Transverse Vibrations of a Beam Using Hamilton's Principle

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Transverse Vibrations of a Beam Using Hamilton's Principle

Рет қаралды 12,561

Күн бұрын

Deriving the equations of motion for the transverse vibrations of an Euler-Bernoulli Beam using Hamilton's Principle.
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Пікірлер: 31

@alibagherinia4851 12 күн бұрын

Very smooth and useful video. Thank you sooo much

@vinayakvarma4887 3 жыл бұрын

Great stuff! If possible, please also try to make video(s) on Rayleigh-Ritz FEM for vibration problems on beams, bars, etc. Thanks for the great content!

@user-oy2by6oz3x 11 ай бұрын

Love from INDIA!!

@arguelar 2 жыл бұрын

Thank you! the explanations helped a lot!

@MV-xm4xg Жыл бұрын

Really really good! Thanks

@birukterefe1517 2 жыл бұрын

This is awesome. Your video was helpful. Thanks.

@singerice3671 9 ай бұрын

Thanks, a good start for me

@sagaravijeetshukla 3 жыл бұрын

I love your videos. Kindly solve some problems which will let us understand how and where to use the Lagrange's and Hamilton's principles deeply and make us able to deal with real life problems in structural dynamics!

@Freeball99 3 жыл бұрын

I have two videos to go and then I will be working many different examples.

@Altekameraden79 2 жыл бұрын

For anyone interested in vibration analysis, RDI Technologies has some videos on KZfaq showing their motion amplification capabilities. Using the RDI cameras to view low frequency resonance on buildings always creeps people out. PS With a voice like that, Freeball should be contracted to narrate mechanics textbooks to an audible source.

@Jeddhu 3 жыл бұрын

Thank You So Much for this video. Really Appreciate it

@themightyquinn100 2 жыл бұрын

Outstanding!

@info-dawg 3 жыл бұрын

You're a god. Thank you for this.

@Omar-sj7wl 3 жыл бұрын

Keep up the great work 👍

@souviksinghrathore5423 3 жыл бұрын

Great to learn... thanks

@pauloedmachado9137 3 жыл бұрын

Great video! As a undergrad engineering student, I've been doing a research on wind turbine towers vibrations basically using all of this. Do you know of any 3D version of those beam vibration equations? Thank you very much

@Freeball99 3 жыл бұрын

What you are asking about is a "frame" element. It will have 6 DOFs at each end node (3 translations and 3 rotations) and will handle bending in two transverse directions plus deformation in the axial direction (like a bar) plus torsion (like a shaft). You should be able to find many examples of frame elements online.

@antonventer8532 Жыл бұрын

Dankie vir die video man. Ek wil net graag weet. Is die delta n derivative of is dit n variable?

@Freeball99 Жыл бұрын

No 𝛿 is the variational operator and not a variable. It is similar to the differential operator, d, but operates on functionals rather than on functions. I have a video explaining it: kzfaq.info/get/bejne/rNd0e7JjyrG8k6c.html

@burakguclu3380 3 жыл бұрын

Thank you sir! Great video. Can you briefly explain again why we equate (delta)W to zero between t1 and t2

@Freeball99 3 жыл бұрын

To be clear, 𝛿W = 0 at t1 and at t2 and 𝛿W is arbitrary between t1 and t2. Consistent with Hamilton's Principle, it is assumed that the state of the system is know at t1 and at t2, therefore the variation of the displacement at these points is zero. Hamilton's Principle is thus used to determine the path that the system takes from t1 to t2.

@mayureshsalunkhe3732 Жыл бұрын

Hello sir! First, thanks for that great video. I have one question: if the beam has a circular cross-section, what method should be used to solve the problem? Is it something we need to solve in polar coordinates?

@Freeball99 Жыл бұрын

The current method makes no assumption as to the shape of the cross-section. Does not require polar coordinates. Changing the shape of the cross-sections should only affect the value of I (the moment of area).

@arsnova69 3 жыл бұрын

Splendid work there! Can you give us a lead why every term of the integral must separately be zero? I understand that δw' is indeed a different entity, but why the other boundary terms?

@Freeball99 3 жыл бұрын

In the case of the integral...since δw is arbitrary, therefore what multiplies it must be zero (this applies to every point along the beam). In the case of the boundary terms, these must be zero independently because each of them only applies at a single point - there is no way that what happens with the integral for every point can always zero out the effects of a single point (the boundaries), so these must go to zero independently.

@arsnova69 3 жыл бұрын

@@Freeball99 thank you! That was really helpful!

@vijayr1485 3 жыл бұрын

How did you know when to switch order of integration?

@Freeball99 3 жыл бұрын

In order to integrate by parts, one needs to consider the derivative that is under the integral sign (in other words, do we have a prime or a dot on the variation?). If the derivative is with respect to time, then we must integrate by parts with respect to time. If the derivative is with respect to x, then we must integrate by parts with respect to x. We are free to switch the order of integration to accommodate this. Ultimately, we re-write the equations with the time integral first, because this is the form of Hamilton's Principle. Hope this makes sense.