Dynamic Characteristics of a Gear System with Double-Teeth Spalling Fault and Its Fault Feature Analysis

Abstract

Tooth spalling is one of the most destructive surface failure models of the gear faults. Previous studies have mainly concentrated on the spalling damage of a single gear tooth, but the spalling distributed over double teeth, which usually occurs in practical engineering problems, is rarely reported. To remedy this deficiency, this paper constructs a new dynamical model of a gear system with double-teeth spalling fault and validates this model with various experimental tests. The dynamic characteristics of gear systems are obtained by considering the excitations induced by the number of spalling teeth, and the relative position of two faulty teeth. Moreover, to ensure the accuracy of dynamic model verification results and reduce the difficulty of fault feature analysis, a novel parameter-adaptive variational mode decomposition (VMD) method based on the ant lion optimization (ALO) is proposed to eliminate the background noise from the experimental signal. First, the ALO is used for the self-selection of the decomposition number K and the penalty factor â of the VMD. Then, the raw signal is decomposed into a set of Intrinsic Mode Functions (IMFs) by applying the ALO-VMD, and the IMFs whose effective weight kurtosis (EWK) is greater than zero are selected as the reconstructed signal. Combined with envelope spectrum analysis, the de-nosing ability of the proposed method is compared with that of the method known as particle swarm optimization-based variational mode decomposition (PSO-VMD), the fixed-parameter VMD, the empirical mode decomposition (EMD), and the local mean decomposition (LMD), respectively. The results indicate that the proposed dynamic model and background elimination method can provide a theoretical basis for spalling defect diagnosis of gear systems.

1. Introduction

Gear systems commonly exist in mechanical equipment, such as wind turbines, shearer, and the motor vehicle [1]. Due to the excessive load, poor lubrication, and other harsh operating conditions, faults often occur in the gear systems. The faults may generate serious economic losses and even casualties. Specifically, damages (e.g., pitting, cracking, and spalling) on the gear teeth are the main causes of system failure, and they usually impact the dynamic characteristics of gear devices [2,3,4,5]. Background noise often occurs in actual fault detection. It is difficult to extract fault feature accurately when background noise is involved [6]. Therefore, it is essential to effectively obtain the dynamic characteristics of gear systems and eliminate the noise for identifying faults because they play an important role in improving reliability and reducing the maintenance cost of gear systems.

Dynamical modeling is a useful way to obtain the dynamic characteristics of gear systems, and there are many studies about the dynamic modeling [7,8,9,10,11,12]. To effectively obtain the dynamic characteristics of gear systems, researchers either increase the degrees of freedom from single to multi degrees, or consider various practical phenomena such as backlash, tooth friction, manufacturing eccentric errors, etc. Kahraman et al. [13] proposed a simplified purely torsional mechanical model of a gear pair, which was supported by rigid mounts with backlash and time-varying mesh stiffness. Mohammed et al. [14] modelled a one-stage reduction gear by using three different dynamic models, which consider the gear mesh friction. Ren et al. [15] considered manufacturing eccentric errors and tooth profile errors, and generalized a bending-torsional-axial coupled dynamical model to investigate the dynamic floating performances. Dynamical modeling of gear systems with faults is also an important topic for understanding the dynamic characteristics of the gear fault. Spalling is one of the most destructive surface failure models of gear faults [16]. Until now, many researchers have modeled the tooth spalling and investigated its effect on dynamic characteristics of the gear systems [17]. According to their research, tooth spalling was regarded as different shapes (e.g., rectangle, circular, and V-shape), and the influence of the spalling on time-vary mesh stiffness (TVMS) of gear was explored [18,19,20]. Luo et al. [21] proposed a curved bottom shape method to model the geometric features of gear tooth spalls and studied the effect of different shapes and severity conditions time-vary mesh stiffness by finite element analysis. Chen et al. [22] presented a new two-dimensional Gaussian distribution model to depict pitting distribution. In their analysis, the overlaps of the pits and pits in the boundary of the tooth surface were considered in the computation of TVMS. Furthermore, Luo et al. [23] built a dynamic model that includes tooth surface roughness changes and geometric deviations because of the pitting and spalling of gear teeth. This model was validated by comparison with experiments under different rotational speed and fault conditions. Abouel-seoud et al. [24] proposed a 12-degree-of-freedom lumped parameter model to simulate the effects of tooth crack, breakage, and spalling on the dynamic response of wind turbine gearbox with spur gears. Jiang et al. [25] proposed a 6-degree-of-freedom approach to investigate the dynamic characteristics of a pair of helical gears under sliding friction with spalling faults. The previously mentioned studies only focus on a single tooth spalling fault. In fact, tooth spalling, which is caused by excessive local Hertzian contact fatigue stress, may distribute over multiple teeth [26,27]. In term of this phenomenon, Liang et al. [28] modeled a gear system with double-teeth spalling fault, and simulated its effect on vibration characteristics. However [28], only the phenomenon where the tooth spalling occurs on two adjacent teeth is considered. According to Reference [28], this paper will further explore the influence of the relative position of two faulty teeth on the dynamic characteristics of gears.

