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Stability investigation of velocity-modulated gear system using a new computational algorithm

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Abstract

By considering the lubricant in gear system, one degree-of-freedom model is set up which incorporates the pinion’s speed and the drag torque as the excitation sources. By introducing a permissible error (\(\varepsilon \)), a new computational algorithm using double-changed time steps is proposed in order to reduce the ill-conditioning arising from the numerical stiffness of the gear system and validated by comparison with Runge–Kutta–Fehlberg integration scheme. Then, the influences of the lubricant on the vibration of the gear system are analyzed. The results obtained in this paper indicate that the proposed numerical algorithm not only improves the accuracy of the solution, but also accelerates the calculation speed of the whole system. And according to the collision feature, the contributions of the lubricant on the system are totally different with different pinion’s speed and drag torque. Next, by introducing the proposed computational algorithm into the Floquet theory, the stability analyses of the gear system are investigated under the different excitation sources, which demonstrates that the excitation sources significantly affect the operating instability regions. In practice, particular instabilities can be minimized by the proper selection of pinion’s speed and drag torque, which can be adjusted according to the working requirements in advance.

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Abbreviations

\(\delta \) :

Dynamical transmission errors

\(\dot{\delta }\) :

Relative speed

\(\ddot{\delta }\) :

Relative acceleration

C :

Nonlinear viscous damping

K :

Nonlinear stiffness

W :

Nonlinear torque

c, k, w :

Intermediate variables for C, K, W

L :

Total backlash

t :

Time in second

\(T_f\) :

Time of one excited period cycle

\(\Delta t\) :

Time interval for any given time \(t_0\)

\(\Delta t_{+k}\) :

Time interval from lubricant contact to solid contact

\(\Delta t_{-k}\) :

Time interval from solid contact to lubricant contact

\(\Delta t_{-,+}^{\max }\) :

Maximum time step for lubricant and solid contact

\(\omega _\mathrm{p}\) :

Excitation frequency

\(\zeta _{1,2}\) :

Critical viscous damping ratio of lubricant and solid

\(k_{1,2}\) :

Stiffness of lubricant and solid

\(I_\mathrm{p,g}\) :

Rotational inertia of the pinion and gear

\(I_\mathrm{eq}\) :

Equivalent mass

\(\theta _\mathrm{p,g}\) :

Rotational displacements of the pinion and gear

\(\dot{\theta }_\mathrm{p,g}\) :

Rotational velocity of the pinion and gear

\(\ddot{\theta }_\mathrm{p,g}\) :

Rotational acceleration of the pinion and gear

\(R_\mathrm{p,g}\) :

Pitch radius of the pinion and gear

\(\vartheta _\mathrm{m}\) :

Mean part of the pinion’s speed

\(\vartheta _\mathrm{p}^i\) :

Amplitude of vibratory part of the ith harmonic for the pinion’s speed

\(\varphi _i\) :

Initial phase of the ith harmonic for the pinion’s speed

\(T_\mathrm{d}\) :

Drag torque

\(\bar{T}_\mathrm{m}\) :

Mean part of the drag torque

\(\bar{T}_\mathrm{p}^j\) :

Amplitude of vibratory part of the jth harmonic for the drag torque

\(\phi _j\) :

Initial phase of the jth harmonic for the drag torque

\(T_\mathrm{m}, T_\mathrm{p}^j\) :

Intermediate variables for \(\bar{T}_\mathrm{m}, \bar{T}_\mathrm{p}^j\)

\(\varepsilon \) :

Small value defining transition area

\(\varepsilon _{1,2}\) :

Perturbation number of the relative displacement and relative velocity

\(\lambda _{1,2}\) :

Eigenvalues for the Jacobian matrix

N :

Initial resolution of the numerical solution

M :

Number of the period

P :

Poincaré map

\({[} \Pi {]}\) :

Jacobian matrix

\(E{[} \,{]}\) :

Expectation operator

\(\gamma \) :

Periods of the solution

\(a_{\pm k} ,b_{\pm k} ,c_{\pm k}\) :

Intermediate variables for \(\Delta t_{\pm k}\)

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Acknowledgements

The authors acknowledge the financial support from National Natural Science Foundation of China (Grant No. 51305378), Jiangsu Provincial Key Laboratory of Automotive Engineering (QC201306), China Postdoctoral Science Foundation funded Project (2016M590643), Jiangsu Provincial Science and Technology Department (BY2015057-25) and the Research Laboratory of Mechanical Vibration (MVRLAB).

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Authors and Affiliations

  1. Department of Automotive Engineering, Yancheng Institute of Technology, Yancheng, 224051, People’s Republic of China

    Fuhao Liu, Liang Zhang & Xuehua Yu

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  1. Fuhao Liu
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Correspondence to Fuhao Liu.

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Liu, F., Zhang, L. & Yu, X. Stability investigation of velocity-modulated gear system using a new computational algorithm. Nonlinear Dyn 89, 1111–1128 (2017). https://doi.org/10.1007/s11071-017-3504-3

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