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Modification of friction for straightforward implementation of friction law

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Abstract

Friction phenomena exist in almost every mechanical device. Due to its complicated nature and influence on the system performance, extensive dynamic simulations are often required in the early system design stage. In this work, a novel approach for eliminating the numerical discontinuity in the classical Coulomb law and its extension is developed. Specifically, the method improves the computation process instead of modifying the Coulomb friction model. The estimated error of this procedure is derived under a simple and idealized model with an externally applied sinusoidal force. Two application examples of a single-body system are used to verify the proposed method, namely, 1-DOF mass-spring system with a moving belt and static ground. Results indicate that the proposed approach reveals the characteristics of the classical Coulomb friction law and its development and eliminates oscillations in the simulated friction force. Furthermore, a 3-DOF multi-body system is simulated to investigate the difference between the LuGre model and the proposed approach.

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Abbreviations

\(A\) :

Amplitude of force \(\boldsymbol{F}_{e}\) (N)

\(C\) :

Constant

\(\boldsymbol{f}\) :

Friction (N)

\(f_{c}\) :

Magnitude of dynamic friction (N)

\(f_{s}\) :

Magnitude of static friction (N)

\(f_{v}\) :

Coefficient of viscosity (N s/m)

\(\boldsymbol{F}_{e}\) :

External force (N)

\(\boldsymbol{I}\) :

Impulse in contact surface (N s)

\(\boldsymbol{I}_{s}\) :

Impulse of maximum stiction in contact surface (N s)

\(J\) :

Principal moments of inertia (kg m2)

\(m\) :

Mass (kg)

\(M\) :

Mass coefficient (kg)

\(\boldsymbol{N}\) :

Normal contact force (N)

\(\boldsymbol{r}\) :

Contact position vector (m)

\(t\) :

Time in the simulation process (s)

\(\Delta t\) :

Fixed or maximum step size in the simulation process (s)

\(\Delta t_{\mathrm{maxf}}\) :

Maximum allowable step size (s)

\(\boldsymbol{v}\) :

Velocity at the contact point (m/s)

\(\boldsymbol{v}_{{0}}\) :

Initial velocity (m/s)

\(\boldsymbol{v}_{r}\) :

Relative tangential velocity (m/s)

\(v_{\varepsilon }\) :

Velocity threshold to distinguish between static friction and dynamic friction (m/s)

\(v_{s}\) :

Stribeck velocity (m/s)

\(\Delta \boldsymbol{v}\) :

Change in velocity of the contact point (m/s)

\(w\) :

Angular frequency of force \(\boldsymbol{F}_{e}\) (rad/s)

\(\alpha \) :

Angle between \(\boldsymbol{r}\) and the friction \(\boldsymbol{f}\) (rad)

\(\varepsilon \) :

Numerical calculation errors

\(\varepsilon_{e}\) :

Numerical calculation errors relative to amplitude \(A\)

\(\delta_{\sigma }\) :

Geometry factor

\(\mu_{k}\) :

Kinetic friction coefficient

\(\mu_{s}\) :

Static friction coefficient

\(\sigma_{0}\) :

Stiffness coefficient (N/m)

\(\sigma_{1}\) :

Damping coefficient (N s/m)

\(\sigma_{2}\) :

coefficient of viscosity (N s/m)

1, 2, 3:

Bodies 1, 2 and 3

\(i\) :

\(i\)th contacting pairs

vr:

Relative velocity

fs:

Static friction

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Acknowledgements

This study was financially supported by the National Natural Science Foundation of China (Grant no. U1637206, 51575340), State Key Laboratory of Mechanical System and Vibration of Shanghai Jiao Tong University (Grant no. MSVZD201912) and Shanghai Aerospace Science and Technology Innovation Fund (Grant no. SAST2017-079).

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Authors and Affiliations

  1. State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai, 200240, China

    Shi Chen & Zhinan Zhang

  2. School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China

    Shi Chen & Zhinan Zhang

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  1. Shi Chen
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Correspondence to Zhinan Zhang.

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Chen, S., Zhang, Z. Modification of friction for straightforward implementation of friction law. Multibody Syst Dyn 48, 239–257 (2020). https://doi.org/10.1007/s11044-019-09694-0

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