Vibration problem: natural frequency=resonance?

[athosanian]

Hi... I am leanring some knowledge about vibration.
For a given vibaraiton system, for example, cantilever beam, we can perform modal anlaysis and found five natural frequency of 3, 6, 8, 10 and 20 Hz (just made up) and five corresponding vibration modes.
after that we give a harmonic force on the system at the mentioned natural frquencies. Will the beam give resonance response at all the natural frequencies ? If I contorl the way of force (such as acting point and direction), is it possible that no resonance occurs for some natural frequencies ?

 

[paisiello2]

Is the beam constrained to move in one direction only? I.e. it cannot twist or move sideways?

Is the harmonic load a single point load?

 

[athosanian]

I consider the beam is a 3D problem, it can be extended, twisted in the 3D space. the applied force on the beam can be any form, but is harmonic.
The resonance effect which is excited by the harmonic loading may not only depend on the frequency, but also on the shape of the force.

 

[paisiello2]

The beam would have a greater response, of course, to any load that was at one of the harmonic frequencies. This amplified response caused by matching frequencies is the definition of "resonance", which I think you would agree with.

So the answer to your first question is yes, it's possible to apply a number of loadings so that the beam could have "resonance" in all frequencies at the same time. And similarly, in answer to your second question, you could have the response so that some of the frequencies would not have "resonance".

However, I am not sure that you could have "resonance" from any of the frequencies by simply having a loading condition have the same shape as one of the mode shapes since the definition is, specifically, that "resonance" is caused by matching frequencies.

 

[athosanian]

it is ture in my knowledge the natural frquency almost equal to resonance frequency.
However, when I perform a vibration analysis on a structure by FEA, I found resonance effect did not happen for some natural frequencies in the harmonic analysis. These natural frquencies are obtained by modal analysis in previous.
For example, I found about 289 natural modes in modal analysis in frequency range of 1-300 khz. But I can only observe 78 resonance peaks in the same frquency range.
What is the reason ?

 

[paisiello2]

But I can only observe 78 resonance peaks in the same frquency range.

I don't quite understand what you mean by "observe"? Maybe you can be more explicit about what exactly you are doing?

 

[athosanian]

I analyze the vibration of a strucuture in ANSYS. I performed modal analysis on it at first and obtained several natural frequencies and modes, for example, a number of 10, in a frequency range of 1-100 kz.
Then I performed harmonic analysis to see the variation of the displacement with frequency in range 1-100Hz. However, only some resonance peaks is apparent in the frequency response curve and the others do not occurs. What is wrong with it ?

 

[paisiello2]

Can you explain explicitly what you did to perform the harmonic analysis?

 

[athosanian]

In harmonic analysis I apply a load on the strucutre at a particular frequency and change the frequency, for example, from 1 to 100Hz in an increment of 1 Hz. for every frequency point I record the displacement of a point on the structure. At last I can have a curve which plots the variation of the displacement with frequency. I expect the number of resonance peaks equal to natural modes from modal analysis. But it seems not.

 

[paisiello2]

But you said the model is in 3 dimensions, correct? Did you change the direction of the load?

 

[athosanian]

No, the amplitude is fixed, only the frequency is changed. In textbook examples are 1D. In 3D the symmetry of the load may have influence on resonance frequencies.

 

[paisiello2]

I am talking about the direction not the amplitude. I don't know what you mean by the symmetry of the load.

Unless you can be more explicit by showing exactly the model and loading you are using, I don't think I really can answer your question.

 

[Randy Beikmann]

athosian, the reason you don't get a "resonance" (a peak in the response, if you will) at every modal frequency is that the vibration response of a structure is from ALL the modes responding to the excitation at the excitation frequency, regardless of their modal frequency. In your example, the 3, 6, 8, 10 and 20 Hz modes would all respond if you applied a 10 Hz force. The vibration response at a given point on the structure is then the sum of the responses from each mode. Some add, some cancel. Some respond more than others (the 10 Hz mode will usually respond more than the others, but not if you excite where it has a node).`The upshot is that a structure with two closely spaced modes will often not have two peaks in the "response amplitude vs. frequency" plot, because the response plots from each overlap.

 

[athosanian]

Randy, thanks a lot ! This is called mode superpositon right ? the response of a structure in a forced vibration can be expanded as a sum of natrual modes weighted by different coefficients.that is

displacement at some point(ω) =∑ coefficient(ω) ×ith mode shape,

I think the coefficient should be a function of the excitation frequency ω, when ω=ωi, the natural frequency of ith mode, usually the ith mode will response more strongly than other modes if the excitation force is not on the nodes.

 

[Randy Beikmann]

athosian, that's right. This is one of the most important concepts in vibration. The actual vibration not only depends on what frequency you excite it with (relative to the natural modal frequencies), but where you excite it.

 

[paisiello2]

And the direction you excite it.

 

[athosanian]

Thanks very much !
by the way, in the software ANSYS is it possible to find the coefficients for each mode in the harmonic analysis ?

 

[Randy Beikmann]

I can't say anything about ANSYS (or another other FEA for that matter!). I've usually done lumped parameter models with maybe 20 degrees of freedom, and done it in Matlab.

 

[athosanian]

Randy, Thanks, I see. Then will the symmetry of a structure and the loads play any role on the vibration resonance ?

 

[Randy Beikmann]

Yes, quite a bit. Symmetry (or lack thereof) of structure affects the natural modes' shapes and frequencies, and symmetry of the applied forces affects which modes are most strongly excited.

 

[athosanian]

is the symmetry of the modes which are most strongly excited as same as the symmetry of the applied forces ?

 

[Randy Beikmann]

In general, the more the overall shape of the applied forces is similar to the mode shape, the more that mode will be excited (all else being equal). So modes that are symmetric (like bending of a pipe) are excited most by tangential forces applied equally to both sides of the cross-section, and asymmetric modes (like torsion) are excited most by tangential forces applied oppositely.

 

[athosanian]

Randy, thanks a lot !
by the way, could you recommend some references about the mode symmetry discussions for me to read ? Thanks again.