## 2 Cantilever Beam Model with Tuned Mass Dampers  
Figure 2 demonstrates a typical cantilever beam with a TMD system that faces base disturbance. This figure shows that the translational and rotational displacement vectors may be described as a vector, where \(\mathbf{x}(t)=\{x_{1},\theta_{1},\ x_{2},\theta_{2},\ldots x_{n-1},\theta_{n-1},x_{n},\theta_{n},x_{TMD}\}\) in \((2n+1)\) dimensions. In this expression, \(x_{n}\) demonstrates the vertical displacement of the beam and the acceleration of the random base excitation is indicated by \(\tilde{x}_{dist}(t)\). [PERSON] [22] provides the element stiffness \(\mathbf{k}\) and mass matrices \(\mathbf{m}\) of the Timoshenko beam. \[\mathbf{k}=\frac{EI}{L^{3}(1+\phi)} \begin{bmatrix}12&6L&-12&6L\\ 6L&L^{2}(4+\phi)&-6L&L^{2}(2-\phi)\\ -12&-6L&12&-6L\\ 6L&L^{2}(2-\phi)&-6L&L^{2}(4+\phi)\end{bmatrix}  \tag{1}\]  
\[\mathbf{m}=\frac{\rho\text{AL}}{(1+\phi)^{2}} \begin{bmatrix}m_{1}&m_{2}&m_{3}&m_{4 }\\ m_{2}&m_{5}&-m_{4}&m_{6}\\ m_{3}&-m_{4}&m_{1}&-m_{2}\\ m_{4}&m_{6}&-m_{2}&m_{5}\end{bmatrix} +\frac{\rho\text{AL}}{(1+\phi)^{2}} \begin{bmatrix} m_{7}&m_{8}&-m_{7}&m_{8}\\ m_{8}&m_{9}&-m_{8}&m_{10}\\ -m_{7}&-m_{8}&m_{7}&-m_{8}\\ m_{8}&m_{10}&-m_{8}&m_{9}\end{bmatrix}  \tag{2}\]  
where,  
\[m_{1}=\frac{13}{35}+\frac{7\phi}{10}+\frac{\phi^{2}}{3},\ m_{2}=\left(\frac{1 1}{210}+\frac{11\phi}{120}+\frac{\phi^{2}}{24}\right)L,\ m_{3}=\frac{9}{70}+ \frac{3\phi}{10}+\frac{\phi^{2}}{6},\]  
\[m_{4}=-\left(\frac{13}{42\beta}+\frac{3\phi}{40}+\frac{\phi^{2}}{24}\right)L, \ m_{5}=\left(\frac{15}{105}+\frac{\phi}{60}+\frac{\phi^{2}}{120}\right)L^{2}, \ m_{6}=-\left(\frac{1}{140}+\frac{\phi}{60}+\frac{\phi^{2}}{120}\right)L^{2},\]  
\[m_{7}=\frac{6}{5},m_{8}=\left(\frac{1}{10}-\frac{\phi}{2}\right)L,m_{9}=\left( \frac{2}{5}+\frac{\phi}{6}+\frac{\phi^{2}}{3}\right)L^{2},m_{10}=\left(-\frac {1}{30}-\frac{\phi}{6}+\frac{\phi^{2}}{6}\right)L^{2} \tag{3}\]  
Here, \(r=\left(\frac{I}{A}\right)\) and \(\phi=\left(\frac{12 EI}{(G\kappa A\tilde{L}^{2})}\right)\). \(E\), \(I,G,\kappa,A,L\), and \(\rho\) represent the modulus of elasticity, moment of inertia, shear modulus, shear correction factor, cross section area, length of the element, and mass density, respectively. The equation of motion is presented below:  
\[\left(\mathbf{M}_{s}+\mathbf{M}_{TMD}\right)\,\tilde{x}(t)+\left(\mathbf{C}_{s}+\mathbf{C}_{TMD }\right)\,\tilde{x}(t)+\left(\mathbf{K}_{s}+\mathbf{K}_{TMD}\right)\,x(t)=-(\mathbf{M}_{s} +\mathbf{M}_{TMD})\,\tilde{x}_{dist}(t) \tag{4}\]  
This equation shows \(\tilde{x}(t)\) and \(\tilde{x}(t)\) as the translational and rotational acceleration and velocity vectors of the cantilever beam, respectively. The mass, damping, and stiffness matrices of the beam are signified by \(\mathbf{M}_{s}\), \(\mathbf{C}_{s}\), and \(\mathbf{K}_{s}\), respectively in \((2n+1)\)\(\times\)\((2n+1)\) dimensions. The mass, damping, and stiffness matrices of the TMD are shown by \(\mathbf{M}_{TMD}\), \(\mathbf{C}_{TMD}\), and \(\mathbf{K}_{TMD}\), respectively. Finally, the impact of the vector in \((2n+1)\) dimensions is represented by \(\mathbf{r}=\{1,0,\ 1,0,\ldots 1\}\). Figure 2: Cantilever beam–TMD model. A different approach is used to describe structural damping, which is taken to be proportional to mass in this study. To achieve the transfer function of the equation of motion, the following equation can be used to show the Fourier transform of the previous equation. \[(K+\mathrm{i}\omega\mathbf{C}-\omega^{2}\mathbf{M})\mathbf{X}(\omega)=-\mathbf{M}\ddot{\mathbf{r}} \ddot{\mathbf{X}}_{dist}(\omega) \tag{5}\]  
In this equation, the transfer function of white noise base excitation is represented by \(\ddot{\mathbf{X}}_{dist}(\omega)\); the disturbance frequency is shown by \(\mathbf{X}(\omega)\); the Fourier transform of the displacement vector in \((2n+1)\) dimensions is indicated by \(\mathbf{X}(\omega)\); and the mass, damping, and stiffness matrices of a cantilever beam regulated by a TMD are represented by \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\), respectively.