## 7 The Harvester Model with Constant Physical Parameters. The Left Reflection Matrix  
As shown in Sections 1-6, for the harvester model with the Timoshenko beam as a substructure, the asymptotic distribution of the vibrational modes has the following important property: the leading asymptotical term does not depend on the properties of the piezoceramic patches. It means that the perturbation induced by these patches is, in some sense, small. At the same time, the entire effect of the energy harvesting takes place exactly due to the physical properties of the piezoelectricity. Therefore, it is important to figure out the level of accuracy in the asymptotical formulas for the spectrum so that the contribution of the piezoelectric patches physical parameters will be revealed. To this end, we consider the same model of the harvester based on the Timoshenko beam substructure with _constant_ (not variable) physical parameters. We derive refined asymptotic approximations for the vibrational modes and show that the second asymptotical term (next to the leading one) contains contribution from the electrical circuit of the model. Similar results can be obtained for the case of the variable structural parameters of the beam. However, preliminary calculations show that the derivation of the formulas is extremely lengthy and the result does not provide any additional insight into the physics of the problem. We consider the following system of equations  
\[\rho\ \ W_{tt}(x,t)=-K\Big{(}\Phi_{x}(x,t)-W_{xx}(x,t)\Big{)}, \tag{161}\]  
\[S\rho\ \ \Phi_{tt}(x,t)=EI\ \ \Phi_{xx}(x,t)-K\Big{(}\Phi(x,t)-W_{x}(x,t)\Big{)}, \tag{162}\]  
equipped with the boundary conditions:  
\[\Phi(0,t)=W(0,t)=0; \tag{163}\]  
\[EI\ \ \Phi_{x}(L,t)=-\theta\wp(t).\qquad K\Big{(}\Phi(x,t)-W_{x}(x,t)\Big{)}= \alpha\ \ W_{t}(L,t),\qquad\alpha>0. \tag{164}\]  
In this section, we use the standard engineering approach and look for the solutions of the boundary-value problem in the form  
\[W(x,t)=e^{i\lambda t}w(\lambda,x),\qquad\Phi(x,t)=e^{i\lambda t}\varphi( \lambda,x). \tag{165}\]  
Substituting these representations into Equations (161) and (162), we obtain a system for \(w\) and \(\varphi\)  
\[-\rho\lambda^{2}w(\lambda,x)+K\Big{(}\varphi^{\prime}(\lambda,x)-w^{\prime \prime}(\lambda,x)\Big{)}=0, \tag{166}\]  
\[-S\rho\lambda^{2}\varphi(\lambda,x)-EI\varphi^{\prime\prime}(\lambda,x)+K \Big{(}\varphi(\lambda,x)-w^{\prime}(\lambda,x)\Big{)}=0. \tag{167}\]  
To eliminate \(w\) from the system and write a new equation in terms of \(\varphi\), we have to complete some preliminary steps. First, we derive from Equation (167) that  
\[w^{\prime}(\lambda,x)=\varphi(\lambda,x)-\lambda^{2}\ \ \frac{S\rho}{K}\ \ \varphi( \lambda,x)-\frac{EI}{K}\ \ \varphi^{\prime\prime}(\lambda,x). \tag{168}\]  
Then, we differentiate Equation (166) once and use Equation (168) to get  
\[-\rho\lambda^{2}w^{\prime}(\lambda,x)+K\Big{(}\varphi^{\prime\prime}(\lambda, x)-w^{\prime\prime\prime}(\lambda,x)\Big{)}=0,\]  
which yields the desired form of the equation for \(\varphi\)  
\[\varphi^{\prime\prime\prime\prime}(\lambda,x)+\lambda^{2}\bigg{(}\frac{S\rho }{EI}+\frac{\rho}{K}\bigg{)}\varphi^{\prime\prime}(\lambda,x)+\frac{\rho}{EI} \bigg{(}\lambda^{4}\frac{S\rho}{K}-\lambda^{2}\bigg{)}\varphi(\lambda,x)=0. \tag{169}\]  
The corresponding characteristic equation is  
\[S\rho\ \ \Phi_{tt}(x,t)=EI\ \ \Phi_{xx}(x,t)-K\Big{(}\Phi(x,t)-W_{x}(x,t)\Big{)}, \tag{162}\]0  
The roots of the biquadratic equation are  
\[ \begin{split}\Big{(}z^{2}\Big{)}_{1,2}&=-\frac{ \lambda^{2}}{2}\bigg{(}\frac{S\rho}{EI}+\frac{\rho}{K}\bigg{)}\pm\sqrt{\frac{ \lambda^{4}}{4}\bigg{(}\frac{S\rho}{EI}+\frac{\rho}{K}\bigg{)}^{2}-\lambda^{4} \frac{S\rho^{2}}{KEI}+\lambda^{2}\frac{\rho}{EI}}\\ &=-\frac{\lambda^{2}}{2}\bigg{(}\frac{S\rho}{EI}+\frac{\rho}{K} \bigg{)}\pm\sqrt{\frac{\lambda^{4}}{4}\bigg{(}\frac{S\rho}{EI}-\frac{\rho}{K }\bigg{)}^{2}+\lambda^{2}\frac{\rho}{EI}}.\end{split}