A novel nonsmooth approach for flexible multibody systems with contact and friction in 3D space

Abstract

For computational multibody system dynamics, contact and friction problems are very complex and important problems. Therefore, this paper proposes a novel nonsmooth method for flexible multibody systems with contact and friction in 3D space. Considering the nonsmooth effect of contact and friction on the state variable of the multibody systems, the proposed method is divided into two parts: (i) the change of state variables under the action of smooth force and (ii) the change of state variables caused by contact and friction. Furthermore, for the contact part, the Newton’s impact law and the classical Coulomb friction model are employed to deal with normal impact contact impulse and tangential friction contact impulse, respectively. In addition, Fischer–Burmeister function and Karush–Kuhn–Tucker conditions are also used to solve the normal impact contact impulse and tangential friction contact impulse. What’s more, since the symplectic discrete format performs well robustness to numerical results, the discrete format of the proposed method is based on the symplectic discreteness, and it is expected to obtain the robust numerical results. Finally, several numerical examples are tested by the proposed nonsmooth method, and the numerical simulation results verify the effectiveness of the proposed approach.

Appendix: Symplectic discrete equations of the multibody system with contact free

According to Eqs. (54) and (55), when the state variables are smooth and continuous, the governing equations are:

$$ {\mathbf{M}}\left( {\mathbf{q}} \right){\mathbf{\dot{\tilde{v}}}} + {{\varvec{\Phi}}}\left( {\mathbf{q}} \right)_{{\mathbf{q}}}^{{\mathbf{T}}} {{\varvec{\uplambda}}}_{b} = {\mathbf{F}}\left( {{\mathbf{q}},{\mathbf{v}},t} \right) $$

(A-1)

$$ {{\varvec{\Phi}}}\left( {\mathbf{q}} \right) = {\mathbf{0}}. $$

(A-2)

In accordance with D’Alembert’s principle, the variational formulation $\delta {\text{S}}$ of the action S in time interval $\left[ {t_{n} ,t_{n + 1} } \right]$ can be demonstrated as:

$$ \delta {\text{S}}{ = }\delta \int_{{t_{n} }}^{{t_{n + 1} }} {\left( {\frac{1}{2}{\mathbf{v}}^{{\mathbf{T}}} {\mathbf{M}}\left( {\mathbf{q}} \right){\mathbf{v}} - {{\varvec{\Phi}}}\left( {\mathbf{q}} \right)^{{\mathbf{T}}} {{\varvec{\uplambda}}}_{b} - {\mathbf{U}}\left( {\mathbf{q}} \right)} \right){\text{d}}t} - \int_{{t_{n} }}^{{t_{n + 1} }} {\delta {\mathbf{q}}^{{\mathbf{T}}} {\mathbf{Q}}\left( {{\mathbf{q}},{\mathbf{v}},t} \right){\text{d}}t}. $$

(A-3)

By the variation of q and v using Eq. (A-3), we have:

$$ \delta {\text{S}} = \delta {\mathbf{q}}^{{\mathbf{T}}} {\mathbf{p}}|_{{t_{n} }}^{{t_{n + 1} }} - \int_{{t_{n} }}^{{t_{n + 1} }} {\delta {\mathbf{q}}\left( {{\mathbf{M}}\left( {\mathbf{q}} \right){\mathbf{\dot{\tilde{v}}}} - \frac{1}{2}{\mathbf{v}}^{{\mathbf{T}}} \frac{{\partial {\mathbf{M}}\left( {\mathbf{q}} \right)}}{{\partial {\mathbf{q}}}}{\mathbf{v}} + {{\varvec{\Phi}}}\left( {\mathbf{q}} \right)_{{\mathbf{q}}}^{{\mathbf{T}}} {{\varvec{\uplambda}}}_{b} + \frac{{\partial {\text{U}}\left( {\mathbf{q}} \right)}}{{\partial {\mathbf{q}}}} + {\mathbf{Q}}\left( {{\mathbf{q}},{\mathbf{v}},t} \right)} \right){\text{d}}t} $$

