What is the ODE for a mass-spring-damper system with dynamic friction?

[David Ren]

I know that the second order differential equation for a mass-spring-damper system, assuming some initial velocity and position and no external force is:
m*a + c*v + k*x = 0
Where a is the second derivative of x, and v is the first derivative.

Say I want to add dynamic friction to my system. I know that the force of such friction will be m*g*mu, where mu is the coefficient of dynamic friction, and its direction will be opposite of the velocity direction.
How do I incorporate this in my equation? I can't look at it simply as an external force (and as such, write m*a + c*v + k*x = m*g*mu) since that would miss the fact the direction of the force changes. I am sure there is an elegant way, I just seem to not find it.

Side note, at first I thought that the damper might actually represent dynamic friction, but then I realized it is proportional to the velocity, and dynamic friction size is dependent only on the force of the normal and friction coefficient. I hope I am correct.

 

[drawoh]

David Ren,

Dynamic friction would be a form of damping would it not?

I recall being told that the the c*y you list above assume fluid damping. Actually, fliud damping should be some factor time velocity squared. The resulting differential equation is not solvable by algebra, although it is easily solved in a spreadsheet.

Your dynamic friction, assuming it is Newtonian, would be a constant value, any time v!=0. This also is easily modelled in a spreadsheet.

--
JHG

 

[GregLocock]

You can write the ODE for a system with friction, but you can't solve it other than numerically. You can approximate it to damping for a given excitation amplitude, but it is not correct, the graph of F vs x is the wrong shape. The equation for friction can be written as -v/(abs(v))*m*g*mu (not too sure about the - the idea is that force opposes the velocity)

hence m*a + c*v +v/(abs(v))*m*g*mu + k*x =0

but abs has no place in a conventional ODE. The world is full of systems for which one can write unsolvable ODEs. By the time you've left uni you'll have probably seen all the solvable ones! Slight exaggeration. An MIT professor's thoughts on that

https://web.williams.edu/Mathematics/lg5/Rota.pdf

ChatGPT will write a script to simulate your equation, but will need a fair bit of coaching to get it right.





Cheers

Greg Locock


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[GregLocock]

Looking at it you can see why it isn't properly solvable - how do handle v/abs(v) when v=0?

Cheers

Greg Locock


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[David Ren]

I see,
So if I understand correctly, the ODE would be:
m*a + c*v + v/(abs(v))*m*g*mu + k*x = 0
And unlike the original ODE, it is unsolvable analytically.
How is it solvable numerically? Does anyone know how I can use MATLAB numerical toolbox to solve it?

 

[GregLocock]

https://au.mathworks.com/support/search.html?q=spring mass&page=1

If you've got simulink that provides a neat way of solving spring mass systems.


Cheers

Greg Locock


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