G.220.127.116.11 Modal Damping
With the EXPLICIT option, you must provide unique modal damping values for some or all modes. Each value can be preceded by a repetition factor (rf*damp) without spaces.
DEFINE DAMPING INFORMATION EXPLICIT 0.03 7*0.05 0.04 - 0.012 END
In the above example, mode 1 damping is .03, modes 2 to 8 are .05, mode 9 is .04, mode 10 (and higher, if present) are 0.012.
If there are fewer entries than modes, then the last damping entered will apply to the remaining modes. This input may be continued to 10 more input lines with word EXPLICIT only on line 1; end all but last line with a space then a hyphen. There may be additional sets of EXPLICIT lines before the END.
The formula used to calculate the damping for modes i = 1 to N per modal frequency based on mass and/or stiffness proportional damping (for CALCULATE) is:
D(i) = (α /2ωi) + (ωiβ /2)
If the resulting damping is greater than MAX, then MAX will be used (MAX=1 by default). If the resulting damping is less than MIN, then MIN will be used (MIN=1.E-9 by default). This is the same damping as D = (αM + βK).
DEFINE DAMPING INFORMATION CALC ALPHA 1.13097 BETA 0.0013926 END
To get 4% damping ratio at 4 Hz and 6% damping ratio at 12 Hz
|D(i) = (α /2ωi) + (ωiβ /2)|
|0.04 = α / 50.266 + 12.567 β|
|0.06 = α / 150.796 + 37.699 β|
|α = 1.13097|
|β = 0.0013926|
However they are determined, the α and β terms are entered in the CALC data above. For this example calculate the damping ratio at other frequencies to see the variation in damping versus frequency.
The damping, due to β times stiffness, increases linearly with frequency; and the damping, due to alpha times mass, decreases parabolicly. The combination of the two is hyperbolic.
Graph of Damping Ration, D(i), versus Natural Frequency, ω with Max. and Min. values applied.
The formula used for EVALUATE (to evaluate the damping per modal frequency) is:
Damping for the first 2 modes is set to dmin from input.
Damping for modes i = 3 to N given dmin and the first two frequencies ω1 and ω2 and the ith modal frequency ωi.
|A1 = dmin / (ω1 + ω2)|
|A0 = A1 * ω1 * ω2|
|D(i) = (A0 / ωi ) + (A1 * ωi )|
If the resulting damping is greater than the dmax value of maximum damping, then dmax will be used.
DEFINE DAMPING INFORMATION EVALUATE 0.02 0.12 END
for dmin = .02 , dmax = .12 and the ωi given below:
|N||100||0.1200 (calculated as .28605 then reset to maximum entered)|