(A-4)

where in the integral term of Eq. (A-4), the terms of $\frac{1}{2}{\mathbf{v}}^{{\mathbf{T}}} \frac{{\partial {\mathbf{M}}\left( {\mathbf{q}} \right)}}{{\partial {\mathbf{q}}}}{\mathbf{v}}$, $\frac{{\partial {\text{U}}\left( {\mathbf{q}} \right)}}{{\partial {\mathbf{q}}}}$ and ${\mathbf{Q}}\left( {{\mathbf{q}},{\mathbf{v}},t} \right)$ represent the inertia force and external force, which are included in ${\mathbf{F}}\left( {{\mathbf{q}},{\mathbf{v}},t} \right)$, so Eq. (A-4) can be expressed as:

$$ \delta {\text{S}} = \delta {\mathbf{q}}^{{\mathbf{T}}} {\mathbf{p}}|_{{t_{n} }}^{{t_{n + 1} }} - \int_{{t_{n} }}^{{t_{n + 1} }} {\delta {\mathbf{q}}\left( {{\mathbf{M}}\left( {\mathbf{q}} \right){\mathbf{\dot{\tilde{v}}}} + {{\varvec{\Phi}}}\left( {\mathbf{q}} \right)_{{\mathbf{q}}}^{{\mathbf{T}}} {{\varvec{\uplambda}}}_{b} - {\mathbf{F}}\left( {{\mathbf{q}},{\mathbf{v}},t} \right)} \right){\text{d}}t}. $$

(A-5)

Due to Eq. (A-1) and Eq. (A-5), we have:

$$ \delta {\text{S } = \text{ }}\delta {\tilde{\mathbf{q}}}_{n + 1}^{{\mathbf{T}}} {\mathbf{p}}_{n + 1} - \delta {\mathbf{q}}_{n}^{{\mathbf{T}}} {\mathbf{p}}_{n}. $$

(A-6)

In this paper, we use center interpolation to approximate the state variables within the integration interval; then, Eq. (A-3) can be replaced as:

$$ \delta {\text{S}} \approx \delta \int_{{t_{n} }}^{{t_{n + 1} }} {\left( {\frac{1}{2}{\hat{\mathbf{v}}}^{{\text{T}}} {\mathbf{M}}\left( {{\hat{\mathbf{q}}}} \right){\hat{\mathbf{v}}} - {{\varvec{\Phi}}}\left( {{\hat{\mathbf{q}}}} \right)^{{\mathbf{T}}} {{\varvec{\uplambda}}}_{b} - {\mathbf{U}}\left( {{\hat{\mathbf{q}}}} \right)} \right){\text{d}}t} - \int_{{t_{n} }}^{{t_{n + 1} }} {\delta {\hat{\mathbf{q}}}^{{\text{T}}} {\mathbf{Q}}\left( {{\hat{\mathbf{q}}},{\hat{\mathbf{v}}},t} \right){\text{d}}t} $$

(A-7)

where in Eq. (A-7), ${\hat{\mathbf{v}}} = \left( {{\tilde{\mathbf{q}}}_{n + 1} - {\mathbf{q}}_{n} } \right)/h$, ${\hat{\mathbf{q}}} = \left( {{\tilde{\mathbf{q}}}_{n + 1} + {\mathbf{q}}_{n} } \right)/2$, $h = t_{n + 1} - t_{n}$.

Then, according to Eq. (A-6), we have:

$$ - {\mathbf{p}}_{n} = \int_{{t_{n} }}^{{t_{n + 1} }} {\frac{{\partial {\text{S}}}}{{\partial {\mathbf{q}}_{n} }}} {\text{d}}t = - {\mathbf{M}}\left( {{\hat{\mathbf{q}}}} \right){\hat{\mathbf{v}}} + \frac{1}{4}h{\hat{\mathbf{v}}}^{{\mathbf{T}}} \frac{{\partial {\mathbf{M}}\left( {{\hat{\mathbf{q}}}} \right)}}{{\partial {\hat{\mathbf{q}}}}}{\hat{\mathbf{v}}} - \frac{1}{2}h{{\varvec{\Phi}}}\left( {{\hat{\mathbf{q}}}} \right)_{{\mathbf{q}}}^{{\mathbf{T}}} {{\varvec{\uplambda}}}_{b} - \frac{1}{2}h\frac{{\partial {\text{U}}\left( {{\hat{\mathbf{q}}}} \right)}}{{\partial {\hat{\mathbf{q}}}}} + \frac{1}{2}h{\mathbf{Q}}\left( {{\hat{\mathbf{q}}},{\hat{\mathbf{v}}},t} \right) $$

