# A Deep Learning-Based Approach for the Identification of a Multi-Parameter BWBN Model
## 2 The BWBN Model
### Problem Definition
By utilizing the normalized variables, the equation of motion of the BWBN model given by Equations (1)-(5) can be rewritten in the modal coordinate [62]:  
\[\ddot{\mu}+2\zeta\omega_{n}\dot{\mu}+\alpha\omega_{n}{}^{2}\mu+(1-\alpha) \omega_{n}{}^{2}\mu_{z}=-\frac{\ddot{\mu}_{gx}}{\phi\mu_{0}} \tag{7}\]  
\[\dot{\mu}_{z}=\frac{h(\mu_{z},\epsilon_{n})}{1+\delta_{\eta}\epsilon_{n}} \big{\{}\dot{\mu}-(1+\delta_{v}\epsilon_{n})|\mu_{z}|^{n}\dot{\mu}[sgn(\dot{ \mu}  
solve the problem through an iterative process with small time increments. \(Newmark-\beta\) is a common algorithm, it is based on a linear acceleration method by introducing two parameters of \(\overline{\beta}\) and \(\overline{\gamma_{i}}\); the bar symbol is used to differentiate from the BWBN model parameters. A major advantage of this method is that it can find the tangent or secant stiffness in each load step. The iterative flow chart [63] is shown in Figure 2. \(\varepsilon\) and \(\varepsilon_{tal}\) denote the calculation error and allowable error respectively in Figure 2. Equations (7)-(10) of the BWBN model are discretized as:  
\[\Delta\ddot{\mu}_{i}+2\zeta\omega_{n}\Delta\dot{\mu}_{i}+\alpha{\omega_{n}}^{ 2}\Delta\mu_{i}+(1-\alpha){\omega_{n}}^{2}\Delta\mu_{zi}=-\frac{\Delta\ddot{ \mu}_{gzi}}{\phi\mu_{0}} \tag{11}\]  
\[{(\dot{\mu}_{z})}_{i+1}=\frac{h(({\mu_{z})}_{i+1},({\varepsilon_{n}})_{i+1})}{ 1+\delta_{\eta}{({\varepsilon_{n}})}_{i+1}}\{\dot{\mu}_{i+1}-(1+\delta_{v}({ \varepsilon_{n}})_{i+1})|({\mu_{z}})_{i+1}|^{n}\dot{\mu}_{i+1}[sgn(\dot{\mu}_ {i+1}({\mu_{z}})_{i+1})\beta+\gamma]\} \tag{12}\]  
\[f(t)=\alpha ku+(1-\alpha)kz \tag{2}\]0  
\[ \begin{array}{l}h(({\mu_{z}})_{i+1},({\varepsilon_{n}})_{i+1})=\\ 1-\zeta_{s}[1-\exp(-p({\varepsilon_{n}})_{i+1})]\exp[-(\frac{({\mu_{z}})_{i+1} sgn({\mu_{z}})_{i+1})-q([1+\delta_{v}({\varepsilon_{n}})_{i+1})(\beta+\gamma)] \gamma_{i}}{(\beta+\delta_{v}({\varepsilon_{n}})_{i+1})(\lambda+\zeta_{s}[1- \exp(-p({\varepsilon_{n}})_{i+1})])})^{2}]\\ \Delta\ddot{\mu}_{i}=\frac{1}{\beta\Delta t^{2}}\Delta\mu_{i}-\frac{1}{\beta \Delta t}\dot{\mu}_{i}-\frac{1}{2\beta}\ddot{\mu}_{i}\\ \text{where}\hskip 28.452756 pt\Delta\dot{\mu}_{i}=\frac{7}{\beta\Delta t} \Delta\mu_{i}-\frac{7}{\beta\Delta t}\dot{\mu}_{i}+(1-\frac{7}{2\beta})\ddot{ \mu}_{i}\Delta t\\ \hskip 28.452756 pt({\mu_{z}})_{i+1}={({\mu_{z}})}_{i}+\Delta t(\dot{\mu}_{z})_ {i+1}\\ \hskip 28.452756 pt\mu_{i+1}={\mu_{i}}+\Delta t\dot{\mu}_{i+1}\end{array}  \tag{15}\]  
Figure 2: The flow chart of Newmark-\(\beta\)[62]. After time-discretization of the nonlinear dynamic differential equations (DDEs), during a very small time increment, the nonlinear system can be approximately treated as a linear dynamic system. Another commonly used iteration algorithm is the Runge-Kutta method [64]. The Runge-Kutta method reduces higher-order differential equations to lower orders. DDEs can be converted into a set of first-order ordinary differential equations (ODEs). The detailed solution process is shown in [62]. To confirm the effectiveness of \(Newmark-\beta\) algorithms, we consider the same input earthquake load, i.e., El Centro record. The normalized solution results of the \(Newmark-\beta\) and Runge-Kutta algorithms are consistent [63]. The single degree of freedom BWBN model has 12 dimensionless characteristic parameters, which play different roles in the hysteresis model. The contributions of different characteristic parameters on hysteresis behavior can be found visually in [62].