What does the viscous damping coefficient depend on?

Question

I’m doing a theoretical calculation involving the damping on an oscillating string, and I found the following relationship, where a certain damping factor $b$ is proportional to $frac{c}{d^2}$ where $c$ is the viscous damping coefficient of the string and $d$ is the diameter of the string.

I was wondering, would it be fair to say that this $b$ value is proportional to $frac{1}{d^2}$ or is it possible that the viscous damping coefficient could depend on the diameter of the string, therefore making it so that this proportionality isn’t true?

Basically, I’m asking what factors the viscous damping coefficient depends on (mass, surface area, etc), and if any of these factors could in turn depend on the diameter itself?

alephzero · Answer

The "viscous damping" considered in a first course on dynamics is basically a mathematical fiction which is easy to analyse and demonstrates the basic ideas of under- and over- damped motion.

Physically real damping has many different causes, with the common factor that they dissipate mechanical energy from the system being modelled. Either they transport mechanical energy away from the system, or the convert it into another form, usually heat.

Most real damping mechanisms are not proportional to velocity. However for lightly damped systems, it is often a good enough approximation to assume an equivalent amount of viscous damping which dissipates energy at the same average rate as the real damping.

In a "real" oscillating string there may be hysteretic damping within the material of the string itself, aerodynamic damping transferring energy (a.k.a. "sound") into the air, and energy losses caused by imperfectly "fixed" boundary conditions. Those three mechanisms usually account for most of the energy dissipation, but neither hysteretic damping nor aerodynamic damping are proportional to velocity.

What does the viscous damping coefficient depend on?

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