Unified method for typical gear failure modeling and stiffness calculation based on the matrix equation

doi:10.1007/s11465-024-0794-2

Front. Mech. Eng.

2024, Vol. 19

Issue (3) : 22 https://doi.org/10.1007/s11465-024-0794-2

Unified method for typical gear failure modeling and stiffness calculation based on the matrix equation

Fanshan MENG¹, Xin ZHANG¹(

), Heng XIA¹, Jiaxu WANG^1,²

¹. School of Mechanical Engineering, Southwest Jiaotong University, Chengdu 610031, China
². State Key Laboratory of Mechanical Transmissions for Advanced Equipment, Chongqing University, Chongqing 400044, China

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Abstract

The failure types in gear systems vary, with typical ones mainly including pitting, cracking, wear, and broken teeth. Different modeling and stiffness calculation methods have been developed for various gear failure types. A unified method for typical gear failure modeling and stiffness calculation is introduced in this study by considering the deviations in the time-varying meshing stiffness (TVMS) of faulty gears resulting from the use of different methods. Specifically, a gear tooth is discretized into a large number of microelements expressed with a matrix, and unified models of typical gear failures are built by adjusting the values of the matrix microelements. The values and positions of the microelements in the tooth failure model matrix have the same physical meaning as the parameter variables in the potential energy method (PEM), so the matrix-based failure model can be perfectly matched with PEM. Afterward, a unified method for TVMS is established. Modeling of healthy and faulty gears with pitting, wear, crack, and broken tooth is performed with the matrix equation, and the corresponding TVMS values are calculated by incorporating the matrix models with PEM. On the basis of the results, the mechanism of typical fault types that affect TVMS is analyzed, and the conclusions are verified through the finite element method. The developed unified method is a promising technique for studying the dynamic response characteristics of gear systems with different failure types because of its superiority in eliminating stiffness deviations.

Keywords gears matrix equation failure modeling TVMS calculation unified method

Corresponding Author(s): Xin ZHANG

Issue Date: 01 July 2024

Cite this article:

Fanshan MENG,Xin ZHANG,Heng XIA, et al. Unified method for typical gear failure modeling and stiffness calculation based on the matrix equation[J]. Front. Mech. Eng., 2024, 19(3): 22.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-024-0794-2
https://academic.hep.com.cn/fme/EN/Y2024/V19/I3/22

Fig.1 Tooth profile curve of an involute spur gear. The BC segment is an involute curve that participates in engagement contact. The CD segment is a transitional curve at the tooth root and does not participate in engagement contact.

Parameter	Pinion	Wheel
Tooth number	21	31
Modulus/mm	1.5	1.5
Mass/kg	0.21	0.3
Density/(kg?m³)	7.85	7.85
Tooth width L/mm	22	22
Hub radius r_int/mm	5	12
Yield strength σ_y/MPa	3.53 × 10⁹	3.53 × 10⁹
Poisson’s ratio ν	0.3	0.3
Elastic modulus E/GPa	206	206
Torque T/(N?m)	200
Addendum coefficient $h a ∗$	1	1
Tip clearance coefficient c^*	0.25	0.25

Tab.1 Gear parameters

Fig.2 Utilizing the matrix equation method to simulate the healthy gear tooth model: (a) discretization of the smooth, healthy tooth surface into multiple microelements, (b) formation of the microelement tooth surface model post-discretization, and (c) establishment of the microelement tooth thickness model.

Fig.3 Gear tooth is simplified to a cantilever beam for meshing stiffness calculations: (a) gear body of the elastic material and gear tooth of the rigid material and (b) correction of the actual load-carrying length of the cantilever beam during meshing transmission.

Fig.4 Tooth model of pitting failure established using a matrix equation: (a) discretized tooth surface microelement model, (b) 2D matrix corresponding to the tooth surface microelement model, (c) generation of single pitting matrix

P S (m, n)

, (d) morphology of a single pit, (e) pitting failure matrix

P (m, n)

, and (f) healthy tooth matrix

H (m, n)

Tab.2 Pitting failure degree, pit number, and size [53]

Fig.5 Effects of pitting failure on the gear tooth dimensions and structure: (a) depiction of pitting failure in the tooth profile direction and (b) assessment of effective contact width and alterations in the tooth surface’s cross-sectional structure resulting from pitting damage at a specific meshing position.

Fig.6 Wear depth of the tooth surface under different working conditions: (a) gears rotating at different speeds and (b) different durations of gear work at the same rotation speed.

Fig.7 Gear wear modeling for stiffness calculation by the matrix equation: (a) change in tooth profile shape with wear, (b) comparison of 3D models of gear teeth before and after tooth wear, and (c) change in tooth thickness before and after tooth surface wear.

