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

Note that, transformation and scaling are two different mathematical operations. In transformation,
the vector is rotated and translated, but preserved in its length. During scaling, both length and
orientation would change, unless, all the components of the vector are equally scaled, which
preserves the orientation.

Let us find the relationship between the original coordinates (x, y), and new coordinates x , y , in
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terms of (α), as shown in [Fig 1.53]. This gives us an important rule of linear algebra, that, when a
matrix operates on a vector, we get a new vector.
From the [Fig 1.53], assuming the magnitude of the vector to be ‘V’, we have:

x = V cos θ + α = V cosθ cosα − Vsinθ sinα
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y = V sin θ + α = V sinθ cosα + Vcosθ sinα
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Rewriting (V cosθ) as ‘x’ and (V sinθ) as ‘y’, we get:

x = V cos θ + α = x cosα − y sinα - - - - (1.45)
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y = V sin θ + α = y cosα + x sinα - - - - (1.46)
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Equations (1.45) and (1.46) can be rewritten in the matrix form to get a relationship between
original and transformed coordinates of vector V .

cosα −sinα x x 1
= - - - - (1.47)
sinα cosα y y 1

transformation matrix original vector = transformed vector

Let us now express the original coordinates in terms of the transformed coordinates.

In order to get ‘x’ in terms of ‘x ’, let us multiply equation (1.45) by ‘cosα’ and equation (1.46) by
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‘sinα’ and add the resulting equations.

x cosα = x cos α − y cosα sinα - - - - (1.48)
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2
y sinα = y cosα sinα + x sin α - - - - (1.49)
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Adding equations (1.48) and (1.49), we get:

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x cosα + y sinα = x cos α − y cosα sinα + y cosα sinα + x sin α
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2
x cosα + y sinα = x cos α + sin α = x cosα + y sinα = x - - - - (1.50)
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1
1
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QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries,
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