A Time-Variant/Invariant Equivalence for the Transient Response of Rotor Blades in Resonance Crossing

The transient response of a time-varying oscillator under a chirp loading is considered in this paper. It is assumed that the natural frequency of the system changes with time during sweeping. Numerical simulations indicate that the maximum response amplitude and the apparent damping ratio are both higher than in the case without time-varying natural frequency. These two specificities of the time-varying problem are caused by the prolongated locking of the excitation frequency to the natural frequency. An asymptotic solution of the problem is derived and demonstrates that, at leading order, the time variant system responds like a time invariant system subjected to another sine sweep, having a slightly lower frequency rate. This suggests a simple equivalence. Previous studies of the time invariant problem indicate that a decrease in the sweeping rate results in increased amplification and apparent damping. This shows that the proposed equivalence is able to capture the main feature of the transient response of the time variant problem.