Dynamical modeling of spur gear with pitting based on image processing tooth surface

Abstract

Reliable dynamic model of gear system with pitting can be utilized to investigate the effects of surface pitting on vibration response and to identify its severity. In this study, a dynamic model with pitting based on image processing is proposed, and pitting is simulated through the discrete tooth surface. The contact points are modeled by a network of linear springs with different stiffness in parallel. The revised mesh stiffness is calculated and a new mesh stiffness calculation method with surface pitting is proposed, which combines pixel matrix with mesh stiffness to improve the accuracy and efficiency of calculation. The dynamic model is validated by comparison with experimental test signals under different rotations and pitting states. As the Hilbert–Huang transform energy spectrum method has been applied for the detection of the vibration signals with small pitting areas, the signal characteristics caused by pitting can be well represented based on the statistical indicators and energy spectrum.

Keywords

Dynamic model surface pitting weak springs stiffness distribution image processing

Highlights

A dynamic model with surface pitting based on image processing is proposed.

A new mesh stiffness calculation method based on surface pitting stiffness matrix is studied.

The proposed model is validated by experimental tests.

The statistical indicators and energy spectrum are induced based on the Hilbert–Huang transform energy spectrum method.

Introduction

Gears are key components for connecting and transmitting power. Due to the tough working conditions or improper gear design, the gear system is prone to failure. Compared with crack and wear, surface pitting is the most typical failure mode and the research on pitting is crucial to fault diagnosis.^1,2 A reliable dynamic model can be utilized to investigate the effects of surface pitting on vibration response and to reduce the costs of diagnosis algorithms. Dynamic simulation of gear transmission with pitting plays an important role in understanding the dynamic behaviors and corresponding vibration features of a gear system.

Research status

At present time, the work on gear pitting dynamic model is focused on two aspects: calculation of time-varying mesh stiffness (TVMS) and dynamic model.³ Many scholars have studied the relationship between TVMS and dynamic response at present, which focused on more accurately reflecting the characteristics of fault to illustrate the dynamic model. Gelman et al.⁴ generate circular pits along the pitch line of a single gear tooth to simulate pitting fault. Ma et al.⁵ studied by comparing the TVMS under different rectangle pitting widths, lengths, and locations with that obtained from the finite element method (FEM). Chaari F et al.⁶ modeled pitting as a rectangular indentation and studied the effects of the spalling damage on the gear mesh stiffness. The influence of pitting on gear TVMS and spectrum characteristics of vibration response are studied. Liang et al.⁷ focused on the influence of different pitting levels and the TVMS. Liu et al.⁸ established the probability distribution model of gear TVMS combining image processing, which improves the accuracy of the calculation. Meng et al.⁹ investigated the vibration of a spur gear pair in the presence of tooth root crack and spalling by analytically defining the gear mesh stiffness. Considering the irregularly shape surface pitting, the previous research mainly studied the TVMS but only consider a single pit,^4–7 which may lose some important signal characteristics and not be sufficient for comprehensive analysis. Research⁸ considered the evolution characteristics of tooth pitting but the essential calculation process still adopted the traditional method. Although the mesh stiffness of complex pitting area is more accurate, the calculation process is time-consuming. Furthermore, the analysis of geometric characteristics is essential in traditional calculation of mesh stiffness,^10,11 especially in different pitting shapes.^12–14

Several studies have examined the analytical model of dynamic response of spur gear with pitting. Yang¹⁵ proposed a new dynamic model to simulate the vibration behaviors of a spur gear transmission with pitting and spalling defects, which considered the effect of gearbox and the experimental validation was performed. Yi¹⁶ studied the nonlinear frequency responses of the geared rotor system with tooth tip chipped and several commonly used statistical indicators are applied to assess the system vibration behaviors. Tiancheng¹⁷ developed a dynamic model of 8-degree-of-freedom spur gear pair under ideal conditions, which considered pitting-crack coupling but lack of experimental verification. I. Dadon^18,19 developed a realistic dynamic model for pitting diagnosis, and the effect of pitting on response is represented by the lack of stiffness. Although many dynamic modeling methods have been investigated, the modeling ideas are generally consistent, that is the responses are affected by the change in the TVMS curve. A key problem ignored by many scholars is that the calculation process of TVMS is a process of transforming the surface pitting problem from three dimensions to two dimensions, which will lead to errors due to huge calculations.

Pitting gear vibration signals often show nonstationary characteristics as gears are always affected by the nonlinear impact. Although traditional signal processing methods such as the time-domain method have been well applied,^20–23 these methods are not effective in small pitting states. Loutridis²⁴ presented a method for monitoring the evolution of gear faults based on the newly developed empirical mode decomposition (EMD) scheme, which can be utilized for system failure prediction. Parey²⁵ modeled a 6-degree-of-freedom gear dynamic model including localized tooth crack defect and decomposed a signal into some individual, monocomponent signals signal by the Hilbert–Huang transform (HHT) method. The EMD method could be applied to nonlinear and nonstationary signal processing perfectly because of the self-adaptive ability.^26–29

Main work of this paper

In this paper, a dynamic model with pitting based on image processing is proposed, on this foundation, the healthy surface area is modeled as springs and pitting area is modeled as weak springs. The revised mesh stiffness is calculated and a new mesh stiffness calculation method with surface pitting is proposed, and the correctness of the method is proved. Furthermore, the dynamic model is validated by experimental data, and the HHT method has been applied for the detection of the vibration signals with small pitting area in gear system. The statistical indicators are induced and the Hilbert energy spectrum is obtained based on HHT. The detailed structure of this paper is shown in Figure 1.

Figure 1.

The structure schematic of this paper.

The practical significance of this research is that the dynamic response of pitting gear is directly solved by the surface spring stiffness matrix based on image processing, and the proposed model is specially for surface pitting with arbitrary shapes. The meshing process of pitting gears need not be analyzed separately, the calculation amount is reduced and the analysis process is simplified. Compared with the time-varying of traditional stiffness,⁸ the calculation method in this paper focuses on the distribution of stiffness to replace the TVMS and the calculation efficiency is greatly improved.

