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Fundamentals of Stress and Vibration
[A Practical guide for aspiring Designers / Analysts] 1. Mathematics for Structural mechanics
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Let us simplify equation (1.75) to compute a relationship between X and X .
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2
1
′
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10 ∗ X 10 ∗ X
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′
′
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X = − 2 = X = − 2 = X = −0.351 X
1
50 − λ 1 50 − 21.49 2 2
2
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Substituting (X = 1), we get the components of the second Eigen vector to be: −0.351
2
1
The Eigen vectors are invariant (unchanged) under transformation.
ͳǤͳͳ ‰‹‡‡”‹‰ —”˜‡•
‰‹‡‡”‹‰ —”˜‡• Šƒ˜‡ ƒŽ™ƒ›• „‡‡ ”‹–‹ ƒŽ –‘ –Š‡ ‹‡ƒ–‹ • ‘ˆ •‡˜‡”ƒŽ ‡ Šƒ‹••ǡ •— Š ƒ•ǡ
‰‡ƒ”•ǡ ƒǯ•ǡ –‡Ž‡• ‘’‡•ǡ ‘‘Ž‹‰ –‘™‡”• ‡– Ǥ
 –Š‹• •‡ –‹‘ǡ ™‡ ™‘—Ž† Ž‘‘ ƒ– –Š”‡‡ ‹’‘”–ƒ– —”˜‡•ǡ ƒ‡Ž›ǣ

ͳǤ › Ž‘‹†
ʹǤ ˜‘Ž—–‡
͵Ǥ ‘‰ƒ”‹–Š‹ •’‹”ƒŽ

Š‡ ƒ–Š‡ƒ–‹ ƒŽ ‡š’”‡••‹‘ ˆ‘” –Š‡ ƒ„‘˜‡ —”˜‡•ǡ ƒ”‡ †‡”‹˜‡† „ƒ•‡† ‘ –Š‡ ˆƒ – –Šƒ–ǡ –Š‡› ƒ”‡
‰‡‡”ƒ–‡† „› ‘‡ ‰‡‘‡–”‹ ‡Ž‡‡– ”‘ŽŽ‹‰ ‘ –Š‡ ‘–Š‡”Ǥ

ͳǤͳͳǤͳ › Ž‘‹†

› Ž‘‹† ‹• ‰‡‡”ƒ–‡† „› –”ƒ ‹‰ ƒ ’ƒ”–‹ —Žƒ” ’‘‹– ‘ ƒ ‹” Ž‡ –Šƒ– ”‘ŽŽ• ’‡”ˆ‡ –Ž› ‘ ƒ •–”ƒ‹‰Š– Ž‹‡ǡ
™‹–Š‘—– •Ž‹’’‹‰ǡ ƒ• •Š‘™ ‹ ȏ ‹‰ ͳǤ͸ͲȐǤ

[Fig 1.60: Cycloidal Motion]

Š‡ –Š‡ ‹” Ž‡ ™‘—Ž† Šƒ˜‡ ”‘ŽŽ‡† –Š”‘—‰Š ƒ ƒ‰—Žƒ” †‹•–ƒ ‡ ǮɅǯǡ „› –Š‡ ‘ ‡’– ‘ˆ ”‘ŽŽ‹‰ǡ –Š‡
†‹•–ƒ ‡ „‡–™‡‡ –Š‡ ‘–ƒ – ’‘‹–• ‹• Ǯ ɅǯǤ

Š‡ › Ž‘‹† ‹• †‡ϐ‹‡† „› –Š‡ ”‡Žƒ–‹‘•Š‹’ „‡–™‡‡ Ǯšǯ ƒ† Ǯ›ǯ ‘‘”†‹ƒ–‡•Ǥ Š‡”‡ˆ‘”‡ǡ ƒ ‡š’”‡••‹‘
ˆ‘” Ǯšǯ ƒ› Ǯ›ǯ ‹ –‡”• ‘ˆ Ǯ ǯ ƒ† ǮɅǯǡ ™‘—Ž† ‰‹˜‡ ƒ ‡“—ƒ–‹‘ ˆ‘” › Ž‘‹†Ǥ

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