According to the physical system of the double-compound axial piston pump, the dynamic model and control model of the double-compound axial piston pump are established.
3.1. Dynamic Model of Axial Piston Pump Body
Each piston chamber is modeled as a capacitive volume model, as shown in
Figure 4, and its pressure is obtained by integrating the expression of time derivative [
9,
29]. Different from the models established in previous studies, the model established in this study regards the leakage of the slipper–swash plate pair and the piston–barrel block pair as coming from the piston chamber, and the leakage of the port plate–barrel block pair as coming from the oil suction chamber and the oil discharge chamber. Therefore, only the slipper–swash plate pair and the piston–barrel block pair are considered in the oil leakage in the capacitive volume model of the piston chamber, and the transient pressure in the piston chamber is expressed as:
where
pp,i is the pressure of piston chamber,
Vp,i is the volume of piston chamber,
E is the bulk modulus of hydraulic oil,
Qpout,i is the outlet flow of piston chamber,
Qpin,i is the inlet flow of piston chamber,
Qlpbb,i is the leakage flow of the piston–barrel block pair, and
Qlssp,i is the leakage flow of the slipper–swash plate pair.
Between the piston chamber and the port plate is regarded as a thin-walled hole. Therefore, the flow between the piston chamber and the inlet and outlet of the port plate is turbulent. The flow
Qpout,i and
Qpin,i through the orifice are expressed as [
9]:
where
pd is the pump outlet pressure,
ps is the pump inlet pressure,
Cd is the flow coefficient,
Apin,i is the flow area between the inlet main groove and the
ith piston chamber, and
Apout,i is the flow area between the outlet main groove and the
ith piston chamber.
The piston makes compound movement with the main shaft, and the volume of the piston chamber is expressed as:
where
dp is the piston diameter,
R is the piston distribution radius,
β is the inclination angle of the port plate,
ω is the rotational speed of the spindle,
V0 is the structure dead volume of the piston chamber, and
V1 is the volume of the piston cavity at zero position.
The flow rate of piston chamber generated by piston movement is expressed as:
The internal kinematic pair leakage of axial piston pump is mainly composed of three parts: piston–barrel block pair leakage, slipper–swash plate pair leakage, and port plate–barrel block pair leakage [
16]. In order to accurately calculate the outlet flow rate, it is necessary to numerically calculate the leakage of the piston–barrel block pair, the slipper–swash plate pair, and the port plate–barrel block pair. The total leakage flow is expressed as:
where
Qlppb is the leakage flow of the port plate–barrel block pair.
The leakage flow of the piston–barrel block pair is obtained as:
where
hp is the diameter gap between the piston and the piston chamber,
ep is the eccentricity distance between the piston and the piston chamber,
lpc,i is the contact length between the piston and the piston chamber,
μp,i is the dynamic viscosity of the piston–barrel block pair gap, and
up,i is the Couette effect of the piston speed on the leakage flow.
The leakage flow of the slipper–swash plate pair is obtained as [
13]:
where
hssp is the gap between slipper and swash plate,
dtp is the diameter of piston damping orifice,
ltp is the length of the piston damping orifice,
rspo is the outer radius of the slipper, and
rspi is the inner radius of the slipper.
The leakage of the port plate–barrel block pair is divided into two parts: the leakage
Qlpbi between the port plate inlet and the barrel block and the leakage
Qlpbo between the port plate outlet and the barrel block. The leakage caused by triangular damping grooves should also be included, but the studies of Bergada et al. [
14,
16] show that such leakage can be negligible compared with the port plate main groove. The leakage flow of port plate–barrel block pair is obtained as [
14]:
where
pc is the pressure of leakage cavity,
hppb is the gap between the barrel block and the port plate,
θin is the angle range of the inlet main groove of the port plate,
θout is the angle range of the outlet main groove of the port plate,
rint1 is the outer radius of the internal port plate,
rint2 is the inner radius of the internal port plate,
rext1 is the inner radius of the external port plate,
rext2 is the outer radius of the external port plate, and
μ is the dynamic viscosity.
The inlet and outlet flow of the pump is the sum of the inlet and outlet flow of all the piston chamber through the port plate. The phase angle of each piston is considered:
Given the working conditions of the inlet and outlet of the pump, the power input of the spindle and the working state of the swash plate, the model can calculate the pressure of the piston chamber according to Equation (1), and then calculate the inlet and outlet flow of each piston chamber by using the equation. Finally, Equations (6), (10) and (11) are used to calculate the leakage flow and the inlet and outlet flow of the piston pump.
The oil pressure in each piston chamber acts on the bottom surface of the piston, and the equivalent force
Fp,i is applied to the middle of the bottom surface of the piston and is perpendicular to it (i.e., along the
z axis). The viscous friction
Fvlpc,i caused by the leakage of the piston–barrel block pair is also applied to the piston along the
z axis.
Therefore, the resultant force acting on each piston is obtained as:
The viscous friction generated by the rotation of each slipper on the swash plate is obtained as:
The torque applied by the piston to the swash plate is obtained as:
where
yp,i is the force arm of the resultant force along the
z axis in the
y direction,
zp,i is the force arm of the resultant force along the
y axis in the
z direction,
Fptz,i and
Fpty,i are the projections of the resultant force
Fpt,i on the
z and
y axis, and
Fvlsz,i and
Fvlsy,i are the projections of the resultant force
Fvls,i on the
z and
y axis.
