# Difference between revisions of "Kelvin's medium"

From Department of Theoretical and Applied Mechanics

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<!--[[Виртуальная лаборатория]]>[[One-dimensional Kelvin medium]] <HR>--> | <!--[[Виртуальная лаборатория]]>[[One-dimensional Kelvin medium]] <HR>--> | ||

− | + | Kelvin's one-dimensional medium is a chain consisting of masses interacting via a pair torque potential. In this example the masses are visualized by rods. | |

− | Kelvin's one-dimensional medium is a chain consisting of | + | The masses interact via the torque potential: |

− | The | ||

::<math> | ::<math> | ||

U = C({\bf n}_{1}\cdot{\bf n}_{2}) | U = C({\bf n}_{1}\cdot{\bf n}_{2}) | ||

</math>, | </math>, | ||

− | where <math>C</math> is | + | where <math>C</math> is the interaction constant, <math>\bf{n}_1</math>, <math>\bf{n}_2</math> are the unit vectors bound to the masses. The interaction torque has the form: |

::<math> | ::<math> | ||

{\bf M}_{1} = {\bf n}_{1}\times\frac{\partial U}{\partial {\bf n}_{1}} = C({\bf n}_{1}\times{\bf n}_{2}) | {\bf M}_{1} = {\bf n}_{1}\times\frac{\partial U}{\partial {\bf n}_{1}} = C({\bf n}_{1}\times{\bf n}_{2}) | ||

</math> | </math> | ||

− | Then the motion equation | + | Then the motion equation for the k-th particle is as follows: |

::<math> | ::<math> | ||

J\ddot{\bf \phi}_{k} = C(({\bf n}_{k}\times{\bf n}_{k+1}) + ({\bf n}_{k}\times{\bf n}_{k-1})) | J\ddot{\bf \phi}_{k} = C(({\bf n}_{k}\times{\bf n}_{k+1}) + ({\bf n}_{k}\times{\bf n}_{k-1})) |

## Revision as of 02:29, 9 June 2016

Kelvin's one-dimensional medium is a chain consisting of masses interacting via a pair torque potential. In this example the masses are visualized by rods. The masses interact via the torque potential:

- ,

where

is the interaction constant, , are the unit vectors bound to the masses. The interaction torque has the form:Then the motion equation for the k-th particle is as follows: