© Sharif University of Technology - CEDRAmech.sharif.edu/~meghdari/files/AD-Session#11.pdf · ©...
Transcript of © Sharif University of Technology - CEDRAmech.sharif.edu/~meghdari/files/AD-Session#11.pdf · ©...
© Sharif University of Technology - CEDRA
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
Purpose:
To Study Inertia Tensor of a Material System.
Topics:
Inertia Tensor of a Mass Particle.
Inertia Tensor of a System of Particles.
Inertia Tensor of a Continuum.
Transfer Theorem.
Principal Values of Inertia Tensor.
INERTIA TENSOR
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
iimvPorvmP
From Chapter-6, we defined the Kinetic States as:
For a Single particle;
Linear Momentum (Momentum):
(7.1)
Moment of Momentum (Angular Momentum):
kjijk
O
i
O
PxH
orrrmPrH
,
, where: “O” is a moment center.
(7.2)
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
)3,2,1(
,
1
1 1
iPP
orvmPP
N
ii
N N
For a System of Particles;
Linear Momentum (Momentum):
(7.3)
)3,2,1,,(
,
1
1 11
kjiPxH
orvrmPrHH
N
kjijki
N NNOO
Moment of Momentum (Angular Momentum):
(7.4)
rWhere; : position vector of the “βth” particle from
the moment center “O”.
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
or, by Theorem-16, for the equivalent mass system, the
Total/Global Momentum and Global Moment of Momentum
of the system of particles is:
N
CvmvmP
1
= (Global Momentum) (7.5)
m1
m2
m3
mβ
Cr
βrO
m, C
βρ
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
m1m2
m3
mβ
Cr
βrO
m, C
βρ
CH
Cv=mP
m, C
Cr
O
O
eq
CCCOHHPrHH (Global M.O.M.) (7.6)
OHWhere; : (M.O.M. about any point “O” in space).
PrH
mPH
CO
eq
NNC
11
(Central M.O.M.)
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
int
.
:11
PoBodyC
rigidm
const
wheremPHNN
C
?C
HNow, let us consider the system of particles shown.
We wish to define the quantity (Central M.O.M.).
To do this, suppose that the system is:
- Locked rigidly, so that the distance between any pair of
particles remain constant throughout the dynamic process.
Therefore: The system’s rotation may be described by a single
angular velocity “ω”.
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
)()()(
i
N NC
i
CmHormH ])([)(
1 1
Hence:
But,
iiux
i
x; where : its origin is at the mass center.
Therefore:
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
jjiijkk
ijjijjkk
iii
xxxx
xxxx
x
)(
)()(
)()()]([
Therefore:
CC
j
C
ij
jji
N
ijkk
C
i
IH
I
xxxxmH )]([1
where:
(7.7)
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
CCC
CCC
CCC
C
ij
III
III
III
I
333231
232221
131211
)]([1
ji
N
ijkk
C
ij
CxxxxmII
≡ (The Inertia Tensor; a 2nd order and symmetrical tensor), or:
C
CC
vmP
IH Compare (for a rigid system): where:
Cvand
CI
: are the velocity properties, and
: plays the role of mass “m” in describing the kinetic
state of a rigid system.
(7.8)
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
Definition: The “Inertia Tensor” actually defines the mass
distribution of a material system.
Consider the particle “m” in
coordinate system {xi} as
shown;
Inertia Tensor of a Particle of Mass “m” about a fixed
point “O”:
(moment center) O
m
x1
x2
x3
x1
x2
x3
r1
r2
r3
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
(moment center) O
m
x1
x2
x3
x1
x2
x3
r1
r2
r3
rrrrm
rrm
rmrvmrPrHO
)()(
)(
)]([
)]([
jiijkk
O
ij
jjiijkk
O
i
xxxxmI
xxxxmH
(7.9)
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
)]([ jiijkk
O
ij xxxxmI (7.9)
1,1)()( 2
1
2
3
2
2
2
1
2
3
2
2
2
111 jiformrxxmxxxxmI O
2,2)()( 2
2
2
3
2
1
2
2
2
3
2
2
2
122 jiformrxxmxxxxmI O
3,3)()( 2
3
2
2
2
1
2
3
2
3
2
2
2
133 jiformrxxmxxxxmI O
Definition: The moments of inertia of the particle “m” about
the axes {xi} are:
2,1)0(212112
jiforxmxxxmI O
3,2)0(323223
jiforxmxxxmI O
1,3)0(131331
jiforxmxxxmI O
Definition: The products of inertia of the particle “m” are:
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
Inertia Tensor of a System of Particles about a fixed
point “O”: A physical property of the material system;
N
jiijkk
O
ijxxxxmI
1
][
N
O
ij
xxxxxx
xxxxxx
xxxxxx
mI1
2
2
2
12313
32
2
1
2
312
3121
2
3
2
2
)()(
)()(
)()(
(7.10)
N
A
jj
A
iiij
A
kk
A
kk
A
ijxxxxxxxxmI
1
)])(())([(
Note: To write the inertia tensor about an arbitrary point “A”
in space, simply do a coordinate shift, as:
(7.11)
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
2x
1x
3x
m1
m2
m3
mβ
Cr
βrO
m, C
βρ
x1
x2
x3
CI
Definition: When the moment center is specifically the
mass center, the inertia tensor is called the Central Inertia
Tensor, “ ”, where:
N
jiijkk
C
ijxxxxmI
1
][
i
x
; (7.12)
: its origin is located at the mass center.
Where:
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
(7.13)
dmxxxxxxxxIm
A
jj
A
iiij
A
kk
A
kk
A
ij]))(())([(
And about any arbitrary moment center like “A” is:
(7.14)
m
jiijkk
O
ijdmxxxxI )(
m
O
m
O
dmxxI
dmxxI
2112
2
3
2
211)(
Inertia Tensor for a Continuum: A generalization of the
inertia tensor for a system of particles where the finite
sums “ ∑ ” are integrally summed “ ∫ ”.
Ex:
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
Transfer Theorem-(18): When the mass, the mass center,
and the central inertia tensor are known, the inertia tensor
of a mass system about any given moment center can be
obtained from:
A
eqI
.where; : inertia tensor about “A” due to a single particle
of mass “m” at the mass center.
CA
eq
A
C
ij
A
j
C
j
A
i
C
iij
A
k
C
k
A
k
C
k
A
ij
III
IxxxxxxxxmI
.
)])(())([(
(7.15)
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
A
ix
)( C
iii xxx
x1
x2
x3
2x
1x
3x
m, C
A
mβ
O
C
ix
ix
A
ix
)( A
i
C
i xx
)( A
ii xx
Proof:
ix : its origin is located at the mass center.
i
x : its origin is located at the fixed point “O”.
i
C
iixxx
i
A
i
C
i
A
iixxxxx
)()()( C
ii
A
i
C
i
A
iixxxxxx
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
0A
ix
However, if the origin “O” is selected as the moment center,
then , and Equation (7.15) reduces to:
N
A
jj
A
iiij
A
kk
A
kk
A
ijxxxxxxxxmI
1
)])(())([(
Note: Inertia Tensor about an arbitrary point “A”
in space, can be computed by a coordinate shift, as:
(7.11)
Substituting the last equation into the Equation (7.11) and
simplifying results in:
)()()( C
ii
A
i
C
i
A
iixxxxxx
C
ij
A
j
C
j
A
i
C
iij
A
k
C
k
A
k
C
k
A
ijIxxxxxxxxmI )])(())([(
(7.15)
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
2
2
2
12313
32
2
1
2
312
3121
2
3
2
2
)()(
)()(
)()(
CCCCCC
CCCCCC
CCCCCC
CO
xxxxxx
xxxxxx
xxxxxx
mII
(7.17)
or:
CO
eq
O
C
ij
C
j
C
iij
C
k
C
k
O
ij
III
IxxxxmI
.
)( (7.16)
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
I
Principal Values of the Inertia Tensor: In rigid body dynamics,
it is often convenient to use a coordinate system fixed to the
rigid body in which the products of inertial are zero. In this
case, the inertial tensor “ ” will be a diagonal matrix such
as:
This coordinate system is then called the Principal Coordinate,
and the moments of inertia about the principal axes are called
the Principal Moments of Inertia, and the three planes formed
by the principal axes are called the Principal Planes.
3
2
1
00
00
00
I
I
I
I (7.18)
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
A Thin Uniform Disk:
C x1
x2
x3
R
2
2
2
2
100
04
10
004
1
mR
mR
mR
IC
m
Example:
x1
x2
x3
C
1200
012
0
000
2
2
ml
mlI
C
A Thin Uniform Rod:
m
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
A Sphere of Radius “R” and Mass “m”:
100
010
001
5
2 2mRIC
More on the Principal Inertia Tensor: If we examine the
moment and product of inertia terms for all possible
orientation of the coordinate axes with respect to a rigid
body for a given origin, we will find in the general case
one unique orientation “{xα}” for which the products of
inertia are zero.
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
O
ijI
OI
OI
O
ij
OII Theorem-19: Given “ ”, there exist three principal
values of “ ”, namely (Principal Moments of Inertial)
which are the eigenvalues of the inertia tensor , and may
be computed as follows:
therefore;
0 EII OO
ij II O
E ij, and let , ( = Unit Matrix = )
(7.19)
O
ij
O
jj
O
ii
O
ij
O
ij
O
ii
IJ
IIIIJ
IJ
whereEquationsticCharacteriJIJIJI
3
2
1
32
2
1
3
)(2
1
:,)(0
(7.20)
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
and the direction cosines of the principal axes of inertia
{ } are the corresponding normalized eigenvectors,
defined as follows:i
x2
x3
x1
O
2x
3x
1x
{ }= Principal Axes
x
1
0)(
ii
iij
OO
ijII
(7.21)
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
313
212
111
1
e
e
e
e
From equations (7.22), we can write:
1
0)(
0)(
0)(
2
13
2
12
2
11
1313312231113
1332121221112
1331122111111
OOOO
OOOO
OOOO
IIII
IIII
IIII
1xExample: For the -axis we can write:
(7.22)
(In the first 3 Equations of (7.22), only 2 are independent, and the 3rd is a
linear combination of the others.)
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
2x 3
xAnd similarly for and axes we have:
}{}{}{
},{}{}{
332313
322212
312111
333231
232221
131211
eTee
oreTee
t
i
ii
(7.23)
323
222
121
2
e
e
e
e
333
232
131
3
e
e
e
e
, and therefore:,
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
Now, let us consider the Moment of Momentum vector
about the point “O”, such that:
}]}{][][{[
}}]{]}{[][][{[
}]{][[}]{[
tO
ij
i
tO
ij
i
O
ij
O
TIT
TTIT
ITI
substituting in equation (7.24) results:
i
OOHTH }]{[}{ (7.24)
i
O
iji
O
OO
IH
IH
}]{[}{
}]{[}{
But:
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
Therefore:
]][][[][tO
ij
O TITI
Equation (7.25) is known as the Rotation Transformation
of Inertia Properties. (same as Eq. 5.54 in Ginsberg Book)
(7.25)
}]}{][][{[
}}]{]}{[][][{[
}]{][[}]{[
tO
ij
i
tO
ij
i
O
ij
O
TIT
TTIT
ITI
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
Theorem-20: For a body with a plane of symmetry, any axis
perpendicular to the plane is a principal axis of inertia at the
point of intersection with that plane. (In other words, if two
coordinate axes form a plane of symmetry for a body, then
all product of inertias involving the coordinate normal to
that plane are zero).
Theorem-21: For a body having two planes of symmetry, the
line of intersection is also the principal axis for any moment
center lying on the line.
Theorem-22: If at least two of the three coordinate planes
are planes of symmetry for a body, then all products of
inertias are zero.
© Sharif University of Technology - CEDRA
© Sharif University of Technology - CEDRA By: Professor Ali Meghdari
For a Matrix [ ]:A
0 IA I ij, ( = Unit Matrix = )Eigenvalues:
0 xIA I ij, ( = Unit Matrix = )
Eigenvectors:
(READ EXAMPLES 5.1 and 5.3 of Ginsburg)