Explicit 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. 

Example

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.

Calculate Damping

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 /2ω_i) + (ω_iβ /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). 

Example:

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.

Evaluate Damping

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 i^th modal frequency ω_i.

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.

Example:

DEFINE DAMPING INFORMATION
EVALUATE 0.02 0.12
END

for dmin = .02 ,  dmax = .12  and the ω_i given below: