## 1 Introduction  
In the rotor mechanism of an aero-engine, gear drive plays an important role as its main transmission mode. The motion and power between any two axes in the space are transferred by the direct contact of the tooth profile, which has the advantages of large transmission power, stable and reliable transmission, high efficiency and long service life in the gear mechanism [1]. Due to the advantages discussed above, gear drive is widely used in the transmission systems of aero-engines. However, because of the harsh working environment of high-speed, high temperature and high alternating load, the central bevel gear has a tendency to fail due to fatigue [2; 3; 4]. The forms of gear failure usually include pitting erosion, tooth-breaking, plastic deformation, etc. [5; 6; 7; 8], among which aviation bevel gear whole-tooth fracture and block drop fault caused by the propagation of the fatigue source at the tooth root due to strong vibrational impacts caused by TWR lead to oil cut off, power cut off, surge stop and even plane crash accidents, which usually occur suddenly and have disastrous consequences. Therefore, it is the response of the gear under TWV conditions that should be studied. However, it is difficult to obtain the dynamic response of the fatigue source at the tooth root due to the working environment and the structure of the bevel gear. Therefore, in view of the above reasons, it is particularly significant and necessary to analyze the characteristics of the bevel gear in aero-engine TWR and stress distribution accurately through the combination of experiments and simulations so as to provide a theoretical basis and fault diagnosis of such accidents in aero-engine gear systems. Some progress has been made in the theoretical research and gear dynamics in relation to gears in aero-engine TWR. [PERSON] et al. [9] studied the vibration characteristics of rotating, cylindrical thin shells under different boundary conditions by using Fourier series expansion, taking into account the effects of centrifugal force, Coriolis force and toroidal tensor caused by rotation, and finally derived the exact frequency expression of any type of rotating thin cylinders with classical boundary conditions. [PERSON] et al. [10] studied the vibration mode and frequency of spur planetary gears in the single vibration mode in a rotating reference frame and a stationary reference frame. They found that the frequency content of motion was different between the rotating carrier and the base, which is helpful when analyzing the experimental measurement results of planetary gears. [PERSON] et al. [11] used thin shell theory to analyze the frequency response of six different forms of mode expansion. He proposed a more concise and accurate mode for the expansion method of TWV and compared the results of the analytical method and numerical method to prove the reliability of the method. [PERSON] et al. [12] studied the self-excited vibration of spur gears based on the energy method, proposed a theoretical method for predicting the occurrence of the self-excited vibration of thin spur gears and described the conditions and loads of the occurrence of the self-excited vibrations. [PERSON] et al. [13] established a three-degree-of-freedom dynamic model for a gear-rotor-bearing system, taking into account the effects of the nonlinearities associated with radial clearances in the radial rolling element bearings and backlash between a spur gear pair. Afterward, they verified this approximate bearing model by comparing the steady-state frequency spectra, a criterion used to classify the steady-state solutions was presented, and the conditions for chaotic, quasi-periodic and subharmonic steady-state solutions were determined. [PERSON] et al. [14] used the incremental harmonic balance (IHB) method to investigate the periodic motions of a nonlinear geared rotor-bearing system. The periodic solutions and subharmonic solutions obtained through the use of the IHB method compared very well with those obtained through numerical integration.