The accuracy of the dynamic model needs to be verified by the experimental signal. However, the experimental signal has the background noise caused by the sensor’s location (e.g., the bearing pedestal or the gearbox housing), which will affect the results of dynamic model accuracy verification. Moreover, the fault features hidden in the experimental signal are also susceptible to the background noise and become weaker, which increase the difficulty of fault feature analysis. It is of great significance for dynamic model validation and fault feature analysis to eliminate the background noise in the experimental signal. To tackle the complex background, several signal processing methods such as Wavelet Transform (WT) [29,30], Empirical Mode Decomposition (EMD) [31,32], and Local Mean Decomposition (LMD) [33,34] have been applied. However, the treatment effect of these methods is often not satisfactory because of the inherent limitations. The WT method needs to empirically choose wavelet basis function. The EMD method decomposes any signal into a set of intrinsic mode functions (IMFs), but its efficiency is restricted by the problems of end effects and mode mixing. The LMD method is confined by the decomposition level. In recent years, a novel adaptive signal processing method called Variational Mode Decomposition (VMD) is presented [35]. By determining the relevant bands of the fault feature and dividing the signal into several IMFs, the VMD can effectively avoid the end effects and mode mixing [36,37]. In addition, Zhang et al. [38] and An et al. [39] proved that the VMD performs better than the EMD. With the merits, many novel signal processing methods around the VMD have been proposed [40,41,42,43]. However, the performance effectiveness of the VMD is mainly affected by the penalty factor â and the decomposed mode number K, which are obtained empirically at present. Ant lion optimization (ALO) is a novel intelligent algorithm, which mainly solves the optimization of the parameters by imitating the hunting mechanism of antlions [44]. Compared with other intelligent algorithms such as a genetic algorithm (GA) and particle swarm optimization (PSO), the ALO has the merit of fewer control variables and high convergence precision in the classical optimization problem [45,46]. With the merit, we adopt the ALO to search the optimal parameters (i.e., the penalty factor â and the decomposed mode number K) in the VMD, which ensures the decomposition performance of the VMD.

Generally, gear system dynamic characteristics and fault feature analysis that considers the double-teeth spalling fault have not been well studied in existing research. To make this gap, a new dynamic model of gear systems considering the double-teeth spalling fault is established in this paper. This dynamic model is validated by various experimental tests, and could supply more realistic dynamic environment for extraction of a gear double-teeth fault feature. Then, the dynamic characteristics considering the effect of the number of spalling teeth, and the relative position of two faulty teeth are analyzed. In addition, to ensure the accuracy of dynamic model verification results and reduce the difficulty of fault feature analysis, a novel parameters-adaptive VMD based on the ALO is proposed to eliminate the background noise in the experimental signal. The optimal parameters K and â of the VMD is obtained by the ALO. Moreover, the raw signal acquired from gear systems is decomposed into several IMFs by using the ALO-VMD method, and the IMFs with effective weight kurtosis (EWK) are summed as a reconstructed signal. Then, envelope spectrum analysis is applied to filter the reconstructed signal and carried out to extract the features of the faults.

The main contributions of this paper are summarized as follows: ① a new dynamical model that considers double-teeth spalling fault is proposed, and the dynamic characteristics considering the effect of the relative position of two faulty teeth are investigated. This investigation is proposed for the first time in previous studies ② by integrating the VMD and the ALO, a novel parameter-adaptive VMD is proposed to eliminate the strong noise. The ALO is used to search for the optimal combination value of VMD’s parameters K and â, which ensures the decomposition effectiveness of the VMD.

The rest of the paper is organized as following. The TVMS calculation method considering the double-teeth spalling fault is introduced in Section 2. In Section 3, the new dynamical model is proposed. In Section 4, the effectiveness of the dynamic model is validated by analyzing the dynamic characteristics of the gear system and some influencing factors are discussed. In Section 5, the background elimination method based on the ALO-VMD is detailed. Conclusions are provided in Section 6.

2. Mesh Stiffness Calculation of Gears

There are five critical points (i.e., B₁, C, P, D, and B₂) in the gear meshing process, as shown in Figure 1. The position B₁ is the contact point where a tooth pair just enters the meshing zone, and the position B₂ is the contact point when the second tooth pair exits the meshing zone. The position P denotes the pitch point where the tooth surface friction force changes its direction. During the meshing process, the single tooth pair is in contact between CD, and the double teeth pairs are in contact between B₂C and DB₁. The mesh stiffness of the gear will change abruptly when the double toothed meshing zone (between CD) and single toothed meshing zone (between B₂C and DB₁) alternate.

2.1. Mesh Stiffness Calculation of Gears with Healthy Teeth

Mesh stiffness is a time-varying parameter that can reflect the meshing conditions of gears. In this paper, the meshing stiffness of one tooth is calculated from defections caused by bending, axial compressive, shear, and contact deflection. According to Reference [47], in order to calculate the deflections of a tooth, the tooth is considered as a non-uniform cantilever beam with an effective length L_e and under applied force F. Meanwhile, the tooth is divided into n segments, as shown in Figure 2.

On the basis of the Timoshenko’s beam theory [48], the stiffness of the tooth bending, shear, and axial compressive are given as follows.

$$\frac{1}{k_{bj}}={\displaystyle {\sum }_{i=1}^n\frac{1}{E_e}}\left(\begin{array}{l}\frac{{L_i}^3+3{L_i}^2{S}_{ij}+3L_i{S_{ij}}^2}{3I_j}{\cos }^2{a}_j-\\ \frac{{L_i}^2{Y}_j+2L_i{Y}_j{S}_{ij}}{2I_j}\cos a_j\sin a_j+\\ \frac{12(1+v)L_i}{5A_j}{\cos }^2{a}_j+\frac{L_i}{A_j}{\sin }^2{a}_j\end{array}\right)$$

(1)

where L_i and a_j represent the thickness of a microelement and the pressure angle of the contact point j, respectively. S_ij is the distance between a micro-element and the contact point in the X direction. Y_j is the corresponding half tooth thickness. A_j and I_j denote the cross-sectional areas and area moments of inertia, respectively. The A_j and I_j could be expressed by Equations (2) and (3).

$$A_j=2B{Y}_j$$

(2)

$$I_j=\frac{1}{12}B{(2Y_j)}^3$$

(3)

Ee represents the effective-Young’s modulus, which can be expressed as:

$$E_e=\{\begin{array}{ll}E& \frac{B}{H_p}<5\\ \frac{E}{1-v^2}& \frac{B}{H_p}\ge 5\end{array}$$

(4)

in which E and v are the Young’s modulus and the Poisson’s ratio, respectively. B is the teeth width, and H_p is the thickness on the pitch circle.

