## 5 Calibration of STMD Using Numerical Optimization
### Objective and Procedure of Optimization  
In the case of the sub-optimal STMD, a clear performance limit was observed and manifested as the smallest achievable value of \(DAF_{max}\) (Figure 7a). Therefore, the first natural objective of optimizing the STMD was to determine the best calibration of the STMD force parameters for the highest possible efficiency, exceeding that of the sub-optimal STMD.  
The main objective of optimization was to determine the STMD force parameters for various \(K\) and \(\mu\), in such a way as to minimize the differences in efficiency between the STMD and the ATMD for a given \(K\). It was also assumed that the optimized STMD must satisfy the condition concerning the amplitude of relative displacement \(X_{d}\), which cannot exceed that for a TMD with the same mass and for the same excitation.  
The numerical optimization was carried out in MATLAB using the unconstrained minimization method implemented as the _fminsearch_ function. The objective function \(J(v)\) to be minimized using the _fminsearch_ function was defined based on the dynamic amplification of the primary structure displacement, as introduced in [42]. This objective function consists of the sum of squared differences between the local peaks, and the local minimum between those peaks, for both the STMD with specific \(\mu\) and \(K\), and the [PERSON]'s TMD with mass ratio \(K\mu\):  
\[J(\mathbf{v})=(P_{sa}(\mathbf{v})-P)^{2}+(Q_{sa}(\mathbf{v})-Q)^{2}+(S_{sa}(\mathbf{v})-S)^{2}, \tag{49}\]  
where \(\mathbf{v}=[K_{cor}\ c_{cor}]\) is a vector of the optimization variables; \(P_{sa}(\mathbf{v})\), \(Q_{sa}(\mathbf{v})\) and \(S_{sa}(\mathbf{v})\) are, respectively, the left and right local peaks and the local minimum in between those peaks of the frequency response \(X_{1}/X_{1 static}\) of the primary structure with the STMD of mass ratio \(\mu\) and control parameter \(K\); while \(P\), \(Q\) and \(S\) are the corresponding quantities for the TMD with mass ratio \(K\mu\).  
Specifically, the objective function (49) assumes a zero value when the two peaks of the frequency response for the STMD are equal and match the value given by (13), and when the local minimum is as for the TMD with mass ratio of \(K\mu\).  
The _fminsearch_ function was run with an initial estimate of the optimization variables \(\mathbf{v}=[1\ 1]\). The maximum number of iterations allowed was set to 400. In each iteration of the optimization, the actual feedback gains \(c_{sa}\) (47), \(k_{sa}\) (48) and the corresponding equivalent model \(c_{eq}\) (44), \(k_{eq}\) (45) are calculated for the current estimate of optimization variables \(K_{cor}\), \(c_{cor}\). Then, the frequency response of the primary structure with the STMD is determined, as well as the current value of the objective function (49). The frequency responses are calculated using (16), but with \(c_{eq}\) and \(k_{eq}\) instead of \(c_{a}\) and \(k_{a}\). This approach allows for rapid frequency response calculations in each iteration, leading to quickly obtained optimization results even for large number of iterations.  
The optimization was conducted for the sample model parameters listed in Section 3.5, for various \(K\), and for \(\mu\) ranging from 0.4% to 3%.