The
Precession and Nutation of a Gyroscope
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The motion of a gyroscope may be modeled by a set of three
coupled differential equations, the three equations can be
solved simultaneously with a 4th order Range Kutta algorithm,
from plots of the results its possible to see the three types
of motion exhibited by a gyroscope.
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(1)
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(2)
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(3)
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| the three types of motion
can be described as cusped, undulatory and looped. I wrote a
Fortran program to model these equations, I will rewrite it
in Java and see if I can write a simple program to graph the
motion as well; when I get the time. This page uses the font
"symbol" to display properly. |
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The instantaneous orientation of the gyroscope is described
by three Eulerian angles y, q
and f. The axis O
h x z are fixed in the rotor with O z
coincident with the spindle. O x y z are the axis that
move so that Oz is coincident with Oz
and Ox remains in the vertical plane ZOz, so that Oy is in
the horizontal plane. Axes OXYZ are axes which are fixed in
a Newtonian frame, with OZ vertical and OXY a horizontal plane.
The angle y is xOz,
so that dy/dt is the angular
velocity of spin of the gyro about its own axis Oz,
or Oz in the frame Oxyz which is itself moving. q
is zOZ and is the inclination of the gyro axis to the
vertical. f is yOY so df/dt
is the rate at which the vertical plane zOZ containing the
axis Oz of the gyro rotates about the vertical OZ. A rate
of change df/dt is called precession
and the oscillation in q and df/dt
is called nutation.
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Figure 1. Suspended gyroscope hanging from point O. |
| the equations 1-3 are derived from
the basic equations for the potential energy and the kinetic
energy. Using Lagrange's equation as an expression of Newtonian
mechanics provides three equations of motion. When integrated
and simplified these three equations of motion can have substitutions
made to produce 1-3. The full derivation can be found in many
books [1] and [2] are particularly good. |
I will write the
program Java program when I get time, I will gradually add the
full derivation of the equations and a simple explanation of
a spinning top. Maybe even the forces in a motorcycle swinging
arm during a turn! |
A gyroscope is a symmetrical flywheel
in a cage, the cage means the flywheel's spatial orientation
can be changed without affecting the rotation of the flywheel.
While spinning, the flywheel possesses angular momentum L. L
is parallel to the spin axis. L is given by :
L=rxp (4) |
Figure 2. Spinning top, direction of vectors shown. |
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Figure 2 shows the vectors r, mg p and L. The
direction of L is found with the right hand rule; put your
hand around the r vector with fingers pointing in the direction
of p.
to be continued.........
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References:
[1]Gray, A, Gyrostatics and Rotational Motion, Macmillan, 1918.
[2]an excellent book on spinning tops written in 1890, I can't
find it at the moment.
[3]Kreysig, E, Advanced Engineering Mathematics, Wiley, 1883
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