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The rattleback, also called a Celt or wobblestone,
is an oblong boat-shaped object which,
when placed on a rough horizontal surface and made to rotate about a vertical
axis, sometimes stops rotating, begins to oscillate (wobble),
then starts rotating in the reverse direction.
As demonstrated by the MotionGenesis™ simulations below
(and documented in technical papers),
the strange behavior is not due to an offset center of mass,
but instead a mis-alignment of the ellipsoid's principal axes of curvature
with the Celt's principal inertia axes.
Because the curved portion of the surface of the rattleback is part of an ellipsoid,
and because the ellipsoid rolls without slip on the rough horizontal surface,
many commercial multi-body programs have serious difficulties when trying
to simulate the motion of this simple system
(due to the rolling, geometry, and/or friction).
The figure to the right is a schematic representation of a rattleback
B
that is in contact with a rough horizontal surface
N
at point
BN of
B.
The curved portion of the surface of
B
is part of an ellipsoid S,
whose principal axes
Sx,
Sy,
Sz
intersect at point
So on
B.
The locus of points of
S
is defined by the equation
sx2 / a2
+ sy2 / b2
+ sz2 / c2
- 1
= 0
where si are the Si (i = x, y, z)
coordinates of a generic point
P of
S, and
a,
b,
c
are semi-diameters of the ellipsoid.
Point Bcm (B's mass center) lies on
Sx, a distance h from
So.
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In formulating equations of motion, it is convenient
to introduce right-handed sets of mutually perpendicular unit vectors
bi and
ni (i = x, y, z),
fixed in
B and
N, respectively, with
bi parallel to
Si (i = x, y, z), and
nx directed vertically upward and
perpendicular to the planar surface of
N in contact with
B.
The orientation of
B in
N is found by first aligning
bi with
ni (i = x, y, z),
and then subjecting B to the
rotations described in magnitude and direction by
q1 bx,
q2 by,
q3 bz.
System Identifiers
| Description |
Symbol |
Value (or initial value) |
| Semi-diameter of ellipsoid |
a |
2 cm |
| Semi-diameter of ellipsoid |
b |
20 cm |
| Semi-diameter of ellipsoid |
c |
3 cm |
| Local gravitational constant |
g |
9.81 m/sec2 |
| Distance between Bcm (B's center of mass) and So |
h |
1 cm |
| Mass of B |
m |
1.0 kg |
| Moment of inertia of B about Bcm for bx |
Ixx |
17 kg*cm2 |
| Moment of inertia of B about Bcm for by |
Iyy |
2 kg*cm2 |
| Moment of inertia of B about Bcm for bz |
Izz |
16 kg*cm2 |
| Product of inertia of B about Bcm for by and bz |
Iyz |
0.2 kg*cm2 |
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| q1 Orientation angle |
q1 |
0.0 degrees |
| q2 Orientation angle |
q2 |
0.5 degrees |
| q3 Orientation angle |
q3 |
-0.5 degrees |
| bx measure of B's angular velocity in N |
wx |
5.0 rad/sec |
| by measure of B's angular velocity in N |
wy |
0.0 rad/sec |
| b3 measure of B's angular velocity in N |
wz |
0.0 rad/sec |
| time |
t |
0 to 5 seconds |
Shown below are two lists of files that analyze the behavior of the rattleback.
The left-files use a free-body analysis and calculate contact forces,
whereas those on the right use Kane's method and do not calculate contact forces.
The file RattlebackKane.1 was created by
running the MATLAB®, C, or Fortran code, and the data in this file were graphed with the
MotionGenesis plotting program.
The graph on the left clearly shows the spin reversal of the rattleback.
The rattleback provides an excellent demonstration of the effect of
product of inertia on motion.
For example, setting the product of inertia Iyz = 0 results
in no spin reversal, as can be seen from the following graph on the right.
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Spin angle q1 shows spin reversal
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Spin angle q1 with no spin reversal
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