## 3 Operator Reformulation of the Problem  
Since we are interested in the asymptotic approximation for the natural frequencies of the system, we place \(y(t)=0\) (where \(y(t)\) is an external force), and arrive at the following system of partial differential equations governing the harvester dynamics:  
\[\Phi_{tt}(x,t)=\frac{1}{S\rho(x)}\Bigg{[}\frac{d}{dx}\Big{(}EI(x)\Phi_{x}(x, t)\Big{)}-K(x)\Big{(}\Phi(x,t)-W_{x}(x,t)\Big{)}\Bigg{]}, \tag{16}\]  
\[W_{tt}(x,t)=-\frac{1}{\rho(x)}\frac{d}{dx}\bigg{(}K(x)\Big{(}\Phi(x,t)-W_{x}( x,t)\Big{)}\bigg{)}, \tag{17}\]  
\[v_{t}(t)=-\frac{\theta}{C}\Phi_{t}(L,t)-\frac{1}{CR}v(t). \tag{18}\]  
The boundary conditions are given in Equation (9) and (10). Let \(\mathcal{H}\) be the set of five-component vector-valued functions \(\mathfrak{F}=\left(\varphi_{0}(x),\varphi_{1}(x),\varphi_{2}(x),\varphi_{3}(x), \varphi_{4}\right)^{T}\) obtained as a closure of smooth functions satisfying the conditions \(\varphi_{0}(0)=\varphi_{2}(0)=0\) in the following energy norm:  
\[ \begin{split}\left\|\mathfrak{F}\right\|_{\mathcal{H}}^{2}=\frac {1}{2}\bigg{\{}\int_{0}^{L}\left[\rho(x)\big{|}\varphi_{3}(x)\big{|}^{2}+K(x) \big{|}\varphi_{0}(x)-\varphi_{2x}(x)\big{|}^{2}+\right.\\ \left.S\rho(x)\big{|}\varphi_{1}(x)\big{|}^{2}+EI(x)\big{|}\varphi_ {0x}(x)\big{|}^{2}\right]dx+\big{|}\varphi_{4}\big{|}^{2}\bigg{\}}.\end{split}  \tag{19}\]  
If we introduce a new vector-valued function \(\mathbb{F}\) by the rule:  
\[\mathbb{F}(x,t)=\Big{(}\Phi(x,t),\;\Phi_{t}(x,t),\;W(x,t),\;W_{t}(x,t),\;v(t) \Big{)}^{T},\]  
then it can be easily checked that the problem in Equations (16)-(18), (9) and (10) can be written as a first order in time linear system in the energy space, i.e.,  
\[\mathbb{F}_{t}(x,t)=i\Big{(}\mathbb{L}\mathbb{F}\Big{)}(x,t), \tag{20}\]  
\[\mathbb{F}(x,t)\Big{|}_{t=0}=\mathbb{F}_{0}(x)\equiv\Big{(}\varphi_{0}^{0}(x),\varphi_{1}^{0}(x),\varphi_{2}^{0}(x),\varphi_{3}^{0}(x),\varphi_{4}^{0} \Big{)}^{T}, \tag{21}\]  
where the superscript \"0\" stands for the notation of the initial state. The operator \(\mathbb{L}\) (the dynamics generator) can be given by the following differential expression:  
\[\mathbb{L}=-i\left[ \begin{array}{cccc}0&1&0&0&0\\ \frac{1}{S\rho(x)}\bigg{[}\frac{d}{dx}\Big{(}EI(x)\frac{d}{dx}\cdot\Big{)}-K(x )\cdot\bigg{]}&0&\frac{K(x)}{S\rho(x)}\frac{d}{dx}.&0&0\\ 0&0&0&1&0\\ -\frac{1}{\rho(x)}\frac{d}{dx}\Big{(}K(x)\cdot\Big{)}&0&\frac{1}{\rho(x)} \frac{d}{dx}\Big{(}K(x)\frac{d}{dx}\cdot\Big{)}&0&0\\ 0&-\frac{\theta}{C}A_{L}&0&0&-\frac{1}{CR}\end{array} \right], \tag{22}\]  
defined on the domain  
\[ \begin{split}\mathcal{D}(\mathbb{L})=\bigg{\{}&\Psi\in \mathcal{H}:\;\psi_{0},\psi_{2}\in H^{2}(0,L),\;\;\psi_{1},\psi_{3}\in H^{1}(0,L),\;\;\psi_{4}\in\mathbb{C},\\ &\psi_{0}(0)=\psi_{2}(0)=0,\;\;\psi_{1}(0)=\psi_{3}(0)=0,\;\;\psi_{0}(L)- \psi_{2x}(L)=0,\;\;EI(L)\psi_{0x}(L)=\theta\psi_{4}\bigg{\}}.\end{split}  \tag{23}\]  
The operation \(A_{L}\) means that \(A_{L}f=f(L)\). **Remark 2**. : _We introduce the imaginary unit \"\(i\)\" into the definition in Equation (22) of the dynamics generator and into Equation (20) for convenience. As is shown below, \(\mathbb{L}\) is a finite-rank perturbation of a selfadjoint operator. Thus, owing to this factor, we deal with a selfadjoint operator rather than with a skew-selfadjoint one._  
**Properties of the coefficients:**  
there exist two constants \(0<C_{0}\leq C_{1}<\infty\) such that  
\[C_{0}\leq\rho(x)\leq C_{1},\;\;C_{0}\leq EI(x)\leq C_{1},\;\;C_{0}\leq K(x) \leq C_{1}\;\;\text{for}\;\;x\in[0,L]; \tag{24}\]  
\[EI\in H^{3}(0,L),\;\;K\in H^{3}(0,L),\;\;\rho\in H^{4}(0,L). \tag{25}\]  
**Derivation of the formula for the adjoint operator \(\mathbb{L}^{*}\). **  
**Lemma 1**. : _The operator \(\mathbb{L}^{*}\) is defined by the differential expression obtained from Equation (22) in which the second column is replaced with the column \(\big{(}1,0,0,0,\theta A_{L}\big{)}^{T}\). This operator is defined on the domain_  
\[