Establishment of dynamic model with pitting defect based on image processing

The extraction process of tooth surface pitting and tooth stiffness model along the contact lines is described in this section. Furthermore, the dynamic model of gear system is presented.

Modeling of the tooth stiffness in the presence of pitting based on image processing

Tooth surface pitting is introduced on a single tooth in pinion as shown in Figure 2(a). In consideration of studying the influence of different pitting degrees and the convenience of subsequent experiments, surface pitting is created near the pitch line of gear tooth by drill. After smoothing by Gaussian filtering in Figure 2(b), the gradient operator is used to extract the pitting characteristics of tooth surface based on OPENCV. Figure 2(d) shows the binary image of surface pitting M (x, y) can be described as

M (x, y) = {\begin{matrix} 0 (x, y) \in Pitting area \\ 255 Other points \end{matrix}

(1)

Therefore, the tooth surface is discretized by pixels and the pitting features are extracted through image processing in Figure 2(e), and the image is composed of m × n pixels.

Figure 2.

The extraction process of tooth surface pitting.

After the discretization of the tooth surface based on image processing, the mesh stiffness at any position on the tooth surface can be easily represented by pixels and the values of mesh stiffness along the contact line can be regarded as the sum of stiffness values of different pixels.

From Figure 3(b) and (c), it can be obtained that the contact plane is discretized into a set of pixel points, and a network of linear springs in parallel is used to model the contact points. The contact points of the healthy area are modeled as springs and the contact points of pitting area are modeled as weak springs. Therefore, the contact stiffness of pitting area will not be considered.

The stiffness of weak springs $\tilde{k}$ is related to the pitting area compared with springs k in healthy area. Thus, the value of mesh stiffness can be expressed as

k_{i row} = {\begin{matrix} \sum_{j = 1}^{l_{1}} k_{i, j} + \sum_{j = l_{2}}^{n} k_{i, j} + \sum_{j = l_{1}}^{l_{2}} {\tilde{k}}_{i, j} & Pitting area \\ \sum_{j = 1}^{n} k_{i, j} & Healthy area \end{matrix}

(2)

where m and n are pixel numbers along tooth profile and width direction as shown in Figure 3(b). i and j are, respectively, the coordinate points along tooth profile and width direction (i = 1,…, m, j = 1, …, n). The position of i is related to the angular displacement of pinion. l₁ and l₂ are, respectively, the distance from the pitting area boundary to the profile as shown in Figure 3(b).

Figure 3.

Modeling of contact tooth based on image processing.

Therefore, the mesh stiffness at any position and the values of stiffness along the contact line can be accurately calculated combined with image processing. The interaction between a gear pair is shown in Figure 3(d), in which the contact area is discretized by pixel points and the mesh stiffness can be described by pixel stiffness $k_{i, j}$ and ${\tilde{k}}_{i, j}$ .

Dynamic model of the gear system with pitting defect based on image processing

The gear system model consists of a spur gear pair, motor, two shafts, load, bearings, and two couplings. It is assumed the lubrication is ideal and the two shafts were assumed to be of finite torsional stiffness and infinite bending stiffness. The interaction between teeth meshing was modeled as shown in Section “Modeling of the tooth stiffness in the presence of pitting based on image processing.” A dynamic model of one spur gear pair including localized tooth pitting has been developed according to the stress characteristics shown in Figure 4.

Figure 4.

Dynamic model of the gear system with surface pitting.

Potential energy related to meshing gears can be deduced from stiffness along each contact line:

\begin{matrix} U = \frac{1}{2} (\sum_{j = 1}^{l_{1}} k_{i, j} + \sum_{j = l_{2}}^{n} k_{i, j}) δ^{2} + \frac{1}{2} \\ [k_{c} {(θ_{1} - θ_{m})}^{2} + k_{c} {(θ_{2} - θ_{b})}^{2}] \\ + \frac{1}{2} (k_{x 1} x_{1}^{2} + k_{y 1} y_{1}^{2}) + \frac{1}{2} (k_{x 2} x_{2}^{2} + k_{y 2} y_{2}^{2}) \end{matrix}

(3)

The kinetic energy can be expressed as

\begin{matrix} T = \frac{1}{2} (I_{m} {\overset{\cdot}{θ}}_{m}^{2} + I_{b} {\overset{\cdot}{θ}}_{b}^{2}) + \frac{1}{2} m_{1} ({\overset{\cdot}{x}}_{1}^{2} + {\overset{\cdot}{y}}_{1}^{2}) + \frac{1}{2} m_{2} ({\overset{\cdot}{x}}_{2}^{2} + {\overset{\cdot}{y}}_{2}^{2}) \\ + \frac{1}{2} (I_{1} {\overset{\cdot}{θ}}_{1}^{2} + I_{2} {\overset{\cdot}{θ}}_{2}^{2}) \end{matrix}

(4)

where m₁ and m₂ are the mass of the pinion and wheel. y₁ and y₂ are the vertical displacements of the pinion and wheel, respectively. θ₁, θ₂, θ_m, and θ_b are the angular displacements of the pinion, wheel, motor, and load, respectively. T_m and T_b are the moments of pinion and wheel, respectively. I₁, I₂, I_m, and I_b are the mass moments of inertia of the pinion, wheel, motor, and load, respectively. R_b1 and R_b2 are the radius of the base circle of the pinion and wheel, respectively. φ indicates the pressure angle.

