| 861 |
|
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
| 862 |
|
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
| 863 |
|
\end{equation} |
| 864 |
< |
Careful choice of coefficient $a_1 \ldot b_m$ will lead to higher |
| 864 |
> |
Careful choice of coefficient $a_1 \ldots b_m$ will lead to higher |
| 865 |
|
order method. Yoshida proposed an elegant way to compose higher |
| 866 |
|
order methods based on symmetric splitting\cite{Yoshida1990}. Given |
| 867 |
|
a symmetric second order base method $ \varphi _h^{(2)} $, a |
| 914 |
|
\end{enumerate} |
| 915 |
|
These three individual steps will be covered in the following |
| 916 |
|
sections. Sec.~\ref{introSec:initialSystemSettings} deals with the |
| 917 |
< |
initialization of a simulation. Sec.~\ref{introSec:production} will |
| 918 |
< |
discusses issues in production run. Sec.~\ref{introSection:Analysis} |
| 919 |
< |
provides the theoretical tools for trajectory analysis. |
| 917 |
> |
initialization of a simulation. Sec.~\ref{introSection:production} |
| 918 |
> |
will discusses issues in production run. |
| 919 |
> |
Sec.~\ref{introSection:Analysis} provides the theoretical tools for |
| 920 |
> |
trajectory analysis. |
| 921 |
|
|
| 922 |
|
\subsection{\label{introSec:initialSystemSettings}Initialization} |
| 923 |
|
|
| 1257 |
|
Using Equation (\ref{introEquation:motionHamiltonianCoordinate}, |
| 1258 |
|
\ref{introEquation:motionHamiltonianMomentum}), one can write down |
| 1259 |
|
the equations of motion, |
| 1259 |
– |
\[ |
| 1260 |
– |
\begin{array}{c} |
| 1261 |
– |
\frac{{dq}}{{dt}} = \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
| 1262 |
– |
\frac{{dp}}{{dt}} = - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
| 1263 |
– |
\frac{{dQ}}{{dt}} = PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
| 1264 |
– |
\frac{{dP}}{{dt}} = - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP}\\ |
| 1265 |
– |
\end{array} |
| 1266 |
– |
\] |
| 1260 |
|
|
| 1261 |
+ |
\begin{eqnarray} |
| 1262 |
+ |
\frac{{dq}}{{dt}} & = & \frac{p}{m} \label{introEquation:RBMotionPosition}\\ |
| 1263 |
+ |
\frac{{dp}}{{dt}} & = & - \nabla _q V(q,Q) \label{introEquation:RBMotionMomentum}\\ |
| 1264 |
+ |
\frac{{dQ}}{{dt}} & = & PJ^{ - 1} \label{introEquation:RBMotionRotation}\\ |
| 1265 |
+ |
\frac{{dP}}{{dt}} & = & - \nabla _Q V(q,Q) - 2Q\Lambda . \label{introEquation:RBMotionP} |
| 1266 |
+ |
\end{eqnarray} |
| 1267 |
+ |
|
| 1268 |
|
In general, there are two ways to satisfy the holonomic constraints. |
| 1269 |
|
We can use constraint force provided by lagrange multiplier on the |
| 1270 |
|
normal manifold to keep the motion on constraint space. Or we can |
| 1344 |
|
\[ |
| 1345 |
|
\hat vu = v \times u |
| 1346 |
|
\] |
| 1347 |
– |
|
| 1347 |
|
Using \ref{introEqaution:RBMotionPI}, one can construct a skew |
| 1348 |
|
matrix, |
| 1349 |
|
\begin{equation} |
| 1350 |
< |
(\mathop \Pi \limits^ \bullet - \mathop \Pi \limits^ \bullet ^T |
| 1350 |
> |
(\mathop \Pi \limits^ \bullet - \mathop \Pi \limits^ {\bullet ^T} |
| 1351 |
|
){\rm{ }} = {\rm{ }}(\Pi - \Pi ^T ){\rm{ }}(J^{ - 1} \Pi + \Pi J^{ |
| 1352 |
|
- 1} ) + \sum\limits_i {[Q^T F_i (r,Q)X_i^T - X_i F_i (r,Q)^T Q]} - |
| 1353 |
|
(\Lambda - \Lambda ^T ) . \label{introEquation:skewMatrixPI} |
| 1376 |
|
|
| 1377 |
|
If there is not external forces exerted on the rigid body, the only |
| 1378 |
|
contribution to the rotational is from the kinetic potential (the |
| 1379 |
