## 4 DE Algorithm
### Selection
This case can be expressed as  
\[\overrightarrow{\textbf{\emph{X}}}_{i}^{g+1}=\begin{cases}\overrightarrow{ \textbf{\emph{U}}}_{i}^{g},\;e\text{\text{\text{\text{\text{\text{\text{ \text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\of the beam are \(\omega_{1}\) = 29.87 rad/s, \(\omega_{2}\) = 187 rad/s, and \(\omega_{3}\) = 526.31 rad/s. The disturbance for design can be determined by considering the PSD function \(S_{dist}(\omega)\) of the constant acceleration \(\ddot{x}_{dist}\) for base excitation as a white noise within the given frequency range. This study considers PSD as \(S_{dist}(\omega)\) = 0.015 m\({}^{2}\)/s\({}^{3}\), and the frequency range is described for the first mode control as 0.2 rad/s \(\leq~{}\omega\leq\) 60 rad/s, for the second mode control as 150 rad/s \(\leq~{}\omega\leq\) 200 rad/s, and the third mode control as 450 rad/s \(\leq~{}\omega\leq\) 650 rad/s.  
To determine the optimum TMD parameters, i.e., stiffness, damping and mass coefficients, the objective function is taken to be the mean square of the absolute accelerations (\(f_{1}\)) and the mean square of the absolute displacement (\(f_{2}\)) which are to be decreased in the frequency domain. The DE algorithm is employed under the given constraints in this minimization process. The constraints are as follows for the stiffness, mass, and damping coefficients: \(0\leq\overline{k}\leq 4\times 10^{4}\) N/m, \(0\leq\overline{m}\leq 20\) kg, and \(0\leq\overline{c}\leq 4\times 10^{3}\) Ns/m for the first mode control, \(0\leq\overline{k}\leq 4\times 10^{6}\) N/m, \(0\leq\overline{m}\leq 20\) kg, \(0\leq\overline{c}\leq 4\times 10^{5}\) Ns/m for the second mode control and \(0\leq\overline{k}\leq 8\times 10^{6}\) N/m, \(0\leq\overline{m}\leq 20\) kg, and \(0\leq\overline{c}\leq 8\times 10^{5}\) Ns/m for the third mode control, respectively. Figure 4 demonstrates the variation in the transfer functions with respect to the design steps of the DE algorithm considering the first, second, and third modes of the beam with respect to \(f_{1}\) and \(f_{2}\) minimization. In this figure, it is evident that the objective function reaches a minimum value through design steps.  
Considering all mode controls, the stiffness coefficient of TMD is the largest for the third mode control, then the second, and then the first. Figure 5 shows the frequency response of the sixth node's vertical absolute value of the absolute acceleration transfer function \(|H_{AA6}(\omega)|\), absolute displacement transfer functions \(|H_{D6}(\omega)|\), and sixth node rotation transfer function \(|H_{R6}(\omega)|\) for \(f_{1}\) and \(f_{2}\) minimization of the first mode control. As seen in Figure 5, the TMD is successfully decreased for all responses of transfer functions. All results of the \(f_{1}\) and \(f_{2}\) objective functions give close response reductions, even though they differ.  
To demonstrate the effectiveness of the TMD design the unit harmonic sinus functions of acceleration considering the first three natural frequencies and white noise base excitation, which has unit standard deviation and zero mean, are used at the cantilever beam as the base excitation. The harmonic excitations have frequencies equal to the first mode natural frequencies of the cantilever beam, \(\omega_{1}\), \(\omega_{2}\), and \(\omega_{3}\), so that the analyses can be performed under unfavorable conditions. All disturbance excitations are applied for 10 s durations, and are shown in Figure 6. In addition, the proposed method has been compared to some basic techniques, which are [PERSON] [29] and [PERSON] et al. [30], to show the advantages of the proposed method.  
Figure 3: Cantilever beam–TMD model with six nodes.  
Figure 4: Variation in the transfer functions with respect to the design steps of the DE algorithm considering: (**a**) first, (**b**) second, and (**c**) third modes of the beam with respect to to \(f_{1}\) and \(f_{2}\) minimization.  
Figure 5: Frequency response of: (**a**) 6 th node vertical absolute acceleration transfer function \(|\mathbf{H}_{AA6}(\omega)|\), (**b**) absolute displacement transfer functions \(|\mathbf{H}_{D6}(\omega)|\), and (**c**) 6 th node absolute rotational transfer function \(|\mathbf{H}_{R6}(\omega)|\) for \(f_{1}\) and \(f_{2}\) minimization of the first mode control.  
Figure 6: Cont.  
The method used by [PERSON] [23] to determine the optimum TMD parameters is based on white noise excitation. The formulation of the mass ratio \(\mu_{i}\) for the \(i^{th}\) mode, optimal tuning frequency \(f_{opt}\), and damping ratio \(\xi_{dopt}\) can be specified as follows:  
\[\mu_{i}=\frac{m_{d}}{\mathbf{\phi}_{i}^{T}M_{s}\mathbf{\phi}_{i}} \tag{23}\]  
Figure 6: Disturbance base excitations: (**a**) 1 sin \(\omega_{1}t\), (**b**) 1 sin \(\omega_{2}t\), (**c**) 1 sin \(\omega_{3}t\), and (**d**) white noise excitation.