According to Lagrange equations for nonconservative systems, the general form of the dynamic equation is

M \overset{\cdot\cdot}{n} + C \overset{\cdot}{n} + K_{t} n = F_{ext}

(5)

where F_ext is the excitation force vector that produced by input and output torques and profile error.¹⁸ The profile error is defined as the deviation of the actual tooth profile and the ideal profile. Vector n is a generalized coordinate and M is a diagonal mass matrix. K_t is a structural stiffness matrix with pitting composed of spring stiffness k_{i, j} and weak spring stiffness ${\tilde{k}}_{i, j}$ in Equation (2), which can be described as

K_{t} = K_{1} + K_{2} + K_{3}

(6)

where K₁ represents the partial spring stiffness matrix, K₂ represents the complementary spring stiffness matrix, and K₃ represents the weak spring stiffness caused by pitting. C is a damping matrix, which can be obtained by the damping ratio ζ. All the matrix n, M, C, F_ext, K₁, K₂K₃, and parameters ζ are given in Appendix A.

Here, e is the tooth profile errors considered in this work. The value of the profile error depends on the gear precision grade.

e = f_{H} \frac{(s (t) - s_{0})}{L} + \frac{f}{2} \sin (2 π n \frac{(s (t) - s_{0})}{L} + ϕ)

(7)

where f_H and f are the profile angle deviation and the profile form deviation, respectively. s indicates the radius of the contact point. n represents the number of sinusoid cycles along the functional profile length and ϕ is the phase component of the sinusoidal term. The detail and derivation can be obtained in Mucchi et al.³⁰

The numerical parameters of the gear system are given in Table 1, and the stiffnesses and the damping coefficients can obtained from Kisssoft.

Table. 1.

Parameters of the spur gear pair.

Parameters	Pinion/Wheel	Parameters
Tooth number	29/100	Radial stiffness of the bearing k_x₁, k_x₂/(N/m)	6.6 × 10⁷
Module (mm)	1.5	Damping coefficient of the bearing c_x₁, c_x₂/(Nms/rad)	1.8 × 10⁵
Pressure angle	20	Torsional stiffness of the coupling k_c/(N/m)	3.3 × 10³
Width of teeth/mm	15	Damping coefficient of the coupling c_c/(Nms/rad)	3.0 × 10²
Mass (pinion/wheel)/kg	0.25/2.43	Young’s modulus E/GPa	209
Addendum height coefficient	1	Poisson’s ratio ν	0.3
Load/N m	11.3	Profile deviation f_H, f/(μm)	30

Calculation of gear mesh stiffness with pitting based on image processing

Stiffness calculation and verification

The stiffness of healthy gear is the basis of stiffness distribution in this paper and the revised energy method³¹ is used in this section. The gear tooth is modeled as a nonuniform cantilever beam,^32–34 as shown in Figure 5. As shown in Figure 3, a₁ denotes the operating pressure angle, a₂ denotes the half tooth angle of the pinion. The total potential energy is composed of the Hertzian contact energy U_h, bending energy U_b, shear energy U_s, axial compressive energy U_a, and fillet foundation energy U_f. These five energies can be written as follows based on mechanics of elastic solids:

U_{h} = \frac{F^{2}}{2 k_{h}}, U_{b} = \frac{F^{2}}{2 k_{b}}, U_{s} = \frac{F^{2}}{2 k_{s}}, U_{a} = \frac{F^{2}}{2 k_{a}}, U_{f} = \frac{F^{2}}{2 k_{f}}

(8)

where F is the resultant force of gear meshing on the contact patch of the engaged tooth pair and k denotes the stiffness in meshing. The tooth stiffness k_m consists of the bending stiffness k_b, shear stiffness k_s, and axial compressive stiffness k_a, respectively.

Figure 5.

Nonuniform cantilever beam model of the tooth with pitting.

To conveniently calculate the stiffness, angular displacement notation is adopted.³⁵ According to the characteristics of the involute and transition curve the bending stiffness k_b, shear stiffness k_s and axial compressive stiffness k_a can be expressed as

{\begin{matrix} \frac{1}{k_{a}} = \int_{0}^{d} \frac{\sin^{2} a_{1}}{E A_{x}} d x = \int_{- a_{1}}^{a_{2}} \frac{R_{b 1} (a_{2} - a) cosa \sin^{2} a_{1}}{E A_{x}} d a \\ \frac{1}{k_{s}} = \int_{0}^{d} \frac{2.4 (1 + ν) \cos^{2} a_{1}}{E A_{x}} d x = \int_{- a_{1}}^{a_{2}} \frac{2.4 R_{b 1} (1 + ν) (a_{2} - a) \cos a \cos^{2} a_{1}}{E A_{x}} d a \\ \frac{1}{k_{b}} = \int_{0}^{d} \frac{{[\cos a_{1} (d - x_{1}) - \sin a_{1} h]}^{2}}{E I_{x}} d x \\ = \int_{- a_{1}}^{a_{2}} \frac{R_{b 1}^{3} {1 + \cos a_{1} [(a_{2} - a) \sin a - \cos a]}^{3} (a_{2} - a) \cos a}{E I_{x}} d a \end{matrix}

(9)

where L, E, and ν are tooth width, Young’s modulus, and the Poisson’ ratio, respectively. I_x and A_x indicate the area moment of inertia and the area of the section. Besides, fillet foundation stiffness k_f can be expressed as

\frac{1}{k_{f}} = \frac{1}{EL} \cos^{2} a_{u} [L^{*} {(\frac{u_{w}}{S_{fw}})}^{2} + M^{*} (\frac{u_{w}}{S_{fw}}) + P^{*} (1 + Q * \tan^{2} a_{u})]

(10)

where a_u, u_w, S_fw, L^*, M^* and Q^* can be found in Yang and Shi.³⁶

Generally, the Hertzian contact stiffness in gears is nonlinear, it is related to the nonlinear contact force. The local contact stiffness k_h can be expressed as Ma and Zeng.³¹

k_{h} = \frac{E^{0.9} L^{0.8} F_{i}^{0.1}}{1.275}, F_{i} = F \times ls r_{i}

(11)

where F_i denotes the meshing force and lsr_i is the load sharing ratio. A revised analytical model⁵ modeled the stiffness as springs connected in parallel and the mesh stiffness k_m at the meshing point is

k_{m, i} = \frac{1}{(1 / (λ_{1 N} k_{f 1})) + (1 / (k_{tooth, i})) + (1 / (λ_{2 N} k_{f 2}))}, (N = 1 or 2)

(12)

k_{tooth, i} = \sum_{i = 1}^{j} \frac{1}{(1 / k_{h, i}) + (1 / k_{a 1, i}) + (1 / k_{a 2, i}) + (1 / k_{s 1, i}) + (1 / k_{s 2, i}) + (1 / k_{b 1, i}) + (1 / k_{b 2, i})}

(13)

where j is the number of contact points, k_f1 and k_f2 denote the fillet foundation stiffness of the pinion and wheel. λ_1N and λ_2N are the correction coefficients which are detailed in Meltzer and Dien²⁶ (Figure 6).

