Rotational stiffness from eigen frequency

[ghylander]

Hi all,

I have this doubt when it comes to estimating a rigid body rotational stiffness from a eigen frequencies analysis.

At first it seemed straightforward to me, using the fundamental dynamics equation w_n = sqrt(K/M), where w_n = f_n · 2 · PI

However, I seem to be reaching a unit disagreement issue that I have not found a satisfactory solution to.

Reverting to the spring equation, F = K · x
In terms of rotation, x would be the angle in radians and F would actually be a moment (so F = N.m).
This results in K having units of N.m/rad, which seems consistent.


Going back to the dynamic equation, it seems correct to me that in terms of rotation, mass is to be replaced by inertia:

w_n [rad/s] = sqrt(K [N.m/rad] / I [kg.m^2])

Solving the RHS results in:

sqrt(1/(rad · s^2 )) = 1/(rad · s)

When on the LHS w_n is in units of rad/s


Alternatively, if one was to define the inertia in terms of [kg.m^2/rad], the RHS simple ends up in Hz



I've tried searching similar discussions, but most seem centred on "linear" stiffness
I also found one published paper (Journal of Sound and Vibration) where a more complex analysis is carried, accounting for mass eccentricities: https://www.sciencedirect.com/science/article/abs/pii/S0022460X20301528

However, it sheds little light on the unit agreement as it uses the dynamics equation and specifies inertia in terms of [kg.m^2], with a final result for the rotational stiffness in terms of [N.m/rad]:





Now, I do realize that at the end of the day radians is a dimensionless unit (hence why rad^2 or sqrt(rad) = rad), but is this something that one just have to live with (sqrt(k/I) = rad/s), or is there something I'm missing?

 

[GregLocock]

What do you think the the dimensions of N are? You need to use a less confusing notation, and I'd use the square of wn to simplify in the first place


The reason that radians is dimensionless is that it is a ratio,it just tells you what proportion of a circle your angle is.

w^2  = N                *m     /I
[T-2]=[(M+1)(L+1)(T-2)][(L+1)][(M-1)(L-2)]

That works.

 

[BEMPE16524]

radian is dimensionless, you don't need to include it in your units balancing.
here is the reason why radian is dimensionless:

One radian is defined as the angle at the center of a circle in a plane that subtends an arc whose length equals the radius of the circle. More generally, the magnitude in radians of a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is,

, where θ is the magnitude in radians of the subtended angle, s is arc length, and r is radius. A right angle is exactly

radians.

 

[MintJulep]

I think this stems from the fundamental ambiguity of Hz. Hz does not define what is being counted each second. Hz is just s^-1.

A radian, even though dimensionless, is tangible.

So, even though rad/s and Hz both have the same unit of s^-1, they are not the same.

 

[ghylander]

Hi all, thanks for your replies.

My question is not so much about "why radians", but "why LHS is radians/s and RHS is 1/rad · s)"

I have formatted the equations, I hope this makes it clearer:



The question is:

Why in equation 1, LHS is in [rad/s^2] while RHS is in [1/rad·s^2], and in equation 2 LHS is in [N·m/rad] and RHS is in [N·m·rad]?
Is it because radians are dimensionless? Can one just simply "flip the radians from top to bottom" (and viceversa)?

 

[BEMPE16524]

again, radian is omitted in dimensional analysis.

 

[ghylander]

again, radian is omitted in dimensional analysis.

So it's fine then to just solve with no regard to the radians, then simply plug it in?

I.e.:

 

[MintJulep]

I've struggled with this exact same thing.

This is the answer: https://community.ptc.com/t5/Mathcad/unit-problem-rad-s-to-Hz-1-s/m-p/308189#M120310

It's not a satisfying answer. Basically, Hz, as a unit is broken, so it's not possible to convert robustly between Hz and rad/s.

You can use 2pi as a magic quasi-conversion to make the number right, but the units don't work.

Wikipedia further helps to confuse or explain. https://en.wikipedia.org/wiki/Angular_frequency

You can't "just flip rad from top to bottom".

@Bembe16524 says to just omit rad from the dimensional analysis, but I think more correctly replace rad with a dimensionless number 1. Do that and the inequalities in your post #5 become equalities.

 

[btrueblood]

If you define 1 Hz = 1 cycle/second,

then define 2pi radians = 1 cycle,

you can get to 1 Hz = 1 cycle/sec * (2pi radians/cycle) = 2p radians/sec

Keep the quasi-units of cycles and radians until you can cancel them...

 

[GregLocock]

Trouble is in this case radians don't cancel. Frankly you can use rpm for the rotational speed and it still works dimensionally. It's not the Hz that's really causing the problems, it is the dimensionless way of counting rotation, whether degrees, radians, revolutions or any other method. The reason for that is that we are really dividing the distance travelled around the circumference by the radius (which is obviously dimensionless) and then multiplying that by an arbitrary number which is also dimensionless. But it is a puzzle.

 

[BEMPE16524]

yup, as mentioned earlier, arc lenght/radius = meter/meter (or whatever unit you have) = 1. that's it.

 

[ghylander]

Circling back to the OP, is this something that one just has to live with?

If I want to solve for the rotational stiffness, it'd best if I forget radians are involved in the equation, then the result will be N.m/rad?

 

[BEMPE16524]

Radians are not "missing" or "ignored"; they are simply dimensionless, and their inclusion in the equations is for clarity rather than for balancing dimensions.