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Investigation on dynamic characteristics of herringbone planetarygear system considering tooth surface friction

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Abstract

The sliding friction on the tooth surface is an important factor affecting the dynamic characteristics of herringbone planetary gear system (HPGS) which is a complex high power density system. A novel coupling dynamic model of HPGS with friction effect is proposed through an iterative numerical scheme under the tribo-dynamic condition, and the dynamic characteristics are discussed considering tooth surface friction. The results show that the dynamic meshing force amplitude increases obviously after considering friction, and rises slowly with the increase of roughness. When the load increases gradually, the main peak of dynamic meshing force response increases obviously, but the frequency of the main resonance peak remains unchanged. With the increase of input torque, the main peak of meshing force increases because the input torque, as the input of the transmission system, is positively correlated with the dynamic characteristics of herringbone planetary gear transmission system. The influence of friction excitation on dynamic characteristics gradually decreases with the increase of input speed. As the internal excitation of the system, the friction excitation will also affect the dynamic characteristics. The main peak of dynamic mesh force increases significantly compared with that without considering friction, and increases slowly with the increase of roughness. With the increase of roughness, the dynamic characteristics of the transmission system increase slightly after friction excitation, so the amplitude of dynamic meshing force increases. However, as the roughness continues to increase, the lubrication state changes from full film lubrication to partial elastohydrodynamic lubrication. When the \({S}_{avg}\) of the tooth surface exceeds 1.2 μm, that is, the rough peak contacts, the friction excitation has little effect on the dynamic characteristics, so the amplitude of the main peak of dynamic meshing remains almost unchanged.

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Abbreviations

HPGS:

Herringbone planetary gear system

EHL:

Elastohydrodynamic lubrication

TVMS:

Time-varying mesh stiffness

\(A\) :

The rough surface area

\({\mathbf{C}}_{b}\) :

The support damping matrix

\({\mathbf{C}}_{m}(t)\) :

The meshing damping matrix

\({c}_{sni}^{\theta }(t)\) :

Time-varying mesh damping of the right and left-side ith sun-planet mesh (\(\theta =l,r\))

\({c}_{rni}^{\theta }(t)\) :

Time-varying mesh damping of the right and left-side ith ring-planet mesh (\(\theta =l,r\))

\(E,v\) :

Yang’s modulus and Poisson ratio

\({E}^{^{\prime}}\) :

Equivalent elastic modulus of gear pair

\({e}_{sni}^{\theta }\) :

Transmission error of sun-planetary gear meshing pair (\(\theta =l,r\))

\({e}_{rni}^{\theta }\) :

Transmission error of ring-planetary gear meshing pair (\(\theta =l,r\))

\(\mathbf{F}(t)\) :

Excitation force caused by transmission error excitation and time-varying stiffness excitation

\({F}_{f}(t)\) :

The total friction force expression in a meshing period

\({F}_{fij}(t)\) :

Tooth surface friction force on the jth contact line on the ith slice at the time of \(t\)

\({F}_{ij}(t)\) :

Contact line load on the jth contact line on the ith slice at the time of \(t\)

\({F}_{fs}^{\theta }(t)\) :

Sliding friction forces exerted by the sun gears on the ith planet gear (\(\theta =l,r\))

\({F}_{sni}^{\theta }(t)\) :

Dynamic mesh forces of the ith sun -planet gear pair (\(\theta =l,r\))

\({F}_{rni}^{\theta }(t)\) :

Dynamic mesh forces of the ith ring-planet gear pair (\(\theta =l,r\))

\({f}_{\Lambda (t)}\) :

Percentage of point contact load distribution in mixed state

\(\mathbf{G}\) :

Gyroscopic matrix

\(L(t)\) :

Length of integrated meshing line

\({L}_{i}^{\theta }(t)\) :

The length of the contact line on the left and right side of the pitch points (\(\theta =l,r\))

\({L}_{ij}(t)\) :

Time-varying contact line length

\({l}_{m},{l}_{n}\) :

Actual meshing line length and theoretical meshing line length

\({l}_{p}^{l}\) :

Left length Left length of single gear pitch

\(\mathbf{M}\) :

Matrix of mass

\({\mathbf{M}}_{\Omega e}\) :

Beam element mass submatrix (\(\Omega =s,r,n\))

\({h}_{c}(t)\) :

Central film thickness

\({H}_{pi}^{\theta }\left(t\right),{H}_{gi}^{\theta }(t)\) :

Torque arms of the contact line on left and right side of the driving gear and the passive gear (\(\theta =l,r\))

\({I}_{\Omega }\) :

The rotational inertia of the component (\(\Omega =s,r,n\))

\({\mathbf{K}}_{b}\) :

The matrix of support stiffness

\({\mathbf{K}}_{m}(t)\) :

The meshing stiffness matrix

\({\mathbf{K}}_{\omega }\) :

The centripetal stiffness matrix

\({\mathbf{K}}_{\Omega e}\) :

