Mechanical impedance and dynamic stiffness of a mass, spring, damper system including Coulomb friction

I’m trying to understand the concepts of mechanical impedance and dynamic stiffness, what do they mean and if/how they differ. Consider the very simple system below:

     
   Image curtesy of Joshua Vaughan  

 

with a dynamic equation of:

$$
begin{cases}
if , dot{x} = 0 , and , f – kx – cdot{x} < f_s , implies , ddot{x} = 0 , ,
else , implies , mddot{x} = f – kx – cdot{x} – f_k , text{sign}(dot{x})
end{cases}
$$

where

• $f$ is the response force
• $f_s$ is the maximum static friction
• $f_k$ is the constant kinetic friction
• $x$ is the input position imposed externally

So to my best understanding so far, in order to calculate the mechanical impedance of the above system we need to do a Laplace transform on above equations, while dynamic stiffness is a Fourier transom of them. So my questions are:

• What are the Laplacian and Fourier transform of the above equation?
• Are dynamic stiffness and mechanical impedance in ay way possible related or similar/identical? (According to this text they are the same)

P.S.1. I think the Lapace transform of the equation of motion should be something like:

$$m left( s^2 X – s x_0 – dot{x}_0 right) = -kX -cleft( sX- x_0 right) -F_f + F $$

however, I have no clue what the term $F_f = f_k mathscr{L} { text{sign}(dot{x}) }$ should be.

P.S.2. shared also here on #vibrations Discord channel.