## 2 The Principle Analysis and Modeling for ALS-EHA
### Modeling Investigation
#### 2.2.2 Load-Sensitive Loop  
Load-sensitive loop consists of pump variable mechanism and PFV, of which output pressure and flow act on the variable hydraulic cylinder to move its piston to push the swash plate to adjust its inclination as shown in Figure 5. the loads acting on the swash plate mainly include the spring pre-tightening force, the hydraulic load, and the inertia force. 1. the force balance equation of the variable piston is as follows:  
\[P_{A}\cdot A_{1}=m_{vt}\ddot{x}_{v}+B_{vx}\dot{x}_{v}+Kx_{v}+F_{0}^{\prime }+F_{v} \tag{7}\]  
where \(m_{vt}\) indicates the total mass of the piston and load, \(B_{vx}\) represents viscous damping coefficient, \(K\) is load spring stiffness, \(F_{0}^{\prime\prime}\) is spring preload, \(F_{v}\) is variable piston force acting on the swash plate, \(P_{A}\) is output pressure of valve, \(A_{1}\), \(x_{v}\) denote the velocity and the area of the piston, respectively. 2. Moment balance equation of swash plate:  
\[L_{f}F_{v}=J_{s}\ddot{y}+B_{s}\dot{y}+M_{T}+M_{0} \tag{8}\]  
Figure 5: Schematic diagram of active load sensitive loop. where \(L_{f}\) denotes the arm of the variable piston force acting on swash plate, \(J_{s}\) denotes moment of inertia of swash plate, \(B_{s}\) denotes viscous damping coefficient, \(M_{T}\), \(M_{0}\) denote the torque generated by the hydraulic load and spring preload acting on the swash plate. \(\gamma\) denotes swash plate inclination. **Remark 1**. : _The average value of the resultant moment of the force acting on the swash plate at high speed is basically 0, so it can be assumed \(M_{T}+M_{0}=0\) during analysis._  
So, formula (8) can be simplified as:  
\[L_{f}F_{v}=J_{s}\ddot{\gamma}+B_{s}\dot{\gamma} \tag{9}\]  
The relationship between the inclination angle of the variable pump swash plate and the displacement of the variable piston is as follows:  
\[x_{v}=L_{f}\tan\gamma \tag{10}\]  
Since \(\gamma\) is generally small in practice, assuming that it can be approximated as  
\[\gamma=\frac{x_{v}}{L_{f}} \tag{11}\]  
The following formula can be derived from Equations (8) to (12)  
\[P_{A}A_{v}=M_{d}\ddot{x}_{v}+B_{d}\dot{x}_{v}+Kx_{v}+F_{0}^{\prime\prime} \tag{12}\]  
where \(M_{d}=m_{t}+\frac{J}{L_{f}^{2}}\), \(B_{d}=B_{c}+\frac{B}{L_{f}^{2}}\) are the total equivalent inertia and damping of load sensitive loop, respectively. According to the relationship between the tilt angle of the swash plate and the displacement, the displacement can be expressed as follows  
\[D_{p}=\bigg{(}1-\frac{\gamma}{\gamma_{\max}}\bigg{)}D_{p\max}=D_{p\max}-K_{p}\gamma \tag{13}\]  
where \(\gamma_{\max}\), \(D_{p\max}\) represent the swash plate inclination maximum and displacement maximum, respectively; \(K_{p}=D_{p\max}/\gamma_{\max}\) is pump displacement gain. Combining formulas (11) to (13) gives:  
\[ \begin{array}{ll}P_{A}A_{1}=&-\frac{M_{d}L_{f}}{K_{p}}\ddot{D}_{p}-\frac{B_ {d}L_{f}}{K_{p}}\ddot{D}_{p}-\frac{KL_{f}(D_{p}-D_{p\max})}{K_{p}}+F_{0}^{ \prime\prime}\\ &=-\frac{M_{d}L_{f}A_{v}}{K_{p}}\ddot{k}_{h}-\frac{B_{d}L_{f}A_{v}}{K_{p}}\ddot {k}_{h}-\frac{KL_{f}A_{v}}{K_{p}}k_{h}+\frac{KL_{f}D_{p\max}}{K_{p}}+F_{0}^{ \prime\prime}\\ &=M_{d}E\ddot{k}_{h}+B_{d}\dot{E}\dot{k}_{h}+KEk_{h}+\frac{KL_{f}D_{p\max}}{K_ {p}}+F_{0}^{\prime\prime}\end{array}  \tag{14}\]  
where \(E\) is equal to \(-\frac{L_{f}A_{v}}{K_{p}}\). Formula (14) shows that the load sensitive loop is a spring mass damping system. Since the inertia of load sensitive loop is very small, it can be simplified as elastic link to analyze the load sensitive characteristic. The DLS characteristics derived from formula (14) after simplification are shown in the Figure 6 below [19]. \[D=\left\{ \begin{array}{ll}D_{p\max}&for\quad 0\leq P_{l}\leq P_{0}\\ D_{p\max}-\frac{D_{p\max}-D_{p\min}}{P_{p\max}-P_{0}}(P_{l}-P_{0})&for\quad P _{0}\leq P_{l}\leq P_{l\max}\end{array} \right. \tag{15}\]  
where \(D_{p\max}\), \(D_{p\min}\) represent the maximum and minimum displacement of pump, \(P_{l\max}\), \(P_{0}\) are the maximum load pressure and the swash plate pre-tightening pressure.