Submatrix of beam element stiffness (\(\Omega =s,r,n\))

\({k}_{s\zeta }^{\theta }\) :

The support stiffness of left and right sun gear (\(\zeta =x,y,z\), \(\theta =l,r\))

\({k}_{as\zeta }\) :

Bending stiffness in x, y direction and axial tension stiffness of sun gear retreating groove (\(\zeta =x,y,z\))

\({k}_{asu}\) :

Torsion stiffness of sun gear retreat groove

\(k_{{sni}}^{\theta } \left( t \right)\) :

Time-varying mesh stiffness of the right and left-side ith sun-planet mesh (θ = l, r)

\(k_{{rni}}^{\theta } \left( t \right)\) :

Time-varying mesh stiffness of the right and left-side ith ring-planet mesh (θ = l, r)

\(k_{b} ,k_{s} ,k_{a} ,k_{h} ,k_{f}\) :

Bending, shear and axial compressive, Hertzian and fillet-foundation and elastic stiffness of the gear tooth

\({p}_{t}\) :

The tooth pitch of the end face

\({P}_{hij}(t)\) :

Hertz pressure of tooth contact point on the jth contact line on the ith slice at the time of t

\({R}_{ij}(t)\) :

Meshing comprehensive radius of curvature on the jth contact line on the ith slice at the time of t

\({r}_{\Omega }\) :

Pitch circle of the component (\(\Omega =s,r,n\))

\({r}_{a\Omega }\) :

Addendum circle of the component (\(\Omega =s,r,n\))

\({r}_{b\Omega }\) :

The base pitch of the component (\(\Omega =s,r,n\))

\(S{R}_{ij}(t)\) :

Slip ratio of tooth surface velocity on the jth contact line on the ith slice at the time of \(t\)

\({S}_{avg}\) :

The root mean square value of the roughness

\({t}_{c}\) :

Time of one mesh circle.

\({T}_{s}\) :

The external load on the sun gear

\({T}_{fpij}(t),{T}_{fgij}(t)\) :

The friction torque of driving gear and passive gear on the jth contact line on the ith slice at the time of \(t\)

\({T}_{fp}(t),{T}_{fg}(t)\) :

The total tooth friction torque of driving gear and passive gear

\({T}_{fs}^{\theta }(t)\) :

Sliding friction torques exerted by the nth planet gear on the sun gears (\(\theta =l,r\))

\(\mathbf{T}\) :

External torque vector

\(z(x,y)\) :

The rough surface height

\({\alpha }_{1}\) :

Viscosity–pressure coefficient of the lubricant

\({{\mu }_{m}}_{ij}(t)\) :

The time-varying friction coefficient in mixed state on the jth contact line on the ith slice at the time of \(t\)

\({{\mu }_{e}}_{ij}(t)\) :

The time-varying friction coefficient of elastohydrodynamic lubrication contact on the jth contact line on the ith slice at the time of \(t\)

\({\mu }_{b}\) :

The average friction coefficient of rough contact

\({{u}_{s}}_{ij}(t),{{u}_{r}}_{ij}(t)\) :

Sliding velocity and entrainment velocity on the jth contact line on the ith slice at the time of \(t\)

\({\eta }_{0}\) :

Ambient viscosity of the lubricant

\({\omega }_{c}\) :

Planet carrier angular rate

\(\Lambda (t)\) :

Film thickness ratios

\({\psi }_{sni},{\psi }_{rni}\) :

Angle made by the plane of action with the vertical y-axis

\({\psi }_{ni}\) :

The relative angle of ith planet gear on the right rotates to the position of the first planet gear

\({\varepsilon}_{p}, {\varepsilon }_{g}\) :

The rough peak height of two contact rough surfaces.

\(\lambda\) :

Constants related to helical gear overlap ratio

\({\xi }_{g}\) :

Gear meshing damping ratio

\(c\) :

Planet carrier

\(n\) :

Planet gear

\(r\) :

Ring gear

\(s\) :

Sun gear

l :

Left side

\(r\) :

Right side

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Acknowledgements

This research is supported by the Foundation of the National Key Research and Development Plan of China (No. 2018YFB2001300), the Foundation of National Natural Science of China (No. 51975078), the Chongqing Science Fund for Distinguished Young Scholars (cstc2021jcyj-jqX0010).

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Authors and Affiliations

  1. School of Mechatronics and Vehicle Engineering, Chongqing Jiaotong University, ChongQing, 400074, China

    Xiangyang Xu, Guosong Jiang, Han Wang & Xihui Liang

  2. Department of Mechanical Engineering, University of Manitoba, Winnipeg, MB, Canada

    Xihui Liang

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Xu, X., Jiang, G., Wang, H. et al. Investigation on dynamic characteristics of herringbone planetarygear system considering tooth surface friction. Meccanica 57, 1677–1699 (2022). https://doi.org/10.1007/s11012-022-01526-4

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