Optimizing Oil Distributor Port for Low-Pulsation Cam-Lobe Hydraulic Motors

High-end equipment always operate in low-speed and heavy-load working environments, highlighting the need for cam-lobe hydraulic motors with excellent speed stability (< 1 r/min) and ultrahigh-power rotary output (> 1 MW). The successful operation of cam-lobe hydraulic motors relies on the circulation supply of high-and low-pressure oil. However, the switching between high-/low-pressure oil controlled by the oil distributor inevitably causes an obvious pressure impact and speed pulsation, which directly reduces the speed stability of hydraulic motors. Therefore, an optimization design approach for the oil distributor port is proposed to minimize the speed pulsation of cam-lobe hydraulic motors. In the proposed approach, a simulation model that links the oil distributor port structural parameters with the hydraulic motor speed pulsation was developed to clarify the effect of the oil distributor structural parameters on speed pulsation. Then, an orthogonal analysis method was used to identify the optimized oil distributor port structural parameters while minimizing the hydraulic motor’s speed pulsation as much as possible. Finally, several experiments were conducted to validate the effectiveness and accuracy of the proposed optimization design approach. The experimental results indicate that the pulsation rate of the hydraulic motor equipped with the optimized oil distributor was 62.5% lower than that of the original motor at a working pressure of 25 MPa, which is consistent with the simulation results using the proposed optimization design approach. The findings of this study offer a feasible and effective approach to guide the design optimization of the oil distributor port for low-pulsation hydraulic motors.


Introduction
Hydraulic actuation plays a critical role in various machinery and equipment, including ships, dredgers, and shield machines [1][2][3][4].Cam-lobe hydraulic motors, which are typical low-speed and high-torque hydraulic motors, are widely used as the core component of engineering machinery because their excellent speed stability allows for position fine-tuning and precise control of the driven devices [5][6][7][8][9].If a cam-lobe hydraulic motor exhibits speed instability under its working conditions, it will easily cause vibration and fatigue fracture of the driven devices, leading to the loss of valuable driven equipment [10,11].Thus, reducing the speed pulsation or instability of cam-lobe hydraulic motors is an important issue.
A large number of scholars have devoted extensive efforts to design low-pulsation hydraulic motors.Some scholars have attempted to develop theoretical or simulation models to obtain the output pulsation of hydraulic motors.A non-linear dynamic theoretical model, output characteristic simulation model based on non-linear friction, and other speed pulsation models have been developed to study the pulsation of hydraulic motors.These previous studies indicate that the output pulsation of hydraulic motors is related to several factors [12][13][14][15][16][17], which mainly consist of outer (e.g., oil pressure fluctuations and load fluctuations) and inner factors (e.g., the unreasonable design of the structure including the cam ring and the oil distributor) [18,19].Regarding external factors, pulsation is related to the hydraulic system.Regarding the inner factors of hydraulic motors, many scholars have focused on optimizing the cam-ring curve to develop low-pulsation hydraulic motors.A series of cam-ring curves have been designed to achieve lower speed pulsation in hydraulic motors, such as a common curve (including parabolic, sine, and equal acceleration), a modified cardioid curve, and a generalized elliptical cam-ring curve [20][21][22][23].Several optimization design methods have been developed to design special cam-ring curves for low-pulsation hydraulic motors, including a genetic-algorithm-based optimization design method for high-order function curves and non-uniform rational spline curves [20,23].Except for the optimization design of the cam-ring curve, the oil distributor controls the flow direction of inputting high-pressure oil and outputting low-pressure oil to drive the hydraulic motor rotation, in which the high-/low-pressure oil switching process generates an obvious pressure impact to induce high-speed pulsation [24][25][26].Therefore, oil distributor optimization is crucial for achieving the low-pulsation and low-speed stability of hydraulic motors; however, few studies have focused on this issue.
In this study, an optimized design approach for oil distributor ports is introduced to reduce the speed pulsation of a cam-lobe hydraulic motor.A hydraulic motor speedpulsation simulation model was developed to link the oil distributor port structural parameters with the hydraulic motor speed pulsation, as described in Section 3. Based on the simulation results, an orthogonal analysis method was used to select the optimized oil distributor port, as described in Section 4. In Section 5, several speed-pulsation experiments on hydraulic motors are described to validate the effectiveness of the proposed optimization design approach.Finally, Section 6 concludes the paper.

