Dual harmonic oscillators connected by shear spring

[samu0034]

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.

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If I sum up the forces on the left mass (m₁), I get...

$$m_1 a_1 = m_1 g - K_1 (\Delta L_1 + x_1) - \gamma (\frac{dx_1}{dt}) + K_3 ((\Delta L_2 + x_2) - (\Delta L_1 + x_1))$$

Where the...

• 1^st term right of the equal sign is the acceleration of $m_1$ due to gravity
• 2^nd term is the resistance of the spring $K_1$ due to the stretch of the added mass and any additional displacement
• 3^rd 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)
• 4^th term is the additional loading supplied by the shear spring due to the differential between the position of $m_1$ and $m_2$ (I'm not at all certain this is formulated correctly)

And there would be a similar equation for the acceleration of $m_2$. I can solve the above equation for $a_1$, and I get...

$$a_1 = \frac{d^2 x_1}{dt^2} = \frac{m_1 g - K_1 (\Delta L_1 + x_1) - \gamma (\frac{dx_1}{dt}) + K_3 ((\Delta L_2 + x_2) - (\Delta L_1 + x_1))}{m_1}$$
$$= g - \frac{K_1}{m_1} (\Delta L_1 + x_1) - \frac{\gamma}{m_1} (\frac{dx_1}{dt}) + \frac{K_3}{m_1} ((\Delta L_2 + x_2) - (\Delta L_1 + x_1))$$

Which, grand scheme of things, is pretty easy to solve numerically. Except that I don't know what $\Delta L_1$ and $\Delta L_2$ 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.

 

[Alfred Cann]

Masses and springs are not enough. You need to add dampers (dashpots) to account for the friction losses (viscosity).
Find an acoustics expert; they deal with systems like this all the time.

 

[samu0034]

Hi Alfred,

Yeah, I know that in order to make it into my "earthquake simulator" I will have to add in some sort of damping term to account for friction, but since this isn't really my area of expertise I'm REALLY starting from first principles. Modeling simple harmonic motion, modeling a "simple" 2 mass - 2 springs-in-series system, and now I'm trying to model this shear loaded springs-in-parallel system. This is all just so I understand how all of the loading works when I add in some sort of frictional resistance.

I do appreciate the advice of seeking out an acoustics expert, that seems like a sound recommendation.

[EDIT] Holy crap, I didn't even notice that pun... terrible... I'm ashamed.

 

[Alfred Cann]

I love your pun. Take credit for it even if it was unconscious.

 

[Alfred Cann]

Suggest you look in the seismology/rheology community. I'd be very surprised if what you are attempting hasn't already been done. Search under 'finite element analysis'