Recent work on multi-regime per cycle machinery has shown that for such systems the transient response of the link deflections, which is governed by the homogeneous solution to a Hill-type differential equation, cannot be neglected in a meaningful simulation. Based on mathematical work by E.T. Whittaker, a new analytical approach for obtaining such transient responses is given. It is achieved by the determination of the characteristic exponent in the stable regions of the Floquet solution to the Hill’s equation involved. This technique is applied to a specific elastic mechanism. Several sample computations show that the transient response may vary from near-periodic to totally aperiodic.