Dynamic viscoelastic rod stability modeling by fractional differential operator

The need to take into account the oscillation of a system is a special feature in the linear problem of the stability of a cantilevered rod under a follower force. Involvement of viscoelastic materials leads to damping of the oscillation hence to overestimation of critical loads. This new problem is solved here by means of an additional term introduced into the constitutive equation and proportional to the fractional time derivative with complex order-besides the inertial one. The effects contributed by the damping ratio, the real part of the order and the corrective role of its imaginary part on the shape of the bifurcation line, on its maximum and on the disposition of the inflection and maximal deflection points on the centerline of the deformed rod during the secondary loss of stability, are discussed.