## 3 Results and Discussion
### Dynamical Mechanical Properties of the Self-Decoupling MR Damper
The performance parameter calculation formulas [40] are given in Equation (1) to Equation (4):  
\[K_{\rm h}=\frac{Q_{\rm max}-Q_{\rm min}}{D_{\rm max}-D_{\rm min}} \tag{1}\]  
\[K_{\rm d}=\frac{K_{\rm d}^{+}-K_{\rm d}^{-}}{2}=\frac{1}{2}\Bigg{(}\frac{Q_{\rm max }-Q_{\rm d}^{+}}{D_{\rm max}}+\frac{Q_{\rm min}-Q_{\rm d}^{-}}{D_{\rm min}} \Bigg{)} \tag{2}\]  
\[Q_{\rm d}=\frac{1}{2}(Q_{\rm d}^{+}-Q_{\rm d}^{-}) \tag{3}\]  
\[h_{\rm eq}=\frac{1}{4\pi}\frac{\Delta W}{W}=\frac{1}{2\pi}\frac{\Delta W}{K_{h }D_{\rm max}^{2}} \tag{4}\]  
where \(Q_{\rm max}\) is the maximum shear force; \(Q_{\rm min}\) is the minimum shear force; \(D_{\rm max}\) is the maximum displacement; \(D_{\rm min}\) is the minimum displacement; \(Q_{\rm d}\)\({}^{+}\) and \(Q_{\rm d}\)\({}^{-}\) are the intersection points of the positive and negative hysteresis curves and shear axis, respectively; \(K_{\rm d}\)\({}^{+}\) and \(K_{\rm d}\)\({}^{-}\) are the positive and negative yield stiffness, respectively; \(h_{\rm eq}\) is the equivalent damping ratio; \(\Delta W\) is the envelope area of the hysteresis curve; and \(W\) is the elastic potential energy. The results of the self-decoupling MR damper calculations are shown in Table 2. The equivalent damping ratio \(h_{\rm eq}\) in large-displacement conditions surpasses that observed in small-displacement scenarios. At an applied displacement of 5 mm, the \(h_{\rm eq}\) values vary between 37.62% and 45.39%. In contrast, at a displacement of 60 mm, the \(h_{\rm eq}\) values range from 55.03% to 57.29%. Thus, it is clear that displacement stands out as the primary factor affecting \(h_{\rm eq}\), while frequency, current, and magnetic field have comparatively minor effects on \(h_{\rm eq}\). Figure 9a illustrates the changes in equivalent stiffness \(K_{\rm h}\) and yield force \(Q_{\rm d}\) as a function of frequency, considering various input currents and a small displacement of 5 mm. The utilization of specific frequency and current input can improve both the equivalent stiffness \(K_{\rm h}\) and the yield force \(Q_{\rm d}\). At an input of \(-2.0\) A, the \(K_{\rm h}\) value ranges from 18.31 kN/mm at 0.1 Hz to 27.56 kN/mm at 2.5 Hz, indicating a change rate of 50.5%. Additionally, the value of \(Q_{\rm d}\) ranges from 82.00 kN at 0.1 Hz to 120.05 kN at 2.5 Hz, indicating a change rate of 46.4%. As the current transitions from \(-2.0\) A to 2.0 A at an applied frequency of 0.1 Hz, the value of \(K_{\rm h}\) increases from 18.31 kN/mm to 51.85 kN/mm, indicating a change rate of 183.2%. Additionally, the value of \(Q_{\rm d}\) ranges from 82.00 kN at \(-2.0\) A to 247.55 kN at 2.0 A, indicating a change rate of 201.9%. The various factors indicate that the self-decoupling MR damper is capable of adapting to changes in applied frequency and displays a stiffness characteristic that is influenced by current input. Figure 9b demonstrates the relationship between equivalent stiffness  
Figure 8: Determination of the shear properties of the self-decoupling MR damper. and yield force \(Q_{\rm d}\) as a function of input currents at a large displacement of 60 mm. The maximum damping force obtained is constrained by the measurement range of the testing machine, which does not exceed 300 kN. Consequently, as the input current rises from \(-2.0\) A to \(2.0\) A, the values of \(K_{\rm h}\) and \(Q_{\rm d}\) appear to stabilize. An earlier investigation into the self-decoupling MR damper utilized a common MRF as the filler, which exhibited a maximum shear yield strength of 54 kPa. Figure 10 shows the comparison of the dynamic mechanical properties for the self-decoupling MR damper filled with this common MRF and the HP-MRF. Figure 10a illustrates that, in comparison to the previous study, the damping force of the self-decoupling MR damper filled with the