## 2 Modeling of TSDs
### Design Principles  
Required design steps follow the line of thought representative for optimally tuning a TMD. Both are passive AMDs, aimed at reducing oscillations, specifically using the rules applicable for accelerations. Displacements will also be decreased. This process starts by identifying and decomposing the main directions of motion. In Figure 2, we show the schematic representation of the TMD and TSD, respectively. These are depicted in-plane on the cross-section of the structure. Mechanically, the TMD acts on each bending mode as a Single-Degree-of-Freedom (SDOF) system. Due to the fact that the mode shapes of the construction are uncoupled (i.e., bending modes neither affect each other, nor the torsional mode), the TMD can be tuned for each direction independently. This results in one-dimensional models. We are led to the total action by superposing the orthogonally functioning oscillators. TSDs follow an analogous principle. We considered two perpendicular sloshing directions, which are presented by a water in motion using a two-dimensional model (with sideways and vertical orientations) for each of the directions. This is actually a beneficial simplification, as it discards potential effects from the waves influencing each other. The key design steps need to address the following major points:  
1. Determine the mass ratio as an upper limit, depending on the maximum weight to be added to the structure;
2. Set a frequency (or tuning) ratio by determining at which frequency the added device should naturally vibrate;
3. Assess the damping ratio in the case of a TSD representing the estimate of the additional damping that could be achieved. Optimizing AMDs in order to affect accelerations is analogous between the TMD and TSD. As the behaviour of TSDs is inherently nonlinear, this approach provide a starting basis for the design. We outline the necessary steps for a TMD in Setup 1. Such steps are applicable for setting up an equivalent linear SDOF mechanical model, with viscous damping. The process needs to take place for each of the dominant directions. Numerical values for the parameters are presented in Table 2. In the case of a TSD, this process is slightly modified, as highlighted in Setup 2. The workflow follows steps proposed by [24; 25]. Figure 2: In-plane schematic representation of TMDs and TSDs. ```
1 Choose the total mass ratio 1 \(\mu_{tot}\leq 1.5\,\%\) 1 2 Calculate the added mass 2 \(m=\mu_{tot}\ m_{S}\) 2 3 Calculate the modal mass ratio \(\mu=\dfrac{m}{m_{S}^{*}}\ \eta^{2}\) 3 4 Calculate the tuning (i.e., frequency) ratio \(\Omega_{opt}=\dfrac{1}{\sqrt{1+\mu}}\) 4 5 Calculate the frequency of the damper based on the target structural frequency 6 Calculate the stiffness \(k=\left(2\ \pi\ f_{opt}\right)^{2}\ m\) 6 7 Calculate the optimal damping ratio \(\zeta_{opt}=\sqrt{\dfrac{3\mu}{4(1+\mu)(2+\mu)}}\) 7 8 Calculate the damping coefficient \(c=2\ \zeta_{opt}\ m\left(2\ \pi\ f_{opt}\right)\) 8
```  
**Setup 2** The steps for optimally tuning a TSD  
Footnote 1: All parameters are for a TDD, unless specified otherwise and appropriately marked by a subscript. 1 The initial displacement can be achieved by a priori determining or assuming a typical value. 2 The damping ratio for a TSD with screens under random excitation, as given by the mathematical model in [25]. 3 The total amount of water mass would typically be inefficient in one container and also difficult to allocate space for, so it needs to be distributed. Footnote 2: All parameters are for a TDD, unless specified otherwise and appropriately marked by a subscript. 1 Here, we use as an upper limit the value typical for cantilever-type structures, as recommended by [26]. 2 \(\eta\) is the value of the mode shape at the AMD.