ENGR-36_Lec-27_Mass_Moment_of_Inertia
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Transcript ENGR-36_Lec-27_Mass_Moment_of_Inertia
Engineering 36
Chp10:
Moment of Interia
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
[email protected]
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
[email protected] • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
Mass Moments of Inertia
The Previously Studied “Area Moment
of Inertia” does Not Actually have True
Inertial Properties
• The Area Version is More precisely Stated
as the SECOND Moment of Area
Objects with Real mass DO have inertia
• i.e., an inertial Body will Resist Rotation by
An Applied Torque Thru an F=ma Analog
T Iα
I Mass Momentof Inertia
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Bruce Mayer, PE
[email protected] • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
Mass Moment of Inertia
The Moment
of Inertia is
the Resistance
to Spinning
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Bruce Mayer, PE
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Linear-Rotational Parallels
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Bruce Mayer, PE
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Mass Moment of Inertia
The Angular acceleration, , about the axis
AA’ of the small mass m due to the
application of a couple is
proportional to r2m.
• r2m moment of inertia of the mass m
with respect to the axis AA’
For a body of mass m the resistance to
rotation about the axis AA’ is
I r12 m r22 m r32 m r 2 m
r 2 dm m ass m om entof inertia
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Bruce Mayer, PE
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Mass Radius of Gyration
Imagine the entire Body Mass
Concentrated into a single Point
Now place this mass a distance k from
the rotation axis so as to create the
same resistance to rotation as the
original body
• This Condition Defines, Physically, the
Mass Radius of Gyration, k
Mathematically
I
I r dm k m or k
m
2
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Ix, Iy, Iz
Similarly, for the
moment of inertia with
respect to the x and z
axes
2
2
I z x 2 y 2 dm
Units Summary
I r 2 dm kg m 2
I x y z dm
Mass Moment of inertia
with respect to the
y coordinate axis r is
the ┴ distance to y-axis
I y r 2 dm z 2 x 2 dm
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• SI
• US Customary Units
lb s 2 2
2
I slug ft
ft lb ft s 2
ft
Bruce Mayer, PE
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Parallel Axis Theorem
x x ' x
The Translation
Relationships
y y ' y
Then Write Ix
z z ' z
y 2 z 2 dm 2 y y dm 2 z z dm y 2 z 2 dm
I x y 2 z 2 dm y y 2 z z 2 dm
0
0
In a Manner Similar to
Consider CENTRIODAL
the Area Calculation
Axes (x’,y’,z’) Which are
• Two Middle Integrals are
Translated Relative to
1st-Moments Relative to
the Original CoOrd
the CG → 0
Systems (x,y,z)
• The Last Integral is
the Total Mass
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Bruce Mayer, PE
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m
Parallel Axis Theorem cont.
Similarly for the Other
two Axes
mx
I y I y m z x
I z I z
So Ix
I x y2 z2 dm 0 0 y 2 z 2 m
I x I x' m y z
so
2
2
I x I x m y z
2
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2
2
2
2
y2
In General for any axis
AA’ that is parallel to a
centroidal axis BB’
I I md
Also the Radius of
Gyration
2
k k d
2
2
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2
Thin Plate Moment of Inertia
For a thin plate of uniform thickness t and
homogeneous material of density , the mass
moment of inertia with respect to axis AA’
contained in the plate
I AA r 2 dm r 2 tdA t r 2 dA t I AA,area
Similarly, for perpendicular axis BB’ which is also
contained in the plate
I BB t I BB,area
For the axis CC’ which is PERPENDICULAR to the
plate note that This is a POLAR Geometry
I CC t J C ,area t I AA,area I BB,area
I AA I BB
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Bruce Mayer, PE
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Polar Moment of Inertia
The polar moment of inertia is an important parameter in
problems involving torsion of cylindrical shafts, Torsion
in Welded Joints, and the rotation of slabs
In Torsion Problems, Define a Moment of Inertia
Relative to the Pivot-Point, or “Pole”, at O
J O r dA
2
Relate JO to Ix & Iy Using The
Pythagorean Theorem
J O r 2 dA x 2 y 2 dA x 2 dA y 2 dA
Iy Ix
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Bruce Mayer, PE
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Thin Plate Examples
For the principal centroidal axes on a
rectangular plate
I AA t I AA,area t 121 a3b 121 taba2 121 ma2
I BB t I BB,area t
1
12
ab3 121 tabb2 121 mb2
ICC I AA,mass I BB,mass 121 m a2 b2
For centroidal axes on a
circular plate
I AA I BB t I AA,area t 14 r 4 14 mr 2
I CC I AA I BB 12 mr 2
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Bruce Mayer, PE
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3D Mass Moments by Integration
• The Moment of inertia of a
homogeneous body is obtained from
double or triple integrations of the form
I r 2 dV
• 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.
