2. Working Principle and Displacement Design of VAPP
Variable displacement asymmetric axial piston pumps are used in hydraulic circuits [
22]. The working principle of their variable displacement is to adjust the inclination angle of the swashplate using the control mechanism, so that it changes the flow rate of the pump outlet, and then obtains different ascending speeds under the same load condition.
The schematic diagram of the VAPP-driven hydraulic cylinder and potential energy recovery is shown in
Figure 1. As the main component of potential energy recovery, VAPP is different from ordinary two-port pumps. It has three distribution windows. An unloading groove is added to the distribution window, which can effectively improve the overall characteristics of VAPP.
It can directly change the inclination of the swashplate so that the distribution windows A and B can obtain different flows to drive the movement of the hydraulic cylinder without other auxiliary components. The distribution window A is connected to the rodless chambers of the hydraulic cylinder, and the distribution windows B and T are connected to the rod chamber and the accumulator of the hydraulic cylinder, respectively. Theoretically, the flow of distribution window A is equal to the sum of the flow of distribution windows B and T.
In the beginning, the load is at the lowest position. The motor drives the VAPP to work, and the VAPP is in pump’s condition at this time. The hydraulic oil enters the rodless chamber of the hydraulic cylinder through the distribution window A to drive the load to lift.
The swash-plate angle of the VAPP is changed by the control mechanism, and the VAPP is in the working state of the hydraulic motor at this time. Hydraulic oil enters the connecting-rod chamber of the cylinder through the distribution window B. At the same time, part of the hydraulic oil is stored by the accumulator through the distribution window T.
When the load rises for the second time, the energy stored in the accumulator is released to speed up the rise of the load and reduce the work performed by the motor, thereby achieving potential energy recovery and utilization. In this VAPP system, the throttling loss is reduced, and the integration of drive and energy recovery can be realized.
The distribution window is shown in
Figure 2. It can be seen from the figure that there are three transition zones in the series asymmetric flow distribution plate, namely the top dead-center TDC, the nominal dead-center NDC and the bottom dead-center BDC [
23].
The VAPP belongs to the conical cylinder axial piston pump in structure, and its structure diagram is shown in
Figure 3. Unlike the structure in which the plungers in the cylindrical plunger pump are evenly distributed along the cylindrical surface, the plunger axis of the conical cylinder axial piston pump forms a certain angle with the cylinder axis and is distributed along the conical surface. Although this renders the movement trajectory of the plunger more complex and increases the difficulty of solving the equation of motion at the same speed, the linear velocity of the pump distribution surface will be relatively low, which enables it to operate at a higher speed. Therefore, a larger displacement of the piston pump can be obtained under the same conditions.
When the conical piston pump moves in the direction of rotation as shown in
Figure 3, the kinematics of the plunger are analyzed at the top dead center, A. During the movement of the plunger, on the one hand, it rotates with the cylinder block. On the other hand, it moves back and forth in the cylinder block. According to the actual movement trajectory of the plunger head, the spatial coordinate system as shown in
Figure 4 is constructed from the spatial position relationship, it can be concluded that the movement trajectory of the plunger head is the intersection point between the cone of the plunger axis and the plane of the swashplate.
In
Figure 4, point
is the apex of the conical surface where the axis of the plunger is located.
OS is the axis of the main shaft of the cylinder. The planes
and
are the bottom surface of the conical surface where the axis of the plunger and swashplate are located, respectively. The arc
is the movement trajectory of the plunger head originally located at the top dead center,
. The length of the line segment
represents the distance from the plunger head at
to the cylinder axis when the swashplate inclination angle is
, and
represents the distance from the plunger head at the top dead center to the cylinder axis when
.
represents the position of the plunger head when the cylinder rotates through the angle
, and
is the projection of point
on the
z-axis.
In
Figure 4, point
is created by connecting point
and point
and extending it. The plane view of the axis of the plunger is shown in
Figure 5a. In the
plane, the vertical line of the
-axis passing through the top dead center,
, and the extension line of
intersect at the point
. As shown in
Figure 5b, it is a schematic diagram of the intersection of the vertical line of the
x-axis passing through the point,
and the moving plane. According to the spatial relationship, the point
and the point
are coincident in spatial geometric relationship.
From the similarity of triangles, the following formula can be obtained:
According to
Figure 4, this formula can be obtained:
According to
Figure 5a, this formula can be obtained:
Substituting Equations (4) and (5) into Equation (6), this formula can be obtained:
Simultaneously, Formulas (3) and (7) are achieved:
According to
Figure 4, it can be known from the geometric relationship:
Substituting Equation (9) into Equation (8) produces:
Incorporating Equation (10) into Equations (1)–(3) produces:
The displacement equation of the plunger in the conical cylinder is:
The relationship between the speed of the plunger relative to the cylinder is:
where:
is the inclination angle of the swashplate.
and
are the rotation angle and angular velocity of the cylinder, respectively.
From the derived kinematic Equation (12) of the plunger of the conical plunger pump, the maximum displacement of the plunger in the cylinder is:
The theoretical displacement
of the conical piston pump can be obtained as:
where:
is diameter and
is the number of plungers, respectively.
Therefore, the instantaneous theoretical flow of the conical plunger pump can be further solved:
where:
is the cross-sectional area of the plunger; is the number of plungers in the oil-discharge area at a certain time. Since the pump has nine plungers in total, which is an odd number, , where represents the angular distance between plunger and plunger.
4. Parallel Parameter-Matching Method Based on Multi-Core CPU
4.1. Rapid Parallel Optimization Method Framework for VAPP Products Based on Multi-Core CPU
The main principle of the rapid parallel optimization method for complex hydraulic products with a multi-core CPU is to dynamically tune the simulation parameters for the CVODE simulation program based on the PSO. The CVODE solver can improve the simulation speed. The CVODE executable simulation program is regarded as the swarm particles of a swarm-intelligence algorithm. Each particle (CVODE executable program) is assigned to different design parameters, and the parameterization process is realized by writing the parameter file [
25].
