Modes of vibration building 7

[slickdeals]

Folks,
I am looking for the elders here to break down vibration modes in understandable terms (the real physics of behavior).

Assuming a building has the first 3 modes as X translation, Y translation and torsion and associated modal mass participations.

Say the first mode has a 80% modal mass participation, what does it physically mean? What will actually happen when this building is excited by a certain wind or seismic force? Will all possible modes shapes be involved in the behavior of the building?

My question is rather stupid, but i really want to physically understand how each mode is excited, and how each mode is participating in the response of the building?

Any layman examples will help.

Thanks

 

[NS4U]

1) Say the first mode has a 80% modal mass participation, what does it physically mean?

Think of the deflected shape of a static simply supported beam under a uniform load. The deflection is max in the middle and zero at the ends. Now think of the first mode of a simply supported beam, it has the same deflected shape as the static case above, except it moves up and down. Considering this dynamic case… at the supports, the beam is NOT moving since it’s restrained. But the middle is… this is said to be something like “80% modal mass participation.” For it to be 100% the entire beam… and thus the entire mass of the beam… would be need to be moving up and down, but that’s not physically possible because of the supports.

2) What will actually happen when this building is excited by a certain wind or seismic force? Will all possible modes shapes be involved in the behavior of the building?

The response of a building will be a combination of all of the modes. Let’s consider the case of a wind gust, which get’s applied to the building then quickly removed. Using a Fourier series analysis, the wind gust can be broken into to several sine waves each with their own specific frequency and amplitude which, when summed together, would give you the exact same excitation as the wind gust.

This is important because you know what the natural frequencies of Mode 1, 2, 3, etc… for the building are. So say for example the dominant part of the wind gust is a 5 Hz sine wave, with an amplitude of 10… you know 5 Hz happens to be the first natural frequency of your building and you also know it’s mode shape, thus you know the response of the building at 5 Hz.

However there is also a strong 7 Hz component of the gust, with an amplitude of 5. Mode 2 occurs at 8 Hz, therefore your excitation (7 Hz) is somewhere in-between Mode 1 and Mode 2. Since this excitation frequency is close to the second mode, Mode 2 will heavily influence the response of the building at 7 Hz HOWEVER Mode 1 will also contribute, as will Mode 3, 4, 5, etc… (the higher modes will contribute less, but contribute nonetheless)

Now when I say a “mode will contribute” I mean that the response of the building to the 7Hz will look something like:
- 74% of the mode 2’s mode shape PLUS
- 15% of mode 1’s mode shape PLUS
- 7% of mode 3’s mode shape PLUS
- 3% of modes 4’s mode shape PLUS
- 1% of all of the other modes shapes

These percentages are called modal participation, adding them up to get the response of the 7Hz excitation is called modal summation. *NOTE that this is different than your first question where 80% of the mass of the beam was contributing to the response. For that case you had 100% modal participation of mode 1 but 80% mass participation.*

So now we have the response of the building due to the 5Hz component and 7 Hz component of the wind gust. Since these frequencies make up the wind gust (which acts simultaneously on the building) these responses must be combined to determine the overall response of the structure due to the wind gust.

So simply put, yes, for a wind gust or seismic event it will be a combination of all of the modes of the building.

Please keep in mind that this is an extremely simple case using arbitrary numbers and in actuality the wind gust is a summation of hundreds of frequencies. There’s a lot more to dynamics and a good text book or graduate class would probably be a lot more helpful to understand the physics as to why this happens.

 

[slickdeals]

Thanks mh819 for breaking it down. Graduate courses/ text books really go into the mathematics of the problem rather than the physics behind it.

 

[NS4U]

True... It took me an entire graduate thesis of experimental and analytical work on the dynamic behavior of buildings before this stuff really started sinking in and making sense.


A couple of texts which I find to be extremely helpful:

deSilva, C. (2000). Vibration Fundamentals and Practice. Boca Raton: CRC Press.

Fahy, F., & Gardonio, P. (2007). Sound and Structural Vibration, Radiation,
Transmission and Response (Second ed.). London: Academic Press.

 

[abusementpark]

mh819,

Thanks for the informative post. Can you elaborate on how one begins to represent winds gusts as a Fourier Series? All I know about applying wind loads is based on ASCE-7, in which you apply a dynamic wind gust as a static loading. Where does one find information about the periodicy of wind gusts and the corresponding magnitude?

 

[Lion06]

I started a thread not too long ago about the frequency of wind. I had to do a dynamic analysis of a structure and the engineer I was working with asked me to come up with that information. I looked pretty extensively online and came up empty (the same as from my post here).
The only thing I ended up finding was in the commentary to ASCE 7 that says if the frequency is above 1Hz, that you can ignore wind resonant response.

I, too, would be interested in knowing how you represent the wind as a Fourier series.

And thanks for the informative post, mh819!

 

[NS4U]

Well I'm still relatively new to practice and I really haven't done anything with ASCE 7 and wind gusts, so someone here might know more, and if so please elaborate, I'm curious myself.

I think that they way to represent a wind gust as fourier series would require knowning the peak velocity of the gust as well as known the how the gust is applied over time. For example, you could treat the gust as a triangularly shaped load vs time. Where the gust starts as zero linearly builds to its peak velocity, then linearly decays back to zero. The period of the gust would be the time it takes for it go from zero-to-zero. Now that the shape of the gust is known a Fourier series can be applied to it. A Fourier series is simply an infinite sum of sine and cosine waves at various frequencies and amplitudes. Such that when enough sines and cosines are added together the shape of these added sin and cos waves will have the exact same shape as the triangular loading of the gust. By doing this Fourier analysis you know what each frequency and amplitude of the sine and cosine waves need to fit this triangular shape are, and then you can go and determine the frequency response of the building. Keep in mind that you could get frequencies ranging from 0.1 Hz to 100+ Hz. Unfortunately though the assumptions ASCE uses and how they break this down into an equivalent static load I’m not sure on.

I found this website which demonstrates the Fourier series using a java applet

http://www.jhu.edu/~signals/fourier2/index.html

You can select the shape of you’re signal, then determine the number of fouier coefficients (the numbers of frequencies to use when fitting the signal) the magnitude spectrum at the bottom shows the magnitude for each frequency. Note how the low frequencies are the most dominant in the spectrum.