(A-8)

$$ {\mathbf{p}}_{n + 1} = \int_{{t_{n} }}^{{t_{n + 1} }} {\frac{{\partial {\text{S}}}}{{\partial {\mathbf{q}}_{n} }}} {\text{d}}t = {\mathbf{M}}\left( {{\hat{\mathbf{q}}}} \right){\hat{\mathbf{v}}} + \frac{1}{4}h{\hat{\mathbf{v}}}^{{\mathbf{T}}} \frac{{\partial {\mathbf{M}}\left( {{\hat{\mathbf{q}}}} \right)}}{{\partial {\hat{\mathbf{q}}}}}{\hat{\mathbf{v}}} + \frac{1}{2}h{{\varvec{\Phi}}}\left( {{\hat{\mathbf{q}}}} \right)_{{\mathbf{q}}}^{{\mathbf{T}}} {{\varvec{\uplambda}}}_{b} - \frac{1}{2}h\frac{{\partial {\text{U}}\left( {{\hat{\mathbf{q}}}} \right)}}{{\partial {\hat{\mathbf{q}}}}} + \frac{1}{2}h{\mathbf{Q}}\left( {{\hat{\mathbf{q}}},{\hat{\mathbf{v}}},t} \right) $$

(A-9)

where in Eqs. (A-8) and (A-9), the terms $\frac{1}{4}{\hat{\mathbf{v}}}^{{\mathbf{T}}} \frac{{\partial {\mathbf{M}}\left( {{\hat{\mathbf{q}}}} \right)}}{{\partial {\hat{\mathbf{q}}}}}{\hat{\mathbf{v}}}$, $\frac{1}{2}\frac{{\partial {\text{U}}\left( {{\hat{\mathbf{q}}}} \right)}}{{\partial {\hat{\mathbf{q}}}}}$ and $\frac{1}{2}{\mathbf{Q}}\left( {{\hat{\mathbf{q}}},{\hat{\mathbf{v}}},t} \right)$ are included in ${\mathbf{F}}\left( {{\hat{\mathbf{q}}},{\hat{\mathbf{v}}},t} \right)$, so Eqs. (62) and (63) can be obtained using Eqs. (A-8) and (A-9):

$$ {\mathbf{p}}_{n} - {\mathbf{M}}\left( {{\hat{\mathbf{q}}}} \right){\hat{\mathbf{v}}} - \frac{1}{2}h{{\varvec{\Phi}}}\left( {{\hat{\mathbf{q}}}} \right)_{{\mathbf{q}}}^{{\mathbf{T}}} {{\varvec{\uplambda}}}_{b} + \frac{1}{2}h{\mathbf{F}}\left( {{\hat{\mathbf{q}}},{\hat{\mathbf{v}}},t} \right) = {\mathbf{0}} $$

(A-10)

$$ {\mathbf{p}}_{n + 1} - {\mathbf{M}}\left( {{\hat{\mathbf{q}}}} \right){\hat{\mathbf{v}}} + \frac{1}{2}h{{\varvec{\Phi}}}\left( {{\hat{\mathbf{q}}}} \right)_{{\mathbf{q}}}^{{\mathbf{T}}} {{\varvec{\uplambda}}}_{b} - \frac{1}{2}h{\mathbf{F}}\left( {{\hat{\mathbf{q}}},{\hat{\mathbf{v}}},t} \right) = {\mathbf{0}}. $$

(A-11)

Proposition 8

Define the state variables at $t_{n}$ and $t_{n + 1}$ as $\pi_{n} = \left[ {{\mathbf{q}}_{n} ,{\mathbf{p}}_{n} } \right]$ and $\pi_{n + 1} = \left[ {{\tilde{\mathbf{q}}}_{n + 1} ,{\mathbf{p}}_{n + 1} } \right]$ , respectively; then, the above transfer process from $\pi_{n}$ to $\pi_{n + 1}$ satisfies the symplectic transfer.

