# An Intelligent Artificial Neural Network Modeling of a Magnetorheological Elastomer Isolator
## 4 Back-Propagation ANN Model  
To make full use of proposed MRE isolator for a vibration control system, its dynamic behavior should be modeled in terms of force-displacement hysteresis loops. In the present work, a back-propagation artificial neural network (BPANN) is utilized for modeling nonlinear performance of MRE isolator. BPANN model has many advantages--e.g., the characteristics of self-learning, strong robust, generalization, and adaptability [19]. The basic principle of BPANN is to put input data through a series of hidden layers and then builds the nonlinear relationship between input and output data [20]. The forward transmission of inputs and the reverse transmission of output error data form the cycles of BPANN. To develop an accurate nonlinear model, the weights in BPANN are optimized based on output error of neurons [21]. For mapping dynamic hysteresis loops of MRE isolator, the input data of BPANN is deformation, velocity, current, and the output data is the force. It is known that velocity is  
Figure 3: Hysteresis loops of MRE isolator in various applied currents: (**a**) \(A\) = 0.05 mm; (**b**) \(A\) = 0.15 mm. Figure 2: Experimental setup for characterizing MRE isolator. calculated by deformation and applied frequency. The neuron number in hidden layer is optimized to eight [17]. The BPANN structure is shown in Figure 4.  
The algorithm of back-propagation neural network [22] is described as follows:  
Firstly, the weights \(w_{ij}^{[l]}\) and the threshold \(v_{j}^{[l]}\) are randomly initialized. Secondly, the output of all layers, \(y_{jp}\), is iteratively computed by Equation (1) after feeding the prepared training samples including input dataset and the output dataset to the neural network. \[y_{jp}^{[IC1]}=f\!\left(\sum\
olimits_{i=1}^{N1}w_{ij}^{[l+1]}y_{ip}^{[l]}+v_{ j}^{[l+1]}\right) \tag{1}\]  
Thirdly, in each layer, the square error is computed. The square root errors of the output layer (\(err_{jp}^{[L]}\)) and the hidden layer (\(err_{jp}^{[l]}\)) are respectively calculated by Equations (2) and (3). \[err_{jp}^{[L]}=f^{\prime}\!\left(y_{jp}^{[L]}\right)\!\left(dp-y_{jp}^{[L]}\right) \tag{2}\]  
\[err_{jp}^{[l]}=f^{\prime}\!\left(y_{j}^{[l]}\right)\!\sum\
olimits_{k=l}^{N_{ l+1}}err_{kp}^{[l+1]}w_{jk}^{[l+1]} \tag{3}\]  
Then, the change in the weights between the input and the output are calculated based on Equations (4) and (5). \[v_{ij}^{[l]}=(n+1)=v_{i}^{[l]}(n)+\eta\cdot err_{jp}^{[l]} \tag{4}\]  
\[w_{ij}^{[l]}=(n+1)=w_{i}^{[l]}(n)+\eta\cdot err_{jp}^{[l]}\cdot y_{ip}^{[l-1]} \tag{5}\]  
The above procedures are repeated until the mean-squared error is less than the threshold value; otherwise, the weight value is stopped iteration and the training procedure of BPNN model is finished. In the NN model, many neuron activation functions are utilized, and the sigmoidal function was used [22]. The overall framework is summarized in flowchart, as shown in Figure 5.  
The experimental data is divided into two groups, one is the training data of BPANN, and the other is testing data. By referring to [PERSON]'s work [17], the training data are one cycle samples in each excitation case (various amplitude and current) as shown in Figure 6; and the testing data are one-cycle samples in a specified condition (one amplitude and current). Figure 4: A basic BPANN model structure for modeling MRE isolator. Figure 2:0 Flow chart of BPANN model structure for modeling MRE isolator. Figure 2:1 Training data: (**a**) displacement, (**b**) velocity, (**c**) current, (**d**) force. The training results of BPANN model are compared with experimental results as shown in Figure 2:2 At minimum displacement amplitude (0.05 mm), the comparison of output force of MRE isolator in all currents is presented in Figure 7a, and Figure 7b illustrates the compared results at maximum displacement amplitude (0.15 mm) under all currents.