The Hertzian contact stiffness is calculated by Equation (5).

$$\frac{1}{k_{hj}}=\frac{4(1-v^2)}{\pi BE}$$

(5)

For a single-tooth-pair meshing duration, the total effective stiffness is defined by the equation below [28].

$$k_{pg}=\frac{1}{\frac{1}{k_h}+\frac{1}{k_{bp}}+\frac{1}{k_{bg}}}$$

(6)

where p and g are pinion and gear, respectively.

For a double-tooth-pair meshing duration, there are two pairs of gears meshing at the same time. Therefore, the total effective stiffness is defined as [28]:

$$k_t=k_{pg1}+k_{pg2}={\displaystyle \sum _{f=1}^2\frac{1}{\frac{1}{k_{h,f}}+\frac{1}{k_{bp,f}}+\frac{1}{k_{bg,f}}}}$$

(7)

where f = 1 for the first pair and f = 2 for the second pair of meshing teeth.

2.2. Mesh Stiffness Calculation of Gears with Tooth Spalling

The calculation method of mesh stiffness of gears with spalling teeth is similar to that of healthy gears. Both of these calculation methods consider the axial compressive stiffness, shear stiffness, bending stiffness, and Hertzian contact stiffness. According Reference [20], spalling is molded as a circular shape and occurred near the gear pitch line of the pinion, as shown in Figure 3.

For a gear tooth with spalling, the value of B, A_j, and I_j are reduced. Here, ΔB, ΔA_j, and ΔI_j represent the reduction of tooth contact width, area, and area moment of inertia of the tooth spalling section, respectively, which are given as follows.

$$\Delta B_j=\{\begin{array}{ll}2\sqrt{r^2-{[u-s_j]}^2},& x_j\in [u-r, u+r]\\ 0,& others\end{array}$$

(8)

$$\Delta A_j=\{\begin{array}{ll}sh\Delta B_j,& x_j\in [u-r,\begin{array}{l}\hfill \end{array}u+r]\\ 0,& others\end{array}$$

(9)

$$\Delta I_j=\{\begin{array}{ll}\frac{1}{12}s\Delta B_j{h}^3+\frac{A_j\Delta A_j{(Y_j-h/2)}^2}{A_j-\Delta A_j},& x_j\in [u-r, u+r]\\ 0,& others\end{array}$$

(10)

where r, h, and s are the radius, depth, and number of spalling points, respectively. u is the distance between the center point of spalling and the tooth root. s_j denotes the distance between the gear contact point and the tooth root. Details of u and s_j could be found [20].

Here, $B_j^{\prime }$, $A_j^{\prime }$, and $I_j^{\prime }$ denote the tooth contact width, area, and area moment of inertia of the tooth spalling section, respectively. They can be expressed as follows.

$$B_j^{\prime }=B-\Delta B_j$$

(11)

$$A_j^{\prime }=A_j-\Delta A_j$$

(12)

$$I_j^{\prime }=I_j-\Delta I_j$$

(13)

Substituting Equations (2) to (4), (8) to (10), (11) to (13) into Equation (1), the stiffness of the tooth bending, shear, and axial compressive of gear with spalling can be obtained as follows.

$$\frac{1}{k_{bj}^{\prime }}={\displaystyle {\sum }_{i=1}^n\frac{1}{E_e}}\left(\begin{array}{l}\frac{{L_i}^3+3{L_i}^2{S}_{ij}+3L_i{S_{ij}}^2}{3(I_j-\Delta I_j)}{\cos }^2{a}_j-\\ \frac{{L_i}^2{Y}_j+2L_i{Y}_j{S}_{ij}}{2(I_j-\Delta I_j)}\cos a_j\sin a_j+\\ \frac{12(1+v)L_i}{5(A_j-\Delta A_j)}{\cos }^2{a}_j+\frac{L_i}{(A_j-\Delta A_j)}{\sin }^2{a}_j\end{array}\right)$$

(14)

Substituting Equations (2) to (4), (8) to (10), (11) to (13) into Equation (5), the Hertzian contact stiffness of gear with spalling can be obtained as follows.

$$\frac{1}{k_{hj}^{\prime }}=\frac{4(1-v^2)}{\pi E(B-\Delta B_j)}$$

(15)

Substituting Equations (14) and (15) into Equation (6), the total effective stiffness is expressed as the following.

$$k_{pg}^{\prime }=\frac{1}{\frac{1}{k_h^{\prime }}+\frac{1}{k_{bp}^{\prime }}+\frac{1}{k_{bg}}}$$

(16)

3. Dynamic Model of a Gear System

As shown in Figure 4, a six-degree of freedom dynamical model that considers the time-vary mesh stiffness, transmission error, and bearing support stiffness is established in which the changes of TVMS are caused by a double-teeth spalling fault [12].

