The broader goal of this article is to re-examine the classical machinery shut down vibration problem in the context of a two degree of freedom nonlinear torsional system that essentially describes a braking system example. In particular, resonant amplifications during deceleration, as excited by a multi-order rotor surface distortion and pad friction regime, are investigated using a nonlinear model, and the order domain predictions are successfully compared with an experiment. Then a quasi-linear model at higher speeds is proposed and analytically solved to obtain closed form expressions for speed-dependent torque as well as its envelope curve. The Hilbert transform is also utilized to successfully calculate the envelope curves of both quasi-linear and nonlinear systems. Finally, the multi-term harmonic balance method is applied to construct semi-analytical solutions of the nonlinear torsional model, and the order domain results are successfully compared with measurements. New analytical solutions provide more insight to the speed-dependent characteristics given instantaneous frequency excitation. (C) 2012 Elsevier Ltd. All rights reserved.