Configuration and Working Principle of the Cam-Lobe Hydraulic Motor
A typical cam-lobe hydraulic motor configuration is illustrated in Figure 1, which mainly consists of an oil distributor, multiple piston assemblies (i.e., pistons and rollers), a cam ring, and a cylinder block.It is worth noting that the cylinder block is fixed, while the cam ring and oil distributor rotate to convert hydraulic energy into output torque and mechanical rotation.Rotation of the cam ring is achieved by the tangential force generated between the cam ring and piston assemblies, which is controlled by periodically supplying high-pressure oil and draining low-pressure oil from the piston chambers through the oil distributor.Thus, the oil distributor plays a critical role in regulating the flow of high-/low-pressure oil in and out of the piston chambers.
To further illustrate the working principle of the oil distributor, the detailed high-/low-pressure oil distribution and switching process is illustrated in Figure 2. The oil distribution process can be described as that when the piston chamber is connected to the high-pressure oil port of the oil distributor through the cylinder block port, high-pressure oil flows into the piston chamber and drives the piston assemblies outward to push the roller from the inner dead center to the outer dead center.Conversely, when the piston chamber is connected to the low-pressure oil port of the oil distributor, the piston assemblies move inward, causing the roller to move from the outer dead center to the inner dead center.After the roller reaches the inner dead center of the next cycle, the piston chamber connects to the next high-pressure oil port of the oil distributor.During the high-/low-pressure oil switching process, the inevitable pressure impact causes the speed pulsations of the hydraulic motor, which is directly related to the oil distributor port configuration.Therefore, optimizing the oil distributor port is crucial for achieving low-speed pulsation and stability in hydraulic motors.

Theoretical Analysis and Oil Distribution Pulsation Simulation Model
To optimize the oil distributor port for low-pulsation hydraulic motors, it is necessary to establish a relationship between the structural parameters of the oil distributor port and the hydraulic motor speed pulsation, that is, an oil distribution pulsation model.The oil distribution pulsation model is primarily determined by the flow area of the oil distribution port, leakage of the oil distribution port, and kinematics of the piston assembly.Thus, in this section, the flow area of the oil distribution port, leakage of the oil distribution port, and kinematics of the piston assembly are analyzed in detail, and the oil distribution simulation model of a single-piston assembly is established based on these three aspects.According to the different positions and motion law of the piston assembly on the cam ring, the oil distribution pulsation model of the entire hydraulic motor can be established by summing the oil distribution simulation model of the single-piston assembly.

Flow Area Theoretical Analysis of the Oil Distribution Port
The basic structural parameters of the oil distributor and cylinder block ports are shown in Figure 3(a), (b), respectively.To reduce the pressure impact on the high-/low-pressure oil-switching process, an arc transition area was designed on the oil distributor port, as shown in Figure 3(a).The corresponding plane expansion view of a high-or low-pressure oil-distribution process is shown in Figure 3(c), in which the oil distributor port gradually moves through the cylinder block port.It is apparent that the flow area of the oil distribution port is equal to the intersection area between the oil distribution port and the cylinder block port, which can be calculated by integrating the contour of the intersection.
To obtain the contour of the intersection, the contours of the oil distributor and cylinder block ports are established.By considering the midpoint of each contour as its origin, two coordinate systems are established, as shown in Figure 3(a), (b).It can be seen that each contour can be established by a quarter of its entirety, as both contours are centrosymmetric shapes.A quarter of the oil distributor port contour is described by function f(x), while a quarter of the cylinder block port is described by function g(x), which can be respectively expressed as follows: where s, d, r 1 , and r 2 are the half-height, width, arc transition radius, and corner radius of the oil distributor port, respectively.l and r are the semi-major and -minor axes of the cylindrical block port, respectively.

