40

Section 5 Mass Moment of Inertia

Center of Gravity The center of gravity, cg, of an object is the balance point of that object. That is, it is the single point at which the object’s weight could be held and be in balance in all directions. . For parts made of homogeneous material, the cg is the three-dimensional, geometric center of the object. For many simple parts, such as a cylinder, the geometric center is apparent.

For complex parts, the location of the center of gravity is not obvious. A common method of determining the center of gravity is to divide the complex part into primary shapes, where the center of gravity of each is apparent. The composite center of gravity can be determined form a weighted average of the coordinates of the individual cg’s.

...

...

321

332211

+++

+++=

partpartpart

partcgpartpartcgpartpartcgparttotalcg mmm

xmxmxmx

...

...

321

332211

+++

+++=

partpartpart

partcgpartpartcgpartpartcgparttotalcg mmm

ymymymy

...

...

321

332211

+++

+++=

partpartpart

partcgpartpartcgpartpartcgparttotalcg mmm

zmzmzmz

Since the acceleration due to gravity will be the same for the entire body, weight

can be substituted for mass in the equations above. Mass Moment of Inertia (IA) Mass moment of intertia is a measure of an object's resistance to rotational acceleration

• It is more difficult to “speed up” a spinning object with a large mass moment of inertia.

• Charts available for common volumes. • Computed relative to an axis. • Strongly influenced by the amount of mass distributed from the axis.

∫= dmrI A2

r

dm

AA

41

Mass Moments of Inertia for Primary Shapes

Rectangular Block: ( )[ ]22

121

hwmI x +=

( )[ ]22

121

lwmI y +=

( )[ ]22

121

lhmI z +=

Cylinder:

[ ]2

21

mrI x =

( )[ ]223121

lrmI y +=

( )[ ]223121

lrmI z +=

Thin Disk:

[ ]2

21

mrI x =

[ ]2

41

mrI y =

[ ]2

41

mrI z =

Slender Rod: 0=xI

[ ]2

121

mlI y =

[ ]2

121

mlI z =

l

h

w

x

z

y

l

r

x

z

y

r

x

z

y

l

x

z

y

42

Radius of Gyration (k): Distance from the reference axis to a point, where a concentrated mass would have the same moment of inertia.

mI

k AA =

Parallel Axis Theorem

To transfer the mass moment of inertia, from one axis to another, the following equation is used..

IA’ = IA ± md2

ü d is the distance between the two axes. ü Add if transfer is away from the centroid. ü Subtract if transfer is towards the centroid.

Composite Bodies:

The mass moment of inertia of a body, comprised of several simple shapes, can be combined as long as there is a single reference axis.

IA (Total) = IA (Body 1) + IA (Body 2) +... Experimental Determination:

The mass moment of inertia of a body, can be determined experimentally by swinging the part as a pendulum and measuring the time to complete one oscillation (∆t).

−

∆

= cgcgcg rgt

mrI2

2π

Hints for Determining the Moment of Inertia:

• When determining the center of gravity, use symmetry as much as possible • Divide a complex shape into a minimum number of primary shapes. • Select a convenient location to place a reference coordinate system. • Use the primary shape equations, which correspond to the reference coordinate

system. Ix in the table may actually be Iz in the analysis. • Make certain that a cutout is tabulated as a negative mass and moment of inertia

...

...

21

2211

+−+

+−+=

−

−−

outcutpartpart

outcutcgoutcuopartcgpartpartcgparttotalcg mmm

zmzmzmz

IA (Total) = IA (Body 1) + IA (Body 2) - IA (cut-out) + ...

• Double check the distance between inertial axes.

rcg

43

Problems: 5-1. The plate shown is made from steel (0.283 lb/in3). Determine the coordinates of

the center of gravity. 4Ans: 5.74, .25, 0 in.

