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Mass Moment of InertiaEngineering Mechanics- Distributed Forces

• Encounter in the engineering problems involving rotational motion of rigid bodies in dynamics

Moment of Inertia of a Mass• Angular acceleration about the axis AA’ of the

small mass Dm due to the application of a couple is related to r2Dm.

r2Dm = moment of inertia of the mass Dm with respect to the axis AA’

• For a body of mass m the resistance to rotation about the axis AA’ is

inertiaofmomentmassdmr

mrmrmrI

2

23

22

21

• The radius of gyration for a concentrated mass with equivalent mass moment of inertia is

mIkmkI 2

Engineering Mechanics- Distributed Forces

12/3/2013

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Moment of Inertia of a Mass

• Moment of inertia with respect to the ycoordinate axis is

dmxzdmrI y222

• Similarly, for the moment of inertia with respect to the x and z axes,

dmyxI

dmzyI

z

x22

22

Engineering Mechanics- Distributed Forces

Parallel Axis Theorem• For the rectangular axes with origin at O and parallel

centroidal axes,

dmzydmzzdmyydmzy

dmzzyydmzyIx2222

2222

22

22 zymII xx

22

22

yxmII

xzmII

zz

yy

• Generalizing for any axis AA’ and a parallel centroidal axis,

2mdII

Engineering Mechanics- Distributed Forces

12/3/2013

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Moments of Inertia of Thin Plates• For a thin plate of uniform thickness t and homogeneous

material of density r, the mass moment of inertia with respect to axis AA’ contained in the plate is

IA A r 2dm t r 2 dA t IA A ,area

• Similarly, for perpendicular axis BB’ which is also contained in the plate,

IB B t IB B ,area

• For the axis CC’ which is perpendicular to the plate,IC C t JC ,area t IA A ,area IB B ,area

IA A IB B

Engineering Mechanics- Distributed Forces

Moments of Inertia of Thin Plates• For the principal centroidal axes on a rectangular plate,

21213

121 mabatItI area,AAAA

21213

121 mbabtItI area,BBBB

22121 bamIII mass,BBmass,AACC

• For centroidal axes on a circular plate,

2414

41 mrrtItII area,AABBAA

221 mrIII BBAACC

Engineering Mechanics- Distributed Forces

12/3/2013

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Mass Moment of Inertia of a Slendor Rod

2zI r dm

2

0

3 2 2

( )

/

( )3 3 3

L

zI x mdx

mm mass unit lengthL

mL mL L mL

2 2

2 12z zcg

L mLI I m

Engineering Mechanics- Distributed Forces

Moments of Inertia of a 3D Body by Integration

• Moment of inertia of a homogeneous body is obtained from double or triple integrations of the form

dVrI 2

• For bodies with two planes of symmetry, the moment of inertia may be obtained from a single integration by choosing thin slabs perpendicular to the planes of symmetry for dm.

Engineering Mechanics- Distributed Forces

12/3/2013

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Mass Moment of Inertia of a Rectangular PrismEngineering Mechanics- Distributed Forces

Find mass moment of inertia about Z axis

Engineering Mechanics- Distributed Forces

Mass Moment of Inertia of a Sphere

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Moments of Inertia of Common Geometric ShapesEngineering Mechanics- Distributed Forces

Sample Problem: Composite Section

Determine the moments of inertia of the steel forging with respect to the xyz coordinate axes, knowing that the specific weight of steel is 7896 kg/m3.

SOLUTION:

• With the forging divided into a prism and two cylinders, compute the mass and moments of inertia of each component with respect to the xyz axes using the parallel axis theorem.

• Add the moments of inertia from the components to determine the total moments of inertia for the forging.

Engineering Mechanics- Distributed Forces

12/3/2013

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Sample Problem: Composite Section

kg0744.01031

kg/m7896

:cylindereach

362

3

mm

Vm

2

222121

222121

kgm5394.0

5.20744.03130744.0

3

xmLamI y

2 2 2 2112

2 2 2 2112

3

0 0744 3 1 3 0 0744 2 5 2

0 837 2kgm

zI m a L m x y

. . .

.

2

2221

2221

kgm5394.0

20744.010744.0

ymmaI x

cylinders :cm2cm,5.2,cm3,cm1 yxLaSOLUTION:• Compute the moments of inertia

of each component with respect to the xyz axes.

Engineering Mechanics- Distributed Forces

Sample Problem: Composite Section

kg1895.0

m10622kg/m7896

:prism36–3

mVm

prism (a = 2 cm, b = 6 cm, c = 2 cm):

2

2212122

121

gmk 6320

2618950

.

.cbmII zx

2

2212122

121

gmk 1260

2218950

.

.acmI y

• Add the moments of inertia from the components to determine the total moments of inertia.

372.02632.0 xI2kgm376.1xI

5394.02126.0 yI2kgm2048.1yI

837.02632.0 zI2kgm306.2zI

Engineering Mechanics- Distributed Forces