| 376 |
|
bead $i$ and origin $O$, the elements of resistance tensor at |
| 377 |
|
arbitrary origin $O$ can be written as |
| 378 |
|
\begin{eqnarray} |
| 379 |
+ |
\label{introEquation:ResistanceTensorArbitraryOrigin} |
| 380 |
|
\Xi _{}^{tt} & = & \sum\limits_i {\sum\limits_j {C_{ij} } } \notag , \\ |
| 381 |
|
\Xi _{}^{tr} & = & \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
| 382 |
|
\Xi _{}^{rr} & = & - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } |
| 383 |
|
U_j + 6 \eta V {\bf I}. \notag |
| 383 |
– |
\label{introEquation:ResistanceTensorArbitraryOrigin} |
| 384 |
|
\end{eqnarray} |
| 385 |
|
The final term in the expression for $\Xi^{rr}$ is correction that |
| 386 |
|
accounts for errors in the rotational motion of certain kinds of bead |
| 475 |
|
calculated in body-fixed frame and converted back to lab-fixed frame |
| 476 |
|
using the rigid body's rotation matrix ($Q$): |
| 477 |
|
\begin{equation} |
| 478 |
< |
\begin{array}{l} |
| 479 |
< |
\mathbf{F}_{f}(t) = Q^{T} \mathbf{F}_{f}^b (t), \\ |
| 480 |
< |
\mathbf{F}_{r}(t) = Q^{T} \mathbf{F}_{r}^b (t). \\ |
| 481 |
< |
\end{array} |
| 478 |
> |
\mathbf{F}_{f}(t) = \left( \begin{array}{l} |
| 479 |
> |
Q^{T} \mathbf{f}_{f}^b (t) \\ |
| 480 |
> |
Q^{T} \tau_{f}^b (t) \\ |
| 481 |
> |
\end{array} \right), \\ |
| 482 |
> |
\mathbf{F}_{r}(t) = \left( \begin{array}{l} |
| 483 |
> |
Q^{T} \mathbf{f}_{r}^b (t) \\ |
| 484 |
> |
Q^{T} \tau_{r}^b (t) \\ |
| 485 |
> |
\end{array} \right). |
| 486 |
|
\end{equation} |
| 487 |
|
Here, the body-fixed friction force $\mathbf{F}_{f}^b$ is proportional to |
| 488 |
|
the body-fixed velocity at the center of resistance $\mathbf{v}_{R}^b$ and |
| 516 |
|
|
| 517 |
|
The equation of motion for $\mathbf{v}$ can be written as |
| 518 |
|
\begin{equation} |
| 519 |
< |
m \dot{\mathbf{v}} (t) = \mathbf{f}_{s} (t) + \mathbf{f}_{f}^l (t) + |
| 519 |
> |
m \dot{\mathbf{v}} (t) = \mathbf{f}_{s}^l (t) + \mathbf{f}_{f}^l (t) + |
| 520 |
|
\mathbf{f}_{r}^l (t) |
| 521 |
|
\end{equation} |
| 522 |
|
Since the frictional force is applied at the center of resistance |
| 525 |
|
frictional torque at the center of mass, $\tau_{f}^b (t)$, is |
| 526 |
|
given by |
| 527 |
|
\begin{equation} |
| 528 |
< |
\tau_{f}^b \leftarrow \tau_{f}^b + \mathbf{r}_{MR} \times \mathbf{f}_{r}^b |
| 528 |
> |
\tau_{f}^b \leftarrow \tau_{f}^b + \mathbf{r}_{MR} \times \mathbf{f}_{f}^b |
| 529 |
|
\end{equation} |
| 530 |
|
where $r_{MR}$ is the vector from the center of mass to the center |
| 531 |
|
of the resistance. Instead of integrating the angular velocity in |
| 532 |
|
lab-fixed frame, we consider the equation of angular momentum in |
| 533 |
|
body-fixed frame |
| 534 |
|
\begin{equation} |
| 535 |
< |
\dot j(t) = \tau_{s} (t) + \tau_{f}^b (t) + \tau_{r}^b(t) |
| 535 |
> |
\dot j(t) = \tau_{s}^b (t) + \tau_{f}^b (t) + \tau_{r}^b(t) |
| 536 |
|
\end{equation} |
| 537 |
|
Embedding the friction terms into force and torque, one can integrate |
| 538 |
|
the Langevin equations of motion for rigid body of arbitrary shape in |