< |
first term of \ref{ introEquation:bodyAngularMotion}). The free |
| 1380 |
< |
rigid body is an example of Lie-Poisson system with Hamiltonian |
| 1382 |
< |
function |
| 1379 |
> |
first term of \ref{introEquation:bodyAngularMotion}). The free rigid |
| 1380 |
> |
body is an example of Lie-Poisson system with Hamiltonian function |
| 1381 |
|
\begin{equation} |
| 1382 |
|
T^r (\pi ) = T_1 ^r (\pi _1 ) + T_2^r (\pi _2 ) + T_3^r (\pi _3 ) |
| 1383 |
|
\label{introEquation:rotationalKineticRB} |
| 1654 |
|
\end{array} |
| 1655 |
|
\] |
| 1656 |
|
, we obtain |
| 1657 |
< |
\[ |
| 1658 |
< |
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - |
| 1657 |
> |
\begin{eqnarray*} |
| 1658 |
> |
m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - |
| 1659 |
|
\sum\limits_{\alpha = 1}^N {\left\{ {\left( { - \frac{{g_\alpha ^2 |
| 1660 |
|
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\int_0^t {\cos (\omega |
| 1661 |
< |
_\alpha t)\dot x(t - \tau )d\tau - \left[ {g_\alpha x_\alpha (0) |
| 1662 |
< |
- \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} \right]\cos |
| 1663 |
< |
(\omega _\alpha t) - \frac{{g_\alpha \dot x_\alpha (0)}}{{\omega |
| 1664 |
< |
_\alpha }}\sin (\omega _\alpha t)} } \right\}} |
| 1665 |
< |
\] |
| 1666 |
< |
\[ |
| 1667 |
< |
m\ddot x = - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
| 1661 |
> |
_\alpha t)\dot x(t - \tau )d\tau } } \right\}} \\ |
| 1662 |
> |
& & + \sum\limits_{\alpha = 1}^N {\left\{ {\left[ {g_\alpha |
| 1663 |
> |
x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} |
| 1664 |
> |
\right]\cos (\omega _\alpha t) + \frac{{g_\alpha \dot x_\alpha |
| 1665 |
> |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t)} \right\}} |
| 1666 |
> |
\end{eqnarray*} |
| 1667 |
> |
\begin{eqnarray*} |
| 1668 |
> |
m\ddot x & = & - \frac{{\partial W(x)}}{{\partial x}} - \int_0^t |
| 1669 |
|
{\sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
| 1670 |
|
}}{{m_\alpha \omega _\alpha ^2 }}} \right)\cos (\omega _\alpha |
| 1671 |
< |
t)\dot x(t - \tau )d} \tau } + \sum\limits_{\alpha = 1}^N {\left\{ |
| 1672 |
< |
{\left[ {g_\alpha x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha |
| 1673 |
< |
\omega _\alpha }}} \right]\cos (\omega _\alpha t) + |
| 1674 |
< |
\frac{{g_\alpha \dot x_\alpha (0)}}{{\omega _\alpha }}\sin |
| 1675 |
< |
(\omega _\alpha t)} \right\}} |
| 1676 |
< |
\] |
| 1678 |
< |
|
| 1671 |
> |
t)\dot x(t - \tau )d} \tau } \\ |
| 1672 |
> |
& & + \sum\limits_{\alpha = 1}^N {\left\{ {\left[ {g_\alpha |
| 1673 |
> |
x_\alpha (0) - \frac{{g_\alpha }}{{m_\alpha \omega _\alpha }}} |
| 1674 |
> |
\right]\cos (\omega _\alpha t) + \frac{{g_\alpha \dot x_\alpha |
| 1675 |
> |
(0)}}{{\omega _\alpha }}\sin (\omega _\alpha t)} \right\}} |
| 1676 |
> |
\end{eqnarray*} |
| 1677 |
|
Introducing a \emph{dynamic friction kernel} |
| 1678 |
|
\begin{equation} |
| 1679 |
|
\xi (t) = \sum\limits_{\alpha = 1}^N {\left( { - \frac{{g_\alpha ^2 |
| 2011 |
|
\begin{array}{l} |
| 2012 |
|
\Xi _P^{tt} = \Xi _O^{tt} \\ |
| 2013 |
|
\Xi _P^{tr} = \Xi _P^{rt} = \Xi _O^{tr} - U_{OP} \Xi _O^{tt} \\ |
| 2014 |
< |
\Xi _P^{rr} = \Xi _O^{rr} - U_{OP} \Xi _O^{tt} U_{OP} + \Xi _O^{tr} U_{OP} - U_{OP} \Xi _O^{tr} ^{^T } \\ |
| 2014 |
> |
\Xi _P^{rr} = \Xi _O^{rr} - U_{OP} \Xi _O^{tt} U_{OP} + \Xi _O^{tr} U_{OP} - U_{OP} \Xi _O^{{tr} ^{^T }} \\ |
| 2015 |
|
\end{array} |
| 2016 |
|
\label{introEquation:resistanceTensorTransformation} |
| 2017 |
|
\end{equation} |
