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The properties and characteristics of torque free gyros with rotational symmetry and changing moments of inertia are the subject of the subsequent discussion. It shall be understood that the symmetry can be expressed by the notation (A=B) which does not presuppose geometric symmetry, where A and B are the principle moments of inertia about x and y axes respectively. We study the case of a torque free gyro upon which no external torque is acting. The equations of motion are derived when the origin of the xyz-coordinate system coincides with the gyro’s mass center c. This study is useful for the satellites, which have rotational symmetry and changed inertia moments, the antennas and the solar power collector systems.

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 [

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

Thus the gyro’s kinetic energy of rotation becomes

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

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

that is

where

Thus the

The

For torque free axi-symmetric gyro, angular velocity

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

then

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

where

We can get the

where the subscript (0) refers to values of time

The components of angular velocity can be shown, see

The angular velocity component

Introducing a floating

The Equations (3.11) and (3.12) show that the component

In the xy-plane, the z-component of the absolute angular velocity of v-axis is

We can find that, for a flattened gyro

Thus, the

The angular velocity

where

The frequency of the angular momentum remains constant since there is no external torque applied to the gyro [

If the z-axis of the rotating

The nutation angle

The components of

The nutation angle remains a constant and the gyro is carry out a steady precession about the angular momentum vector, since

Also from the figure

Also for the angle

The motion of a torque free gyro with rotational symmetry and changing moment of inertia can be visualized by imaging a space cone and body cone as shown see

We can obtain the relation between the precession and the spin as follows

For the elongated gyro

For the flattened gyro

The system (2.4) is integrated to obtain the angular velocities and the angular momentum,

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 [

This project is supported by Institute of Scientific Research and Islamic Heritage Revival, Umm Al-Qura University, Saudi Arabia.

Ismail, A.I. and El-Haiby, F.D. (2016) Torque Free Axi- Symmetric Gyros with Changing Moments of Inertia. Applied Mathematics, 7, 1934- 1942. http://dx.doi.org/10.4236/am.2016.716159