Page 55 - C:\Users\trainee\AppData\Local\Temp\msoEAA3.tmp

Document Title
Fundamentals of Stress and Vibration 2. Engineering Mechanics Chapter
[A Practical guide for aspiring Designers / Analysts]

collecting the i , j , k terms from the above expression, we get:
dm i α y − α xy − α xz + α z - - - - (2.35)
2
2
z
y
x
x
dm j −α xy + α x + α z − α yz - - - - (2.36)
2
2
x
y
y
z
dm k α x − α zx − α zy + α y - - - - (2.37)

2
2
z
x
y
z
Casting the equations (2.35), (2.36) and (2.37) in the matrix form we have:
2
2
dT x dm y + z dm(−xy) dm(−xz) α x
2
2
dT = dm(−xy) dm(x + z ) dm(−yz) α
y
y
dT z dm(−zx) dm(−zy) dm x + y α z
2
2
Integrating the above expression on both sides, we have:
dT dm y + z dm(−xy) dm(−xz)
2
2
x α
2 2 x
y

dT = dm(−xy) dm(x + z ) dm(−yz) α
y
α z

2
2
dT z dm(−zx) dm(−zy) dm x + y

dm y + z dm(−xy) dm(−xz)
2
2
T α
x x
2
2
= T = dm(−xy) dm(x + z ) dm(−yz) α
y
y

T z α z
2
2
dm(−zx) dm(−zy) dm x + y

The inertia tensor for an object in a 3 dimensional space is given as:
dm y + z dm(−xy) dm(−xz)
2
2
I −I −I
xx I xy xz 2 2
yz
Inertia tensor = −I yx yy −I = dm(−xy) dm(x + z ) dm(−yz)
−I zx −I zy I zz
2
2
dm(−zx) dm(−zy) dm x + y

The inertia tensor has perfect symmetry about the principal diagonal. Therefore, it is a symmetric
matrix and has 3 Eigen values, meaning, 3 principal inertias.
The non-diagonal terms are called cross product of inertia. These terms come about either due to
the orientation as seen in the above discussion, or, due to asymmetry of the component.
For symmetric components, such as, discs, shafts, cubes, etc., the principal diagonal terms are the
principal inertia terms, as the non-diagonal terms are zero (attributed to symmetry).
QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries, Page 55 age 55
P
Copyright Diary No – 9119/2018-CO/L