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Modeling and Theoretical Investigation of Nonlinear Torsional Characteristics for Double-Helical Star Gearing System in GTF Gearbox

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Abstract

A improved torsional nonlinear dynamic model of double-helical star gearing system of Geared Turbofan (GTF) aero-engine is established, considering multiple nonlinear parameters and the effect of tool withdrawal groove that connect helical gears on both sides of double-helical gear. The dynamic responses of the system are solved by Runge–Kutta numerical integration method. The results reveal that the double-helical star gearing system exhibits abundant torsional nonlinear behaviors, which were illustrated by the evolution curve of dynamic meshing force and load sharing coefficient under varying rotational speed, gear backflash and width of groove, with either side of helical gear pairs in double-helical gear system are considered and illustrated, respectively, which can be hardly seen in previous research and worth being analyzed for better understanding of undesirable torsional dynamic motion for the double-helical gear system and therefore provide a reference for the design and control of gear system.

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Abbreviations

\({L}_{t}^{L},\) \({L}_{t}^{R},\) \({L}_{c}\) :

Gear width of left, right sided helical gear, width of tool withdrawal groove

\({k}_{ij},\) \({c}_{ij}\) :

Stiffness, damping influence coefficient for tool withdrawal groove

\({R}_{\mathrm{inner}},\) \({R}_{\mathrm{outer}}\) :

Inner and outer radii of rotational component

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

Helix angle in base circle of gear

\({R}_{c},\) \({r}_{c}\) :

Radius of outer, inner circle of groove

\({D}_{c},\) \({d}_{c}\) :

Diameter of outer, inner circle of groove

\({I}_{s},\) \({l}_{s}\) :

Area moment of inertia, torsional length

\({k}_{s},\) \({c}_{s}\) :

Torsional stiffness, torsional damping of rotational component

\({Z}_{s}\), \({Z}_{p}\), \({Z}_{r}\) :

Gear tooth number of sun gear, star gear, and ring gear

\({L}_{sp}\), \({L}_{rp}\) :

Contact length for external, and internal gear meshing pair

\({k}_{sp}\), \({c}_{sp}\), \({k}_{rp}\), \({c}_{rp}\) :

Meshing stiffness and damping of sun–star, and star–ring gear meshing pair

\({k}_{p}\), \({c}_{p}\), \({k}_{g}\), \({c}_{g}\) :

Torsional stiffness, damping of pinion and gear

\({b}_{sp}\), \({b}_{rp}\) :

Gear backlash of sun–star, and star–ring gear meshing pair

\({\alpha }_{sp}\),\({ \alpha }_{pr}\) :

Meshing angle of the external and internal meshes

\({\delta }_{sp}\), \({\delta }_{rp}\) :

Relative displacement of the external, internal gear pair along the meshing line

\({\theta }_{s}, {\theta }_{p}, {\theta }_{r}\) :

Angular displacement of sun gears, star gear and ring gear

\({e}_{sp}\), \({e}_{rp}\) :

Comprehensive meshing error of external, internal gear meshing pair

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

Transverse contact ratio, axial contact ratio of helical gear meshing pair

\({\xi }_{sp}\), \({\xi }_{pr}\), \({\xi }_{t}\) :

Meshing damping ratio of external, internal gear meshing pair, torsional damping ratio

\({r}_{bs}\), \({r}_{bp},\) \({r}_{br}\) :

Radius of base circle for sun gear, star gear and ring gear

\({J}_{s}\), \({J}_{p},\) \({J}_{r}\) :

Torsional inertia for sun gear, star gear and ring gear

\({\lambda }_{sp},\) \({\lambda }_{rp},\) \({\lambda }_{sr}\) :

Coefficient of meshing phase for external, internal, and external-internal gear pair

\({F}_{sp},\) \({F}_{rp}\) :

Meshing force of sun–star, star–ring gear meshing pair

\({F}_{km},\) \({F}_{cm}\) :

Elastic meshing force and viscous meshing force component

\({\mathrm{LSC}}_{sp},\) \({\mathrm{LSC}}_{rp}\) :

Load sharing coefficient of external, internal gear meshing pair

\({T}_{\mathrm{in}}\), \({T}_{\mathrm{out}}\) :

Input torque, output torque

\(L, R\): Left-side helical gear:

Right-side helical gear of double-helical gear

\(p, g\): Pinion (driving gear):

Gear (driven gear)

\(s, p, r\): Sun gear:

Star gear, ring gear

\(sp, rp\): External gear pair (sun–star gear pair):

Internal gear pair (star-ring gear pair)

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Acknowledgements

This work is supported by the National Key R&D Program of China (Grant No. 2018YFB2001300); National Natural Science Foundation of China (Grant No. 51775265); the China Scholarship Council (202006830075).

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Authors and Affiliations

  1. College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Siyu Wang & Rupeng Zhu

  2. National Key Laboratory of Science and Technology on Helicopter Transmission, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Siyu Wang & Rupeng Zhu

  3. School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore, 639798, Singapore

    Siyu Wang

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Correspondence to Siyu Wang or Rupeng Zhu.

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Wang, S., Zhu, R. Modeling and Theoretical Investigation of Nonlinear Torsional Characteristics for Double-Helical Star Gearing System in GTF Gearbox. J. Vib. Eng. Technol. 10, 193–209 (2022). https://doi.org/10.1007/s42417-021-00371-1

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