Application of the HB–AFT method to the primary resonance analysis of a dual-rotor system

This paper focuses on the primary resonance analysis of a dual-rotor system having two rotor unbalance excitations of different rotating speeds and being connected by an inter-shaft ball bearing. Due to the complex excitation condition and the complicated nonlinear bearing forces of the inter-shaft bearing, the general analytical methods, e.g., the multiple scales method or the harmonic balance method, are failed to give the theoretical solutions. Thus, the harmonic balance–alternating frequency/time domain (HB–AFT) method is formulated to deal with this problem. The basic idea of the method is using the inverse discrete Fourier transform and the discrete Fourier transform, instead of the direct analytical relationship between the supposed solutions of the system and the nonlinear forces, to construct the harmonic expressions of the nonlinear forces, which is the so-called alternating frequency/time domain technique. By using the HB–AFT method, therefore, a Newton– Raphson iteration procedure can be performed to demonstrate the approximate solutions of the system. Accordingly, the frequency responses of the system affected by some critical parameters, such as rotating speed ratio, unbalances of both the inner and outer rotors, and clearance of the inter-shaft bearing, are analyzed, respectively. A variety of phenomena including double resonance peaks, biperiodic and quasi-periodic behaviors, and resonance hysteresis phenomenon are obtained, which are discussed in detail through diagrams for separated frequency responses with different frequency components and rotors’ orbits for different combinations of system parameters. Most prominently, for a relatively small unbalance of rotor as well as a relatively large clearance of the inter-shaft bearing, the resonance hysteresis phenomena are more obvious. The results obtained are also compared with the direct numerical simulation results, and the comparisons show good agreements. In addition, the methodology formulated in this paper is a general approach, which can be applied to other engineering systems with multi-frequency excitations.