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

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:

$$F = \frac{3y_sEI}{x_s^3} - cEI\theta_t$$

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

I have seen some suggestions that simple damping of cantilever 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.

The suggestion that "simple damping of cantelever beams is done by applying viscosity to the rate of angle change with respect to time" is correct.

However, the expression you provided

$$F = \frac{3y_sEI}{x_s^3} - cEIθ_t$$

where: $$θ = \frac{FL^2}{2EI}$$

is not appropriate.

The reasons are:

1. you use angle not angle rate .
2. for different points in the beam the angle and the corresponding rate will be different, so you'd have to use calculus to get the correct expression.
• Thanks NMech. By $θ_t$ I meant the derivative of $θ$ by time which would be the angle rate, right? If I'm only applying pressure to one point on the beam at $x_s$, and using the angle rate at this point, would this then be valid? Or would it still be a poor method? I figure one way to solve this would be to instead solve the beam as a finite difference model using something like $F = ρAy_{tt} = -EI(y_{xxxx} + c*y_{xxxxt})$ but I was hoping for a simpler solution just based on the fact that I only need one point to apply pressure to and measure from. Is there an easier approach?
– mike
Feb 1 at 4:47
• @mike On the way to find a feasible solution, you need to define the physical model (which you did), and the goal you wanted it to achieve (which I don't think it is clear). I am quite confused on what the "damper" is expected to do in such setup.
– r13
Mar 3 at 16:42