| 113 |
|
\subsubsection{\label{introSection:equationOfMotionLagrangian}\textbf{The |
| 114 |
|
Equations of Motion in Lagrangian Mechanics}} |
| 115 |
|
|
| 116 |
< |
For a holonomic system of $f$ degrees of freedom, the equations of |
| 117 |
< |
motion in the Lagrangian form is |
| 116 |
> |
For a system of $f$ degrees of freedom, the equations of motion in |
| 117 |
> |
the Lagrangian form is |
| 118 |
|
\begin{equation} |
| 119 |
|
\frac{d}{{dt}}\frac{{\partial L}}{{\partial \dot q_i }} - |
| 120 |
|
\frac{{\partial L}}{{\partial q_i }} = 0,{\rm{ }}i = 1, \ldots,f |
| 231 |
|
,q_f ,p_1 , \ldots ,p_f )$, with a unique set of values of $6f$ |
| 232 |
|
coordinates and momenta is a phase space vector. |
| 233 |
|
|
| 234 |
+ |
%%%fix me |
| 235 |
|
A microscopic state or microstate of a classical system is |
| 236 |
|
specification of the complete phase space vector of a system at any |
| 237 |
|
instant in time. An ensemble is defined as a collection of systems |
| 283 |
|
{\rho (q,p,t)dq_1 } ...dq_f dp_1 } ...dp_f }}. |
| 284 |
|
\label{introEquation:unitProbability} |
| 285 |
|
\end{equation} |
| 286 |
< |
With the help of Equation(\ref{introEquation:unitProbability}) and |
| 287 |
< |
the knowledge of the system, it is possible to calculate the average |
| 286 |
> |
With the help of Eq.~\ref{introEquation:unitProbability} and the |
| 287 |
> |
knowledge of the system, it is possible to calculate the average |
| 288 |
|
value of any desired quantity which depends on the coordinates and |
| 289 |
|
momenta of the system. Even when the dynamics of the real system is |
| 290 |
|
complex, or stochastic, or even discontinuous, the average |
| 307 |
|
thermodynamic equilibrium. |
| 308 |
|
|
| 309 |
|
As an ensemble of systems, each of which is known to be thermally |
| 310 |
< |
isolated and conserve energy, the Microcanonical ensemble(NVE) has a |
| 311 |
< |
partition function like, |
| 310 |
> |
isolated and conserve energy, the Microcanonical ensemble (NVE) has |
| 311 |
> |
a partition function like, |
| 312 |
|
\begin{equation} |
| 313 |
|
\Omega (N,V,E) = e^{\beta TS} \label{introEquation:NVEPartition}. |
| 314 |
|
\end{equation} |
| 315 |
< |
A canonical ensemble(NVT)is an ensemble of systems, each of which |
| 315 |
> |
A canonical ensemble (NVT)is an ensemble of systems, each of which |
| 316 |
|
can share its energy with a large heat reservoir. The distribution |
| 317 |
|
of the total energy amongst the possible dynamical states is given |
| 318 |
|
by the partition function, |
| 322 |
|
\end{equation} |
| 323 |
|
Here, $A$ is the Helmholtz free energy which is defined as $ A = U - |
| 324 |
|
TS$. Since most experiments are carried out under constant pressure |
| 325 |
< |
condition, the isothermal-isobaric ensemble(NPT) plays a very |
| 325 |
> |
condition, the isothermal-isobaric ensemble (NPT) plays a very |
| 326 |
|
important role in molecular simulations. The isothermal-isobaric |
| 327 |
|
ensemble allow the system to exchange energy with a heat bath of |
| 328 |
|
temperature $T$ and to change the volume as well. Its partition |
| 338 |
|
Liouville's theorem is the foundation on which statistical mechanics |
| 339 |
|
rests. It describes the time evolution of the phase space |
| 340 |
|
distribution function. In order to calculate the rate of change of |
| 341 |
< |
