An enhanced multi-term harmonic balance solution for nonlinear period- β dynamic motions in right-angle gear pairs

Junyi Yang, Tao Peng, Teik C. Lim

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


A nonlinear, time-varying dynamic model for right-angle gear pair systems is formulated to analyze the existence of sub-harmonics and chaotic motions. This pure torsional gear pair system is characterized by its time-varying excitation, clearance, and asymmetric nonlinearities as well. The period-1 dynamic motions of the same system were obtained by solving the dimensionless equation of gear motion using an enhanced multi-term harmonic balance method (HBM) with a modified discrete Fourier transform process and the numerical continuation method presented in another paper by the authors. Here, the sub-harmonics and chaotic motions are studied using the same solution technique. The accuracy of the enhanced multi-term HBM is verified by comparing its results to the solutions obtained using the more computational intensive direct numerical integration method. Due to its inherent features, the enhanced multi-term HBM cannot predict the chaotic motions. However, the frequency ranges where chaotic motions exist can be predicted using the stability analysis of the HBM solutions. Parametric studies reveal that the decrease in drive load or the increase of kinematic transmission error (TE) can result in more complex gear dynamic motions. Finally, the frequency ranges for sub-harmonics and chaotic motions, as a function of TE and drive load, are obtained for an example case.

Original languageEnglish (US)
Pages (from-to)1237-1252
Number of pages16
JournalNonlinear Dynamics
Issue number2
StatePublished - Apr 2014
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mechanical Engineering
  • Aerospace Engineering
  • Ocean Engineering
  • Applied Mathematics
  • Electrical and Electronic Engineering
  • Control and Systems Engineering


  • Asymmetric mesh
  • Gear backlash
  • Multi-term harmonic balance
  • Nonlinear gear dynamics
  • Right-angle gear pairs


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