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Effects of tooth plastic inclination deformation due to spatial cracks on the dynamic features of a gear system

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Abstract

Spatial cracks may occur in the gear tooth fillet region due to the presence of non-uniform load distribution, local non-homogeneity of material quality, and potential misalignment of gear shafts and bearings, among other things. Previous researchers have discussed the tooth plastic inclination deformation due to a uniform depth planar crack. To the best of the authors’ knowledge, the non-uniform depth spatial crack has not been included in previous analyses. This paper presents a tooth plastic inclination model to analytically calculate the three-dimensional distributions of tooth plastic deformations due to spatial cracks based on the slicing principle. In addition, their influence on the dynamic performance of a spur gear pair was studied by building six-degrees-of-freedom model. The simulated results reveal that the non-uniformity of the crack depth along the tooth width direction could lead to uneven dynamic load distributions on the cracked tooth flank, which may excite a tilting motion of the gears. The larger the tooth plastic inclination angle, the higher the amplitude of the tilting motion that would be excited.

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Abbreviations

2D:

Two dimensional (or 2 dimensional)

3D:

Three dimensional (or 3 dimensional)

6DOFs:

Six degrees of freedom (or 6 degrees of freedom)

DOF:

Degree of freedom

FEA:

Finite element analysis

GMS:

Gear mesh stiffness

LOA:

Line of action

OLOA:

Off-line of action

STE:

Static transmission error

TC:

Tooth crack

A, B, C, D, E, \(E'\), \(E''\) :

Point labels defined in Fig. 4

T, P, Q :

Point labels defined in Fig. 2

F :

Total tooth contact force

\({{\varvec{F}}}_{\mathbf{0}}\), \({{\varvec{F}}}_{\mathbf{1}}\) :

External excitations due to static load \(T_{j}\) and tooth profile deviations \(e_{i}\)

\(H_{i}\) :

Contact function at \(M_{i}\)

M :

Moment about Y-axis due to total contact force F

M, C, K :

Overall system mass, viscous damping ,and stiffness matrices

\({{\varvec{K}}}_{{{\varvec{g}}}}\), \({{\varvec{K}}}_{{{\varvec{b}}}}\) :

Stiffness matrices regarding to bearing and gear pair

\(R_{b}, R_{f}\) :

Radii of the gear base circle and reference circle, respectively

T :

Transform matrix between the U–V coordinate and R–S coordinate system

\(T_{j} (j=1, 2)\) :

Torques applied on the driving (\(j=1\)) and driven (\(j=2\)) gear, respectively

V \(_{i}\) :

Structure vector at \(M_{i}\)

W, \(W_{c}\) :

Length of the tooth width and the crack along tooth width

a, b, c, f, \(\alpha \) :

Geometries of the rack in Fig. 1

\(c_{i}\) :

Distance of \(M_{i}\) to the tooth center along Z-axis

\(e_{ji}\) :

Tooth profile deviation error at \(M_{i}\) for jth gear

\(e_{w}(E)\) :

Distance between points \(E'\) and \(E''\), i.e., plastic deformation at point E

\(f_{i}\) :

Contact force at \(M_{i}\)

\(k_{i}, k_{ci}\) :

Cell stiffness and local contact stiffness at \(M_{i}\)

\(k_{g}\) :

Global stiffness

\(l_{0}\), \( l_{2}\), \(l_{w}\) :

Crack location at the two ends of tooth and variation of crack location along tooth width in Figs. 3 and 7

m :

Gear module

\(m_{j}\), \(I_{j}\), \(I_{pj}\) :

Mass, transverse moment of inertia, and polar moment of inertia of the jth gear

\(k_{Bx}\), \(k_{B\theta y}\), \(k_{B\theta z}\) :

Radial stiffness and rotational stiffness along Y- and Z-axes of the bearing support

q :

DOF vector of the gear pair

\(q_{0}\), \(q_{2}\), \( q_{w}\) :

Crack depth at the two ends of tooth and variation of crack depth along tooth width as shown in Figs. 3 and 5

r, s :

Coordinates of a point in the local R-axis and S-axis

u, v :

Coordinates of a point in the global U-axis and V-axis

x, y, z :

Translational motions along X-, Y-, and Z-axes, respectively

\(\alpha \) :

Gear pressure angle

\(\alpha _{c}\) :

Crack angle with respect to the V-axis

\(\alpha _{P}\) :

Mesh angle of the point P on the tooth profile

\(\alpha _{0}\) :

Angle between the tooth centerline and the line OA in Fig. 2

\(\delta _{i}\) :

Normal approach at \(M_{i}\)

\(\delta _{w}\) :

Width of tooth thin piece

\(\varepsilon _{i}\) :

Elastic deformation at \(M_{i}\)

\(\theta _{x}\), \(\theta _{y}\), \(\theta _{z}\) :

Rotational motions along X-, Y-, and Z-axes, respectively

\(\theta _{p0}\), \(\theta _{p}\) :

Initial tooth plastic inclination angle and inclination angle along tooth width

\(\theta _{T}\) :

Acute angle between the U-axis and R-axis

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Acknowledgements

The authors acknowledge the financial support of the China Scholarship Council (201306050004) and are grateful for the facility resources and support provided by the Natural Sciences and Engineering Research Council of Canada (203023-06).

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Authors and Affiliations

  1. Department of Mechanical and Materials Engineering, Queen’s University, Kingston, ON, K7L 3N6, Canada

    Wennian Yu & Chris K. Mechefske

  2. Bharti School of Engineering, Laurentian University, Sudbury, ON, P3E 2C6, Canada

    Markus Timusk

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  1. Wennian Yu
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  2. Chris K. Mechefske
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Correspondence to Wennian Yu.

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Yu, W., Mechefske, C.K. & Timusk, M. Effects of tooth plastic inclination deformation due to spatial cracks on the dynamic features of a gear system. Nonlinear Dyn 87, 2643–2659 (2017). https://doi.org/10.1007/s11071-016-3218-y

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