Proof

According to [48,49,50], if ${\mathbf{S}}_{{\mathbf{s}}}$ is the transfer matrix from $\pi_{n} = \left[ {{\mathbf{q}}_{n} ,{\mathbf{p}}_{n} } \right]$ to $\pi_{n + 1} = \left[ {{\tilde{\mathbf{q}}}_{n + 1} ,{\mathbf{p}}_{n + 1} } \right]$, and ${\mathbf{S}}_{{\mathbf{s}}}$ satisfies the following relationship, then the transfer process is symplectic:

$$ {\mathbf{S}}_{{\mathbf{s}}}^{{\mathbf{T}}} J{\mathbf{S}}_{{\mathbf{s}}}^{{\mathbf{T}}} = J $$

(A-12)

where

$$ J = \left[ {\begin{array}{*{20}c} {\mathbf{0}} & {\mathbf{I}} \\ { - {\mathbf{I}}} & {\mathbf{0}} \\ \end{array} } \right] $$

(A-13)

Define $f_{1}$ is equal to Eq. (A-10), $f_{2}$ is equal to Eq. (A-11) and $f = \left[ {f_{1} ,f_{2} } \right]^{{\mathbf{T}}}$. As ${\mathbf{S}}_{{\mathbf{s}}}$ is the transfer matrix of $\pi_{n}$ and $\pi_{n + 1}$, the ${\mathbf{S}}_{{\mathbf{s}}}$ has the following formula:

$$ {\mathbf{S}}_{{\mathbf{s}}} = \frac{{\partial \pi_{n + 1} }}{{\partial \pi_{n} }} = \left( {\frac{\partial f}{{\partial \pi_{n + 1} }}} \right)^{ - 1} \frac{\partial f}{{\partial \pi_{n} }}. $$

(A-14)

Then, according to Eqs. (A-10), (A-11) and (A-14), we have:

