Stability of discretized elastic columns with periodic retarded follower forces

The objective of this work is to present a stability analysis for discretized elastic columns under the influence of periodically-varying nonconservative follower forces whose follower angle is retarded, i.e., depends on a previous position of the system. One- and two-degree-of-freedom systems under the simultaneous influence of both parametric excitation and time-delay, whose effects on such systems have previously been studied separately, are considered. By employing an orthogonal polynomial approximation, the so-called "infinite-dimensional Floquet transition matrix" associated with the time-periodic differential-delay system is approximated. The stability criteria that all the eigenvalues (Floquet multipliers) of this matrix must lie within the unit circle is then applied. The stability charts for different combinations of the remaining system parameters are shown, and the previously-reported results for the special cases where either the parametric excitation or the time-delay vanishes are verified. Two cases, when the parametric forcing period is equal to or twice the delay period are taken into consideration in this work.