Rotating brushes are commonly used in cleaning air ducts. A very simple model simulating the behaviour of a typical brush bristle is presented. The model consists, in effect, of a combination of a polynomial trial solution, the point collocation method, Simpson's integration rule and a Mathcad code. The main interest from the cleaning point of view is in determining the contact force and the contact angle between the bristle tup and the duct surface. The effect of the degree n of the polynomial trial solution - n=4, n=6, n=8 - is studied. With typical data, case n=6 already seems to give results which are accurate enough for practical purposes. The application of Simpson's integration rule in the formulation is described in some detail.