Non-smooth numerical solution for Coulomb friction, rolling and spinning resistance of spheres applied to flexible multibody system dynamics

Abstract

The general motion of a sphere in a mechanism in contact with a rigid planar surface under rolling, sliding and spinning friction is studied in the context of non-smooth contact dynamics. The equations of motion are solved by the non smooth generalized–$\alpha$ implicit time integration scheme, where the position and velocity level constraints are satisfied exactly without requiring to define any particular value for a penalty parameter. The geometrical properties of the spheres are described by a rigid-body formulation with translational and rotational degrees of freedom. The robustness and the performance of the proposed methodology is demonstrated by different examples, including both flexible and/or rigid elements.

Appendices

Appendix A

Algorithm 1

Modified non-smooth GGL generalised-$\alpha$ time integration scheme

Appendix B

Hessian matrix for sliding resistance at position level for the different contact conditions,

$$\begin{aligned} \varDelta \boldsymbol {F}_{S}^{p} = \textstyle\begin{cases} \begin{aligned} &\begin{Bmatrix} \boldsymbol{\mathit{0}} \\ -\frac{k_{p}^{2}}{p_{p}}\varDelta \nu _{N} \\ -\frac{k_{p}^{2}}{p_{p}}\varDelta \boldsymbol{\nu }_{T} \end{Bmatrix} \\ &\quad \xi _{N} < 0 \quad \text{Gap} \\ &\begin{Bmatrix} -\varDelta g_{N\boldsymbol {q}}^{T}\xi _{N} - g_{N \boldsymbol {q}}^{T}\varDelta \xi _{N} - \mu \varDelta \xi _{N} \boldsymbol {g}^{T}_{T\boldsymbol {q}}\boldsymbol{\tau }_{p} - \mu \xi _{N} \varDelta \boldsymbol {g}^{T}_{T\boldsymbol {q}} \boldsymbol{\tau }_{p}- \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \mu \xi _{N} \boldsymbol {g}^{T}_{T \boldsymbol {q}}\varDelta \boldsymbol{\tau }_{p} \\ -k_{p} \varDelta g_{N} \\ \frac{k_{p}}{p_{p}}(-k_{p}\varDelta \boldsymbol{\nu }_{T} +\mu \varDelta \xi _{N} \boldsymbol{\tau }_{p} + \mu \xi _{N}\varDelta \boldsymbol{\tau }_{p}) \end{Bmatrix} \\ &\quad \|\boldsymbol{\xi }_{T}\| \geq \mu \xi _{N} \quad \text{Slip} \\ &\begin{Bmatrix} -\varDelta g_{N\boldsymbol {q}}^{T}\xi _{N} - g_{N \boldsymbol {q}}^{T}\varDelta \xi _{N} - \varDelta \boldsymbol {g}^{T}_{T \boldsymbol {q}}\boldsymbol{\xi }_{T} - \boldsymbol {g}^{T}_{T \boldsymbol {q}}\varDelta \boldsymbol{\xi }_{T} \\ -k_{p}\varDelta g_{N} \\ -k_{p}\varDelta \boldsymbol {g}_{T} \end{Bmatrix} \\ &\quad \|\boldsymbol{\xi }_{T}\| < \mu \xi _{N} \quad \text{Stick} \end{aligned} \end{cases}\displaystyle \end{aligned}$$

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Hessian matrix for sliding resistance at velocity level for the different contact conditions,

$$\begin{aligned} \varDelta \boldsymbol {F}_{S}^{v} = \textstyle\begin{cases} \begin{aligned} &\begin{Bmatrix} \boldsymbol{\mathit{0}} \\ -\frac{k_{v}^{2}}{p_{v}}\varDelta \varLambda _{N} \\ -\frac{k_{v}^{2}}{p_{v}}\varDelta \boldsymbol{\varLambda }_{T} \\ 0 \\ \boldsymbol{\mathit{0}}\end{Bmatrix} &\sigma _{N} < 0 \quad \text{Gap} \\ &\begin{Bmatrix} - g_{N\boldsymbol {q}}^{T}\varDelta \sigma _{N} - \mu \varDelta \sigma _{N} \boldsymbol {g}^{T}_{T\boldsymbol {q}} \boldsymbol{\tau }_{v} - \mu \sigma _{N} \boldsymbol {g}^{T}_{T \boldsymbol {q}}\varDelta \boldsymbol{\tau }_{v} \\ -k_{v} \varDelta \mathring{g}_{N} \\ \frac{k_{v}}{p_{v}}(-k_{v}\varDelta \boldsymbol{\varLambda }_{T} + \mu \varDelta \sigma _{N}\boldsymbol{\tau }_{v} + \mu \sigma _{N}\varDelta \boldsymbol{\tau }_{v}) \\ 0 \\ \boldsymbol{\mathit{0}}\end{Bmatrix} & \|\boldsymbol{\sigma}_{T}\| \geq \mu \sigma _{N} \quad \text{Slip} \\ &\begin{Bmatrix} -g_{N\boldsymbol {q}}^{T}\varDelta \sigma _{N} - \boldsymbol {g}^{T}_{T \boldsymbol {q}}\varDelta \boldsymbol{\sigma}_{T} \\ -k_{v}\varDelta \mathring{g}_{N} \\ -k_{v}\varDelta \mathring{\boldsymbol{g}}_{T} \\ 0 \\ \boldsymbol{\mathit{0}}\end{Bmatrix} & \|\boldsymbol{\sigma}_{T}\| < \mu \sigma _{N} \quad \text{Stick} \end{aligned} \end{cases}\displaystyle \end{aligned}$$

