Spur gear crack modelling and analysis under variable speed conditions using variational mode decomposition

Highlights

• Realistic gear tooth model with parabolic crack was developed under variable speed.

• The Variational Mode Decomposition was applied to filter out non-stationarities.

• A new feature SPI was proposed to quantify severity levels.

• The simulation model was validated by exhaustive experiments with seeded defects.

• Random Forest based feature importance study is performed to prove efficiency of SPI.

Abstract

Cracks in gear are the most common mode of failure. The presence of cracks leads to a reduction in the time-varying mesh stiffness and eventually affects the vibration properties of the gear system. The non-stationarities induced due to the speed variation complicate the fault diagnosis process. A gear tooth crack model for spur gear with a simplified crack profile is proposed to mimic the natural path of crack propagation. The vibration response simulated using a six-degrees of freedom lumped parameter model is exploited with Short-Time Fourier Transform and the Variational Mode Decomposition. The effectiveness of the Variational Mode Decomposition method to filter out the non-stationarities in the signals subjected to low and very low amplitude speed fluctuations is addressed. A new state feature (Side-band Power Index) that does not require the measured speed profile (using tacho pulse input) is proposed for severity level classification. A feature importance study based on the Random Forest classifier is performed to prove the effectiveness of the proposed state feature amongst the conventional state features. The simulation model is validated with several experiments. The article aims to provide a complete diagnostic solution for the spur gear systems subjected to variable speed conditions.

Introduction

Gear is a key machinery element found in various engineering applications. In most of these applications, gears are subjected to harsh operating conditions and excessive service loads, leading to fault development. The gear faults account for around 60% of gearbox failures [1], with the fatigue cracks being the most common failure mode [2]. The presence of crack leads to an overall reduction in the gear mesh stiffness. The mesh stiffness being the primary source of parametric excitation [3], the vibration response of the system is affected. The severity level of the cracks may be assessed with the quantification of reduction in the mesh stiffness, and the vibration response of the system can be simulated with dynamic simulation [4], [5]. The simulated vibration response may be utilized for further diagnostic studies. The simulation method may be either a finite element method (FEM) or an analytical method (AM) [6]. The ease of computation and shorter process time over FEM gives an extra edge to AM.

Evaluation of the time-varying mesh stiffness (TVMS) is a crucial aspect in AM, as it reflects the health condition of the gear teeth in contact. Over the years, various spur gear tooth crack models are reported for the TVMS evaluation. Yang and Lin [7] proposed the mesh stiffness evaluation based on the potential energy method. The first potential energy-based model proposed by Yang and Lin [7] comprised bending stiffness, axial stiffness, and the Hertzian contact stiffness. Later, the shear stiffness was included by Tian et al. [8], followed up with the inclusion of the tooth-fillet foundation effect by Sainsot et al. [9]. Recently, Sainsot et al. [10] reported a detailed analysis of the tooth contact deflection methodologies addressing a simplified thin slice contact model and validated it with 2D, 3D finite element models. Cooley et al. [11] reported a study on the force-deflection relationship associated with the spur gear cracks. The evaluation of TVMS is also affected by manufacturing errors, tooth profile modifications [12], tip relief modifications [13], and time-varying backlash [14]. The effect of long symmetric tip relief on the high contact ratio spur gears was analyzed by Snchez et al. [15]. Further, Pleguezuelos et al. [16] extended the study and developed the analytical relations for load-sharing ratio and quasi-static transmission error in spur gears. The effect of gear tooth profile modifications was analyzed for standard spur gears and high contact ratio spur gears. Dai et al. [17] proposed an analytical model for spur gear mesh stiffness evaluation with non-linear contact stiffness and addendum modifications.

Various gear tooth crack models are reported in the literature for different types of crack. Based on the crack generation mechanism, two major gear crack types are surface cracks and root fatigue cracks (most common). The loss of lubrication film accelerates the process of surface crack generation. Yang et al. [18] developed a tooth surface crack model for the mesh stiffness evaluation and analyzed the gear system dynamics. Li et al. [19] reported a gear tooth scuffing model for spur gears and represented the meshing of gear teeth with a line contact of varying radii of curvature. As the gear root bending stress increases, the gears are more prone to develop root cracks. Dogan et al. [20] studied the relationship between bending stress and the fatigue crack propagation life. The gear root stress dependency and fatigue crack were numerically evaluated following the variation in the rim thickness and backpressure. The models reported by Yang and Lin [7], Tian et al. [8], Mohammed and Rantatalo [21], and Zhou et al. [22] considered a linear crack profile that does not resemble the natural path of crack propagation. In addition, most of these models considered the gear tooth as a cantilever beam starting from the base circle and ignored the region between the base circle and the root circle. The gear tooth initiates from the root circle, and the tooth profile of this section is not an involute curve. The cutting tool tip geometry governs the curve trajectory of this section. The contribution of this region changes the overall mesh stiffness behaviour, and the models reported earlier lacked to comment on it.

Chaari et al. [23] proposed a model with crack initiating from root circle and considered the overall reduction in the gear tooth thickness for meshing stiffness evaluation. Mohammed and Rantatalo [21] reported a spur gear tooth crack model with crack profile varying along the tooth width. However, the models proposed ignored the cracks propagating into the gear tooth body beyond the gear root circle. Liang et al. [24] proposed a gear tooth crack model for the planetary gear set with a straight line crack starting from the gear root circle. Recently, Wang and Shao [25] proposed a crack path function as a combination of a half sine-wave and a straight line. However, the crack propagation path tends to have a curvature in the root region, as reported in [26]. Thus, it is required to develop a gear crack model to address the shortcomings mentioned above.

The systems reported in the literature are subjected to constant operating conditions even though the real-life structures operate under varying operating conditions [27], [28]. Significantly, limited literature is reported in this area. Over the past decade, Mahgoun et al. [27] and recently, Sharma and Parey [29] studied gear systems under variable operating conditions. The literature reported focuses on the speed variation over 10–20% of the mean speed. However, it fails to address speed fluctuations of less than 5% of the mean speed (low amplitude speed fluctuations). These low amplitude speed fluctuations create a series of intra-cycle shocks and are of great practical importance for performance-based critical applications such as wind turbines, helicopter transmission, and precision machine tools. These conditions may lead to increased fatigue load and reduced degree of reliability in the wind turbines and helicopter transmission. The precision machine tools may witness an increase in manufacturing errors, reduced tool life, and weak control over tolerance levels. The speed fluctuations may arise from the variable load, varying voltage, or the prime mover. Therefore, the study of low amplitude varying speed conditions is crucial.

Varying speed conditions induce the non-stationarities in the signal, making the signal processing tedious. The conventional time-domain (TD) analysis techniques, such as Time Synchronous Averaging (TSA) [30] contributes to the suppression of the noise and asynchronous components of the signal. In contrast, the frequency-domain (FD) technique involves Fourier transform and provides information about dominant frequency components. These methods cannot track non-stationarities and handle instantaneous frequency components [31]; hence, these are inadequate for the analysis. Therefore, the time-frequency domain (TFD) analysis methods were developed. The TFD methods such as Short Time Fourier transform (STFT) [21], Wavelet transform (WT) [32], Wigner Ville Distribution (WVD) [33] are capable of handling the complex and non-stationary signals. In the last decade, Empirical Mode Decomposition (EMD) [34] method has been developed to decompose the multi-component signal into multiple mono-components called Intrinsic Mode Functions (IMFs). The EMD is based on empirical considerations; however, it has few shortcomings, prominent among these are mode mixing, stopping criteria for extraction of modes, and lack of mathematical justification [35], [36]. An adaptive Variational Mode Decomposition (VMD) method was developed [37] to overcome these signal processing challenges. The VMD decomposes the signal into multiple modes named Variational Mode Functions (VMFs). The method provides concurrent extraction of modes with optimized center frequency for each mode [37]. The VMD technique is utilized to identify the masked fault transients and filter out non-stationary behaviour.

Despite the availability of multiple gear crack models, there is a necessity to develop a gear tooth crack model with a realistic crack propagation path. A model with better control over crack parameters and imitating the early stages of the crack is needed. Also, the effect of low to very low amplitude speed fluctuations (i.e., less than 5% of the mean speed) on gear dynamics has not been addressed yet. From the diagnosis perspective, the VMD capabilities need to be studied in terms of filtering out the masking effect resulting from the low amplitude speed fluctuations. It is required to develop a fault diagnosis feature without tachometer pulse inputs under variable speed conditions. The feature should be independent of the operating speed profile.

A comprehensive diagnostic methodology is proposed to address the shortcomings mentioned above. The study attempts to provide a complete solution for the efficient diagnosis of early gear cracks under low and very low amplitude speed fluctuations occurring unintentionally in day-to-day operating machines. The novel contributions of the present study are as follows;

• A simplified gear tooth crack model for spur gears with better control over the gear crack driving parameters is proposed. Existing literature fails to provide a gear tooth crack model with reduced complications. These complications are majorly associated with the gear tooth crack profile definition, crack initiation points, and crack propagation path. These attributes have a significant impact on the modelling of the natural path of crack propagation. The proposed model provides a set of robust parameters governing the natural path of crack propagation.

• Most of the real-life systems are subjected to operating speed fluctuations conditions of a highly non-linear nature. These fluctuations have a direct impact on the overall gear life cycle. No studies have been reported so far addressing the effect of these conditions on the gear system dynamics. Therefore, the study of the effect of low amplitude non-linear speed fluctuations (i.e., less than 5% of the mean speed) on the gears system dynamics is of great practical importance for real-life systems. The current work is the first attempt wherein two different cases of low amplitude non-linear speed fluctuations are studied. The two cases of speed fluctuations are 5% and 0.5% speed variation about the mean speed (i.e., low and very low amplitude speed fluctuations, respectively).

• A novel condition monitoring strategy for spur gear crack analysis under variable speed conditions is developed. The study being the first of its kind, there is no comprehensive methodology available to analyze and quantify the severity of gear cracks under variable speed conditions. The challenge is to identify and filter out the non-stationarities aroused from the speed fluctuations, which is achieved through the STFT and the multi-mode signal decomposition method known as Variational mode decomposition.

• A new condition monitoring feature Side-band Power Index (SPI), is proposed to assess the severity levels and quantify the non-stationarities resulting from the fluctuating speed conditions without using tachometer pulse inputs. Also, a feature importance study for identifying the severity classification capabilities of SPI is performed. The feature importance study is based on the Random Forest classifier. Important features are identified amongst the SPI, the existing features such as time-domain features, frequency-domain features, time-frequency domain features, and special gear features.

The article is structured as follows: Section 2 discusses the gear tooth crack modelling for spur gears and the TVMS calculations. The proposed gear dynamic simulation model is discussed in Section 3, and Section 4 discusses the signal processing methodologies, i.e., signal analysis using the STFT and the VMD. Section 5 demonstrates the experimental setup for varying speed conditions. Section 6 discusses the results, and the effectiveness of the proposed methodology is analyzed through a feature importance study in Section 7. The major conclusions drawn from the present work are reported in Section 8.