Why is the natural frequency of a column at the buckling load?

In summary, the conversation discusses the concept of natural frequency and its relationship to the critical buckling load of a uniform column. It is explained that when the natural frequency reaches zero, the column can vibrate with infinite amplitude without requiring additional energy input. However, in practice, there are destabilizing forces that can cause the column to buckle. Various sources and studies are referenced for further information on this topic.
  • #1
theBEAST
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So apparently the natural frequency is zero for uniform column with axial load when P is equal to the critical buckling load. Could anyone please explain theoretically why this is the case.
 
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  • #2
Ignoring any damping forces acting on the column, when it is vibrating at its natural frequency, the total energy (kinetic energy + internal strain energy) is constant.

As you increase the compressive force and the natural frequency gets lower, you need less energy to make it vibrate with a given amplitude, because the velocity is smaller and therefore the KE is smaller.

When the frequency reaches zero, in theory you don't have to supply extra energy to start it "vibrating" with "infinite" amplitude (but infinitely slowly, of course!).

In practice, there is always some destabilizing force (caused by geometrical tolerances, off-center loads, etc) that can do work (force x distance) to start the "vibration".

Note, the above only applies to starting the "vibration" (or buckling). Once it has started, the behaviour is nonlinear and the motion is not "infinitely slow", as is obvious in real life when things buckle!
 
  • #3
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1. Why is the natural frequency of a column at the buckling load different from other loads?

The natural frequency of a column is dependent on its length, material properties, and boundary conditions. At the buckling load, the column experiences a critical point where small disturbances can lead to large deformations and ultimately failure. This results in a change in the stiffness of the column, causing a shift in its natural frequency.

2. How does the natural frequency of a column at the buckling load affect its stability?

The natural frequency of a column at the buckling load can indicate the potential instability of the column. If the excitation frequency of the column matches its natural frequency, it can lead to resonance and cause significant vibrations, which can ultimately lead to failure. Therefore, it is essential to consider the natural frequency of a column when designing for stability.

3. Is the natural frequency of a column at the buckling load constant?

No, the natural frequency of a column at the buckling load is not constant and can vary depending on different factors such as the column's length, material properties, and boundary conditions. It can also change if any external loads or disturbances are applied to the column.

4. How can the natural frequency of a column at the buckling load be calculated?

The natural frequency of a column at the buckling load can be calculated using mathematical equations, such as the Euler buckling formula or the Rayleigh-Ritz method. These equations take into account the column's length, material properties, and boundary conditions to determine its natural frequency at the buckling load.

5. Can the natural frequency of a column at the buckling load be altered?

Yes, the natural frequency of a column at the buckling load can be altered by changing its length, material properties, or boundary conditions. For example, increasing the stiffness of the column can result in a higher natural frequency, while increasing its length can decrease the natural frequency. Altering the column's boundary conditions, such as adding supports, can also change its natural frequency at the buckling load.

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