'Loss modulus' related to 'dashpot damping coeffecient'

[Speedy]

Material Damping Question!!

Does anybody know how to relate the Loss Modulus of a material to the damping coeffecient (dashpot model)?

The Loss Modulus has units equivalent to stress (N/m2) while the damping coeffecient has N.s/m units.


Appreciate any help,
Speedy

 

[ivymike]

I've actually wondered the same thing recently... how does one get to a % critical damping value for a given material, based on widely available information about the material? For example, what would be an appropriate % critical damping to apply for a rod made of magnesium, loaded in compression / tension?

 

[GregLocock]

The fraction of critical damping is system dependent.

The critical damping for a SDOF system is given by c=2*sqrt(k*m). Typical values for homogenous /steel/ structures are 0.1-0.2 %.

The damping factor of metals is affected by their microstructure, so it depends which Mg alloy you use, I'd guess around 0.5%-1%. Beranek says that some Mg alloys have up to 10%, which I find hard to believe, and the damping factor is stress dependent, which I can believe.

As soon as you use these materials in real structures the damping rapidly increases - it is rare to see a low frequency mode in a complex steel structure with a damping factor of less than 0.5%.

Speedy - the loss modulus is the imaginary part of the complex Young's modulus, so you just use that in your geometry to give the imaginary stiffness, and then you need to differentiate it in the frequency domain to give your dashpot, I think.




Cheers

Greg Locock

 

[ivymike]

I'd heard that Mg had one of the highest levels of internal damping available in an engineering material, but it was just an anecdote. (Perhaps this is why it is often used to reduce noise and vibration?)

In valvetrain dynamic analyses, it is not uncommon to use damping values in a pushrod model as high as 7% to 10% critical, but this figure includes a significant amount of damping at the interfaces at both ends.

 

[ivymike]

oh, I suppose I should add that the % critical damping used in the pushrod example is based upon the natural frequency of the pushrod mass vibrating on the effective stiffness of one end of the pushrod.

 

[Speedy]

Thanks for all that.

There is probably a simple answer to this but ……Just to elaborate on my question;

Consider a steady state forced damped vibration.

In the damper material, the viscous stresses are a product of the displacement and the loss modulus. These stresses lag the elastic stresses ( and deflection) by 90 deg. By plotting both stresses, from the area under the stress curves, you can work out the % of energy absorbed per cycle. For different materials , this can be derived from the Tan Delta values ( = loss modulus / storage modulus). Since the tan delta of a material is independent of frequency ( at least values I have found are not specified for frequency), the % of energy absorbed per cycle is independent of frequency. Is this correct?

Now consider the dashpot model. The units are N.s/m. The viscous stresses are proportionate to the linear velocity. The linear velocity, however is proportional to the angular velocity or frequency. (Vmax = r.w, ‘r’ is max deflection and ‘w’ is angular velocity or frequency). So at higher frequencies there is an increase in the viscous or damped stresses (as is the case with dashpots). If the max deflection remains the same, the elastic stresses are unchanged. Therefore, the % of energy absorbed per cycle increases with frequency.

The two approaches just don’t agree.

Speedy.

 

[GregLocock]

Admittedly it depends which material you use, but here's the stiffness and loss angle for a simple rubber bush

f(Hz) Phase(deg) K(N/mm)
5 2.04 10559

50 2.99 10973

100 3.56 11226

250 5.36 13008

So as the frequency increases the modulus AND the phase angle increase. Interestingly it still falls short of a dashpot in parallel with a spring.

I'm not surprised the two approaches don't agree, this is part of the reason why FEA is inherently useless for predicting the actual system response for complex assemblies at high frequencies - it can't handle discrete damping elements. To get round this we use small FEA models linked in a building block system called VSIGN. A reasonable model of a hydraulic engine mount would have about 60 parameters, for one preload.

Cheers

Greg Locock