Physics Revision Notes – Thermodynamics Page 1 of 20
Dynamics – Rigid Body Dynamics
Introduction
A rigid body is a many-particle system in which the distance between
particles is fixed. The location of all particles is described by 6
coordinates – 3 spatial and 3 angular.
The velocity is determined by v, the velocity of the CoM and w , the
angular velocity.
The basic two equations of angular motion are
= F
MR 0
The centre of mass moves as if it were a single particle under the
action of a force F0.
J = G0
The rate of change of angular momentum is equal to the total applied
couple.
Other basic equations:
o The velocity v of a particle at a distance r from an axis around
which a rotation at speed w is happening is
v = w ´r
o For similar reasons:
dJ
= w ´J
dt
o Angular speeds are additive. To if frame 1 is rotating with
w1 wrt 2 with respect to frame 2, which is rotating with w2 wrt 3
with respect o frame 3, then
w1 wrt 3 = w1 wrt 2 + w2 wrt 3
Relating J and w
If the body is rotating at w , the total angular momentum is given by
© Daniel Guetta, 2008
, Physics Revision Notes – Thermodynamics Page 2 of 20
J = år ´p
= å r ´ m(w ´ r )
= å m éër 2w - (w ⋅ r )r ùû
= å m éër 2w - (wx x + wyy + wz z )r ùû
In detail
æ m(y 2 + z 2 ) -å mxz ö÷÷
ççå -å mxy
çç ÷÷
J = ç -å mxy
çç
ç å m(x + z ) -å myz ÷÷÷÷ w
2 2
÷
çç - mxz
å -å myz
è å m(x 2 + y 2 )÷÷ø
I
J = Iw
[The non-diagonal elements are fairly easy to derive. The diagonal ones
should actually have x2 + y2 + z2, because one of the terms is always
knocked out by the second term in the sum]. In other words, J is
proportional to w , but not necessarily parallel to it.
The off-axes elements are rather hard to understand – they correspond
to the fact that looking at a particle at a given instant, it’s impossible
to tell exactly around which axis it’s moving.
Also, we can find the kinetic energy
T = å 12 m [(w ´ r ) ⋅ (w ´ r )]
= å 12 m [w ⋅ r ´ (w ´ r )]
T = 21 w ⋅ J
The couple is then given by
G = J = w ´ J
Note that I must be specified with its origin and with its set of axes.
Properties of I
I is a symmetric tensor. It therefore has three real eigenvalues and
three perpendicular eigenvectors.
With respect to the eigenvector basis:
© Daniel Guetta, 2008
, Physics Revision Notes – Thermodynamics Page 3 of 20
æI ö
çç 1 ⋅ ⋅ ÷÷
ç ÷
I ¢ = çç ⋅ I 2 ⋅ ÷÷÷
çç ÷
çç ⋅ ⋅ I ÷÷÷
è 3ø
J a = I a wa [No sum]
T = 12 I a wa2 [Sum]
The eigenvector axes are called the principal axes, and the Is are called
the principal moments of inertia.
An alternative way to think of this is that the principal axes are ones
around which objects are “happy” to rotate without any torque being
applied.
In w -space, surfaces of constant T form an ellipsoid, with axes of
length µ I a-1/ 2 . Also, in w -space:
grad T = I a wa = J
So J is perpendicular to surfaces of constant T at w .
We can classify the principal axes as follows:
o Spherical tops – all the I are equal, and J = I w , with I scalar.
The body is isotropic with the same I about any axis (eg: sphere,
cube).
o Symmetrical tops – I 1 = I 2 ¹ I 3 . e3 axis is unique, but e1 and e2
are any two mutually perpendicular vectors perpendicular to e3
(eg: lens, cigar).
o Asymmetrical tops – all Is different, and axes are unique.
Consider any two Is:
I 1 + I 2 = å m(y 2 + z 2 + x 2 + z 2 ) = I 3 + 2å mz 2 ³ I 3
So no I can be larger than the sum of the other two. Furthermore, if
z = 0 for every particle (ie: if we have a lamina), then
I 3 = I1 + I 2
Consider an axis at a distance a away from a principal axis and parallel
to it, and let r be the distance of each particle from the principal axis.
