BASIC MODELING 4

[shankargovind]

Hi,
Iam very much a starter as far as Modal analysis is concerned. We have a problem concerning the FE modeling of a particular structure for Modal analysis. We have modeled a mass (200 kgs) using a mass element , but we have not specified the mass moments of inertia of that mass. How will it affect the results?. What role does the Mass MOI play in determining the natural frequencies?.

Thanks,

SG

 

[vgarzani]

Validating FE is fun, but be ready (and open) to learn stuff. Regarding the mass, clearly the closer to a sphere it is the less it will matter. I used a lumped mass to represent a cooling tower gear box and fan - I clearly had to iterate with the I-yy (y being up). Be able to develop a matrix of inner products of empirical mode shapes with your FE analytical mode shapes. Some packages do this for you and call it a MAC. Last tidbit - be ready to cut your stiffnesses in mechanical and welded connections by as much as half of the original estimate. This is great stuff, most of the time there isn't enough time to pursue it.

 

[vgarzani]

Correction, I-xx and I-zz were the interresting terms. Doesn't change the message - just wanted to correct the mistake.

 

[brad]

Good points by vgarzani.

Depending on your problem, MOI can have little effect, or a tremendous effect on your results. Obviously, any time there is significant rotation of your structure, discounting the MOI introduces significant error.
One possible way to get around this issue is to, instead of one mass of 200kg, use several discrete masses which more accurately describe the shape.

For example, take a 200 kg beam which goes from x=0 to x=10. The easiest description of this with one mass element would be a single element at x=5 of 200 kg. However, one could instead use 50 kg elements at x=1.25,3.75,6.25,8.75. These four masses could be connected to each other via rigid elements.

For this example of the 10 m beam with m=200, the moments of inertia (Iyy and Izz) about its centerpoint (x=5) would be:
(1/12)m * L^2 = (1/12)*200*10^2 = 1,667 kg-m^2
The "single mass" approach would give Iyy=Izz=0.
and
(using the parallel axis theorem)
The "four-mass" approach would give Iyy=Izz= 1562.5 kg-m^2

(I'm doing the math on the fly and the equations from memory, so if I've made mistakes, please somebody correct me).

As you can see, with only four masses the discretized answer has converged to something nearly equal to the "real" answer in this case. A proper distribution which is consistent with the continuous geometry could fairly quickly net a nearly equivalent "global" inertial response. Of course, modeling it this way does not capture compliance of the particular mass structure, but I am presuming that you do not need this effect (as you have already ignored it in your model).

I hope this is clear. Good luck.
Brad

 

[GregLocock]

I'll creep out of the woodwork at this point and say that if you don't know what the polar moments are you might as well try and get them right by assuming that the mass is evenly distributed across the length of each component, and sice that's a bad guess you might as well say the mxx= M*Rx*Rx, where Rx is 1/2 the length in the x dimenison, and so on for the other axes. Then multiply that by a fudge factor for the compactness of the shape (1 for a rectangle, 0.5 for a sphere) .

You will be astonished and amazed by the accuracy of this horrible method.

It doesn't give you values for Mxy etc, but I'm sure you can figure out an equivalent solution.

Cheers

Greg Locock

 

[brad]

As I read Greg's post, and then re-read the other posts, I realize that I may have slightly misunderstood the intent of the original posting. I initially presumed that shankargovind was using a code which could not describe moments of inertia, only masses. My earlier response was describing a way in which to effectively capture MOI if this limitation were in fact there.

Upon re-reading, it sounds as if you just do not have the moments of inertia (as opposed to not being able to define them). If your code is capable of defining MOI (and that is basically the price of entry for commercial codes), you should try to approximate these for the sake of the analysis. I haven't yet tried Greg Locock's approach, but it seems reasonable at first look (I may have to throw this in my bag of tricks).

If you approximated your structure as basic shapes, calculated MOI, and then used parallel axis theorem, that should also yield a reasonable set of values.

 

[Dmitry]

Hi All

I feel like the initial question was about more simple problem then the answers are. It was about behavior of the model.
If modeled structure is I-beam, then the bigger inertia moment – the bigger rigidity and so – natural frequencies.
If cross-section of modeled structure is square like, then the bigger inertia moment – the bigger rigidity and mass. So increase of rigidity can be compensated by mass increase and natural frequencies can remain even the same.
The higher natural frequencies – the bigger should be excitation level in order to maintain the same amplitude of vibration.

Regards,
D.Semenov

 

[brad]

Dmitry,
You are referencing the area moment of inertia, a cross-sectional property which helps determine stiffness of the structure. Your answer, though correct, was not what shankargovind had asked.

shankargovind specifically worded the question regarding the mass moment of inertia, the rotational corollary to mass.

Brad

 

[tsankar]

Yes, I was asking about the Mass moments only.

Brad,
I would like to know how the mass moments affect the natural frequencies (On the same lines as what Dmitry has explained regarding the area moments).Also I have another doubt. Suppose I model a part as a lumped mass considering the mass moments also and then try another approach by including the actual CAD model of the part in the analysis model, how can we expect the results to vary?.

Thanks,
Shankar

 

[brad]

tsankar,
Regarding the first question, the mass moments of inertia will lower rotational frequencies of the structure.
Just as omega=(k/m)^.5, for rotations omega=(k(r)/I)^.5

where k(r) is rotational stiffness and I is mass moment of inertia.

As in translation, when the "mass" term (in this case MOI) increases, the frequency will decrease.

Note that even if point MOI's are not specified, there is still a total MOI for the structure, because masses distributed across a structure still contribute to the global mass matrix (just as forces result in moments, masses result in global MOI's).

Your second question: If you model a point mass with MOI's (and any non-zero products of inertia), this will have the same effect on the global structure as a discretized model of the complex geometry.

Two things, however may vary:
1) Modeling the structure discretely may give a different stiffness, as you would've (presumably) added element stiffnesses in the discretized structure that aren't there in the point-mass assumption.
2) The discretized structure may also have its own local modes (due to the local masses and stiffnesses), which would not occur with the point-mass assumption.

If your intent is to capture global effects from this part, and you do not care about the part itself, then describing this as a point-mass and appropriate springs would probably suffice. Just make sure that you capture all values of M and K matrices in your simplified description.

Brad

 

[hessian]

Timoshenko and Young, Vibration, (out of print but findable) has Timoshenko's derivation of solutions for continuous beams including rotation effects. It also allows you to compare the results between the two formulations (with/without). As I recall, the effect is quite small on the frequency of the lower modes. Of course if you added a rigidly connected massive object offset from the elastic centerplane of the beam, the effects could range from large to zero, depending on the location of the mass relative to the nodal and maxima points on a given mode shape. For shear waves of short wavelength (high modal number), the effect of rotation on the simple beam alone would increase, decreasing the translatory amplitude of the beam response. It all depends on what you care about.