## 3 Theoretical Formulation
### Discrete Dynamic Equation
=\begin{bmatrix}\mathcal{T}_{0}(x_{0})&\mathcal{T}_{1 }(x_{0})&\mathcal{T}_{2}(x_{0})&\ldots&\mathcal{T}_{N-1}(x_{0})\\ \mathcal{T}_{0}(x_{1})&\mathcal{T}_{1}(x_{1})&\mathcal{T}_{2}(x_{1})&\ldots& \mathcal{T}_{N-1}(x_{1})\\ \mathcal{T}_{0}(x_{2})&\mathcal{T}_{1}(x_{2})&\mathcal{T}_{2}(x_{2})&\ldots& \mathcal{T}_{N-1}(x_{2})\\ \vdots&\vdots&\vdots&The above equation can be abbreviated as \(\bar{\mathbf{w}}=\mathbf{\mathcal{T}}_{q}\mathbf{a}_{1}\). Considering Equations (13) and (23), it holds that  
\[W_{q}=\bar{\mathbf{q}}^{\mathrm{T}}\bar{\mathbf{V}}\bar{\mathbf{w}}=\bar{\mathbf{q}}^{\mathrm{T}} \bar{\mathbf{V}}\mathbf{\mathcal{T}}_{q}\mathbf{a}_{1}=\bar{\mathbf{q}}^{\mathrm{T}}\bar{\mathbf{V }}\mathbf{\mathcal{T}}_{q}\mathbf{\Gamma}_{\mathrm{F1}}\mathbf{w}=\mathbf{F}_{\mathrm{equal}}^ {\mathrm{T}}\mathbf{w} \tag{26}\]  
where \(\mathbf{F}_{\mathrm{equal}}=(\bar{\mathbf{q}}^{\mathrm{T}}\bar{\mathbf{V}}\mathbf{\mathcal{T} }_{q}\mathrm{F}_{\mathrm{F1}})^{\mathrm{T}}\) is the equivalent point load applied at the points \((x_{k},w_{k})\). Define the generalized coordinates for the beam system as follows \(\mathbf{\eta}\):  
\[\mathbf{\eta}^{\mathrm{T}}=\left[\mathbf{w}^{\mathrm{T}},\mathbf{\alpha}^{\mathrm{T}}\right] \tag{27}\]  
The governing equation of motion can be derived by employing [PERSON]'s equation  
\[\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial\mathcal{L}}{\partial\eta_{i}}- \frac{\partial\mathcal{L}}{\partial\eta_{i}}+\frac{\partial D}{\partial\eta_{ i}}=\mathbf{F}_{\mathrm{equal}_{i}},\quad i=1,2,\dots,2N \tag{28}\]  
where \(\eta_{i}\) is the \(i\)-th entry of the column vector \(\mathbf{\eta}\), and the obtained discrete governing equation is given by  
\[\mathcal{M}\bar{\mathbf{\eta}}+\mathbf{C}\bar{\mathbf{\eta}}+\mathbf{K}\mathbf{\eta}=\mathcal{F} \tag{29}\]  
\[\mathcal{K}= \begin{bmatrix}\mathbf{Q}_{1}{}^{\mathrm{T}}\mathbf{V}_{h_{4}}\mathbf{Q}_{1}+ \mathbf{V}_{h_{5}}&-\mathbf{Q}_{1}{}^{\mathrm{T}}\mathbf{V}_{h_{4}}\\ -\mathbf{V}_{h_{4}}\mathbf{Q}_{1}&\mathbf{Q}_{1}{}^{\mathrm{T}}\mathbf{V}_{h_{3}}\mathbf{Q}_{1}+ \mathbf{V}_{h_{4}}\end{bmatrix}  \tag{30}\]  
Boundary conditions, such as pinned-pinned (PP), clamped-clamped (CC), clamped-pinned (CP), and clamped-free (CF), are handled by a projection matrix method in reference [28]. All linear boundary conditions can be written in the following form:  
\[\mathbf{\beta}^{\mathrm{T}}\mathbf{\eta}=\mathbf{0} \tag{31}\]  
where \(\mathbf{\beta}\) is a \(2N\) column vector. For example, for a beam clamped at the left end, both the displacement and the slope must be zero, i.e.,  
\[w|_{x=0}=0,\quad\alpha|_{x=0}=0 \tag{32}\]  
This can be written in a discrete form as  
\[ \begin{bmatrix}\mathbf{e}_{1}{}^{\mathrm{T}}\\ \mathbf{e}_{N+1}{}^{\mathrm{T}}\end{bmatrix} \mathbf{\eta}=\mathbf{0} \tag{33}\]  
where \(\mathbf{e}_{i}\) is a \(2N\) column vector whose \(i\)-th element is unity and all other elements are zero, i.e., \(\mathbf{\beta}_{1}=\mathbf{e}_{1},\ \mathbf{\beta}_{2}=\mathbf{e}_{N+1}\). Similarly, for a beam pinned at the right end, the displacement must be zero, i.e.,  
\[\mathbf{e}_{N}{}^{\mathrm{T}}\mathbf{\eta}=\mathbf{0} \tag{34}\]  
This equation indicates that the \(N\)-th element of \(\mathbf{\eta}\), which is the value of the displacement \(w\) at the right end, is zero. Figure 2: The work carried out by the non-uniformly distributed moving load. If there are \(m\) boundary conditions in the form of Equation (31), they can be written in a matrix form, as follows:  
\[ \begin{bmatrix}\beta_{1}^{\text{T}}\\ \vdots\\ \beta_{m}^{\text{T}}\end{bmatrix} \boldsymbol{\eta}=\boldsymbol{B}\boldsymbol{ \eta}=\boldsymbol{0} \tag{35}\]  
Every row of the matrix \(\boldsymbol{B}\) represents a boundary condition. Now, the problem evolves into solving the equation \(\mathcal{M}\boldsymbol{\hat{\eta}}+\mathcal{C}\boldsymbol{\hat{\eta}}+ \mathcal{K}\boldsymbol{\eta}=\mathcal{F}\) which satisfies \(\boldsymbol{B}\boldsymbol{\eta}=\boldsymbol{0}\). The projection matrix method introduced in reference [28] is an effective way to impose these boundary conditions.