How to analytically derive the equation of the amplitude of the force being transmitted to the ground of a damped, forced vibration system

Question

For this question, I really have tried to produce some working but the entire thing is just deeply confusing me.

I know my equations for the displacement of this mass will be as follows:

I have successfully derived these equations from the freebody diagram of the mass, and understand where these equations come from. However, when it comes to determining the force being applied to the ground from these equations, I have no clue. How would I go about doing this, as I am completely lost.
I was previously under the assumption that Xf was simply the force being transmitted to the ground, but surely that can't possible be the case.
Any help would be greatly appreciated, thanks

JAlex · Accepted Answer

The ground force is the sum of the spring and dashpot forces
$$ F_{rm ground} = k x  + c dot{x} $$
The is because the spring and damper are just force members (equal and opposite forces on their ends) since they are assumed massless.
This means the free body diagram should yield the following
$$ P(t) - F_{rm ground} = m ddot{x} $$
which you use to find a solution.

How to analytically derive the equation of the amplitude of the force being transmitted to the ground of a damped, forced vibration system

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