
y x
y
Simplifying the above expression, we get: log y − k log x = c = log = c
e
e
e
x k
Upon further simplification, we get: y = x e - - - - (1.20)
k c
’’Ž›‹‰ –Š‡ ‹‹–‹ƒŽ ‘†‹–‹‘• ‘ˆ ȋš α ͳȌ ƒ† ȋ› α ͳȌ –‘ ‡“—ƒ–‹‘ ȋͳǤʹͲȌǡ ™‡ ‰‡–ǣ

0
c
1 = e ,we know that e = 1 , therefore c = 0 .

QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries, Page 19