# Cantilevered Beam Oscillation

I'm trying to model the damped oscillation of a cantilevered beam for a project I'm doing. Really I want to know how "stable" this system is going to be given an initial displacement. My main problem is I'm not familiar with typical damping coefficients and material properties (or even how to estimate them). Furthermore, I don't know if my system is going to be under, over, or critically damped.

The beam is made out of aluminum that's L in length with a rectangular cross-section b and h. The relative magnitudes of L, b, and h pretty much make the system look like a diving board. Now if I displace the very end of my diving board some δy how long will it take to effectively die out.

I'm interested in displacements on the order of microns or less. So far I've modeled my system as a damped harmonic oscillator with L = 500 mm, b = 100 mm, h = 50 mm.

$Y(t) = A e^{-\alpha t} cos(\omega t)$
where
$\omega = \sqrt{k/\omega - \alpha^2}$
$\alpha = \frac{C}{2I}$
and C is some damping coefficient in $\frac{Nm \cdot sec}{rad}$, I is the moment of inertia in $kgm^2$, and k is the spring constant in $\frac{Nm}{rad}$

Any ideas on how to estimate k and C for this rectangular beam made of aluminum? There's no physical damper, and my best guesses as to how the energy is dissipated would be through friction (like bending paper clip back and forth). Am I even modeling this correctly?

SteamKing
Staff Emeritus
Homework Helper
If you were to displace the free end of the cantilever some small distance delta, how much force would be required? That would give you a handle on k.

AlephZero