Coulomb's Damping Equation for Machine Vibration 3

[m_ridzon]

I am working on a large industrial machine on frictional skid rails (i.e., 1 dimension of movement is possible on the rails). As its motor starts up, it begins to vibrate the machine around 20Hz; i.e., movement occurs on the frictional rails. The phenomenon is similar to what is shown in the YouTube video HERE. This is quite similar to Coulomb's damping scenario (images below).

"r" is the number of half cycles that elapse before motion ceases. However, the equation assumes a spring participates in the situation (hence the variable "k"). It also assumes there is an initial displacement, x[sub]0[/sub] applied to the assembly. Neither the spring, nor the initial displacement occur in my scenario. Does anyone know how or if Coulomb's equation is modified to account for my scenario?

(note: yes, I understand an imbalance in the rotating assembly is likely the cause of the vibration and should be looked at. However, for all intents and purposes, we can assume in this conversation that the source of the vibration is there to stay. The structure and its motor are huge. So even a tiny imbalance may lead to large energy addition to the structure. With a free DOF along the skids, the energy may be enough to excite movement.)

 

[electricpete]

Coming up with a model that predicts horizontal motion in the way you describe as a reaction to unbalance for example (in similar way described in my 19 Feb 21 00:26 post) is doable. Coming up with the RIGHT model and the RIGHT coefficients to plug into that model would be downright impossible with what we have been given. But since op seems hell bent on going down that path for whatever reason (education / entertainment ?), here's my thoughts to get started with the simplest model I can imagine

You probably can't solve it analytically (ex laplace transform methods) because overcoming the static coefficient of friction represents a non-linearity. You need a numberical ODE solver (like ODE45 in matlab) which I presume most engineers have access to or can program themselves. Here's the simplest formulation I can come up with (just a starting point).

State variables Xv, Xh, Vv, Vh
where X is displacement, V is velocity, subscript h for horizontal, subscript v for vertical

The time derivatives of the state variables are as follows:
Xv' = Vv
Xh' = Vh
Vv' = [-K*Xv + Fub*cos(w*t)]/M
Vh' = [Fub*sin(w*t) +Friction]/M

where
Fub = Force magnitude (unbalance)
Ffriction = {K*Xv-M*g}*{Vh/|Vh|}{Mu_Static + (Mu_Dynamic-MuStatic)*f(|Vh|}}
...{K*Xv-M*g} represents the normal force.
...{Vh/|Vh|} gives the proper sign
...The third term in curly brackets { } will return either Mu_Static or Mu_Dynamic… ideally which one is returned depends on whether Vh = 0 or not but such an abrupt transition would cause instabilities in your numerical algorithm so it needs to be a smooth transition. The function be defined to satisfy conditions something like: f(0)=0, f'(0)=0, and f(TW)=1 f'(TW)=0 where TW is transition width and no abrupt changes in f or f'. Wide transition width favors numerical stability. Narrow transition width favors accuracy. You can build functions like this yourself. Edit: try using f(|Vh|)=0.5*(1-cos(pi*Vh/TW)) for |Vh|< TW, and f=1 for |Vh|> TW.

I may have made a sign error along the way somewhere… no guarantees. This is just first thought at how op could build a model for pure educational/entertainment purposes with no hope to do anything useful.

Edit - I didn't see Greg's post. I imagine he solved it with ode solver of some type in similar fashion to what I described. Also I didn't particularly notice until just now that op elects to ignore static coefficient of friction, that simplifies the problem to the point that it's linear and in theory it could be solved analytically.


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(2B)+(2B)' ?

 

[GregLocock]

No, that was just a time based simulation. Often I linearise friction as damping, but that is a pretty terrible solution as it locks the analysis to a particular velocity. It also gives the wrong harmonic structure for any frequency domain work.



Cheers

Greg Locock


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[electricpete]

Thanks Greg. I didn't say you used analytical solution. I think you misunderstood me because I had two separate ideas mixed into the same paragraph:

1 - You probably used an ode solver. (i.e. time domain).
2 - op was allowing a simplified linear model of friction, which in theory permits analytical solution (a correction to my earlier comment that analytical solution wouldn't be possible).

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(2B)+(2B)' ?

 

[GregLocock]

Got it. I think calling my spreadsheet an ODE solver is a bit high falutin, but I do wish that universities spent more time on numerical solutions than higher math.

Cheers

Greg Locock


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[m_ridzon]

Very interesting problem, but you could probably write a dissertation on modelling and solving it analytically.

Yup, I'm starting to see that and would agree. Thus, I'm going to stop running down this rabbit hole and revert to FEA instead.

A few plots for your amusement. SDOF only

Very interesting. Thanks for sharing!

Coming up with a model that predicts horizontal motion in the way you describe as a reaction to unbalance for example (in similar way described in my 19 Feb 21 00:26 post) is doable. Coming up with the RIGHT model and the RIGHT coefficients to plug into that model would be downright impossible with what we have been given. But since op seems hell bent on going down that path for whatever reason

I stopped reading your post here. I didn't come to this forum for snark, but for an educated discussion about the situation. Nowhere did I assertively indicate the problem MUST be solved analytically. In fact, to my surprise, I found myself more focused on keeping people in the thread on track, rather than discussing a wobbly YouTube bench grinder setting on a rubber floor mat (I felt like I was herding cats). I realize there are many unknowns, too many at this point. I realize it's a complicated problem. I merely thought I could apply the K.I.S.S. principal (keep it simple stupid) and uncover some slick approximate analytical method to roughly guide my troubleshooting. I wasn't hoping for a detailed, exact analytical equation to nail the situation perfectly. FEA will likely be the path forward.

 

[electricpete]

Coming up with a model that predicts horizontal motion in the way you describe as a reaction to unbalance for example (in similar way described in my 19 Feb 21 00:26 post) is doable. Coming up with the RIGHT model and the RIGHT coefficients to plug into that model would be downright impossible with what we have been given. But since op seems hell bent on going down that path for whatever reason

I stopped reading your post here. I didn't come to this forum for snark, but for an educated discussion about the situation

It wasn't snark. I was putting the information I was about to provide into full context. I wouldn't provide such information without the disclaimer that I didn't expect it to lead to a reliable model / prediction.

If you put your pride aside and read further into my post, what I did was put the problem into a format suitable for solving by time domain simulation (ode solver) as Greg did. I made some assumptions along the way which I was prepared to discuss (*).... there is certainly room for tweaks to improve it. I spent about an hour organizing that because I was under the impression you wanted to try a numerical model and it seems like a good starting point to me. Low and behold, you liked the pretty pictures from Greg... but you went out of your way to explicitly reject my comments about how you might go about doing something similar yourself.

Humility is a virtue, whether in solving machinery problems or in asking for advice.
No worries. It's an interesting problem / discussion.

* Edit - discussion of my model for posterity:
[ul]
[li]My K represented whatever stiffness / flexibility is below the modal mass[/li]
[li]Xv = 0 is the position where the mass rests on the spring under its own weight.[/li]
[li]The model assumes the machine never loses contact with the ground below it... you could certainly adapt the model to cover the case where the machine momentarily lifts off the support... this would give an adjustment to the Vv and Vh terms which applies when Xv is above a certain value [/li]
[/ul]

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