$\rho$, we begin from Equation(\ref{introEquation:deltaN}). If we |
| 342 |
< |
consider the two faces perpendicular to the $q_1$ axis, which are |
| 343 |
< |
located at $q_1$ and $q_1 + \delta q_1$, the number of phase points |
| 344 |
< |
leaving the opposite face is given by the expression, |
| 341 |
> |
$\rho$, we begin from Eq.~\ref{introEquation:deltaN}. If we consider |
| 342 |
> |
the two faces perpendicular to the $q_1$ axis, which are located at |
| 343 |
> |
$q_1$ and $q_1 + \delta q_1$, the number of phase points leaving the |
| 344 |
> |
opposite face is given by the expression, |
| 345 |
|
\begin{equation} |
| 346 |
|
\left( {\rho + \frac{{\partial \rho }}{{\partial q_1 }}\delta q_1 } |
| 347 |
|
\right)\left( {\dot q_1 + \frac{{\partial \dot q_1 }}{{\partial q_1 |
| 377 |
|
|
| 378 |
|
Liouville's theorem states that the distribution function is |
| 379 |
|
constant along any trajectory in phase space. In classical |
| 380 |
< |
statistical mechanics, since the number of particles in the system |
| 381 |
< |
is huge, we may be able to believe the system is stationary, |
| 380 |
> |
statistical mechanics, since the number of members in an ensemble is |
| 381 |
> |
huge and constant, we can assume the local density has no reason |
| 382 |
> |
(other than classical mechanics) to change, |
| 383 |
|
\begin{equation} |
| 384 |
|
\frac{{\partial \rho }}{{\partial t}} = 0. |
| 385 |
|
\label{introEquation:stationary} |
| 432 |
|
\label{introEquation:poissonBracket} |
| 433 |
|
\end{equation} |
| 434 |
|
Substituting equations of motion in Hamiltonian formalism( |
| 435 |
< |
\ref{introEquation:motionHamiltonianCoordinate} , |
| 436 |
< |
\ref{introEquation:motionHamiltonianMomentum} ) into |
| 437 |
< |
(\ref{introEquation:liouvilleTheorem}), we can rewrite Liouville's |
| 438 |
< |
theorem using Poisson bracket notion, |
| 435 |
> |
Eq.~\ref{introEquation:motionHamiltonianCoordinate} , |
| 436 |
> |
Eq.~\ref{introEquation:motionHamiltonianMomentum} ) into |
| 437 |
> |
(Eq.~\ref{introEquation:liouvilleTheorem}), we can rewrite |
| 438 |
> |
Liouville's theorem using Poisson bracket notion, |
| 439 |
|
\begin{equation} |
| 440 |
|
\left( {\frac{{\partial \rho }}{{\partial t}}} \right) = - \left\{ |
| 441 |
|
{\rho ,H} \right\}. |
| 1737 |
|
\end{equation} |
| 1738 |
|
which is known as the Langevin equation. The static friction |
| 1739 |
|
coefficient $\xi _0$ can either be calculated from spectral density |
| 1740 |
< |
or be determined by Stokes' law for regular shaped particles.A |
| 1740 |
> |
or be determined by Stokes' law for regular shaped particles. A |
| 1741 |
|
briefly review on calculating friction tensor for arbitrary shaped |
| 1742 |
|
particles is given in Sec.~\ref{introSection:frictionTensor}. |
| 1743 |
|
|
| 1770 |
|
\end{equation} |
| 1771 |
|
In effect, it acts as a constraint on the possible ways in which one |
| 1772 |
|
can model the random force and friction kernel. |
| 1771 |
– |
|
| 1772 |
– |
\subsection{\label{introSection:frictionTensor} Friction Tensor} |
| 1773 |
– |
Theoretically, the friction kernel can be determined using velocity |
| 1774 |
– |
autocorrelation function. However, this approach become impractical |
| 1775 |
– |
when the system become more and more complicate. Instead, various |
| 1776 |
– |
approaches based on hydrodynamics have been developed to calculate |
