Difference between revisions of "Kelvin's medium"

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[[ru:Одномерная среда Кельвина]]
 
<!--[[Виртуальная лаборатория]]>[[One-dimensional Kelvin medium]] <HR>-->
 
<!--[[Виртуальная лаборатория]]>[[One-dimensional Kelvin medium]] <HR>-->
 
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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 solids interacting via a pair torque potential. In this example, the solids are visualized by rods.  
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The masses interact via the torque potential:
The solids interact via the torque potential:
 
 
::<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 an interaction constant, <math>\bf{n}_1</math>, <math>\bf{n}_2</math> are the unit vectors bound to the solids. The interaction torque has the form:
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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 of the k-th particle is as follows:
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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}))

Latest revision as of 18:51, 18 January 2017

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:

[math] U = C({\bf n}_{1}\cdot{\bf n}_{2}) [/math],

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] {\bf M}_{1} = {\bf n}_{1}\times\frac{\partial U}{\partial {\bf n}_{1}} = C({\bf n}_{1}\times{\bf n}_{2}) [/math]

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

[math] J\ddot{\bf \phi}_{k} = C(({\bf n}_{k}\times{\bf n}_{k+1}) + ({\bf n}_{k}\times{\bf n}_{k-1})) [/math]