Elastic Velocity Damping Model for Inelastic Structures

An alternative inherent damping model for inelastic vibrations is presented in which the damping matrix is represented by $[\tilde{c}]=[k_T]{[k]}^{-1}[c]{[k]}^{-1}[k_T]$, where $[k]$ is the initial stiffness matrix, $[k_T]$ is the tangent stiffness matrix, and $[c]$ is the initial damping matrix. The model is based on consideration of a dissipation function that depends only on the elastic components of velocity as opposed to the total velocity, which includes elastic and plastic components. For elastic loading and unloading, the model reduces to the standard viscous model as the matrix $[\tilde{c}]$ becomes equal to $[c]$. The damping matrix $[c]$ can be represented by Rayleigh, modal, or Caughey damping matrices based on initial structural properties. The model allows for appropriate representation of the modal damping ratios as a function of frequency in the limiting linear case and complies with the recommendations from some prominent researchers that inherent damping should be included mostly during loading and unloading phases. The approach can be formulated at a total system level as well as at an element-by-element level.