| 1777 |
– |
the friction coefficients. The friction effect is isotropic in |
| 1778 |
– |
Equation, $\zeta$ can be taken as a scalar. In general, friction |
| 1779 |
– |
tensor $\Xi$ is a $6\times 6$ matrix given by |
| 1780 |
– |
\[ |
| 1781 |
– |
\Xi = \left( {\begin{array}{*{20}c} |
| 1782 |
– |
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
| 1783 |
– |
{\Xi _{}^{tr} } & {\Xi _{}^{rr} } \\ |
| 1784 |
– |
\end{array}} \right). |
| 1785 |
– |
\] |
| 1786 |
– |
Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are translational friction |
| 1787 |
– |
tensor and rotational resistance (friction) tensor respectively, |
| 1788 |
– |
while ${\Xi^{tr} }$ is translation-rotation coupling tensor and $ |
| 1789 |
– |
{\Xi^{rt} }$ is rotation-translation coupling tensor. When a |
| 1790 |
– |
particle moves in a fluid, it may experience friction force or |
| 1791 |
– |
torque along the opposite direction of the velocity or angular |
| 1792 |
– |
velocity, |
| 1793 |
– |
\[ |
| 1794 |
– |
\left( \begin{array}{l} |
| 1795 |
– |
F_R \\ |
| 1796 |
– |
\tau _R \\ |
| 1797 |
– |
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
| 1798 |
– |
{\Xi ^{tt} } & {\Xi ^{rt} } \\ |
| 1799 |
– |
{\Xi ^{tr} } & {\Xi ^{rr} } \\ |
| 1800 |
– |
\end{array}} \right)\left( \begin{array}{l} |
| 1801 |
– |
v \\ |
| 1802 |
– |
w \\ |
| 1803 |
– |
\end{array} \right) |
| 1804 |
– |
\] |
| 1805 |
– |
where $F_r$ is the friction force and $\tau _R$ is the friction |
| 1806 |
– |
toque. |
| 1807 |
– |
|
| 1808 |
– |
\subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shape}} |
| 1809 |
– |
|
| 1810 |
– |
For a spherical particle, the translational and rotational friction |
| 1811 |
– |
constant can be calculated from Stoke's law, |
| 1812 |
– |
\[ |
| 1813 |
– |
\Xi ^{tt} = \left( {\begin{array}{*{20}c} |
| 1814 |
– |
{6\pi \eta R} & 0 & 0 \\ |
| 1815 |
– |
0 & {6\pi \eta R} & 0 \\ |
| 1816 |
– |
0 & 0 & {6\pi \eta R} \\ |
| 1817 |
– |
\end{array}} \right) |
| 1818 |
– |
\] |
| 1819 |
– |
and |
| 1820 |
– |
\[ |
| 1821 |
– |
\Xi ^{rr} = \left( {\begin{array}{*{20}c} |
| 1822 |
– |
{8\pi \eta R^3 } & 0 & 0 \\ |
| 1823 |
– |
0 & {8\pi \eta R^3 } & 0 \\ |
| 1824 |
– |
0 & 0 & {8\pi \eta R^3 } \\ |
| 1825 |
– |
\end{array}} \right) |
| 1826 |
– |
\] |
| 1827 |
– |
where $\eta$ is the viscosity of the solvent and $R$ is the |
| 1828 |
– |
hydrodynamics radius. |
| 1829 |
– |
|
| 1830 |
– |
Other non-spherical shape, such as cylinder and ellipsoid |
| 1831 |
– |
\textit{etc}, are widely used as reference for developing new |
| 1832 |
– |
hydrodynamics theory, because their properties can be calculated |
| 1833 |
– |
exactly. In 1936, Perrin extended Stokes's law to general ellipsoid, |
| 1834 |
– |
also called a triaxial ellipsoid, which is given in Cartesian |
| 1835 |
– |
coordinates by\cite{Perrin1934, Perrin1936} |
| 1836 |
– |
\[ |
| 1837 |
– |
\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
| 1838 |
– |
}} = 1 |
| 1839 |
– |
\] |
| 1840 |
– |
where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately, |
| 1841 |
– |
due to the complexity of the elliptic integral, only the ellipsoid |
| 1842 |
– |
with the restriction of two axes having to be equal, \textit{i.e.} |
| 1843 |
– |
prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved |
| 1844 |
– |