Then:
I = å m(r + a ) ⋅ (r + a ) = I 0 + Ma 2 + 2 ( mr )
å
⋅ a = I 0 + Ma 2
= 0 when r measured
relative to C of M
© Daniel Guetta, 2008
Dynamics – Rigid Body Dynamics
Introduction
A rigid body is a many-particle system in which the distance between
particles is fixed. The location of all particles is described by 6
coordinates – 3 spatial and 3 angular.
The velocity is determined by v, the velocity of the CoM and w , the
angular velocity.
The basic two equations of angular motion are
= F
MR 0
The centre of mass moves as if it were a single particle under the
action of a force F0.
J = G0
The rate of change of angular momentum is equal to the total applied
couple.
Other basic equations:
o The velocity v of a particle at a distance r from an axis around
which a rotation at speed w is happening is
v = w ´r
o For similar reasons:
dJ
= w ´J
dt
o Angular speeds are additive. To if frame 1 is rotating with
w1 wrt 2 with respect to frame 2, which is rotating with w2 wrt 3
with respect o frame 3, then
w1 wrt 3 = w1 wrt 2 + w2 wrt 3
Relating J and w
If the body is rotating at w , the total angular momentum is given by
© Daniel Guetta, 2008
, Physics Revision Notes – Thermodynamics Page 2 of 20
J = år ´p
= å r ´ m(w ´ r )
= å m éër 2w - (w ⋅ r )r ùû
= å m éër 2w - (wx x + wyy + wz z )r ùû
In detail
æ m(y 2 + z 2 ) -å mxz ö÷÷
ççå -å mxy
çç ÷÷
J = ç -å mxy
çç
ç å m(x + z ) -å myz ÷÷÷÷ w
2 2
÷
çç - mxz
å -å myz
è å m(x 2 + y 2 )÷÷ø
I
J = Iw
[The non-diagonal elements are fairly easy to derive. The diagonal ones
should actually have x2 + y2 + z2, because one of the terms is always
knocked out by the second term in the sum]. In other words, J is
proportional to w , but not necessarily parallel to it.
The off-axes elements are rather hard to understand – they correspond
to the fact that looking at a particle at a given instant, it’s impossible
to tell exactly around which axis it’s moving.
Also, we can find the kinetic energy
T = å 12 m [(w ´ r ) ⋅ (w ´ r )]
= å 12 m [w ⋅ r ´ (w ´ r )]
T = 21 w ⋅ J
The couple is then given by
G = J = w ´ J
Note that I must be specified with its origin and with its set of axes.
Properties of I
I is a symmetric tensor. It therefore has three real eigenvalues and
three perpendicular eigenvectors.
With respect to the eigenvector basis:
© Daniel Guetta, 2008
, Physics Revision Notes – Thermodynamics Page 3 of 20
æI ö
çç 1 ⋅ ⋅ ÷÷
ç ÷
I ¢ = çç ⋅ I 2 ⋅ ÷÷÷
çç ÷
çç ⋅ ⋅ I ÷÷÷
è 3ø
J a = I a wa [No sum]
T = 12 I a wa2 [Sum]
The eigenvector axes are called the principal axes, and the Is are called
the principal moments of inertia.
An alternative way to think of this is that the principal axes are ones
around which objects are “happy” to rotate without any torque being
applied.
In w -space, surfaces of constant T form an ellipsoid, with axes of
length µ I a-1/ 2 . Also, in w -space:
grad T = I a wa = J
So J is perpendicular to surfaces of constant T at w .
We can classify the principal axes as follows:
o Spherical tops – all the I are equal, and J = I w , with I scalar.
The body is isotropic with the same I about any axis (eg: sphere,
cube).
o Symmetrical tops – I 1 = I 2 ¹ I 3 . e3 axis is unique, but e1 and e2
are any two mutually perpendicular vectors perpendicular to e3
(eg: lens, cigar).
o Asymmetrical tops – all Is different, and axes are unique.
Consider any two Is:
I 1 + I 2 = å m(y 2 + z 2 + x 2 + z 2 ) = I 3 + 2å mz 2 ³ I 3
So no I can be larger than the sum of the other two. Furthermore, if
z = 0 for every particle (ie: if we have a lamina), then
I 3 = I1 + I 2
Consider an axis at a distance a away from a principal axis and parallel
to it, and let r be the distance of each particle from the principal axis.
Then:
I = å m(r + a ) ⋅ (r + a ) = I 0 + Ma 2 + 2 ( mr )
å
⋅ a = I 0 + Ma 2
= 0 when r measured
relative to C of M
© Daniel Guetta, 2008
