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Fundamentals of Stress and Vibration 1. Mathematics for Structural mechanics
[A Practical guide for aspiring Designers / Analysts]
šƒ’Ž‡ ͵ǣ ϐ‹† –Š‡ ƒš‹— ƒ† ‹‹— ˜ƒŽ—‡• ‘ˆ –Š‡ ˆ— –‹‘ a cos θ + b sin θ .

‘Ž—–‹‘ ͵ǣ ‹ ‘”†‡” –‘ ϐ‹† –Š‡ ƒš‹— ƒ† ‹‹— ˜ƒŽ—‡ǡ ™‡ ™‘—Ž† ƒ‡ —•‡ ‘ˆ –Š‡ ˆƒ –
–Šƒ– Ǯ•‹Ʌǯ ˜ƒ”‹‡• „‡–™‡‡ ƒš‹— ȋΪͳȌ ƒ† ‹‹— ȋ-ͳȌǤ

‘ ƒ’’Ž› –Š‡ ƒ„‘˜‡ ˆƒ – Ž‡– —• †‹˜‹†‡ ƒ† —Ž–‹’Ž› –Š‡ ‡š’”‡••‹‘ ȋƒ ‘• Ʌ Ϊ „ •‹ ɅȌ ™‹–Š a + b ,
2
2
— Š –Šƒ–ǡ ™‡ ‘—Ž† —•‡ –Š‡ ˆ‘”—Žƒ ˆ‘” •‹ȋ Ϊ ȌǤ Š‡”‡ˆ‘”‡ǡ ™‡ Šƒ˜‡ǣ

a cos θ + b sin θ a b
2
2
2
2
∗ a + b = cos θ + sinθ a + b
2
2
2
a + b 2 a + b 2 a + b 2
”‘ –Š‡ ȏ ‹‰ ͳǤͻȐǡ ™‡ Šƒ˜‡ǣ

[Fig 1.9: by Pythagoras theorem]

= sinΦ cos θ + cos Φ sin θ a + b = sin Φ + θ ∗ a + b - - - - (1.6)
2
2
2
2
Since we know that the value of sin θ varies from (+1) to (-1),
The maximum value of equation (1.6) = 1 ∗ a + b
2
2
The minimum value of equation (1.6) = −1 ∗ a + b
2
2

QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries, Page 11