Controlling friction at the wheel-rail interface is indispensable for extending track life, minimising wheel-flange wear, improving fuel efficiency, reducing noise and lateral forces. A particular implementation of friction modifier system consists of a stick-tube assembly, attached through a bracket which is suspended from the railway bogie frame. Inside the tube, a set of interlocking solid sticks resides with one end pressed against the tread or flange of the wheel, and the other end against a constant force tape spring. Rubbing action at the stick-wheel interface and the action of the spring results in a gradual transfer of friction modifier film to the wheel and thence to the rail through the wheel-rail contact. This results in effective friction management between the wheel and the rail. Friction modifier systems can experience unstable friction-induced vibrations due to a complex set of in situ contact conditions. Stability prediction is important for efficient functioning of friction control systems. This dissertation contributes a stability analysis procedure in frequency domain based on Frequency Response Functions (FRFs) of the wheel and the applicator-bracket subsystems. The stability analysis yields stability maps delineating stable and unstable regions of operation in the design parameter space defined by speed of train, angle of applicator, and friction coefficient. Stability characteristics of three bracket designs are compared using experiments and finite element models. Results are summarised in the form of stability diagrams indicating the operating conditions that will lead to unstable vibrations. This methodology can easily incorporate design changes to the bracket and/or applicator, thus facilitating a rapid comparison of different designs for their stability characteristics even before they are built.