BIFURCATION OF A ROTATING SHAFT WITH UNSYMMETRICAL STIFFNESS
具有非轴对称刚度转轴的分岔

In this paper 1/2 subharmonic resonance and bifurcation in a rotating shaft with an unsymmetrical stiffness are studied. By means of the Hamilton's principle the nonlinear differential equations of motion of the shaft are derived in the rotating rectangular coordinate system. Trans forming the equations of motion from rotating coordinate system into stationary coordinate system and introducing a complex variable we obtain the equation of motion in complex variable forms in which the stiffness coefficient varies periodically with time. It represents a nonlinear oscillation system under parametric excitation. By applying the method of multiple scales we obtain the averaged equations, the bifurcation response equations and local bifurcating set. Via the theory of singularity we analyze the stability of constant solutions and obtain bifurcation response curves. This study exhibits that the shaft with unsymmetrical stiffness possess an unstable range in neigh borhood of the rotating speed. The asymmetry and the external damping have great influence on the shaft stability and local bifurcation. As asymmetry increases, the width of the instability region increases. However, the external damping can reduce the width of the unstable region. This study still exhibits that the rotating shaft has rich bifurcation phenomena. The size of bifurcation region is related to the asymmetry of the shaft stiffness. When asymmetry of stiffness vanishes the bifurcation regions disappear. In this study we may see that degenerate bifurcation of codimension two in the shaft system above may occur.