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Dynamic modelling and analysis of cracked gear system with tip relief based on proposed variable-angle deformation energy integration method

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Abstract

Due to ignoring the effects of the change of the tooth attachment position caused by the cracks, traditional time vary mesh stiffness (TVMS) calculation models and dynamic simulations for cracked gears will lose their precision in the body crack case. To address this shortcoming, a new analytical TVMS calculation model of cracked gear considering tip relief (TR) is developed based on a proposed variable-angle deformation energy integration method. On this basis, a dynamic model of the gear system for the analysis of fault vibration characteristics is established. The effectiveness and accuracy of the proposed TVMS calculation model are verified by the finite element method. A comprehensive investigation is finally carried out to reveal the effects of the parameters of TR, load and crack on the TVMS and dynamic characteristics of the cracked gears. The study results indicate that the proposed models can meet the accurate TVMS calculation and dynamic simulation for both the tooth- and body-cracked gears, and the influences of the tooth attachment position change caused by the crack cannot be ignored. This study could provide a systematic methodology and meaningful reference for the dynamic modelling, simulation and fault diagnosis of gear systems with crack failures.

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Data availability

The datasets generated and analysed during the current study are available from the corresponding author on reasonable request.

Abbreviations

subscript i :

Subscript i = 1, 2, 3 represents Zone I (i = 1), II (i = 2) or III (i = 3), respectively

subscript λ :

Subscript λ = p, g represents driving gear (λ = p) or driven gear (λ = g), respectively

α 1 :

The angle of the meshing force

α 2 :

Half of the tooth angle corresponding to the base circle

α p :

The included angle of the action line and the x coordinate in the fixed coordinate system in Fig. 8

A xi :

The area of the micro-section dxi

c λx/c λy :

Bearing damping of gear λ in x/y direction

c m :

Total meshing damping

C n :

Amount of tooth TR

d 1 :

The horizontal distance between the meshing position and the intersection of tooth profile and base circle

d 2 :

The horizontal distance between the intersections of tooth profile with dedendum circle and base circle

d 3 :

The horizontal distance between the crack tip and the intersection of tooth profile and dedendum circle

e pg :

The relative error excitation displacement of the driven gear relative to the driving gear

\({\dot{e}}_{\mathrm{pg}}\) :

The relative error excitation velocity of the driven gear relative to the driving gear

E :

Gear elasticity modulus

E kj :

The relative profile error of any tooth pair j relative to the maximum deformation tooth pair k

E λj /E λk :

The tooth profile error of gear λ of any tooth pair j or the maximum deformation tooth pair k

F :

Meshing force

F n :

Normal force

F ai/F bi :

Axial compression/shear force decomposed by F corresponding to the micro-section dxi

f r :

Rotation frequency of driving gear

f m :

Meshing frequency

G :

Shear modulus of gear

h F ( xi ) :

The effective moment arm of Fai on the micro-section dxi

h r :

The vertical distance between the intersections of the addendum circle and tooth profile and the centerline of the gear tooth

h x3 :

The length of the effective micro-section dx3 in Zone III

I xi :

Area of inertia of the micro-section dxi

J λ :

Rotational inertia of gear λ

k m :

Total TVMS modified by the tooth errors

K :

TVMS of single tooth pair

K b/K s/K a /K f :

Stiffness of bending/shear/axial compression/fillet foundation

K bλ/K sλ/K aλ/K fλ :

Stiffness of bending/shear/axial compression/fillet foundation of gear λ

K fFEM :

TVMS of single tooth pair calculated by the FE model

K h :

Hertz contact stiffness

k λx/k λy :

Bearing stiffness of gear λ in the x/y direction

l :

The effective moment arm of Fb3 on the micro-section dx3

L n :

Length of tooth TR

m :

Module of gear

m λ :

Mass of gear λ

M i :

Total bending moment of the micro-section dxi

n :

The first (n = 1) and second (n = 2) tooth pair

p a :

Principal stress pa of the beam section NN in Fig. 3a

q :

Crack depth

R b :

Base circle radius

R bλ :

Base circle radius of gear λ

R g :

Dedendum circle radius

R o :

Addendum circle radius

T r :

Rotation period of the driving gear

T λ :

Torque of gear λ

U b /U s /U a :

Deformation energy of bending/shear/axial compression

W :

Tooth width

x i :

Distance of the effective micro-section dxi from the base circle (i = 1 or 2) or dedendum circle (i = 3)

\(x_{\lambda } /\dot{x}_{\lambda } /\ddot{x}_{\lambda } {\kern 1pt}\) :

Lateral displacement/velocity/acceleration of gear λ along the x-direction

\(y_{\lambda } /\dot{y}_{\lambda } /\ddot{y}_{\lambda } {\kern 1pt}\) :

Lateral displacement/velocity/acceleration of gear λ along the y-direction

\(\theta_{\lambda } /\dot{\theta }_{\lambda } /\ddot{\theta }_{\lambda }\) :

Angular displacement/velocity/acceleration of gear λ

β :

F Decomposition angle for the effective micro-section dx3 in Zone III

δ j :

Tooth deformation of the jth tooth pairs

δ λ :

The meshing point displacement of gear λ along the line of action

θ :

Position angle of the effective micro-section dx3 in Zone III

θ f :

Half of the tooth angle corresponding to the dedendum circle

φ :

Crack initiation position angles

ϕ :

Meshing position ϕ

τ a :

The shear stress of the beam section NN in Fig. 3a

σ a :

The tensile stress of the beam section NN in Fig. 3a

ν :

Crack angle

ω λ :

Rotation speed of gear λ

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Acknowledgements

The authors are grateful for the National Natural Science Foundation of China (Grant No. 52035002), the Graduate Research and Innovation Foundation of Chongqing, China (Grant No. CYB21014) and China Scholarship Council (202106050062).

Funding

This study is financially supported by the National Natural Science Foundation of China (Grant No. 52035002) and the Graduate Research and Innovation Foundation of Chongqing (Grant No. CYB21014).

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Authors and Affiliations

  1. State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing, 400044, China

    Lantao Yang, Desheng Zou, Liming Wang & Yimin Shao

  2. Centre for Efficiency and Performance Engineering, University of Huddersfield, Huddersfield, HD1 3DH, UK

    Lantao Yang, Xiuquan Sun, Fengshou Gu & Andrew Ball

  3. Creative Computing Institute, University of the Arts London, London, WC1V 7EY, UK

    David Mba

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  1. Lantao Yang
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Correspondence to Yimin Shao.

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Yang, L., Zou, D., Sun, X. et al. Dynamic modelling and analysis of cracked gear system with tip relief based on proposed variable-angle deformation energy integration method. Nonlinear Dyn 111, 4141–4172 (2023). https://doi.org/10.1007/s11071-022-08077-z

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