# Semi-Active System of Vehicle Vibration Damping
## 2 MR Damper  
A magnetorheological damper (MR) has a varying damping force that changes due to the changing current values. In such a damper, an enclosed coil generates the magnetic field. Then, the change in the friction force is induced by means of controlling the intensity of the magnetic field of the damper coil. One feature of magnetorheological fluid is its changing viscosity ensued by the magnetic field. Resulting from variations in the viscosity of the MR fluid, a change in the damping forces takes place. Thus, changes in the current intensity in the electronic system enable controlling changes in the dissipated energy. The developed MR damper was subjected to tests. Figure 1 shows the original design of the magnetorheological damper (MR-IP damper), developed at the Institute of Vehicles, Warsaw University of Technology (Instytut Pojazdow Politechniki Warszawskiej). The damper was filled with an MR fluid (3), the viscosity of which varies according to the magnetic field (5), generated by the coil (1), and enclosed in the piston (6). During the piston motion, the fluid is pressed through the orifices (2) in the piston. In the discussed solution, the housing (4) does not move; only the piston gets displaced and (6) mounted onto the piston rod (7). At that point, the change in the current intensity affects the change in the damping force acting on the damper's piston rod. Figure 1: Diagram of the magnetorheological MR-IP damper: 1—coil, 2—orifices, 3—magnetorheological fluid, 4—housing, 5—magnetic field, 6—piston, 7—piston rod. The developed MR-IP damper design served the purpose of conducting the property tests. As a result of the experimental research, the dissipative characteristic of the MR damper were determined. Figure 2 illustrates the conducted tests on the plane: force-displacement and force-velocity, where the coil of the MR damper was supplied with the 2 A current, and with no supply (0 A). Experimental research was conducted with the kinematic excitation at the frequency of 0.8 Hz with 0.18 m amplitude. The diagrams show the change in the frictional force value in the case of tests conducted under no-supply conditions (0 A). The frictional force value at the level of 500 N was obtained, whereas in the case of the tests under the condition of the coil supply 2 A, the frictional force value was at the level of 1500 N. As far as the conducted tests are concerned, it has been postulated that controlling the frictional force in dampers is possible in the course of changing the supply current. Then, during the varying current intensity in the range of 0-2 A, controlling the frictional force value will be possible. In the shown force-velocity characteristics, a hysteresis loop can be observed with regard to the tests conducted under no-supply conditions, as well as with the 2 A current. Figure 2 shows only the limit values of the frictional forces. The change in the friction forces in the range of 0-2 A is not linearly dependent on the current changes, which should be taken into account in the algorithm for selecting the friction forces. The test results served the purpose of developing the model of the controlled MR damper. In the subject literature, the Bouc-Wen model [28; 29; 30] is most frequently used to model this type of damper. In this work, a model in the form of a rheological structure was used, presented in Figure 3, originally proposed by [PERSON]. This structure shows the viscoplastic features of the MR damper. The discussed model is characterized by four parameters that are relatively easy to determine on the basis of the experimental research results. Figure 3: Rheological model of the MR damper; T\({}_{0}\), c\({}_{0}\), c\({}_{1}\), k —model parameters, x, y —coordinates of the model. Figure 2: Dissipative characteristic of the MR-IP damper, (a) plane: force–displacement, (b) plane: force–velocity. The mathematical model describing the MR damper is as follows:  
\[F=\ c_{1}(\dot{x}-\dot{y})+k(x-y) \tag{1}\] \[(c_{0}+c_{1})\dot{y}+\tau T_{0}=c_{1}\dot{x}+k(x-y)\] (2) \[\tau\in\left\{