Analysis of Steady-State Oscillations in an Autonomous Oscillator under Multi-Frequency Excitation

Abstract

The analysis of the behavior of an oscillator under multi-frequency excitation is considered in the paper. The investigation is based on the phase macromodel. The paper shows that three steady-state modes can exist in oscillator under multi-frequency excitation. The synchronized (locked) mode can be defined as the coincidence of the oscillator fundamentals with the excitation fundamentals in the region of sufficiently large excitation magnitude. The unsynchronized (unlocked) mode exists outside the synchronized region and its spectrum contains additional intrinsic fundamental besides the excitation ones. Singular points mode in some isolated points outside the synchronized region is characterized by the equality of the number of the oscillator fundamentals with the number of the excitation fundamentals. Performed numerical experiments confirmed the appearance of bifurcation points while transition of oscillator into the synchronization mode. The existence of singular points outside the synchronization region and their isolated character was also experimentally demonstrated. The problems of finding a steady-state solution of the phase equation of an excited oscillator by the Harmonic Balance (HB) method are considered. It is shown that main difficulties are connected with the presence of linear term in the steady-state solution. A transformation is proposed to provide the formation of HB equations for the phase micromodel in a standard form. Additional difficulties of HB simulations of synchronized oscillator phase equations are discussed.