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

Now, let us consider a ‘zyx’ transformation, meaning, the first rotation is about the ‘z-axis’, the second
rotation is about the ‘y-axis’ and the final rotation is about the ‘x-axis’.

This is mathematically represented as follows:

x
Rotation Rotation Rotation y
y
x
z
z
x x′
= R R R y = y′
y
x
z
z z′
1 0 0 cosθ 0 −sinθ cos θ sin θ 0 x x ′
′
= 0 cos θ sinθ 0 1 0 − sinθ cos θ 0 y = y - - - - (1.59)
0 − sin θ cos θ sinθ 0 cosθ 0 0 1 z z ′

Equation (1.59) could be used to transform the basis vectors as well:

1 0 0 cos θ 0 −sinθ cos θ sin θ 0 e 1 e ′ 1
′

0 cosθ sinθ 0 1 0 − sinθ cos θ 0 e 2 = e - - - - (1.60)
2
0 − sin θ cos θ sin θ 0 cos θ 0 0 1 e 3 e ′
3
Note: the above transformation matrices are valid, given that, the vector length is preserved.

Let us now consider a situation that involves shear distortion transformation.

šƒ’Ž‡ ͳǣ †‡”‹˜‡ –Š‡ –”ƒ•ˆ‘”ƒ–‹‘ ƒ–”‹š ”‡“—‹”‡† –‘ –”ƒ•ˆ‘” –Š‡ „Ž‘  •Š‘™ ‹ ȏ ‹‰
ͳǤͷͺȐ –‘ –Š‡ ‘‡ •Š‘™ ‹ ȏ ‹‰ ͳǤͷͻȐǤ Ž•‘ ‘‡– ‘ –Š‡ †‡–‡”‹ƒ– ‘ˆ –Š‡ –”ƒ•ˆ‘”ƒ–‹‘
ƒ–”‹šǤ

[Fig 1.58: Square block] [Fig 1.59: Deformed square block]

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