Stability of nonlinear oscillations predicted by harmonic-linearisation technique

An analytic method of establishing the stability of oscillations which occur under both forced and limit-cycle conditions in nonlinear systems of any order is presented. The method allows the rate at which oscillations diverge or converge to be calculated. it is shown that the normally used graphical argument for assessing the stability of limit cycles is an approximate solution of the theory, obtained by neglecting the frequency and damping dependence of the nonlinearity. The theory is also used to derive a general stability bound on forced oscillations occurring in nonlinear systems. Application of the theory to a relay system stabilised by integral control indicates that stability is always predicted correctly and that divergence of stable oscillations is calculated with sensible accuracy. The jump phenomenon in a ferroresonant circuit is also examined, and a formula is derived for the stability boundaries. Digital-computer modelling is used to verify the predicted exponential change in sinusoidal output for step changes in input amplitude and frequency.