Here are some assumptions: (1) the bearings at both ends of the shaft have the same stiffness and damping, (2) the gearbox casing is a rigid body, and (3) the shaft of the pinion and gear are parallel to each other. The equations of the dynamic model are given by Equation (17).

$$\{\begin{array}{l}{m}_p{\ddot{x}}_p+c_{bp}{\dot{x}}_p+k_{bp}{x}_p+F_{pg}\cos {\phi }_{pg}=0\\ m_p{\ddot{y}}_p+c_{bp}{\dot{y}}_p+k_{bp}{y}_p+F_{pg}\sin {\phi }_{pg}=0\\ I_p{\ddot{\theta }}_p+r_{bp}{F}_{pg}=T_M\\ m_g{\ddot{x}}_g+c_{bg}{\dot{x}}_g+k_{bg}{x}_g-F_{pg}\cos {\phi }_{pg}=0\\ m_g{\ddot{y}}_g+c_{bg}{\dot{y}}_g+k_{bg}{y}_g-F_{pg}\sin {\phi }_{pg}=0\\ I_g{\ddot{\theta }}_g-r_{bg}{F}_{pg}=-T_L\end{array}$$

(17)

F_pg denotes the total mesh force, and it is expressed by:

$$\begin{array}{ll}{F}_{pg}=& c_m\left[\begin{array}{l}{r}_{bp}{\dot{\theta }}_p-r_{bg}{\dot{\theta }}_g-({\dot{x}}_p-{\dot{x}}_g)\cos {\phi }_{pg}-\\ ({\dot{y}}_p-{\dot{y}}_g)\sin {\phi }_{pg}-{\dot{e}}_{pg}\end{array}\right]+\\ & k_m\left[\begin{array}{l}{r}_{bp}{\theta }_p-r_{bg}{\theta }_p-(x_p-x_g)\cos {\phi }_{pg}-\\ (y_p-y_g)\sin {\phi }_{pg}-e_{pg}\end{array}\right]\end{array}$$

(18)

where r_bp and r_bg are the base circle radius of pinion and gear, I_p and I_g are the mass moments of inertia of the pinion and gear, m_p and m_g are the masses of the pinion and gear, k_bp and k_bg are the bearing stiffness of the pinion and gear, respectively, c_bp and c_bg are the bearing damping of the pinion and gear, respectively, T_M is the input torque, T_L is the load torque, the time-varying meshing damping c_m is mentioned in Reference [49], and e_pg denotes the transmission error, which can be obtained by the equation below.

$$e_{pg}(t)=e_m+e_r\sin (2\pi f_m+\varphi )+\frac{\Delta f_f}{\cos \alpha }+\frac{\Delta f_{pb}}{\cos \alpha }$$

(19)

where e_m is the constant of the mesh error on the pitch circle, and e_r is the amplitude of mesh error on the pitch circle. a is the pressure angle, f_m is the mesh frequency of the gear, and ϕ is the phase. Δf_f is tooth error and Δf_pb is the base pitch error, as shown in Figure 5.

4. Dynamic Simulation and Experimental Verification

The simulated vibration signal is compared with the experimental measurement signal, and the dynamic simulation results are verified under conditions of a different faulty tooth number, and relative position of faulty teeth. The simulation parameters are detailed in

Table 1. Parameters for the dynamic simulation.

Parameters	Pinion	Gear
Number of teeth	16	24
Pressure angle (deg)	20
Teeth module (mm)	4.5
Teeth width (mm)	14
Mass (kg)	0.676	1.084
Mass moment inertia (kg·m2)	0.000407	0.001168
Young’s modulus (N/mm2)	2.06 × 105
Poisson’s ratio	0.3
Bearing damping (N·s/m)	4.86 × 103	6.16 × 103
Bearing stiffness (N/m)	3.5 × 109	3.5 × 109
Torque (N·m)	23.5	35.3

.

The gear test-rig consists of a controller, a three-phase motor, a test gearbox, and a load applied by a clutch, as shown in Figure 6. Table 1 indicated the performance parameters of the gear and bearing. The data acquisition system is B&K-3560C, and the experimental vibrations signal is measured along the meshing line of the gear teeth by the sensor mounted on the bearing pedestal. The sampling frequency is 65,536 Hz, and the sampling time is 1 s. Since the pinion is a faulty gear, the vibration signal a_py of the pinion in the direction of the meshing line is collected.

The test-rig presented in Figure 6 is the standard FZG test-rig of STRAMA-MPS. This FZG test-rig consists of a controller, a three-phase motor, a test gearbox, and a load applied by a clutch. Table 1 indicated the performance parameters of the gear and bearing. The data acquisition system is BRUEL&KJAER-3560C, and the experimental vibrations signal is measured along the meshing line of the gear teeth by the sensor mounted on the bearing pedestal. The sampling frequency is 65536 Hz, and the sampling time is 1 s. Since the pinion is a faulty gear, the vibration signal a_py of the pinion in the direction of the meshing line is collected.

4.1. Effect of the Number of Spalling Faulty Teeth on Dynamic Characteristics of Gears

In this paper, we assume that the rotational speed of pinion is n = 1500 r/min, the rotational frequency of pinion is f_s = 25 Hz, the rotating period of pinion is t_s = 0.04 s, the meshing frequency is f_m = 400 Hz, and the meshing period is t_m = 0.0025 s. Three types of the spalling faulty tooth number are considered, namely health pinion, pinion with single tooth spalling, and pinion with adjacent double teeth spalling.

