This paper was accepted for publication in the journal Journal of Sound and Vibration and the definitive published version is available at http://dx.doi.org/10.1016/j.jsv.2015.01.017The paper aims at studying the effects of the inherent fractional constitutive law of viscoelastic bearings used as devices for tuned mass dampers. First, the proper constitutive law of the viscoelastic supports is determined by the local constitutive law. Then, the characteristic force-displacement relationship at the top of the bearing is found. Taking advantage of the whole bearing constitutive laws, the tuning of the mass damper is proposed by defining the damped fractional frequency, which is analogous to the classical damped frequency. The effectiveness of the optimal tuning procedure is validated by a numerical application on a system subjected to a Gaussian white noise