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

dT

T = −sinθ i + cosθ j = = − cosθ i + sinθ j = −r - - - - (1.44)
dθ
The facts expressed in equations (1.43) and (1.44), are used in deriving accelerations for curvilinear
motion and many other rigid body dynamic situations.

ͳǤͻ ”ƒ•ˆ‘”ƒ–‹‘

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

ͳǤͻǤͳ ‡ –‘” ”ƒ•ˆ‘”ƒ–‹‘

Consider a vector V , in the ‘x-y’ plane, passing through the origin, as shown in [Fig 1.53].

[Fig 1.53: Rotation of a vector]

The vector V has an initial orientation of angle (θ), with respect to positive ‘x-axis’.

The vector V is rotated by an angle (α) in the anti-clockwise direction. Let us derive a matrix,

relating (x, y) to x , y .
1
1

QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries, Page 47