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Fundamentals of Stress and Vibration 1. Mathematics for Structural mechanics
[A Practical guide for aspiring Designers / Analysts]
ͳǤͷ ‡ –‘” Ž‰‡„”ƒ

‡ –‘”• ‘ˆ •ƒ‡ ’Š›•‹ ƒŽ ƒ–—”‡ ƒ”‡ ƒ††‡† ƒ† •—„–”ƒ –‡†ǡ ƒ†ǡ ˜‡ –‘”• ‘ˆ †‹ˆˆ‡”‡– ’Š›•‹ ƒŽ
ƒ–—”‡ ƒ”‡ —–—ƒŽŽ› —Ž–‹’Ž‹‡†Ǥ Š‡•‡ ‘’‡”ƒ–‹‘•ǡ –‘‰‡–Š‡”ǡ ‹• –‡”‡† ƒ• Dz˜‡ –‘” ƒŽ‰‡„”ƒdzǤ

ͳǤͷǤͳ ‡ϐ‹‹–‹‘ ‘ˆ ˜‡ –‘”• ƒ† • ƒŽƒ”•

‡ •–”ƒ–‡‰‹œ‡ Ž‡ƒ”‹‰ ƒ–Š‡ƒ–‹ • ˆ‘” ‡ Šƒ‹ • ˜‹ƒ ˜‡ –‘”• ƒ† „ƒ•‹ ƒŽ —Ž—• ƒ’’Ž‹ ƒ–‹‘•Ǥ

‡ –‘” “—ƒ–‹–‹‡• ƒ”‡ –Š‘•‡ –Šƒ– ƒ”‡ Šƒ”ƒ –‡”‹œ‡† „› „‘–Š †‹”‡ –‹‘ ƒ† ƒ‰‹–—†‡Ǥ ‡ –‘”•
ˆ‘ŽŽ‘™ǡ Žƒ™ ‘ˆ ˜‡ –‘” ƒ††‹–‹‘ǡ ˜‡ –‘” —Ž–‹’Ž‹ ƒ–‹‘ ƒ† —Ž–‹’Ž‹ ƒ–‹‘ „› ƒ • ƒŽƒ”Ǥ ‡– —•
—†‡”•–ƒ† ‡ƒ Š ‘‡ ‘ˆ –Š‡ ‹ †‡–ƒ‹ŽǤ
ͳǤͷǤʹ –”‘†— –‹‘ –‘ ‡ –‘”• ƒ† ƒ’’Ž‹ ƒ–‹‘•

‡ –‘”• ƒ”‡ „”‘ƒ†Ž› Žƒ••‹ϐ‹‡† ƒ• ’‘Žƒ” ƒ† ƒš‹ƒŽ ˜‡ –‘”•Ǥ Š‡ ’‘Žƒ” ˜‡ –‘”• ƒ”‡ –Š‘•‡ ™Š‘•‡
†‹”‡ –‹‘ ‹• ƒ–—”ƒŽ –‘ ˆ‘ŽŽ‘™Ǥ ‘” ‡šƒ’Ž‡ǡ –Š‡ †‹”‡ –‹‘ ‘ˆ –Š‡ ˜‡Ž‘ ‹–› ‘ˆ –Š‡ ™Š‡‡Ž –”ƒ˜‡ŽŽ‹‰

ƒŽ‘‰ ƒ •–”ƒ‹‰Š– Ž‹‡ –‘™ƒ”†• ƒ•– ˆ”‘ ‡•–ǡ ‹• ‘„˜‹‘—• ƒ† ‹• ‹†‹ ƒ–‡† „›ǣ V = vn ,
™Š‡”‡ –Š‡ —‹– ˜‡ –‘” (n ) ‹†‹ ƒ–‡• –Š‡ †‹”‡ –‹‘ ˆ”‘ ‡•– –‘ ƒ•– ƒ† Ǯ˜ǯ ‹• –Š‡ ƒ‰‹–—†‡ ‘ˆ

–Š‡ ˜‡Ž‘ ‹–› ˜‡ –‘” –Šƒ– ‹•ǡ •’‡‡†Ǥ ‡ ‡ ‡šƒ’Ž‡• ˆ‘” ’‘Žƒ” ˜‡ –‘”• ƒ”‡ Ž‹‡ƒ” ˜‡Ž‘ ‹–›ǡ Ž‹‡ƒ”
V,
ƒ ‡Ž‡”ƒ–‹‘ǡ Ž‹‡ƒ” ‘‡–—ǡ ˆ‘” ‡ ƒ† ‘–Š‡” Ž‹‡ƒ” ˜‡ –‘”•Ǥ

š‹ƒŽ ˜‡ –‘”• ƒ”‡ –Š‘•‡ ™Š‘•‡ †‹”‡ –‹‘ ‘‹ ‹†‡• ™‹–Š –Š‡ ƒš‹• ‘ˆ ”‘–ƒ–‹‘Ǥ ‘•‹†‡” ƒ ™Š‡‡Ž
”‘ŽŽ‹‰ ‘ ƒ ”‘—‰Š Š‘”‹œ‘–ƒŽ •—”ˆƒ ‡ ™‹–Š ƒ‰—Žƒ” ˜‡Ž‘ ‹–›Ǥ

[Fig a: Polar and Axial vectors]

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

QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries, Page 23