Journal of Mechanical Science and Technology

, Volume 33, Issue 11, pp 5117–5127 | Cite as

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

  • Lin HanEmail author
  • Houjun Qi


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.


Helical gear Multi-fault Dynamics Frequency analysis 



Tooth width


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


Meshing damping


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

dov, doh

Variables describing center’s locations of spalling defect


Elastic modulus


Normal force between meshing teeth

Ff1, Ff2

Friction forces acting on pinion and gear


Meshing force between pinion and gear


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

Fk(ik, j)

Friction force of ith segment of kth tooth of pinion at jth meshing instant

fk(ik, j)

Load per unit of length of contact line at ith segment of kth tooth of pinion at jth meshing instant


Auxiliary variable for calculating coefficient of friction

fm, fn

Meshing and rotating frequency


Height of spalling defect


Height of local tooth breakage defect

I1, I2

Moment of inertia of pinion and gear


Equivalent moment of inertia of gear pair


Auxiliary variable for calculating fk(ik,j)

ka, kb, kh, kf, ks

Stiffness components for healthy tooth

ksa, ksb, kss

Stiffness components for defected tooth


Meshing stiffness of gear pair


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


Meshing stiffness of single tooth-pair


Meshing stiffness of gear pair, = km


Length of spalling defect


Distance between points A and P in plane of action


Mass of pinion and gear, respectively


Auxiliary variable for calculating km


Rotating speed of pinion


Hertzian contact pressure


Effective radius of curvature at contact point


Radii of base circles of pinion and gear


Surface roughness parameter


Driving and loading torque

Tf1, Tf2

Friction torque action on pinion and gear


Entraining velocity

wb, ws

Width of local breakage, spalling


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

yi, , ÿ

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


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


Variable used in integration operation

α′1, α2, αs1, αs2

Auxiliary variables for calculating stiffness terms


Helix angle at basis circle


Total contact ratio of helical gear pair


Coefficient of friction


Viscosity of lubricant

θi, θ̇i, θ̈i

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


Damping ratio

\(\overline G \)

Auxiliary variables for calculating fk(ik, j)


Width of each sliced segment of tooth


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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.


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Copyright information

© KSME & Springer 2019

Authors and Affiliations

  1. 1.Tianjin Key Laboratory of High-speeding Cutting and Precision MachiningTianjin University of Technology and EducationTianjinChina

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