Stator Curvature Optimization and Analysis of Axial Hydraulic Vane Pumps

Abstract

Aiming at the problem of large vibration of a high-subside stator inner cavity curve vane pump, the force analysis of the vane at the transition curve is carried out, and the functional relationship between the vane turning angle θ, the large arc radius R, the small arc radius r and the sliding friction is established. The particle swarm algorithm is used to optimize the solution of the objective function, and the optimized parameter values are brought into the MATLAB simulation program to obtain the optimized stator curve profile diagram. The dynamic performance of the vane pump before and after optimization is simulated using ADAMS. The results show that: the acceleration of the vane pump slide is significantly reduced; the friction between the slide and the slide groove is significantly reduced; the contact force between the slide and the stator is significantly reduced; the impact vibration of the optimized vane pump is significantly reduced; and the dynamic performance of the vane pump is improved.

1. Introduction

A vane pump is a type of displacement pump that works by regularly changing the volume of a sealed working chamber. Compared to other structured pump types, the vane pump has the benefits of a compact structure, uniform flow rate and low rotor inert. It is widely employed in petrochemical, aerospace, metallurgical and other fields [1,2]. Figure 1 depicts the structure of a double-acting vane pump. The pump rotor is concentrically attached to the stator; the vanes are radially installed in the rotor slots; and the rotor glides back and forth in the places as it rotates with the main shaft. The cavity is formed by the rotor, vane, stator inner cavity surface and pump end face. In the first half of the rotation of the rotor, a vacuum is formed in the pump and the liquid is sucked in; when the volume of the cavity is reduced from the maximum value to the minimum value, the liquid medium is pressed out. The media transfer is completed by periodic adjustments in the volume cavity [3].

With the trend toward low noise and low pulsation in vane pumps, the classic vane pump’s long-term vibration, noise, durability and other lingering difficulties must be addressed [4], particularly noise and vibration. The pulsation of the pump output and pressure is propagated to all components of the hydraulic system, including the hydraulic hose and valves [5,6].

The noise is mainly created by the pressure pulsation, the stator curve structure and the working chamber pressure rise in three aspects [7], whereas the vibration is primarily caused by the flow pulsation caused by leakage and the impact vibration between the vane and the stator curve’s two elements. As a result, a proper design of the stator curve structure and the construction of the vane pump’s force model are critical to reducing noise and vibration and increasing the pump’s dynamic performance. The dynamic performance of the pump has been explored in a few related articles.

One of the origins of vibration in vane pumps is the imbalance forces, which increase the amplitude as the manufacturing errors rise [8]. Numerical approaches have been demonstrated to predict pump vibrations, particularly those generated by hydrodynamic forces. This technique, which may be used in the initial design phase [9], can reduce the computational effort and enhance development efficiency. Pump vibration can also be caused by the guiding vane of an axial flow pump. It was discovered that an appropriate number of guide vanes has a role in suppressing the pressure fluctuations and vibroacoustics of axial pumps by comparing the radiated noise field stimulated by diffusers with the varied numbers of vanes [10]. Experiments have also revealed that different vane leading edge forms alter vane pump vibrations, and a method for determining the degree of cavitation behavior has been developed. As a result, a measurement system with a pump and sensors was designed to successfully eliminate the cavitation phenomena when the pump is operated [11]. To forecast the vibration characteristics of a multistage rotor system, a novel axial dynamic model integrating the transient force of the balance disk was presented, which can serve as a reference for the axial vibration model of a multistage pump rotor system [12]. Meanwhile, a two-way iterative fluid–structure coupling-based technique is presented to analyze the vibration characteristics of axial flow pumps. The fluid–structure coupling interaction has received less attention. The apparent time-frequency law of fluid pressure pulsation and structure vibration is given [13].