Figure 6.

Revised mesh stiffness analytical model.

A finite element model using Solid185 elements is established to calculate the TVMS, as is shown in Figure 7. Considering the computational efficiency, only three teeth are established and other teeth are assumed to have no effect on the mesh stiffness. Due to the gear deformation, the TVMS can be calculated by the rotational displacement of the master node of pinion. The comparison between finite element model and theoretical results is shown in Figure 8.

Figure 7.

Finite element model of meshing gears.

Figure 8.

TVMS of healthy gear: (a) theoretical model and (b) finite element model.

The distribution of surface spring stiffness under different pitting states

The previous section has shown that the feature of surface pitting could be extracted combined with OPENCV based on the image processing technology and the TVMS has been calculated by energy method. This section takes the experimental gear as the research object (Figure 9) to calculate the TVMS with surface pitting.

The state of tooth surface pitting is expressed by the ratio of pitting area to tooth surface area. It is shown in Figure 9 that there are three kinds of surface pitting in different states (15% pitting, 20% pitting, and 25% pitting) on the tooth surface. The healthy surface and irregular shape pit in Figure 3 are also considered.

Figure 9.

Gear with different pitting states: (a) healthy, (b) irregular shape pitting, (c) 15% pitting, (d) 20% pitting, and (e) 25% pitting.

A discretization of the tooth surface is performed in both length and width directions based on Section “Modeling of the tooth stiffness in the presence of pitting based on image processing.” (The discrete size of pitting surface is m = 219 pixels and n = 1000 pixels, the following Figures 10 –12 are scaled-down diagrams for convenience of understanding and analysis.) The TVMS of healthy gear can be discretized by pixels along the tooth width as shown in Figure 11(a). Taking irregular shape pitting as the analysis object, Figure 11(b) shows the diagram of pixel value distribution matrix P_p1 of tooth surface without pitting area. Furthermore, Figure 11(c) shows the diagram of pixel value distribution matrix P_p2 of pitting area and the surface pitting is modeled as weak springs according to Section “Modeling of the tooth stiffness in the presence of pitting based on image processing,” the values of pixels are calculated by the ratio r_pit of the pitting stiffness k_pitting and the healthy stiffness k_healthy.

Figure 10.

Diagram of stiffness distribution calculation process (irregular shape pitting).

Figure 11.

Diagram of distribution of stiffness with irregular shape pitting.

Figure 12.

TVMS extraction.

It is worth noting that the pitting stiffness k_pitting only includes bending stiffness k_{b_pit}, shear stiffness k_{s_pit}, axial compressive stiffness k_{a_pit}, and fillet foundation stiffness k_f. The calculation process of k_{tooth_pit} can be found in Ma et al.⁵

As a result, the values of pixel are related to Equation (9) change accordingly. The calculation process of pixel matrix P_p1 and P_p2 is shown in Figure 11. Therefore, the values of spring and weak spring stiffness (k and $\tilde{k}$ ) in Appendix A can be calculated as follows:

{\begin{matrix} K_{p 1} = K_{healthy} * P_{p 1} \\ K_{p 2} = K_{healthy} * P_{p 2} \end{matrix}

(14)

where the matrix K_p1 represents the stiffness distribution of spring k and the matrix K_p2 represents the stiffness distribution of weak spring $\tilde{k}$ . Thus, the mesh stiffness of the whole surface can be described as (K_p1 + K_p2).

Figure 13.

Calculation process of pixel matrix.

In the view of the above calculation, the pitting surface stiffness distribution can be obtained. Figure 12(a) presents the mesh stiffness matrix (K_p1 + K_p2) of the whole surface with irregular shape pitting in Figure 9(b). It can be found in Figure 12(b) that the stiffness characteristics of irregular shape pitting can be clearly displayed from the top view of distribution. It is worth noting that it takes less than 40 s to calculate the stiffness matrix of pitting surface based on the TVMS of healthy gear in one mesh cycle at a time.

The results in Figure 12 can be further used in the subsequent dynamic calculation, and the values of K₁, K₂, and K₃ in Appendix. A can be determined. Therefore, the dynamic model based on image processing presented in Sections “Modeling of the tooth stiffness in the presence of pitting based on image processing” and “Dynamic model of the gear system with pitting defect based on image processing” is solved by the Newmark method.

Method comparison

To verify the accuracy of the model, Section “Method comparison” proposes a method to calculate TVMS based on Section “The distribution of surface spring stiffness under different pitting states.” The TVMS matrix K_TVMS can be expressed as

K_{TVMS} = diag (K_{healthy} {(P_{p 1} + P_{p 2})}^{T})

(15)

Furthermore, the TVMS matrix K_TVMS with different pitting states as shown in the blue line in Figure 13.

Although it is no longer necessary to calculate the TVMS, the TVMS with different pitting states is calculated as shown in Figure 14 to verify the results. It can be found that the amplitude of variation of TVMS gradually increases with the evolution of pitting and extends from the single tooth meshing area to the double tooth meshing area.

Figure 14.

Comparison of TVMS.

Compared with the traditional energy method,⁸ there is a big difference of stiffness between the initial and end of the pitting area. Figure 14(a) to (d) shows the maximum error of stiffness values can reach 10% when the gear is at the edge of pitting area. The mesh stiffness curve changes suddenly in Figure 14, which is due to the inappropriate simplification of pitting area and this will introduce unnecessary impact components into the dynamic response.

