Springs with Dashpot: Understanding Motion Equations

[wololo]

Homework Statement

Homework Equations

∑F=ma
F=-kx
x = Ae-γt/2cos(ωt + ∅)

The Attempt at a Solution

I have looked up this old thread but don't understand how to get the equation of motion from :
ma1=-kx1-k(x1-x2) - bv
ma2=-kx2-k(x2-x1) - bv

https://www.physicsforums.com/threads/system-of-2-masses-3-springs.648400/

 

[haruspex]

Is your problem what to do with v? How do you think that relates to x1 and x2?
But there is something wrong with the signs. The forces the dashpot exerts on the two masses must be in opposite directions, so they can't both be -bv.

 

[wololo]

What i got so far:
ma1=-kx1-k(x1-x2)-bv <==> bv=-kx1-k(x1-x2)-ma1
ma2=-kx2+k(x1-x2)-bv <==> bv=-kx2+k(x1-x2)-ma2
Substitute bv
-kx1-k(x1-x2)-ma1=-kx2+k(x1-x2)-ma2
m(a2-a1)=-kx2+kx1+k(x1-x2)+k(x1-x2)
m(a2-a1)=3k(x1-k2)

This relation is only in terms of x1-x2 not x1+x2 and also the equation of motion should be ma1+ma2=m(a1+a2) but i can't get that since ma has the same sign on both sides of the equation. Does anyone have advice on how to get to b) ? thanks

 

[haruspex]

What i got so far:
ma1=-kx1-k(x1-x2)-bv <==> bv=-kx1-k(x1-x2)-ma1
ma2=-kx2+k(x1-x2)-bv <==> bv=-kx2+k(x1-x2)-ma2
Substitute bv
-kx1-k(x1-x2)-ma1=-kx2+k(x1-x2)-ma2
m(a2-a1)=-kx2+kx1+k(x1-x2)+k(x1-x2)
m(a2-a1)=3k(x1-k2)

This relation is only in terms of x1-x2 not x1+x2 and also the equation of motion should be ma1+ma2=m(a1+a2) but i can't get that since ma has the same sign on both sides of the equation. Does anyone have advice on how to get to b) ? thanks

You don't seem to have addressed either of my comments in post #2. Do I need to explain them more?

 

[rude man]

The forces the dashpot exerts on the two masses must be in opposite directions, so they can't both be -bv.

The problem states that " .. v is the relative velocity of its two ends." By itself, this statement is of course ambiguous. But the additional statement that the dashpot force "opposes the motion", when considered for each mass separately, implies that F for the left mass (mass 1) has to be -b (d/dt)(x1 - x2) and for the #2 mass, -b (d/dt)(x2 - x1), thus resolving the ambiguity.

 

[haruspex]

The problem states that " .. v is the relative velocity of its two ends." By itself, this statement is of course ambiguous. But the additional statement that the dashpot force "opposes the motion", when considered for each mass separately, implies that F for the left mass (mass 1) has to be -b (d/dt)(x1 - x2) and for the #2 mass, -b (d/dt)(x2 - x1), thus resolving the ambiguity.

Ok, you are saying that the two v's are equal and opposite. But it does not look as though wololo understood that. (And it strikes me as a misleading way to have written the equations.)

 

[rude man]

Ok, you are saying that the two v's are equal and opposite. But it does not look as though wololo understood that. (And it strikes me as a misleading way to have written the equations.)

Agreed.

 

[wololo]

I considered the force by the dashpot bv as being opposite for m1 and m2 but when I put them together to get the total force they cancel themselves. Also the force caused by the middle spring cancels itself (since it points on opposite direction at m1 and m2). The equation of motion I get is then solely expressed in terms of x1+x2 . I know that if if I considered the middle spring force as being equal for both m1 and m2, I would get an equation that depends on x1-x2, but I don't see how this could be true since a spring exerts a force in different direction at it's ends. Am I correct in assuming that in order to solve my two equations I need to put them together and find the total force exerted on the system? Perhaps the way I attempt to solve these equations is wrong and stops me from obtaining an equation that depends on both y1 and y2. Thanks for helping me clarify the way the dashpot works but I am still at a lost on how to successfully answer part b. Any ideas? Here is how I proceeded:

ma1=-kx1-k(x1-x2)-b(d/dt)(x1-x2)
ma2=-kx2+k(x1-x2)-b(d/dt)(x2-x1) <==> (rearrange) ma2=-kx2+k(x1-x2)+b(d/dt)(x1-x2)

m(a1+a2)=-kx1-kx2-k(x1-x2)+k(x1-x2)-b(d/dt)(x1-x2)+b(d/dt)(x1-x2)
Ftot=-k(x1+x2) (all the other terms cancel)

 

[haruspex]

I considered the force by the dashpot bv as being opposite for m1 and m2 but when I put them together to get the total force they cancel themselves.

They only cancel because of the way you are combining the two equations.
Assuming you corrected the sign (so that they are now opposite) if you add them the bv terms cancel. What else could you do?
Please also answer my other question: what equation directly relates v to x1, x2 and time?
Oh, and before all that, you need to define clearly what v means.

