Measuring the natural frequency of a spring-mass system with the graph 
On a graph of a system under a external force y = distance and x = time where the external force start at t = 0, it's easy to find the driving frequency.
$$F = \frac{\omega}{2\pi}, \omega = \frac{2\pi}{T}$$ and we can get $T$ easily with the steady state part of the graph.
However, is there a way to find the natural frequency?
Maybe by finding where $w_d = w_0$, which is the resonance frequency.
I made a graph. Can I consider steady-state amplitude as the driving force amplitude?
 A: If this is real data (or if you have real data) the first part of the graph may represent what is called the transient regime. In this regime the motion is a superposition of two motions, one with the natural frequency and another with the driving frequency. After some time the motion with natural frequency dies out.
So, if you have enugh data for the transient regime to do a decent Fourier analysis you may be able to see two peaks in the spectrum, one for each of the two frequencies.
But I am not sure it will work for this curve.
A: Obtaining the natural frequency is difficult in the presence of a sinusoidal driving force.
The unforced frequency of oscillation, $f$, is not the natural frequency $f_{N}$. But they are related by the damping ratio, $\zeta$.$$f = f_{N} \sqrt{1-\zeta^{2}}$$
One way is to change the frequency of the drive until you obtain a resonant peak in amplitude. The frequency obtained this way is the same as the unforced frequency of oscillation, $f$, then if $\zeta$ is small, the natural frequency is approximately the same.
If you can, apply an impulse, or displace the system then release it. The cycle period of the resulting decaying sinusoid is used to calculate $f$. To find $\zeta$, apply logarithmic decrement. Then calculate the natural frequency.
The natural frequency is not an oscillation. It is a parameter of a second order system along with the damping ratio. For $\zeta <0.1$ the difference between the actual oscillation frequency and the natural frequency may be insignificant.
This discussion is for $2^{nd}$ order systems. Higher order systems can have many natural frequencies.