5-2. The plate shown is made from steel (0.283 lb/in3). Determine the coordinates of

the center of gravity. 4Ans: -1.81, .14, .38 in. 5-3. Calculate the mass moment of inertia and the radius of gyration about a centroidal

longitudinal axis of a shaft that weighs 5 lb and has a diameter of 0.625 in. 4Ans: 0.000632 lb in s2

5-4. Calculate the mass moment of inertia and the radius of gyration about a centroidal

longitudinal axis of a shaft that has a mass of 100 kg and a diameter of 50 mm. 4Ans: 0.03125 kg m2

5-5. A solid cylinder is 2 ft in diameter, 3 ft long and weighs 48 lbs. Determine the

mass moment of inertia about its centroidal axial axis. 4Ans: 8.944 lb in s2

5-6. A solid cylinder is 2 ft in diameter, 3 ft long and weighs 48 lbs. Determine the

mass moment of inertia about a centroidal axis, perpendicular to its length. 4Ans: 17.89 lb in s2

5-7. A slender rod, 14 in long, rotates about an axis perpendicular to its length and 3

inches from its center of gravity. Knowing that the rod weighs 2 lb, determine its mass moment of inertia about that axis.

4Ans: 0.131 lb in s2

d

d

δ

6” 3”

12”

3” ∅2”

0.5” x

z

y

x

.125 in

.75 in 1.75 in

∅.375 in

y

3.25 in

.75 in

z

∅.375 in

44

5-8. A slender rod, 0.4 m long, rotates about an axis perpendicular to its length and 0.12 m from its center of gravity. Knowing that the rod has a mass of 6 kg, determine its mass moment of inertia about that axis.

4Ans: 0.1664 kg m2 5-9. Determine the mass moment of inertia of the plate about the rotation axis. The

plate is made from steel, with a density of 0.283 lb/in3. 4Ans: 0.036 lb in s2

5-10. A flywheel can be considered as being composed of a thin disk and a rim. The rim

weighs 300 lbs and has diameters of 24 in and 30 in. The disk weighs 50 lbs. Determine the mass moment of inertia about the centroidal axis which the flywheel rotates.

4Ans: 40.76 lb in s2 5-11. The cross section of a steel (0.283 lb/in3)

flywheel is shown. The web of the flywheel consists of a solid plate 1 in. thick. Determine the mass moment of inertia of the flywheel with respect to the axis of rotation

4Ans: 6.63 lb in s2

4 in 3.5 in

2 in 0.5 in

30 in 24 in

A

A

Section A-A

6 in

1 in

2 in. 16 in.

12 in. 4 in.

4 in.

6 in.

1 in.

45

5-12. The cross section of a small steel flywheel is shown. The rim and hub are connected by eight spokes (two of which are shown in the cross section). Each spoke has a cross sectional area of 160 mm2. Determine the mass moment of inertia of the flywheel with respect to the axis of rotation. The density of steel is 7850 kg/m3.

4Ans: 0.211 kg m2 5-13. A shaft of a shredder has four cutter blades welded to the shaft. Each blade has a

mass of 1 kg. The shaft has a mass of 4.5 kg. Determine the mass moment of inertia about the centroidal (z) axis of the shaft.

4Ans: 0.0161 kg m2 5-14. Determine the mass moment of inertia of the plate about an axis through its center

of gravity. The plate is made from steel, with a density of 0.283 lb/in3. 4Ans: 0.0044 lb in s2

60 mm

12 mm

180 mm 120 mm

30 mm

800 mm

Z

Z

30 mm 440

380 60

100

20

4.0 in

2 in 0.5 in

0.75 in

0.75 in

46

5-15. The section of sheet steel is 2mm thick anbd is cut and bent into the machine component shown. The density is 7850 kg/m3. Determine the mass moment of inertia relative to the (a) x-axis, (b) y-axis, (c) z-axis.

5-16. Determine the moment of inertia of the steel link (ρ = 0.283 lb/in3) with respect to

the y axis. 4Ans: 0.000191 lb in s2 5-17. Determine the moment of inertia of the steel link (ρ = 0.283 lb/in3) with respect to

the y axis. 4Ans: 0.00024 lb in s2 5-18. Experiments reveal that the moment

of inertia relative of the 2.2 kg connecting arm is 0.80 kg m2 and 0.40 kg m2, relative to axis A and B, respectively. Determine the location of the center of gravity.

4Ans: rB=55 mm

x

120 mm

120 mm

150 mm 150 mm

70 mm

70 mm

y

z

275 mm

A

B rB

.125 in

.75 in 1.75 in

∅.375 in y

3.25 in

.75 in

∅.375 in

.125 in

.75 in

1.75 in

∅.375 in -typ.

y

3.25 in