The influence of the shape of the pitting area on stiffness can be well reflected without the complicated calculation of the gear meshing process, and the speed of calculation is greatly improved. The computational time for the conventional model^5,8 may take several minutes, whereas this model requires less than 40 s in one mesh cycle for computation of the stiffness matrix of pitting surface.

Dynamic response and model validation

The validation of the proposed dynamic model is based on the comparison of the simulated vibration signals with the experimentally measured signals both in the time and frequency domains under different rotation speeds and pitting states.

Description of the experimental setup

The one-stage reduction gearbox (Gearbox Dynamics Simulator) with different pitting shown in Figure 15 is to validate the theoretical results. The parameters of the gearbox are the same as Table 1. The experimental bench consists of three-phase AC motor, a pair of spur gears, a power supply unit with the speed control electronics (Lenze SMVector), and the uTekL data acquisition system. The brake type used in the test rig is an electromagnetic brake. The input shaft speed is from 500 to 1500 rpm and the load is 11.3 N m. The pitting is manufactured near the pitch line of the driven gear as shown in Figure 8. The vibration signals are measured by PCB three-axis 356A32 accelerometers of which the location as shown in Figure 16. Point I and point II are used to acquire signals by accelerometers. The sampling interval is 1/12,800 s corresponding to a sampling frequency of 12,800 Hz.

Figure 15.

Gear pitting experimental test rig.

Figure 16.

The location of PCB accelerometers.

The experimental and dynamic signals under healthy state

Taking the vibration signals of the pinion at 500 rpm, the meshing frequency f_m is 241.6 Hz and the rotational frequency f_z is 8.3 Hz. As shown in Figures 17 and 18, the vibration characteristics of experimental and dynamic signals can be found. As the same as the theoretical frequency spectrum, the 3f_m is primary and the vibration response of the experimental signal is very close to the theoretical results both time and frequency domains.

Figure 17.

The experimental signal under healthy state at 500 rpm: (a) time history and (b) frequency spectrum.

Figure 18.

The dynamic signal under healthy state at 500 rpm: (a) time history and (b) frequency spectrum.

As observed in Figures 19 and 20, the vibration characteristics of pinion vary with the increase of the shaft speed. The primary frequency changes from 3f_m frequency to f_m frequency with the increase in speed.

Figure 19.

The experimental signal under healthy state at 1500 rpm: (a) time history and (b) frequency spectrum.

Figure 20.

The dynamic signal under healthy state at 1500 rpm: (a) time history and (b) frequency spectrum.

The experimental and dynamic signals under different pitting states

In this experiment, a gear tooth with surface pitting is tested. Compared with Figures 17 and 18, Figure 21 shows the existence of impulse under 25% pitting state and the frequency of impulse is Δt = 0.04 s (the shaft speed is 1500 rpm). Time history shows impulse will be caused by the decrease of stiffness. And the vibration response of the experimental signal with pitting is very close to the theoretical results both time and frequency domains.

Figure 21.

The experimental signal under 25% pitting state at 1500 rpm: (a) time history and (b) frequency spectrum.

As shown in Li and Ding,³⁷ for the convolution between the natural frequency evoked by the dynamic mesh stiffness, some spurious resonant frequencies would appear, which makes the resonant frequency bandwidth largely amplified. Figures 21 and 22 can clearly observe sidebands f_z = 25 Hz caused by the surface pitting, which is the rotational frequency of the pitting gear. In general, the vibration response of the experimental signal agrees with the dynamic response.

Figure 22.

The dynamic signal under 25% pitting state at 1500 rpm: (a) time history and (b) frequency spectrum.

The comparisons between the experimental and dynamic signals with 20% and 15% pitting are shown in Figures 23 –26. It can be observed that the magnitudes of the impulse vibrations in the time domain increase with the increase in pitting severity. All the results show the amplitude of primary frequency f_m increase and the amplitude of 2f_m decrease gradually with the increase in pitting severity. And the amplitude of sideband frequency increases with the evolvement of pitting in the frequency domain. In addition, the experimental and dynamic results indicate surface pitting has a great influence on the frequency between f_m and 2f_m.

Figure 23.

The experimental signal under 20% pitting state at 1500 rpm: (a) time history and (b) frequency spectrum.

Figure 24.

The dynamic signal under 20% pitting state at 1500 rpm: (a) time history and (b) frequency spectrum.

Figure 25.

The experimental signal under 15% pitting state at 1500 rpm: (a) time history and (b) frequency spectrum.

Figure 26.

The dynamic signal under 15% pitting state at 1500 rpm: (a) time history and (b) frequency spectrum.

As shown in Figures 23 and 25, the magnitudes of the impulse vibrations have no obvious change when the increase in pitting from 15% to 20%, which shows the impulse will be covered by external noise. Besides, Figure 25(b) shows the sideband frequency caused by pitting is difficult to distinguish from the noise frequency.

Analysis of signal characteristics of gear system with pitting

Time-domain characteristics of pitting gear vibration signal

As mentioned in Section “Introduction,” among various signal-processing techniques, a number of simple signal metrics based on the time domain waveform still have widespread applications in mechanical fault detection. All the indexes (root-mean-square, Kurtosis, and Crest factor) above¹ have been used in this paper to further analyze the pitting gear signal as shown in Figure 27. Each pitting degree corresponds to five groups of statistical indicators to improve the accuracy of results.

Figure 27.

The statistical indicators under different pitting severity.

Figure 27 shows the relationship between the three statistical indicators and the pitting severity, which indicates some values of statistical indicators under 20% pitting is less than the 15% pitting values and even smaller than healthy gear with the increase in the pitting degree.

However, the above three statistical indicators reach the maximum value when the pitting area reaches 25%. The reason for this phenomenon may be the vibration signal is disturbed by external noise during the experiment, which makes the impulse caused by pitting inconspicuous. Therefore, it is difficult to distinguish the surface pitting degree according to these statistical indicators when the pitting area is small.

