## 2 Chang-Veerarajan Method  
In structural dynamic analysis, the equation of motion for the SDOF system based on the discrete mathematical model, is expressed as  
\[m\ddot{u}+c\dot{u}+ku\ =\ f, \tag{1}\]  
where \(f\ =\) external force and \(k,\ c,\ m\ =\) stiffness, viscous damping coefficient and mass, respectively; and \(\ddot{u},\ i\) and \(u\ =\) acceleration, velocity and displacement, respectively. Although many integration methods are used to solve Equation (1), a novel family of SDIM is presented, since it can have the desired numerical properties. Herein, for brevity and due to the authors [PERSON] and [PERSON], this method will be referred to as CVM.  
In general, the SDIMs can be derived from an eigen-based theory and, thus, the detailed derivations will not be elaborated, herein [32]. As a result, the general formulation of CVM can be written as:  
\[\begin{array}{l}(1+\alpha)ma_{i+1}-\alpha ma_{i}+cv_{i+1}+kd_{i+1}\ =\ f_{i+1}\\ d_{i+1}\ =\ d_{i}-B_{1}\Omega_{i}^{2}d_{i}+B_{2}(\Delta t)v_{i}+B_{3}(\Delta t)^{2} a_{i}+p_{i+1},\\ (\Delta t)v_{i+1}\ =\ (\Delta t)v_{i}+\gamma_{1}(\Delta t)^{2}a_{i}+\gamma_{2}( \Delta t)^{2}a_{i+1}\end{array} \tag{2}\]  
where \(a_{i},\ v_{i}\) and \(d_{i}\ =\) acceleration, velocity and the nodal displacement at the end of the \(i\)-th time step, respectively. Many tests have been tried for the coefficients, until they satisfy the desired numerical properties. Finally, the coefficients are found to be:  
\[\begin{array}{l}B_{1}\ =\ \frac{1}{D}\beta_{3},\ B_{2}\ =\ \frac{1}{D}[(1+\alpha) \beta_{1}+(\beta_{1}\gamma_{2}-\beta_{3})2\xi\Omega_{0}],\\ B_{3}\ =\ \frac{1}{D}[(1+\alpha)\beta_{2}+\alpha\beta_{3}+(\beta_{2}\gamma_{2}- \beta_{3}\gamma_{1})2\xi\Omega_{0}],\ p_{i+1}\ =\ \frac{1}{D}\beta_{3}(F_{i+1}-F_{i}) \end{array} \tag{3}\]Here, \(\xi\) is a viscous damping ratio and \(\Omega_{0}~{}=~{}\omega_{0}(\Delta t)\); \(\omega_{0}~{}=~{}\sqrt{k_{0}/m}\) is the natural frequency of the system, determined from the initial stiffness of \(k_{0}\cdot p_{i+1}\) is the loading dependent term. In addition  
\[D~{}=~{}(1+\alpha)+\gamma_{2}2\xi\Omega_{0}+\beta_{3}\Omega_{0}^{2}. \tag{4}\]  
For computational efficacy, it is very important to rewrite \(\xi\Omega_{0}\) and \(\xi\Omega_{0}^{2}\) in terms of step size and the initial structural properties for an SDIM. Thus, based on the theory of structural dynamics [34], the relations \(\Omega_{0}^{2}~{}=~{}(\Delta t)^{2}(k_{0}/m)\) and \(c_{0}~{}=~{}2\xi\omega_{0}m\) can be obtained. After substituting these relations into Equations (3) and (4), they become:  
\[\begin{array}{l}B_{1}~{}=~{}\frac{1}{D}\beta_{3}m\\ B_{2}~{}=~{}\frac{1}{D}[(1+\alpha)\beta_{1}m+(\beta_{1}\gamma_{2}-\beta_{3})( \Delta t)c_{0}]\\ B_{3}~{}=~{}\frac{1}{D}\{[(1+\alpha)\beta_{2}+\alpha\beta_{3}]m+(\beta_{2} \gamma_{2}-\beta_{3}\gamma_{1})(\Delta t)c_{0}\}\\ p_{i+1}~{}=~{}\frac{1}{D}\beta_{3}(f_{i+1}-f_{i})\\ D~{}=~{}(1+\alpha)m+\gamma_{2}(\Delta t)c_{0}+\sigma\beta_{3}(\Delta t)^{2}k_ {0}\end{array} \tag{5}\]  
From the formulation of CVM, it is clear that it is a structure-dependent, explicit method. Notice that the coefficients of \(B_{1}\), \(B_{2}\) and \(B_{3}\) depend only on the initial properties of the structure and step size. Hence, they will remain invariant and, thus, there is no need to re-compute these coefficients during an entire step-by-step integration process. This helps the methods to become computationally efficient.