The stability of a family of non-trivial equilibrium orientations to the attracting centre of a gyrostat with an elastic rod

Stabilization of the non-trivial relative equilibria of a gyrostat with an elastic element in a circular orbit

“…where, in turn, the quantities are determined using the solutions of the corresponding transcendental equation (compare with the situation considered previously in Ref. 7). For our further analysis it will be extremely fruitful to introduce the new variables q n (t) ≡ (c nn ) 1/2q n (t) and, in accordance with representation (1.1), the new functions n (r) ≡ (c nn ) −1/2 n (r), ( n , m ) = −2 n ␦ nm .…”

“…By increasing the stiffness of the elastic element, the inequality in (2.6) can also be satisfied. For example, 7 if the elastic element of the system is a rod and its flexural deformations correspond to Kirchhoff's hypotheses, then 2 n = EI␤ 4 n /, where ␤ n are the corresponding roots of the equation (it can be shown that { −1 n } ∈ l 1 ) 12 and EI is the flexural stiffness of the rod. As mentioned earlier the Krylov beam functions are used as n (r).…”