- #1
samu0034
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Hello everyone, looking around I have faith that the members of this forum will be able to point me in the right direction, and I apologize if it's more basic than I'm giving it credit for.
I'm an experimental researcher in rock mechanics, but I've always been fascinated by elasto-dymamic numerical modelling of earthquake rupture nucleation and propagation. To that end, I'm trying to build an earthquake simulator, which in my vision is basically an inter-connected sheet of spring-sliders, each connected to the one ahead/behind it with a spring of stiffness equivalent to the Young's Modulus, and the the slider on either side by a spring equivalent to the Shear Modulus. Not being a master of this sort of thing though, I'm starting from first principles, and this problem which I've posed to myself has me stumped… Below is a diagram of what I'd like to model, and below that is a brief description of my own progress on the whole thing. A point in the right direction is what I'm hoping for.
Also, please feel free to move this thread if this is the incorrect place to pose such a question.
http://imageshack.us/photo/my-images/593/screenshot20130612at411.png/
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If I sum up the forces on the left mass (m1), I get...
[tex]m_1 a_1 = m_1 g - K_1 (\Delta L_1 + x_1) - \gamma (\frac{dx_1}{dt}) + K_3 ((\Delta L_2 + x_2) - (\Delta L_1 + x_1))[/tex]
Where the...
And there would be a similar equation for the acceleration of [itex]m_2[/itex]. I can solve the above equation for [itex]a_1[/itex], and I get...
[tex]a_1 = \frac{d^2 x_1}{dt^2} = \frac{m_1 g - K_1 (\Delta L_1 + x_1) - \gamma (\frac{dx_1}{dt}) + K_3 ((\Delta L_2 + x_2) - (\Delta L_1 + x_1))}{m_1}[/tex]
[tex]= g - \frac{K_1}{m_1} (\Delta L_1 + x_1) - \frac{\gamma}{m_1} (\frac{dx_1}{dt}) + \frac{K_3}{m_1} ((\Delta L_2 + x_2) - (\Delta L_1 + x_1))[/tex]
Which, grand scheme of things, is pretty easy to solve numerically. Except that I don't know what [itex]\Delta L_1[/itex] and [itex]\Delta L_2[/itex] are.
My research group isn't really focused on this sort of thing, so I don't have anyone to bounce questions like this off of. Any help this community can provide will be greatly appreciated. Maybe I'm over thinking this, maybe I'm missing something entirely. I feel like it's a bit of both.
I'm an experimental researcher in rock mechanics, but I've always been fascinated by elasto-dymamic numerical modelling of earthquake rupture nucleation and propagation. To that end, I'm trying to build an earthquake simulator, which in my vision is basically an inter-connected sheet of spring-sliders, each connected to the one ahead/behind it with a spring of stiffness equivalent to the Young's Modulus, and the the slider on either side by a spring equivalent to the Shear Modulus. Not being a master of this sort of thing though, I'm starting from first principles, and this problem which I've posed to myself has me stumped… Below is a diagram of what I'd like to model, and below that is a brief description of my own progress on the whole thing. A point in the right direction is what I'm hoping for.
Also, please feel free to move this thread if this is the incorrect place to pose such a question.
http://imageshack.us/photo/my-images/593/screenshot20130612at411.png/
Uploaded with ImageShack.us
If I sum up the forces on the left mass (m1), I get...
[tex]m_1 a_1 = m_1 g - K_1 (\Delta L_1 + x_1) - \gamma (\frac{dx_1}{dt}) + K_3 ((\Delta L_2 + x_2) - (\Delta L_1 + x_1))[/tex]
Where the...
- 1st term right of the equal sign is the acceleration of [itex]m_1[/itex] due to gravity
- 2nd term is the resistance of the spring [itex]K_1[/itex] due to the stretch of the added mass and any additional displacement
- 3rd term is a viscous damping term (which I'm happy to ignore as once everything else is solved I presume it'd be trivial to add this back in)
- 4th term is the additional loading supplied by the shear spring due to the differential between the position of [itex]m_1[/itex] and [itex]m_2[/itex] (I'm not at all certain this is formulated correctly)
And there would be a similar equation for the acceleration of [itex]m_2[/itex]. I can solve the above equation for [itex]a_1[/itex], and I get...
[tex]a_1 = \frac{d^2 x_1}{dt^2} = \frac{m_1 g - K_1 (\Delta L_1 + x_1) - \gamma (\frac{dx_1}{dt}) + K_3 ((\Delta L_2 + x_2) - (\Delta L_1 + x_1))}{m_1}[/tex]
[tex]= g - \frac{K_1}{m_1} (\Delta L_1 + x_1) - \frac{\gamma}{m_1} (\frac{dx_1}{dt}) + \frac{K_3}{m_1} ((\Delta L_2 + x_2) - (\Delta L_1 + x_1))[/tex]
Which, grand scheme of things, is pretty easy to solve numerically. Except that I don't know what [itex]\Delta L_1[/itex] and [itex]\Delta L_2[/itex] are.
My research group isn't really focused on this sort of thing, so I don't have anyone to bounce questions like this off of. Any help this community can provide will be greatly appreciated. Maybe I'm over thinking this, maybe I'm missing something entirely. I feel like it's a bit of both.
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