The damping matrix, [C], may be used in transient as well as substructure generation:

In its most general form, the damping matrix is composed of the following components:

(14–22)

where:

= structural damping matrix

= mass matrix multiplier (input on ALPHAD)

= structural mass matrix

= stiffness matrix multiplier (input on BETAD)

= structural stiffness matrix

= number of materials with MP,ALPD input

= mass matrix multiplier for material (input as ALPD on MP)

= portion of structural mass matrix based on material

= number of elements with mass proportional material damping input (TB,SDAMP,,,,ALPD)

= number of sections in an element with mass proportional material damping input

= mass proportional material damping factor for section point with material (TB,SDAMP,,,,ALPD); see Material Damping in the Mechanical APDL Material Reference

= portion of element structural mass matrix based on section

= number of materials with MP,BETD input

= stiffness matrix multiplier for material (input as BETD on the MP)

= portion of structural stiffness matrix based on material

= number of elements with stiffness proportional material damping input (TB,SDAMP,,,,BETD)

= number of sections in an element with stiffness proportional material damping input

= stiffness proportional material damping factor for section point with material (TB,SDAMP,,,,BETD); see Material Damping in the Mechanical APDL Material Reference.

= portion of element structural stiffness matrix based on section

= number of elements with specified damping

= element damping matrix

= number of elements with Coriolis or gyroscopic damping

= element Coriolis or gyroscopic damping matrix; see Rotating Structures in the Mechanical APDL Theory Reference

= constant structural damping coefficient (input with DMPSTR)

= number of materials with MP,DMPR input

= constant structural damping coefficient for material j (input with MP,DMPR)

= frequency for the calculation of equivalent viscous damping (DMPSFreq on TRNOPT)

Element damping matrices are available for:

, the structural stiffness matrix, may include plasticity and/or large-deflection effects (that is, may be the tangent matrix). In the case of a rotating structure, it may also include spin softening or rotating damping effects. Stiffness matrices generated by contact elements are not included in the damping matrix generation.

For the special case of thin-film fluid behavior, damping parameters may be computed for structures and used in a subsequent structural analysis (see Extraction of Modal Damping Parameter for Squeeze Film Problems).

For damped modal analysis (ANTYPE,MODAL with MODOPT,QRDAMP or MODOPT,DAMP), the damping matrix is complex and is written as:

(14–23)

where:

= the imaginary number

= the circular frequency

= the structural damping matrix defined in Equation 14–22. The equivalent damping terms based on are not included in a damped modal analysis.

= constant structural damping coefficient (input with DMPSTR command)

= number of materials with MP,DMPR input

= constant structural damping coefficient for material j (input with MP,DMPR)

= portion of structural stiffness matrix based on material j

= number of elements with specified imaginary stiffness matrix

= imaginary element stiffness matrix

The damping matrix () used in harmonic analyses (ANTYPE,HARM with Method = FULL, AUTO, or VT on the HROPT command) is composed of the following components:

(14–24)

where:

= structural damping matrix

= mass matrix multiplier (input on ALPHAD)

= structural mass matrix

= stiffness matrix multiplier (input on BETAD)

= constant structural damping coefficient (input on DMPSTR)

= excitation circular frequency

= structural stiffness matrix

= number of materials with MP,ALPD input

= mass matrix multiplier for material (input as ALPD on MP)

= portion of structural mass matrix based on material

= number of elements with mass proportional material damping input (TB,SDAMP,,,,ALPD)

= number of sections in an element with mass proportional material damping input

= mass proportional material damping factor for section point with material (input on TB,SDAMP,,,,ALPD); see Material Damping in the Mechanical APDL Material Reference

= portion of element structural mass matrix based on section

= number of materials with MP,BETD, MP,DMPR, or TB,SDAMP,,,,STRU input

= stiffness matrix multiplier for material (input as BETD on MP)

= constant structural damping coefficient for material (input as DMPR on MP)

= structural damping coefficient for material (TB,SDAMP,,,,STRU)

= portion of structural stiffness matrix based on material

= number of elements with stiffness proportional material damping input (TB,SDAMP,,,,BETD)

= number of sections in an element with stiffness proportional material damping input

= stiffness proportional material damping factor for section point with material (TB,SDAMP,,,,BETD); see Material Damping in the Mechanical APDL Material Reference.

