Rattleback Simulation

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.

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 horiztonal surface, many commercial multi-body programs have serious difficulties when trying to simulate the motion of this simple system.

Rattleback schematic

The figure above 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 S1, S2, S3 intersect at point So on B. The locus of points of S is defined by the equation


s12 / a2  +  s22 / b2  +  s32 / c2  -  1   =   0

where s1 are the Si (i=1,2,3) coordinates of a generic point P of S, and a, b, c are semi-diameters of the ellipsoid. Point Bo, the mass center of B, lies on S1, a distance h from So. In formulating equations of motion, it is convenient to introduce dextral sets of mutually perpendicular unit vectors bi and ni (i=1,2,3), fixed in B and N, respectively, with bi parallel to Si (i=1,2,3), and n1 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=1,2,3), and then subjecting B to the rotations described in magnitude and direction by q1b1, q2b2, q3b3.

System Identifiers
Description Symbol 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 from Bo, the mass center of B, to So h 1 cm
Mass of B m 1.0 kg
Central moment of inertia of B parallel to b1 I11 17 kg*cm2
Central moment of inertia of B parallel to b2 I22 2 kg*cm2
Central moment of inertia of B parallel to b3 I33 16 kg*cm2
Central product of inertia of B for b2 and b3 I23 0.2 kg*cm2
q1 Orientation angle   q1 0.0 degrees (initial value)
q2 Orientation angle   q2 0.5 degrees (initial value)
q3 Orientation angle   q3 -0.5 degrees (initial value)
b1 measure number of the angular velocity of B in N   w1 5.0 rad/sec (initial value)
b2 measure number of the angular velocity of B in N   w2 0.0 rad/sec (initial value)
b3 measure number of the angular velocity of B in N   w3 0.0 rad/sec (initial value)
Time t 0 to 5 seconds

Shown below are two lists of files relevant to analyzing the behavior of the rattleback. The files on the left use a Newton-Euler analysis, whereas those on the right use Kane's method. Examination of these files reveals that it is easier and more efficient to employ Kane's method than it is to perform a Newton-Euler analysis.

Description F=ma Fr + Fr* = 0
Autolev input file rattlebackNewton.al rattlebackKane.al
Autolev responses rattlebackNewton.all rattlebackKane.all
Matlab code created by Autolev rattlebackNewton.m rattlebackKane.m
C code created by Autolev rattlebackNewton.c rattlebackKane.c
Fortran code created by Autolev rattlebackNewton.f rattlebackKane.f
C/Fortran input file rattlebackNewton.in rattlebackKane.in

The file rattlebackKane.1 was created by running the Matlab, C, or Fortran code, and the data in this file were graphed with Autolev's 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 I23 to 0 results in no spin reversal, as can be seen from the following graph on the right.

Rattleback Spin Angle q1 showing Spin Reversal Rattleback Spin Angle with Reversal

Rattleback Spin Angle q1 with no Spin Reversal Rattleback Spin Angle no Reversal