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Research on nonlinear characteristics of herringbone planetary gear transmission system considering double-sided meshing impact

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Abstract

Aiming at the problem of the increase of vibration and noise caused by the double-sided meshing impact of herringbone gears. A dynamic model of double-sided meshing impact is established in consideration of the gear error, meshing phase, time-varying mesh stiffness (TVMS), time-varying damping and backlash in the herringbone planetary gear transmission system (HPGTS). The double-sided meshing process and mechanism of the herringbone gear are studied. The effects of excitation frequency and backlash on nonlinear characteristics of the HPGTS are analyzed by phase trajectories, Poincaré sections, the largest Lyapunov exponent (TLE) spectrum and bifurcation diagrams. The results indicate that the variation of excitation frequency will directly change the mesh state of the herringbone planetary gears. Within the excitation frequency range of the value, the system will experience the alternating variation of periodic motion and chaotic motion with the increase of excitation frequency. The phenomenon of the back-side meshing will gradually disappear as the excitation frequency increases to a certain value, and the drive-side meshing state will become stable. The transmission system will gradually transition to chaotic motion with increased backlash. Reducing the backlash while ensuring that the efficiency of the transmission can decrease the vibration and noise caused by the double-sided meshing impact of the system. The research results provide a theoretical reference for the stable operation of the system under double-side impact.

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Data availability

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

Abbreviations

\(b\) :

The tooth width

\(\overline{b}\) :

The dimensionless backlash

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

The displacement scale

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

Back clearance of tooth

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

Backlash

B :

Single edge tooth width

BL s–n :

The meshing plane when the left gear engages in back-side meshing

BR s–n :

The meshing plane when the right gear engages in back-side meshing

\(\overline{c}\) :

Dimensionless forms of damping

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

The support damping matrix

\({C}_{{\text{m}}}\) :

The meshing damping matrix

\({c}_{{\text{s}}\varsigma }\) :

Supporting damping of sun gear in \(\varsigma \) direction

\({c}_{{\theta }_{\varsigma {\text{s}}}}\) :

Torsional damping of the sun gear in \(\varsigma \) direction

\({e}_{spn}(t)\) :

Static transmission error on the nth s–p meshing pair

\({e}_{rpn}(t)\) :

Static transmission error on the nth r–p meshing pair

E :

Error amplitude

\({E}_{{\text{k}}}\) :

Impact kinetic energy

\({f}_{cs}\) :

Impact force of tooth surface

\({F}_{bs}\) :

The maximum impact force

\({F}_{bssp}\) :

The meshing impact force of s–p meshing pair in back-side meshing

\({F}_{cssp}\) :

The meshing impact force of s–p meshing pair in drive-side meshing

\({F}_{{\text{e}}}\left(t\right)\) :

The internal excitation force

\({F}_{{\text{m}}}\left(t\right)\) :

The dynamic meshing force which consider double-sided meshing

\({F}_{s}\left(t\right)\) :

Double-sided meshing impact excitation

\({G}_{{\text{g}}}\) :

The gyro matrix

\({I}_{\varsigma {\text{s}}}\) :

The moment of inertia of the sun gear in the direction of \(\varsigma \left( {\varsigma = x,y,z} \right)\)

\({J}_{l}(l=\mathrm{1,2})\) :

Instantaneous moment of inertia of driving gear and driven gear

\(\overline{k}\) :

Dimensionless forms of stiffness

\({k}_{{\text{s}}\varsigma }\) :

Supporting stiffness of sun gear in \(\varsigma \) direction

\({k}_{{\theta }_{\varsigma {\text{s}}}}\) :

Torsional stiffness of the sun gear in \(\varsigma \) direction

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

The support stiffness matrix

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

The centripetal stiffness matrix

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

Average meshing stiffness of the external meshing pair

\({l}_{LR}\) :

Width of recess

L s–n :

The meshing plane when the left gear engages in drive-side meshing

\({m}_{{\text{e}}}\) :

Equivalent mass

\({m}_{{\text{Ls}}}\) :

The mass of the left sun gear

\({m}_{{\text{n}}}\) :

Normal module

\(M\) :

The mass matrix of the system

\({N}_{1}{N}_{2}\) :

Theoretical meshing line

\({N}_{1}^{\prime}{N}_{2}^{\prime}\) :

The meshing line of the back-side meshing

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

Base pitch

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

Planetary gear

P :

The meshing node

\({q}_{{\text{s}}}\) :

Flexibility of single tooth pair of meshing gear at the back-side meshing point \({E}^{\prime}\)

\({r}_{aj}\)(\(j=r,s\)):

Tooth top circle radius of the ring gear and sun gear

\({r}_{bj}\)(\(j=r,s,p)\) :

