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Symmetry breaking and chaos-induced imbalance in planetary gears

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Abstract

The goal of the present paper was a complete analysis of the dynamic scenario of planetary gears. A lumped mass two-dimensional model is adopted; the model takes into account: time-varying stiffness; nonsmooth nonlinearity due to the backlash, i.e., teeth contact loosing; and bearing compliance. The time-varying meshing stiffness is evaluated by means of a nonlinear finite element model, which allows an accurate evaluation of global and local teeth deformation. The dynamic model is validated by comparisons with the most authoritative literature: linear natural frequencies and nonlinear response. The dynamic scenario is analyzed over a reasonable engineering range in terms of rotation speed and torque. The classical amplitude–frequency diagrams are accompanied by bifurcation diagrams, and for specific regimes, the spectral and topological properties of the response are discussed. Periodic, quasiperiodic and chaotic regimes are found; nonsmooth bifurcations lead period one to period two trajectories. It is found that the bearing compliance can influence the natural frequencies combination magnifying the modal interactions due to internal resonances and greatly enlarging the chaotic regions. It is evidenced that the chaotic response indices a symmetry breaking in the dynamical systems. The physical consequence is that the planetary gearbox under investigation, which is perfectly balanced for each position, can suffer of a big dynamic imbalance when chaotic regimes take place; such imbalance gives rise to alternate and unexpected high-level stresses on bearings.

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Abbreviations

\(x,y\) :

\(x,y\)-translational coordinate (\(\upmu \hbox {m}\))

\(\varTheta \) :

Rotational coordinate (rad)

\(k_{s}\) :

Translational sun bearing stiffness (equal for \(x,y\)) (N/m)

\(\varPsi _{n}\) :

Angular position of the \(n\)th-planet with respect to the \(x\)-axis (rad)

\(\alpha _{s}\) :

Working pressure angle for sun–planet mesh (rad)

\(\alpha _{r}\) :

Working pressure angle for planet–ring mesh (rad)

\(r_{bs},r_{bn},r_{br}\) :

Base radii of (sun, \(n\)th-planet, ring) (m)

\(k_{sn}\) :

Mesh stiffness of sun–\(n\)th-planet (N/m)

\(k_{su}\) :

Torsional sun bearing stiffness (\(\hbox {N}\,\hbox {m/rad}\))

\(k_{r}\) :

Translational ring bearing stiffness (equal for \(x,y\)) (N/m)

\(k_{rn}\) :

Mesh stiffness of \(n\)th-planet–ring (N/m)

\(r_{c}\) :

Carrier radius (sun–planet center distance) (m)

\(k_{p}\) :

Translational planet bearing stiffness (equal for \(x,y\)) (N/m)

\(k_{pu}\) :

Rotational planet bearing stiffness (in this paper \(k_{pu}=0\)) (\(\hbox {N}\,\hbox {m/rad}\))

\(k_{c}\) :

Translational carrier bearing stiffness (equal for \(x,y\)) (N/m)

\(k_{cu}\) :

Rotational carrier bearing stiffness (\(\hbox {N}\,\hbox {m/rad}\))

\(M_{s}\) :

Sun mass (kg)

\(\theta _{s}\) :

Rotational coordinate of sun (rad)

\(C_{s}\) :

Translational sun bearing–damping (equal for \(x,y\)) (\(\hbox {N}\,\hbox {s/m}\))

\(C_{sn}\) :

Mesh damping of sun–\(n\)th-planet (\(\hbox {N}\,\hbox {s/m}\))

\(b_{s}\) :

Sun–planet backlash (m)

\(I_{s}\) :

Sun moment of inertia (\(\hbox {kg}\,\hbox {m}^{2}\))

\(C_{su}\) :

Torsional sun bearing–damping (\(\hbox {N}\,\hbox {m}\,\hbox {s/rad}\))

\(M_{r}\) :

Ring mass (kg)

\(\theta _{r}\) :

Rotational coordinate of ring (rad)

\(C_{r}\) :

Translational ring bearing–damping (equal for \(x,y\)) (\(\hbox {N}\,\hbox {s/m}\))

\(C_{rn}\) :

Mesh damping of \(n\)th-planet–ring (\(\hbox {N}\,\hbox {s/m}\))

\(\rho _{i}\) :

Dimensionless damping for the \(i\)th mode

\(b_{r}\) :

Planet–ring backlash (m)

\(I_{r}\) :

Ring moment of inertia (\(\hbox {kg}\,\hbox {m}^{2}\))

\(C_{ru}\) :

Torsional ring bearing–damping (\(\hbox {N}\,\hbox {m}\,\hbox {s/rad}\))

\(z\) :

Gear numbers of teeth

\(s\) :

Sun

\(n\) :

\(n\)th-planet

\(c\) :

Carrier

\(r\) :

Ring (annulus)

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Acknowledgments

The authors would like to thank the Lab INTERMECH MO.RE. (HIMECH District, Emilia Romagna Region) for supporting the research.

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Authors and Affiliations

  1. Department of Engineering “Enzo Ferrari”, University of Modena and Reggio Emilia, Via P. Vivarelli 10/1, 41125, Modena, Italy

    Asma Masoumi, Francesco Pellicano & Marco Barbieri

  2. Department of Mechanical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran

    Farhad S. Samani

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  1. Asma Masoumi
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  2. Francesco Pellicano
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Correspondence to Francesco Pellicano.

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Masoumi, A., Pellicano, F., Samani, F.S. et al. Symmetry breaking and chaos-induced imbalance in planetary gears. Nonlinear Dyn 80, 561–582 (2015). https://doi.org/10.1007/s11071-014-1890-3

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