# Current Trends in Fluid Viscous Dampers with Semi-Active and Adaptive Behavior
## 3 Semi-Active Fluid Viscous Dampers
### Rheological Fluid Viscous Dampers
#### 3.1.3 Modeling of ER and MR Dampers  
Popular models used to represent the behavior of MR and ER dampers are the Bingham model [126; 127; 128; 129; 130] and the Bouc-Wen model [131; 132; 133], largely used in many studies. However, other models suitable for reproducing the response of both MR and ER devices are available in the literature, as reviewed by [PERSON] et al. [46] and [PERSON] et al. [134]. The Bingham model is characterized by a friction element in parallel with a viscous damper (Figure 16a); the damping force is given by the expression of Equation (4):  
\[F(t)=C_{D}\dot{x}+f_{c}sgn(\dot{x})+f_{0} \tag{4}\]  
where \(f_{c}\) is the frictional force related to the fluid yield stress and \(f_{o}\) is the offset of the force to account for the nonzero mean that can be observed in the measured force due to the presence of an accumulator [129]. This model represents the force-velocity relationship through a bilinear curve. Figure 16b shows that the Bingham model is not able to perfectly describe the real behavior of the rheological damper, especially at low velocities [129]. Figure 14: MR damper developed by Lord Corporation [46]: (**a**) section of the device; (**b**) fluid path in the piston head and coil placement [121]. Figure 15: MR damper proposed by [PERSON] et al. [106]. The Bouc-Wen model [135, 136] is frequently adopted for its versatility and capability of representing the wide variety of hysteretic behaviors typical of both MR and ER dampers; moreover, it allows control of the intensity of the device's output force [134]. The scheme of the Bouc-Wen model is shown in Figure 17a, where the output force of the damper is described by Equation (5):  
\[F(t)=C_{D}\dot{x}+k_{0}(x-x_{0})+\alpha z \tag{5}\]  
where \(k_{0}\) is the stiffness induced by the accumulator, \(x_{0}\) is the initial displacement of the spring, and \(z\) is an evolutionary variable governed by the law reported in Equation (6):  
\[\dot{z}=-\gamma\big{|}\dot{x}\big{|}z\big{|}z\big{|}^{n-1}-\beta\dot{x}\big{|}z \big{|}^{n}+A\dot{x} \tag{6}\]  
The parameters \(\gamma\), \(A\), and \(\beta\) allow the shape of the hysteresis loop to be controlled, while \(n\) controls the smoothness of the force-displacement curve; all these parameters can be calibrated through experimental evidence [46]. By changing the parameters of Equation (6), the Bouc-Wen model is able to describe the linearity in the unloading phase and the smoothness of the transition between the pre-yield and post-yield region [129]. Figure 17b shows the comparison between the force-velocity loops of the Bouc-Wen model (the outer curve) and the one derived from experimental evidence (the inner curve). Like the Bingham model, the Bouc-Wen model is also not suitable for describing the behavior of devices in the regions at low velocities. To overcome this issue, [PERSON] et al. [129] developed a refined model consisting of a Bouc-Wen model that works in parallel with a spring and in series with a viscous dashpot, as represented in Figure 18a. The force generated by the model is expressed by Equation (7):  
\[F_{D}=\alpha z+C_{D}\big{(}\dot{x}-\dot{y}\big{)}+k_{0}(x-y)+k_{1}(x-x_{0}) \tag{7}\]  
Figure 16: (**a**) Bingham model (adapted from [46]); (**b**) force-velocity response of the Bingham model (\(f_{c}=670\)\(N\), \(C_{D}=50 Ns/\)cm, \(f_{0}=-95\)\(N\)) [129]. Figure 17: (**a**) Bouc–Wen model (adapted from [46]); (**b**) force–velocity response of the Bouc–Wen model [129]. where \(k_{1}\) allows the stiffness at large velocity to be controlled and \(\dot{y}\) is given by Equation (8):  
\[\dot{y}=\frac{1}{C_{D}+C_{1}}\big{[}\alpha z+C_{D}\dot{x}+k_{1}(x-y)\big{]} \tag{8}\]  
where \(C_{1}\) is the damping coefficient that controls the behavior of the device at low velocity. Figure 18b represents the predicted force-velocity loops and the experimental data [129]. As can be seen, the two curves are almost overlapping, showing that the proposed model is able to reproduce the behavior of the rheological damper quite well.