Natural Frequency Consistent Unit 6

[Eser]

Hi Everbody!

I have a general question about dynamic analysis. If we want to have the natural frequency in Hz and we have the distances in mm and the Young modulus in MPa what should be the consistent unit of material densities?

Thanks in advance,

Eser

 

[UcfSE]

That seems pretty simple to have to post to figure it out...

MPa = N/mm^2, length = mm, Hz = s^-1 shouldn't have anything to do with density units. Work from there back to mass/volume.

 

[ThomasH]

I don't want to be a pain but:

1 N = 1 kg * m/s^2, so there i a connection.

 

[JStephen]

Just change the distances to meters and the units will be consistent. And be careful that angular frequency and natural frequency aren't confused.

 

[Drej]

All units matter in FE, especially in dynamic analyses, where you are dealing with the unit of time. If your fundamental length unit is mm your density must be in Tonnes/mm^3 and your acceleration in mm/s^2 to obtain your frequencies in Hz. Always make sure in dynamic analyses that your unit of TIME is consistent:

F = m*a
F = Rho*V*a
F = (Tonnes/mm^3)*(mm^3)*(mm/s^2) = (T*mm/s^2) = kg*(1000)*m*(0.001)/s^2 = kg*m/s^2 = N

a = local gravitational acceleration
Rho = density
V = volume
m = mass

To check your time unit (and hence frequency) check the units in:

omega = (1/2*pi)*root(k/m)

pi = 3.1414592 etc
k = stiffness (force/length)
m = mass

Cheers,

-- drej --

 

[corus]

Density should be in Tonne/mm^3 as the frequency is related to the stiffness/mass which is related to the Young's Modulus(N/mm^2)/(density.Volume). As Thomas said 1N = 1Kg.m/s^2 so density has to be 1000.kg/mm^3 or equivalently tonne/mm^3
For americans working in lbs and feet I'd have no idea what the answer would be.

corus

 

[ThomasH]

Now I have a little more time to explain.

My recommendation is SI units (or consistent US units).

m, meter is SI (mm is not)
s, seconds
Pa, not MPa

and so on.

The only SI unit with a prefix is kg (kilo is a prefix). There has actually been discussions about renaming kg to something else, just to loose the prefix because it is inconsistent.

Since you are "only" dealing with natural frequencies you can fix the units with resonable effort. When doing transient (time-stepping) or harmonic analysis so the load varies with time (or frequency) you have to be careful with the units. It can be done but SI units always works.

Hence, I use SI units when dealing with dynamics. The only inconvinience are all the digits in Pa instead of MPa but I let the postprocessor handle that, not the solver.

Good Luck

Thomas

 

[GregLocock]

No, that is not the only inconvenience. Real programs suffer from rounding errors, and solving matrices is far more accurate if all the terms are of roughly the same magnitude. Therefore, unfortunately, the choice of units will impact on the numerical accuracy of the solution.

Cheers

Greg Locock

 

[jhardy1]

GregLocock,

A philosophical question, based on your comment that "the choice of units will impact on the numerical accuracy of the solution":

When comparing two terms of a stiffness matrix (or mass matrix, or load vector, or whatever), if one stiffness DOF term is exactly 100,000 times greater than another DOF term when I use SI units (say), won't it still be EXACTLY 100,000 times greater when I use pound/inch units or atomic mass units / cubits, etc? If it is 100,000 times stiffer, it is 100,000 times stiffer.

That is, while I agree that the absolute magnitude of terms can vary enormously, depending on the unit set you select, shouldn't the relative magnitudes be the same in any consistent unit set?

 

[GregLocock]

No, consider using SI and using tonne mm

In the mass matrix for a car, in SI the mass will be of the order of 1000 kg, and the moments of inertia will be of the order of 1000 kg m^2, so it'll be a nice matrix to crunch.

In tonne mm, the mass will be about 1, but the moment of inertia will be about 10^6. So you've already lost 5 decimal figures of precision, or about 17 bits.

I must confess I must have the details of this wrong, as this discussion usually results in people recommending tonne mm, not SI!





Cheers

Greg Locock

 

[jhardy1]

GregLocock,

Not trying to be pedantic (and I could have this totally wrong, anyway!)- but ...

Your maths is fine, but my point is that you don't add the mass terms to the mass-moment-of-inertia terms. Instead, these terms get multiplied by other terms to derive yet further terms. They don't get directly added together (or subtracted from each other) until they have been multiplied by other terms to derive terms with the same units (force, moment, stiffness, displacement, rotation, acceleration, etc). If one force (moment, displacement, ...) is x times bigger than another force (moment, ...) in SI units, it would still be x times bigger in any other consistent unit set.

I thought that when solving matrices, the problem arises when you add or subtract terms of significantly different magnitude, leading to numeric truncation or round-off. Multiplying and dividing terms with very different magnitudes shouldn't directly lead to round-off, as long as the resultant doesn't overflow or underflow the precision of the computer.

You can certainly generate round-off induced problems when the stiffness of adjacent elements is of significantly different magnitude (e.g. joining a very small element directly to a very large element, without a suitable intermediate graded mesh), because the equivalent stiffness terms are directly added and subtracted in the solution process, but the ratio of stiffnesses should be the same whatever unit system you adopt.