Meanwhile, other researchers optimized pump vibration and noise, leading to developing a multi-objective experimental and response surface-based optimization strategy for improving centrifugal pump hydroacoustic performance. The ideal combination of parameters for the synergistic optimization of hydraulic performance and noise was established using a multiple regression response surface model, which might result in a 3.03 percent gain in hydraulic efficiency and a 3.01 percent reduction in noise [14]. The results show that the radiation noise has typical dipole characteristics, and its directionality varies with the flow rate. A hybrid numerical method was proposed to obtain the streamwise noise source for the radiation noise of centrifugal pumps; the finite element method was used to obtain the acoustic simulation results, and the numerical method was verified by experiments [15]. Simulating the force situation of the components under various dynamic conditions using computational fluid dynamics (CFD) software COMSOL, some scholars proposed a new type of cylindrical vane pump by deriving the displacement calculation formula based on the design structure of the cylindrical vane pump, deriving the theoretical stator internal surface equation and establishing a three-dimensional model and using CFD software COMSOL to simulate the force situation of the components under various dynamic conditions [16]. The candidate transition curve functions for the stator with triangular and polynomial forms have also been discussed, and the V-A-J method (which converts the shape optimization task into a parameter value optimization task) has been used to find the best candidate for the “velocity criterion” and “acceleration criterion” curve equations. Without considering the effect of fluid movement, the dynamic features of the transition curves were investigated, and the findings revealed that different stator curves are acceptable for various applications [17].

Although these studies aid in understanding the source of vibration and noise in the vane pump, there is less research in the form of a high-dimensional curve stator curve of the vane pump generated by vibration optimization. This work is based on a balanced eight-dimensional stator curve vane pump, the vane of the vane pump force analysis. Few people consider the effect of the ratio of large to small arc radius of the stator cavity on the performance of the transition curve. Especially in the process of constructing the derivation of higher order equations, the difference between the radius of large and small arcs directly affects the dynamic characteristics of the curve as the coefficient of the transition curve, which in turn affects the motion form of the vane, the contact state. Therefore, a reasonable stator curve is crucial to the performance of the vane pump. In this paper, we will combine the vane force analysis and consider the design requirements to optimize the large and small arc radius ratio of the stator inner cavity curve of double acting sliding vane pump by intelligent algorithm to get the optimal large and small arc radius ratio under the constraint condition.

The vibration noise during the operation of the vane pump and the stator inner cavity transition curve part of the wear are significant, as shown in Figure 2, and the size of the arc part has almost no scratch, according to the experiments. Each force on the slider is related to the octant of the stator curve, and the boundary condition of the octant is set with R − r and the angle δ of the transition curve section; in this vane pump design, R/r = 39/32 = 1.21875, and the stator inner cavity transition curve part of the wear is serious. The stator inner cavity length and short radius ratio must be optimized, as indicated by the red outlined area in Figure 2.

2. Establishment of the Blade Force Model’s Objective Function

Because there are apparent cut marks on the stator inner cavity surface transition curve of the pump after the experiment, consider the sliding friction between the vane and the wall as the objective function of optimization, and the value of R and r is the optimized result when the friction force is reduced to the minimum value.

The force analysis of the blade in the transition curve section is shown in Figure 3 [18].

In the transition curve section, the blades are exposed to centrifugal force F_i, spring force F_t, inertia force F_m, friction force F_f between the stator cavity and the stator, Coe-type inertia force F_k, viscous friction force F_η, contact reaction force F_n1, F_n2 of the rotor and hydraulic pressure F_p1, F_p2, F_p3, F_p4, F_p5. Each force’s angular function is illustrated below.

(1)

The centrifugal force of the blade as a function of the rotation angle [19] is shown in Equation (1).

$$F_i\left(\theta \right)=\{\begin{array}{cc}{m}_v{\omega }^2\left(R-\lambda C\right)\hfill & \theta \in \left[0,20°\right]\&\left[160°,200°\right]\&\left[340°,360°\right]\hfill \\ m_v{\omega }^2\left(\rho \left(\theta \right)-\lambda C\right)\hfill & \theta \in \left[20°,70°\right]\&\left[110°,160°\right]\&\left[200°,250°\right]\&\left[290°,340°\right]\hfill \\ m_v{\omega }^2\left(r-\lambda C\right)\hfill & \theta \in \left[70°,110°\right]\&\left[250°,290°\right]\hfill \end{array}$$

(1)

where λc is the blade center of mass to the stator cavity distance, m_v is the rotating blade center mass, ω is the spindle angular velocity, R is the stator cavity large arc radius, r is the stator cavity small arc radius, ρ(θ) is the vector diameter of the blade.

