torsional natural frequency - beams 1

[40818]

Hi, with simple structures such as beams (fixed,simply supported, cantilevers etc) its pretty well documented how to calculate the natural frequencies of the beams for different end restraints.
Now, from FE analysis on many real structures, i am also getting an abundance of torsional modes, and i have no method of calculating these frequencies by hand, and havn't managed to find any simple equations to use.
Does anybody have any information on how to calculate torsional frequencies for the common beam solutions.

Thanks.

 

[Twoballcane]

Well the K for free rotation is K=(GJ)/L where G is the shear modulas and J is the polar moment of inertia which is J=Ix +Iy of the beam. From there you should be able to find the Fn in rotation

Tobalcane
"If you avoid failure, you also avoid success."

 

[GregLocock]

I rather imagine crankshaft designers have a good handle on this.

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Greg Locock

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[electricpete]

We have K and J which roughly play the role of K and M for familiar systems.

If you have a beam clamped at one end and supporting J at the other end, of course the resonant frequency is w = sqrt(K/J).

If the beam is not massless but instead has inertia j, you can calculate as above, except subsitute
Jeffective = J + 0.3 *j (similar to the more familiar Meffective = M+0.24*m)

If you have a disk on each end of a massless shaft (not clamped at either end), the resonant frequency can be calculated using:
Jeffectve = J1*J2/(J1+J2)

Formula for torsional equivalent of 2-mass/2-spring system is shown in Harris' Shock and Vib Handbook page 1.11 (too long for me to reproduce... but analogous to the formula for linear vibration 2mass/2spring system.


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[electricpete]

Harris' Shock and Vib Handbook Chapter 10 describes mechanical impedance analysis for lumped mass spring systems. I believe it would not take much thought to apply an analogous approach for lumped torsional systems. Series and parallel elements are combined in a similar manner that we combine electrical components to develop a single expression for equivalent impedance of the entire circuit. Then look at either the poles or zero's of the final impedance expression to determine resonant frequency. If we had defined impedance as torque / angularvelocity , then I believe it is the zero's that would typically represent resonant frequencies. Impedance analysis is a very powerful technique which imo gives a lot of intuition (especially if your mind is used to thinking about lumped electrical circuits).

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[electricpete]

".. then I believe it is the zero's that would typically represent resonant frequencies. "
Actually, it is not always the zero's. Could be either one. Need to look closely at the specific system and transfer function to interpret the poles and zero's.

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[GregLocock]

Aaaagh. I know you do it just to annoy me, but man that old electrical/mechanical analogy thing drives me mad.

Anyway. Holzer method, for shaft/flywheel systems, or Rayleigh Ritz if we are feeling funky.





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Greg Locock

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[electricpete]

Thanks for your interest Greg ;-). Since you obviously can't wait to hear more, I will share electricpete's version of the impedance analysis using the electrical/mechanical vibration analogy.

You need only 6 simple rules in order to solve any sinsoidal steady state lumped mechanical vibration problem using superior ;-) electrical circuit impedance analysis techniques:

1 - mechancial force F plays the role of electrical current i

2 - mechanical velocity difference V plays the role of electrical voltage difference v

3 - Let mechanical impedance be defined as Z=F/V
(this is reciprocal of Harris, but provides a direct analogy to electrical impedance).

4 - A mechanical spring (k) acts like an electrical inductor of inductance L=1/k
The impedance is Zk = j*w/*k

5 - A mechanical damper (c) acts like an electrical resistor of resistance R = 1/c
The impedance is Zc = 1/c

6 - A mechanical mass (m) acts like an electrical capacitor of capacitance C =m **
The impedance is Zm = 1/(j*w*k)
** one terminal of this mass/capacitance is connected to ground.

This analogy also preserves the definition of instantaneous power.
Pmech(t) = V(t)*F(t) vs Pelec(t) = v(t) i(t)

Also if you look carefully at the max stored energy in the spring and mass elements, you will find they are exactly what is predicted by electical calc of Einductor = 0.5*L*i^2 and Ecapacitor = 0.5*C*v^2.

