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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?

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    $\begingroup$ The damping coefficient is mostly depend on the oil viscosity $\endgroup$
    – Eli
    Commented Mar 22, 2019 at 13:37

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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.

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  • $\begingroup$ So would the $1/d^2$ proportionality then be affected by these other forces? Can I say that b is proportional to $1/d^2$? $\endgroup$
    – Jack Ceroni
    Commented Mar 21, 2019 at 21:45
  • $\begingroup$ There isn't necessarily even anything that looks like "b". For example, the standard model of hysteretic damping has the equation of motion (in the frequency domain) as $(k(1 + i \eta) - \omega^2 m)x = f$, not $(k + i \gamma \omega - \omega^2m)x = f$. The hysteretic damping term isn't multiplied by $\omega$, like the viscous damping term. $\endgroup$
    – alephzero
    Commented Mar 21, 2019 at 21:57
  • $\begingroup$ But theoretically, if there was something that looked like b, would it be proportional to $1/d^2$? $\endgroup$
    – Jack Ceroni
    Commented Mar 21, 2019 at 23:13

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