Leakage Analysis of the Piston Assembly Based on Oil Distribution
Leakage of the piston assembly occurs at two locations: the switching interface between the oil distributor and cylinder block (i.e., oil distribution pair) and the sliding interface between the piston assembly and cylinder block port (i.e., piston pair).Leakage can be considered as an eccentric annular flow with linear pressure reduction, considering the fabrication error and thermal deformation of the oil distributor.Based on the basic eccentric annular flow equation [27], the leakage of the oil distribution pair (q 1 ) can be expressed as: (1) where α is the radian of the leakage port (i.e., the highpressure oil distribution port), R a is the average radius (R a = 0.5r a + 0.5r b , where r a and r b are the radii of the oil distributor and cylinder block center port, respectively), h is the concentric gap (h = r b -r a .),μ is the dynamic viscosity, L is the length between the oil distributor port and the outer sealing belt, Δp is the leakage pressure difference, and ε is the eccentricity ratio (ε = e/h, where e is the eccentric distance).
Similarly, the leakage in the piston pair (q 2 ) is analyzed in accordance with the leakage in the oil distribution pair, and can be expressed as: where R a2 is the average radius (R a2 = 0.5R 2 + 0.5r 2 , where R 2 and r 2 are the radii of the piston chamber and piston, respectively), h 2 is the concentric gap (h 3 = R 2 −r 2 .),L 3 is the leakage length, Δp 2 is the leakage pressure difference, and ε 2 is the eccentricity ratio (ε 2 = e 2 /h 2 , where e 2 is the eccentric distance).
Thus, the total leakage of the piston assembly (q X ) is the sum of the leakages in the oil distributor pair (q 1 ) and piston pair (q 2 ), and is given as follows:

Kinematics Analysis of the Piston Assembly Based on the Oil Distribution
The kinematics of the piston assembly are mainly determined by the cam-ring curve; a constant acceleration and constant deceleration cam-ring curve were used in this study.As shown in Figure 1, the piston assembly moves from the inner dead center to the outer dead center (upward movement) and then moves to the inner dead center in the next cycle (downward movement) in each acting cycle.During this upward and downward movement, the piston assembly has an equal and opposite velocity.After the piston assembly reaches the inner dead center of the next cycle, the piston assembly repeats the above motion.During the movement of the piston assembly from the inner dead center to the outer dead center, a degree of velocity is introduced to constrain the motion of the piston assembly, as illustrated in Figure 4.
The velocity of the piston assembly during the piston assembly movement from the inner dead center to the outer dead center can be mathematically expressed as follows: (3) (5) where k 1 and k 2 are the coefficients expressed as follows: where (ϕ) is the rotation angle of the hydraulic motor, h io is the difference in radius between the outer dead center and the inner dead center of the cam-ring curve, ( 0, ϕ 1 ) and ( ϕ 4 , ϕ 5 ) are the zero-speed zone, ( ϕ 1 , ϕ 2 ), ( ϕ 2 , ϕ 3 ), and ( ϕ 3 , ϕ 4 ) are the acceleration, constant-speed, and deceleration zones, respectively.

Oil Distribution Pulsation Simulation Model
By coupling the flow area theoretical analysis of the oil distribution port (Section 3.1), leakage analysis of the piston assembly based on oil distribution (Section 3.2), and kinematics analysis of the piston assembly based on oil distribution (Section 3.3), the oil distribution pulsation simulation model of a single-piston assembly can be obtained, as shown in Figure 5(a).According to the initial positions and motion law of the piston assemblies, the oil distribution pulsation model of the hydraulic motor can also be established by summing the oil distribution simulation models of all piston assemblies, as shown in Figure 5(b).The oil distribution pulsation simulation model can be used to calculate the speed pulsation of the hydraulic motor during the subsequent optimization.( 6) ,

Oil Distributor Port Optimization for Low-Pulsation Hydraulic Motors
To reduce the speed pulsation of the cam-lobe hydraulic motor, an optimization design approach for the oil distributor port was developed based on the orthogonal analysis method.The orthogonal analysis method was used to evaluate the influence of the oil distributor port parameters on the speed pulsation of the hydraulic motor, thereby obtaining the optimal oil distributor port structural parameters.The orthogonal analysis process and oil distributor port parameter results are described in this section.