Figure 7a, Figure 8a, and Figure 9a present the failure states of tooth surface spalling, respectively. It can be clearly seen that the spalling is near the pitch line. This paper we set that r = 0.75 mm, h = 1 mm, and s = 3, where r, h, and s are the radius, depth, and numbers of spalling point, respectively. According to the method described in Section 2, the mesh stiffness is simulated. It can be observed from Figure 7b that the mesh stiffness curve is periodically changing during the meshing process. The mesh stiffness decreases when the spalling faulty tooth participates in an engagement. The frequency of mesh stiffness reduction is related to the number of teeth with spalling fault, as shown in Figure 8b and Figure 9b.

The simulated vibration signal and the experimental measurement signal under a different number of spalling faulty tooth are displayed in Figure 10, Figure 11 and Figure 12. The simulated results are close to the experimental results when the pinion is in a healthy state, as shown in Figure 10. There are no clear impulse vibrations in the time domain. Only tooth meshing frequency (TMF) and its harmonics are included in the frequency domain. In the frequency domain, the TMF is equal to the meshing frequency of gears, and the remaining harmonics are multiples of the TMF, as marked in Figure 10b,d.

Compared with the healthy gear, the spalling fault can generate the periodic impulse vibration, and sidebands are spaced by the rotational frequency of the faulty gears in the frequency domain. These features are successfully captured by the simulated signal. Figure 11 shows the simulated and experimental signal results of the pinion with single tooth spalling fault. As shown in Figure 11, the periodic impulse vibration can be recognized in the time domain, and the period of the impulse vibration is 0.04 s, which is equal to the rotating period of the pinion. The sideband appears in the frequency domain, and the spacing of the sideband is equal to the rotating frequency of the pinion. Figure 12 shows the simulated and experimental signal results of the pinion with an adjacent double tooth spalling fault. Compared with the time domain in Figure 11a, Figure 12a shows that the impulse vibration was excited twice with an interval of 0.0025 s in one rotational period of the pinion. However, impulse vibration in the experimental signal is submerged by the background noise, as shown in Figure 11c and Figure 12c. In the frequency domain, the maximum value of the sideband of Figure 12 is higher than that in Figure 11. The results of frequency domain analysis show that the fault feature in a case of double-teeth spalling are more clear than that in the case of single tooth spalling. However, due to the strong background noise, the fault feature frequencies in the experimental signal are not clear, and may even be submerged by the noise frequencies, as shown in Figure 11d and Figure 12d.

4.2. Effect of the Relative Position of the Faulty Teeth on Dynamic Characteristics of Gears

Based on the simulated and experimental parameters in Section 4.1, in this paper, we assume that the number of healthy teeth between two spalling faulty teeth is H. Three relative positions (H = 0, H = 1, and H = 2) of the two faulty teeth are considered. Figure 13a, Figure 14a, and Figure 15a show the failure states of tooth surface spalling in three relative positions. Furthermore, when the two spalling fault teeth are meshed separately, the mesh stiffness decreases, and the interval time of the reductions is related to the number of healthy teeth between the two spalling fault teeth, as shown in Figure 14b and Figure 15b.

The simulated and experimental signals both in time and frequency domain at the three relative positions of the double faulty teeth are presented in Figure 12, Figure 15 and Figure 16. As shown in Figure 12a, Figure 15a and Figure 16a, impulse vibration excited twice in one rotational period of the pinion. In addition, the distance between two impulse vibrations are 0.0025 s, 0.005 s, and 0.0075 s, which can be expressed by (H + 1)t_m. However, the impulse vibration is not clear in Figure 12c, Figure 15c and Figure 16c. One possible explanation is that these features are submerged by the background noise in the experimental signal.

Moreover, as presented in Figure 12d, Figure 15d and Figure 16d, the maximum amplitudes of the sidebands at the three relative positions are 0.0019 m/s², 0.0015 m/s², and 0.0011 m/s², respectively. Among them, the amplitude of the sidebands at the adjacent positions is the highest, which is close to the simulated results. It can be concluded that, when the two teeth with spalling fault are adjacent, the features of the spalling fault are the most clear. In addition, compared with Figure 15b and Figure 15d, as well as Figure 16b and Figure 16d, the strong background noise in the experimental signal also affects the identification of fault features, and then affects the accuracy of the verification results of the dynamic model and experimental signal.

5. Background Noise Elimination Method

As mentioned in Section 4, the periodic impulse of the experimental signal caused by the fault is submerged by the background noise, and the corresponding fault feature frequencies are difficult to identify in the frequency domain. In this section, a novel parameter-adaptive variational mode decomposition method is proposed to eliminate the background noise, while avoiding the disadvantage of choosing decomposed parameters manually by human experience in VMD.

5.1. The Principle of ALO-VMD Method

5.1.1. VMD Method

VMD is an adaptive signal decomposition method, which can adaptively decompose signals into a series of components with a separate property. The essence of VMD is to effectively solve the constrained variational problems, as described below.

$$\{\begin{array}{l}\underset{\left\{u_k\right\},\left\{{\omega }_k\right\}}{\min }\left\{{\displaystyle \sum _k{\Vert {\partial }_t[(\delta (t)+\frac{j}{\pi t})\cdot u_k(t)]e^{-j{\omega }_k}\Vert }_2^2}\right\}\\ s.t.\begin{array}{c}{\displaystyle \sum _k{u}_k=f}\end{array}\end{array}$$

(20)

where u_k is the k-th signal component, and ω_k is the corresponding center-frequency of the k-th signal component.

