  This paper examines the oscillatory behaviour of complex viscoelastic systems
with power law-like relaxation behaviour. Specifically, we use the fractional
Maxwell model, consisting of a spring and fractional dashpot in series, which
produces a power-law creep behaviour and a relaxation law following the
Mittag-Leffler function. The fractional dashpot is characterised by a parameter
beta, continuously moving from the pure viscous behaviour when beta=1 to the
purely elastic response when beta=0. In this work, we study the general
response function and focus on the oscillatory behaviour of a fractional
Maxwell system in four regimes: stress impulse, strain impulse, step stress,
and driven oscillations. The solutions are presented in a format analogous to
the classical oscillator, showing how the fractional nature of relaxation
changes the long-time equilibrium behaviour and the short-time transient
solutions. We specifically test the critical damping conditions in the
fractional regime, since these have a particular relevance in biomechanics.