Anyway, I might have to have a bit of a think about it!

 

[johnhors]

It is very important to have a well conditioned stiffness matrix, since from a simplistic point of view the stiffness matrix has to be effectively inverted in the solution, and is thus very sensitive to round off errors and ill-conditioning. The mass matrix however is then basically used as multiplier of the inverted stiffness matrix (the flexibility matrix), a far less intense mathematical operation. Large disparities between leading diagonal terms on the mass matrix are quite acceptable, whilst similar disparities in the stiffness matrix would be a killer.

 

[hacksaw]

how is it that SI units have been elevated to the status of religious mantra. units are units and everyone has a calulator or these days a PDA, etc.

there are at least a half-dozen versions of SI in use, so why bash customary units. use the units that match the scale of the problem at hand

there is a time and place for rigorous application of SI, beyond that whatever units are convenient justify their use...

 

[johnhors]

Hacksaw

You seem to have missed the point. Many, many dynamic and natural frequency calculations over the years have gone wrong simply because the analyst had not used a consistent set of units. The original question of this thread was answered by Drej, but even this consistent set of units has caused confusion. I would always advise anyone to use SI units with modal and dynamic calculations i.e. Newtons, metres, kilograms and seconds, thus ensuring that they are working with a consistent set. What are your other five versions of SI?

 

[ThomasH]

Hacksaw

You got me a bit confused, half-dozen versions of SI units?? Can you give me two versions and I would be impressed. Note that Pa (pascal) is strictly speaking SI, while MPa is not (M is just a prefix).

The test "constumary" units can also be missleading. Costumary where I work (europe) is SI while in the UK feet etc is sometimes called Imperial units.

Use the units you feel comfortable with but the connection between the units and "the scale of the problem" I don't understand your point. I work with meters, if I work with feet because the design is smaller (problem scaled down) I loose my normal references.

I don't bash "costumary" units but use them either, I've worked with SI since grammer school.

 

[corus]

Two versions of SI units would be to use either metres or millimetres.
For some preprocessors it's better not to use metres as they have problems dealing with 3 decimal places when your dimensions are only of a few millimetres. There is no problem in using mm as a unit of length providing you use the units of density desribed above. It just takes a little thought.

corus

 

[ThomasH]

corus

I understand what you are saying but strictly speaking only meter is base SI unit while "milli" is a prefix, in my opinion that does not mean two versions of the SI system.

I think that is the basic problem in this discussion. That's also what I meant when saying that kg (mass) is in som respects a "problem", it's the only SI unit where the base unit has a prefix. It's no problem to use mm as a length unit but then you have to make sure "your" units are consistent all the way. If you use SI base units the consistency is "automatic".

Say you want to make a thermal analysis. Common unit Watt (W):
1 W = 1 J/s = 1 m^2 * kg * s^(-3). (I picked that from the webpage below.) If you want to the analysis using mm instead of m? Fine, just make sure that you are consistet.

I have seen and heard so many discussions over the years "if I use cm for length, what should I use for mass?" and so on. As for Imperial units I would say that "feet" and "inches" are equally important but, as far as I know, inches are seldom used for length in a dynamic analysis. In stress analysis it is, as you know, much easier to be consistent.

You can find mor information at for example:

http://physics.nist.gov/cuu/Units/

Regards

Thomas

 

[UcfSE]

I thought SI was grams, and MKS was kilograms.

 

[JStephen]

SI is kilograms as the standard unit of mass.

There was the old CGS system- centimeter-gram-second.

There are various versions of "the metric system" around. When having the SI system pounded into my head repeatedly in school (20 years ago), it was always impressed upon me how logical it was, how orderly, etc. Then I go to work and discover in the real world, the countries that are "metric" are using stuff like kgf/cm^2. Sheesh.

The basis of the meter was that it was one ten-millionth of the distance from pole to equator. Only it's not exactly, so that leaves it with no real basis.

The standard of mass is the kilogram. Derived by using the units of length and the density of water. Only it SHOULD have been that one cubic meter of water is one gram, rather than than a cubic centimeter. So the logical system loses it's logic. And the standard mass should be a gram, rather than a kilogram.

The unit of temperature measurement SHOULD have been tied to the other units such that heating one gram of water one degree required one joule. Instead, the temperature scale was chosen arbitrarily, and you have to throw in conversion factors there.

And I still don't know what dyne is.

 

[ThomasH]

Is the temerature scale arbitrary?

0 degrees Celsius, freezing point for water at sea level.
100 degrees Celsius, boiling point for water at sea level.

And then you have the steps in the scale. Now, strictly speaking Celcius is not SI but for Kelvin the difference is that 0 K is absolute freexing point. That is about -273.15 degrees Celsius. Note that there is no "degrees" Kelvin only Kelvin. Since the steps in Kelvin and Celcius are identical they are often used for the same applications. Farenheit, thats a completely different story.

A question: Do you consider pounds force and kips (kilo pounds) as one or two units? If you consider them as two that would explain to me why you persist in saying there are several SI systems. To me, kilo is a prefix, and they are the same. "milli", "centi" and so on are a concinient way of handling digits, nothing more and nothing less. "kilo" in "kilograms" is another story mentioned in a previous post.

Regards

Thomas