= portion of element structural stiffness matrix based on section

= number of elements with specified damping

= element damping matrix

= number of elements with viscoelastic damping

= element viscoelastic damping matrix (input as TB,PRONY)

= number of elements with Coriolis or gyroscopic damping

= element Coriolis or gyroscopic damping matrix; see

= number of elements with specified imaginary stiffness matrix

= imaginary element stiffness matrix

The input exciting frequency, , is defined in the range between and via:

= beginning frequency (input as FREQB on HARFRQ command)

= end frequency (input as FREQE on HARFRQ command)

Substituting Equation 14–24 into the harmonic response equation of motion (Equation 15–60) and rearranging terms yields:

(14–25)

The complex stiffness matrix in the first row of the equation consists of the normal stiffness matrix augmented by the structural damping terms given by , , ,, and which produce an imaginary contribution. Structural damping is independent of the forcing frequency, , and produces a damping force proportional to displacement (or strain). The terms , , and are damping coefficients and not damping ratios.

The second row consists of the usual viscous damping terms and is linearly dependent on the forcing frequency, , and produces forces proportional to velocity.

Viscoelastic damping (see Harmonic Viscoelasticity) introduces a contribution to the complex stiffness matrix via the loss moduli. Note that the stresses are also computed using the loss moduli, whereas in the case of structural damping, which is a phenomenological model, the stresses are computed only using the real material properties and g is not used in the stress calculations.

For mode-superposition based analyses:

the damping matrix is not explicitly calculated, but rather the damping is defined directly in terms of a damping ratio . The damping ratio is the ratio between actual damping and critical damping.

The damping ratio for mode i is the combination of:

(14–26)

where:

= constant modal damping ratio (input on DMPRAT)

= modal damping ratio for mode shape i (see below)

= circular natural frequency associated with mode shape

= natural frequency associated with mode shape i

= mass matrix multiplier (input on ALPHAD)

= stiffness matrix multiplier (input on BETAD)

The modal damping ratio can be defined for each mode directly via the MDAMP command (undamped modal analyses only).

Alternatively, for the case where multiple materials are present whose damping ratios are different, an effective mode-dependent damping ratio can be defined in the modal analysis if material-dependent damping is defined and the element results are calculated during the expansion (MXPAND,,,,YES). This effective damping ratio is computed from the ratio of the strain energy in each material in each mode using:

(14–27)

where:

Nm = number of materials

= damping ratio for material j (input as DMPR on MP command); see note below

= strain energy contained in mode i for material j

= displacement vector for mode i

= stiffness matrix of part of structure of material j

These mode-dependent (and material-dependent) ratios, , will be carried over into the subsequent mode-superposition or spectrum analysis. Note that any manually-defined damping ratios (MDAMP) will overwrite those computed in the modal analysis via Equation 14–27.

For harmonic analyses (ANTYPE,HARM with HROPT,MSUP), constant structural damping may also be included. In this case, the harmonic equation of motion in modal coordinates (Equation 15–75) is:

(14–28)

where:

= complex modal coordinate

= natural circular frequency of mode i

= fraction of critical damping for mode i as given in Equation 14–26

= constant structural damping coefficient (input on DMPSTR command)

= complex force in modal coordinates

14.3.4.1. Specialization for Mode-Superposition Analyses Following a QR Damp Modal Analysis

Equation 14–26 presumes Rayleigh damping (see the derivation in Mode-Superposition Method). With Rayleigh (or proportional) damping, the damping in the modal basis leads to the diagonal entries of Equation 14–26.

(14–29)

For QR Damp modal analyses (MODOPT,QRDAMP), the damping may be from non-Rayleigh sources such as element damping or Coriolis damping. The damping in the modal basis is no longer diagonal, and the full matrix must be retained in the mode-superposition transient or harmonic analysis:

(14–30)

where:

= damping matrix in the modal basis

= matrix of the mode shapes (see QR Damped Method)

= damping matrix from Equation 14–22. The equivalent damping terms based on are not included in the [C] matrix when calculating [Cm]

= constant structural damping coefficient (input on DMPSTR command)

= number of materials with MP,DMPR input

= constant structural damping coefficient for material j (input as DMPR on MP command)

= number of elements with specified damping

= diagonal matrix with the terms + on the diagonal

= in a harmonic analysis, , while in a transient analysis, is the frequency used for the calculation of equivalent viscous damping (DMPSFreq on TRNOPT).