## 1 Introduction  
Intershaft bearings, a type of rolling bearing, differ from traditional rolling bearings because both the inner and outer rings rotate. To obtain a compact structure, intershaft bearings are widely used in dual-rotor equipment where they play a significant role such as in gas turbines. Additionally, their functional reliability directly affects the life and safety of the equipment. The gas turbine study in this paper is aimed at gas turbines which have intershaft bearings without considering application situations. Intershaft bearings are installed close to the combustion chamber, so the operating temperatures are very high, and large bearing clearances are designed for operating at such temperatures. Faults occur frequently in intershaft bearings because of the above reasons and the high rotational speeds. Moreover, their variable load makes their working conditions even worse. When compared with the faults in other types of bearings, faults in an intershaft bearing lead to greater damage and economic losses [1]. In order to prevent serious faults in dual-rotor equipment, it is very necessary to study the condition monitoring and fault diagnosis of intershaft bearings. Researches on fault diagnosis for rotating machinery mainly focus on dynamic analysis and signal processing. For the former, dynamic analysis is used for recognizing and predicting a fault. A method based on dynamics and probability for calculating failure probability was investigated by [PERSON] [2]. Then, [PERSON] et al. proposed semidiscretization methods instead of time domain analysis in vibration prediction. They proved that Chebyshev or homotopy can be useful to solve the basic equation more efficiently in dynamic analysis [3; 4]. However, this paper focuses on the latter one. Vibration analysis is a widely used method for rolling-element bearing (REB) condition monitoring and fault diagnosis, and the corresponding vibration signal processing methods are constantly being improved. With a fixed bearing housing, a traditional REB installed in a piece of equipment with a single rotor has an outer ring connected to the solid bearing housing. In this case, the vibration signal can be measured in the bearing housing, which is directly linked to the REB with a single rotor. Naturally, the fault features can be extracted effectively using the existing processing methods. However, as for the intershaft bearings in dual-rotor equipment (as shown in Figure 1), fault diagnosis has several technical challenges as compared with traditional REBs. (1) As both the inner and outer rings of intershaft bearings rotate, no positions for direct measurements of vibration can be selected. The vibration must be indirectly measured, while traditional vibration of REBs can be directly measured because the outer ring is fixed; (2) As the red dotted line shown in Figure 1, elastic supports and a variety of thin-wall struts are included in the transfer path of the indirectly measured signal, so the weak signal of the intershaft bearings is buried in strong vibration and interference; (3) The inner and outer rings of the intershaft bearing rotate at different high speeds, thus the fault feature is relevant to two different and varied rotating speeds. However, as for traditional bearings, the fault feature is relevant to one rotating speed. One of the most common methods for analyzing the vibration signals of REBs is resonance demodulation, which is based on envelope analysis. However, because it is difficult to extract fault features directly from vibration signals through envelope analysis, it is necessary to filter the signal before analyzing it. This method is known as resonance demodulation. In 1974, [PERSON] and [PERSON] [5] applied resonance demodulation to fault diagnosis of bearings for the first time, achieving good results. However, this extraction method for fault features is mainly applicable to signals mixed with a small amount of noise.