By adding Lagrangian multiplier λ and penalty factor â into Equation (20), the constrained variational problem is rewritten as Equation (21).

$$\begin{array}{ll}L({\{\mathrm{u}}_k\}, \{{\omega }_k\}, \lambda )=& \widehat{\alpha }{\displaystyle \sum _k{\Vert {\partial }_t[(\delta (t)+\frac{j}{\pi t})\cdot u_k(t)]e^{-j{\omega }_k}\Vert }_2^2}\\ & +{\Vert f(t)-{\displaystyle \sum _k{u}_k(t)}\Vert }_2^2\\ & +\langle \lambda (t),f(t)-{\displaystyle \sum _k{u}_k(t)}\rangle \end{array}$$

(21)

Equation (21) can be processed by the alternate direction method of multipliers (ADMM) [24]. Subsequently, the signal components ${\hat{\mu }}_k$ and its corresponding center-frequencies ${\omega }_k$ are updated by Equations (22) and (23).

$${\widehat{u}}_k^{n+1}(\omega )=\frac{\widehat{f}(\omega ){\displaystyle \sum _{i<k}{\widehat{u}}_i^{n+1}}(\omega )-{\displaystyle \sum _{i>k}{\widehat{u}}_i^n}(\omega )+\frac{{\lambda }^n(\omega )}{2}}{1+2\widehat{\alpha }{(\omega -{\omega }_k^n)}^2}$$

(22)

$${\omega }_k^{n=1}=\frac{{\displaystyle {\int }_0^{\infty }\omega {\left|{\widehat{u}}_k^{n+1}(\omega )\right|}^{2}d\omega }}{{\displaystyle {\int }_0^{\infty }{\left|{\widehat{u}}_k^{n+1}(\omega )\right|}^{2}d\omega }}$$

(23)

After each updating, the Lagrangian multiplier λ is also replaced by Equation (24).

$${\widehat{\lambda }}^{n+1}(\omega )={\widehat{\lambda }}^n(\omega )+\tau (\widehat{f}(\omega )-{\displaystyle \sum _k{\widehat{u}}_i^{n+1}}(\omega ))$$

(24)

The above process is updated repeatedly until the stop conditions are met, as shown in the equation below.

$${\displaystyle \sum _k\frac{{\Vert {\widehat{u}}_k^{n+1}-{\widehat{u}}_k^n\Vert }_2^2}{{\Vert {\widehat{u}}_k^n\Vert }_2^2}<\epsilon },$$

(25)

where $\tau$ denotes the noise-tolerance and the tolerance of convergence criterion ε is usually set as 10⁻⁶. For more details on VMD [35].

5.1.2. Ant Lion Optimization Method

Ant lion optimization (ALO) is one of the swarm intelligence algorithms, which solves the functional optimization problem by imitating the hunting mechanism of ant lions. The principle of the ALO algorithm is introduced as below [44].

To imitate the random walk of ants inside the search space, a normalized equation is applied in Equation (26).

$$X_i^{Iter}=\frac{(X_i^{Iter}-a_i)\times (d_i-c_i^{Iter})}{(d_i^{Iter}-a_i)}+c_i$$

(26)

where a_i is the minimum of random walk of the i-th variable, d_i is the maximum of random walk in the i-th variable, $c_i^{Iter}$ is the minimum of the i-th variable at the Iter-th iteration, and $d_i^{Iter}$ indicates the maximum of the i-th variable at the Iter-th iteration.

As mentioned in Reference [33], the random walk of ants is affected by the ant lions’ traps. This behavior can be modeled by Equations (27) and (28).

$$c_i^{Iter}=Antlio{n}_j^{Iter}+c^{Iter}$$

(27)

$$d_i^{Iter}=Antlio{n}_j^{Iter}+d^{Iter}$$

(28)

Therefore, c^Iter is the minimum of all the variables at the Iter-th iteration, d^Iter indicates the vector including the maximum of all variables at the Iter-th iteration, and $Antlio{n}_j^{Iter}$ shows the position of the selected j-th ant lion at the Iter-th iteration.

According to the roulette wheel, the ALO algorithm selects the fittest ant lion to catch the ants. Once the ant is in the trap, the ant lions throw the sand toward the center of the hole. The process is described mathematically by Equations (29) and (31).

$$c^{Iter}=\frac{c^{Iter}}{I}$$

(29)

$$d^{Iter}=\frac{d^{Iter}}{I}$$

(30)

$$I=\{\begin{array}{ll}1& I\mathrm{ter}\le 0.1Ite{r}_{\max }\\ {10}^{\gamma }\frac{Iter}{Ite{r}_{\max }}& Iter>0.1Ite{r}_{\max }\end{array}$$

(31)

As such, Iter is the current iteration, Iter_max is the maximum number of iterations, and γ is a constant defined based on the current iteration. The value of γ can be found in Reference [44].

Assumed by the ALO algorithm, the hunting stage is ended when the fitness value of ant is greater than that of the ant lion. To increase the probability of catching new ants, the ant lion is then expected to replace its position to the latest position of the hunted ant, as defined by Equation (32).

$$Antlio{n}_j^{Iter}=An{t}_i^{Iter}$$

(32)

where $Antlio{n}_j^{Iter}$ shows the position of selected j-th ant lion at the Iter-th iteration, and $An{t}_i^{Iter}$ indicates the position of i-th ant at the Iter-th iteration.

