# Cantilevered Beam Oscillation

## Main Question or Discussion Point

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?

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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
Homework Helper
There are several different sources of damping in this sort of system. The main ones will be
1. Hysteretic damping in the material.
2. Aerodynamic damping as the structure moves the surrounding air about
3. Friction at the "joint" where the structure is "rigidly" clamped.

As a rule of thumb, 1 + 2 usually give between about 2% and 5% of critical damping.
3 can be anywhere between neglible and surprisingly large (even approaching critical), but it you have a clamping system that has good geometrical tolerances (flat parallel surfaces, etc). a high clamping load, and the surfaces are completely free of oil and grease etc, it should be negligible.

The best way to estimate the effective stiffness and mass of the beam is from an undamped vibration model (this is standard theory, Google should find plenty of references). Note, the effective mass is not the same as the total mass of the beam, because different parts of the beam are moving by different amounts.