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Hessian matrix for rolling resistance at velocity level for the different contact conditions,

$$\begin{aligned} \varDelta \boldsymbol {F}^{v}_{R}(\dot{\boldsymbol{\varPhi }}) = \textstyle\begin{cases} \begin{aligned} & \begin{Bmatrix} \boldsymbol{0} \\ 0 \\ \boldsymbol{0} \\ 0 \\ -\frac{k_{v}^{2}}{p_{v}}\varDelta \boldsymbol{\chi }_{T} \end{Bmatrix} &\sigma _{N} < 0\\ & \begin{Bmatrix} \boldsymbol{0} \\ - \boldsymbol{\omega }_{T\boldsymbol {q}}^{T}\varDelta \boldsymbol{\eta }_{T} \\ \boldsymbol{0} \\ 0 \\ -k_{v}\varDelta \boldsymbol{\omega }_{T} \end{Bmatrix} & \quad \|\boldsymbol{\eta }_{T}\|< \rho \sigma _{N} \\ & \begin{Bmatrix} -\rho \varDelta \sigma _{N}\boldsymbol{\omega }_{T\boldsymbol {q}}^{T} \boldsymbol{\tau }_{TW}-\rho \sigma _{N}\boldsymbol{\omega }_{T \boldsymbol {q}}^{T}\varDelta \boldsymbol{\tau }_{TW} \\ 0 \\ \boldsymbol{0} \\ 0 \\ \frac{k_{v}}{p_{v}}\left [-k_{v} \varDelta \boldsymbol{\chi }_{T} + \rho \varDelta \sigma _{N} \boldsymbol{\tau }_{TW}+ \rho \sigma _{N}\varDelta \boldsymbol{\tau }_{TW}\right ] \end{Bmatrix} & \quad \|\boldsymbol{\eta }_{T}\|\geq \rho \sigma _{N}\end{aligned} \end{cases}\displaystyle \end{aligned}$$

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Hessian matrix for spinning resistance at velocity level for the different contact conditions,

$$\begin{aligned} \varDelta \boldsymbol {F}^{v}_{D}= \textstyle\begin{cases} \begin{aligned} & \begin{Bmatrix} \boldsymbol{0} \\ 0 \\ \boldsymbol{0} \\ -\frac{k_{v}^{2}}{p_{v}}\varDelta \chi _{N} \\ \boldsymbol{0} \end{Bmatrix} &\sigma _{N} < 0\\ & \begin{Bmatrix} - \omega _{N\boldsymbol {q}}^{T} \varDelta \eta _{N} \\ 0 \\ \boldsymbol{0} \\ -k_{v}\varDelta \omega _{N} \\ \boldsymbol{0} \end{Bmatrix} & \qquad \|\eta _{T}\|< \gamma \sigma _{N} \\ & \begin{Bmatrix} -\gamma \varDelta \sigma _{N}\omega _{N\boldsymbol {q}}^{T} \text{sgn}(\eta _{N}) \\ 0 \\ \boldsymbol{0} \\ \frac{k_{v}}{p_{v}}\left [-k_{v}\varDelta \chi _{N} + \gamma \varDelta \sigma _{N}\text{sgn}(\eta _{N})\right ] \\ \boldsymbol{0} \end{Bmatrix} & \qquad \|\eta _{T}\|\geq \gamma \sigma _{N} \end{aligned} \end{cases}\displaystyle \end{aligned}$$

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