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Fundamentals of Stress and Vibration
[A Practical guide for aspiring Designers / Analysts] 1. Mathematics for Structural mechanics
—„•–‹–—–‹‰ –Š‡ ˜ƒŽ—‡ ‘ˆ Ǯ ǯ ‹ ‡“—ƒ–‹‘ ȋͳǤʹͲȌǡ ™‡ ‰‡– –Š‡ ”‡Žƒ–‹‘•Š‹’ „‡–™‡‡ Ǯ›ǯ ƒ† Ǯšǯ –‘ „‡ǣ

k
y = x − this is parabolic equation of degree k .
‘ ’Ž‘– –Š‡ —”˜‡ǡ Ž‡– —• ƒ••—‡ ȋ α ͵Ȍ ƒ• •Š‘™ ‹ ȏ ‹‰ ͳǤͳ͸Ȑ

3
[Fig 1.16: parabola of form [y = x ]]

‘–‡ǣ –Š‡ •‘Ž—–‹‘ –‘ ƒ› †‹ˆˆ‡”‡–‹ƒŽ ‡“—ƒ–‹‘ ‹• ƒŽŽ ƒ„‘—– ‡•–ƒ„Ž‹•Š‹‰ ƒ ”‡Žƒ–‹‘•Š‹’ „‡–™‡‡ –Š‡
†‡’‡†‡– ƒ† ‹†‡’‡†‡– ˜ƒ”‹ƒ„Ž‡•Ǥ

ͳǤͶǤ͹ ʹ ‘”†‡” ‹ˆˆ‡”‡–‹ƒŽ “—ƒ–‹‘•
†

 •‡˜‡”ƒŽ ’”ƒ –‹ ƒŽ ‡‰‹‡‡”‹‰ •‹–—ƒ–‹‘•ǡ •— Š ƒ•ǡ ˆ”‡‡ ˜‹„”ƒ–‹‘ǡ „— Ž‹‰ǡ ‡– Ǥǡ –Š‡ ʹ ‘”†‡”
†
†‹ˆˆ‡”‡–‹ƒŽ ‡“—ƒ–‹‘ ‹• ‘ˆ”‘–‡†Ǥ

šƒ’Ž‡ ͳǣ ϐ‹† –Š‡ ”‡Žƒ–‹‘•Š‹’ „‡–™‡‡ Ǯ›ǯ ƒ† Ǯ–ǯ ˆ”‘ –Š‡ †‹ˆˆ‡”‡–‹ƒŽ ‡“—ƒ–‹‘ ȋͳǤʹͳȌǡ ‰‹˜‡
–Š‡ ‹‹–‹ƒŽ ‘†‹–‹‘ ȋ› α ͲȌ ƒ– ȋ– α ͲȌǤ
2
d y
+ ky = 0 - - - - (1.21)
dt 2
‹ ‡ –Š‡ ‡“—ƒ–‹‘ ‹• ƒ ʹ ‘”†‡” Š‘‘‰‡‘—• †‹ˆˆ‡”‡–‹ƒŽ ‡“—ƒ–‹‘ǡ ‹– ‹• ‡ƒ•› –‘ ’”‘’‘•‡ ƒ •‘Ž—–‹‘Ǥ
†
– ƒ „‡ ‘„•‡”˜‡† –Šƒ–ǡ —’‘ †‹ˆˆ‡”‡–‹ƒ–‹‰ ƒ •‹—•‘‹†ƒŽ ‘” ƒ ‘•‹—•‘‹†ƒŽ ˆ— –‹‘ –™‹ ‡ǡ ‹ –Š‡
‰‹˜‡ †‹ˆˆ‡”‡–‹ƒŽ ‡“—ƒ–‹‘ǡ ™‡ ‰‡– „ƒ  –‘ –Š‡ •ƒ‡ ˆ— –‹‘ǡ „—– ™‹–Š ƒ ‡‰ƒ–‹˜‡ •‹‰Ǥ

‡– —• ƒ••—‡ –Š‡ •‘Ž—–‹‘ –‘ –Š‡ †‹ˆˆ‡”‡–‹ƒŽ ‡“—ƒ–‹‘ –‘ „‡ ‘ˆ –Š‡ ˆ‘”ǣ y = a sinct - - - - (1.22)
Š‡ Š‘‹ ‡ ‘ˆ •‘Ž—–‹‘ ƒŽ•‘ ‹• ‹ Ž‹‡ ™‹–Š –Š‡ ‹‹–‹ƒŽ ‘†‹–‹‘ •—‰‰‡•–‡†Ǥ

dy
Differentiating equation . , we get: = ca cos ct
dx

2
d y
2
Differentiating again, we get: = − c a sinct
dx 2
QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries,
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