## 2 Basic Formulations  
For structural dynamic systems with damping and subjected to external loads, the ODE in a discrete form can be derived by using the general form of [PERSON]'s variational principle [49]:  
\[\int_{t_{1}}^{t_{2}}\delta(T-V)\mathrm{d}t+\int_{t_{1}}^{t_{2}}\delta W \mathrm{d}t=0 \tag{1}\]  
\[T=\frac{1}{2}\dot{\mathbf{u}}^{\mathrm{T}}M\dot{\mathbf{u}},\quad V=\frac{1}{2}\mathbf{u} ^{\mathrm{T}}\mathbf{K}\mathbf{u} \tag{2}\]  
\[\delta W=\dot{\mathbf{u}}^{\mathrm{T}}C\delta\mathbf{u}+\mathbf{f}^{\mathrm{T}}\delta\mathbf{u}, \tag{3}\]  
where \(T\), \(V\), and \(\delta W\) stand for the kinetic energy, strain energy, and virtual work of damping forces and external forces, respectively; \(\mathbf{M}=[\mathbf{M}_{ij}]_{n\times n}\), \(\mathbf{K}=[\mathbf{K}_{ij}]_{n\times n}\), and \(\mathbf{C}=[\mathbf{C}_{ij}]_{n\times n}\) are the mass, stiffness, and damping symmetrical matrices, respectively; \(n\) is the number of spatial DOFs of the system; \(\mathbf{u}(t)\)and \(\mathbf{f}(t)\) are the displacement vector and the prescribed load vector corresponding to spatial nodes, respectively; and a time derivative is denoted by a dot over the letter. \[\mathbf{u}^{\rm T}=[u_{1}\quad u_{2}\quad u_{3}\quad\cdots\quad u_{n}],\,\mathbf{f}^{ \rm T}=[f_{1}\quad f_{2}\quad f_{3}\quad\cdots\quad f_{n}] \tag{4}\]  
Using the variational principle, Equation (1) yields the ODE of discretized system as  
\[\mathbf{M}\ddot{\mathbf{u}}+\mathbf{C}\dot{\mathbf{u}}+\mathbf{Ku}=\mathbf{f}. \tag{5}\]  
For the clarity of derivations, the time evolutions of a spatial node \"\(i\)\" are written as  
\[ \begin{array}{ccccc}\mathbf{u}_{i}^{\rm T}=[&u_{i1}&u_{i2}&u_{i3}&\cdots&u_{i(N-1 )}&u_{iN}\ ]\\ &\mathbf{f}_{i}^{\rm T}=[&f_{i1}&f_{i2}&\cdots&f_{iN}\ ]\end{array}  \tag{6}\]  
for the time sampling points 1 to \(N\), and \(N\) is the number of time sampling points. Note that \(u_{ij}\) and \(f_{ij}\) are the values of \(u_{i}\) and \(f_{i}\) at time sampling point \(t_{j}\). For a Multiple DOF dynamic system (Equation (5)), each spatial DOF is the function of time; its derivative with respect to time can be given through DQ rule as  
\[\dot{\mathbf{u}}_{i}=\mathbf{Au}_{i}, \tag{7}\]  
where \(\mathbf{A}=[\mathbf{A}_{jk}]_{N\times N}\) is the explicit weighting coefficients matrix [46]. It is noteworthy that the weighting coefficients are independent of variables, and dependent only on coordinates of time sampling points. [PERSON] [50] solved Equation (5) by using DQM. However, Equation (1) is discretized using the DQ and Gauss-Lobatto quadrature rules in this work. For the development of the present TFEM, all terms of Equation (1) should be dealt with by the DQ and Gauss-Lobatto quadrature rules, that is, \(\dot{\mathbf{u}}\) is expressed in terms of \(\mathbf{u}\) based on the DQ rule and the integrals are fulfilled by using the Gauss-Lobatto quadrature rule; then we have  
\[\int_{t_{1}}^{t_{2}}\delta T\mathrm{d}t=\int_{t_{1}}^{t_{2}}\dot{\mathbf{u}}^{\rm T }\mathbf{M}\delta\dot{\mathbf{u}}\mathrm{d}t=\overline{\mathbf{U}}^{\rm T}\overline{\mathbf{M} }\delta\overline{\mathbf{U}} \tag{8}\]  
\[\int_{t_{1}}^{t_{2}}\delta V\mathrm{d}t=\int_{t_{1}}^{t_{2}}\mathbf{u}^{\rm T} \mathbf{K}\mathbf{s}\mathbf{u}\mathrm{d}t=\overline{\mathbf{U}}^{\rm T}\overline{\mathbf{K}} \delta\overline{\mathbf{U}} \tag{9}\]  
\[ \begin{array}{ll}\int_{t_{1}}^{t_{2}}\delta W\mathrm{d}t&=\int_{t_{1}}^{t_{ 2}}(\dot{\mathbf{u}}^{\rm T}\mathbf{C}\mathbf{s}+\mathbf{f}^{\rm T}\mathbf{s}\mathbf{u})\mathrm{d}t\\ &=\left(\overline{\mathbf{U}}^{\rm T}\overline{\mathbf{C}}+\overline{\mathbf{F}}^{\rm T} \overline{\mathbf{E}}\right)\delta\overline{\mathbf{U}}\end{array} , \tag{10}\]  
where  
\[\overline{\mathbf{U}}=\mathrm{cs}(\mathbf{U}),\ \ \ \overline{\mathbf{F}}=\mathrm{cs}(\mathbf{F}) \tag{11}\]  
\[\mathbf{U}^{\rm T}=\left[ \begin{array}{ccccc}\mathbf{u}_{1}&\mathbf{u}_{2}&\cdots&\mathbf{ u}_{n}\end{array} \right],\ \ \ \mathbf{F}^{\rm T}=\left[ \begin{array}{ccccc}f_{1}&f_{2}&\cdots&f_{n}\end{array}