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Fundamentals of Stress and Vibration
[A Practical guide for aspiring Designers / Analysts] 1. Mathematics for Structural mechanics

[Fig 1.69: Summation of the first four harmonics leading to sawtooth wave]

Š‡ •ƒ™–‘‘–Š ™ƒ˜‡ Šƒ• ƒ ”‡•‡„Žƒ ‡ –‘ ‡›„‘ƒ”† ™ƒ˜‡ˆ‘” ™Š‹ Š Šƒ• •‡˜‡”ƒŽ ‡›• ‰‹˜‹‰ ‘Ǧ
„‹ƒ–‹‘ ‘ˆ •‡˜‡”ƒŽ Šƒ”‘‹ •Ǥ Š‡ ‘„‹ƒ–‹‘ ‘ˆ •‡˜‡”ƒŽ Šƒ”‘‹ • ‹• ƒŽ•‘ ƒŽŽ‡† Šƒ”‘‹ †‹•Ǧ
–‘”–‹‘Ǥ Š‹• ‘ —”• ‹ ‡Ž‡ –”‹ ƒŽ ‹” —‹–• †—‡ –‘ ‘-Ž‹‡ƒ” ”‡Žƒ–‹‘•Š‹’ „‡–™‡‡ —””‡– ƒ†
˜‘Ž–ƒ‰‡Ǥ
‘”ƒŽŽ› –Š‡ ‘—”‹‡” •‡”‹‡• ‹• ƒ–Š‡ƒ–‹ ƒŽŽ› ‰‹˜‡ „› ƒ› ’‡”‹‘†‹ ‡š ‹–ƒ–‹‘ ™‹–Š ˆ”‡“—‡ › ™
”ƒ†Ȁ ǡ ƒ• ƒ •—ƒ–‹‘ ‘ˆ ƒ –”‹‰‘‘‡–”‹ •‡”‹‡•Ǥ ‡ ‡ Ǯ™ǯ ‹• –Š‡ ˆ—†ƒ‡–ƒŽ ˆ”‡“—‡ ›ǡ ‹–
”‡’”‡•‡–• –Š‡ ƒš‹— ‡‡”‰›Ǥ ‹–Š Š‹‰Š‡” Šƒ”‘‹ • ‡‡”‰› ”—• †‘™Š‹ŽŽǤ ‘™‡˜‡”ǡ –Š‡ ϐ‹”•–
ˆ‡™ Šƒ”‘‹ • ‘–ƒ‹ ƒš‹— ‡‡”‰›

n=∞
a
f t = + a cosnωt + b sinnωt
2 n n
n=1
π
1
a = f x cos nωt dt
n
π
−π
π
1
b = f x sin nωt dt
n
π
−π

π
1
a = f t dt
π
−π

QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries,
Page 72