# A Deep Learning-Based Approach for the Identification of a Multi-Parameter BWBN Model
## 2 The BWBN Model  
### Problem Definition  
The BWBN model can reproduce the behavior of general nonlinear hysteretic systems. By incorporating a pinching function in the conventional Bouc-Wen model, the BWBN model captures the strength, stiffness, and pinching degradation effects. Figure 1 shows a single degree of freedom BWBN model. Based on the D'Alembert principle, and assuming that the system is subjected to a horizontal earthquake excitation, the equation of an SDOF system with a BWBN restoring force can be described by Equation (1),  
\[m\ddot{u}+c\dot{u}+f(t)=-m\ddot{u}_{gx} \tag{1}\]  
where the restoring force has the form:  
\[f(t)=\alpha ku+(1-\alpha)kz \tag{2}\]  
In this equation \(u\) is the relative translational displacement, \(\dot{u}\) and \(\ddot{u}\) are velocity and acceleration responses, respectively, \(k\), \(m\) and \(c\) are stiffness, mass and viscous damping  
Figure 1: BWBN hysteretic model of SDOF System [62]. coefficients, respectively, \(\ddot{u}_{gx}\) is the ground acceleration in the x-direction, and \(\alpha\) is the ratio of post-yield to initial stiffness. The restoring force is composed of the elastic restoring force \(\alpha ku\) and the nonlinear restoring force \((1-\alpha)kz\). The hysteretic displacement \(z\) is given by [62]:  
\[\dot{z}=\frac{h(z,\epsilon)}{1+\delta_{\eta}\epsilon}[\dot{u}-(1+\delta_{v} \epsilon)(\beta|\dot{u}||z|^{n-1}z+\gamma\dot{u}|z|^{n})] \tag{3}\]  
where \(\beta\) and \(\gamma\) are loading and unloading control parameters, \(n\) controls the yield point smoothness, and \(\delta_{v}\) and \(\delta_{\eta}\) are the strength and stiffness degradation parameters, respectively. \(\epsilon\) is the hysteresis energy, and \(h(z,\epsilon)\) is the pinching function. The hysteresis energy dissipated from \(t=0\) to the current time \(T\) is:  
\[\epsilon=(1-\alpha)k\int\limits_{0}^{T}\dot{u}zdt \tag{4}\]  
The pinching function \(h(z,\epsilon)\) is given by [62]:  
\[h(z,\epsilon)=1-\zeta_{s}[1-\exp(-p\epsilon)]\exp[-(\frac{z\text{sgn}(\dot{u} )-q[(1+\delta_{v}\epsilon)(\beta+\gamma)]^{\
icefrac{{1}}{{n}}}}{(\psi+\delta _{\psi}\epsilon)(\lambda+\zeta_{s}[1-\exp(-p\epsilon)])})^{2}] \tag{5}\]  
where \(\zeta_{s}\) is the total slip, \(p\) is the pinch slope, \(q\) is initial slip, \(\psi\) is the pinch size, \(\delta_{\psi}\) is the pinch rate, \(\lambda\) is the pinch intensity, and \(sgn(\cdot)\) is the signum function. The BWBN model contains 12 characteristic parameters which can be divided into four categories: four shape parameters \(\alpha,\beta,\gamma\) and \(n\), one strength degradation parameter \(\delta_{v}\), one stiffness degradation parameter \(\delta_{\eta}\) and six pinch parameters \(\zeta_{s},p,q,\psi,\delta_{\psi}\) and \(\lambda\). The relative translational displacement \(u\) and hysteretic displacement \(z\) are normalized to dimensionless variables [7]:  
\[\mu=u/u_{y},\mu_{z}=z/u_{y},\phi=u_{y}/u_{0}=f_{y}/f_{0} \tag{6}\]  
where \(u_{0}\) is the peak-value displacement in yield stage,\(f_{0}\) is the peak-value resisting force in yield stage. \(u_{y}\) is the displacement of yield point, and \(f_{y}\) is the force at the yield point. \(\Phi\) is the normalized yield strength. The yield point is the intersection point between the elastic stage and the yield stage of backbone curve.