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

OD x
′
x = OB + BC , OB = = and BC = AB sin θ
cos θ cos θ
x
′
AB = y − x tanθ , therefore, BC = y − x tan θ sinθ and x = + y − x tan θ sin θ
cosθ
2
x sin θ
′
Simplifying for (x’) we get: x = + y sinθ − x
cosθ cos θ
x x
′
′
2
2
= x = 1 − sin θ + y sinθ = x = cos θ + y sinθ
cos θ cosθ
′
Therefore, we have: x = x cosθ + y sin θ - - - - (1.55)
sinθ
′
′
′
y = AB cos θ = y = y − x tanθ cos θ = y = y cos θ − x cos θ
cos θ
Therefore, we have: y = y cos θ − x sin θ - - - - (1.56)
′
Rewriting equation (1.55) and (1.56) in the matrix from, gives us a relationship between the
original and transformed coordinates.

cosθ sinθ x x′
= - - - - (1.57)
−sinθ cosθ y′

The equation (1.57) could be recast in terms of the basis vectors (using dot product), that is, (e – for
original coordinate system) and (e’ – for the transformed coordinate system).

′
′
e ∙ e e ∙ e x x′
1 1 1 2 = - - - - (1.58)
′
′
e ∙ e 1 e ∙ e 2 y′
2
2
In a 3D space, equation (1.57) could be represented as follows:
cosθ sinθ 0 x x ′
′
− sinθ cos θ 0 y = y
0 0 1 z z ′

QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries, Page 51