# A Comparative Study of Differential Quadrature Methods for METE Nanobeam Vibrations
## 2 Formulation of the Problem
\[\sigma_{ij}=\int_{V}\alpha(|x^{\prime}-x|,\tau)\left[C_{ijkl} \varepsilon_{kl}(x^{\prime})-\varepsilon_{kij}E_{k}(x^{\prime})-\lambda_{ij} \Delta T\right]dx^{\prime}, \tag{1}\]3 \[\sigma_{ij}=\int_{V}\alpha(|x^{\prime}-x|,\tau)\left[C_{ijkl} \varepsilon_{kl}(x^{\prime})-\varepsilon_{kij}E_{k}(x^{\prime})-\lambda_{ij} \Delta T\right]dx^{\prime}, \tag{1}\]4  
\[\sigma_{ij}=\int_{V}\alpha(|x^{\prime}-x|,\tau)\left[C_{ijkl} \varepsilon_{kl}(x^{\prime})-\varepsilon_{kij}E_{k}(x^{\prime})-\lambda_{ij} \Delta T\right]dx^{\prime}, \tag{1}\]5 \[\sigma_{ij}=\int_{V}\alpha(|x^{\prime}-x|,\tau)\left[C_{ijkl} \varepsilon_{kl}(x^{\prime})-\varepsilon_{kij}E_{k}(x^{\prime})-\lambda_{ij} \Delta T\right]dx^{\prime}, \tag{1}\]6  
\[\sigma_{ij}=\int_{V}\alpha(|x^{\prime}-x|,\tau)\left[C_{ijkl} \varepsilon_{kl}(x^{\prime})-\varepsilon_{kij}E_{k}(x^{\prime})-\lambda_{ij} \Delta T\right]dx^{\prime}, \tag{1}\]7  
\[\sigma_{ij}=\int_{V}\alpha(|x^{\prime}-x|,\tau)\left[C_{ijkl} \varepsilon_{kl}(x^{\prime})-\varepsilon_{kij}E_{k}(x^{\prime})-\lambda_{ij} \Delta T\right]dx^{\prime}, \tag{1}\]8  
The boundary conditions are derived by assuming zero electric and magnetic potential at the ends of the nanobeam, as outlined in the reference [35]. In the case of clamped (C) boundary conditions,  
\[\sigma_{ij}=\int_{V}\alpha(|x^{\prime}-x|,\tau)\left[C_{ijkl} \varepsilon_{kl}(x^{\prime})-\varepsilon_{kij}E_{k}(x^{\prime})-\lambda_{ij} \Delta T\right]dx^{\prime}, \tag{1}\]9  
In the case of hinged (H) boundary conditions,  
\[D_{i}=\int_{V}\alpha(|x^{\prime}-x|,\tau)\left[\varepsilon_{ikl }\varepsilon_{kl}(x^{\prime})-\varepsilon_{ik}E_{k}(x^{\prime})+p_{i}\Delta T \right]dx^{\prime}, \tag{2}\]0  
\[D_{i}=\int_{V}\alpha(|x^{\prime}-x|,\tau)\left[\varepsilon_{ikl }\varepsilon_{kl}(x^{\prime})-\varepsilon_{ik}E_{k}(x^{\prime})+p_{i}\Delta T \right]dx^{\prime}, \tag{2}\]1  
\[D_{i}=\int_{V}\alpha(|x^{\prime}-x|,\tau)\left[\varepsilon_{ikl }\varepsilon_{kl}(x^{\prime})-\varepsilon_{ik}E_{k}(x^{\prime})+p_{i}\Delta T \right]dx^{\prime}, \tag{2}\]2  
\[D_{i}=\int_{V}\alpha(|x^{\prime}-x|,\tau)\left[\varepsilon_{ikl }\varepsilon_{kl}(x^{\prime})-\varepsilon_{ik}E_{k}(x^{\prime})+p_{i}\Delta T \right]dx^{\prime}, \tag{2}\]3 \[D_{i}=\int_{V}\alpha(|x^{\prime}-x|,\tau)\left[\varepsilon_{ikl }\varepsilon_{kl}(x^{\prime})-\varepsilon_{ik}E_{k}(x^{\prime})+p_{i}\Delta T \right]dx^{\prime}, \tag{2}\]4  
where \(w,\theta_{x},\phi\), and \(\psi\) denote the transverse displacement, rotation, electric potential, and magnetic potential, respectively. The parameter \((e_{0}a)\) is a scale coefficient that incorporates the influence of small-scale effects. The shear correction factor k is set to \(5/6\), which is a commonly used value for macroscale beams [31; 32; 33]. \[D_{i}=\int_{V}\alpha(|x^{\prime}-x|,\tau)\left[\varepsilon_{ikl }\varepsilon_{kl}(x^{\prime})-\varepsilon_{ik}E_{k}(x^{\prime})+p_{i}\Delta T \right]dx^{\prime}, \tag{2}\]5  
Here, \(N_{E},N_{M},N_{T}\), and \(N_{P}\) denote the normal forces caused by the external electrical potential \(V_{0}\), the magnetic potential \(A_{0}\), the temperature variation \(\Delta T\), and the mechanical potential \(P_{0}\), respectively. The coefficients \(\bar{\lambda}_{1},\bar{e}_{31}\), and \(\bar{q}_{31}\) are associated with thermal, piezoelectric, and piezomagnetic properties, respectively [41]. The current approach, employing Equation (19), represents a simplified treatment of the thermal environment, neglecting the complexities inherent in transient heat transfer processes. Generalized thermoelasticity theories, such as Lord-Shulman (LS) or Green-Lindsay (GL) models, offer a more realistic representation by incorporating time-dependent heat conduction effects and relaxation times. These models account for the finite speed of heat propagation, a phenomenon absent in the classical coupled thermoelasticity theory implicitly used in Equation (19).