(2)

The spring force on the blade as a function of the angle of rotation is shown in Equation (2).

$$F_t\left(\theta \right)=\{\begin{array}{cc}k\left(\mu -15\right)\hfill & \theta \in \left[0,20°\right]\&\left[160°,200°\right]\&\left[340°,360°\right]\hfill \\ k\left(\mu -\rho \left(\theta \right)+24\right)\hfill & \theta \in \left[20°,70°\right]\&\left[110°,160°\right]\&\left[200°,250°\right]\&\left[290°,340°\right]\hfill \\ k\left(\mu -8\right)\hfill & \theta \in \left[70°,110°\right]\&\left[250°,290°\right]\hfill \end{array}$$

(2)

where k is the spring stiffness factor, and μ is the spring free length.

(3)

The inertia force on the blade as a function of the angle of rotation is shown in Equation (3).

$$F_m(\theta )=\{\begin{array}{cc}m{\omega }^{2}R=\frac{m{\pi }^2{n}^2}{900}R\hfill & \theta \in \left[0,20°\right]\&\left[160°,200°\right]\&\left[340°,360°\right]\hfill \\ \frac{m{\pi }^2{n}^2}{900}a\left(\theta \right)\hfill & \theta \in \left[20°,70°\right]\&\left[110°,160°\right]\&\left[200°,250°\right]\&\left[290°,340°\right]\hfill \\ m{\omega }^{2}r=\frac{m{\pi }^2{n}^2}{900}r\hfill & \theta \in \left[70°,110°\right]\&\left[250°,290°\right]\hfill \end{array}$$

(3)

where m is the blade mass, n is the spindle speed, a(θ) is the radial acceleration of the blade in the transition curve section.

(4)

The frictional force between the blade and the inner cavity of the stator as a function of the rotation angle is shown in Equation (4).

$$F_{\mathrm{trip}}\left(\theta \right)=\{\begin{array}{cc}m{\omega }^2\left[R-\frac{L}{2}\right]·\cos \left(90°-\alpha \right)·f\hfill & \theta \in \left[0,20°\right]\&\left[160°,200°\right]\&\left[340°,360°\right]\hfill \\ m{\omega }^2\left[\rho \left(\theta \right)-\frac{L}{2}\right]·\cos \left(90°-\alpha \right)·f\hfill & \theta \in \left[20°,70°\right]\&\left[110°,160°\right]\&\left[200°,250°\right]\&\left[290°,340°\right]\hfill \\ m{\omega }^2\left[r-\frac{L}{2}\right]·\cos \left(90°-\alpha \right)·f\hfill & \theta \in \left[70°,110°\right]\&\left[250°,290°\right]\hfill \end{array}$$

(4)

where L is the blade length, α is the pressure angle formed by the blade and the stator cavity surface, and ƒ is the friction coefficient.

(5)

The Coe-type inertia force oonthe blade as a function of the rotation angle [20] is shown in Equation (5).

$$F_k\left(\theta \right)=\{\begin{array}{cc}2m{\omega }^2\left[R-\frac{L}{2}\right]\hfill & \theta \in \left[0,20°\right]\&\left[160°,200°\right]\&\left[340°,360°\right]\hfill \\ 2m\omega V\left(\theta \right)·\sin \left[\omega ,V\left(\theta \right)\right]\hfill & \theta \in \left[20°,70°\right]\&\left[110°,160°\right]\&\left[200°,250°\right]\&\left[290°,340°\right]\hfill \\ 2m{\omega }^2\left[r-\frac{L}{2}\right]\hfill & \theta \in \left[70°,110°\right]\&\left[250°,290°\right]\hfill \end{array}$$

(5)

where V(θ) is the radial velocity of the blade in the transition curve section.

(6)

The viscous friction force on the blade as a function of the rotation angle is shown in Equation (6).

$$F_{\eta }\left(\theta \right)=\{\begin{array}{cc}\eta \omega \left(R-\frac{L}{2}\right)\left[\frac{A_{vr1}}{h_{vr}}-\frac{A_{vs}}{h_{vs}}\right]\hfill & \theta \in \left[0,20°\right]\&\left[160°,200°\right]\&\left[340°,360°\right]\hfill \\ \eta V\left(\theta \right)\left[\frac{A_{vr}}{h_{vr}}-\frac{A_{vs}}{h_{vs}}\right]\hfill & \theta \in \left[20°,70°\right]\&\left[110°,160°\right]\&\left[200°,250°\right]\&\left[290°,340°\right]\hfill \\ \eta \omega \left(r-\frac{L}{2}\right)\left[\frac{A_{vr2}}{h_{vr}}-\frac{A_{vs}}{h_{vs}}\right]\hfill & \theta \in \left[70°,110°\right]\&\left[250°,290°\right]\hfill \end{array}$$

(6)

where η is the oil viscosity, A_b is the actual contact area between the bottom of the blade and the oil according to the design size of the blade, A_vs is the actual contact area between the blade and the end plate, A_vr1 is the actual contact area between the large circular arc section blade and the rotor, A_vr2 is the actual contact area between the small circular arc section blade and the rotor, the actual contact area between the blade and the rotor in the stator transition curve section $A_{vr}=65\ast \left[46-\rho \left(\theta \right)\right]$. h_vr is the clearance between the blade and the rotor, h_vs is the clearance between the blade and the end plate.