Orthogonal Analysis of the Oil Distributor Port Parameters
Orthogonal analysis is a scientific optimization analysis method that was used in this study to select the optimal oil distributor port parameters [28][29][30][31][32].The four main oil distributor port structural parameters should be analyzed and determined, including the half-height (s), width (d), radius of the arc transition part (r 1 ), and radius of the elliptical groove (r 2 ), as indicated in Figure 3.The next step was to define the parameter values as factor levels, where one parameter value corresponds to a one-factor level for each structural parameter in Table 1.The parameter values are selected for structural parameters while considering the rationality of geometric constraints and parameters, including the number of oil distributor ports and distance between the oil distributor ports, among others.It is apparent that Factors A(s) and B(d) have five Levels, whereas Factors C(r 1 ) and D(r 2 ) have three Levels.Therefore, it is necessary to establish a mixed orthogonal analysis, as the levels of each factor are not equal; a special standard orthogonal analysis is presented in Table 2.In addition, it is worth noting that the pulsation rate values in Table 2 were calculated using the oil distribution pulsation simulation model (Figure 5(b)).The simulation parameters of the structure and working conditions are listed in Table 3.

Orthogonal Analysis Results for the Oil Distributor Port Parameters
The orthogonal analysis results for the oil distributor port parameters were divided into range and variance analyses.Range analysis can be regarded as an intuitive analysis that is used to determine three factors (including the primary factor, the optimal level of the factor, and the optimal level combination of factors).The range analysis results are presented in Table 4.In this study, the R-value was used to determine the primary factor by determining the influence of different factors on the evaluation index.The larger the R-value, the greater the influence on the pulsation rate.As can be seen in Table 4, C(r 1 ) has the greatest influence on the pulsation rate, followed by A(s), B(d), and D(r 2 ).To directly determine the influence and trend of the factors on the evaluation index, the relationship between the factor level and evaluation index was established.The pulsation rate at each factor level is shown in Figure 6.From an optimization perspective, the pulsation rate should be as small as possible.The optimal level of each factor was as follows: Factor A(s) was Level 4, Factor B(d) was Level 5, Factor C(r 1 ) was Level 3, and Factor D(r 2 ) was Level 2. However, range analysis cannot provide an accurate quantitative estimate of the factors in the order of importance.Thus, a variance analysis was introduced to determine whether there was a bias in the results, which was used to ensure the accuracy of the orthogonal analysis design, as presented in Table 5.In this study, the P-value was used to judge the significance of the different factors.The variance analysis results also  show that Factor C(r 1 ) has the greatest influence on the results because a P-value of 0.000** indicates significant differences in the pulsation rate.The variance analysis results were consistent with those of the range analysis.
According to the range and variance analyses, Factor C (i.e., the radius of the arc transition part, r 1 ) is the structural parameter with the greatest influence on speed pulsation.Finally, the optimized oil distributor port

Experimental Validation of the Optimized Oil Distributor Port
To validate the effectiveness of the optimization design approach, cam-lobe hydraulic motors with different oil distributors were used to test the speed pulsation.The accuracy of the hydraulic motor speed pulsation simulation model and the influence of r 1 on the hydraulic motor pulsation were validated, as discussed in this section.

Validation of the Hydraulic Motor Speed Pulsation Simulation Model
Figures 7 and 8 show the hydraulic system schematic and a photograph of the test rig, respectively, which were used to test the speed pulsation of the hydraulic motor.In the hydraulic system of the test rig, the pump delivered high-pressure oil and the directional control valve controlled the high-pressure oil flow into ports A and C of the tested hydraulic motor.The rotation of the tested hydraulic motor was controlled by periodically supplying high-pressure oil and draining low-pressure oil from the piston chambers via the oil distributor.The loading hydraulic motor was identical to that of the tested motor, which was loaded in the form of torsion.A speed sensor was used to measure the speed pulsation of the hydraulic motor and collect the speed pulsation signals using a data acquisition device.
To validate the hydraulic motor speed pulsation simulation model, three types of oil distributors with different arc transition part radii (r 1 = 0.007, 0.008, and 0.009 m) were designed and manufactured, as shown in Figure 9.These three oil distributors were installed sequentially in the tested hydraulic motor, as shown in Figure 8.Using a working pressure of 25 MPa, the speed pulsation of the hydraulic motor with three different oil distributors was measured, and the experimental speed pulsation rate was obtained by taking the average of the measured speed data and the standard error of the pulsation rate.The simulation speed pulsation rates of these three oil distributors were calculated using a hydraulic motor speed pulsation simulation model at a working pressure of 25 MPa.
Figure 10 shows a comparison between the simulated and experimental speed pulsation rates of the three oil distributors under a working pressure of 25 MPa.The comparison results indicate that the simulated speed pulsation rate was approximately consistent with the experimental speed pulsation rate, with a maximum error of less than 0.2%.This error may   be attributed to the machining error and cushioning effect of the coupling, which can be negligible and within an acceptable range.Thus, the hydraulic motor speed pulsation simulation model is determined to be accurate and can be used to analyze the speed pulsation and optimize other structures to reduce the pulsation of hydraulic motors.

