An Efficient Uncertainty Quantification Approach for Robust Design of Tuned Mass Dampers in Linear Structural Dynamics

The application of tuned mass dampers (TMDs) to high-rise buildings or slender bridges can significantly decrease the dynamical vibrations due to external excitation, such as wind or earthquake loads. However, the individual properties of a TMD such as mass, stiffness and damping have to be designed carefully with respect to the dynamical properties of the investigated structure. In real-world structures, the influence of uncertain system properties might be critical for the performance of a TMD and thus the whole structure. Therefore, the design under uncertainty of such systems is an important issue, which is addressed in the current paper. For our investigations, we consider linear single-degree-of-freedom (SDOF) systems, where analytical formulas for the deterministic design already exist, and linear multi-degree-of-freedom (MDOF) systems, where a time integration and numerical optimization algorithms are usually applied to obtain the optimal TMD parameters. If the numerical optimization should be coupled with a sampling-based uncertainty quantification method, such as Monte Carlo sampling, the design procedure would require the evaluation of a coupled double-loop approach, which is very demanding from the computation point of view. Therefore, we focus the following paper on an efficient analytical uncertainty quantification approach, which estimates the mean and scatter from a Taylor series expansion. Additionally, we introduce an efficient mode decomposition approach for MDOF systems with multiple TMDs, which estimates the maximum displacements using a modal analysis instead of a demanding time integration. Different optimal design problems are formulated as single- or multi-objective optimization tasks, where the statistical properties of the maximum displacements are considered as safety margins in the optimization goal functions. The application of numerical optimization algorithms is straightforward and not limited to specific algorithms. As numerical examples, we investigate an SDOF system with single TMD and a multi-story frame with multiple TMDs. The presented procedure might be interesting for the design process of structures, where the dynamical vibrations reach a critical threshold.