Thread: Partial differential equation - spring and damper. Need help with setup and solution

1. Partial differential equation - spring and damper. Need help with setup and solution

I have a system of a point mass m connected to a base with one spring and one damper in parallel. There is 0 gravity, 0 air resistance. The base has a constant acceleration towards the point mass m which of course is free to move away from the the base. Speed at time 0 is 0 for both objects. Inertia of the point mass is I guess the only thing allowing the base to get closer than initial distance.

I do not need to plot continuous curve, but I need to know the relative change in position between the base and the point mass at a given time.

m=100kg
damper coefficient c=variable
spring constant k=variable
constant acceleration a= 6g= 6x9.81m/s^2
time = 0.25s

my formula looks something like this
mx''+ cx' + kx = c*a*t + 1/2*k*a*t2

My particulate solution looks something like this: Xp(t)=1/2*a*t2 - m*a/k

This is pretty much as far as I have come. I do have a homogenous solution also, but it depends greatly on the value of k and c as you do know.

Could anyone help me to assess this?

2. Okey, I have found the need to complicate things a bit. I cannot use constant acceleration for the base. I need to use a pyramid function based on time. Time runs from 0 to 180 milliseconds (0 to 0.180 seconds)

Now my equation looks like this:

. . .$m\, x''(t)\, +\, c\, x'(t)\, +\, k\, x(t)$

. . . . . . .$=\, c\, t\, \left(-\dfrac{2000}{3}\, \times\, \dfrac{(t\, -\, 0.09)^2}{\big|0.09\, -\, t\big|}\, +\, 60\right)\, +\, 0.5\, k\, t^2\, \left(-\dfrac{2000}{3}\, \times\, \dfrac{(t\, -\, -0.09)^2}{\big|0.09\, -\, t\big|}\, +\, 60\right)$

Can anyone out there help me solve it for an overdamped situation? I do not need solution for underdamped or critically damped at this moment.

The other thing I am considering is if I need to add gravity load also? The acceleration is the actual acceleration measured on the base, and I have initially thought to consider only internal system forces thus neglecting gravity.

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