Why are there modes in cantilever beam oscillation equations

[actualalias]

I'm doing an experiment measuring the relationship between length of a cantilever beam and period of oscillation when I twang it on one end, but I can't seem to understand the equation. The equation for measuring frequency is given here:https://www.hindawi.com/journals/amse/2013/329530/
but I don't understand how the mode has anything to do with it. I am just twanging a cantilever beam by my hand and measuring the period, so in that case would it be resonating in its fundamental frequency or all modes at once? Someone told me that the higher frequencies die out quickly leaving only the fundamental frequency, but is this true, and can someone please give me a link that can teach me this? Thanks so much.

 

[Andy Resnick]

Ibut I don't understand how the mode has anything to do with it. I am just twanging a cantilever beam by my hand and measuring the period, so in that case would it be resonating in its fundamental frequency or all modes at once? Someone told me that the higher frequencies die out quickly leaving only the fundamental frequency, but is this true, and can someone please give me a link that can teach me this? Thanks so much.

The idea is that the motion of the beam, that is solutions to equations 3-8, can be more easily analyzed by 'decomposing the motion into normal modes' (equation 9). In some sense this is similar to a Fourier decomposition of a time-varying signal In general, many modes may be simultaneously excited, each with their own frequency and damping rate.

Can't think of an alternative link right now, the paper seemed clear enough. What are you struggling with?

 

[actualalias]

The idea is that the motion of the beam, that is solutions to equations 3-8, can be more easily analyzed by 'decomposing the motion into normal modes' (equation 9). In some sense this is similar to a Fourier decomposition of a time-varying signal In general, many modes may be simultaneously excited, each with their own frequency and damping rate.

Can't think of an alternative link right now, the paper seemed clear enough. What are you struggling with?

I just need to do an experiment to investigate the relationship between the period/frequency of oscillation of a beam and the oscillating length, and I thought there would be a direct relationship. After graphing my results I found that the period was proportional to the oscillating length squared, which is consistent with the equation. However I don't understand the modes part, as I just twanged the beam to find the period. I filmed the oscillations and it sure looked like it was oscillating in the first mode, but I can't just ignore the modes. The reason I need a link is because I want to cite the information I'm getting. Thanks again.

Edit: I should mention I don't know how to solve differential equations and equations 3-9 don't make any sense at all to me. Also someone told me the higher modes die out because of the faster energy dissipation, leaving only the fundamental mode. Is this true? Can someone provide me a source for this? Thanks so much.

 

[Spinnor]

How did you "twang" the beam, did you deflect the end and then release it? If so I think that method of getting the beam in motion favors most of the vibration energy being in the first mode, though I can't right now think of the argument for my hunch?

 

[Nidum]

In first mode the dynamic deflection curve of a uniform cantilever is similar in shape to the static deflection curve for the beam as loaded by it's own weight .

At any instant the dynamic deflection curve approximates to an ordinate scaled version of the static deflection curve .

A small point load at the outer end of the beam distorts the shape of the self weight static defection curve but only by a small amount .

In the 'twanging' procedure a point load is applied to the end of the beam and then rapidly removed .

When the point load is removed the cantilever beam is already quite near to being in the correct deflected shape to enter first mode vibration preferentially to other modes .

To initiate vibration in higher modes needs a more complicated hold and release procedure . Basically the beam is held at two or more locations so as to get an approximation to a higher mode deflection curve and then released at all points simultaneously . Bit academic since with real beams the higher modes will die away very rapidly .

 

[Andy Resnick]

Edit: I should mention I don't know how to solve differential equations and equations 3-9 don't make any sense at all to me. Also someone told me the higher modes die out because of the faster energy dissipation, leaving only the fundamental mode. Is this true? Can someone provide me a source for this? Thanks so much.

Ah- then this thread should probably get changed to 'B', rather than 'I'.

I'm not sure how to start- using a differential equation to describe motion is required when you are given, for example, an initial state of motion and then you need to determine how the system evolves.

In terms of your 'plucked' cantilever, this may be of some help:

http://www-personal.umd.umich.edu/~jameshet/IntroLabs/IntroLabDocuments/150-12 Waves/Waves 5.0.pdf

A related discussion, concerning tuning forks:
http://physics.stackexchange.com/questions/266008/tuning-fork-clanging-mode-boundary-conditions

 

[Spinnor]

There are two pieces of information online relevant to this problem.

1. Beam deflection with an end force, static,

Deflection = Cx^2(3L-x) (Tried to copy and paste a Wiki image but can't, I can in the draft though?)

https://en.wikipedia.org/wiki/Deflection_(engineering)

2. Beam deflection, vibrating beam, different modes,

,

http://iitg.vlab.co.in/?sub=62&brch=175&sim=1080&cnt=1

We guess the first expression, for small displacements, nearly equals the second expression for n = 1? The first expression is more properly equal to an infinite sum of the second expression for n = 1, 2, 3, ... ? Or by some symmetry arguments one can eliminate some of the n's?

Thanks!