The method of multiple scales is used to study the nonlinear response of a relief valve under a constant static pressure and a sinusoidal dynamic pressure that can be generated by variations in the pneumatic transmission lines. Under certain conditions, large vibrations may occur, causing unsatisfactory performance as well as undesirable noise. In this theoretical analysis, the valve is modeled by a continuous spring fixed at one end and attached to a valve ball at the other end. The ball rests on a valve seat having nonlinear spring characteristics. The method of multiple scales is used to determine the non-linear response of the valve for the cases of harmonic, higher-harmonic, subharmonic and combination resonances. Conditions are determined for the onset and stability of large-amplitude oscillations. / M.S.