Fig.8 Tooth model with uniform cracks. (a) The crack does not touch the centerline of the tooth, and (b) the crack passes through the centerline along the tooth thickness.

Fig.9 Gear crack modeling by the matrix equation. (a) Spatial crack damage gear tooth model, (b) model of the crack-damaged gear tooth with dx-width, (c) depth of uniform penetrating-type cracks, (d) depth of nonuniform cracks, and (e) crack depth for every microelement width in the tooth width direction.

Fig.10 Matrix equation for modeling gears with crack failure. (a) 3D model of crack-damaged gear, (b) healthy gear tooth model, (c) uniform penetration crack model, (d) nonuniform crack model, and (e) nonuniform vertical tooth crack model generated by the matrix equation.

Fig.11 Two types of broken tooth fault models established by the matrix equation. 3D model of (a, c) uniform penetration-type broken-tooth damage and (b, d) nonuniform broken-tooth damage.

Fig.12 TVMS at different pitting damage degrees. (a) Total stiffness results and (b–d) local zoomed-in results of the stiffness curves. The black, green, blue, and red lines correspond to the TVMS of the healthy gear, sun gear with slight pitting, sun gear with moderate pitting, and sun gear with severe pitting, respectively (the pit data corresponding to each damage degree are shown in Tab.2).

Fig.13 TVMS under different working conditions. (a) The black, brown, green, blue, and red lines correspond to pinion rotation speeds of 0, 1000, 2000, 3000, and 4000 r/min, respectively. (b) The black, brown, green, blue, and red lines correspond to pinion rotation durations of 0, 1 × 10⁵, 2 × 10⁵, 3 × 10⁵, and 4 × 10⁵ s, respectively.

Fig.14 TVMS under different forms and degrees of crack damage: penetrating cracks with (a) different depths and (b) different propagation angles; (c) nonuniform cracks with different depths; (d) nonuniform cracks with different depths and extension angles.

Fig.15 TVMS at different levels of broken tooth degree: (a) uniform broken tooth and (b) nonuniform broken tooth. The black, brown, green, blue, and red lines correspond to 0%, 25%, 50%, 75%, and 100% (in the direction away from the tooth tip) of the broken region of the gear tooth, respectively, as a percentage of the complete meshing length.

Fig.16 Finite element models of the pinion gear with different failure types: (a) pinion and wheel gear meshing transmission, (b) healthy gear, (c) pitting damage gear, (d) crack damage gear, (e) wear damage gear, and (f) broken tooth damage gear.

Meshing gear pair	Point M			Point N
Meshing gear pair	Analytical/(10⁶ $N ? m − 1$ )	FEM/(10⁶ $N ? m − 1$ )	Error/%	Analytical/(10⁶ $N ? m − 1$ )	FEM/(10⁶ $N ? m − 1$ )	Error/%
Fig.17(a)	455.0	467.8	2.81	253.0	266.0	5.14
Fig.17(b)	454.3	464.6	2.27	247.4	258.1	4.32
Fig.17(c)	451.4	464.1	2.81	237.9	245.0	2.98
Fig.17(d)	441.5	450.8	2.11	249.5	260.0	4.21
Fig.17(e)	454.8	469.9	3.32	252.4	263.7	4.48

Tab.3 Comparison of stiffness results between points M and N

Fig.17 Comparison of TVMS between the proposed method and FEM for different faulty gears. (a) Healthy gear, (b) pitting damage gear, (c) wear damage gear, (d) crack damage gear, and (e) broken tooth damage gear.

Meshing gear pair	Point M				Point N
Meshing gear pair	Healthy/(10⁶ $N ? m − 1$ )	Error/%	Crack/(10⁶ $N ? m − 1$ )	Error/%	Healthy/(10⁶ $N ? m − 1$ )	Error/%	Crack/(10⁶ $N ? m − 1$ )	Error/%
FEM [63]	540	0	480	0	350	0	310	0
3D-LTCA [63]	545	0.93	488	1.67	352	0.57	315	1.61
Proposed method	559	3.52	520	8.33	320	8.57	302	2.58

Tab.4 Comparison of the stiffness results of FEM, 3D-LTCA, and the proposed method

Fig.18 Comparison of TVMS under four typical gear failures. The black, green, red, blue, and magenta lines correspond to the stiffness of the healthy gear, pitting damage gear, crack damage gear, wear damage gear, and broken tooth damage gear, respectively.

Fig.19 Effect of faulty gears on different stiffness values: (a) nonlinear Hertz contact stiffness, (b) bending stiffness, (c) shear stiffness, and (d) compression stiffness. The black, green, red, blue, and magenta lines correspond to the stiffness of the healthy gear, pitting damage gear, crack damage gear, wear damage gear, and broken tooth damage gear, respectively.

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