A gyro is a body of rotation which is set spinning at a large angular velocity around its axis of symmetry. The most important practical applications of gyros are met in devices for measuring the orientation or maintaining the stability of airplanes, spacecrafts and submarine vehicles in general. Various gyros are used as sensors in inertial guidance systems. Most textbooks in introductory mechanics explain the mysterious behavior of a spinning gyro by using Lagrange equations and severe mathematics   . Other textbooks   treat the problem of torque-induced precession of a top based on Euler equations, which are referred to the non-inertial reference frame rotating together with the body. The problem of the torque free inertial rotation of a symmetrical top is discussed, in particular, in  and illustrated by a simulation computer program (free rotation of an axially symmetrical body)  . For our problem, we consider a gyro body with xyz-coordinate system fixed of it, such that the z-axis of the body is the axis of symmetry and the inertia tensor is assumed to take the form
Here A and C are the principal moments of inertia in the x and z directions.
The rate of change of the inertia tensor with respect to time takes the form
We note that the inertia products remain zeros.
Assume that the angular velocity of the satellite or the gyros is
and the angular momentum is
where and are the unit vectors in the and directions.
Thus the gyro’s kinetic energy of rotation becomes
2. Equations of Motion
Applying Euler’s equations of motion and putting the applied torque equal zero, we get
For the considered problem (mentioned above) we apply the angular momentum principle to get
3. Components of Angular Momentum
The Equation (2.4c) can be integrated to give
We can conclude that the z-component of the angular moment is constant, that is
The angular velocity is obtained by multiplying the Equations (2.4a) by and (2.4b) by and adding the resulted equations, we get
that is or (3.4)
Thus the―component of angular momentum is constant
The and can be shown, see Figure 1.
For torque free axi-symmetric gyro, angular velocity, angular momentum,
Figure 1. The angular momentum components.
and the gyro’s symmetry axis lie in one plane  .
Since the Equations (3.1) and (3.5) represent the angular momentum components, it can be deduced that the nutation angle remains constant when the inertia moments change as it when the inertia moments does not change.
For the components of angular velocity (3.4), we introduce the auxiliary frequency defined by
The two Equations (2.4a) and (2.4b) can be combined to yield
The solution of this differential equation can be obtained as
If the constant, and in order to find the value of constant we employ the Equation (3.5) to get and then
We can get the―component of the angular velocity at any instant by using
where the subscript (0) refers to values of time
The components of angular velocity can be shown, see Figure 2.
The angular velocity component has a relative angular velocity in the xy- plane.
Introducing a floating coordinate system, we can express the angular velocity () by using the or the coordinate system.
The Equations (3.11) and (3.12) show that the component and consequently v-axis and u-axis are perpendicular, such that u-axis rotates with a relative angular velocity where
In the xy-plane, the z-component of the absolute angular velocity of v-axis is
Figure 2. The components of angular velocity.
We can find that, for a flattened gyro, the v-axis rotates faster than the x-axis, and for elongated gyro the v-axis rotates more slowly than the x-axis.
Thus, the coordinate system moves with an angular velocity
The angular velocity of the gyro body and the angular velocity of the floating coordinate system are related by
where is the spin of the gyro.
4. Euler Frequencies
The frequency of the angular momentum remains constant since there is no external torque applied to the gyro  , the direction of the angular momentum vector may be used to define as space-fixed coordinate axis L which assigned to Z-axis and is called the precession axis  .
If the z-axis of the rotating coordinate system is the symmetry axis of the gyro, the v-axis lies in a plane formed by the Z- and z- axes, and we write
The nutation angle is the angle between the z-axis and Z-axis where Z-axis and L are coincide, see Figure 3.
The components of for elongated axis symmetric gyro are obtained from the shape, so
The nutation angle remains a constant and the gyro is carry out a steady precession about the angular momentum vector, since
(4.4) we obtain
Also from the figure
Figure 3. Euler’s angles.
Also for the angle between the z-axis and the angular velocity axis, we have
We can obtain the relation between the precession and the spin as follows
For the elongated gyro we find that and and have the same sign then the precession is direct.
For the flattened gyro we find that and have the opposite signs, then the precession is retrograde.
The system (2.4) is integrated to obtain the angular velocities and the angular momentum,
Figure 4. Space cone and body cone for an elongated gyro.
Figure 5. Space cone and body cone for a flattened gyro.
then Euler’s angles are deduced. The motions are classified into two cases:
1) the elongated gyro.
2) the flattened gyro.
For each case, we investigate the equations of motions, the precession, the nutation and the spin for these motions in detailed by using the analytical techniques and the illustrated shapes. The obtained results can be applied on the satellites  with rotational symmetry and changed inertia moments, the antennas   and the solar power collector systems  .
This project is supported by Institute of Scientific Research and Islamic Heritage Revival, Umm Al-Qura University, Saudi Arabia.
 Bruno, A.D. (2007) Analysis of the Euler-Poisson Equations by Methods of Power Geometry and Normal Form. Journal of Applied Mathematics and Mechanics, 71, 168-199.