Radius of the ring gear, sun gear, and planetary gear

\({r}_{bl}(l=\mathrm{1,2})\) :

Base circle radius of the driving gear and driven gear

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

Trajectory circle radius of the planetary gear rotary shaft \({z}_{ipn}\) around the planetary carrier rotary shaft \({Z}_{ic}\)

\({r}_{{\text{hl}}}(l=\mathrm{1,2})\) :

Inner hole radius of driving gear hubs and driven gear hubs

R s–n :

The meshing plane when the right gear engages in drive-side meshing

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

Impact time

\({T}_{{\text{m}}}\) :

The meshing period

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

Input torque for the sun gear

\(T(t)\) :

The force vector composed of the equivalent force obtained by dividing the driving torque and the resistance torque with the radius of the corresponding component

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

Back-side meshing impact velocity

\({v}_{l}(l=\mathrm{1,2})\) :

Circumferential velocities of the driving gear and driven gear

\(\overline{x}\) :

Dimensionless displacement

\(\dot{\overline{x}}\) :

Dimensionless velocity

\(\ddot{\overline{x}}\) :

Dimensionless acceleration

\(\alpha \) :

Normal pressure angle

\({\alpha }_{{\text{al}}}(l=\mathrm{1,2})\) :

Tooth top circle pressure angle of the driving gear and driven gear

\({\alpha }_{{\text{p}}n}\) :

Position phase angle of the planetary gear n

\({\alpha }_{jpt}\)(\(j=r,s)\) :

Meshing angles of the internal meshing face and the external meshing face

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

End face pressure angle

\({\beta }_{ib}\left(i=L,R\right)\) :

Helical angle of the left or right single helical gear of herringbone gear

\(\Delta {f}_{\Sigma }\) :

Gear shape error

\({\Delta t}_{sn}\) :

Time difference between the nth s–p meshing pair and the first s–p meshing pair

\(\Delta {\varphi }_{l}(l=\mathrm{1,2})\) :

Geometric clearance angle of the driving gear and driven gear when the gear is back-side meshed

\({\delta }_{12}\) :

Vibration displacement projection of the two gears on the mesh line

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

Impact deformation between the two meshing gear

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

End face coincidence degree

\({\theta }_{l}(l=\mathrm{1,2})\) :

Angle of the driving gear and driven from the node \(P\) to the position of the back contact point \({E}^{\prime}\)

\({\lambda }_{jn}\)(\(j=r,s)\) :

Phase coefficient between the nth r–p meshing pair and the first r–p meshing pair

\({\lambda }_{rs}^{(n)}\) :

Phase coefficient between the r–p meshing pair of the nth planetary gear and the corresponding s–p meshing pair

\({\lambda }_{sn}\) :

Phase coefficient between the nth s–p meshing pair and the first s–p meshing pair

\(\rho \) :

Material density of gear

\({\phi }_{n}\) :

Angle between the nth planetary gear and the first planetary gear

\({\phi }_{jp}\)(\(j=r,s)\) :

End face meshing angle of the r–p meshing pair and the s–p meshing pair

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

Angle between the action plane of s–p mesh pair and y-axis

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

Carrier angular velocity

\(\Omega \) :

Dimensionless excitation frequency

\(c\) :

Carrier

\(i\) :

Left or right side of herringbone gear (\(i\)= L, R)

\(j\) :

Ring gear or sun gear (\(j\)=r, s)

\(l\) :

The driving gear or driven gear (\(l \)= 1, 2)

\(n\) :

The nth planetary gear

\({\text{p}}\) :

Planetary gear

\({\text{r}}\) :

Ring gear

\({\text{s}}\) :

Sun gear

\(x\) :

\(x\) direction

\(y\) :

\(y\) direction

\(z\) :

\(z\) direction

L:

Left side of herringbone gear

R:

Right side of herringbone gear

\(\varsigma \) :

Translational along x, y and z axis

1:

The driving gear

2:

The driven gear

\({\text{b}}\) :

Back-side meshing

\(c\) :

Drive-side meshing

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Funding

This research is supported by the Foundation of National Natural Science of China (51975078), the Chongqing Science Fund for Distinguished Young Scholars (cstc2021jcyj-jqX0010), the Foundation of the Chongqing Basic Research and Frontier Exploration Project (cstc2022ycjh-bgzxm0130).

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Authors and Affiliations

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

    Xiangyang Xu, Hongwei Ge, Hui Wu & Hanjie Jia

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  1. Xiangyang Xu
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  2. Hongwei Ge
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Correspondence to Xiangyang Xu.

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Xu, X., Ge, H., Wu, H. et al. Research on nonlinear characteristics of herringbone planetary gear transmission system considering double-sided meshing impact. Nonlinear Dyn 112, 3195–3215 (2024). https://doi.org/10.1007/s11071-023-09201-3

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