There are several damping model options available within the program. The traditional approach has been to use a Rayleigh or Proportional damping model where the structures damping matrix C is given by

C = aM + bK

where M and K are the mass and stiffness matrices for the structure. The coefficients a and b are computed to give the required levels of viscous damping at two different frequencies, most commonly those of the first and second modes of free vibration.

Rayleigh damping may be modelled proportional to the tangent stiffness matrix or the initial stiffness matrix.

A means of matching the required amount of damping at a greater number of modes is provided by a damping model proposed by Caughey. The method suggested by Wilson and Penzien makes use of the properties of orthogonality of the mode shapes with respect to the mass, damping and stiffness matrices. Given the ith mode shape of free-vibration f_i such that

where m_i is the generalised mass for the ith mode. In a similar manner the generalised damping can be computed from

If the fraction of critical damping l is specified for each mode then the damping matrix C can be obtained using the inverse of the modal matrix to transform back from the generalised damping matrix.

In Ruaumoko there are incorporated three different versions of this damping model. The first assumes a linear variation of the fraction of critical damping with frequency where the user specifies the damping required at any two modes. If the same fraction of critical damping is specified at both modes then constant damping is provided at all frequencies. The second version assumes that the damping is constant at frequencies below the first nominated mode with a linear variation of the fraction of critical damping to the fraction of critical damping specified at the second specified mode and then remains constant to the highest frequency mode in the structure. A third model, similar to the second, but having linear interpolations used between up to 9 intermediate modes nominated in between the first and last specified modes.

The program also has a further Rayleigh damping model using either the initial or tangent stiffness matrix where different a and b may be specified for each member in the structure. This not meant to imply great generality but to enable different structures, say, in carrying out a pounding study between a steel frame and a reinforced concrete frame to have different levels of damping in each of the frames. It may also be used in foundation-structure interaction studies where different levels of damping may be expected in the structure and in the foundation model.

The program also has specific translational and rotational damping members or dashpots which may have linear or non-linear force-velocity relationships. The properties may also differ in the positive and negative directions. These members are designed to represent special damping devices inserted in the structure.

Damping is also associated with contact members but only acts whilst the contact members are actually in contact. Damping can also be associated with the ground or foundation members.

Note that the damping only affects the structure during the time-history analysis and has no effect on the static analysis or on the frequency and mode shape analysis.