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The Precession and Nutation of a Gyroscope


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.

 
(1)
(2)
(3)
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.

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.


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.

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.........

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
       

 

 
     
  Cobbled together on the {cobbled}
by
Richard

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