Mind blown by mechanical systems (2nd order)

[hihiip201]

Homework Statement

B, K, M

Homework 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

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 homework solution suggest otherwise.

 

[HallsofIvy]

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

 

[hihiip201]

Do you not understand what a "damper" is? It is a way of introducing friction into the equation, so "damping" (reducing) the motion. Often it is thought of a some sort of hydraulic piston moving through water, but ordinary friction as an object moves over the floor will do. The whole point is, and I am sure that this is given in your text, that we can take the friction force to be proportional to, and in opposite direction to, the speed- the faster the object moves, the more the friction opposes the motion. The friction force is typically $\alpha |v|$ where $\alpha$ is some negative number. As long as your object is moving in one specific direction, you can remove the absolute value and put the sign into the constant.

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).

 

[hihiip201]

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

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.

 

[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.