Proportional viscous damping model for matching damping ratios

A proportional viscous damping model is proposed to address the limitations of existing models that are either inaccurate or computationally costly in matching damping ratios. The proposed model allows easy curve-fitting to a given damping ratio distribution with negligible error and is remarkably accurate for matching a constant damping ratio within a practical range of structural frequencies, which is commonly considered in seismic response history analysis of large-scale structures for simulating 'un-modeled' damping. The resultant curve is always positive and is zero only at zero and infinite frequencies, thereby avoiding spurious damping forces and undamped response in higher vibration modes that would otherwise cause significant errors in structural forces. The model combines several bell-shape basis functions, with each parameterized by the frequency and damping ratio at its peak, to form a user-specified damping ratio curve. Three methods are suggested to determine the coefficients of the proposed model: exact, linear least squares, and nonlinear least squares curve fitting. In the first method, the matrix for determining the model coefficients is always well-conditioned and symmetric positive definite. Several graphs and formulas are provided to help users determine the required number of basis functions, resultant maximum residuals, and optimized coefficients for matching a constant damping ratio within a practical range of frequencies. Two response history analysis examples showcase the performance of the proposed model compared against existing proportional damping models.