The application of the HHT energy spectrum method on the pitting gear signal

In this study, an HHT energy spectrum method is applied for pitting gear signals. Hilbert–Huang transform includes EMD and corresponding Hilbert transform.³⁸ Each signal could be decomposed into a number of intrinsic mode functions (IMFs) with different frequencies.³⁹ The process is summarized as follows.

HHT is applied to the experimental signal with 20% pitting, the gear pitting can introduce modulations into the vibration signals.

Select the sensitive IMF containing fault features. The frequency analyses are performed for each IMF and compared with dynamic signals. The statistical indicators in Section “Time-domain characteristics of pitting gear vibration signal” are calculated for the most sensitive IMF.

The energy signal is used for monitoring the evolution of gear fault based on IMFs. Therefore, taking the most sensitive IMF as the main signal to further analysis. The theoretical and experimental signals are compared based on the index E(t).

Figure 28(a) shows the result of decomposition based on the vibration signal of 20% pitting degree and the instantaneous signal frequencies from IMF1 to IMF6 as shown in Figure 28(b). The effect of pitting on the vibration signal is stronger in the first six IMFs, so the instantaneous signal frequencies of these are given.

Figure 28

HHT on the experimental signal: (a) the intrinsic mode functions of vibration acceleration experimental signal (20% pitting) and (b) the instantaneous frequency of vibration acceleration experimental signal (20% pitting).

Each segment of the signal contains 6 rotation periods (40 ms). The impulse of IMF6 under this pitting state can be identified clearly in Figure 28(b). The instantaneous frequency of IMF6 fluctuates between the harmonic meshing frequency of 725 and 1450 Hz, which is the same as the dynamic results that the surface pitting has a great influence on the frequency between f_m and 2f_m as shown in Figure 29. These features are exactly the same with the gear with pitting vibration signal characteristics and coincide with the real condition.

Figure 29.

IMF6 of dynamic signal.

At the moment, when the pitting tooth is engaged in power transmission, the instantaneous frequency is significantly lowered because of the abrupt change in stiffness. Therefore, the instantaneous frequency is more sensitive to frequency modulation phenomena, and the characteristics of pitting gear signals can be better identified by the HHT method when the area of pitting is small.

Hence, the intrinsic mode IMF6 is considered to extract pitting features from experimental vibration signals. Figure 30 shows three indicators above of IMF6 signals versus pitting severity. It can be seen from the plot that the RMS, K, and C values increase steadily compared with Figure 27.

Figure 30

The statistical indicators under different pitting severity based on HHT.

When pitting occurs in gear, the energy of the gear fault vibration signal would change strongly in some frequency bands, but in other frequency bands the energy maybe changes weakly.⁴⁰ Therefore, taking the IMF6 experimental signal as the main signal to further analysis.

Thus, the index E(t) of component IMF6 can be defined as

E (t) = \int_{- \infty}^{\infty} H^{2} (ω, t) d ω

(16)

where the Hilbert spectrum H (ω, t) offers a measure of amplitude distribution with the change in each frequency and time. ω indicates the instantaneous frequency. Figure 31 shows the comparison between the index E(t) of experimental and dynamic signals under 20% pitting state, from which we know that the signal has some periodical characteristics.

Figure 31.

Comparison of index E(t) of 20% pitting gear vibration signal: (a) the index E(t) of experimental signal under 20% pitting state and (b) the index E(t) of dynamic signal under 20% pitting state.

The vibration energy comparison between the energy spectrum of experimental and dynamic signals from Figure 31 shows good agreement. And the energy of the pitting gear vibration signal mainly concentrates in a certain frequency band, which can be used to further identify the pitting gear severity.

Figure 32 further shows the vibration energy as detailed above versus pitting states. Obviously, the vibration energy gradually increases as the increase in pitting severity. In general, the tendency of the experimental signal index E(t) approximatively agrees with the theoretical results.

Figure 32.

Comparison of the index E(t) between experimental and dynamic signal.

Conclusion

This paper proposed a dynamic model with pitting based on image processing. First, the proposed model enables us to simulate tooth surface pitting more accurately especially in the initial and end of the pitting area with an accuracy improvement of 10% compared to traditional methods at the edge of the pitting area. The revised mesh stiffness of healthy gear is calculated and validated by FEM with stiffness error of less than 5%. Furthermore, the proposed model also demonstrated faster calculation speeds, with less than 40 s required to calculate the stiffness matrix of pitting in one mesh cycle. Moreover, the effectiveness of the proposed model is validated by comparing the responses obtained from experimental test rigs under different rotation speeds and pitting states. Statistical indicators and index E(t) are induced based on HHT-energy spectrum method. The characteristics caused by surface pitting can be well represented which demonstrates the method’s sensitive to the impact caused by surface pitting.

However, the proposed method still requires further improvements. For instance, the fillet foundation stiffness should be revised by considering the structure coupling effect, and the effect of irregular pitting shape on the dynamic response signal should be investigated. Therefore, a further investigation is required to reasonably quantify the extent of tooth surface irregular pitting, as well as to extract fault signals completely.