exactly. Introducing an elliptic integral parameter $S$ for prolate, |
| 1845 |
– |
\[ |
| 1846 |
– |
S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
| 1847 |
– |
} }}{b}, |
| 1848 |
– |
\] |
| 1849 |
– |
and oblate, |
| 1850 |
– |
\[ |
| 1851 |
– |
S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
| 1852 |
– |
}}{a} |
| 1853 |
– |
\], |
| 1854 |
– |
one can write down the translational and rotational resistance |
| 1855 |
– |
tensors |
| 1856 |
– |
\[ |
| 1857 |
– |
\begin{array}{l} |
| 1858 |
– |
\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ |
| 1859 |
– |
\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ |
| 1860 |
– |
\end{array}, |
| 1861 |
– |
\] |
| 1862 |
– |
and |
| 1863 |
– |
\[ |
| 1864 |
– |
\begin{array}{l} |
| 1865 |
– |
\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ |
| 1866 |
– |
\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
| 1867 |
– |
\end{array}. |
| 1868 |
– |
\] |
| 1869 |
– |
|
| 1870 |
– |
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shape}} |
| 1871 |
– |
|
| 1872 |
– |
Unlike spherical and other regular shaped molecules, there is not |
| 1873 |
– |
analytical solution for friction tensor of any arbitrary shaped |
| 1874 |
– |
rigid molecules. The ellipsoid of revolution model and general |
| 1875 |
– |
triaxial ellipsoid model have been used to approximate the |
| 1876 |
– |
hydrodynamic properties of rigid bodies. However, since the mapping |
| 1877 |
– |
from all possible ellipsoidal space, $r$-space, to all possible |
| 1878 |
– |
combination of rotational diffusion coefficients, $D$-space is not |
| 1879 |
– |
unique\cite{Wegener1979} as well as the intrinsic coupling between |
| 1880 |
– |
translational and rotational motion of rigid body, general ellipsoid |
| 1881 |
– |
is not always suitable for modeling arbitrarily shaped rigid |
| 1882 |
– |
molecule. A number of studies have been devoted to determine the |
| 1883 |
– |
friction tensor for irregularly shaped rigid bodies using more |
| 1884 |
– |
advanced method where the molecule of interest was modeled by |
| 1885 |
– |
combinations of spheres(beads)\cite{Carrasco1999} and the |
| 1886 |
– |
hydrodynamics properties of the molecule can be calculated using the |
| 1887 |
– |
hydrodynamic interaction tensor. Let us consider a rigid assembly of |
| 1888 |
– |
$N$ beads immersed in a continuous medium. Due to hydrodynamics |
| 1889 |
– |
interaction, the ``net'' velocity of $i$th bead, $v'_i$ is different |
| 1890 |
– |
than its unperturbed velocity $v_i$, |
| 1891 |
– |
\[ |
| 1892 |
– |
v'_i = v_i - \sum\limits_{j \ne i} {T_{ij} F_j } |
| 1893 |
– |
\] |
| 1894 |
– |
where $F_i$ is the frictional force, and $T_{ij}$ is the |
| 1895 |
– |
hydrodynamic interaction tensor. The friction force of $i$th bead is |
| 1896 |
– |
proportional to its ``net'' velocity |
| 1897 |
– |
\begin{equation} |
| 1898 |
– |
F_i = \zeta _i v_i - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }. |
| 1899 |
– |
\label{introEquation:tensorExpression} |
| 1900 |
– |
\end{equation} |
| 1901 |
– |
This equation is the basis for deriving the hydrodynamic tensor. In |
| 1902 |
– |
1930, Oseen and Burgers gave a simple solution to Equation |
| 1903 |
– |
\ref{introEquation:tensorExpression} |
| 1904 |
– |
\begin{equation} |
| 1905 |
– |
T_{ij} = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij} |
| 1906 |
– |
