## 3 Results
### Experimental Validation of the Proposed MR Damper Model  
In order to validate the MR model (1), a series of experimental data were obtained for a commercial MR damper. The most common method for determining the behavior of an MR damper is to excite it with a sinusoidal displacement in a series of test cases with varying frequencies of excitation, piston displacements, and applied currents. The experimental data were obtained from our laboratory using the damper test machine and setup depicted in Figure 4.  
In the test machine, a hydraulic actuator was used to drive the MR damper through sinusoidal displacement cycles with amplitudes of 10, 15, and 20 mm, within a frequency range of 2-6 Hz, and under the application of magnetizing currents ranging from 0 to 1.8 A.  
The damper stroke was positioned at its center before the test was initiated in order to avoid extreme damper stroke positions. The test machine is equipped with a displacement sensor to measure the displacement \(z_{def}\) and velocity \(\dot{z}_{def}\) of the MR damper piston, as well as a load cell to measure the output force \(F_{md}\). The signals for displacement and force are sampled at a rate of 0.5 kHz.  
After processing the experimental data, the MR damper model parameters introduced in (1) were adjusted and are presented in Table 2. The Microsoft Excel Solver tool was employed in this context.  
Figures 5 and 6 illustrate the comparison of the response curves for the experimental and estimated MR damper behavior corresponding to piston displacement amplitudes of 15 and 20 mm, with excitation frequencies of 3 Hz and 6 Hz, respectively. As observed, within the range of small velocities, the force variation exhibits significant hysteretic behavior, whereas at higher velocities, the force varies almost linearly with velocity. With an increase in current, the force required to yield the fluid also increases, leading to behavior characteristic of a plastic material in parallel with a viscous damper. The model's efficacy in estimating the MR damper force is evidenced by two key metrics: the consistently low estimation error, which always remains below 0.1 kN, and the high correlation coefficient, which exceeds 0.995 in every case.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline \(a_{0}\) & 236.3 N \\ \(a_{1}\) & 655.8 N/A \\ \(a_{2}\) & 4.67 s/m \\ \(a_{3}\) & 343.1 N s/m \\ \(V_{0}/X_{0}\) & 6.05 s\({}^{-1}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Experimental MR damper parameters.  
Figure 4: Experimental MR damper setup.  
Figure 7 shows a comparison of the proposed damper model to existing literature alternatives, such as the Bingham and bi-viscous approaches. In the presented example, an experiment with an amplitude of the piston displacement of 20 mm and a current of 1.8 A at a frequency of excitation of 6 Hz, the maximum estimation errors for the Bingham and bi-viscous models are 1.67 and 0.25 kN, respectively. In contrast, the estimation error for the proposed model remains below 0.1 kN in all cases, which serves to illustrate its superiority.  
Figure 5: Experimental and estimated MR damper curves for an amplitude of the piston displacement of 15 mm and different levels of applied current at a frequency of excitation of 3 Hz: (**a**) Deflection vs. Force, (**b**) Deflection Rate vs. Force, (**c**) Estimation Error, (**d**) Correlation.  
Figure 6: Experimental and estimated MR damper curves for an amplitude of the piston displacement of 20 mm and different levels of applied current at a frequency of excitation of 6 Hz: (**a**) Deflection vs. Force, (**b**) Deflection Rate vs. Force, (**c**) Estimation Error, (**d**) Correlation.