ON THE LOCAL STABILITY ANALYSIS OF THE APPROXIMATE HARMONIC-BALANCE SOLUTIONS

Periodic response of nonlinear oscillators is usually determined by ap proximate methods. In the ''steady state'' type methods, first an appr oximate solution for the steady state periodic response is determined, and then the local stability of this solution is determined by analyz ing the equation of motion linearized about this predicted ''solution' '. An exact stability analysis of this linear variational equation can provide erroneous stability type information about the approximate so lutions. It is shown that a consistent stability type information abou t these solutions can be obtained only when the linearized variational equation is analyzed by approximate methods, and the level of accurac y of this analysis is consistent with that of the approximate solution s. It is demonstrated that these consistent stability results do not i mply that the approximate solution is qualitatively correct. It is als o shown that the difference between an approximate and the next higher order stability analysis can be used to ''guess'' the role of higher harmonics in the periodic response. This trial and error procedure can be used to ensure the qualitatively correct and numerically accurate nature of the approximate solutions and the corresponding stability an alysis.