Mechanics Rigid Body Motion 3, Lecture Notes - Physics, Study notes of Mechanics

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Mechanics

Physics 151

Lecture 10

Rigid Body Motion

(Chapter 5)

What We Did Last Time!

Found the velocity due to rotation

!

Used it to find Coriolis effect

!

Connected

with the Euler angles

!

Lagrangian

translational and rotational parts

!

Often possible if body axes are defined from the CoM

!

Defined the inertia tensor

!

Calculated angular momentumand kinetic energy

s

r

d

d

dt

dt

×

ω

(

)

2

j

k

i

i

jk

ij

ik

I

m

r

x x

L

I

ω

2

T

I

ω

I

ω

Principal Axes!

Consider a rigid body with body axes

x-y-z

!

Inertia tensor

I

is (in general) not diagonal

!

But it can be made diagonal by

!

Rotate

x-y-z

by

R

New body axes

x’-y’-z’

!

In

x’-y’-z’

coordinates

D

I

RIR

ω

R

ω

D

L

RLRI

ω

RIRR

ω

I

ω

RR

One can choose a set of

body axes that make the

inertia tensor diagonal

Principal Axes

How do I

find them?

Finding Principal Axes!

Consider unit vectors

n

1

n

2

n

3

along principal axes

!

To find the principal axes and principal moments:

!

Express

I

in in any body coordinates

!

Solve eigenvalue equation

!

Eigenvectors point the principal axes

!

Use them to re-define the body coordinates to simplify

I

!

You can often find the principal axes by just looking atthe object

i

i

i

I

In

n

1

2

3

0

0

0

0

0

0

D

I

I

I









=

=













I

RIR

!

n

i

is an eigenvector of

I

with eigenvalue

I

i

I

r

λ

I

1

2

3

I

I

I

Euler’s Equation of Motion

!

Special cases:

!!

1

1

1

1

1

1

2

2

2

2

2

2

3

3

3

3

3

3

I

I

N

d

I

I

N

dt

I

I

N

×

1

1

2

3

2

3

1

2

2

3

1

3

1

2

3

3

1

2

1

2

3

I

I

I

N

I

I

I

N

I

I

I

N

Euler’s equation of motion for rigid body

with one point fixed

2

3

1

1

1

I

N

ω =

2

3

I

I

1

1

1

I

N

ω =

Torque-Free Motion

!

No linear force

Conservation of linear momentum

!

No torque

Conservation of angular momentum

!

Try

N

= 0 in Euler’s equation of motion

!

We will do something more intuitive (hopefully)

!

Geometrical trick by L. Poinsot

1

1

2

3

2

3

2

2

3

1

3

1

3

3

1

2

1

2

I

I

I

I

I

I

I

I

I

Integrating these equation will give usenergy and angular momentumconservation

Inertia Ellipsoid

I

n

ρ

x

y

z

1

I

2

I

3

I

ρ

Inertia ellipsoid represents

the moment of inertia of a

rigid body in all directions

2

2

2

1

1

2

2

3

3

I

I

I

Usefulness of this

definition will become

apparent soon …

Inertia Ellipsoid!

Inertia along axis

n

is

!

F

is a function (like potential) defined in the

space

!

F

ρ

Inertia ellipsoid

!

Normal of the ellipsoidgiven by the gradient

!

Using

I

n

ρ

I

n In

2

2

2

1

1

2

2

3

3

F

I

I

I

ρ

ρ

I

ρ

F

= 1

ρ

F

ρ

F

ρ

I

ρ

I

T

ρ

n

ω

F

T

ρ

L

Surface of the

inertia ellipsoid is

perpendicular to

L

Invariable Plane!

Inertia ellipsoid touches the invariable plane

!

Distance between the center and the plane is

!

Touching point = instantaneous axis of rotation

!

Tip of the

ρ

vector is momentarily at rest in space

!

i.e. it’s not sliding, butrolling without slippingon the invariable plane

T

L

Determined by the initial conditions

ρ

d

dt

T

ρ

ω

ω

ω

ρ

Invariable Plane!

Inertia ellipsoid rolls on the invariable plane

!

Rotation of ellipsoid gives the rotation of the body

!

Direction of

ρ

gives the direction of

ω

in space

!

Touching point draws curves

!

Curve drawn on the ellipsoid = polhode

!

Curve drawn on the invariable plane= herpolhode

!

Let’s examine a fewsimple cases

Simple Cases!

Initial axis is close to one of the principal axes

!

Assume

I

1

I

2

I

3

!

Stable rotation around

I

1

and

I

3

!

Not so obvious around

I

2

!

If

1

3

2

is constant

!

Small deviation leads to instability

1

I

3

I

1

1

2

3

2

3

2

2

3

1

3

1

3

3

1

2

1

2

I

I

I

I

I

I

I

I

I

Simple Cases

!

Since

I

1

I

2

I

3

, distance

allows a polhode that

wraps around the inertia ellipsoid

!

Rotation around a principal axis is stable except for theone with the intermediate moment of inertia

2

I

Precession!

I

1

I

2

turns the Euler’s equation of motion to

!

precesses around the

I

3

axis

!

Draws the body cone

1

1

2

3

1

3

1

2

3

1

3

1

3

3

I

I

I

I

I

I

I

ω

3

is constant

Consider it as a given initial condition

1

2

2

1

3

1

3

1

I

I

I

1

2

cos

sin

A

t

A

t

3

ω

A

Rotation Under Torque!

We introduce torque

!

Things get messy

!

Consider a spinning top

!

Define Euler angles

1

1

2

3

2

3

1

2

2

3

1

3

1

2

3

3

1

2

1

2

3

I

I

I

N

I

I

I

N

I

I

I

N

z

y

x

φ