Mechanical APDL Structural Analysis Guide

3. Supported by the QR Damped eigensolver when the complex mode shapes are requested (Cpxmod =
   YES on the MODOPT command). All other types of damping are ignored.

The damping ratios may be retrieved using *GET,,MODE,,DAMP. They are calculated for the following
analyses:

• Spectrum analysis


• Damped modal analysis


• Mode-superposition transient and harmonic analysis

After a modal analysis (ANTYPE,MODAL) using the unsymmetric (MODOPT,UNSYM), damped (MOD-
OPT,DAMP) or QR Damped (MODOPT,QRDAMP) method, the modal damping ratios are deduced from
the complex eigenvalues using Equation 14.222 in the Mechanical APDL Theory Reference. These frequen-
cies appear in the last column of the complex frequencies printout.

1.4.1. Alpha and Beta Damping (Rayleigh Damping)

Alpha damping and Beta damping are used to define Rayleigh damping constants α and β. The
damping matrix [C] is calculated by using these constants to multiply the mass matrix [M] and stiffness
matrix [K]:

[C] = α[M] + β[K]

The ALPHAD and BETAD commands are used to specify α and β, respectively, as decimal numbers.
The values of α and β are not generally known directly, but are calculated from modal damping ratios,
ξi. ξi is the ratio of actual damping to critical damping for a particular mode of vibration, i. If ωi is the

natural circular frequency of mode i, α and β satisfy the relation

ξi = α/2ωi + βωi/2

In many practical structural problems, alpha damping (or mass damping) may be ignored (α = 0). In
such cases, you can evaluate β from known values of ξi and ωi, as

β = 2 ξi/ωi

Only one value of β can be input in a load step, so choose the most dominant frequency active in that
load step to calculate β.

To specify both α and β for a given damping ratio ξ, it is commonly assumed that the sum of the α
and β terms is nearly constant over a range of frequencies (see Figure 1.1: Rayleigh Damping (p. 6)).
Therefore, given ξ and a frequency range ω 1 to ω 2 , two simultaneous equations can be solved for α

and β:

α ξ

ω ω

ω ω

=

+

1 2

1 2

β

ξ

ω ω

=

+ 

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Damping