Validation of the Influence of the Arc Transition Part Radius (r 1 ) on the Hydraulic Motor Pulsation
To validate the influence of r 1 on hydraulic motor pulsation, the above three oil distributors with different arc transition part radii (r 1 = 0.007, 0.008, and 0.009 m) were used again.Figure 11 shows the speed pulsation of the hydraulic motor with different oil distributors, which was measured at different operation conditions with different inlet pressures (i.e., the pressure was varied from 5 to 25 MPa).The results show that the speed pulsation rate decreases as r 1 increases, which is consistent with the results discussed in Section 4. At a working pressure of 25 MPa, the speed pulsation rate was approximately 2% for r 1 = 0.007 m, while it was approximately 0.77% for r 1 = 0.009 m.Thus, the influence of r 1 on the hydraulic motor pulsation was validated.

Validation of the Effectiveness of the Optimization Design Approach for the Oil Distributor Port
To validate the effectiveness of the optimization design approach for the oil distributor port, original and optimized oil distributors were manufactured, as depicted in Figure 12(a), (b) respectively.The original and optimized oil distributors were installed in turn in the hydraulic motor, the speed pulsation of the hydraulic motor was measured, and the results were processed as described in Section 5.1.Figure 13 presents the speed pulsation rate of the hydraulic motors with the original and optimized oil distributors under working pressures ranging from 5 to 25 MPa.As shown in Figure 13, the speed pulsation rate of the hydraulic motor equipped with the optimized oil distributor was less than that of the hydraulic motor equipped with the original oil distributor, and the former maintained a low-speed pulsation rate (no more than 1%) under the given operating conditions.At a working pressure of 25 MPa, the speed pulsation rate of the hydraulic motor equipped with the original oil distributor was approximately 2%, whereas that of the hydraulic motor equipped with the optimized oil distributor decreased significantly to only 0.75%, indicating a 62.5% reduction.Consequently, the proposed optimization design approach for the oil distributor port was validated to be effective.

Conclusions
(1) A hydraulic motor speed pulsation simulation model was established and validated experimentally.The simulation speed pulsation results were consistent with the experimental results, with maximum error of less than 0.2%.(2) The arc transition radius r 1 was identified as the primary oil distributor port structural parameter with the greatest influence on hydraulic motor pulsation.
Furthermore, as r 1 increases, the hydraulic motor speed pulsation decreases.(3) An optimized design approach for an oil distributor port was proposed and validated.The speed pulsation rate of a hydraulic motor equipped with the optimized oil distributor decreased to 0.75% at a working pressure of 25 MPa, indicating a 62.5% reduction compared with that of the original oil distributor.

Figure 1 Figure 2
Figure 1 Configuration and working principle of the cam-lobe hydraulic motor

Figure 3
Figure 3 Shape and motion of the oil distribution port: (a) Oil distributor port, (b) Cylinder block port, (c) Oil-distribution process

Figure 4
Figure 4 Velocity degree of the piston assembly under constant acceleration and constant deceleration cam-ring curves

Figure 5
Figure 5 Simulation model: (a) Oil distribution pulsation simulation model for a single-piston assembly, (b) Oil distribution pulsation model of the hydraulic motor

Figure 6
Pulsation rate at each factor level

Figure 8
Figure 8 Test rig and hydraulic motor

Figure 13
Figure 13 Pulsation comparison between the original and optimized oil distributors

Table 1
Factor level of the orthogonal analysis

Table 2
Orthogonal analysis design, L25, and pulsation rate

Table 3
Parameters of the structure and working conditions

Table 4
Range analysis results

Table 5
Variance analysis results