# Transverse Vibration of Functionally Graded Tapered Double Nanobeams Resting on Elastic Foundation
## 3 Problem Formulation
### Solution Procedure  
The CSCM is employed to transform the governing differential equations of transverse motion and boundary conditions into algebraic equations. For thin structures modeled such as Euler-Bernoulli beams, two boundary conditions are imposed at each edge, and as there is only one equation for each beam; hence, the CSCM will be used for one of the boundary conditions only. To resolve this issue, these conditions are satisfied by expressing the transverse deflections at the boundaries and their adjacent points in terms of the transverse deflections of other points along the beam's span. For example, if the ends of nano beam 1 are clamped, the boundary conditions are given as:  
\[W1(0)=\frac{\partial W1(0)}{\partial X}=W1(1)=\frac{\partial W1(1)}{\partial X }=0. \tag{27}\]  
Using the Chebyshev collocation method, these conditions are discretized as:  
\[ \begin{array}{c}W1_{1}=0.\\ D1_{1,1}^{1}W1_{1}+D1_{1,2}^{1}W1_{2}+D1_{1,N}^{1}W1_{N}+D1_{1,N+1}^{1}W1_{N+ 1}=-\sum_{k=3}^{N-1}D1_{1,k}^{1}W1_{k}.\\ W1_{N+1}=0.\\ D1_{N+1,1}^{1}W1_{1}+D1_{N+1,2}^{1}W1_{2}+D1_{N+1,N}^{1}W1_{N}+D1_{N+1,N+1}^{1 }W1_{N+1}=-\sum_{k=3}^{N-1}D1_{N+1,k}^{1}W1_{k}.\end{array}  \tag{28}\]  
From Equation (27), the deflections \(W1_{2}\) and \(W1_{N}\) are expressed in terms of the deflections at all other points in the domain as:  
\[W1_{2}=\frac{1}{Det}\Bigg{(}-D1_{N+1,N}^{1}\sum_{k=3}^{N-1}D1_{1,k}^{1}W1_{k} +D1_{1,N}^{1}\sum_{k=3}^{N-1}D1_{N+1,k}^{1}W1_{k}\Bigg{)}, \tag{29}\]  
\[W1_{N}=\frac{1}{Det}\Bigg{(}D1_{N+1,2}^{1}\sum_{k=3}^{N-1}D1_{1,k}^{1}W1_{k} -D1_{1,2}^{1}\sum_{k=3}^{N-1}D1_{N+1,k}^{1}W1_{k}\Bigg{)}, \tag{30}\]  
where \(Det\) is given by  
\[Det=\Big{(}D1_{1,2}^{1}\times D1_{N+1,N}^{1}\Big{)}-\Big{(}D1_{1,N}^{1}\times D 1_{N+1,2}^{1}\Big{)}.\]Consequently, the Chebyshev collocation differentiation matrices are modified as:  
\[ \begin{array}{c}\overline{D1}_{i,k}^{n}=\sum_{k=3}^{N-1}D1_{i,k}^{n}+D1_{i,2}^ {n}\frac{1}{D1}(-D1_{N+1,N}^{1}\sum_{k=3}^{N-1}D1_{1,k}^{1}+D1_{1,N}^{1}\sum_{k =3}^{N-1}D1_{N+1,k}^{1})+\\ D1_{i,N}^{n}\frac{1}{D\ell}\Big{(}D1_{N+1,2}^{1}\sum_{k=3}^{N-1}D1_{1,k}^{1}W1_ {k}-D1_{1,2}^{1}\sum_{k=3}^{N-1}D1_{N+1,k}^{1}W1_{k}\Big{)},\end{array}  \tag{31}\]  
for  
\[i=3,4,\ \cdots,\ N-1.\]  
In case beam-2 has simply supported edges, the boundary conditions are expressed as:  
\[W2(0)=\frac{\partial^{2}W2(0)}{\partial x^{2}}=W2(1)\frac{\partial^{2}W2(1)}{ \partial x^{2}}=0. \tag{32}\]  
The boundary conditions given in Equation (32) are discretized using the CSCM as in Equation (28), except that each differentiation matrix \(D^{1}\) is substituted with the differentiation matrix \(D^{2}\). Using the proposed method, Equations (25) and (26) are given as:  
\[\left[ \begin{array}{c}LHS1\\ LHS2\end{array} \right]\left\{ \begin{array}{c}W1\\ W2\end{array} \right\}=\Omega^{2}\Bigg{[} \begin{array}{c}RHS1\\ RHS2\end{array} \Bigg{]}\Bigg{\{} \begin{array}{c}W1\\ W2\end{array} \Bigg{\}}, \tag{33}\]  
where  
\[LHS1=\Big{(}\Big{[} \begin{array}{cc}1&0\end{array} \Big{]}LHS11\Big{)}+ \Big{(}\Big{[} \begin{array}{cc}0&1\end{array} \Big{]}LHS12\Big{)}. \tag{34}\]  
\[LHS2=\Big{(}\Big{[} \begin{array}{cc}1&0\end{array} \Big{]}LHS21\Big{)}+ \Big{(}\Big{[} \begin{array}{cc}0&1\end{array} \Big{]}LHS22\Big{)}. \tag{35}\]  
\[RHS1=\Big{(}\Big{[} \begin{array}{cc}1&0\end{array} \Big{]}RH11\Big{)}. \tag{36}\]  
\[RHS2=\Big{(}\Big{[} \begin{array}{cc}0&1\end{array} \Big{]}RH21\Big{)}. \tag{37}\]  
\[LHS11=\Big{(}R11\times\overline{D1}^{2}\Big{)}+\Big{(}R12\times\overline{D1}^ {3}\Big{)}+\Big{(}R13\times\overline{D1}^{4}\Big{)}+\Big{(}K\,I\Big{)}-\mu^{2 }\Big{(}K\,\overline{D1}^{2}\Big{)}. \tag{38}\]  
\[LHS12=(-K\,I)+\mu^{2}\Big{(}K\,\overline{D2}^{2}\Big{)}. \tag{39}\]  
\[LHS21=(-K\,I)+\mu^{2}\Big{(}K\,\overline{D1}^{2}\Big{)}. \tag{40}\]  
\[LHS22=\Big{(}R21\times\overline{D2}^{2}\Big{)}+\Big{(}R22\times\overline{D2}^ {3}\Big{)}+\Big{(}R23\times\overline{D2}^{4}\Big{)}+\Big{(}K\,I\Big{)}-\mu^{2 }\Big{(}K\,\overline{D2}^{2}\Big{)}+ \tag{41}\]  
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