Coupled Harmonic Balance based approach for the non-linear dynamics of spur gear pairs 169 Conclusions A new approach to the analyses of spur gear pairs is presented in this paper. The already well-established Harmonic Balance Method in its coupled static-dynamic formulation is extended to the equation of motion of a lumped parameters model of a spur gear pair. A solution scheme is proposed making use of the Alternating Frequency-Time method. The AFT allows to compute the non-linear contact forces moving temporarily to the time domain via a IFFT of the generalized displacements. Subsequently the Fourier coefficient of the contact forces are obtained with the FFT of the single period time signal of the contact forces. A contact model is required to express the contact force as a function of the generalized displacements and its derivative. A basic penalty-based contact model is introduced defining a time varying gap between the teeth. The formulation obtained allows to simulate the system in the frequency domain without the need for any preliminary static analysis. In the results section, multiple non-linear responses are presented comparing the responses amplitudes obtained with the Newmark method and with the proposed methodology. The comparison holds for all the frequency range considered. Contact force, mesh stiffness and one period responses are also compared at specific frequencies. HBM and DTI show overlapping results for all the cases. Coupled HBM proved to be an effective alternative to DTI for the simulation of simple lumped parameters models. The results obtained are not close quantitatively to the experimental behavior of the modeled gears, however other studies (see [8]) have shown that lumped parameters model can simulate real components as long as an accurate contact model is introduced. The proposed methodology allows for the incorporation of various contact models without affecting the overall framework. The authors plan to leverage this flexibility in future work by introducing more realistic contact models to enhance the accuracy of the simple lumped parameter model in relation to actual spur gear pairs. References 1. Harris, S.L. “Dynamic Loads on the Teeth of Spur Gears”. Proceedings of the Institution of Mechanical Engineers, 172(1):87–112 (1958). 2. Fisher, A. “Factors in calculating the load-carrying capacity of helical gears”. Machinery, 98:545–552 (1961). 3. O¨ zgu¨ven, H.N. and Houser, D.R. “Mathematical models used in gear dynamics—a review”. Journal of sound and vibration, 121(3):383–411 (1988). 4. Nakata, T. “The dynamic load on gear caused by the varying elasticity of the mating teeth”. In Proc 6ˆ¡ th¿ Japan National Congress for Applied Mechanics, volume 493 (1956). 5. Lund, J.W. “Critical Speeds, Stability and Response of a Geared Train of Rotors”. Journal of Mechanical Design, 100(3):535–538 (1978). 6. Parker, R., Vijayakar, S., and Imajo, T. “Non-linear dynamic response of a spur gear pair: modelling and experimental comparisons”. Journal of Sound and Vibration, 237(3):435–455 (2000). 7. Ambarisha, V.K. and Parker, R.G. “Nonlinear dynamics of planetary gears using analytical and finite element models”. Journal of sound and vibration, 302(3):577–595 (2007). 8. Dai, X., Cooley, C.G., and Parker, R.G. “An Efficient Hybrid Analytical-Computational Method for Nonlinear Vibration of Spur Gear Pairs”. Journal of Vibration and Acoustics, 141(1):011006 (2019). 9. O¨ zgu¨ven, H. and Houser, D. “Dynamic analysis of high speed gears by using loaded static transmission error”. Journal of Sound and Vibration, 125(1):71–83 (1988). 10. Blankenship, G. and Kahraman, A. “Steady state forced response of a mechanical oscillator with combined parametric excitation and clearance type non-linearity”. Journal of Sound and Vibration, 185(5):743–765 (1995). 11. Velex, P. and Ajmi, M. “Dynamic tooth loads and quasi-static transmission errors in helical gears–approximate dynamic factor formulae”. Mechanism and Machine Theory, 42(11):1512–1526 (2007). 12. Zucca, S. and Firrone, C.M. “Nonlinear dynamics of mechanical systems with friction contacts: Coupled static and dynamic Multi-Harmonic Balance Method and multiple solutions”. Journal of Sound and Vibration, 333(3):916–926 (2014). 13. Cameron, T.M. and Griffin, J.H. “An Alternating Frequency/Time Domain Method for Calculating the Steady-State Response of Nonlinear Dynamic Systems”. Journal of Applied Mechanics, 56(1):149–154 (1989). 14. Newmark, N.M. “A method of computation for structural dynamics.”. Journal of the engineering mechanics division, 85(3):67–94 (1959). 15. Hotait, M. and Kahraman, A. “Experiments on the relationship between the dynamic transmission error and the dynamic stress factor of spur gear pairs”. Mechanism and Machine Theory, 70:116–128 (2013).
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