To maintain the best solution for the optimization process, the fittest ant lion is selected as an elite, which can affect the random walk of all ants. That is, the random walk of each ant is guided by the selected ant lion and the elite ant lion. Therefore, the average of two randomly walks are used to represent the repositioning of a given ant, which can be modeled by Equation (33).

$$An{t}_i^{Iter}=\frac{R_A^{Iter}+R_E^{Iter}}{2}$$

(33)

where $R_A^{Iter}$ is the random walk around the roulette wheel selected ant lion, and $R_E^{Iter}$ is the random walk around the elite ant lion.

5.1.3. The Proposed ALO-VMD Method

The basic idea of the proposed ALO-VMD method is to obtain the optimal parameters (i.e., the modal component number K and penalty factors â) by the ALO algorithm, so as to ensure the decomposition capability of the VMD. As per the above discussion, a reasonable fitness function is an important factor in the ALO algorithm. To further ensure the optimal decomposition effect, the maximum average weighted kurtosis (MAWK) is setting as the fitness function in the ALO, and it is defined by the equation below.

$$\{\begin{array}{l}f=argmax\left\{WK{I}_i\right\}\\ \begin{array}{ll}s.t.& K_{\min }\le K\le K_{\max }\\ & {\widehat{a}}_{\min }\le \widehat{a}\le {\widehat{a}}_{\max }\end{array}\end{array},$$

(34)

where K and â is the optimal parameters. WKI_i (i = 1, 2, …, K) represents the weighted kurtosis index value of the components from the VMD decomposition, which can be calculate by the equation below [51].

$$WK=KI·C,$$

(35)

$$KI=\frac{(1/N){\displaystyle {\sum }_{n=0}^N{x}^4(n)}}{{[(1/N){\displaystyle {\sum }_{n=1}^{N-1}{x}^2(n)}]}^2},$$

(36)

$$C=\frac{E[(x-\overline{x})(y-\overline{y})]}{E[{(x-\overline{x})}^2]E[{(y-\overline{y})}^2]}$$

(37)

where KI is the kurtosis index of signal sequence x(n), N is the signal length, C is the correlation coefficient between signals x and y, and E(·) represents the mathematical expectation.

The flow chart of the proposed ALO-VMD method is illustrated in Figure 17. First, the initialization parameters of ALO-VMD are initialized. Second, the original signal is decomposed by the VMD method. Third, the average weighted kurtosis (AWK) value of each components is calculated and the maximum average weighted kurtosis (MAWK) is replaced if the average weighted kurtosis index value is greater than the current optimal fitness value. Finally, if the number of iterations is less than Iter_max, let Iter = Iter + 1, and the positions of each ant are updated. Otherwise, the iteration ends and the optimal parameters are saved.

5.2. The Framework of the Background Noise Elimination Method

The framework of the feature extraction method is shown in Figure 18. It divides into four steps.

Step 1: Signal collection. Obtaining the original vibration signal by using the test rig, which is shown in Section 4.

Step 2: Signal decomposition. According to the fitness function and ALO algorithm, the optimal parameters of VMD are automatically determined. Then, the original signal is decomposed into a series of IMF components by a parameter-adaptive VMD method.

Step 3: Signal reconstruction. Combining the advantages of kurtosis index and correlation coefficient, an efficient weighted kurtosis (EWK) index is used to obtain the reconstructed signal, as shown in Equation (38). The intrinsic mode functions (IMF) component with EWK index greater than zero is selected as a sensitive component, and the rest are noisy components [52].

$$EWK=WK-\frac{1}{K}{\displaystyle \sum _{j=1}^{K}WK} i,j=1 to K$$

(38)

Step 4: Reconstructed signal demodulation. The reconstructed signal is demodulated by the Hilbert envelope, and the faulty feature frequencies are detected from the envelope spectrum.

5.3. Background Elimination and Comparative Analysis

According to the conclusion given in Section 4, the double-teeth spalling fault feature are the most clear when the two teeth are adjacent. For the sake of discussion, the experimental signal when the two spalling faulty teeth are adjacent is processed, and the experimental conditions are listed in

Table 2. The information of the experimental conditions.

Fault Gear	Speed (r/min)	Sampling Frequency (Hz)	Fault Feature Frequency (Hz)
Pinion	1.500	65.536	25

.

Due to the background noise, the periodic impulse cannot be recognized in the time-domain signal, and the faulty feature frequency is also not clear, as shown in Figure 12. In other words, the background noise in the experimental signal leads to the recognition effect of the double-teeth spalling fault feature, and then affects the accuracy of dynamic model validation. Therefore, the identification of double-teeth spalling fault feature from the noisy signal is of great significant.

First, the proposed ALO-VMD method is applied to process the experimental signal. To verify the optimization performance of ALO, the standard particle swarm optimization (PSO) is tested and compared. Here, the population of the ants is set to 30, and the maximum iterations is set to 10, the range of decomposition layer K = [2,11], and the range of penalty factor â = [50,8000]. Figure 19 and

Table 3. Comparison of optimization performance between ant lion optimization (ALO) and particle swarm optimization (PSO).

Parameters	Methods
	ALO	PSO
(K,a)	(3,1626)	(3,3580)
Iterations	4	7
Optimal fitness	1.606	1.576

show the comparison results of ALO and PSO. The results show that the optimum speed and optimization accuracy of the ALO algorithm is better than that of the PSO algorithm. Then, the optimal parameters K = 3 and â = 1626 are obtained by ALO.