(7)

The contact reaction force oonthe blade by the rotor is shown in Equation (7) as a function of the rotation angle.

$$F_N\left(\theta \right)=\{\begin{array}{l}{F}_{N1}=\frac{\xi ·L_1\left(f\cos \alpha -\sin \alpha \right)+\left(L_1-L/2\right)F_k\left(\theta \right)}{L-L_1}\\ F_{N2}=\frac{\xi ·L\left(f\cos \alpha -\sin \alpha \right)+L/2·F_k\left(\theta \right)}{L-L_1}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}\theta \in \left[0,360°\right]$$

(7)

where L₁ is the length of the blade extending outside the rotor slot, L is the length of the blade, and ξ is the coefficient of the contact reaction force on the blade by the rotor.

(8)

The hydraulic pressure on the blade as a function of the turning angle is shown in Equation (8).

$$F_p\left(\theta \right)=\{\begin{array}{c}{F}_{p1}=a_1·\sin \left(b_1\theta +C_1\right)+\hfill \\ a_2·\sin \left(b_2\theta +C_2\right)+a_3·\sin \left(b_3\theta -C_3\right)\hfill \\ F_{p2}=a_4·\sin \left(b_4\theta -C_4\right)+\hfill \\ a_5·\sin \left(b_5\theta +C_5\right)+a_6·\sin \left(b_6\theta +C_6\right)\hfill \\ F_{p3}=a_7·\sin \left(b_7\theta -C_7\right)+\hfill \\ a_8·\sin \left(b_8\theta -C_8\right)+a_9·\sin \left(b_9\theta +C_9\right)\hfill \\ F_{p4}=a_{10}·\sin \left(b_{10}\theta +C_{10}\right)+\hfill \\ a_{11}·\sin \left(b_{11}\theta +C_{11}\right)+a_{12}·\sin \left(b_{12}\theta +C_{12}\right)\hfill \\ F_{p5}=41840158.12N\hfill \end{array}\begin{array}{c}\theta \in \left[20°,70°\right]\&\left[110°,160°\right]\&\\ \begin{array}{c}\begin{array}{c}\end{array}\end{array}\begin{array}{c}\end{array}\left[200°,250°\right]\&\left[290°,340°\right]\end{array}$$

(8)

where the blade is subjected to the various hydraulic component forces by design of experiment (DOE) simulation results for fitting the way to establish the individual component forces as follows. The function equation of the hydraulic pressure on the slide is detailed in Appendix A.

The blade rotation angle as a function of the hydraulic pressure at the wall of the pressure chamber is shown in Equation (9).

$$F_{p1}\left(\theta \right)=a_1·\sin \left(b_1·\theta +c_1\right)+a_2·\sin \left(b_2·\theta +c_2\right)+a_3·\sin \left(b_3·\theta +c_3\right)$$

(9)

$$\begin{array}{cc}{a}_1=9.277\times {10}^5\hfill & \left(-4.131\times {10}^8,4.15\times {10}^8\right)\hfill \\ b_1=0.0597\hfill & \left(-8.295,8.415\right)\hfill \\ c_1=0.255\hfill & \left(-403.3,403.8\right)\hfill \\ a_2=5.412\times {10}^5\hfill & \left(-4.329\times {10}^8,4.34\times {10}^8\right)\hfill \\ b_2=0.088\hfill & \left(-11.26,11.43\right)\hfill \\ c_2=2.063\hfill & \left(-482.6,486.7\right)\hfill \\ a_3=4.692\times {10}^4\hfill & \left(-1.088\times {10}^6,1.182\times {10}^6\right)\hfill \\ b_3=0.2742\hfill & \left(-1.396,1.944\right)\hfill \\ c_3=-1.033\hfill & \left(-52.72,50.65\right)\hfill \end{array}$$

