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Fundamentals of Stress and Vibration 1. Mathematics for Structural mechanics
[A Practical guide for aspiring Designers / Analysts]
ͳǤͳͶ ’–‹‹œƒ–‹‘

‡•‹‰ ’–‹‹œƒ–‹‘

‡•‹‰ ‹• ƒ –”ƒ†‡ „‡–™‡‡ ƒ› ’ƒ”ƒ‡–‡”• –‘ ƒ Š‹‡˜‡ ƒ •‹‰Ž‡ ‘” —Ž–‹’Ž‡ ‘„Œ‡ –‹˜‡•Ǥ Š‡
’ƒ”ƒ‡–‡”• ƒ› „‡ ”‡Žƒ–‡† ‘” ‹ϐŽ—‡ ‡ ‡ƒ Š ‘–Š‡” Ž‹‡ƒ”Ž› ‘” ‘- Ž‹‡ƒ”Ž›Ǥ

‘”†‹‰Ž›ǡ ƒ–Š‡ƒ–‹ ƒŽ –‡ Š‹“—‡• ƒ”‡ †‡˜‡Ž‘’‡† •–ƒ”–‹‰ ˆ”‘ •‹’Ž‡ †‹ˆˆ‡”‡–‹ƒŽ ƒŽ —Ž—•ǡ
ƒ‰”ƒ‰‹ƒ —Ž–‹’Ž‹‡” –‘ ‘-Ž‹‡ƒ” ’”‘‰”ƒ‹‰ –‡ Š‹“—‡•Ǥ Š‡•‡ –‡ Š‹“—‡• Šƒ˜‡ „‡‡
†‡˜‡Ž‘’‡† ƒ† ƒ”‡ ƒ˜ƒ‹Žƒ„Ž‡ ‹ ƒ› ‘‡” ‹ƒŽ •‘ˆ– ™ƒ”‡• ȋ‘– †‹• —••‡† ‹ –Š‹• „‘‘ȌǤ

‘†ƒ›ǡ „‡ ‹– ’‡–”‘Ž‡—ǡ ƒ—–‘ǡ ƒ˜‹ƒ–‹‘ ‘” ‡˜‡ ‡†‹ ƒŽ ‹†—•–”›ǡ ™‡‹‰Š– ƒ† •‹œ‡ ‹‹‹œƒ–‹‘ ƒ”‡
”‹–‹ ƒŽǡ ˆ”‘ –Š‡ ˜‹‡™ ’‘‹– ‘ˆ ’‡”ˆ‘”ƒ ‡ ƒ† ‡”‰‘‘‹ •Ǥ

’–‹‹œƒ–‹‘ ‡–Š‘†• ƒ”‡ „”‘ƒ†Ž› Žƒ••‹ϐ‹‡† ‹–‘ –Š‡ ˆ‘ŽŽ‘™‹‰ǣ
ͳǤ ”ƒ†‹–‹‘ƒŽ ‰”ƒ†‹‡– „ƒ•‡† ‡–Š‘†•
ʹǤ –‘ Šƒ•–‹ ‡–Š‘† ȋ”‘„—•– ‘’–‹‹œƒ–‹‘Ȍ
͵Ǥ ‘ ‰”ƒ†‹‡– ‡–Š‘†

”ƒ†‹–‹‘ƒŽ ‰”ƒ†‹‡– „ƒ•‡† ‡–Š‘†ǣ ™Š‡ ƒ ’”‘„Ž‡ ‹• ™‡ŽŽ †‡ϐ‹‡† ƒ† –Š‡”‡ ‹• ƒ ”‡Žƒ–‹‘•Š‹’
„‡–™‡‡ †‡•‹‰ ‘„Œ‡ –‹˜‡ ƒ† †‡•‹‰ ˜ƒ”‹ƒ„Ž‡•Ǥ

‘” ‡šƒ’Ž‡ǡ ‹ˆ †‡ϐŽ‡ –‹‘ ‘ˆ ƒ ƒ–‹Ž‡˜‡” ‹• –‘ „‡ ‘’–‹‹œ‡†Ȁ‹‹‹œ‡†ǡ –Š‡ǡ –Š‡”‡ ‹• ƒ †‡ϐ‹‹–‡ ”‡Ǧ
Žƒ–‹‘•Š‹’ „‡–™‡‡ –Š‡ †‡ϐŽ‡ –‹‘ǡ †‡•‹‰ ‘„Œ‡ –‹˜‡ ƒ† ‘–Š‡” ’ƒ”ƒ‡–‡”• •— Š ƒ•ǡ ˆ‘” ‡ǡ ‰‡‘‡–”›
ƒ† ƒ–‡”‹ƒŽǡ –Šƒ– ‹•ǣ

FL 3
δ =
3EI

Where, F = is the applied force

L = is the length of the beam (geometric parameter)

E = is the material modulus

I = is the area moment of inertia (geometric parameter)

δ = is the deflection of the cantilever

Š‡ • ƒŽƒ” —Ž–‹’Ž‹‡” Ǯ͵ǯ ‹• ƒ ‘•–ƒ– ƒ† ‹• †‡’‡†‡– ‘ –Š‡ ‡† ‘†‹–‹‘• ‘ˆ –Š‡ ‡„‡”Ǥ
Š‡ †‡ϐŽ‡ –‹‘ ‘ˆ –Š‡ ƒ–‹Ž‡˜‡” ‘—Ž† „‡ ‹‹‹œ‡† „› •‡Ž‡ –‹‰ –Š‡ ƒ’’”‘’”‹ƒ–‡ ˜ƒŽ—‡• ‘ˆ Ǯ ǯǡ Ǯ ǯǡ
Ǯ ǯ ƒ† Ǯ ǯǤ Š‹• ‘—Ž† „‡ †‘‡ ‡‹–Š‡” „› –”‹ƒŽ ƒ† ‡””‘”ǡ ‘”ǡ „› ”‡Žƒ–‹‰ –Š‡ ˜ƒ”‹ƒ–‹‘ ‘ˆ –Š‡•‡
’ƒ”ƒ‡–‡”• –‘ –Š‡ ˜ƒ”‹ƒ–‹‘ ‹ ǮɁǯǤ
Š‹• ”‡Žƒ–‹‘•Š‹’ ‘—Ž† „‡ ƒ Š‹‡˜‡† —•‹‰ —Ž–‹˜ƒ”‹ƒ„Ž‡ ƒŽ —Ž—•Ǥ ‘™‡˜‡”ǡ ‹ˆ –Š‡ ƒˆ‘”‡‡–‹‘‡†
’ƒ”ƒ‡–‡”• ™‡”‡ –‘ ˜ƒ”› •–ƒ–‹•–‹ ƒŽŽ›ǡ –Š‡ǡ ƒ –”ƒ†‹–‹‘ƒŽ ‰”ƒ†‹‡– „ƒ•‡† ‡–Š‘† ƒ‘– „‡ ƒ’’Ž‹‡†Ǥ
Š‡”‡ˆ‘”‡ǡ Ž‡– —• Ž‘‘ ƒ– ƒ‘–Š‡” ‡–Š‘† ‘ˆ ‘’–‹‹œƒ–‹‘ ƒŽŽ‡† Dz•–‘ Šƒ•–‹ ‡–Š‘†dz ‘” Dz”‘„—•– ‘’Ǧ
–‹‹œƒ–‹‘dzǤ

QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries, Page 65