$$ \left\{ {\begin{array}{*{20}l} {\frac{{\partial f_{1} }}{{\partial {\mathbf{q}}_{n} }} = \frac{1}{h}{\mathbf{M}}\left( {{\hat{\mathbf{q}}}} \right) - \frac{{\partial {\mathbf{M}}\left( {{\hat{\mathbf{q}}}} \right)}}{{\partial {\hat{\mathbf{q}}}}}{\hat{\mathbf{v}}} + \frac{1}{8}h{\hat{\mathbf{v}}}^{{\mathbf{T}}} \frac{{\partial^{2} {\mathbf{M}}\left( {{\hat{\mathbf{q}}}} \right)}}{{\partial {\hat{\mathbf{q}}}^{2} }}{\hat{\mathbf{v}}} - \frac{1}{4}h\left( {{{\varvec{\Phi}}}\left( {{\hat{\mathbf{q}}}} \right)_{{\mathbf{q}}}^{{\mathbf{T}}} {{\varvec{\uplambda}}}_{b} } \right)_{{{\hat{\mathbf{q}}}}} + \frac{1}{4}h\frac{{\partial {\mathbf{F}}\left( {{\hat{\mathbf{q}}},{\hat{\mathbf{v}}},t} \right)}}{{\partial {\hat{\mathbf{q}}}}}} \hfill \\ {\frac{{\partial f_{1} }}{{\partial {\tilde{\mathbf{q}}}_{n + 1} }} = - \frac{1}{h}{\mathbf{M}}\left( {{\hat{\mathbf{q}}}} \right) + \frac{1}{8}h{\hat{\mathbf{v}}}^{{\mathbf{T}}} \frac{{\partial^{2} {\mathbf{M}}\left( {{\hat{\mathbf{q}}}} \right)}}{{\partial {\hat{\mathbf{q}}}^{2} }}{\hat{\mathbf{v}}} - \frac{1}{4}h\left( {{{\varvec{\Phi}}}\left( {{\hat{\mathbf{q}}}} \right)_{{\mathbf{q}}}^{{\mathbf{T}}} {{\varvec{\uplambda}}}_{b} } \right)_{{{\hat{\mathbf{q}}}}} + \frac{1}{4}h\frac{{\partial {\mathbf{F}}\left( {{\hat{\mathbf{q}}},{\hat{\mathbf{v}}},t} \right)}}{{\partial {\hat{\mathbf{q}}}}}} \hfill \\ {\frac{{\partial f_{1} }}{{\partial {\mathbf{p}}_{n} }} = {\mathbf{I}}} \hfill \\ {\frac{{\partial f_{1} }}{{\partial {\mathbf{p}}_{n + 1} }} = {\mathbf{0}}} \hfill \\ {\frac{{\partial f_{2} }}{{\partial {\mathbf{q}}_{n} }} = \frac{1}{h}{\mathbf{M}}\left( {{\hat{\mathbf{q}}}} \right) - \frac{1}{8}h{\hat{\mathbf{v}}}^{{\mathbf{T}}} \frac{{\partial^{2} {\mathbf{M}}\left( {{\hat{\mathbf{q}}}} \right)}}{{\partial {\hat{\mathbf{q}}}^{2} }}{\hat{\mathbf{v}}} + \frac{1}{4}h\left( {{{\varvec{\Phi}}}\left( {{\hat{\mathbf{q}}}} \right)_{{\mathbf{q}}}^{{\mathbf{T}}} {{\varvec{\uplambda}}}_{b} } \right)_{{{\hat{\mathbf{q}}}}} - \frac{1}{4}h\frac{{\partial {\mathbf{F}}\left( {{\hat{\mathbf{q}}},{\hat{\mathbf{v}}},t} \right)}}{{\partial {\hat{\mathbf{q}}}}}} \hfill \\ {\frac{{\partial f_{2} }}{{\partial {\tilde{\mathbf{q}}}_{n + 1} }} = - \frac{1}{h}{\mathbf{M}}\left( {{\hat{\mathbf{q}}}} \right) - \frac{{\partial {\mathbf{M}}\left( {{\hat{\mathbf{q}}}} \right)}}{{\partial {\hat{\mathbf{q}}}}}{\hat{\mathbf{v}}} - \frac{1}{8}h{\hat{\mathbf{v}}}^{{\mathbf{T}}} \frac{{\partial^{2} {\mathbf{M}}\left( {{\hat{\mathbf{q}}}} \right)}}{{\partial {\hat{\mathbf{q}}}^{2} }}{\hat{\mathbf{v}}} + \frac{1}{4}h\left( {{{\varvec{\Phi}}}\left( {{\hat{\mathbf{q}}}} \right)_{{\mathbf{q}}}^{{\mathbf{T}}} {{\varvec{\uplambda}}}_{b} } \right)_{{{\hat{\mathbf{q}}}}} - \frac{1}{4}h\frac{{\partial {\mathbf{F}}\left( {{\hat{\mathbf{q}}},{\hat{\mathbf{v}}},t} \right)}}{{\partial {\hat{\mathbf{q}}}}}} \hfill \\ {\frac{{\partial f_{2} }}{{\partial {\mathbf{p}}_{n} }} = {\mathbf{0}}} \hfill \\ {\frac{{\partial f_{2} }}{{\partial {\mathbf{p}}_{n + 1} }} = {\mathbf{I}}} \hfill \\ \end{array} } \right. $$

(A-15)

According to Eqs. (A-14) and (A-15), one can have the transfer matrix ${\mathbf{S}}_{{\mathbf{s}}}$ and the transfer matrix ${\mathbf{S}}_{{\mathbf{s}}}$ satisfies Eq. (A-12); therefore, the transfer process from $\pi_{n}$ to $\pi_{n + 1}$ is symplectic.□