Appendix
A. Electromechanical energies analysis of main piezoelectric cantilever beam
If \(Y_{s}\) and \(Y_{p}\) are the \(young^{'}s\) modulus of substrate and piezoelectric materials, \(e_{31}\) and \(\varepsilon _{33}\) are piezoelectric and permittivity constants of piezoelectric material. Also, V(t) is the output voltage across piezoelectric material. Then, the stress developed on substrate (\(T_{s}\)) and piezoelectric layer \(Y_{p}\) and strain \((\varepsilon _{m})\) are expressed as below.
$$\begin{aligned}&T_{s}=Y_{s}\,\varepsilon _{m} \end{aligned}$$
(33)
$$\begin{aligned}&T_{p}=Y_{p}\,\varepsilon _{m}-\frac{e_{31}V(t)}{t_{p}} \end{aligned}$$
(34)
$$\begin{aligned}&\varepsilon _{m}=\frac{\partial D_{T}^{(m)}(x_{o},z,t)}{\partial x_{o}}\nonumber \\&\quad =\frac{\partial D^{(m)}(x_{o},t)}{\partial x_{o}}-z\frac{\partial ^{2}E^{(m)}(x_{o},t)}{\partial x_{o}^{2}} \end{aligned}$$
(35)
In addition, according to [3] the transverse and longitudinal displacement with respect to the moving base of the beam can be expressed as following absolute and uniform convergent series of the eigenfunctions.
$$\begin{aligned}&E^{(m)}(x_{o},t)=\sum _{i}^{N}\alpha _{i}^{(m)}(t)\,\theta _{i}^{(m)}(x_{o)} \end{aligned}$$
(36)
$$\begin{aligned}&D^{(m)}(x_{o},t)=\sum _{i}^{N}\beta _{i}^{(m)}(t)\,\eta _{i}^{(m)}(x_{o)} \end{aligned}$$
(37)
Here, \(\theta _{i}^{(m)}(x_{o)}\) and \(\eta _{i}^{(m)}(x_{o)}\) are the assumed mode shape functions. Also, \(\alpha _{i}^{(m)}(t)\) and \(\beta _{i}^{(m)}(t)\) are modal coordinates of the clamped-free main beam for the \(i_{th}\) mode. Hence, by substituting the expressions of stress, strain, longitudinal and transverse displacement of the main beam in the following standard expressions, the strain, kinetic and electrical energy of the main beam can be calculated.
$$\begin{aligned}&SE^{(m)}=\frac{1}{2}\int _{V_{s}}T_{s}\varepsilon _{m}dV_{s}\nonumber \\&\quad +\frac{1}{2}\int _{V_{p}}T_{p}\varepsilon _{m}dV_{p} \end{aligned}$$
(38)
$$\begin{aligned}&KE^{(m)}={} \frac{1}{2}\int _{V_{s}}\rho _{s\,}\nonumber \\&\quad [\,\left( \frac{\partial D_{T}^{(m)}(x_{o},z,t)}{\partial t}\right) ^{2}\nonumber \\&\qquad +\left( \frac{\partial E_{T}^{(m)}(x_{o},t)}{\partial t}\right) ^{2}]\,dV_{s} \nonumber \\&\qquad +\frac{1}{2}\int _{V_{p}}\rho _{p\,}[\,\left( \frac{\partial D_{T}^{(m)}(x_{o},z,t)}{\partial t}\right) ^{2}\nonumber \\&\qquad +\left( \frac{\partial E_{T}^{(m)}(x_{o},t)}{\partial t}\right) ^{2}]\,dV_{p} \end{aligned}$$
(39)
