## 3 Optimization of Static Stiffness Characteristics of Magnetorheological Suspension
### Static Stiffness  
When the load of the connecting rod of the magnetorheological suspension is F, the resetting force formed by the liquid in the upper and lower liquid chambers is in balance with the load F so that it can be obtained as follows:  
\[F=A_{s}(\Delta p_{2}-\Delta p_{1}) \tag{15}\]  
where \(A_{s}\)--the effective area of the extrusion plate; mm\({}^{2}\)  
\(\Delta p_{2}\)--the increased flow rate of the liquid chamber during the operation of magnetorheological suspension; m\({}^{3}\)/s  
\(\Delta p_{1}\)--the reduced flow rate in the upper fluid chamber during the operation of magnetorheological Suspension; m\({}^{3}\)/s  
The flow rate is calculated by the formula [17]  
\[\Delta p_{2}=\frac{A_{s}\Delta x_{p}\beta_{e}}{V_{2}} \tag{16}\]  
\[\Delta p_{1}=-\frac{A_{s}\Delta x_{p}\beta_{e}}{V_{1}} \tag{17}\]  
where \(\Delta x_{p}\)--displacement in the direction of \(x_{p}\); mm  
\(\beta_{e}\)--modulus of elasticity; MPa  
Therefore, the formula for calculating the overall stiffness of magnetorheological Suspension can be obtained, as shown in Equation (18). \[F=\beta_{e}A_{s}^{2}\bigg{(}\frac{1}{V_{1}}+\frac{1}{V_{2}}\bigg{)}\Delta x_ {p} \tag{18}\]  
As can be seen from Equation (18), the reset force is proportional to the displacement of the upper extruded plate, so the compressed magnetorheological Liquid is equivalent to a linear hydraulic spring, and its stiffness is the hydraulic spring stiffness. Figure 8: Static stiffness–displacement curve. The expression of static stiffness is shown in Equation (19). \[K_{h}=\beta_{e}A_{s}^{2}\bigg{(}\frac{1}{V_{1}}+\frac{1}{V_{2}}\bigg{)} \tag{19}\]  
where \(V_{1}\)--the volume of the upper liquid chamber; mm\({}^{3}\)  
\(V_{2}\)--the volume of the lower liquid chamber; mm\({}^{3}\)  
Due to the actual engineering specifications, the upper extruded plate is fully submerged in the magnetorheological fluid during the operation of the magnetorheological suspension. The effective area expression of the upper extruded plate can be defined as follows:  
\[A_{s}=A_{d}+A_{d}-A_{r}+A_{c}=2A_{d}+A_{c}-A_{r} \tag{20}\]  
\[A_{d}=\pi\bigg{(}\frac{d}{2}\bigg{)}^{2} \tag{21}\]  
\[A_{r}=\pi\text{k}_{1}{}^{2} \tag{22}\]  
\[A_{c}=d\pi\text{k}_{2} \tag{23}\]  
where \(A_{c}\)--the side area of the extruded plate; mm\({}^{2}\)  
\(A_{d}\)--bottom area of the upper extrusion plate; mm\({}^{2}\)  
\(A_{r}\)--bottom area of connecting rod; mm\({}^{2}\)  
\(d\)--Diameter of the upper extrusion plate; mm  
\(k_{1}\)--\(k_{2}\) actual engineering constant. The volume expression of the upper fluid chamber is  
\[\Delta p_{2}=\frac{A_{s}\Delta x_{p}\beta_{e}}{V_{2}} \tag{16}\]0  
The volume expression of the lower fluid chamber is  
\[\Delta p_{2}=\frac{A_{s}\Delta x_{p}\beta_{e}}{V_{2}} \tag{16}\]1  
In the formula, \(S_{1}\)--the upper liquid chamber fixed bottom area;mm\({}^{2}\)  
\(S_{2}\)--the fixed bottom area of the lower liquid chamber; mm\({}^{2}\)  
\(h_{1}\)--height of upper liquid chamber; mm  
\(h_{2}\)--lower liquid chamber height; mm  
According to the above formula, the static stiffness expression of suspension can be derived as follows:  
\[\Delta p_{2}=\frac{A_{s}\Delta x_{p}\beta_{e}}{V_{2}} \tag{16}\]2  
In order to better simulate and obtain more ideal optimization results, combined with the actual engineering and consult the relevant data, it is necessary to introduce the correction coefficient \(b\). Then, the static stiffness expression is as follows [18]:  
\[\Delta p_{2}=\frac{A_{s}\Delta x_{p}\beta_{e}}{V_{2}} \tag{16}\]3  
The optimal static stiffness is related to the diameter of the upper extrusion plate, the height of the upper cavity volume, the height of the lower cavity volume, and other factors. According to the optimizable part, the optimized design variables are set as the diameter of the upper extruded plate, the height of the upper liquid chamber, and the height of the lower liquid chamber of the suspension device.