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Document Title
Fundamentals of Stress and Vibration Chapter Title
[A Practical guide for aspiring Designers / Analysts] 2. Engineering Mechanics
The rod is given an angular acceleration ‘α’, about the vertical axis. Consider an element on the rod
with coordinates (x, y). This element experiences a linear acceleration αx . Therefore, the linear
inertia of the element is given by dmαx .
The linear inertia has a moment about both ‘x and y-axis’, given by:
About the ‘x-axis’ = dmαx ∗ y
About the ‘y-axis’ = dmαx ∗ x
Therefore, if we sum up the contributions of all the elements along the length of the rod, we would
get two inertia terms:
2
1) dmαx = α dmx
2
2) dmαxy = α dmxy
The first term gives the inertia of the rod about the ‘y-axis’. However, the second term is called the
cross product of inertia, as, it contains two coordinates (x and y).
If the angular acceleration were given about the ‘x-axis’, as shown in [Fig 2.37], then the two inertia
terms would be:
2
2
1) dmαy = α dmy
2) dmαyx = α dmyx
These terms have the same explanations as before.
[Fig 2.37: an inclined rod accelerated about the 'x-axis']
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