The General Free Gyrostat (continued)

Abstract

In this lecture the angular velocity components of a gyrostat are expressed as functions of time. Again, the case of torque-free motion of an unsymmetrical gyrostat with three non-zero components h_1,2,3 of relative angular rotor momentum is considered. Then, two integrals of motion are available:

$$$$

(1)

$$$$

(2)

Hence, of the three differential equations of motion only one is needed to determine together with (1) and (2) ω_1,2,3 as functions of time. We select the third one which reads

$$$$

(3)

This mathematical problem was solved independently by two scientists late last century. In 1889 Wangerin [1] (mathematician at Halle, Germany) published his solution and in 1898 the Italian Volterra, without knowing Wangerin’s paper, discovered a completely different way to solve it. From a purely mathematical point of view Volterra’s method is more appealing. His final equations for ω_1,2,3(t) display a beautiful symmetry which reflects the cyclical symmetry of the differential equations of motion. Furthermore, only Volterra succeded in integrating also the kinematical (Poisson) differential equations in addition to the Eqs. (1), (2), (3) so that the attitude of the gyrostat in inertial space is given explicitely as a function of time. Vol-terra’s formalism, however, has properties which render it rather inconvenient for applications where numerical calculations of the functions ω_1,2,3 (t) are needed. This is due to the fact that these fucntions are expressed in terms of complex variables and that these expressions are so complicated that an analytical separation of real and imaginary parts cannot be achieved (the main problem is presented by a complex modulus of elliptic functions). In this respect Wangerin’s approach is simpler because his formalism employs only real quantities. For this reason only Wangerin’s method will be presented here.