# How does one mathematically derive the damping coefficient of a theoretical viscous dashpot?

I am very well aware of how to get the damping coefficient experimentally by observing a system in action.

Given the dimensions and fluid properties of a theoretical viscous fluid dashpot, how does one calculate the damping coefficient?

I found this website, which seems to have a calculator that does just that, but I cannot find where they get their formula: http://www.tribology-abc.com/calculators/damper.htm

I want to design an damper with a specific damping coefficient, and would love to be pointed in the right direction.

• Welcome New contributor ProbablyAProfessional! One of the criteria for a good question here is that the question shows sufficient prior research. After reading your question, I Googled "How to design a dashpot" and immediately found A STUDY OF THE CHARACTERISTICS OF DASHPOTS: SOME DESIGN CRITERIA FOR HYDRAULIC SHOCK ABSORBERS. Perhaps you've read this and didn't find what you're looking for. Regardless, I'm down-voting this question because it does not show any research effort. – Alfred Centauri Sep 19 '18 at 1:39
• If you have the calculator, why do you need a formula? Insert what values you must have, then adjust the variable values until you get the desired coefficient. – sammy gerbil Sep 19 '18 at 10:25
• The reason is that this was the only calculator I’ve ssen like this and writing in my master’s thesis “results were found via this online calculator that doesn’t list its equation” isn’t good form. – ProbablyAProfessional Sep 19 '18 at 13:10
• In my answer, I showed you the derivation of their equation. What part don’t you understand? – Chet Miller Sep 19 '18 at 14:23
• Apologies Chester, I was responding to Sammy Gerbil when they were asking why I needed the formula when I had an online calculator. You were very helpful. – ProbablyAProfessional Sep 19 '18 at 18:38

It seems pretty straightforward how they derived the hidden formula for the damper constant in their idealized design. The volumetric flow rate of fluid displaced by the piston would be $$Q=VA$$where V is the piston velocity and A is the cross sectional area of the piston. This is also the volumetric flow rate of fluid through the small tube of diameter d. The force on the piston is $$F=A\Delta P$$where $\Delta P$ is the pressure difference across the piston. The relationship between the pressure drop and the volumetric flow rate through the small tube is determined by the Hagen Poiseulle equation for laminar flow in a tube: $$\Delta P=\frac{128 QL}{\pi d^4}\mu$$where L is thickness of the piston and $\mu$ is the fluid viscosity. Just combine these equations to get F/V.