# Natural period of vibration

1. May 17, 2012

### svishal03

Hi,

I had been reading the book on 'Dynamics of Structures' authored by Prof.Anil K Chopra of University of Berkeley.

In his book, the author has provided the natural period of vibration of some buildings like:

1) Transamerical building of San Francisco- California which is 2.90 seconds for vibration in north-south direction which was obtained by forced vibration tests

My question is when under a forced vibration the building will not vibrate 'naturally'. I mean the vibration characteristics during a forced vibration are not now only dependent on the natural characteristics of the system . Right?

Hence, the tiem period obtained (time required to complete one cycle of vibration) is not just the property of natural characteristics of the building but also the force applied.

In such a case how natural period was obtained experimentally?

Is it that they make the building go to resonance and instead measure the time period of the applied force during such an experiment?

2. May 17, 2012

### haruspex

I would assume that the experimenters tuned forcing period until resonance detected, so there's no reason for it to misrepresent the natural period. Do you have any info to the contrary?

3. May 18, 2012

### svishal03

I was just trying to ascertain that, what I'm thinking is right.

So, you mean, the natural period of vibration is detected (or obtained) by measuring the forcing period corresponding to the detection of resonance?

4. May 18, 2012

### haruspex

Yes, but I am only guessing as to the procedure.

5. May 18, 2012

### sophiecentaur

I have a feeling that the way to measure the natural period is to record its movements continually. Analysing all the data would, I think, give a peak at the natural frequency. Measurements would need to be over a very long period.
This is how the natural period of an organ pipe reveals itself (in a much shorter time) because, when it's excited with random air movements (the turbulence of the air from the blower) it forms its note.

6. May 18, 2012

### HallsofIvy

A structure with natural frequency $\omega$ satisfies the differential equation $y''+ \omega^2 y= 0$ which has general solution $y= C_1cos(\omega t)+ C_2 sin(\omega t)$. If there is a forcing function of the form $A sin(\alpha t$, then the differential equation becomes $y''+ \omega^2 y= A sin(\alpha)$. The general solution to that is the previous $C_1 cos(\omega t)+ C_2 sin(\omega t)$ plus any one solution to the entire equation. IF $\omega\ne \alpha$, then we can look for a "specific" solution to the entire equation of the form $y= C sin(\alpha t)$. Then $y'= C\alpha cos(\alpha t)$ and $y''= -C\alpha^2 sin(\alpha t)$ so, putting that into the equation, we have $-C\alpha^2 sin(\alpha t)+ \omega^2 Csin(\alpha t)= C(\omega^2- \alpha^2)sin(\alpha t)= A sin(\alpha t)$ so that we have
[tex]C= \frac{A}{\omega^2- \alpha^2)[/itex].

Of course, if $\alpha= \omega$ that cannot be used. Instead, we have a solution of the form "$Cx sin(\alpha x)$". That is "resonance". At that particular frequency, $\omega$, the forcing term causes larger and larger oscillations until the building collapses. That is, the frequency at which resonance occurs is the natural frequency.

7. May 18, 2012

### sophiecentaur

Surely you need to include damping in this analysis. All real resonators have loss mechanisms and buildings can be no exception.

8. May 18, 2012

### AlephZero

Applying a sine-wave force and varying the frequency till you get the maximum response is one way to measure this, but in fact you can use almost any type of force input and the corresponding motion of the structure to find the mode frequencies. To understand how to do that needs quite a lot of math that tends to be covered in university courses on "digital signal processing" and "control systems", rather than "dynamics".

This is a short overvew paper: http://www.ias.ac.in/sadhana/Pdf2000June/Pe871.pdf and Ewins's "Modal Testing" book is a good basic introduction.

9. May 18, 2012

### Studiot

My guess would be that they used seismometer methods.

That is they dropped a heavy weight as the forcing function and measured the building response using seismometers.

It should be noted that HOI offered a simple equation for a single degree of freedom of vibration appropriate to taking the building as a single body. this establishes the principle only.
Buildings, in reality, fall into the many degrees of freedom category and need matrix methods of analysis since there are systems of differential equations involved.

10. May 18, 2012

### sophiecentaur

I should have thought that the natural resonances of buildings would be such low frequencies that impulses such as explosions or dropping heavy weights would not have the best frequency content. It would be possible to use either the wind as a natural exciting force or large eccentric masses rotated by a motor and located at various heights in the building.

11. May 18, 2012

### Studiot

Have you heard of the deflectograph?

12. May 18, 2012

### sophiecentaur

How relevant is that technique to horizontal oscillations, though?

13. May 18, 2012

### Studiot

P waves can run North South.

Certainly pavements, especially asphalt ones, exhibit considerable damping that you mentioned earlier.

However I said I was guessing the excitation method and someone else said we don't have enough information.

Seismology provides a quick, simple, cheap, reliable method of both excitation and measurement. Of course you would refrain from applying explosions to buildings, hence the dropped weight.

How would you go about excitations and measurement?

14. May 18, 2012

### AlephZero

Applying an impulsive load to a structure to excite it works fine at any frequency, provided you can apply enough impulse at the right place (i.e. somewhere on the structure that will actually move) without doing any structural damage. I agree dropping a heavy weight on the ground wouldn't be very effective way to excite a building though. Swinging one of the counterweights in an elevator shaft might be a better idea. Or using part of the building's vibration damping system to create a disturbance rather than cancel it.

The problem with using something like wind excitation is that unless you know the applied forces you can only get a very limited amount of information. A vibration frequency on its own is not much use unless you also know the corresponding mode shape.

15. May 18, 2012

### Studiot

Why on the ground?

Why not on the first or fiftieth floor?

16. May 18, 2012

### the_emi_guy

There are a number of ways civil engineers address the issue of building resonance. They include simulation, continuous long term monitoring of ambient motion, and direct measurement with excitation (the topic of this thread). Continuous long term monitoring of ambient will reveal ground resonance as well as building resonance. This is important because during an earthquake both the ground and the building are vibrating.
Anyway, on the topic of this thread, go to the website of Anco corporation (www.ancoengineers.com). They make the machine that you are interested in. It is called an eccentric mass vibrator. Looks like you can get one from 4 ton up to 1000 ton!

17. May 19, 2012

### sophiecentaur

A relatively cheap method, I should imagine, but very long term.
I remember reading somewhere about a building technique for high rise in earthquake areas which decouples lateral movements of the ground from the building (mounting the foundations on rollers of some sort?) to reduce the excitation on the building structure by lateral ground motion.