The viscous friction torque between the port plate and the barrel block is obtained as:
Therefore, the spindle torque of piston pump is obtained as:
In the actual hydraulic system, hydraulic oil always contains some gas, which is air in most cases. In a hydraulic system, when the pressure somewhere of the liquid is lower than saturation pressure (or air separation pressure)
Psat, the air dissolved in the liquid is released, and a large number of bubbles are produced in the liquid, which is called aeration (or air separation). Aeration is very important in hydraulic system simulation. In addition, there is a separate phenomenon called cavitation. When the fluid pressure decreases to a certain value, the fluid evaporates and produces a large amount of vapor, which is called cavitation. This happens when the pressure reaches saturated vapor pressure. Generally, the chemical properties of the fluid are not pure, so cavitation does not occur under a single pressure, but within a certain pressure range [
19,
30]. Therefore, the pressure at the beginning of cavitation is called high-saturation vapor pressure
PHvap, and the pressure at the completion of cavitation is called low-saturation vapor pressure
PLvap.
When the pressure of hydraulic oil is higher than the saturation pressure, the air is completely dissolved in the hydraulic oil, as shown in
Figure 5. When the pressure of hydraulic oil is lower than the saturation pressure, the air begins to separate out, and part of the air in the hydraulic oil is dissolved and partially free. When the pressure of hydraulic oil is lower than the high-saturation vapor pressure, the hydraulic oil begins to evaporate to produce vapor. When the pressure continues to decrease to low-saturation vapor pressure, the hydraulic oil is completely vaporized, and only air and vapor exist. Aeration and cavitation can not be ignored for axial piston pump, so these phenomena are considered in the double-compound axial piston pump model. The modeling of fluid properties, as described in IMAGING [
30], takes these phenomena into account.
3.2. Model of Swash Plate Variable Displacement Control
The diagram of the linkage mechanism is obtained by simplifying the structure of the double-compound axial piston pump controller, showing the pin position points on the linkage mechanism, as shown in
Figure 6. The power control process and negative flow control process are obtained by analyzing the principle of the power control and negative flow control of the double-compound axial piston pump, as shown in
Figure 7 and
Figure 8.
In the process of power control, according to the motion relationship between the power control valve, the servo valve, and linkage mechanism, the motion equation of servo valve is expressed as:
where
pd1 is the load pressure of the front pump,
pd2 is the load pressure of the rear pump,
pf is the power control pilot pressure,
Apcv1 is the effective area of the load pressure of the front pump acting on the power control valve,
Apcv2 is the effective area of the load pressure of the rear pump acting on the power control valve,
Af is the effective area of the power control pilot pressure acting on the power control valve,
Fpcv0 is the spring preload of the power control valve,
Fsv0 is the spring preload of the servo valve,
xsv is the displacement of the servo valve,
mpcv is the mass of power control valve core, and
msv is the mass of servo valve core.
J1 is the moment of inertia of the power control link around O
1;
Jf7 is the moment of inertia of feedback fork around O
7;
k3 =
l1/
l2,
k4 =
l3/
l4,
l1 is the distance between O
1 and O
6;
l2 is the distance between O
1 and O
4;
l3 is the distance between O
4 and O
7;
l4 is the distance between O
3 and O
7;
cpcv is the damping coefficient of the power control valve;
csv is the damping coefficient of the servo valve;
ksv is the spring stiffness of the servo valve; and
kpcv is the spring stiffness of the power control valve.
where
xpcv is the displacement of the power control valve core,
x0 is the distance difference between the zero position of the double springs of the power control valve,
k1 is the spring stiffness of the outer spring of the power control valve,
k2 is the spring stiffness of the inner spring of the power control valve, and
xlim is the maximum displacement of the power control valve core.
According to the motion relationship between the actuator piston, linkage mechanism, and servo valve, the feedback motion equation of servo valve is expressed as:
where
pap is the pressure of the large cavity of the actuator piston,
Ad is the action area of the large cavity of the actuator piston,
Ax is the action area of the small cavity of the actuator piston,
Fsp is the force of the swash plate on the actuator piston,
xcvf is the feedback displacement of the servo valve,
map is the mass of the actuator piston,
Jf4 is the moment of inertia of feedback fork around O
4, and
cap is the damping coefficient of the actuator piston.
The flow continuity equation between the servo valve and actuator piston is expressed as:
where
xap is the displacement of the actuator piston,
kq is the flow gain of the servo valve,
kc is the flow–pressure coefficient of the servo valve,
Vap is the volume of the large cavity of the actuator piston, and
wsv is the flow area of the servo valve.
The motion equation of the swash plate is expressed as:
where
Lsp is the acting force arm of the actuator piston to the swash plate,
Jsp is the moment of inertia of the swash plate, and
csp is the damping coefficient of the swash plate.
In the process of negative flow control, according to the motion relationship between the negative flow control valve, the servo valve, and linkage mechanism, the motion equation of servo valve is expressed as:
where
pncv is the negative flow control pilot pressure;
Ancv is the effective area of the negative flow control pilot pressure acting on the negative flow control valve;
Fncv0 is the spring preload of the negative flow control valve;
mncv is the mass of the negative flow control valve core;
J2 is the moment of inertia of the negative flow control link around O
2;
k5 =
l5/
l6,
l5 is the distance between O
2 and O
5;
l6 is the distance between O
2 and O
4;
cncv is the damping coefficient of the negative flow control valve; and
kncv is the spring stiffness of the negative flow control valve.
The feedback equation of the servo valve, the flow continuity equation between the servo valve and the actuator piston, and the motion equation of the swash plate in the negative flow control process are the same as those in the power control process.