Nonlinear Stability of Differential Surge Chambers

Abstract

Surge oscillations in a differential surge chamber have two degrees of freedom and are characterized by nonlinearities. The linear approximation provides stability criteria for small perturbations around the equilibrium state, and the Thoma stability criterion applies to the differential chamber. Large surge oscillations are investigated with direct numerical integration on two phase planes. The analysis indicates that the system reveals itself as a supercritical Hopf bifurcation—the exchange of stability from an asymptotically stable spiral to an unstable spiral approaching a stable limit cycle. The total chamber area is the controlling parameter. The bifurcation point corresponds to the Thoma criterion. For a surge chamber with an‐area larger than the Thoma value, an unstable limit cycle may exist around the equilibrium state on each phase plane, and it defines the domain of asymptotic stability. If the chamber area is less than the Thoma value, the case of soft self‐excitation with a stable limit cycle inside and an unstable one outside may occur, Stability in the large and instability in the small dominate the system.