Footnotes

Appendix A. Matrix

n = {x_{1}, y_{1}, θ_{1}, x_{2}, y_{2}, θ_{2}, θ_{m}, θ_{b}}^{T}

M = diag [m_{1}, m_{1}, I_{1}, m_{2}, m_{2}, I_{2}, I_{m}, I_{b}]

F_{ext} = [\begin{matrix} - \sum k_{i} e_{i} \sin α \\ - \sum k_{i} e_{i} \cos α \\ - \sum k_{i} e_{i} R_{b 1} \\ \sum k_{i} e_{i} \sin α \\ \sum k_{i} e_{i} \cos α \\ - \sum k_{i} e_{i} R_{b 2} \\ T_{m} \\ T_{b} \end{matrix}]

K_{1} = [\begin{matrix} k_{x 1} + \sum_{j = 1}^{l_{1}} \sin^{2} α k_{i, j} & \sum_{j = 1}^{l_{1}} \sin α \cos α k_{i . j} & \sum_{j = 1}^{l_{1}} R_{b 1} \sin α k_{i, j} & - \sum_{j = 1}^{l_{1}} \sin^{2} α k_{i, j} & - \sum_{j = 1}^{l_{1}} \sin α \cos α k_{i, j} & \sum_{j = 1}^{l_{1}} R_{b 2} \sin α k_{i, j} & 0 & 0 \\ k_{y 1} + \sum_{j = 1}^{l_{1}} co s^{2} α k_{i, j} & \sum_{j = 1}^{l_{1}} R_{b 1} \cos α k_{i, j} & - \sum_{j = 1}^{l_{1}} \cos α \sin α k_{i, j} & - \sum_{j = 1}^{l_{1}} co s^{2} α k_{i, j} & \sum_{j = 1}^{l_{1}} R_{b 2} \cos α k_{i, j} & 0 & 0 \\ k_{c} + \sum_{j = 1}^{l_{1}} R_{b 1}^{2} k_{i, j} & - \sum_{j = 1}^{l_{1}} R_{b 1} \sin α k_{i, j} & - \sum_{j = 1}^{l_{1}} R_{b 1} \cos α k_{i, j} & \sum_{j = 1}^{l_{1}} R_{b 1} R_{b 2} k_{i, j} & 0 & - k_{c} \\ k_{x 2} + \sum_{j = 1}^{l_{1}} \sin^{2} α k_{i, j} & \sum_{j = 1}^{l_{1}} \sin α \cos α k_{i, j} & - \sum_{j = 1}^{l_{1}} R_{b 2} \sin α k_{i, j} & 0 & 0 \\ sym & k_{y 2} + \sum_{j = 1}^{l_{1}} co s^{2} α k_{i, j} & - \sum_{j = 1}^{l_{1}} R_{b 2} \cos α k_{i, j} & 0 & 0 \\ k_{c} + \sum_{j = 1}^{l_{1}} R_{b 2}^{2} k_{i, j} & - k_{c} & 0 \\ k_{c} & 0 \\ k_{c} \end{matrix}]

K_{2} = [\begin{matrix} \sum_{j = l_{2}}^{n} \sin^{2} α k_{i, j} & \sum_{j = l_{2}}^{n} \sin α \cos α k_{i . j} & \sum_{j = l_{2}}^{n} R_{b 1} \sin α k_{i, j} & - \sum_{j = l_{2}}^{n} \sin^{2} α k_{i, j} & - \sum_{j = l_{2}}^{n} \sin α \cos α k_{i, j} & \sum_{j = l_{2}}^{n} R_{b 2} \sin α k_{i, j} & 0 & 0 \\ \sum_{j = l_{2}}^{n} co s^{2} α k_{i, j} & \sum_{j = l_{2}}^{n} R_{b 1} \cos α k_{i, j} & - \sum_{j = l_{2}}^{n} \cos α \sin α k_{i, j} & - \sum_{j = l_{2}}^{n} co s^{2} α k_{i, j} & \sum_{j = l_{2}}^{n} R_{b 2} \cos α k_{i, j} & 0 & 0 \\ \sum_{j = l_{2}}^{n} R_{b 1}^{2} k_{i, j} & - \sum_{j = l_{2}}^{n} R_{b 1} \sin α k_{i, j} & - \sum_{j = l_{2}}^{n} R_{b 1} \cos α k_{i, j} & \sum_{j = l_{2}}^{n} R_{b 1} R_{b 2} k_{i, j} & 0 & 0 \\ \sum_{j = l_{2}}^{n} \sin^{2} α k_{i, j} & \sum_{j = l_{2}}^{n} \sin α \cos α k_{i, j} & - \sum_{j = l_{2}}^{n} R_{b 2} \sin α k_{i, j} & 0 & 0 \\ sym & \sum_{j = l_{2}}^{n} co s^{2} α k_{i, j} & - \sum_{j = l_{2}}^{n} R_{b 2} \cos α k_{i, j} & 0 & 0 \\ \sum_{j = l_{2}}^{n} R_{b 2}^{2} k_{i, j} & 0 & 0 \\ 0 & 0 \\ 0 \end{matrix}]

K_{3} = [\begin{matrix} \sum_{j = l_{2}}^{n} \sin^{2} α {\tilde{k}}_{i, j} & \sum_{j = l_{2}}^{n} \sin α \cos α {\tilde{k}}_{i . j} & \sum_{j = l_{2}}^{n} R_{b 1} \sin α {\tilde{k}}_{i, j} & - \sum_{j = l_{2}}^{n} \sin^{2} α {\tilde{k}}_{i, j} & - \sum_{j = l_{2}}^{n} \sin α \cos α {\tilde{k}}_{i, j} & \sum_{j = l_{2}}^{n} R_{b 2} \sin α {\tilde{k}}_{i, j} & 0 & 0 \\ \sum_{j = l_{2}}^{n} co s^{2} α {\tilde{k}}_{i, j} & \sum_{j = l_{2}}^{n} R_{b 1} \cos α {\tilde{k}}_{i, j} & - \sum_{j = l_{2}}^{n} \cos α \sin α {\tilde{k}}_{i, j} & - \sum_{j = l_{2}}^{n} co s^{2} α {\tilde{k}}_{i, j} & \sum_{j = l_{2}}^{n} R_{b 2} \cos α {\tilde{k}}_{i, j} & 0 & 0 \\ \sum_{j = l_{2}}^{n} R_{b 1}^{2} {\tilde{k}}_{i, j} & - \sum_{j = l_{2}}^{n} R_{b 1} \sin α {\tilde{k}}_{i, j} & - \sum_{j = l_{2}}^{n} R_{b 1} \cos α {\tilde{k}}_{i, j} & \sum_{j = l_{2}}^{n} R_{b 1} R_{b 2} {\tilde{k}}_{i, j} & 0 & 0 \\ \sum_{j = l_{2}}^{n} \sin^{2} α {\tilde{k}}_{i, j} & \sum_{j = l_{2}}^{n} \sin α \cos α {\tilde{k}}_{i, j} & - \sum_{j = l_{2}}^{n} R_{b 2} \sin α {\tilde{k}}_{i, j} & 0 & 0 \\ sym & \sum_{j = l_{2}}^{n} co s^{2} α {\tilde{k}}_{i, j} & - \sum_{j = l_{2}}^{n} R_{b 2} \cos α {\tilde{k}}_{i, j} & 0 & 0 \\ \sum_{j = l_{2}}^{n} R_{b 2}^{2} {\tilde{k}}_{i, j} & 0 & 0 \\ 0 & 0 \\ 0 \end{matrix}]

