# Mind blown by mechanical systems (2nd order)

1. Feb 1, 2013

### hihiip201

1. The problem statement, all variables and given/known data

B, K, M

2. Relevant equations

1. xs(t) -----spring ----mass-----damper-----fixed, derive DE for x of mass

given :2. F - > M -----spring-------damper ---- fixed in series, derive the DE for velocity of spring

3. The attempt at a solution

1. ma = -k(x-xs) - B(v)

But I don't understand, why aren't we taking the natural length of the spring into account?

2. no idea, I have the solution and it says that damping force is B(v(t) - vk) , but I have no idea what v has anything to do with damper, the relative velocity vb should just be vk - 0 to me but the textbook and the hw solution suggest otherwise.

2. Feb 1, 2013

### HallsofIvy

Staff Emeritus
Velocity relative to what? You haven't said what "vk" means.

3. Feb 1, 2013

### hihiip201

I do believe I understand what a damper is, but if you look at my system (indicated in series). I can't see what v(t) of mass has anything to do with the damper.

since the damping force = B(relative velocity between each end)

in this case, right end is fixed, left end is the spring's velocity, so I just can't see why Damping force fb which is equal to fk in this case is not B(vk).

4. Feb 1, 2013

### hihiip201

oh gosh....maybe that's the key, my TA didn't say anything about the reference frame of the velocity, and multiple times they have displacement that have different references...

but i do believe v is the velocity of mass relative to the inertial frame of wall. and vk should be the spring velocity of the spring(right hand side of spring, left hand side of damper) relative to the inertia frame of the wall.

5. Feb 1, 2013

### hihiip201

Got it! vk is actually just v of the right side of spring - vm. so that makes sense that vm cancel out.

but the reason why they used vk is because they want to replace it with the elemental equation 1/k * dfk/dt in the de to develop a 2nd order differential equation for the velocity.

thanks HallsofIvy.

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