# G.17.3.3.2 Modal Damping

## 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_{i}β /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

Mode | Hz | Rad/sec | Damp Ratio |
---|---|---|---|

1 | 4.0 | 25.133 | 0.04 |

3 | 12.0 | 75.398 | 0.06 |

D(i) = (α /2ω_{i}) + (ω_{i}β /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.

Mode | Hz | Rad/sec | Damp Ratio |
---|---|---|---|

1 | 4.0 | 25.133 | 0.040 |

3 | 12.0 | 75.398 | 0.060 |

2 | 12.0664 | 0.05375 | |

8 | 50.2655 | 0.04650 | |

20 | 120.664 | 0.09200 | |

4.5 | 28.274 | 0.03969 |

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.

## 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 ω_{1} and ω_{2} and the i^{th} modal frequency ω_{i}.

A_{1} = dmin / (ω_{1} + ω_{2}) |

A_{0} = A_{1} * ω_{1} * ω_{2} |

D(i) = (A_{0} / ω_{i} ) + (A_{1} * ω_{i} ) |

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:

Mode | ω_{i} |
Damping Ratio |
---|---|---|

1 | 3 | 0.0200 |

2 | 4 | 0.0200 |

3 | 6 | 0.0228568 |

N | 100 | 0.1200 (calculated as .28605 then reset to maximum entered) |