# Research on Bifurcated Origami Hydraulic Dampers for Real Road Vibration Loads
## 2 Materials and Methods
### Damping Force Analysis of Bifurcated Origami Hydraulic Damper  
To investigate the mechanical properties of the bifurcated origami hydraulic damper, we verified the damper on the test stand and simplified the analytical model, as shown in Figure 8.  
For the purposes of discussion, let \(D\) represent the center diameter of the origami hydraulic damper, \(d_{0}\) is the orifice hole leading from the damper to the oil tube, and \(d_{t}\) is the diameter of the oil tube. The pressure at the top surface of the damper is denoted as \(P\), while \(P_{0}\) is the pressure at the orifice hole at the bottom of the damper. Additionally,  
Figure 8: Experimental setup of the bifurcated origami hydraulic damper. Figure 7: Bifurcated origami hydraulic tanks. (**a**) Bifurcated origami tanks. (**b**) the frame of bifurcated origami tank. (**c**) Assembly completed for the two bifurcated origami tanks. (**d**) Overall view of the test device. and \(v_{0}\) indicate the pressure and flow rate at the orifice hole's outlet, respectively, whereas \(P_{1}\) and \(v_{1}\) represent the pressure and flow rate at the ends of several oil tubes, analogous to \(P_{2}\) and \(v_{2}\). The flow rate of the hydraulic oil through an orifice hole in the body of the bifurcated origami hydraulic damper can be determined using the following equation [30]:  
\[Q=cA_{1}\sqrt{\frac{2}{\rho}(P-P_{0})}, \tag{2}\]  
where \(c\) is the flow coefficient, which is assumed to be 0.61, \(A_{1}\) is the cross-sectional area of the orifice hole, and \(\rho\) is the density of the internal oil. \[Q=\
u_{o}A_{1} \tag{3}\]  
By substituting this into Equations (2) and (3), the following equation was obtained:  
\[\
u_{o}A_{1}=cA_{1}\sqrt{\frac{2}{\rho}(P-P_{0})} \tag{4}\]  
According to Equation (4), the pressure inside the bifurcated origami hydraulic damper can be expressed using the following equation:  
\[P=P_{o}+\frac{\rho v_{o}^{2}}{2c^{2}} \tag{5}\]  
The hydraulic oil flows from the orifice hole through the oil tube to each branch hydraulic tank, and the pressure loss in this process can be calculated using the Darcy-Weisbach equation [30]:  
\[P_{o}-P_{1}-P_{2}=\frac{\lambda\rho}{2d_{t}}\Big{(}L_{1}v_{1}^{2}+L_{2}v_{2} ^{2}\Big{)} \tag{6}\]  
Here, \(\lambda\) is the friction coefficient of the oil tube, and L denotes the length of each oil tube. The pressure drop resulting from the sudden expansion of the cross-sectional area when the hydraulic fluid enters each branch origami tank from the tubes can be calculated as follows [30]:  
\[P_{1}=k_{c}\frac{\rho v_{1}^{2}}{2} \tag{7}\]  
\[P_{2}=k_{c}\frac{\rho v_{2}^{2}}{2}, \tag{8}\]  
where \(k_{c}\) is the loss factor. By reorganizing Equations (6)-(8), the following equation can be obtained:  
Figure 8:0  
The relationship between the flow rate at the top of each branch of the origami hydraulic damper and that at the lower part of the orifice hole can be expressed as follows:  
Figure 8:1  
Figure 8:2As shown in Figure 8, the inner diameter of the branch hydraulic tank oil tube is \(d_{t}\). The ratio relationship of the flow rate of each branch is set as shown in Equation (11). Figure 8:3  
Substituting Equation (12) into Equation (11) yields the following equation:  
Figure 8:4  
Summarizing Equations (13), (9), and (5) yields the following equation:  
Figure 8:5  
Substituting Equation (10) into Equation (14) yields the following equation:  
Figure 8:6  
The average cross-sectional area of the bifurcated origami hydraulic damper is given by Equation (16):  
Figure 8:7  
The force is expressed by the following equation, based on the actual use of the bifurcated origami hydraulic damper:  
Figure 8:8  
Equation (17) shows that a nonlinear relationship exists between the force and velocity of the bifurcated origami hydraulic damper.