## Basic Principle  
### Governing Equations and Iterative Schemes  
The nonlinear dynamic problems can be usually attributed to the following nonlinear second-order ordinary differential equations (ODEs), which are named dynamic equations hereinafter:  
\[\left\{\begin{array}{l}\mathbf{M}\mathbf{\ddot{u}}+\mathbf{f}_{\mathrm{d}}( \mathbf{\dot{u}})+\mathbf{f}_{\mathrm{s}}(\mathbf{u})-\mathbf{f}_{\mathrm{ext} }(t)=0\ \,\ \ 0<t\leq t_{f}\\ \mathbf{u}(0)=\mathbf{u}_{0}\,\ \mathbf{\dot{u}}(0)=\mathbf{\dot{u}}_{0}\end{array}\right. \tag{1}\]  
where \(\mathbf{u},\ \mathbf{\dot{u}},\ \mathbf{\ddot{u}}\) are the nodal displacement vector, velocity vector, and acceleration vector respectively, \(\mathbf{M}\) is the mass matrix, \(\mathbf{f}_{\mathrm{d}}\) is the damping force vector, \(\mathbf{f}_{\mathrm{s}}\) is the resilience vector, \(\mathbf{f}_{\mathrm{ext}}\) is the vector of externally applied forces, and \(\mathbf{u}_{0},\ \mathbf{\dot{u}}_{0}\) are the known displacement and velocity vectors at the initial time, respectively. The time domain is (0, \(t_{f}\)].  
Considering time \(t\) in the time interval \([t_{n},t_{n+1}]\), when \(\mathbf{f}_{\mathrm{d}}\) and \(\mathbf{f}_{\mathrm{s}}\) are the nonlinear functions related to \(\mathbf{u}\) and \(\mathbf{\dot{u}}\), the Newton-Raphson method can be used to transform the nonlinear second-order ODEs (Equation (1)) into linear equations as follows:  
\[\left\{\begin{array}{l}\mathbf{M}\mathbf{\ddot{u}}^{(k+1)}+\mathbf{\overline {C}}^{(k)}\mathbf{\dot{u}}^{(k+1)}+\mathbf{\overline{K}}^{(k)}\mathbf{u}^{(k+ 1)}\ =\mathbf{f}_{\mathrm{ext}}-\mathbf{f}_{\mathrm{int}}{}^{(k)}\ =\mathbf{f}\\ \mathbf{u}^{(k+1)}(t_{n})=\mathbf{u}_{n},\ \mathbf{\dot{u}}^{(k+1)}(t_{n})= \mathbf{\dot{u}}_{n}\end{array}\right. \tag{2}\]  
where  
\[\mathbf{\overline{C}}^{(k)}=\left.(\frac{\partial\mathbf{f}_{\mathrm{d}}}{ \partial\mathbf{\dot{u}}}\right)\Big{|}_{\mathbf{\ddot{u}}^{(k)}},\ \mathbf{\overline{K}}^{(k)}=\left.(\frac{\partial\mathbf{f}_{\mathrm{s}}}{ \partial\mathbf{u}})\right|_{\mathbf{u}^{(k)}} \tag{3}\]  
\[\mathbf{f}_{\mathrm{int}}{}^{(k)}=\left.(\mathbf{f}_{\mathrm{d}}{}^{(k)} \right|_{\mathbf{\ddot{u}}^{(k)}}-\mathbf{\overline{C}}^{(k)}\mathbf{\dot{u} }^{(k)})+\left.(\mathbf{f}_{\mathrm{s}}{}^{(k)}\right|_{\mathbf{u}^{(k)}}- \mathbf{\overline{K}}^{(k)}\mathbf{u}^{(k)}) \tag{4}\]  
In Equations (2)-(4) \(\mathbf{u}^{(k)}\), \(\mathbf{\dot{u}}^{(k)}\), and \(\mathbf{\ddot{u}}^{(k)}\) are solutions of the independent variable time \(t\) obtained in the \(k\)-th step of iteration, where the tangent damping \(\mathbf{\overline{C}}^{(k)}\), the tangent stiffness \(\mathbf{\overline{K}}^{(k)}\), and the internal force \(\mathbf{f}_{\mathrm{int}}^{(k)}\) are all functions of time \(t\). Therefore, through [PERSON]'s method, the process of solving nonlinear Equation (1) is transformed into an iterative solution of linear Equation (2). For convenience, the superscript \(k\) standing for the \(k\)-th iteration step is omitted in Equation (2), and Equation (2) is rewritten as follows:  
\[\left\{\begin{array}{l}\mathbf{M}\mathbf{\ddot{u}}+\mathbf{\overline{C}} \mathbf{\dot{u}}+\mathbf{\overline{K}}\mathbf{u}\ =\mathbf{f}\\ \mathbf{u}(t_{n})=\mathbf{u}_{n},\ \mathbf{\dot{u}}(t_{n})=\mathbf{\dot{u}}_{n} \end{array}\right. \tag{5}\]