Dynamics responses analysis in frequency domain of helical gear pair under multi-fault conditions

This work aims to investigate dynamic characteristics of a helical gear pair under multiple faults condition, through model-based dynamic analysis. Two kinds of faults, tooth spalling and local breakages are considered. Firstly, influences of each kind of fault on meshing stiffness of gear pair with one or more faulted teeth were revealed. Then an 8 degree of freedom lumped-parameter model was developed to obtain dynamic responses. Furthermore, frequency spectrum analysis of the responses was conducted and compared with those under healthy condition. Results indicate that spalling or local breakage introduce different affects into mesh stiffness, and consequently displacement & velocity amplitudes of pinion appear with different waveforms, especially for the case only tooth breakages happen. This work could provide useful instructions for fault detection diagnosis of geared transmissions under multiple fault conditions.

b :

Tooth width

b _i :

Constant coefficients used for calculating coefficient of friction (i = 1,2,...,9)

c _m :

Meshing damping

c _ij :

Supporting damping of pinion and gear respectively(i = x, y, z, and j = 1,2)

d _ov, d _oh :

Variables describing center’s locations of spalling defect

E :

Elastic modulus

F :

Normal force between meshing teeth

F _f1, F _f2 :

Friction forces acting on pinion and gear

F _m :

Meshing force between pinion and gear

F _ij :

i = x,y,z, force acting on gears along i-direction, j = 1,2 representing pinion and gear

F _k(i _k, j):

Friction force of i^th segment of k^th tooth of pinion at j^th meshing instant

f _k(i _k, j):

Load per unit of length of contact line at i^th segment of k^th tooth of pinion at j^th meshing instant

f :

Auxiliary variable for calculating coefficient of friction

f _m, f _n :

Meshing and rotating frequency

h _s :

Height of spalling defect

h _b :

Height of local tooth breakage defect

I ₁, I ₂ :

Moment of inertia of pinion and gear

I _e :

Equivalent moment of inertia of gear pair

I _v :

Auxiliary variable for calculating f_k(i_k,j)

k _a, k _b, k _h, k _f, k _s :

Stiffness components for healthy tooth

k ^s _a, k ^s _b, k ^s _s :

Stiffness components for defected tooth

k _m :

Meshing stiffness of gear pair

k _ij :

i = x, y, z, supporting stiffness along i-direction, j = 1, 2 representing pinion and gear

k _single :

Meshing stiffness of single tooth-pair

k _total :

Meshing stiffness of gear pair, = km

l _s :

Length of spalling defect

l _AP :

Distance between points A and P in plane of action

m ₁,2 :

Mass of pinion and gear, respectively

n :

Auxiliary variable for calculating k_m

n ₁ :

Rotating speed of pinion

P _h :

Hertzian contact pressure

R :

Effective radius of curvature at contact point

r _b1,2 :

Radii of base circles of pinion and gear

S :

Surface roughness parameter

T ₁,2 :

Driving and loading torque

T _f1, T _f2 :

Friction torque action on pinion and gear

V _e :

Entraining velocity

w _b, w _s :

Width of local breakage, spalling

x _i,ẋ _i,ẍ _i :

Vibration displacement,velocity and acceleration of pinion (i = 1 or p) and gear (i = 2)

y _i, ẏ, ÿ :

Vibration displacement,velocity and acceleration of pinion (i = 1 or p) and gear (i = 2)

z _i,ż _i,z̈ _i :

Vibration displacement,velocity and acceleration of pinion (i = 1 or p) and gear (i = 2)

α :

Variable used in integration operation

α′ ₁, α ₂, α _s1, α _s2 :

Auxiliary variables for calculating stiffness terms

β _b :

Helix angle at basis circle

ε :

Total contact ratio of helical gear pair

μ :

Coefficient of friction

ν ₀ :

Viscosity of lubricant

θ _i, θ̇ _i, θ̈ _i :

Angular dispacement, velocity and acceleration of pinion for i = 1 and gear for i = 2

ζ :

Damping ratio

\(\overline G \) :

Auxiliary variables for calculating f_k(i_k, j)

Δl :

Width of each sliced segment of tooth

This work was supported by the project funded by Tianjin Municipal Education Commission (Grant No. JWK1601), Natural Science Foundation of Tianjin (Grant No. 18JCQNJC75200) and Innovation Team Training Plan of Tianjin Universities and colleges (Grant No. TD13-5096), China.

Authors and Affiliations

1. Tianjin Key Laboratory of High-speeding Cutting and Precision Machining, Tianjin University of Technology and Education, Tianjin, 300222, China

   Lin Han & Houjun Qi

Corresponding author

Correspondence to Lin Han.

Recommended by Associate Editor Cheolung Cheong

Lin Han is a Lecturer of School of Mechanical Engineering, Tianjin University of Technology and Education, Tianjin, China. He received his Ph.D. degree in mechanical engineering from Tianjin University in 2014. His research interests include gear transmission dynamics, fault diagnosis of rotating machineries.

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