The product performance parameters can be extracted from the output files generated by the COVDE executable program. The fitness value of the swarm particles can be calculated; at the same time, it can be judged whether the product performance constraints are met. Different from the traditional serial optimization simulation method, each COVDE executable program is a DOS program, which can be started and run through the ShellExecute function, and a process is automatically generated. If multiple DOS programs are started, the process allocation rights belong to the operating system, and there is no need for human intervention.
Running a simulation of multiple DOS processes can make full use of multi-core processors, and data interaction with the module can be conducted until the iteration-termination condition is met. Given the poor human–computer interaction of professional hydraulic simulation software, the visual interaction module is also a necessary part. The mainframe structure of the multi-core CPU parallel optimization method is shown in
Figure 18.
4.2. Processing of PSO Multi-Objective Functions and Constraints in Parallelism Simulation
Multi-objective optimization is when multiple objectives need to be achieved and, due to the inherent conflict between objectives, the optimization of one objective is at the expense of the degradation of other objectives, so it is difficult to obtain a unique optimal solution; instead, it is made by coordinating and compromising to render the overall goal as optimal as possible. According to the linear combination method, multiple objective functions form a comprehensive objective function [
26].
Through the above comparative simulation analysis, it can be seen that the equivalent mass of load, initial charging pressure P0, and volume of the accumulator V0, the angle of swashplate in the descending stage, the displacement of VAPP, and the motor speed will all have an impact on the operation efficiency and energy consumption. The larger the displacement, the higher the working efficiency, but it is not conducive to the potential energy-recovery system to form a motor condition in the descending stage. If the swashplate inclination is too large in the descending stage, it may affect the energy recovery, but if it is too small, the load will descend too slowly. Therefore, the parameter matching of the asymmetric pump potential energy-recovery system should take into account low energy consumption and high operation efficiency. The multi-objective optimization method provides an effective way of parameter matching the asymmetric pump potential energy-recovery system.
Design variables: asymmetric pump displacement , accumulator charging pressure , initial volume and the swashplate inclination in the descending stage.
Objective function 1: lifting–lowering–re-lifting total energy consumption E→min.
Objective function 2: the swashplate inclination in the descending stage→max.
Convert each sub-objective function to take a value in the range of 0 to 1.
In Formulas (24) and (25), the energy-consumption lower bound value of the total energy-consumption objective function is and the upper bound value , the angle lower bound value is and the upper bound value of the swashplate-angle objective function is .
The objective function of the asymmetric pump potential-energy-recovery system:
In Formula (26), is the energy consumption weighting coefficient and is the swashplate inclination weighting coefficient.
Constraint 1: Ensure that there is a motor condition in the descending stage, .
Constraint 2: Lifting–lowering–re-lifting time does not exceed a certain value, .
The single-model parallel optimization mechanism of VAPP’s excavator boom energy-recovery system is designed as shown in
Figure 19. Due to the high model complexity, running multiple executable programs in parallel can improve simulation efficiency. The parameter-assignment module based on the PSO assigns different parameters to multiple executable programs running at the same time. The parameter-assignment module based on the particle-swarm algorithms reassigns different parameters according to the response and constraints of the objective function. The process of updating the parameters iterates continuously until the decision condition is satisfied.
It is not difficult to find from
Figure 20 that along with the increasing iteration times, the fitness value of the optimal objective function and the total energy consumed descended gradually.
Under optimized parameter conditions, the changes of load position and motor energy consumption are shown in
Figure 21.
Table 3 shows the displacement of VAPP and related component parameters required for the optimized 7T-position excavator-boom energy-recovery system.
The simulation time comparison is shown in
Table 4. It takes 610 min to simulate with SimulationX software, and 55 min to run the CVODE executable. Compared with the SimulationX simulation platform, the efficiency of the .exe simulation program running on an eight-core CPU is increased by more than 80 times.
According to the above method, the simulation of different working conditions is carried out and the appropriate pump displacement (rounded) is selected according to the difference in its minimum equivalent mass. The appropriate swashplate angle is selected in the descending stage, and the appropriate accumulator parameters are selected.
Table 5 shows the required pump displacement, the swashplate angle in the descending stage, the initial pressure of the accumulator, the volume parameters of the accumulator, and the energy-saving rate for the 12T, 20T, and 30T excavator booms after optimization.
Under the premise of optimized displacement of VAPP, the influence of the swashplate angle on the energy-saving rate and the completion of the lifting–lowering–re-lifting time of the 12T, 20T, and 30T excavator booms was analyzed, as shown in
Figure 22.
It can be seen from the above figure that the smaller the angle of the swashplate in the descending stage, the easier it is to form the motor working condition, which is beneficial to the recovery of energy. The smaller the swashplate angle, the longer it takes to complete the lift–descent–re-raise. After optimization, it can be concluded that the minimum swashplate angles set by VAPP in the descending stage of the booms of 7T, 12T, 20T, and 30T excavators are: 15.8°, 15.4°, 8.9°, and 8.4°, respectively. When the swashplate angles are less than or equal to these, the excavator boom of each tonnage can achieve energy recovery in the descending stage, and the VAPP must have a motor working condition.
The energy-recovery rate of a common closed-loop boom potential-energy-recovery system is 14.8% [
27]. The energy-recovery rate of the optimized closed boom potential-energy-recovery system driven by VAPP is better than that of the ordinary closed boom potential-energy-recovery system.
According to the above-optimized displacement of VAPP required for each tonnage, the relevant values are designed, and the recommended values are given as shown in
Table 6.