Question

In simple harmonic motion, k is obviously (natural frequency)^2*m, but it is just assumed that this is the case with friction, forcedness of the oscillator and damping added- how is this justified?

Answer 1

Solve this differential equation with some initial condition.
\[ m \ddot{x} + c \dot{x} + k x = 0\]
if over-damped you will get this

if critically damped

if damped but follows harmonic motion you will get this

Answer 2

um-damped case, c = 0

and your natural angular frequency (of the system) is
sqrt(k/m)