# TMD Damping for Structures with Uncertain Modal Parameters
## 1 Introduction  
The Tuned Mass Damper (TMD) is a very well-known vibration absorber that is installed in various structures such as bridges and tall buildings to mitigate structural vibrations [1, 2]. The principles and optimum tuning rules for minimum structural displacement and harmonic excitation are described in [PERSON]'s book [3], which is based on the undamped dynamic vibration absorber invented by [PERSON] [4] and includes the damping tuning proposed by [PERSON] [5]. Since then, numbers of papers have been published dealing with the optimum design of TMD natural frequency and damping for various optimization functions, i.e., the minimization of structural displacement, velocity, and acceleration, for harmonic and white noise excitations, for damped structures and base excited structures [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. Common to all these studies is that the parameters of the target structural mode, i.e., eigenfrequency and modal mass, are assumed to be known quantities. However, this is not the case in reality. The modal parameters may significantly vary over time due to temperature effects on the structure, time-varying life loads, the relationship between structural damping and vibration amplitude, and unknown boundary conditions [20, 21, 22, 23, 24, 25, 26, 27]. The reported variations of the observed eigenfrequencies reach up to \(\pm 9\%\)[21] and change with temperature and load as described in [23] for a bell tower. The difference between the actual modal mass, which can only be determined with the use of numerous accelerometers to ensure the high spatial resolution of the mode shape of interest, and the modal mass derived from simple harmonic mode shape functions or those computed by the finite element model, may be up to \(12\%\) according to [27], who analyzed a conveyer belt bridge.  
If modal parameters are expected to significantly change over time due to the aforementioned effects, or if they are simply unknown because they are not precisely measured, the following two concepts exist: Passive TMDs can be replaced by controlled TMDs [28; 29; 30; 31; 32], where natural frequency and damping are adjusted to the actual structural modal properties in real-time, or the damping of the passive TMD is selected to be a greater-than-optimal value. The latter approach is often adopted in practice; although, theoretical evidence is missing so far. This paper aims to answer the following questions: In which cases the concept of increased TMD damping is beneficial and by how much the actual TMD damping should be selected greater than the optimum solution, which is valid when modal parameters are precisely known. To achieve this goal, the vibrations of the primary structure with TMD are computed for typical TMD mass ratios, for reasonable uncertainty ranges of eigenfrequency and modal mass, and for different levels of increased TMD damping ratios. Mistuning of the TMD natural frequency and TMD damping ratio is not considered because the precise tunings of TMD natural frequency and TMD damping ratio are feasible with appropriate methods. The natural frequency can be optimized by adjusting the pendulum length or the number of springs, the oil damper viscous coefficient can be optimized by a bypass valve in the cylinder, etc. The evaluation of the results reveals when and how much increased TMD damping is beneficial depending on TMD mass ratio and the deviation between nominal and actual modal structural properties.