R_{ij}^T }}{{R_{ij}^2 }}} \right). |
| 1907 |
– |
\label{introEquation:oseenTensor} |
| 1908 |
– |
\end{equation} |
| 1909 |
– |
Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$. |
| 1910 |
– |
A second order expression for element of different size was |
| 1911 |
– |
introduced by Rotne and Prager\cite{Rotne1969} and improved by |
| 1912 |
– |
Garc\'{i}a de la Torre and Bloomfield\cite{Torre1977}, |
| 1913 |
– |
\begin{equation} |
| 1914 |
– |
T_{ij} = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I + |
| 1915 |
– |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma |
| 1916 |
– |
_i^2 + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} - |
| 1917 |
– |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. |
| 1918 |
– |
\label{introEquation:RPTensorNonOverlapped} |
| 1919 |
– |
\end{equation} |
| 1920 |
– |
Both of the Equation \ref{introEquation:oseenTensor} and Equation |
| 1921 |
– |
\ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij} |
| 1922 |
– |
\ge \sigma _i + \sigma _j$. An alternative expression for |
| 1923 |
– |
overlapping beads with the same radius, $\sigma$, is given by |
| 1924 |
– |
\begin{equation} |
| 1925 |
– |
T_{ij} = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 - |
| 1926 |
– |
\frac{2}{{32}}\frac{{R_{ij} }}{\sigma }} \right)I + |
| 1927 |
– |
\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right] |
| 1928 |
– |
\label{introEquation:RPTensorOverlapped} |
| 1929 |
– |
\end{equation} |
| 1930 |
– |
|
| 1931 |
– |
To calculate the resistance tensor at an arbitrary origin $O$, we |
| 1932 |
– |
construct a $3N \times 3N$ matrix consisting of $N \times N$ |
| 1933 |
– |
$B_{ij}$ blocks |
| 1934 |
– |
\begin{equation} |
| 1935 |
– |
B = \left( {\begin{array}{*{20}c} |
| 1936 |
– |
{B_{11} } & \ldots & {B_{1N} } \\ |
| 1937 |
– |
\vdots & \ddots & \vdots \\ |
| 1938 |
– |
{B_{N1} } & \cdots & {B_{NN} } \\ |
| 1939 |
– |
\end{array}} \right), |
| 1940 |
– |
\end{equation} |
| 1941 |
– |
where $B_{ij}$ is given by |
| 1942 |
– |
\[ |
| 1943 |
– |
B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
| 1944 |
– |
)T_{ij} |
| 1945 |
– |
\] |
| 1946 |
– |
where $\delta _{ij}$ is Kronecker delta function. Inverting matrix |
| 1947 |
– |
$B$, we obtain |
| 1948 |
– |
|
| 1949 |
– |
\[ |
| 1950 |
– |
C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
| 1951 |
– |
{C_{11} } & \ldots & {C_{1N} } \\ |
| 1952 |
– |
\vdots & \ddots & \vdots \\ |
| 1953 |
– |
{C_{N1} } & \cdots & {C_{NN} } \\ |
| 1954 |
– |
\end{array}} \right) |
| 1955 |
– |
\] |
| 1956 |
– |
, which can be partitioned into $N \times N$ $3 \times 3$ block |
| 1957 |
– |
$C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$ |
| 1958 |
– |
\[ |
| 1959 |
– |
U_i = \left( {\begin{array}{*{20}c} |
| 1960 |
– |
0 & { - z_i } & {y_i } \\ |
| 1961 |
– |
{z_i } & 0 & { - x_i } \\ |
| 1962 |
– |
{ - y_i } & {x_i } & 0 \\ |
| 1963 |
– |
\end{array}} \right) |
| 1964 |
– |
\] |
| 1965 |
– |
where $x_i$, $y_i$, $z_i$ are the components of the vector joining |
| 1966 |
– |
bead $i$ and origin $O$. Hence, the elements of resistance tensor at |
| 1967 |
– |
arbitrary origin $O$ can be written as |
| 1968 |
– |
\begin{equation} |
| 1969 |
– |
\begin{array}{l} |
| 1970 |
– |
\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\ |
| 1971 |
– |
\Xi _{}^{tr} = \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