{\begin{matrix} c_{i, j} = \frac{ξ}{2 \sqrt{m_{e} k_{i, j}}} \\ C = C_{1} + C_{2} \\ m_{e} = \frac{m_{1} m_{2}}{m_{1} + m_{2}} \end{matrix}

C_{1} = [\begin{matrix} c_{x 1} + \sum_{j = 1}^{l_{1}} \sin^{2} α c_{i, j} & \sum_{j = 1}^{l_{1}} \sin α \cos α c_{i . j} & \sum_{j = 1}^{l_{1}} R_{b 1} \sin α c_{i, j} & - \sum_{j = 1}^{l_{1}} \sin^{2} α c_{i, j} & - \sum_{j = 1}^{l_{1}} \sin α \cos α c_{i, j} & \sum_{j = 1}^{l_{1}} R_{b 2} \sin α c_{i, j} & 0 & 0 \\ c_{y 1} + \sum_{j = 1}^{l_{1}} co s^{2} α c_{i, j} & \sum_{j = 1}^{l_{1}} R_{b 1} \cos α c_{i, j} & - \sum_{j = 1}^{l_{1}} \cos α \sin α c_{i, j} & - \sum_{j = 1}^{l_{1}} co s^{2} α c_{i, j} & \sum_{j = 1}^{l_{1}} R_{b 2} \cos α c_{i, j} & 0 & 0 \\ c_{c} + \sum_{j = 1}^{l_{1}} R_{b 1}^{2} c_{i, j} & - \sum_{j = 1}^{l_{1}} R_{b 1} \sin α c_{i, j} & - \sum_{j = 1}^{l_{1}} R_{b 1} \cos α c_{i, j} & \sum_{j = 1}^{l_{1}} R_{b 1} R_{b 2} c_{i, j} & 0 & - c_{c} \\ c_{x 2} + \sum_{j = 1}^{l_{1}} \sin^{2} α c_{i, j} & \sum_{j = 1}^{l_{1}} \sin α \cos α c_{i, j} & - \sum_{j = 1}^{l_{1}} R_{b 2} \sin α c_{i, j} & 0 & 0 \\ sym & c_{y 2} + \sum_{j = 1}^{l_{1}} co s^{2} α c_{i, j} & - \sum_{j = 1}^{l_{1}} R_{b 2} \cos α c_{i, j} & 0 & 0 \\ c_{c} + \sum_{j = 1}^{l_{1}} R_{b 2}^{2} c_{i, j} & - c_{c} & 0 \\ c_{c} & 0 \\ c_{c} \end{matrix}]

C_{2} = [\begin{matrix} \sum_{j = l_{2}}^{n} \sin^{2} α c_{i, j} & \sum_{j = l_{2}}^{n} \sin α \cos α c_{i . j} & \sum_{j = l_{2}}^{n} R_{b 1} \sin α c_{i, j} & - \sum_{j = l_{2}}^{n} \sin^{2} α c_{i, j} & - \sum_{j = l_{2}}^{n} \sin α \cos α c_{i, j} & \sum_{j = l_{2}}^{n} R_{b 2} \sin α c_{i, j} & 0 & 0 \\ \sum_{j = l_{2}}^{n} co s^{2} α c_{i, j} & \sum_{j = l_{2}}^{n} R_{b 1} \cos α c_{i, j} & - \sum_{j = l_{2}}^{n} \cos α \sin α c_{i, j} & - \sum_{j = l_{2}}^{n} co s^{2} α c_{i, j} & \sum_{j = l_{2}}^{n} R_{b 2} \cos α c_{i, j} & 0 & 0 \\ \sum_{j = l_{2}}^{n} R_{b 1}^{2} c_{i, j} & - \sum_{j = l_{2}}^{n} R_{b 1} \sin α c_{i, j} & - \sum_{j = l_{2}}^{n} R_{b 1} \cos α c_{i, j} & \sum_{j = l_{2}}^{n} R_{b 1} R_{b 2} c_{i, j} & 0 & 0 \\ \sum_{j = l_{2}}^{n} \sin^{2} α c_{i, j} & \sum_{j = l_{2}}^{n} \sin α \cos α c_{i, j} & - \sum_{j = l_{2}}^{n} R_{b 2} \sin α c_{i, j} & 0 & 0 \\ sym & \sum_{j = l_{2}}^{n} co s^{2} α c_{i, j} & - \sum_{j = l_{2}}^{n} R_{b 2} \cos α c_{i, j} & 0 & 0 \\ \sum_{j = l_{2}}^{n} R_{b 2}^{2} c_{i, j} & 0 & 0 \\ 0 & 0 \\ 0 \end{matrix}]

Acknowledgements

The authors would like to express the heartfelt thanks to the National Key R&D Program of China (Grant No. 2018YFB2001300).

Declaration of conflicting interests

The author(s) declared the following potential conflicts of interest with respect to the research, authorship, and/or publication of this article: We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service, and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled “Dynamical Modeling of Spur Gear with Pitting based on Image Processing Tooth Surface.”

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the National Key R&D Program of China (Grant No. 2018YFB2001300).

ORCID iD

Rupeng Zhu

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