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Document Title
Fundamentals of Stress and Vibration Chapter Title
[A Practical guide for aspiring Designers / Analysts] 2. Engineering Mechanics

Š‡ ”‘† ‹• ‘”‹‡–‡† ƒ† ”‘–ƒ–‡† ‹ ƒŽŽ ’‘••‹„Ž‡
’Žƒ‡• ƒ† ƒš‡•Ǥ ‡– —• ‘™ ‘ŽŽ‡ – ƒ† –ƒ„—Žƒ–‡
–Š‡ ‹‡”–‹ƒ –‡”•ǣ

••—‹‰ –Šƒ–ǡ ƒ ‡Ž‡‡–ƒŽ ƒ•• ȋ†Ȍ ‹•
Ž‘ ƒ–‡† ƒ– ‘‘”†‹ƒ–‡• ȋšǡ ›ǡ œȌ ƒ• •Š‘™ ‹ ȏ ‹‰
ʹǤͶʹȐǤ

[Fig 2.42: an elemental mass in a 3D space given acceleration about all 3 axes]

Let us now use the matrix form of vector product to find α × r .

i j k

α × r = α x α y α , where r = xi + yj + zk
z
x y z
Expanding the matrix, we get:

α × r = i α z − α y − j α z − α x + k α y − α x
x
y
z
y
z
x
Substituting the above expression of α × r in equation (2.34), we get:

dT = dm r × i α z − α y − j α z − α x + k α y − α x
y
x
x
z
z
y
Rewriting the above vector product in the matrix form, we get:
i j
k

dT = dm x y z
α z − α y − α z − α x α y − α x
x
z
y
z
y
x
Expanding the above matrix, we get:
dm i y α y − α x − z α x − α z − j x α y − α x − z α z − α y
y
x
x
x
z
y
y
z

+ k x α x − α z − y α z − α y
z
y
x
z

Page 54 QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries,