© sharif university of technology - cedramech.sharif.edu/~meghdari/files/ad-session#11.pdf · ©...

© 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


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)


)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”.


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


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.)


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”.


)()()(

i

N NC

i

CmHormH ])([)(

1 1

Hence:

But,

iiux

i

x; where : its origin is at the mass center.

Therefore:


jjiijkk

ijjijjkk

iii

xxxx

xxxx

x

)(

)()(

)()()]([

Therefore:

CC

j

C

ij

jji

N

ijkk

C

i

IH

I

xxxxmH )]([1

where:

(7.7)


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)


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


(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)


)]([ 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


3,3)()( 2

3

2

2

2

1

2

3

2

3

2

2

2


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:


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)


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:


(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:


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)


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


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)


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)


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)


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


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.


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)


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)


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.)


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:,


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:


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


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.


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)

© sharif university of technology - cedramech.sharif.edu/~meghdari/files/ad-session#11.pdf · ©...

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