When this gyrostatic chain is made to revolve with angular velocity n in relative equilibrium as a plane polygon passing through Oz the axis of rotation, each gyrostatic case moves as if its axis produced was attached to Oz by a flexure joint. The instantaneous axis of resultant angular velocity bisects the angle π − θ, if the axis of the case makes an angle θ with Oz, and, the components of angular velocity being n about Oz, and −n about the axis, the resultant angular velocity is 2n cos½ (π − θ) =2n sin½θ; and the components of this angular velocity are
(1) −2n sin ½θ sin ½θ = −n (1 − cos θ), along the axis, and
(2) −2n sin ½θ cos ½θ = −n sin θ, perpendicular to the axis of the case. The flexure joint behaves like a pair of equal bevel wheels engaging.
The component angular momentum in the direction Ox is therefore
L = −An sin θ cos θ − Cn (1 − cos θ) sin θ + K sin θ,
(3)
and Ln is therefore the couple acting on the gyrostat.
If α denotes the angle which a connecting link makes with Oz, and T denotes the constant component of the tension of a link parallel to Oz, the couple acting is
Ta cos θk (tan αk+1 + tan αk) − 2Tα sin θk,
(4)