Street sweeping is an important public service, as it has an impact on aesthetics and public health. Typically, sweeping vehicles have a gutter brush that sweeps the debris that lies in the road gutter. As most of the debris is located in the gutter, the effective operation of the gutter brush is important. The aim of this work is to study the performance of a type of gutter brush, the cutting brush, through a 3D dynamic (transient), large deflection finite element model developed by the authors. In this brush model, the brush mounting board is modelled as fixed, and, consequently, inertia forces are applied to the bristle, which is modelled as a beam element. In order to simulate the interaction with the road surface, this is rotated, translated, and raised. Bristle-road contact is modelled through a flexible-to-rigid contact pair. Particularly, the concept of a cutting brush rotating at variable speed is explored through the finite element analysis of a constrained horizontal cutting brush. This analysis helps to understand the behaviour of oscillatory cutting brushes for different frequencies of brush oscillation. It is concluded that, for a horizontal cutting brush, oscillations have an impact on bristle dynamics, and its performance may be improved by varying the brush rotational speed at certain frequencies.