[Solved]More confusion with random excitations

reiteres

Hi!

Could someone give me an explanation to the following problem.

I think I don't understand the operator DYNA_ALEA_MODAL correctly.

I prepared a simple example for hand calculation. (I modified testcase sdld101a):

Now let's reconsider the one-mass spring system.

X---∧∨∧∨--(m)

P1 k P2

m = 100 (mass)

k = 10⁶ (stiffness)

damping ξ₀ = 0.0 (for easier computing)

⇒ m x''(t) + k x(t) = 0

Ansatz: x = X exp(j ω₀ t)

⇒ (-ω₀² m + k) = 0

⇒ ω₀ = ± √(k/m) = √(10⁶/100) = 100

Consider now an harmonic source term: F(t) = F exp(j ω t)

Then we have the classic

m x''(t) + k x(t) = F(t) = F exp(j ω t) | : m

x''(t) + k/m x(t) = F/m exp(j ω t)

(remember that k/m = ω₀²)

⇔

x''(t) + ω₀² x(t) = F/m exp(j ω t)

Ansatz: x(t) = X exp(j ω₀ t)

(-ω² + ω₀²) X exp(j ω t) = F/m exp(j ω t) | exp(j ω t)

⇒ X = (F/m)/(ω₀² - ω²)

This should be well known.

Now to the main question:

In DYNA_ALEA_MODAL I provide the operator a PSD profile with coefficients |F(ω)|² with ω = 2π f for each frequency f,

and give a random signal F as input on the point P1 (which is fixed).

In EXCIT I tell aster that DERIVATION=2 (i.e. I provide a signal in means of acceleration).

Then I measure the output PSD profile in 3 flavours Accleration (=DSP_0), Velocity (=DSP_1), displacement (=DSP_2)

I realize that by telling REPONSE that DERIVATION = 2,1,0.

From what I have calculated above, if I give |F(ω)|² as input I should get |X(ω)|² = |F(ω)|²/(ω₀² - ω²)² as output.

So if we get near the eigenfrequence we should get resonance for Accel, Velocity and Displacement.

Howver that is what I measure:

FREQ DSP_0 DSP_1 DSP_2 0.00000E+00 1.00000E+00 1.00020E+00 1.00020E+00 5.00000E+00 1.23100E+00 9.93041E-01 1.00022E+00 1.00000E+01 2.73011E+00 9.79344E-01 1.00033E+00 1.50000E+01 8.00971E+01 8.38417E-01 1.00179E+00 2.00000E+01 2.98152E+00 1.04387E+00 9.99655E-01 2.50000E+01 4.64411E-01 1.02152E+00 9.99864E-01

3.00000E+01 1.53419E-01 1.01482E+00 9.99922E-01

I can see resonance in the case that EXCIT gets DERIVATION=2 and in REPONSE DERIVATION=2

but not in the other cases, in fact I get nearly the same back as I put in.

Actually WHAT is the response signal in terms of physical meaning?

Please help!

Example attached.

Attached file:
sdld101a.comm.zip, 15343kb

reiteres

Finally I answered my own question:

The movement computed above is RELATIVE movement not absolut movement

So changing MOUVEMENT='ABSOLU' toMOUVEMENT='RELATIF' in REST_SPEC_PHYS

and considering that the DERIVATION level correct in/output by factors of ω² gave finally the answer.

reiteres

Ok sorry for unmarking this, but one question remains for me:

I still couldn't answer why there is no resonance for displacement and velocity.

Edit: The Mysterie is solved. Exploring the PSD profiles further I got this, for relative values:

FREQ DSP_0 DSP_1 DSP_2


 0.00000E+00  0.00000E+00  1.00000E-08  1.00000E-08
 1.00000E+00  1.57092E-05  3.97920E-07  1.00794E-08
 2.00000E+00  2.57434E-04  1.63022E-06  1.03235E-08
 3.00000E+00  1.35715E-03  3.81967E-06  1.07504E-08
 4.00000E+00  4.54604E-03  7.19704E-06  1.13939E-08
 5.00000E+00  1.19910E-02  1.21495E-05  1.23100E-08
 6.00000E+00  2.74457E-02  1.93113E-05  1.35878E-08
 7.00000E+00  5.75232E-02  2.97363E-05  1.53720E-08
 8.00000E+00  1.14300E-01  4.52382E-05  1.79046E-08
 9.00000E+00  2.20996E-01  6.91098E-05  2.16120E-08
 1.00000E+01  4.25499E-01  1.07780E-04  2.73011E-08
 1.10000E+01  8.36434E-01  1.75100E-04  3.66557E-08
 1.20000E+01  1.73565E+00  3.05309E-04  5.37052E-08
 1.30000E+01  4.01872E+00  6.02339E-04  9.02807E-08
 1.40000E+01  1.16993E+01  1.51197E-03  1.95401E-07
 1.50000E+01  6.31977E+01  7.11474E-03  8.00971E-07
 1.60000E+01  9.00959E+03  8.91467E-01  8.82075E-05
 1.70000E+01  6.55437E+01  5.74478E-03  5.03519E-07
 1.80000E+01  2.10033E+01  1.64204E-03  1.28374E-07
 1.90000E+01  1.12359E+01  7.88388E-04  5.53188E-08
 2.00000E+01  7.43494E+00  4.70823E-04  2.98152E-08
 2.10000E+01  5.52030E+00  3.17076E-04  1.82123E-08
 2.20000E+01  4.40156E+00  2.30357E-04  1.20558E-08
 2.30000E+01  3.68169E+00  1.76292E-04  8.44144E-09
 2.40000E+01  3.18607E+00  1.40111E-04  6.16156E-09
 2.50000E+01  2.82736E+00  1.14589E-04  4.64411E-09
 2.60000E+01  2.55761E+00  9.58360E-05  3.59106E-09
 2.70000E+01  2.34852E+00  8.16031E-05  2.83543E-09
 2.80000E+01  2.18242E+00  7.05120E-05  2.27818E-09
 2.90000E+01  2.04779E+00  6.16779E-05  1.85769E-09

3.00000E+01 1.93679E+00 5.45106E-05 1.53419E-09

I can confrim these values by hand calculation

Now considering that |X_abs|² = |F|²(1 + ω²/(ω₀² - ω²))².

Considering that ω²/(ω₀² - ω²) ≪ 1 this explains why we are near to the amplitudes.

Also the finer grid brought light to my concerns:

FREQ DSP_0 DSP_1 DSP_2


 0.00000E+00  1.00000E+00  1.00020E+00  1.00020E+00
 1.00000E+00  1.00794E+00  9.98739E-01  1.00020E+00
 2.00000E+00  1.03235E+00  9.97448E-01  1.00020E+00
 3.00000E+00  1.07504E+00  9.96095E-01  1.00021E+00
 4.00000E+00  1.13939E+00  9.94642E-01  1.00021E+00
 5.00000E+00  1.23100E+00  9.93041E-01  1.00022E+00
 6.00000E+00  1.35878E+00  9.91230E-01  1.00023E+00
 7.00000E+00  1.53720E+00  9.89124E-01  1.00025E+00
 8.00000E+00  1.79046E+00  9.86593E-01  1.00027E+00
 9.00000E+00  2.16120E+00  9.83443E-01  1.00029E+00
 1.00000E+01  2.73011E+00  9.79344E-01  1.00033E+00
 1.10000E+01  3.66557E+00  9.73710E-01  1.00038E+00
 1.20000E+01  5.37052E+00  9.65359E-01  1.00046E+00
 1.30000E+01  9.02807E+00  9.51517E-01  1.00060E+00
 1.40000E+01  1.95401E+01  9.23744E-01  1.00088E+00
 1.50000E+01  8.00971E+01  8.38417E-01  1.00179E+00
 1.60000E+01  8.82075E+03  3.77982E+00  9.81304E-01
 1.70000E+01  5.03519E+01  1.15733E+00  9.98581E-01
 1.80000E+01  1.28374E+01  1.08269E+00  9.99284E-01
 1.90000E+01  5.53188E+00  1.05694E+00  9.99530E-01
 2.00000E+01  2.98152E+00  1.04387E+00  9.99655E-01
 2.10000E+01  1.82123E+00  1.03593E+00  9.99730E-01
 2.20000E+01  1.20558E+00  1.03059E+00  9.99780E-01
 2.30000E+01  8.44144E-01  1.02673E+00  9.99816E-01
 2.40000E+01  6.16156E-01  1.02381E+00  9.99843E-01
 2.50000E+01  4.64411E-01  1.02152E+00  9.99864E-01
 2.60000E+01  3.59106E-01  1.01968E+00  9.99880E-01
 2.70000E+01  2.83543E-01  1.01815E+00  9.99894E-01
 2.80000E+01  2.27818E-01  1.01686E+00  9.99905E-01
 2.90000E+01  1.85769E-01  1.01577E+00  9.99914E-01

3.00000E+01 1.53419E-01 1.01482E+00 9.99922E-01

especially this line:

1.60000E+01 8.82075E+03 3.77982E+00 9.81304E-01

Everything seems to be understandable now.

I close this now as solved :)