# A Feasibility Study of a Novel Piezo MEMS Tweezer for Soft Materials Characterization
## 3 Modeling Piezoelectric Actions on the Microbeams
### Dynamics Model  
To determine the value of the viscous damping \(C_{ST}\), a dynamic test has been conceived by applying a variable harmonic excitation voltage \(V(t)=V_{0}sin(\omega t)\). By referring to \(\delta L_{int}\), \(\delta L_{in}\), \(\delta L_{a}\) and \(\delta L_{ST}\) as the virtual works of internal, inertial, actuator and ST forces, respectively, and to \(\delta L_{in_{m}}\) as the virtual work of the inertial forces of the clamp teeth, the virtual work principle can be written as:  
\[\delta L_{int}=\delta L_{in}+\delta L_{a}+\delta L_{in_{m}}+\delta L_{ST} \tag{9}\]  
where (the quantities over signed by a tilde are virtual quantities):  
\[\left\{ \begin{array}{l}\delta L_{int}=E_{b}I_{b}\int_{0}^{L}\frac{\partial^ {2}w}{\partial x^{2}}\frac{\partial^{2}\widetilde{w}}{\partial x^{2}}dx\\ \\ \delta L_{in}=-\rho S\int_{0}^{L}\frac{\partial^{2}\widetilde{w}}{\partial t^{2 }}\widetilde{w}\ dx\\ \\ \delta L_{a}=M_{a}\frac{\partial\widetilde{w}}{\partial x}\Big{|}_{x=L}\\ \\ \delta L_{in_{m}}=-m\frac{\partial^{2}w_{L}}{\partial t^{2}}\widetilde{w}_{L} \\ \\ \delta L_{ST}=F_{ST}\ \widetilde{w}_{L}^{\ \widetilde{}}\end{array} \right. \tag{10}\]  
with (see Figure 6):  
\[F_{ST}=K_{ST}w_{L}+C_{ST}\ \dot{w}_{L} \tag{11}\]The viscous damping coefficient \(C_{ST}\) can be identified by measuring the amplitude \(|w_{L}|\) in correspondence of the \(i\)-th vibration mode of the structure that has been excited by means of the piezoelectric elements, where the flexural modes of the beams are obtained as  
\[\varphi(x)=B_{1}\sin(\lambda x)+B_{2}\cos(\lambda x)+B_{3}\sinh(\lambda x)+B_{4} \cosh(\lambda x) \tag{12}\]  
with  
\[\left\{ \begin{array}{l}\varphi(0)=0\\ \left.\frac{\partial\varphi(x)}{\partial x}\right|_{x=0}=0\\ \left.\frac{\partial^{2}\varphi(x)}{\partial x^{2}}\right|_{x=L}=0\\ EI\frac{\partial^{3}\varphi(x)}{\partial x^{3}}\right|_{x=L}=K_{ST}\varphi(L)- \omega^{2}m\varphi(L)\end{array} \right. \tag{13}\]  
By substituting Equation (12) in (13), a system of four equations in four unknowns (\(B_{1}\), \(B_{2}\), \(B_{3}\), \(B_{4}\)) is obtained. The eigenfrequencies are obtained setting to zero the determinant of the matrix coefficient (with \(\omega^{2}=\lambda^{4}\frac{EI}{\rho^{3}}\)) and then the eigenmodes can be calculated. The \(i\)-th flexural mode can be excited by applying a potential function  
\[V(t)=V_{0}\,\sin(\omega_{i}t) \tag{14}\]  
where \(\varphi_{i}(x)\) and \(\omega_{i}\) are the \(i\)-th flexural mode and its related frequency, respectively. In these conditions the displacement \(w(x,t)\) and the virtual displacement \(\hat{w}(x,t)\) can be written as:  
\[\left\{ \begin{array}{l}w(x,t)=A_{i}(t)\varphi_{i}(x)\\ \\ \widetilde{w}\ (x,t)=\varphi_{i}(x)\end{array} \right. \tag{15}\]  
By substituting Equations (15) and (10) in (9), the following equation is obtained:  
\[ \begin{array}{l}E_{b}I_{b}A_{i}(t)\int_{0}^{L}\frac{\partial^{2}\varphi_{i }(x)}{\partial x^{2}}^{2}dx=-\rho S\ \ddot{A_{i}}\ (t)\int_{0}^{L}\varphi_{i}(x)^{2}dx-m\ \ddot{A_{i}}(t)\ \varphi_{i}(L)^{2}+\\ +M_{a}\frac{\partial\varphi_{i}(x)}{\partial x}\Big{|}_{x=L}-K_{ST}A_{i}(t) \varphi_{i}(L)^{2}-C_{ST}\ \dot{A_{i}}(t)\ \varphi_{i}(L)^{2}\end{array}  \tag{16}\]  
Figure 6: Scheme for the calculation of the stiffness and viscous damping of the soft tissue. assuming (see Equation (1)):  
\[\left\{ \begin{array}{l}M=\rho S\int_{0}^{L}\varphi_{i}(x)^{2}dx+m\varphi_{i}(L)^ {2}\\ \\ C=C_{ST}\varphi_{i}(L)^{2}\\ \\ K=E_{l}b_{l}\int_{0}^{L}\frac{\partial^{2}\varphi_{i}(x)}{\partial x^{2}}^{2} dx+K_{ST}\varphi_{i}(L)^{2}\\ \\ Q=\frac{\psi}{6+\psi}E_{a}cT_{b}d_{31}\left.\frac{\partial\varphi_{i}(x)}{ \partial x}\right|_{x=L}V_{0}\end{array} \right.