Robust design optimization of tuned mass dampers in structural dynamics using efficient reduced order models : Calculation files and result data

In the study, efficient methods for the optimal design of Tuned mass damper parameters for linear SDOF and MDOF systems by considering parameter uncertainty are presented. 
In a first step, a perturbation approach for the uncertainty propagation, where the system responses are linearized by a Taylor series expansion with respect to the random system parameters is introduced. This approach is derived and investigated for an SDOF systems first, whereby the optimal parameters are obtained by numerical optimization. 
As excitation, a harmonic excitation, which could vary within a defined frequency range, and the maximum value of the corresponding dynamic amplification function is considered as design criterion.
Here, the amplification function of the displacements is chosen exemplary similar as in early studies by Den Hartog. However, other performance criteria could be considered in the approach in the same manner.
Within the optimization objective the safety margin from the estimated mean and standard deviation are considered within a variance-based robustness evaluation approach. The linearization approach may consider continuous scalar random numbers with arbitrary distribution type, as long as the covariance matrix is known. The numerical examples in the data set will focus on independent and normally distributed random numbers. A Latin hypercube Sampling approach} utilized as benchmark methods to investigate the accuracy of the presented analytical approach.

In a second part of the anylsis, a linear MDOF systems with multiple TMDs is investigated. The damping of the main system is considered as modal damping, which a allows a decoupled modal analysis of the individual vibration modes. No further restrictions are made w.r.t. the system as long as mass, stiffness and damping matrix can be defined.
For efficiency reasosn, a decoupled stationary solution of the displacements of the individual DOFs of the main system and the relative displacements of the TMDs is developed. With help of this very efficient approach, the maximum displacements could be obtained for given range of harmonic excitations similarly to the SDOF system. The novel analytical uncertainty propagation approach based on a linearization of the maximum displacements w.r.t. the random system parameters is applied and an optimal design under the consideration of a defined safety margin could is obtained. As numerical example a three-story frame with two TMDs is investigated.

The mechanical model and the dynamical analysis including the uncertainty quantification approach were implemented in MATLAB Version 2024a and are in this data set. As optimization algorithms the simplex Nelder-Mead method was chosen for single-objective optimization and Non-dominated Sort Genetic Algorithm (NSGA II) for multi-objective optimization. For both algorithms the implementation in the Ansys optiSLang optimization software package was used. 

The optimization is performed with Ansys optiSLang 2023R1. The full project files (opf and opd) and results are available in the data set.