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.

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

This paper focuses on the nonlinear response characteristics of a dual-rotor system with a breathing transverse crack in the hollow shaft of the high-pressure rotor (rotor 1). A finite element model of the system is set up, and the motion equations of the system are formulated, in which the unbalance excitations of the rotor 1 and rotor 2 (low-pressure rotor) and the time-varying stiffness of the cracked shaft are considered. By using the harmonic balance method, the motion equations are analytically solved to obtain the dynamic responses of the two rotors. Accordingly, the effects of the crack depth and location on the vibration amplitudes are discussed in detail. The results indicate that when a transverse crack appears, it may bring super-harmonic responses to the rotor system, and the resonance peaks at the second, third and even fourth subcritical whirling speeds of the two rotors can be observed. The deeper the crack is, the larger the resonances amplitudes are, especially when the crack is located in the middle of the shaft or around the disks. In addition, the super-harmonic responses of rotor 1 where the crack located, can also be observed in rotor 2, which means that the crack signals can be detected in the entire system. Moreover, the numerical computations are carried out by using the Newmark-
 $$\beta $$
 method, which shows great agreement with the previous analytical results. The results obtained in this paper will contribute to the modeling and the fault diagnosis of dual-rotor systems with hollow-shaft crack.

Nonlinear response analysis for a dual-rotor system with a breathing transverse crack in the hollow shaft

Bifurcation and stability analysis of a nonlinear rotor system subjected to constant excitation and rub-impact

This paper focuses on the local bifurcation characteristics of an aircraft cracked rotor system mainly for the 2:1 and 3:1 super-harmonic resonances induced by the maneuver load. The motion equations of the system are formulated with the consideration of the nonlinear stiffness of the Duffing type and the breathing of a transverse crack on the shaft, as well as the maneuver load induced by the climbing and diving flight of the aircraft. By using the multiple scales method, the motion equations are analytically solved to obtain the bifurcation equations for 2:1 and 3:1 super-harmonic resonances, respectively. Furthermore, the two-state variable singularity method is employed to analyze the local bifurcation characteristics of the system affected by crack coefficient and maneuver load. For each case, two curves of hysteresis set dividing $$K-G$$
 parameter plane into three regions are demonstrated. Accordingly, bifurcation modes for different parameter combinations from the three regions and the two curves are obtained. The approach in this paper will provide an effective and convenient way to analyze the bifurcation characteristics of dynamical systems. The results in this paper will contribute to a better understanding of the effect of the maneuver load on the response and bifurcation characteristics of aircraft cracked rotor systems.

Bifurcation analysis for 2:1 and 3:1 super-harmonic resonances of an aircraft cracked rotor system due to maneuver load

In this paper, an investigation on the nonlinear vibration, especially on the super-harmonic resonances, in a cracked rotor system is carried out to provide a novel idea for the detection of crack faults in rotor systems. The motion equations of the system are formulated with the consideration of the additional excitation from an inertial environment as well as the forced excitation of the rotor unbalance. By using the harmonic balance method, the analytical solutions of the equations with four orders of harmonic exponents are obtained to analyze the nonlinear response of the system. Then through numerical calculations, the vibration responses affected by system parameters including the inertial excitation, the forced excitation, the crack and damping factors are investigated in detail. The results show that the occurrence of the super-harmonic resonances of the rotor system is due to the interaction between crack breathing and the inertial excitation. Correspondingly, the super-harmonic responses are significantly affected by the inertial excitation and the crack stiffness (or depth). The rotor unbalance, however, does not make apparent effects on the super-harmonic responses. Consequently, the super-harmonic resonances peaks can be viewed as an identification signal of the crack fault due to the application of the inertial excitation. By utilizing the inertial excitation, the super-harmonic response signals in rotor systems with early crack faults can be amplified and detected more easily.

Super-harmonic responses analysis for a cracked rotor system considering inertial excitation

In this paper, the dynamical simulation of a nonlinear aircraft rotor system in Herbst maneuvering flight is carried out, though which the load control is discussed for safety purposes. The equatio...

Hou Lei

Papers

Dynamical simulation and load control of a Jeffcott rotor system in Herbst maneuvering flight

Constant-excitation caused response in a class of parametrically excited systems with two degrees of freedom