$$\begin{aligned}&EE=-\frac{1}{2}\int _{V_{p}}\left( e_{31}\varepsilon _{m}\frac{V(t)}{t_{p}}\right. \nonumber \\&\quad \left. -\varepsilon _{33}(\frac{V(t)}{t_{p}})^{2}\right) dV_{p} \end{aligned}$$
(40)
Here, mass(m), stiffness (k),damping (\(\zeta\))and force (f) matrices are defined as below:
$$\begin{aligned}&[U_{1}^{KE^{(m)}(1)}\,U_{2}^{KE^{(m)}(1)}\,U_{3}^{KE^{(m)}(1)}]\nonumber \\&\quad =[U_{1}^{KE^{(m)}(3)}\,U_{2}^{KE^{(m)}(3)}\,U_{3}^{KE^{(m)}(3)}]\nonumber \\&\quad =Y_{s}\,[U_{s1}\,U_{s2}\,U_{s3}] \end{aligned}$$
(41)
$$\begin{aligned}&[U_{1}^{KE^{(m)}(2)}\,U_{2}^{KE^{(m)}(2)}\,U_{3}^{KE^{(m)}(2)}]\nonumber \\&\quad =Y_{s}\,[U_{s1}\,U_{s2}\,U_{s3}]+Y_{p}\,[U_{p1}\,U_{p2}\,U_{p3}] \end{aligned}$$
(42)
$$\begin{aligned}&m_{ij}^{\alpha \alpha (m)}\nonumber \\&\quad =\sum _{r=1}^{3}\int _{r}\left( U_{3}^{KE^{(m)}(r)}\theta _{i}^{'\,(m)}\theta _{j}^{'\,(m)}\right. \nonumber \\&\qquad \left. +U_{1}^{KE^{(m)}(r)}\theta _{i}^{(m)}\theta _{j}^{(m)}\right) dx_{o} \end{aligned}$$
(43)
$$\begin{aligned}&m_{ij}^{\alpha \beta (m)}=\sum _{r=1}^{3}\int _{r}U_{2}^{KE^{(m)}(r)}\nonumber \\&\quad \theta _{i}^{'\,(m)}\eta _{j}^{(m)}dx_{o} \end{aligned}$$
(44)
$$\begin{aligned}&m_{ij}^{\beta \beta (m)}=\sum _{r=1}^{3}\int _{r}U_{1}^{KE^{(m)}(r)}\nonumber \\&\quad \eta _{i}^{(m)}\eta _{j}^{(m)}dx_{o} \end{aligned}$$
(45)
$$\begin{aligned}&f_{i}^{(m)}=\sum _{r=1}^{3}\int _{r}U_{1}^{KE^{(m)}(r)}\nonumber \\&\quad \theta _{i}^{(m)}\frac{\partial E_{b}^{(m)}(x_{o},t)}{\partial t}dx_{o} \end{aligned}$$
(46)
$$\begin{aligned}&[U_{s1}\,U_{s2}\,U_{s3}]=\int \int _{A_{s}}[1\,z\,z^{2}]\,dydz \end{aligned}$$
(47)
$$\begin{aligned}&{[}U_{p1}\,U_{p2}\,U_{p3}]=\int \int _{A_{p}}[1\,z\,z^{2}]\,dydz \end{aligned}$$
(48)
$$\begin{aligned}&{[}U_{1}^{SE^{(m)}(1)}\,U_{2}^{SE^{(m)}(1)}\,U_{3}^{SE^{(m)}(1)}]\nonumber \\&\quad =[U_{1}^{SE^{(m)}(3)}\,U_{2}^{SE^{(m)}(3)}\,U_{3}^{SE^{(m)}(3)}]\nonumber \\&\quad =Y_{s}\,[U_{s1}\,U_{s2}\,U_{s3}] \end{aligned}$$
(49)
$$\begin{aligned}&[U_{1}^{SE^{(m)}(2)}\,U_{2}^{SE^{(m)}(2)}\,U_{3}^{SE^{(m)}(2)}]\nonumber \\&\quad =Y_{s}\,[U_{s1}\,U_{s2}\,U_{s3}]+Y_{p}\,[U_{p1}\,U_{p2}\,U_{p3}] \end{aligned}$$
(50)
$$\begin{aligned}&[H_{p1}\,H_{p2}]=\int \int _{A_{p}}\frac{e_{31}}{t_{p}}\,[1\,z]\,dydz \end{aligned}$$
(51)
$$\begin{aligned}&k_{ij}^{\beta \beta (m)}=\sum _{r=1}^{3}\int _{r}U_{1}^{SE^{(m)}(r)}\nonumber \\&\quad \eta _{i}^{'\,(m)}\eta _{j}^{'\,(m)}dx_{o} \end{aligned}$$
(52)
$$\begin{aligned}&k_{ij}^{\alpha \beta (m)}=\sum _{r=1}^{3}\int _{r}U_{2}^{SE^{(m)}(r)}\nonumber \\&\quad \theta _{i}^{''\,(m)}\eta _{j}^{'\,(m)}dx_{o} \end{aligned}$$
(53)
$$\begin{aligned}&k_{ij}^{\alpha \alpha (m)}=\sum _{r=1}^{3}\int _{r}U_{3}^{SE^{(m)}(r)}\nonumber \\&\quad \theta _{i}^{''\,(m)}\theta _{j}^{''\,(m)}dx_{o} \end{aligned}$$
(54)
$$\begin{aligned}&\zeta _{i}^{\beta }=\int _{L_{p}}H_{p2}\theta _{j}^{''\,(m)}dx_{o} \end{aligned}$$
(55)
$$\begin{aligned}&\zeta _{i}^{\alpha }=\int _{L_{p}}H_{p1}\eta _{j}^{'\,(m)}dx_{o} \end{aligned}$$
(56)
B. Electromechanical energies analysis of straight beam
Therefore, by incorporating the effect of vibrating clamped end and proof mass in Eqn. (4-5), the expressions of the strain \((SE^{s})\) and Kinetic \((KE^{s})\) energy have been obtained. Next, the mass (m), stiffness (k) and force (f) matrices for the straight beam are illustrated below:
$$\begin{aligned}&{[}U_{1}^{SE^{(s)}}\,U_{2}^{SE^{(s)}}\,U_{3}^{SE^{(s)}}]\nonumber \\&\quad =Y_{s}\int \int _{A_{s}}[1\,z\,z^{2}]\,dydz \end{aligned}$$
(57)
$$\begin{aligned}&{[}U_{1}^{KE^{(s)}}\,U_{2}^{KE^{(s)}}\,U_{3}^{KE^{(s)}}]\nonumber \\&\quad =\rho _{s}\int \int _{A_{s}}[1\,z\,z^{2}]\,dydz \end{aligned}$$
(58)
$$\begin{aligned}&m_{ij}^{\alpha \alpha (s)}=\int _{0}^{l_{s}}\left( U_{3}^{KE^{(s)}}\theta _{i}^{'\,(s)}\theta _{j}^{'\,(s)}\right. \nonumber \\&\quad \left. +U_{1}^{KE^{(s)}}\theta _{i}^{(s)}\theta _{j}^{(s)}\right) dx_{s}+M_{2}\theta _{i}^{(s)}(l_{s})\nonumber \\&\quad \theta _{j}^{(s)}(l_{s})+I_{2}\theta _{i}^{'\,(s)}(l_{s})\,\theta _{j}^{'\,(s)}(l_{s}) \end{aligned}$$
(59)
$$\begin{aligned}&m_{ij}^{\alpha \beta (s)}=\int _{0}^{l_{s}}U_{2}^{KE^{(s)}}\theta _{i}^{'\,(s)}\eta _{j}^{(s)}dx_{s} \end{aligned}$$
(60)
$$\begin{aligned}&m_{ij}^{\beta \beta (s)}=\int _{0}^{l_{s}}U_{1}^{KE^{(s)}}\eta _{i}^{(s)}\eta _{j}^{(s)}dx_{s} \end{aligned}$$
(61)
$$\begin{aligned}&m_{ij}^{\alpha \alpha (sm)}=\int _{0}^{l_{s}}U_{1}^{KE^{(s)}}\theta _{i}^{(s)}\nonumber \\&\quad (\,\theta _{j}^{(m)}(L)+x_{s}\theta _{j}^{'\,(m)}(L))dx_{s}+M_{2}\theta _{i}^{(s)}(l_{s})(\theta _{j}^{(m)}(L)+l_{s}\theta _{j}^{'\,(m)}(L)) \end{aligned}$$
(62)
$$\begin{aligned}&m_{ij}^{\alpha \alpha (ms)}={} \int _{0}^{l_{s}}[U_{1}^{KE^{(s)}}(\theta _{i}^{(m)}(L)\nonumber \\&\qquad +x_{s}\theta _{i}^{'\,(m)}(L))\,(\theta _{j}^{(m)}(L)\nonumber \\&\qquad +x_{s}\theta _{j}^{'\,(m)}(L))]dx_{s}+M_{2}(\theta _{i}^{(m)}(L) \nonumber \\&\qquad +l_{s}\theta _{i}^{'\,(m)}(L))\,(\theta _{j}^{(m)}(L)+l_{s}\theta _{j}^{'\,(m)}(L)) \end{aligned}$$
(63)
$$\begin{aligned}&k_{ij}^{\alpha \alpha \,(s)}=\int _{0}^{l_{s}}U_{3}^{SE^{(s)}}\nonumber \\&\quad \theta _{i}^{''\,(s)}\theta _{j}^{''\,(s)}dx_{s} \end{aligned}$$
(64)
$$\begin{aligned}&k_{ij}^{\alpha \beta \,(s)}=\int _{0}^{l_{s}}U_{2}^{SE^{(s)}}\nonumber \\&\quad \theta _{i}^{''\,(s)}\eta _{j}^{'\,(s)}dx_{s} \end{aligned}$$
(65)
$$\begin{aligned}&k_{ij}^{\beta \beta \,(s)}=\int _{0}^{l_{s}}U_{1}^{SE^{(s)}}\nonumber \\&\quad \eta _{i}^{'\,(s)}\eta _{j}^{'\,(s)}dx_{s} \end{aligned}$$
(66)
$$\begin{aligned}&f_{i}^{(s)}=\int _{0}^{l_{s}}U_{1}^{KE^{(s)}}\nonumber \\&\quad \theta _{i}^{(s)}\frac{\partial E_{b}^{(m)}(x_{s}+L,\,t)}{\partial t}dx_{s}\nonumber \\&\qquad +M_{2}\theta _{i}^{(s)}(l_{s})\frac{\partial E_{b}^{(m)}(x_{s}+L,\,t)}{\partial t}|_{x_{s}=l_{s}} \end{aligned}$$
(67)
$$\begin{aligned}&f_{i}^{\sim (s)}={} \int _{0}^{l_{s}}U_{1}^{KE^{(s)}}(\theta _{i}^{(m)}(L)\nonumber \\&\quad +x_{s}\theta _{i}^{'\,(m)}(L))\frac{\partial E_{b}^{(m)}(x_{s}+L,\,t)}{\partial t}dx_{s}+M_{2}(\theta _{i}^{(m)}(L) \nonumber \\&\quad +l_{s}\theta _{i}^{'\,(m)}(L))\frac{\partial E_{b}^{(m)}(x_{s}+L,\,t)}{\partial t}|_{x_{s}=l_{s}} \end{aligned}$$
(68)
C. Electromechanical energies analysis of L-shaped beams
The transverse and axial displacement appeared in XY and YZ beam can be represented by convergent series of eigenfunctions [3] as shown below:
$$\begin{aligned}&E^{(k)}(x_{h},t)=\sum _{i}^{N}\alpha _{i}^{(k)}(t)\,\theta _{i}^{(k)}(x_{h}) \end{aligned}$$
(69)
$$\begin{aligned}&D^{(k)}(x_{h},t)=\sum _{i}^{N}\beta _{i}^{(k)}(t)\,\eta _{i}^{(k)}(x_{h}) \end{aligned}$$
(70)
$$\begin{aligned}&E^{(k)}(x_{v},t)=\sum _{i}^{N}\gamma _{i}^{(k)}(t)\,\phi _{i}^{(k)}(x_{v}) \end{aligned}$$
(71)
Here, \(\theta _{i}^{(k)}(x_{h)}\), \(\eta _{i}^{(k)}(x_{h)}\) and \(\phi _{i}^{(k)}(x_{v})\)are the assumed mode shape functions. Also, \(\alpha _{i}^{(m)}(t)\), \(\beta _{i}^{(m)}(t)\) and \(\gamma _{i}^{(k)}(t)\) are modal coordinates of the L-shaped beam for the \(i_{th}\) mode. Therefore, with the use of Eqn. (23-25) in Eqn. (4-5), the simplified expressions of strain and kinetic energies have been derived. Again, the mass (m), stiffness (k) and force (f) matrices for the L-shaped beams are expressed below:
$$\begin{aligned}&{[}U_{1}^{SE^{(k)}}\,U_{2}^{SE^{(k)}}\,U_{3}^{SE^{(k)}}]\nonumber \\&\quad =Y_{s}\int \int _{A_{s}}[1\,z\,z^{2}]\,dydz \end{aligned}$$
(72)
$$\begin{aligned}&{[}U_{1}^{KE^{(k)}}\,U_{2}^{KE^{(k)}}\,U_{3}^{KE^{(k)}}]\nonumber \\&\quad =\rho _{s}\int \int _{A_{s}}[1\,z\,z^{2}]\,dydz \end{aligned}$$
(73)
$$\begin{aligned}&m_{ij}^{\beta \beta (k)}=\int _{0}^{l_{k}}U_{1}^{KE^{(k)}}\eta _{i}^{(k)}\eta _{j}^{(k)}dx_{k} \end{aligned}$$
(74)
$$\begin{aligned}&m_{ij}^{\alpha \alpha (k)}=\int _{0}^{l_{k}}\left( U_{3}^{KE^{(k)}}\right. \nonumber \\&\quad \theta _{i}^{'\,(k)}\theta _{j}^{'\,(k)}\nonumber \\&\qquad +U_{1}^{KE^{(k)}}\theta _{i}^{(k)}\nonumber \\&\quad \left. \theta _{j}^{(k)}\right) dx_{k}+M_{k}\theta _{i}^{(k)}(l_{v})\nonumber \\&\quad \theta _{j}^{(k)}(l_{v})+I_{k}\theta _{i}^{'\,(k)}(l_{v})\,\theta _{j}^{'\,(k)}(l_{v}) \end{aligned}$$
(75)
$$\begin{aligned}&m_{ij}^{\alpha \beta (s)}=\int _{0}^{l_{k}}U_{2}^{KE^{(k)}}\nonumber \\&\quad \theta _{i}^{'\,(k)}\eta _{j}^{(k)}dx_{k} \end{aligned}$$
(76)
$$\begin{aligned}&m_{ij}^{\gamma \alpha (s)}=\int _{0}^{l_{k}}U_{2}^{KE^{(k)}}\nonumber \\&\quad \theta _{i}^{'\,(k)}\psi _{ij}^{(k)}dx_{k} \end{aligned}$$
(77)
$$\begin{aligned}&\psi _{ij}^{(k)}=\int _{0}^{l_{v}}\phi _{i}^{'\,(k)}\phi _{j}^{'\,(k)}dx_{v} \end{aligned}$$
(78)
$$\begin{aligned}&m_{ij}^{\gamma \beta (k)}=\int _{0}^{l_{k}}U_{1}^{KE^{(k)}}\eta _{i}^{(k)}\psi _{ij}^{(k)}dx_{k} \end{aligned}$$
(79)
$$\begin{aligned}&m_{ij}^{\gamma \gamma (k)}=\int _{0}^{l_{k}}U_{1}^{KE^{(k)}}\nonumber \\&\quad \phi _{i}^{(k)}\phi _{j}^{(k)}dx_{k}\nonumber \\&\quad +M_{k}\phi _{i}^{(k)}(l_{v})\,\phi _{j}^{(k)}(l_{v})\nonumber \\&\quad +I_{k}\phi _{i}^{'\,(k)}(l_{v})\,\phi _{j}^{'\,(k)}(l_{v}) \end{aligned}$$
(80)
$$\begin{aligned}&m_{ij}^{\alpha \gamma (k)}=\int _{0}^{l_{k}}U_{1}^{KE^{(k)}}\nonumber \\&\quad \theta _{i}^{(k)}\phi _{j}^{(k)}dx_{k}\nonumber \\&\qquad +M_{k}\theta _{i}^{(k)}(l_{v})\,\phi _{j}^{(k)}(l_{v})\nonumber \\&\qquad +I_{k}\theta _{i}^{'\,(k)}(l_{v})\,\phi _{j}^{'\,(k)}(l_{v}) \end{aligned}$$
(81)
$$\begin{aligned}&q_{i}^{(k)}=\int _{0}^{l_{k}}U_{1}^{KE^{(k)}}\phi _{i}^{(k)}\nonumber \\&\quad \frac{\partial E_{b}^{(m)}(x_{k}+L,\,t)}{\partial t}dx_{k}\nonumber \\&\qquad +M_{k}\phi _{i}^{(k)}(l_{v})\frac{\partial E_{b}^{(m)}(x_{k}+L,\,t)}{\partial t}|_{x_{k}=l_{v}} \end{aligned}$$
(82)
$$\begin{aligned}&m_{ij}^{\gamma \alpha \,(km)}=\int _{0}^{l_{k}}U_{1}^{KE^{(k)}}\nonumber \\&\quad \phi _{i}^{(k)}(\,\theta _{j}^{(m)}(L)\nonumber \\&\qquad +x_{k}\theta _{j}^{'\,(m)}(L))dx_{k}\nonumber \\&\qquad +M_{k}\phi _{i}^{(k)}(l_{v})(\theta _{j}^{(m)}(L)+l_{v}\theta _{j}^{'\,(m)}(L)) \end{aligned}$$
(83)
$$\begin{aligned}&m_{ij}^{\alpha \alpha \,(km)}=\int _{0}^{l_{k}}U_{1}^{KE^{(k)}}\nonumber \\&\quad \theta _{i}^{(k)}(\,\theta _{j}^{(m)}(L)\nonumber \\&\qquad +x_{k}\theta _{j}^{'\,(m)}(L))dx_{k}+M_{k}\theta _{i}^{(k)}(l_{v})(\theta _{j}^{(m)}(L)\nonumber \\&\qquad +l_{v}\theta _{j}^{'\,(m)}(L)) \end{aligned}$$
(84)