| 1972 |
– |
\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j \\ |
| 1973 |
– |
\end{array} |
| 1974 |
– |
\label{introEquation:ResistanceTensorArbitraryOrigin} |
| 1975 |
– |
\end{equation} |
| 1976 |
– |
|
| 1977 |
– |
The resistance tensor depends on the origin to which they refer. The |
| 1978 |
– |
proper location for applying friction force is the center of |
| 1979 |
– |
resistance (reaction), at which the trace of rotational resistance |
| 1980 |
– |
tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of |
| 1981 |
– |
resistance is defined as an unique point of the rigid body at which |
| 1982 |
– |
the translation-rotation coupling tensor are symmetric, |
| 1983 |
– |
\begin{equation} |
| 1984 |
– |
\Xi^{tr} = \left( {\Xi^{tr} } \right)^T |
| 1985 |
– |
\label{introEquation:definitionCR} |
| 1986 |
– |
\end{equation} |
| 1987 |
– |
Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
| 1988 |
– |
we can easily find out that the translational resistance tensor is |
| 1989 |
– |
origin independent, while the rotational resistance tensor and |
| 1990 |
– |
translation-rotation coupling resistance tensor depend on the |
| 1991 |
– |
origin. Given resistance tensor at an arbitrary origin $O$, and a |
| 1992 |
– |
vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
| 1993 |
– |
obtain the resistance tensor at $P$ by |
| 1994 |
– |
\begin{equation} |
| 1995 |
– |
\begin{array}{l} |
| 1996 |
– |
\Xi _P^{tt} = \Xi _O^{tt} \\ |
| 1997 |
– |
\Xi _P^{tr} = \Xi _P^{rt} = \Xi _O^{tr} - U_{OP} \Xi _O^{tt} \\ |
| 1998 |
– |
\Xi _P^{rr} = \Xi _O^{rr} - U_{OP} \Xi _O^{tt} U_{OP} + \Xi _O^{tr} U_{OP} - U_{OP} \Xi _O^{{tr} ^{^T }} \\ |
| 1999 |
– |
\end{array} |
| 2000 |
– |
\label{introEquation:resistanceTensorTransformation} |
| 2001 |
– |
\end{equation} |
| 2002 |
– |
where |
| 2003 |
– |
\[ |
| 2004 |
– |
U_{OP} = \left( {\begin{array}{*{20}c} |
| 2005 |
– |
0 & { - z_{OP} } & {y_{OP} } \\ |
| 2006 |
– |
{z_i } & 0 & { - x_{OP} } \\ |
| 2007 |
– |
{ - y_{OP} } & {x_{OP} } & 0 \\ |
| 2008 |
– |
\end{array}} \right) |
| 2009 |
– |
\] |
| 2010 |
– |
Using Equations \ref{introEquation:definitionCR} and |
| 2011 |
– |
\ref{introEquation:resistanceTensorTransformation}, one can locate |
| 2012 |
– |
the position of center of resistance, |
| 2013 |
– |
\begin{eqnarray*} |
| 2014 |
– |
\left( \begin{array}{l} |
| 2015 |
– |
x_{OR} \\ |
| 2016 |
– |
y_{OR} \\ |
| 2017 |
– |
z_{OR} \\ |
| 2018 |
– |
\end{array} \right) & = &\left( {\begin{array}{*{20}c} |
| 2019 |
– |
{(\Xi _O^{rr} )_{yy} + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} } \\ |
| 2020 |
– |
{ - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz} + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} } \\ |
| 2021 |
– |
{ - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx} + (\Xi _O^{rr} )_{yy} } \\ |
| 2022 |
– |
\end{array}} \right)^{ - 1} \\ |
| 2023 |
– |
& & \left( \begin{array}{l} |
| 2024 |
– |
(\Xi _O^{tr} )_{yz} - (\Xi _O^{tr} )_{zy} \\ |
| 2025 |
– |
(\Xi _O^{tr} )_{zx} - (\Xi _O^{tr} )_{xz} \\ |
| 2026 |
– |
(\Xi _O^{tr} )_{xy} - (\Xi _O^{tr} )_{yx} \\ |
| 2027 |
– |
\end{array} \right) \\ |
| 2028 |
– |
\end{eqnarray*} |
| 2029 |
– |
|
| 2030 |
– |
|
| 2031 |
– |
|
| 2032 |
– |
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
| 2033 |
– |
joining center of resistance $R$ and origin $O$. |
