The rattleback, also called a Celt or wobblestone,
is an oblong boatshaped 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 misalignment 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 multibody 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
B_{N} of
B.
The curved portion of the surface of
B
is part of an ellipsoid S,
whose principal axes
S_{x},
S_{y},
S_{z}
intersect at point
S_{o} on
B.
The locus of points of
S
is defined by the equation
s_{x}^{2} / a^{2}
+ s_{y}^{2} / b^{2}
+ s_{z}^{2} / c^{2}
 1
= 0
where s_{i} are the S_{i} (i = x, y, z)
coordinates of a generic point
P of
S, and
a,
b,
c
are semidiameters of the ellipsoid.
Point B_{cm} (B's mass center) lies on
S_{x}, a distance h from
S_{o}.


In formulating equations of motion, it is convenient
to introduce righthanded sets of mutually perpendicular unit vectors
b_{i} and
n_{i} (i = x, y, z),
fixed in
B and
N, respectively, with
b_{i} parallel to
S_{i} (i = x, y, z), and
n_{x} 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
b_{i} with
n_{i} (i = x, y, z),
and then subjecting B to the
rotations described in magnitude and direction by
q_{1} b_{x},
q_{2} b_{y},
q_{3} b_{z}.
System Identifiers
Description 
Symbol 
Value (or initial value) 
Semidiameter of ellipsoid 
a 
2 cm 
Semidiameter of ellipsoid 
b 
20 cm 
Semidiameter of ellipsoid 
c 
3 cm 
Local gravitational constant 
g 
9.81 m/sec^{2} 
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 b_{x} 
Ixx 
17 kg*cm^{2} 
Moment of inertia of B about Bcm for b_{y} 
Iyy 
2 kg*cm^{2} 
Moment of inertia of B about Bcm for b_{z} 
Izz 
16 kg*cm^{2} 
Product of inertia of B about Bcm for b_{y} and b_{z} 
Iyz 
0.2 kg*cm^{2} 



q_{1} Orientation angle 
q1 
0.0 degrees 
q_{2} Orientation angle 
q2 
0.5 degrees 
q_{3} Orientation angle 
q3 
0.5 degrees 
b_{x} measure of B's angular velocity in N 
wx 
5.0 rad/sec 
b_{y} measure of B's angular velocity in N 
wy 
0.0 rad/sec 
b_{3} 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 leftfiles use a freebody 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.
Spin angle q_{1} shows spin reversal

Spin angle q_{1} with no spin reversal





