Structural components are susceptible to seismic loading, and thus, understanding their numerical response under seismic loading is necessary. Currently, there are two main approaches for modeling the vibrations dampening: the first method accounts for the large energy dissipation caused by yield or damage in major structural elements, while the second method accounts for the remaining unmodeled mechanisms such as joint frictions and micro-cracks that are not easily accounted for using material models. Notably, the second approach commonly uses viscous damping models despite the availability of non-viscous damping models because it is considered sufficiently accurate for simulating seismic responses. For instance, viscous models allow the uncoupling of the modal response, maintain the mode and frequencies shapes, and simplifies calibration measurements, thus producing a high degree of accuracy. Nevertheless, the second component (un-modeled damping mechanisms) has not been extensively explored in literature due to numerous challenges, such as the difficulty matching the user-specified damping ratio, leading to poor results.
To address the limitations of the existing models, University of Canterbury Professor Chin-Long Lee from the Department of Civil and Natural Resources Engineering, proposed a new proportional viscous damping model for matching the damping ratios. The work is currently published in the research journal, Engineering Structures.
In his work, Professor Lee first assessed the performance of the existing damping models, after which he proposed a proportional model combining several parameterized bell-shaped basis functions for forming user-specified damping ratio curves. Three methods: exact, linear least-squares, and nonlinear least-squares curve fittings, were employed to determine the model coefficients. Lastly, two response history analysis examples were conducted to validate the feasibility of the proposed model compared to the existing models.
The author reported that the proposed model could fit, with high accuracy, a set of given damping ratio data distribution using the three curve fitting methods. The exact method, in particular, allows determination of coefficients with a well-conditioned matrix to produce smooth fitting curves with minimal oscillations. Moreover, the fitting curves are always positive and zero at both zero and infinite frequencies to avoid spurious damping forces and undamped responses associated with significant errors prevalent in higher vibration modes. This means that it is easier to correctly recover the undamped response at zero frequency and zero periods unlike in the existing models. It was worth noting that the accuracy in matching the damping ratio increased proportionately with the number of basis functions. Furthermore, several graphs and formulas are provided to help determine the appropriate number of basis functions, optimized coefficients, and maximum residuals necessary for matching the constant damping ratio in practical earthquake simulations.
Compared to Rayleigh and Caughey models, the proposed model is suitable for matching damping ratios, with remarkably negligible errors, over a broad range of frequencies considered in practical seismic response history analysis for simulating un-modeled damping. The two response analysis examples successfully validated its remarkable performance.
In summary, Professor Lee developed a viscous damping model for matching damping ratios. The model exhibited excellent performance with high computation efficiency compared to the existing models. Additionally, the implementation is simplified and less costly. In a statement to Advances in Engineering, he noted that the study results would advance numerical response history analysis of large-scale structural components subjected to seismic loading, thus aiding effective strategies for protecting the structural components from earthquakes.
Lee, C. (2020). Proportional viscous damping model for matching damping ratios. Engineering Structures, 207, 110178.