## 1 Introduction
However, this method imposes stringent requirements on the accuracy of the two resampling procedures involved. [PERSON] et al. proposed a rotating speed isolation method to extract the instantaneous rotating frequency based on the short time-frequency transformation (STFT) spectrogram; the fault characteristic was detected in the angular domain for fault diagnosis [19]. Once the fault is weak, the speed estimation accuracy may be affected. [PERSON] et al. presented a parametric method for the estimation of the instantaneous speed of rotating machines; the proposed approach exhibited good estimating behavior with computational simplicity [20]. However, the speed estimation may degrade when the harmonic components in the vibration signal are weak; further, some parameters should be determined before the estimation. [PERSON] et al. studied an algorithm to automatically estimate the rotational speed from the vibration generated by a gear pair; Singular Spectrum Decomposition was used for decomposing the vibration signal into the mono-component signals, and a non-quadratic phase coupling analysis was applied in order to detect the amplitude modulations [21]. Above all, many rotational frequency estimation methods have been proposed and applied for fault diagnosis of rotational machinery, but they still face the problem of large amount of computation and difficulty in improving the accuracy. Furthermore, the estimation accuracy is deeply dependent on the extraction of the harmonic components, the starting point selection of the IF, the resolution of the TFR, the noise components, etc. In this paper, a new in situ time-varying gravity acceleration decomposition method (GAD) is proposed for rotational speed estimation, and it is applied in the order tracking scene for fault diagnosis of a gearbox under non-stationary working conditions. The core idea of the proposed method is to acquire the time-varying gravity acceleration component by using an in situ MEMS accelerometer during the rotation of the shaft or disc. The corresponding decoupling method GAD and instantaneous rotational phase calculation method are presented based on the linear-phase FIR filter design [22], CEEMDAN [23], Hilbert transformation [24], etc. The correctness and effectiveness of the proposed method are verified by a motor-shaft-disc experimental setup with a local in situ accelerometer. The results show that both the average and instantaneous rotational speed can be well estimated. Furthermore, the GAD method is applied in order tracking for fault diagnosis of a gearbox, and four kinds of vibration-based estimation methods are compared. The superiority of the proposed method in the detection of tooth spalling faults under non-stationary working conditions is verified. Overall, the main contributions of the proposed method can be outlined as follows:  
1. A new gravity acceleration decomposition method is proposed for rotational speed estimation. 2. The time-varying gravity acceleration component is sensed by a tangentially mounted in situ MEMS accelerometer. 3. The proposed method yields better results in terms of order tracking than traditional vibration-based tacholess methods for fault diagnosis of a gearbox. The rest of this paper is organized as follows. The underlying estimation principle of the instantaneous rotational speed based on the decoupled gravity acceleration component of the local in situ accelerometer is introduced in Section 2. The rotational speed estimation method based on CEEMDAN and Hilbert transformation is presented in Section 3. Then, the rotational speed estimation principle is verified in Section 4 based on a kinematic simulation of a rotor with the local in situ accelerometer. Then, six tests are conducted on an established motor-shaft-disc experimental setup in Section 5. Finally, the conclusions are drawn in Section 6.