Derivation is available upon request.

As an example of the power of the method, the linear-motion version of the Jeq =J1*J2/(J1+J2) cited above for two masses on the end of a shaft with no connection to ground is effortlessly derived as shown in attached.


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[electricpete]

One more obvious (?) correction in bold for Zm:
The impedance is Zm = 1/(j*w*m)


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[electricpete]

I'm guessing the original poster was looking for the torsional equivalent of the beam frequency equations for combinations of free, clamped and pinned boundary conditions.

For torsional, there is no analogy to pinned. Harris' Shock and Vib Handbook section 7 has solutions for simple circular beam with boundary conditions: free/clamped. There might be enough info in there to figure out the solutions for free/free and clamped/clamped, but I'm not sure.

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[Tmoose]

I think Den Hartog's vibration book by Dover has some tables with torsional resonant freq formulae

 

[electricpete]

Applied Structural and Mechanical Vibrations by Gatti and Ferrari, Chapter 8, gives torsional frequencies for cylindrical rods:

clamped/free
wn = (2n-1)*pi *sqrt(G/rho) / (2*L)

free/free:
wn = n*pi *sqrt(G/rho) / L

I don't think these can be translated for non-circular cross sections simply by substituting the material properties G and rho.


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[electricpete]

Sorry, one more correction in bold to my earlier discussion.
3 - Let mechanical impedance be defined as Z=v/f

Using that definition, the derivation of the mechanical impedances and their electrical equivalencies:

Spring:
f = k x
f = k (v/jw)
Zk = v/f = j*w / k
By comparison to ZL = j*w*L, we find L = 1/k

Mass
f = m a
f = m* (jw * v)
Zm = v/f = 1/ (jw*m)
By comparison to ZC = 1/(j*w*C), we find C = m

Damping
f = c v
Zc = v/f = 1/C
By comparison to ZR = R, we find R = 1/c
(ok, I think I'm done).

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[electricpete]

[tmoose]I think Den Hartog's vibration book by Dover has some tables with torsional resonant freq formulae [/tmoose]
Yes - page 431.

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[electricpete]

Den Hartog looks to have addressed only 2 of the three possible cases.

Rao's Mechanical Vibrations 3rd edition page 520 addresses all three cases of boundary conditions for cylindrical rods:

FIXED/FREE: wn=(2*n+1)*pi*c/(2*L) n=0,1,2...
FREE/FREE: wn = n*pi*c/L, n=0,1,2
FIXED/FIXED: wn=n*pi*c/L, n=1,2,3...*
(*note the FIXED/FIXED has different choice of n)
where c = sqrt(G/rho)

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[GregLocock]

In greg world

velocity-velocity

mass=mass

damping=damping

spring=spring

derivation available on request.

(grins)


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Greg Locock

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[electricpete]

The interesting result is that the first non-zero freq mode of the free/free is the same frequency as fixed/fixed which might surprise you since we expect the fixed/fixed to be stiffer.

But as Greg has explained before, the very first mode of that free/free is really 0 hz. It corresponds to constant speed rotation of the whole beam. (or we have the same phenomenon for longitudinal vib of free/free beam where 0 hz corresponds to translatio of the beam). Rao communicates the same thing by telling us that n starts at 0 which computes to 0 for the first free free mode.

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[GregLocock]

I still rate that result as mysterious, even though I think the explanation is correct - it 'feels' wrong.



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Greg Locock

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[electricpete]

In greg world
velocity-velocity
mass=mass
damping=damping
spring=spring

There is something to be said for that.

I have yet to see anyone mechanical who said: "Wow - that is really neat.". But for me when I read it and started using it, my reaction was that it really was really neat. Somehow it clicks better. It is no longer equations on paper but graphical understandable representation and vector relationships.

To each his own, I guess.

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[electricpete]

I still rate that result as mysterious, even though I think the explanation is correct - it 'feels' wrong.

Too many posts in too short a time for me to figure out which you're talking about.... the electrical thing or the 0-frequency thing?

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