• The moment of inertia with respect to a
particular axis for a COMPOSITE body
may be obtained by ADDING the
moments of inertia with respect to the
same axis of the components.
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Bruce Mayer, PE
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Common Geometric Shapes
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Bruce Mayer, PE
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Example 1
SOLUTION PLAN
Determine the moments
of inertia of the steel
forging with respect to
the xyz coordinate
axes, knowing that the
specific weight of steel
is 490 lb/ft3 (0.284 lb/in3)
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• With the forging divided
into a Square-Bar 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.
Bruce Mayer, PE
[email protected] • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
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Bruce Mayer, PE
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Example 1 cont.
Referring to the
Geometric-Shape Table
for the Cylinders
•
•
•
•
Then the Axial (x)
Moment of Inertia
For The Symmetrically
Located Cylinders
m
V
g
490lb/ft 1 3in
1728in ft 32.2 ft s
3
3
2
3
m 0.0829lb s 2 ft
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a = 1” (the radius)
L = 3”
xcentriod = 2.5”
ycentriod = 2”
3
2
I x I x my 2 12 m a2 my 2
1
2
0.0829121 2 0.0829122 2
2.59103 lb ft s 2
Bruce Mayer, PE
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Example 1 cont.2
Now the Transverse
(y & z) Moments of Inertia
I y I y mx 2 121 m 3a 2 L2 mx 2
121 0.0829 3121 123 0.0829 212.5
2
2
2
dz
4.17 103 lb ft s 2
d z2
I z 121 m 3a 2 L2 m x 2 y 2
121 0.0829 3121 123 0.0829 212.5 122
2
2
2
2
6.48103 lb ft s 2
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Bruce Mayer, PE
[email protected] • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
Example 1 cont.3
Referring to the
Geometric-Shape Table
for the Block
• a = 2”
• b = 6”
• c = 2”
Then the Transverse (x
& z ) Moments of Inertia
For The Sq-Bar
m
V
g
490lb/ft 2 2 6in
1728in ft 32.2 ft s
3
3
3
3
m 0.211lb s 2 ft
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2
I x I z 121 m b 2 c 2
1
12
0.211 122 2
6 2
12
4.88103 lb ft s 2
Bruce Mayer, PE
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Example 1 cont.4
Add the moments of
inertia from the
components to
determine the total
moment of inertia.
I x 4.88 103 2 2.59 103
1
12
I y 9.32 103 lb ft s 2
0.211
2 2
12
I x 10.06 103 lb ft s2
I y 0.977 103 2 4.17 103
And the Axial (y)
Moment of Inertia
I y 121 m c 2 a 2
2 2
12
I z 4.88 103 2 6.48 103
I z 17.84 103 lb ft s2
0.977103 lb ft s 2
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Bruce Mayer, PE
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T = Iα
When you take ME104 (Dynamics) at
UCBerkeley you will learn that the Rotational
Behavior of the CrankShaft depends on its
Mass Moment of inertia
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Bruce Mayer, PE
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WhiteBoard Work
Some Other
Mass
Moments
For the
Thick Ring
2
2
Router
Rinner
Iz m
2
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Bruce Mayer, PE
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WhiteBoard Work
Find MASS
Moment of
Inertia
for Prism
About the y-axis in this case
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Bruce Mayer, PE
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Engineering 36
Appendix
dy
µx µs
sinh
dx
T0 T0
Bruce Mayer, PE
Registered Electrical & Mechanical Engineer
[email protected]
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Bruce Mayer, PE
[email protected] • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
WhiteBoard Work
Find MASS
Moment of
Inertia
for Roller
About axis AA’ in this case
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Bruce Mayer, PE
[email protected] • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
Mass Moment of Inertia
Last time we discussed the “Area
Moment of Intertia”
• Since Areas do NOT have Inertial
properties, the Areal Moment is more
properly called the “2nd Moment of Area”
Massive Objects DO physically have
Inertial Properties
• Finding the true “Moment of Inertia” is very
analogous to determination of the 2nd
Moment of Area
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Bruce Mayer, PE
[email protected] • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx