# A Comparative Study of Differential Quadrature Methods for METE Nanobeam Vibrations
## 2 Formulation of the Problem  
Consider a METE nanobeam of dimensions \(L\times h\) subjected to various external stimuli. The beam is exposed to an electric voltage, \(\phi(x,z,t)\), a magnetic potential, \(\psi(x,z,t)\), a temperature change, \(\Delta T\), and a mechanical potential, \(P_{0}\). Moreover, the beam is supported by a nonlinear elastic foundation, which is defined by the stiffness coefficients \(K_{1}\), \(K_{2}\), and \(K_{3}\), representing linear and nonlinear characteristics, respectively, as depicted in Figure 1.  
Employing [PERSON]'s nonlocal elasticity theory, the governing equations for a homogeneous, nonlocal piezoelectric solid in the absence of body forces are presented as follows [9]:  
\[\sigma_{ij}=\int_{V}\alpha(|x^{\prime}-x|,\tau)\left[C_{ijkl} \varepsilon_{kl}(x^{\prime})-\varepsilon_{kij}E_{k}(x^{\prime})-\lambda_{ij} \Delta T\right]dx^{\prime}, \tag{1}\] \[D_{i}=\int_{V}\alpha(|x^{\prime}-x|,\tau)\left[\varepsilon_{ikl }\varepsilon_{kl}(x^{\prime})-\varepsilon_{ik}E_{k}(x^{\prime})+p_{i}\Delta T \right]dx^{\prime}, \tag{2}\]  
Figure 1: Schematic of METE nanobeam on nonlinear foundation. \[\sigma_{ij,j}=\rho\dot{a}_{i,j},D_{i,j}=0, \tag{3}\]  
\[\varepsilon_{ij}=\frac{1}{2}\big{(}u_{i,j}+u_{j,j}\big{)},E_{i}=-\phi_{i}, \tag{4}\]  
Additionally, the differential form of the integral constitutive relations is as follows [8; 9]:  
\[\sigma_{ij}-(e_{0}a)^{2}\times\
abla^{2}\sigma_{ij}=C_{ijkl}\varepsilon_{kl}- \varepsilon_{kj}E-\lambda_{ij}\Delta T, \tag{5}\]  
\[D_{i}-(e_{0}a)^{2}\times\
abla^{2}D_{i}=e_{kl}\varepsilon_{kl}+\varepsilon_{ ik}E_{k}+p_{i}\Delta T, \tag{6}\]  
Within this framework, \(D,E,C,e,\rho,\varepsilon,\sigma,\rho\), and \(\in\) correspond to the electric displacement, electric field, elastic constant, piezoelectric constant, mass density, strain, stress, electric constant, pyroelectric constant, and dielectric constants, respectively. These parameters exhibit variability contingent upon the specific material type. The function \(\alpha(|x^{\prime}-x|,\tau)\) characterizes nonlocal attenuation and is embedded into the constitutive formulas at the point of reference x. In this context, \(|x^{\prime}-x|\) represents the Euclidean distance. \(\
abla^{2}\) denotes the Laplace operator. \((\tau=\mathrm{e}_{0}a/L)\), where \(a\) is an internal characteristic length and \(e_{0}\) is a nondimensional material constant denoting the scale coefficient that clarifies the size effect on the behavior of structures at the nanoscale. Numerical simulations based on the lattice dynamics [6; 19] or experimental techniques can be used to find the value \(e_{0}\). The nonlocal constitutive Equations (5) and (6) can be roughly represented as follows, as shown in Figure 1:  
\[\sigma_{xx}-(e_{0}a)^{2}\times\frac{\partial^{2}\sigma_{xx}}{ \partial x^{2}}-C_{11}\varepsilon_{xx}+e_{31}E_{z}+\lambda_{1}\Delta T=0 \tag{7}\]  
\[\sigma_{ij}=\int_{V}\alpha(|x^{\prime}-x|,\tau)\left[C_{ijkl} \varepsilon_{kl}(x^{\prime})-\varepsilon_{kij}E_{k}(x^{\prime})-\lambda_{ij} \Delta T\right]dx^{\prime}, \tag{1}\]0  
\[\sigma_{ij}=\int_{V}\alpha(|x^{\prime}-x|,\tau)\left[C_{ijkl} \varepsilon_{kl}(x^{\prime})-\varepsilon_{kij}E_{k}(x^{\prime})-\lambda_{ij} \Delta T\right]dx^{\prime}, \tag{1}\]1  
\[\sigma_{ij}=\int_{V}\alpha(|x^{\prime}-x|,\tau)\left[C_{ijkl} \varepsilon_{kl}(x^{\prime})-\varepsilon_{kij}E_{k}(x^{\prime})-\lambda_{ij} \Delta T\right]dx^{\prime}, \tag{1}\]2  
where \(\varepsilon_{xx}=\frac{\partial w}{\partial x}+\left(\frac{\partial\theta}{ \partial x}\right)^{2}+z\frac{\partial\phi}{\partial x}\), \(\quad\gamma_{xz}=\frac{\partial\theta}{\partial x}+\psi\). The governing equations of motion are obtained using [PERSON]'s principle, as outlined in the reference [31].