Simple cantilever beam deflection - what is the simplest way to add a damper?

Question

I am looking at a simple cantilever beam deflection:

I understand the general expression for deflection/force would be:

$y_s = \frac{Fx_s^3}{3EI}$

$F_p = \frac{3y_sEI}{x_s^3}$

If you were going to add viscous damping to the bending of the beam, would it be as simple as:

$F = \frac{3y_sEI}{x_s^3} - cEIθ_t$

Where the equation for the angle of deflection is $θ = \frac{FL^2}{2EI}$?

I have seen some suggestions that simple damping of cantelever beams is done by applying viscosity to the rate of angle change with respect to time. Is that generally correct?

I have had some strange behaviors trying this so I'm not sure what the ideal simple solution is.

Thanks for any help or answers/ideas for either question. It is appreciated.

Answer 1

Viscosity is a fluid's resistance to shear, so modeling its effect depends on some fluid parameters. I don't know much about fluid dynamics, so that's all I can say there.

However, it's pretty simple to include some basic damping effects. Simply apply a distributed force similar to $F_d(x, t)= -k\frac{\partial y(x, t)}{\partial t} $, where $k$ is some constant. (This expression holds only in the small deflection limit.)

The intuition here is that the parts of the beam moving faster are resisted upon more. How this distributed force affects $F_p$ depends on the boundary conditions of the cantilever, but I think you can take it from here.