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

2
d y
Substituting the values of ‘y’ and in equation . , we get:
dx 2
2
2
− c a sinct + k a sin ct = 0 = −c + k asinct = 0
‹ ‡ ȋ•‹ –Ȍ ‹• œ‡”‘ ‘Ž› ˆ‘” •’‡ ‹ϐ‹ ‹•–ƒ ‡• ‘ˆ ȋ–Ȍǡ ȋ- Ϊ Ȍ ‰‘‡• –‘ œ‡”‘Ǥ
ʹ
2
Therefore,we have: k − c = 0 = c = K
—„•–‹–—–‹‰ –Š‡ ˜ƒŽ—‡ ‘ˆ Ǯ ǯ ‹ ‡“—ƒ–‹‘ ȋͳǤʹʹȌǡ ™‡ ‰‡– –Š‡ ”‡Žƒ–‹‘•Š‹’ „‡–™‡‡ Ǯ›ǯ ƒ† Ǯ–ǯ –‘ „‡ǣ

y = a sin kt

Š‡ —”˜‡ ‹• •‹—•‘‹†ƒŽ ƒ• •Š‘™ ‹ ȏ ‹‰ ͳǤͳ͹Ȑ

[Fig 1.17: sine curve of the form y = a sin kt]

šƒ’Ž‡ ʹǣ ‘•‹†‡” ‡“—ƒ–‹‘ ȋͳǤʹ͵Ȍǡ ™Š‹ Š ‹• –Š‡ –›’‹ ƒŽ ‡š’”‡••‹‘ ˆ‘” †ƒ’‡† ˜‹„”ƒ–‹‘Ǥ
•–ƒ„Ž‹•Š –Š‡ ”‡Žƒ–‹‘•Š‹’ „‡–™‡‡ Ǯ›ǯ ƒ† Ǯ–ǯ ˆ‘” –Š‡ †‹ˆˆ‡”‡–‹ƒŽ ‡“—ƒ–‹‘ ȋͳǤʹ͵ȌǤ

2
d y dy
c 1 + c 2 + c y = 0 - - - - (1.23)
3
dt 2 dt
bt
The solution for equation . is of the form: y t = pe + qe
at
Š‡”‡ǡ Ǯƒǯ ƒ† Ǯ„ǯ ƒ”‡ –Š‡ ”‘‘–• ‘ˆ –Š‡ “—ƒ†”ƒ–‹ ‡“—ƒ–‹‘ ȋͳǤʹͶȌǣ

2
c m + c m + c = 0 - - - - (1.24)
2
3
1
Š‡ •‘Ž—–‹‘ ’”‘’‘•‡† ‹• ‹ –Š‡ ‡š’‘‡–‹ƒŽ ˆ‘” ƒ† †‹ˆˆ‡”‡–‹ƒ–‹‰ –Š‡ ‡š’‘‡–‹ƒŽ ˆ‘” ‰‹˜‡•
–Š‡ ‡š’‘‡– –‡” —Ž–‹’Ž‹‡† „› ƒ ‘•–ƒ–ǡ ‡ƒ‹‰ǡ •—„•–‹–—–‹‰ –Š‡ •‘Ž—–‹‘ ‹ –Š‡ †‹ˆˆ‡”‡–‹ƒŽ
‡“—ƒ–‹‘ ȋͳǤʹ͵Ȍǡ ™‡ —•– ‰‡– œ‡”‘ǡ •ƒ–‹•ˆ›‹‰ –Š‡ †‹ˆˆ‡”‡–‹ƒŽ ‡“—ƒ–‹‘Ǥ

‡– —• ‘™ ϐ‹† –Š‡ ”‘‘–• ‘ˆ –Š‡ “—ƒ†”ƒ–‹ ‡“—ƒ–‹‘ ȋͳǤʹͶȌǤ

2
2
2
c 2 c − 4c c c 2 c − 4c c c 2 c − 4c c
3 1
3 1
3 1
− ± 2 , where, a = − + 2 and b = − − 2
2c 1 2c 1 2c 1 2c 1 2c 1 2c 1

QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries, Page 21