# Should damping ratio increase or decrease with increase in mass?

I'm currently doing a small project in college for structures being damped. We're adding weights to a structure and measuring the damping affect. We've calculated the damping ratio and coeffecients, however, we currently have damping ratio increasing with mass, which I do not believe is correct following the below equation:

$$\zeta=\frac{c}{2\sqrt{mk}}$$

The above equation, assuming $$k$$ and $$c$$ are constant, suggests that $$\zeta$$ decreases as mass increases.

However, the results we are getting are entirely different, and we do not have a constant value for the damping coefficient:

We got the initial value of damping ratio by interpreting the graphs by using the following equation and process:

$${\sqrt{\frac{\left(\ln \left(\frac{x_n}{x_0}\right)\right)}{4(\pi n)^2+ \left(\ln \left(\frac{x_n}{x_0}\right)\right)}}} = \zeta$$

Where the above was rearranged from:

$$\Large{\frac{x_n}{x_0}= e^{\frac{-2\pi n \zeta }{\sqrt{1-\zeta^2}}}}$$

where $$n=$$ the number of full wave cycles between $$x_0$$ and $$x_n$$

Ultimately, I'm curious if the damping ratio should decrease with increase in mass, and should the damping coefficient stay constant?

Edit 2: Graphs for Unweighted vs 4.004 damping

• Is the following the protocol you followed? : (1) Find $\zeta$ using the formula that relates the ratio of two amplitudes to the damping ratio : To be more specific about your calculation, as an example, the $n$ for the two amplitudes marked in red circles in your plot is $1$, right? ; (2) Find the critical damping coefficient using the already-known value of the mass and spring constant ; (3) Use $\zeta$ and $c_{\text{critical}}$ to find the damping coefficient $c$. – Ajay Mohan Dec 10 '19 at 12:48
• Yeah that's what we done exactly, n is one. – Shaun Cockram Dec 10 '19 at 14:23

I assume that your structure is a simple single spring mass damper system? For such a system the assumption of decreasing dampingratio $$\zeta$$ for increasing mass $$m$$ is correct. You need more dampingforce to stop a heavier system, therefore resulting in a smaller $$\zeta$$.
Edit1: Added image that shows the expected plot for different $$\zeta$$ values and different masses. The given data is flipped. My assumption are $$c=2.5 kg/s$$ and $$k=162~kg/s^2$$.