## 1 Introduction  
Rotor is a key element of rotary machine and its working condition affects the reliability and safety of whole mechanical system. Accurate identification of rotor faults is very helpful for reducing economic losses and ensuring production safety [1; 2; 3; 4]. As the vibration signal of rotors carries plentiful fault information, a lot of rotor fault diagnosis approaches were developed based on vibration signal analysis [5; 6]. Due to the non-stationary and non-linear characteristics of rotor vibration signals, classical signal analysis methods based on Fourier transform fail to extract the fault features of rotors accurately [7]. In recent years, the application of time-frequency analysis to rotor failure detection has attracted considerable attention as the time-frequency analysis approaches are able to reflect the characteristic information in both time domain and frequency domain [8]. Among these developed time-frequency analysis approaches, wavelet transform (WT) is one of the most effective one for its high time-frequency resolution [9; 10]. However, an appropriate basis function should be selected or designed in every use of WT to ensure a satisfactory result, which reduces the adaptability of WT [11]. Considering the drawback of WT,Huang et al. developed a novel adaptive time-frequency technique called Hilbert-Huang transform (HHT) [12], which is the combination of empirical mode decomposition (EMD) and Hilbert transform (HT). EMD is used to adaptively decompose a multi-component signal into several intrinsic mode functions (IMFs) that carry the useful information of different frequency bands. Then the instantaneous amplitude (IA) and instantaneous frequency (IF) of each IMF are calculated by HT. Consequently, the HHT time-frequency spectrum (TFS) can be obtained by integrating the IAs and IFs of all IMFs. HHT has an extensive application in detecting mechanical faults owing to its priorities of extracting non-stationary characteristics of signals. The HHT TFS demonstrates the energy-frequency-time distribution of a signal. For different mechanical faults, the corresponding energy-frequency-time distribution varies greatly. To implement the automatic identification of different mechanical fault types, it is of great significance to exploit an effective index to measure the differences of the corresponding energy-frequency-time distribution. Shannon Entropy is a satisfactory indicator for measuring the complexity and regularity of a signal and the diversity of any distribution [13; 14; 15]. Therefore, some entropy-based indicators were developed to evaluate the fault features of rotating machinery. For example, the Teager energy entropy ration (TEER) was proposed as an impulsive feature evaluation index to select the optimal sub-band component obtained by performing wavelet transform decomposition on the vibration signal of rolling bearings in [16]. The TEER index is the combination of the Teager energy entropy in time domain and frequency domain. However, it cannot reflect the information of the time-frequency domain. [PERSON] et al. proposed a time-frequency entropy (TFE) method by combing the merits of entropy and HHT for gear fault diagnosis [17]. The TFE gives an indicator that can estimate the information of time-frequency domain. However, the TFE has two main shortcomings. One is originated from EMD. EMD is troubled by mode mixing [18], which makes the obtained HHT TFS cannot accurately reflect the time-frequency information of a signal. On the other hand, the TFE index is a measure of the whole information of the HHT TFS, it is not able to extract the local energy-frequency-time distribution of the TFS, which leads the calculation of TFE to be relatively complex. In conclusion, the TFE can be improved from two aspects: (a) Improve the HHT method to obtain a more accurate TFS; (b) Develop a novel index which can reflect the local characteristics of the TFS.