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Fundamentals of Stress and Vibration
[A Practical guide for aspiring Designers / Analysts] 1. Mathematics for Structural mechanics
Let us interpret the numerator (ax bx + ay by) , by relating vectors to physical quantities as follows:
Consider the displacement and force vectors to be ( d ) and ( F ) respectively, as shown in [Fig 1.29].
[Fig 1.29: Scalar product]
ǣ
d = d i + d j and F = F i + F j
x
y
x
y
The work done in the ‘x’ direction is: W = F ∙ d
x
x
x
The work done in the ‘y’ direction is: W = F . d
y
y
y
The total work done is given by:
W + W = F . d + F . d
x
y
y
y
x
x
If we were to assume, vectors ( F ) and ( d ) are equivalent to vectors ( a ) and ( b ) (as discussed in
addition and subtraction of vectors), we get:
W + W = a . b + a . b
y
y
x
x
y
x
Further by projecting ( F ) onto ( d ) as shown in [Fig 1.30], the work done is given by:
Work done = F d cosθ Force in the direction of displacement ∗ displacement
QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries,
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