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Fundamentals of Stress and Vibration 1. Mathematics for Structural mechanics
[A Practical guide for aspiring Designers / Analysts]
˜ƒŽ—‡ ȋ͵Ǥͳ ‹• ˆ‘—† ˆ”‘ ‘”ƒŽ †‹•–”‹„—–‹‘ –ƒ„Ž‡ ˆ‘” ƒ ˆƒ‹Ž—”‡ ’”‘„ƒ„‹Ž‹–› Ž‡•• –Šƒ ͲǤͲͲͳȌǤ ›
™Š‡”‡ –Š‡ ˜ƒŽ—‡ ‘ˆ †‹ƒ‡–‡” ‹• ‘’—–‡†Ǥ
‘–‡ǣ ‡•‹–‹˜‹–› ‹ †‡•‹‰ ‘–‡š– ‘—Ž† ƒŽ•‘ „‡ ™‹–Š ”‡•’‡ – –‘ –‘Ž‡”ƒ ‡ •–ƒ  —’ ™Š‡”‡ ‡”–ƒ‹
–‘Ž‡”ƒ ‡• ȋ‰‡‘‡–”‹ ‘” †‹‡•‹‘ƒŽȌǡ ˆ‘” ‹•–ƒ ‡ ƒˆˆ‡ – –Š‡ ˆ— –‹‘ƒŽ‹–› ‘” ’‡”ˆ‘”ƒ ‡ ‘ˆ ƒ
‡ Šƒ‹• ƒ›„‡ •‡•‹–‹˜‡ –‘ ˆ‡™ –‘Ž‡”ƒ ‡•Ǥ ”‘„—•– †‡•‹‰ •Š‘—Ž† „‡ ƒ†‡ ‹•‡•‹–‹˜‡ –‘ –Š‡
–‘Ž‡”ƒ ‡•Ǥ
‘-‰”ƒ†‹‡– ‡–Š‘† ‘” †‡”‹˜ƒ–‹˜‡ ˆ”‡‡ ȋ‘– †‹• —••‡† ‹ –Š‹• „‘‘Ȍǣ

Š‹• ‡–Š‘† ‹• ƒ’’Ž‹‡† ™Š‡ –Š‡ ƒƒŽ›–‹ ƒŽ ”‡Žƒ–‹‘•Š‹’ „‡–™‡‡ †‡•‹‰ ‘„Œ‡ –‹˜‡ ˆ— –‹‘ ƒ† –Š‡
†‡•‹‰ ˜ƒ”‹ƒ„Ž‡• ‹• ‘– ‘™ ‘” ‹– ‹• ‘ •‘‘–Š Š‹‰ŠŽ› †‹• ”‡–‡ ‘” ‘‹•›Ǥ  •— Š ƒ•‡• †‡”‹˜ƒ–‹˜‡
‡˜ƒŽ—ƒ–‹‘ ‹• ‘ˆ ‘ —•‡ ƒŽ•‘ ‹– ƒ› ‘– „‡ ’‘••‹„Ž‡ –‘ ‡˜‡ ‘•–”— – ‘„Œ‡ –‹˜‡ ˆ— –‹‘Ǥ

‘” ‡šƒ’Ž‡ǡ ƒ”„‘ ‹†— –‹‘ ‹ ‰‡ƒ”• •ƒ› –‘ „‡ ‘–”‘ŽŽ‡† ™‹–Š ‹ ͲǤͳ ‘ˆ ƒ —”ƒ › ‡‡†• †ƒ–ƒ
‘ˆ –‘‘ ƒ› ˜ƒ”‹ƒ„Ž‡•Ǥ ‡ ‹– ‰”‡‡ †‹‡•‹‘•ǡ –‡’‡”ƒ–—”‡ǡ –‹‡ ‘ˆ ‡š’‘•—”‡ǡ ˆ—”ƒ ‡ ‡ˆϐ‹ ‹‡ ›
ƒ† ƒ› ‘”‡Ǥ Š‡ ‘’–‹— ’ƒ”ƒ‡–‡”• ‡‡†‡† ‘—Ž† „‡ Ž‘ ƒ–‡† ‘Ž› „› •–”— –—”‡† ”ƒ†‘
•‡ƒ” ŠǤ ‡‡”ƒŽŽ›ǡ •‹–—ƒ–‹‘• Š‘•‡ ˆ‘” †‡”‹˜ƒ–‹˜‡ ˆ”‡‡ •‹–—ƒ–‹‘•ǡ Šƒ˜‡ —Ž–‹’Ž‡ Ž‘ ƒŽ ‹‹ƒǤ
Š‘—‰Š •‘‡ †ƒ–ƒ •—”ˆƒ ‡• ƒ› Ž‘‘ •‘‘–Š ‹– ƒ› Šƒ˜‡ ƒ› †‹• ”‡–‡ ˆ‡ƒ•‹„Ž‡ •’ƒ ‡•Ǥ Š‘—‰Š
–Š‡ •‡ƒ” Š ƒ› ‘– ‰‹˜‡ ƒŽ™ƒ›• –”—‡ ‰Ž‘„ƒŽ ‘’–‹ƒŽ •‘Ž—–‹‘ „—– ’”‘˜‹†‡• ’”ƒ –‹ ƒŽ •‘Ž—–‹‘Ǥ

šƒ’Ž‡ ͳǣ ‘’–‹‹œ‡ –Š‡ •‡ –‹‘ ’”‘’‡”–› ȋš› Ȍ ‰‹˜‡ –Šƒ– ‹– ‹• ‹• ”‹„‡† ‹ ƒ ‹” Ž‡ǡ ƒ• •Š‘™ ‹
͵
ȏ ‹‰ ͳǤ͸͸ȐǤ

[Fig 1.66: a section inscribed in a circle]

3
Solution 1: The function to be optimized is xy
2
2
2
For a circle, by Pythagoras theorem, we have: x + y = d
Simplifying the above expression we get y = d − x
2
2
3
In order to optimize for xy let us substitute the value of ‘y’ in xy . Therefore, we get:
3
3
3
xy = d − x 2 ∗ x
2
2
QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries, Page 67