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Fundamentals of Stress and Vibration 1. Mathematics for Structural mechanics
[A Practical guide for aspiring Designers / Analysts]
ͳǤͳͷǤͳ ‘—”‹‡” ”ƒ•ˆ‘”

 ‡‰‹‡‡”‹‰ǡ ƒ› •‹–—ƒ–‹‘• ‡š‹•– ™Š‡”‡ –‹‡ †‘ƒ‹ †ƒ–ƒ ȋƒ• •Š‘™ ‹ ȏ ‹‰ ͳǤ͹ͲȐȌ †‘‡• ‘–
ƒ‡ — Š •‡•‡ǡ ”ƒ–Š‡” –Š‡ ˆ”‡“—‡ › †‘ƒ‹ †ƒ–ƒ †‘‡• ȋƒ• •Š‘™ ‹ ȏ ‹‰ ͳǤ͹ͳȐȌǤ

[Fig 1.70: Force Vs Time] [Fig 1.71: Force Vs Frequency]

šƒ’Ž‡ǣ ‘•‹†‡” ƒ ‘„‹‡† ˜‹„”ƒ–‘”› Ž‘ƒ†‹‰ ‘ˆ ƒ ‘’‘‡– †—‡ –‘ –Š”‡‡ •‘—” ‡• ‘ˆ ‡š ‹Ǧ
–ƒ–‹‘Ǥ
‡”‡ ƒ ‡‰‹‡‡” ‹• ‹–‡”‡•–‡† ‹ –Š‡ ƒ‰‹–—†‡ ‘ˆ –Š‡ ‡š ‹–ƒ–‹‘ ˆ‘” ‡ ƒ– ˜ƒ”‹‘—• ˆ”‡“—‡ ‹‡• –‘
ƒ””› ‘—– †›ƒ‹ •–”‡•• ƒƒŽ›•‹•Ǥ ‘” ‡šƒ’Ž‡ǡ ƒ Šƒ‡” –‡•– ǡ –Š‡ ”‡•’‘•‡ ‹• ƒ’–—”‡† ‹ –Š‡
–‹‡ †‘ƒ‹Ǥ Š‡ ƒ–—”ƒŽ ‘” ”‡•‘ƒ– ˆ”‡“—‡ › ‹• ‹†‡–‹ϐ‹‡† „› Ž‘ ƒ–‹‰ –Š‡ ’‘‹– ‘ˆ ƒš‹—
”‡•’‘•‡ ‹ –Š‡ ˆ”‡“—‡ › †‘ƒ‹Ǥ
‘—”‹‡” –”ƒ•ˆ‘” Šƒ• •‡˜‡”ƒŽ ƒ’’Ž‹ ƒ–‹‘•Ǥ ‡ ‹– ‹ƒ‰‡ ’”‘ ‡••‹‰ǡ ‘‹•‡ ϐ‹Ž–‡”‹‰ǡ •‘Ž˜‹‰ ’ƒ”–‹ƒŽ
†‹ˆˆ‡”‡–‹ƒŽ ‡“—ƒ–‹‘• ‡– Ǥ  ‡ Šƒ‹ ƒŽ ‡‰‹‡‡”‹‰ ™‡ —•‡ ȋ ƒ•– ‘—”‹‡” ”ƒ•ˆ‘”•Ȍ –‘
•–—†› •‹‰‹ϐ‹ ƒ– ȋϐ‹Ž–‡”‡†Ȍ ”‡•’‘•‡ ‘ˆ •›•–‡• •ƒ› ’—’ ”‹’’Ž‡ǡ –‘ ‘˜‡”– –Š‡ ˜‹„”ƒ–‹‘ ‡ƒ•—”‡Ǧ
‡– ‹ –Š‡ –‹‡ †‘ƒ‹ –‘ ˆ”‡“—‡ › †‘ƒ‹ ƒ† •‘ ‘Ǥ
ƒ–Š‡ƒ–‹ ƒŽŽ›ǡ –Š‡ ‘—”‹‡” –”ƒ•ˆ‘” ‹• ”‡’”‡•‡–‡† ƒ• ˆ‘ŽŽ‘™•ǣ

To get the frequency domain F ω , the time domain function [f(t)] is input to the Fourier
transformation function (ℱ). Therefore, we get:

∞
F ω = ℱ f t = f t e −iωt dt
−∞

The above equation is used to get frequency domain data and the inverse could be taken to get
back to time domain as follows:

f t = ℱ −1 F ω = 1 ∞ f ω e iωt dω
2π −∞

”ƒ –‹ ‹‰ ‡ Šƒ‹ ƒŽ ‡‰‹‡‡”• ‰‡‡”ƒŽŽ› —•‡ ȋ ƒ•– ‘—”‹‡” ”ƒ•ˆ‘”•Ȍ –‘ ƒ’–—”‡ ‘Ž› –Š‡
•‹‰‹ϐ‹ ƒ– ”‡•’‘•‡• —•‹‰ –‹‡ †‘ƒ‹ †ƒ–ƒǤ ‘” ‡šƒ’Ž‡ǡ ƒ Šƒ‡” ƒ› Š‹– ƒ ƒ–‹Ž‡˜‡”
‘–‹—‘—•Ž› ƒ– ˜ƒ”‹‘—• ƒ‰‹–—†‡•Ǥ Š‡ †‡ϐŽ‡ –‹‘ ‘ˆ –Š‡ ƒ–‹Ž‡˜‡” ™‹–Š –‹‡ ƒ „‡ ”‡ ‘”†‡†Ǥ

QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries, Page 73