Full Harmonic Analysis with Material Damping

[meshparts]

Hello,

I have posted here a simple model of a spindle system, composed of a beam, a spring and a point mass. The rotation of the spindle is transformed in a translation of the spring using CE's. On the other end of the spring is the point mass.

For this model I am doing a Full Harmonic Analysis with Material Damping for the spindle. The harmonic load is a moment on the left end of the spindle.

The Frequency Response of the System is computed as the quotient between the translation of the mass and the rotation of the spindle.

The problem is, that the Amplitude of the frequency response is bigger for high material damping then for low material damping! I always thought, it should be vice versa: low damping, high amplitude. This holds for a simple Sping-Mass system. So why not for my little more complex model?

So can you help me finding an explanation for this weird effect? Thanks in advance!

Regards,
Alex

fini
/clear
/prep7

mm=1/1000
pi=3.14
h=10*mm

et_mass=etyiqr(0,16)
et,et_mass,mass21

et_beam=etyiqr(0,16)
et,et_beam,beam188,0,1,0

et_spring=etyiqr(0,16)
et,et_spring,combin14,,1

mp_steel=mpinqr(0,0,16)
mp,ex,mp_steel,210e9
mp,dens,mp_steel,7800
mp,nuxy,mp_steel,0.3

!----------------------------------------------------------
!mp,damp,mp_steel,0.0025 ! LOW DAMPING -> LOW AMPLITUDE!!!
mp,damp,mp_steel,0.25 ! HIGH DAMPING -> HIGH AMPLITUDE!!!
!----------------------------------------------------------

mp_dummy=mpinqr(0,0,16)
mp,damp,mp_dummy,0.0

rc_mass=rlinqr(0,16)
r,rc_mass,225

rc_spring=rlinqr(0,16)
r,rc_spring,500.0e6,2000

sc_spindel=sectinqr(0,16)
sectype,sc_spindel,beam,csolid
secdata,32*mm/2

n,4, 0.0, 0.0 ! first spindle node
n,5, 245.0e-3, 0.0 ! midle spindle node
n,6, 490.0e-3, 0.0 ! last spindle node
n,7, 245.0e-3, 100.0e-3 ! first spring node
n,8, 345.0e-3, 100.0e-3 ! spring and mass node

type,et_spring
mat,mp_dummy
real,rc_spring
e,7,8

type,et_mass
real,rc_mass
e,8

type,et_beam
mat,mp_steel
secnum,sc_spindel
e,4,5
e,5,6

d,4,ux,,,,,uy,uz,roty,rotz ! Left Spindle end can only rotate about the x-axis (ROTX)
d,5,uy,,,,,uz,roty,rotz ! Middle Spindle end can only rotate and translate (UX and ROTX)
d,7,uy,,,,,uz,rotx,roty,rotz ! Spring can only translate in x
d,8,uy,,,,,uz,rotx,roty,rotz ! Spring can only translate in x

! Contrint equation: translate rotation of spindle in translation of spring
ce,1,0,7,ux,-1,5,rotx,h/(2*pi),5,ux,1

fini
/solu
alls
antype,harmic
hropt,full

f,4,mx,1
/esha,1
/pbc,all,,1
eplo

harfrq,0,2000
nsubst,500
kbc,1

eqslv,sparse
solve
finish

/post26
file,,rst

nsol,2,4,rot,x
nsol,3,8,u,x
quot,4,3,2,,,,,1,h/(2*pi)/2

plvar,4

/axlab,x,'Freq (Hz)'
/axlab,y,'Amplitude (dB)'
/gropt,logx,on  
/gropt,logy,on
/xrange,10,1000
/yrange,default,,1  
/repl

 

[cbrn]

Hi,
I haven't had the time to run your input file yet, but here's my little thought on the problem:
the beam subjected to an harmonic torque is simply a generalized stiffness connecting two points: the application point of the torque and the connection point with the linear spring. Then, the CEs you have set are a "transmission ratio" between the torsional DOF of the first part of the vibrating system and the translational DOF of the second part (the point mass connected by a spring). To cut the story short, you have a 2-DOF semi-definite system, where the only "problem" is attributing the distributed mass / polar inertia of the beam to the appropriate point. But then, you can easily get the response of the system by hand-calcs (well, not so easily, but "sufficiently easily"...). You would then have a comparison term for your FE simulation.

And, by the way, there could be a physical explanation of the strange behaviour (verified that all the setup of the problem was correct...): the damping lowers the peak response but enlarges the freq range where the amplification ratio is > 1, and causes a "freq shift" in the response maximum. Thus, depending on the original "distance" between the excitation frequency and one of the resonance frequencies of the system, the damping could throw you towards the new resonant freq, and give, for this freq, a higher response (though the peak response AT the resonance freq would be lower than before).

I'm really interested on yours and other's comments about your post !!!

Regards

 

[meshparts]

Hello cbrn,

Thank you very much for your answer!

I also thought of writing the analytical equations of the system or at least of a lumped mass version of the system. I will work on that.

I don't think I understand your explanation about the strange damping-amplitude behavior. Could you be more specific?

In the mean time, a have found out, that there is an "optimal material damping" (c_opt) where the amplitude is minimal (A_min)! If I set the material damping lower or higher then the c_opt then the amplitude is always higher then the A_min! (see below)

mp,damp,mp_steel,0.000025 ! HIGH AMPLITUDE (100 dB)
mp,damp,mp_steel,0.00025 ! LOWER AMP (30 dB)
mp,damp,mp_steel,0.0025 ! LOWEST AMP (10 dB)
mp,damp,mp_steel,0.25 ! HIGH AMPLITUDE (100 dB)
mp,damp,mp_steel,2.5 ! HIGH AMPLITUDE (>100 dB)

If there is no error in my model, then this is a very interesting damping effect. Also interesting: the resonance frequency is growing with the material damping.

I hope you and other experts on this forum can help solving this problem.

Regards,
Alex

 

[cbrn]

Hi,
- damping makes natural freq grow: yes, expected behaviour; I don't remember if this is a 100% general statement, but I think so.
- say you have an excitation at 100 Hz and, with "small" damping, your natural frequency is 110 Hz. Say that the relative acceleration amplitude at the natural freq is 100% (our reference level) and that the response decay is steep enough to bring the relative accel amplitude at 100 Hz to 20%. Now, let's increase the damping: first of all, the natural frequency will shift up: say it will become 115 Hz. Say also that, at this 115 Hz, the accel amplitude has been reduced to 50%; but the response decay is now lower, say that instead of the previous 8%/Hz you now have 3%/Hz: in 15 Hz difference, this makes now 45% decay, wrt the new relative amplitude of 50%: this makes a relative amplitude, wrt the original reference, of 22.5%, greater than the 20% we had with the "low" damping...

Regards

 

[meshparts]

Hi cbrn,

now I'm getting it!

There are some points, I am agree with an some others I'm not...

1) If you will take a spring-mass system as an example, then the natural damped freq. is always lower then the undamped natural freq. I always thought, that must hold for all systems. But now I see, I was wrong.

2) I am comparing the amplitudes at the resonance freq. If I define a very high damping, the natural damped freq. will grow (like you said) but amplitude at the resonance will also grow!!! This is my dilemma. On the other hand if I choose a very low damping, then the freq. will sink and the amplitude will grow. Only for an specific middle damping, the amplitude will be very low!

My question is: is this phenomenon correct? If yes, then I think the explanation could lie in the vibration absorbing effect of the spring-mass system in my model.

This could also be important in the process of designing machining tools: Very high damping is not always good!

Regards
Alex