So I decided to calculate damping coefficient. I will make a mass on a string oscillate in a water (somehow) and then using camera will plot the graph of time versus altitude.
Now I know that the maximum altitude points together forms another line (exponential), which slope, when linearised, will give me the damping coefficient. My question is – where could I use this coefficient, what would be its practical value and is there anyway I could compare it with something just so it doesn't look completely random?
Best Answer
It sounds like you are dealing with a second-order system based on the oscillatory nature you described. The governing equation for your system is simply,
$$ \frac{1}{\omega_n^2}\ddot{y} + \frac{2 \zeta }{\omega_n} \dot{y} + y = K F(t) $$
where $y$ is the displacement, $\omega_n$ is the natural frequency, $\zeta$ is the damping coefficient, $K$ is the sensitivity, and $F(t)$ is the forcing input function to your system, respectively. The value of $\zeta$ is going to govern the response of your system. For instance, here is a classic chart that demonstrates the response of a second-order system from a unit step input.
Notice the varying degree of behavior based on the different damping coefficients $\zeta$ for each system. If $\zeta = 0$, the system is undamped and the step input will simply send the system into undamped free oscillation. If $0 < \zeta < 1$ we call the system underdamped and we observe prolonged oscillatory motion as the system returns to the value of the step input. If $\zeta = 1$ the system is critically damped, and there are no observed oscilations from a step input. Lastly, if $\zeta > 1$ then the system is overdamped. From a step input, you can easily determine the systems damping coefficient as you mentioned. It is termed the logarithmic decrement method and is given by,
$$ \zeta = \frac{\ln (y_1/y_2)}{\sqrt[]{4\pi^2 + \ln\left(y_1/y_2\right)^2}} $$
It is not entirely clear what your goal is with this setup. Are you setting up a pendulum system in water? That was my take away, but I may have misunderstood your description. If so, then I would anticipate a damping coefficient $\zeta$ to be less than 1, but probably on the high end near unity. Say $\zeta \approx 0.7-0.9$. Water is a fairly viscous fluid and the viscous dissipation will be large, but probably not enough to critically damp or even overdamp the system.
If you are considering oscillating the mass up and down in the water, then you will most likely be providing some form of a sinusoidal input. In this case the response of your system for different values of $\zeta$ will look similar to these classic forced vibration charts.
The main takeaway for you is how $\zeta$ changes important features of the behavioral response like the magnitude ratio (amplitude response of your system) and the phase shift. However, this is all dependent on what is your goal of your system.