What is the stability of a cantilevered beam subjected to damped oscillation?

In summary: Ideally, you can also do a finite element analysis if you have access to that software.In summary, the system is stable if the displacement is within the range of displacements that the material is able to damp. If the displacement is too large, the system will over-damp and eventually fail.
  • #1
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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.

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

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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  • #2
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.
 
  • #3
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.
 

FAQ: What is the stability of a cantilevered beam subjected to damped oscillation?

1. What is a cantilevered beam oscillation?

A cantilevered beam oscillation is a type of mechanical vibration that occurs in a beam that is anchored at one end and free to move at the other. This type of oscillation is caused by external forces acting on the beam, such as a force applied at the free end or a disturbance in the surrounding environment.

2. What factors affect the oscillation of a cantilevered beam?

The oscillation of a cantilevered beam is affected by several factors, including the mass and stiffness of the beam, the magnitude and direction of the external forces, and the damping properties of the material. Additionally, the length and thickness of the beam can also impact its oscillation behavior.

3. How is the natural frequency of a cantilevered beam calculated?

The natural frequency of a cantilevered beam can be calculated using the formula f = 1/(2π) * √(k/m), where f is the natural frequency, k is the stiffness of the beam, and m is the mass of the beam. This formula assumes a single degree of freedom and neglects any damping effects.

4. How does the damping affect the oscillation of a cantilevered beam?

Damping is the dissipation of energy in a vibrating system, and it can greatly affect the oscillation of a cantilevered beam. Higher levels of damping can reduce the amplitude of the oscillation, while lower levels of damping can result in a longer duration of oscillation. Damping can also cause the natural frequency of the beam to shift.

5. What are some real-world applications of cantilevered beam oscillation?

Cantilevered beam oscillation has many practical applications, including in engineering and structural design, where it is used to study the behavior of bridges, buildings, and other structures under external forces. It is also commonly used in sensors and actuators, such as in the atomic force microscope, where the oscillation of a cantilevered beam is used to measure surface properties at the nanoscale.

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