$$\begin{aligned}&m_{ij}^{\alpha \alpha \,(mm)}={} \int _{0}^{l_{k}}[U_{1}^{KE^{(k)}}(\theta _{i}^{(m)}(L)\nonumber \\&\quad +x_{k}\theta _{i}^{'\,(m)}(L))\,(\theta _{j}^{(m)}(L)\nonumber \\&\quad +x_{k}\theta _{j}^{'\,(m)}(L))]dx_{k}\nonumber \\&\quad +M_{k}(\theta _{i}^{(m)}(L)+l_{v}\theta _{i}^{'\,(m)}(L))\nonumber \\&\quad (\theta _{j}^{(m)}(L)+l_{v}\theta _{j}^{'\,(m)}(L)) \end{aligned}$$
(85)
$$\begin{aligned}&k_{ij}^{\beta \beta \,(k)}=\int _{0}^{l_{k}}U_{1}^{SE^{(k)}}\nonumber \\&\quad \eta _{i}^{'\,(k)}\eta _{j}^{'\,(k)}dx_{k} \end{aligned}$$
(86)
$$\begin{aligned}&k_{ij}^{\alpha \alpha \,(k)}=\int _{0}^{l_{k}}U_{3}^{SE^{(k)}}\nonumber \\&\quad \theta _{i}^{''\,(k)}\theta _{j}^{''\,(k)}dx_{k} \end{aligned}$$
(87)
$$\begin{aligned}&k_{ij}^{\alpha \beta \,(k)}=\int _{0}^{l_{k}}U_{2}^{SE^{(k)}}\nonumber \\&\quad \theta _{i}^{''\,(k)}\eta _{j}^{'\,(k)}dx_{k} \end{aligned}$$
(88)
$$\begin{aligned}&k_{ij}^{\gamma \beta \,(k)}=\int _{0}^{l_{k}}U_{1}^{SE^{(k)}}\nonumber \\&\quad \eta _{i}^{'\,(k)}\gamma _{i}^{'\,(k)}\gamma _{j}^{'\,(k)}dx_{k\gamma } \end{aligned}$$
(89)
$$\begin{aligned}&k_{ij}^{\gamma \alpha \,(k)}=\int _{0}^{l_{k}}U_{2}^{SE^{(k)}}\nonumber \\&\quad \theta _{i}^{''\,(k)}\phi _{i}^{'\,(k)}\phi _{j}^{'\,(k)}dx_{k} \end{aligned}$$
(90)
$$\begin{aligned}&k_{ij}^{\gamma \gamma \,(k)}=\int _{0}^{l_{k}}U_{1}^{SE^{(k)}}\nonumber \\&\quad \phi _{i}^{'\,(k)}\phi _{j}^{'\,(k)}\phi _{q}^{'\,(k)}\phi _{r}^{'\,(k)}dx_{k} \end{aligned}$$
(91)
$$\begin{aligned}&f_{i}^{(k)}=\int _{0}^{l_{s}}U_{1}^{KE^{(k)}}\theta _{i}^{(k)}\nonumber \\&\quad \frac{\partial E_{b}^{(m)}(x_{k}+L,\,t)}{\partial t}dx_{k}+M_{k}\theta _{i}^{(k)}(l_{v})\nonumber \\&\quad \frac{\partial E_{b}^{(m)}(x_{s}+L,\,t)}{\partial t}|_{x_{k}=l_{v}} \end{aligned}$$
(92)
$$\begin{aligned}&f_{i}^{\sim (k)}={} \int _{0}^{l_{k}}U_{1}^{KE^{(k)}}(\theta _{i}^{(m)}(L)\nonumber \\&\quad +x_{k}\theta _{i}^{'\,(m)}(L))\frac{\partial E_{b}^{(m)}(x_{k}+L,\,t)}{\partial t}dx_{k}+M_{k}(\theta _{i}^{(m)}(L) \nonumber \\&\quad +l_{v}\theta _{i}^{'\,(m)}(L))\frac{\partial E_{b}^{(m)}(x_{k}+L,\,t)}{\partial t}|_{x_{s}=l_{v}} \end{aligned}$$
(93)
$$\begin{aligned}&P=\left( \frac{dD^{(k)}(x_{v},t)}{dt}\right) ^{2}\nonumber \\&\quad =\frac{1}{4}\sum _{i}^{N}\sum _{j}^{N}\sum _{q}^{N}\nonumber \\&\quad \sum _{t}^{N}(\gamma _{i}\,\gamma _{j}^{'}\nonumber \\&\qquad +\gamma _{i}^{'}\,\gamma _{j})(\gamma _{q}\,\gamma _{t}^{'}\nonumber \\&\qquad +\gamma _{q}^{'}\,\gamma _{t})\chi _{ij}\chi _{qt} \end{aligned}$$
(94)
$$\begin{aligned}&\chi _{ij}=\int _{0}^{x_{v}}\phi _{i}^{'\,(k)}(x_{v})\phi _{j}^{'\,(k)}(x_{v})dx_{v} \end{aligned}$$
(95)
$$\begin{aligned}&\frac{dD^{(k)}(x_{v},t)}{dt}=\frac{1}{2}\sum _{i}^{N}\nonumber \\&\quad \sum _{j}^{N}(\gamma _{i}\,\gamma _{j}^{'}+\gamma _{i}^{'}\,\gamma _{j})\chi _{ij} \end{aligned}$$
(96)
D. Electromechanical equation : Displacement (\(\alpha\)), force (F) and coupling (\(\zeta\)) vectors, also mass (M) and stiffness (K) matrices
$$\begin{aligned}&\alpha =\left[ \alpha _{j}^{(m)}\,\beta _{j}^{(m)}\,\gamma _{j}^{(m)}\right. \nonumber \\&\quad \alpha _{j}^{(s)}\,\beta _{j}^{(s)}\,\gamma _{j}^{(s)}\nonumber \\&\quad \left. \alpha _{j}^{(k)}\,\beta _{j}^{(k)}\,\gamma _{j}^{(k)}\right] ^{T} \end{aligned}$$
(97)
$$\begin{aligned}&F=\begin{bmatrix}-\frac{\partial }{\partial t}f_{r}^{(m)}-\frac{\partial }{\partial t}(f_{r}^{(s)}+f_{r}^{\sim (s)})-2\,\frac{\partial }{\partial t}(f_{r}^{(k)}+f_{r}^{\sim (s)})\\ 0\\ 2\,\frac{\partial }{\partial t}q_{r}^{(k)} \end{bmatrix} \end{aligned}$$
(98)
$$\begin{aligned}&\zeta =\left[ -\zeta _{r}^{\alpha }\,\zeta _{r\,}^{\beta }0\,0\,0\,0\,0\,0\,0\right] \end{aligned}$$
(99)
Also, the mass and stiffness matrices can illustrate as below:
$$\begin{aligned}&M=\begin{bmatrix}m_{rj} &{} -m_{rj}^{\alpha \beta (m)} &{} 0 &{} m_{rj}^{\alpha \alpha (s)} &{} -m_{rj}^{\alpha \beta (s)} &{} 0 &{} 2\,m_{rj}^{\alpha \alpha (k)} &{} -m_{rj}^{\alpha \beta (k)} &{} 2\,m_{rj}^{\alpha \gamma (k)}\\ -m_{rj}^{\alpha \beta (m)} &{} m_{rj}^{\beta \beta (m)} &{} 0 &{} -m_{rj}^{\alpha \beta (s)} &{} m_{rj}^{\beta \beta (s)} &{} 0 &{} -m_{rj}^{\alpha \beta (k)} &{} m_{rj}^{\beta \beta (k)} &{} 0\\ 2\,m_{rj}^{\gamma \alpha (km)} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 2\,m_{rj}^{\gamma \gamma (k)} \end{bmatrix} \end{aligned}$$
(100)
Where,
$$\begin{aligned}&m_{rj}=m_{rj}^{\alpha \alpha (m)}+2\,m_{rj}^{(sm)}\nonumber \\&\quad +m_{rj}^{(ms)}+2\,m_{rj}^{\alpha \alpha (km)}+2\,m_{rj}^{\alpha \alpha (m)} \end{aligned}$$
(101)
$$\begin{aligned}&K=\begin{bmatrix}k_{rj}^{\alpha \alpha (m)} &{} -k_{rj}^{\alpha \beta (m)} &{} 0 &{} k_{rj}^{\alpha \alpha (s)} &{} -k_{rj}^{\alpha \beta (s)} &{} 0 &{} 2\,k_{rj}^{\alpha \alpha (k)} &{} 2\,k_{rj}^{\alpha \beta (k)} &{} 0\\ -k_{rj}^{\alpha \beta (m)} &{} k_{rj}^{\beta \beta (m)} &{} 0 &{} -k_{rj}^{\alpha \beta (s)} &{} k_{rj}^{\beta \beta (s)} &{} 0 &{} -2\,k_{rj}^{\alpha \beta (k)} &{} 2\,k_{rj}^{\beta \beta (k)} &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -2\,k_{rj}^{\gamma \alpha (k)} \end{bmatrix} \end{aligned}$$
(102)
E. Solution of electromechanical equations
$$\begin{aligned}&\alpha =\left[ \alpha _{1}\,\alpha _{2}.....\alpha _{N}\right] ^{T} \end{aligned}$$
(103)
$$\begin{aligned}&\beta =\left[ \beta _{1}\,\beta _{2}.....\beta _{N}\right] ^{T} \end{aligned}$$
(104)
$$\begin{aligned}&\zeta ^{\sim \,\alpha }=\left[ \zeta _{1}^{\alpha }\,\zeta _{2}^{\alpha }.......\zeta _{N}^{\alpha }\right] ^{T} \end{aligned}$$
(105)
$$\begin{aligned}&\zeta ^{\sim \,\beta }=\left[ \zeta _{1}^{\beta }\,\zeta _{2}^{\beta }.......\zeta _{N}^{\beta }\right] ^{T} \end{aligned}$$
(106)
$$\begin{aligned}&\varDelta ^{\alpha \alpha }={} -\omega ^{2}\left( m^{\alpha \alpha (m)}+2m^{(sm)}+m^{(ms)}\right. \nonumber \\&\qquad +2m^{\alpha \alpha (km)}+2m^{\alpha \alpha (m)}+m^{\alpha \alpha (s)}\nonumber \\&\quad \left. +2m^{\alpha \alpha (m)}\right) \nonumber \\&\qquad +\left( k^{\alpha \alpha (m)}+k^{\alpha \alpha (s)}\right. \nonumber \\&\qquad \left. +2k^{\alpha \alpha (k)}\right) +j\omega \left[ j\omega C_{p}+\frac{1}{R_{l}}\right] ^{-1}\zeta ^{\alpha }\left( \zeta ^{\sim \alpha }\right) ^{T} \end{aligned}$$
(107)
$$\begin{aligned}&\varDelta ^{\alpha \beta }=-\omega ^{2}\left( m^{\alpha \beta (m)}+m^{\alpha \beta (s)}+m^{\alpha \beta (k)}\right) \nonumber \\&\quad +\left( k^{\alpha \beta (m)}+k^{\alpha \beta (s)}+2k^{\alpha \beta (k)}\right) \nonumber \\&\quad +j\omega \left[ j\omega C_{p}+\frac{1}{R_{l}}\right] ^{-1}\zeta ^{\alpha }\left( \zeta ^{\sim \beta }\right) ^{T} \end{aligned}$$
(108)
$$\begin{aligned}&\varDelta ^{\alpha \gamma }=2\omega ^{2}m^{\alpha \gamma (k)} \end{aligned}$$
(109)
$$\begin{aligned}&\varDelta ^{\beta \alpha }=-\omega ^{2}\left( m^{\alpha \beta (m)}+m^{\alpha \beta (s)}+m^{\alpha \beta (k)}\right) \nonumber \\&\quad +\left( k^{\alpha \beta (m)}+k^{\alpha \beta (s)}\right. \nonumber \\&\quad \left. +2k^{\alpha \beta (k)}\right) +j\omega \left[ j\omega C_{p}+\frac{1}{R_{l}}\right] ^{-1}\zeta ^{\beta }\nonumber \\&\quad \left( \zeta ^{\sim \alpha }\right) ^{T} \end{aligned}$$
(110)
$$\begin{aligned}&\varDelta ^{\beta \beta }=-\omega ^{2}\left( m^{\beta \beta (m)}+m^{\beta \beta (s)}\right. \nonumber \\&\quad \left. +m^{\beta \beta (k)}\right) +\left( k^{\beta \beta (m)}\right. \nonumber \\&\quad \left. +k^{\beta \beta (s)}+2k^{\beta \beta (k)}\right) \nonumber \\&\quad +j\omega \left[ j\omega C_{p}+\frac{1}{R_{l}}\right] ^{-1}\zeta ^{\beta }\left( \zeta ^{\sim \beta }\right) ^{T} \end{aligned}$$
(111)
$$\begin{aligned}&\varDelta ^{\gamma \alpha }=2\omega ^{2}m^{\gamma \alpha (km)} \end{aligned}$$
(112)
$$\begin{aligned}&\varDelta ^{\gamma \gamma }=2\left[ \omega ^{2}m^{\gamma \gamma (k)}+k^{\gamma \alpha (k)}\right] \end{aligned}$$
(113)
