Mode shapes question

[Akeee]

Hi guys,
I don't have a clear understanding of how each vibration motion can be decomposed into modal shapes, I mean how each shape form added together can output a single final vibration motion shape. For example lets say that i have a 2 MDOF system and i shake a little the base so that system will exercise vibration, how that simple vibration caused by my shaking base/ground can be composed by 2 made shapes (first whew all the masses goes in on side and second where one mass goes in one direction an the other in opposite direction)? I can see clear that the first shape is the one that forms the "real" deformed shape,how that second S mode participate here? Or that composition of modal shapes refers to composition of the total time / motions from the start of my excitation until the end of it ? Say it's a simple ground push so the 2 MDOF system will start vibrate in the first model shape and at the end of the motion (due damping) when the natural frequency decrease start kicking in the second modal shape ? I mean in the t second of system motion (lets say t=2.5 s) that shape in that moment is composed of that 2 basic mode shape ? Its very hard to image this, it would help me a lot if someone can explain it clear, i would apreciate a lot.
Thank you.

 

[bridgebuster]

Perhaps this old thread will help:

http://www.eng-tips.com/viewthread.cfm?qid=211145

 

[GregLocock]

It can be called modal superposition. It is a defining property of linear dynamic systems. A google search on the obvious phrase has many hits, none of which I feel like reading on a Sunday morning.

Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376

http://eng-tips.com/market.cfm?

 

[Akeee]

Thank you guys!

 

[KootK]

I have trouble getting a "feel" for this stuff too. It's tempting to just say "It's this way because the mathematics say so". And the mathematics do say so. It's hard for many of us to really be convinced by that on an intuitive level however.

I find it best to ditch the fancy math and go back to the fundamental concepts and definitions associated with energy balance. To that end, consider the following, simplified case:

- Six story structure with six concentrated masses and six degrees of freedom.
- No damping at all.
- An imposed, instantaneous seismic motion consisting of a single jump to the right followed by the complete cessation of ground motion. Basically, free vibration after an initial, imposed displacement.

1) Start with what a mode shape / frequency pair is fundamentally. A valid mode shape, oscillated at its natural frequency, will result in a perfect balance between the kinetic energy of the oscillating masses (1/2 mv^2) and the strain energy embodied in the deformed structure. All of the kinetic energy in the system at peak mass velocity / zero displacement is exchanged for internal strain energy at zero mass velocity / peak displacements. And the energy balance is maintained at all instants in between (conservation of energy).

2) A structure will have as many mode shapes / frequency pairs as it has masses / degrees of freedom.

3) A composite vibratory motion comprised of several superimposed mode shapes vibrating at their respective natural frequencies will also satisfy the energy balance described above for individual mode shapes.

4) Now apply our simplified seismic load to our six story structure. Initially, the structure will be displaced into a shape that will probably look a bit first mode-ish but, in general, will not exactly match the first mode shape. Imagine our building having a soft story at grade or a large piece of equipment on the roof etc.

5) If the building simply oscillated in and out of its originally imposed displaced shape, Newtonian laws of motion would be violated because kinetic energy and internal strain energy would not remain in balance throughout the cycles of oscillation. This is a result of the original displaced shape not being a fundamental mode shape.

6) In order to maintain a continual balance between kinetic energy and internal strain energy, the building will instead vibrate all of it's mode shapes concurrently, each scaled appropriately to produce the required energy balance.

The scenario just described is a hypothetical example of free vibration. If we instead consider vibration induced by a forcing function (EQ), then we introduce the possibility of resonance induced dynamic amplification. That phenomenon will result in the system vibration being dominated by the modes that have fequencies closely matching the frequency content of the forcing function.


I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.