As the result, three components and their corresponding EWK are obtained, as shown in

Table 4. The efficient weighted kurtosis (EWK) value of each component.

Component	u1	u2	u3
EWK	0.286	0.605	−0.891
Selection	√	√

. Since the EWK values of u1 and u2 are larger than zero, they are determined as sensitive components and selected as a reconstructed signal, while other modes are noisy modal components.

Subsequently, the reconstructed signal is demodulated by the Hilbert envelope and the envelope spectrum are shown in Figure 20. Compared with Figure 12c, the periodic impulse vibration, which was excited by the spalling fault can be clearly observed in Figure 20a. The period of the impulse vibration is 0.04 s, which is equal to the rotating period of the pinion. In addition, in a rotation period of the pinion, the impulse vibration excited twice, and the interval time of the two impulse vibrations is close to the single tooth meshing period. From the envelope spectrum, as shown in Figure 20b, the faulty feature frequency fs and its multiplications (2fs, 3fs, 4fs, and 5fs) can be clearly observed in the spectrum of the reconstructed signal. The results are consistent with the previous simulated results mentioned in Section 4, and it is demonstrated that the fault feature is extracted effectively by the proposed ALO-VMD method.

For comparisons, the PSO-VMD, the fixed-parameter VMD, the EMD, and the LMD are employed to process the same signal, and the results are displayed in Figure 21. The optimal K and â in PSO-VMD are 3 and 3580, respectively, and the K and â in the fixed-parameter VMD are set as 4 and 2000, respectively. Similar to the ALO-VMD method, the reconstructed signal of other methods is also selected according to the value of the EWK index. For clarity, the visible abscissa range of the spectrum is set to 0–300 Hz.

As shown in Figure 21, only the faulty feature frequency fs and its multiplication 2fs can be identified in the spectrums of the LMD, and three fault feature frequencies fs, 2fs, and 4fs are displayed by the EMD. In contrast, five fault frequencies are existing in the envelope spectrums of fixed-parameter VMD, PSO-VMD, and the proposed ALO-VMD. However, the amplitude of the fault frequencies shown in ALO-VMD are larger than shown in other methods.

To further evaluate the effectiveness of the five methods, two quantitative indexes, namely, the signal-noise-ratio (SNR) and the root-mean-square error (RMSE), are used [53], and described as follows.

$$SNR=10\log \left\{\frac{{\displaystyle {\sum }_{i=1}^N{x}^2}(i)}{{\displaystyle {\sum }_{i=1}^N{(x(i)-\hat{x}(i))}^2}}\right\}$$

(39)

$$RMSE=\sqrt{\frac{1}{N}{\displaystyle {\sum }_{i=1}^N{(x(i)-\hat{x}(i))}^2}}$$

(40)

where $x\left(i\right)$ is the original signal, $\widehat{x}\left(i\right)$ is the mean value of $x\left(i\right)$, and N denotes the length of signal $x\left(i\right)$.

As shown in

Table 5. The efficient weighted kurtosis (EWK) value of each component.

Method	ALO-VMD	PSO-VMD	Fixed-Parameter VMD	EMD	LMD
SNR	10.464	9.916	9.854	9.217	9.101
RMSE	1.577	1.669	1.772	1.964	2.004

, the signal-noise-ratio (SNR) of the reconstructed signal processed by ALO-VMD is larger than other methods, and the root-mean-square error (RMSE) of the reconstructed signal processed by ALO-VMD is smaller than other methods. It is concluded that the proposed ALO-VMD method has better performance in noise reduction for the double-teeth spalling fault than other methods.

6. Conclusions

In this paper, a dynamic model of a gear system with double-teeth spalling fault is presented with considering the influence of TVMS, transmission error, and bearing support stiffness. The effectiveness of the proposed dynamic model is validated by analyzing experimental results. The effect of the number of spalling teeth, and the relative position of the two faulty teeth are investigated. Both of the experimental tests and dynamic simulation results have shown that the double-teeth spalling fault will generate impulse vibration twice in one rotating period of the fault gear, and the interval of the two impulses is related to the number of healthy teeth, which can be expressed as (H + 1)t_m. Moreover, the features of double-teeth spalling fault are significant when the two spalling teeth are adjacent.

In addition, a novel parameter-adaptive VMD method based on the ALO is proposed to eliminate the background noise from the experimental signal for ensuring the accuracy of dynamic model verification results and reducing the difficulty of fault feature analysis. The optimal parameters K and â is obtained by the ALO, and the raw signal obtained from experiments is decomposed by using the ALO-VMD. According to the EWK index, several signal components whose value is greater than zero are summed as a reconstructed signal. The reconstructed signal is then demodulated by the Hilbert envelope, and the fault features are analyzed by the envelope spectrum. The results confirm that, (1) the ALO has a good optimization ability and can provide the VMD with the optimal parameters K and â, (2) by comparing with the PSO-VMD, fixed-parameter VMD, EMD, and LMD, it can be found that the proposed ALO-VMD method is more effective in eliminating the noise than other methods.

Compared with References [19,22,23], the effect of tooth fillet foundation deflection is not considered in the proposed gear model. As reported in Reference [54], an equation for the fillet foundation deflection is valid only when the normal stress at the tooth root is linearly distributed and confined usable for large gears. However, due to the limitation of experimental conditions, the width of the fillet foundation is greater than that of teeth of the spalling pinion in this paper. This may be a limitation of the proposed model that needs to be further studied. Moreover, the spalling effects of random distribution are also worthy of further investigation.