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An original numerical calculation method for impacting contact model of spur gear under lightly loaded condition

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Abstract

In this study, an impacting contact model of spur gear under lightly loaded condition is constructed by considering the pinion and the wheel as rigid body, in which the lubricant is modelled by a linear spring damper combination. The entire motion of the gear system is fell into three phases, i.e.,: meshing state, transient state and non-meshing state. An original numerical calculation method is proposed for this model, and its validation is confirmed by using the Co-simulation of Adams-Matlab/Simulink. Instead of using the Jacobian matrix, the proposed numerical method could effectively avoid the ill condition in the iteration on data resulting from the nonlinearity of gear backlash. The results in this study verified that the algorithm is useful not only can increase the accuracy, but also significantly decrease the calculation time of the entire system. And the original numerical calculation method is employed to illustrate and quantify the influences of the excitation frequency on the impacting contact model of spur gears.

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Abbreviations

\( x \) :

Dynamical transmission errors

\( \dot{x} \) :

Relative speed

\( \ddot{x} \) :

Relative acceleration

\( L \) :

Length of gear backlash

\( t \) :

Time in second

\( \Delta t_{0} \) :

Time interval for any given time \( t_{0} \)

\( T_{f} \) :

Time of one excitation period

\( \Delta t_{m} \) :

Time interval from lubricant contact to solid contact

\( t_{n} \) :

Time interval for the current step \( n{th} \)

\( \Delta t_{ \hbox{max} } \) :

Maximum time-step

\( \Delta t_{ \hbox{min} } \) :

Minimum time-step

\( I_{p,w} \) :

Rotational inertia of pinion and gear

\( \theta_{p,w} \) :

Rotational displacements of pinion and gear

\( \dot{\theta }_{p,w} \) :

Rotational velocity of pinion and gear

\( \ddot{\theta }_{p,w} \) :

Rotational acceleration of pinion and gear

\( R_{p,w} \) :

Base radius of pinion and gear

\( T_{w} \) :

Drag torque

\( T_{wm} \) :

Mean part of drag torque

\( T_{pw}^{j} \) :

Amplitude of vibratory part of the \( j{\text{th}} \) harmonic for drag torque

\( \emptyset_{j} \) :

Initial phase of the \( j{\text{th}} \) harmonic for drag torque

\( T_{p} \) :

Driving torque

\( T_{pm} \) :

Mean part of driving torque

\( T_{pp}^{i} \) :

Amplitude of vibratory part of the \( j{\text{th}} \) harmonic for driving torque

\( \vartheta_{i} \) :

Initial phase of the \( j{\text{th}} \) harmonic for driving torque

\( \omega \) :

Excitation frequency

\( \Pi \) :

Number of engine cylinders

\( \in_{{\Delta t}} \) :

Short time duration of impact

\( e \) :

Coefficient of restitution

\( a_{1,2,3} \) :

Intermediate variables for \( \Delta t_{m} \)

\( c,k,f_{m,p} \) :

Intermediate variables for government equation

\( \varepsilon \) :

Precision tolerant

\( N \) :

Initial resolution of numerical solution

\( K_{d,c} \) :

Lubricant stiffness of driving/coast side

\( C_{d,c} \) :

Lubricant stiffness driving/coast side

NCM:

Numerical calculation method

DTE:

Dynamic transmission error

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Acknowledgements

The authors acknowledge the financial support from National Natural Science Foundation of China (Grant Nos. 51305378, 51605412), Jiangsu Provincial Science and Technology Department (BK20161312, BZ2018052), Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 17KJB460016), Postdoctoral Research Foundation of China (2016M590643).

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Authors and Affiliations

  1. School of Mechanical and Automotive Engineering, Qingdao University of Technology, Qingdao, 266520, China

    Fuhao Liu, Liang Zhang & Hanjun Jiang

  2. College of Automobile Engineering, Yancheng Institute of Technology, Yancheng, 224051, Jiangsu, China

    Fuhao Liu & Xuehua Yu

  3. Jiangsu Tailong Decelerator Machinery Co., Ltd., Taixing, 225400, Jiangsu, China

    Jielu Zhang

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  1. Fuhao Liu
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  2. Liang Zhang
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Correspondence to Fuhao Liu.

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Liu, F., Zhang, L., Jiang, H. et al. An original numerical calculation method for impacting contact model of spur gear under lightly loaded condition. Meccanica 55, 1435–1451 (2020). https://doi.org/10.1007/s11012-020-01173-7

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