Difference between revisions of "Kelvin's medium"
From Department of Theoretical and Applied Mechanics
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| − | + | 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:
