Help with simplifing an equation in terms of 2 variables

[fb360]

Homework Statement

Hey. I need help simplifying and factoring a differential equation in terms of v and p (velocity(xdot) and position(x) respectively). I need the final answer to be in this form:
a = ( )v + ( )p.
This is so i can put the governing equation in a state-space and eventually use it as a TF and manipulate it for a 2D system.

Definitions:
m = mass
a = acceleration = xdotdot
v = velocity = xdot
p = position = x
b = constant
k = constant
c = constant

Homework Equations

ma + (bv + kp) = ((cv)/(v^2 + p^2)^.5))

The Attempt at a Solution

a = (-bv*((v^2+p^2)^.5)-kp*((v^2+p^2)^.5)+cv) / (m*((v^2+p^2)^.5))

 

[haruspex]

Are you sure about that equation? ##(v^2 + p^2)^.5## makes no physical sense. The dimensions don't match.

 

[fb360]

Are you sure about that equation? ##(v^2 + p^2)^.5## makes no physical sense. The dimensions don't match.

Yeah. Thats an obvious square root brother. It's how we denote a square root in software that does not allow it... i.e (x^.5 or x^(1/2))

How does it make no physical sense? Do you not understand how to express a square root in terms of a positive exponent value? (c = (a^2 + b^2)^.5)? Recognize this?

the dimensions match perfectly, infact it's a simple Mass-Spring-Damper with an added phase term (that's where the square root comes from including xdot and x)

e;
The problem is, is that there may be no solution. In fact, this would match with my findings, as the equations seems to be non-linear and non-factorable. I was just hoping some math wiz could prove me wrong so I could put it into a simple SS in MATLAB and manipulate it to find the x,y parameters

e;;
The (cv)/(v^2 + p^2)^.5 come from turning the added phase term into an inverse sin to find the angle the c is a constant and v is xdot, and that is the numerator (opposite). The adjacent is the value c; c = (v^2 + p^2)^.5). Its actually what we call a "phase oscillator" mass-spring-damper "system". We can tune the force to be added in phase (90) or out of phase (180); 90 causes amplification and 180 causes dampening

e;;; My fear is that there is no linear factor of a in terms of v and p, and that I need to manipulate the system from the get go. Pl3ase prove me wrong

 

[jackarms]

I think he's saying that the square root seems to give you units of [itex]\sqrt{\frac{m^{2}}{s^{2}} + m^{2}}[/itex]. Are you sure everything works out dimensionally?

EDIT: And the physical part comes from the fact that you're adding a position squared to a velocity squared, which doesn't really make sense.

 

[fb360]

I think he's saying that the square root seems to give you units of [itex]\sqrt{\frac{m^{2}}{s^{2}} + m^{2}}[/itex]. Are you sure everything works out dimensionally?

Yes, see the attached.

e;
I'm saying that this equation has been proven, and I'm trying to work off of it. It's not my own, but they have proven that it works, so I will stand by their assertion

 

[haruspex]

How does it make no physical sense? Do you not understand how to express a square root in terms of a positive exponent value?

As jackarms has explained, I did not say it makes no algebraic sense; I said it makes no physical sense. It is always wrong to add two physical quantities that have different dimensions. You can't add an area to a volume, etc.

I'm saying that this equation has been proven, and I'm trying to work off of it. It's not my own, but they have proven that it works, so I will stand by their assertion

The equation is not "proven". It's an equation they chose to model, and it gave them results they found interesting. The fact remains that the equation they present is nonsense. You can see this by changing the time units from, say, seconds to hours. The model then behaves differently. The results should not depend on the units you use. At a guess, the authors are engineers, not physicists.
The equation can be rescued by introducing one more constant, making the surd ##\sqrt{\dot x^2 + (ωx)^2}##. The constant ω has dimension of 1/time.
So back to the ODE. Looks nasty. The plot suggests something like x = Ae^-λt sin(ωt), but I couldn't get it to work.

 

[fb360]

As jackarms has explained, I did not say it makes no algebraic sense; I said it makes no physical sense. It is always wrong to add two physical quantities that have different dimensions. You can't add an area to a volume, etc.

The equation is not "proven". It's an equation they chose to model, and it gave them results they found interesting. The fact remains that the equation they present is nonsense. You can see this by changing the time units from, say, seconds to hours. The model then behaves differently. The results should not depend on the units you use. At a guess, the authors are engineers, not physicists.
The equation can be rescued by introducing one more constant, making the surd ##\sqrt{\dot x^2 + (ωx)^2}##. The constant ω has dimension of 1/time.
So back to the ODE. Looks nasty. The plot suggests something like x = Ae^-λt sin(ωt), but I couldn't get it to work.

I don't understand anything you said, nor did I read it, as I posted the solution and how it makes physical sense, even though you suggest it possibly cannot.

I understand it can be made more complicated; that's not the point.
I need it to be linearized, and put in the form I gave: (a = ( )v + ( )p)
I understand it might not make sense to you, but it has been PHYSICALLY PROVEN; thus you ARE WRONG. i cannot emphasize this enough... do the math I gave you, or say you cannot...

The fact remains that the equation they present is nonsense. You can see this by changing the time units from, say, seconds to hours.

LOLL read the paper...

e;
I gave you a paper accepted by ieee etc...

If you can't figure it out just say it is non-linear... No physical-sense LOL. Dude.
e;;
These guys HAVE MADE ROBOTS that correlate to their assertion AND THEY ALL WORK.
Please, prove their correct to be wrong; Seeing as how they already have videos and physical proof...

e;;; I'm not trying to start a "who's smartest" contest... I just want to linearize this equation I posted. I want to be able make it "semi-simple" so that i can turn it into a 2-D analysis and not 1D...

 

[scurty]

e;;; I'm not trying to start a "who's smartest" contest... I just want to linearize this damn hairy-*** equation I posted. I want to be able make it "semi-simple" so that i can turn it into a 2-D analysis and not 1D...

Good luck. I'd be surprised if anyone helped you out now since you seem to keep insulting people who are trying to help you. None of the helpers get paid for helping, we do it on our free time. For what it is worth, haruspex brought up the issue of the units not working out correctly because there is a chance that you copied the problem down wrong. That happens more than you would think and it solves many issues quickly.

 

[ZapperZ]

Closed, pending moderation and actions.

In case it has slipped everyone's mind, please reread the PF rules that ALL of you had agreed to. If you are dead set on telling others to read and understand these sources or what you had written in here, then see the irony in the fact that you had not understood and abide by the rules that you had agreed to!

Zz.