 

[rude man]

ma1=-kx1-k(x1-x2)-b(d/dt)(x1-x2)
ma2=-kx2+k(x1-x2)-b(d/dt)(x2-x1) <==> (rearrange) ma2=-kx2+k(x1-x2)+b(d/dt)(x1-x2)

These equations are correct, except why "rearrange" them? Solve for x1 and x2 for part (a).

Ftot=-k(x1+x2) (all the other terms cancel)

why are you adding the forces on m1 and m2?
Do part (a); then part (b) and (c) follow without too much trouble.

 

[wololo]

They only cancel because of the way you are combining the two equations.
Assuming you corrected the sign (so that they are now opposite) if you add them the bv terms cancel. What else could you do?
Please also answer my other question: what equation directly relates v to x1, x2 and time?
Oh, and before all that, you need to define clearly what v means.

v is the relative speed between both ends of the dashpot, so v2-v1. Speed is equal to the derivate of position in regards to time, dx/dt, in this case dx will be the relative variation of x between both ends of the dashpot so bv=bd(x2-x1)/dt

These equations are correct, except why "rearrange" them? Solve for x1 and x2 for part (a).
why are you adding the forces on m1 and m2?
Do part (a); then part (b) and (c) follow without too much trouble.

If I develop develop F1 and F2 separately, assuming y1=x1+x2 and y2=x1-x2, I get:

F1=-kx1-ky2-b(d/dt)(y2)
F2=-kx2+ky2+b(d/dt)(y2)

I don't see how I can express them in terms of y1

Could it be that the dashpot only exerts a force on the system as a whole, such that

Ftot=F1+F2-bv

Which would give
Ftot=(-kx1-ky2)+(-kx2+ky2)-b(d/dt)(y2)
Ftot=-ky1-b(d/dt)(y2)
?

 

[rude man]

v is the relative speed between both ends of the dashpot, so v2-v1. Speed is equal to the derivate of position in regards to time, dx/dt, in this case dx will be the relative variation of x between both ends of the dashpot so bv=bd(x2-x1)/dt

No, as I pointed out before, the two v's are different for the two equations, and haruspex has also pointed out that they can't be the same for both equations.

I don't see how I can express them in terms of y1

First, what are F1 and F2 in terms of x1 and x2?
Then, try adding the two equations (in post 10) together.
Then, try subtracting the second equation from the first.

 

[wololo]

First, what are F1 and F2 in terms of x1 and x2?

Post 8 (ma1=F1 and ma2=F2):

ma1=-kx1-k(x1-x2)-b(d/dt)(x1-x2)
ma2=-kx2+k(x1-x2)-b(d/dt)(x2-x1)

Then, try adding the two equations (in post 10) together.

I already added the two equations together in post 8:

m(a1+a2)=-kx1-kx2-k(x1-x2)+k(x1-x2)-b(d/dt)(x1-x2)+b(d/dt)(x1-x2)
=-k(x1+x2) (all the other terms cancel)

Then, try subtracting the first equation from the second

-kx1-k(x1-x2)-b(d/dt)(x1-x2)-(-kx2+k(x1-x2)-b(d/dt)(x2-x1))
=-k(x1-x2)-2k(x1-x2)-2b(d/dt)(x1-x2)
=-3k(x1-x2)-2b(d/dt)(x1-x2)

I must be forgetting some relation since both you and Haruspex seem to be trying to point me towards something, though I evidently cant't tell what . Would one of you please be willing to point out what I am missing? I added and substracted the equations and still can't express them in terms of x1-x2 and x1+x2. I really am at a loss. Thanks!

 

[rude man]

Post 8 (ma1=F1 and ma2=F2):

Again: what are F1 and F2 in terms of x1 and x2? Saying F1 = ma1 does not answer that question. This is the first thing you need to get right.

I already added the two equations together in post 8:.

Right, and you got F1 + F2 = -k(x1 + x2). So - one mo' time - what are F1 and F2 in terms of x1 and x2?
EDIT: I just noticed, part (a) does not ask you to solve for x1 and x2. Good thing, because that's a real mess. But in part (b) you substitute y1 = x1 + x2 and y2 = x1 - x2, that makes solving for y1 very easy and then y2 is similarly easily obtained. Finally, solve for x1(t→∞) and x2(t→∞) from y1(t→∞) and y2(t→∞).

 

[rude man]

Post 8 (ma1=F1 and ma2=F2):
I already added the two equations together in post 8:

Yes, that was good. Sorry i asked you why you did it. I was concentrating on solving for x1 and x2 separately which would be a big mess - see my previous post. You got F1 + F2 = -k(x1+x2) (all the other terms cancel) which is right. So can you replace x1+x2 with one of the two y's?

-kx1-k(x1-x2)-b(d/dt)(x1-x2)-(-kx2+k(x1-x2)-b(d/dt)(x2-x1))
=-k(x1-x2)-2k(x1-x2)-2b(d/dt)(x1-x2)
=-3k(x1-x2)-2b(d/dt)(x1-x2)

equals what? Can you replace x1-x2 with the other y?

I must be forgetting some relation since both you and Haruspex seem to be trying to point me towards something, though I evidently cant't tell what . Would one of you please be willing to point out what I am missing? I added and substracted the equations and still can't express them in terms of x1-x2 and x1+x2. I really am at a loss. Thanks!

see my previous post.