The blade rotation angle as a function of the hydraulic pressure on the circular surface of the oil pressure chamber is shown in Equation (10).

$$F_{p2}\left(\theta \right)=a_4·\sin \left(b_4·\theta +c_4\right)+a_5·\sin \left(b_5·\theta +c_5\right)+a_6·\sin \left(b_6·\theta +c_6\right)$$

(10)

$$\begin{array}{cc}{a}_4=4.109\times {10}^6\hfill & \left(-1.808\times {10}^{11},1.808\times {10}^8\right)\hfill \\ b_4=0.1298\hfill & \left(-443.2,443.4\right)\hfill \\ c_4=-1.872\hfill & \left(-1.493\times {10}^4,1.492\times {10}^4\right)\hfill \\ a_5=9.939\times {10}^6\hfill & \left(-4.512\times {10}^{11},4.512\times {10}^{11}\right)\hfill \\ b_5=0.1598\hfill & \left(-1068,1068\right)\hfill \\ c_5=0.2701\hfill & \left(-3.534\times {10}^4,3.534\times {10}^4\right)\hfill \\ a_6=6.122\times {10}^6\hfill & \left(-6.32\times {10}^{11},6.32\times {10}^{11}\right)\hfill \\ b_6=0.177\hfill & \left(-625.6,626\right)\hfill \\ c_6=2.845\hfill & \left(-2.063\times {10}^4,2.063\times {10}^4\right)\hfill \end{array}$$

The blade rotation angle as a function of the liquid pressure at the wall of the suction chamber is shown in Equation (11).

$$F_{p3}\left(\theta \right)=a_7·\sin \left(b_7·\theta +c_7\right)+a_8·\sin \left(b_8·\theta +c_8\right)+a_9·\sin \left(b_9·\theta +c_9\right)$$

(11)

$$\begin{array}{cc}{a}_7=1.615\times {10}^6\hfill & \left(-1.544\times {10}^{10},1.545\times {10}^{10}\right)\hfill \\ b_7=0.1021\hfill & \left(-85.82,86.03\right)\hfill \\ c_7=-1.336\hfill & \left(-3091,3089\right)\hfill \\ a_8=3.58\times {10}^6\hfill & \left(-1.156\times {10}^{11},1.156\times {10}^{11}\right)\hfill \\ b_8=0.131\hfill & \left(-370.9,371.2\right)\hfill \\ c_8=0.8403\hfill & \left(-1.161\times {10}^4,1.161\times {10}^4\right)\hfill \\ a_9=2.279\times {10}^6\hfill & \left(-1.311\times {10}^{11},1.311\times {10}^{11}\right)\hfill \\ b_9=0.1435\hfill & \left(-274.4,274.7\right)\hfill \\ c_9=3.598\hfill & \left(-8224,8231\right)\hfill \end{array}$$

The blade rotation angle as a function of the fluid pressure at the circular arc of the suction chamber is shown in Equation (12).

$$F_{p4}\left(\theta \right)=a_{10}·\sin \left(b_{10}·\theta +c_{10}\right)+a_{11}·\sin \left(b_{11}·\theta +c_{11}\right)+a_{12}·\sin \left(b_{12}·\theta +c_{12}\right)$$

(12)

$$\begin{array}{cc}{a}_{10}=7.522\times {10}^5\hfill & \left(2.657\times {10}^5,1.239\times {10}^6\right)\hfill \\ b_{10}=0.03875\hfill & \left(-0.002936,0.08044\right)\hfill \\ c_{10}=0.3424\hfill & \left(-1.531,2.215\right)\hfill \\ a_{11}=1.541\times {10}^5\hfill & \left(-1.575\times {10}^6,1.883\times {10}^6\right)\hfill \\ b_{11}=0.219\hfill & \left(-0.8047,1.243\right)\hfill \\ c_{11}=0.309\hfill & \left(-29.13,29.75\right)\hfill \\ a_{12}=2.349\times {10}^5\hfill & \left(-1.531\times {10}^6,2.001\times {10}^6\right)\hfill \\ b_{12}=0.3297\hfill & \left(-0.2796,0.9391\right)\hfill \\ c_{12}=0.2197\hfill & \left(-18.26,18.7\right)\hfill \end{array}$$

The contact angle and rotation angle function equations are as follows. MATLAB was used to fit the function equation of the blade contact angle α and the rotation angle θ by sampling ten coordinate points [21]. After comparing various types of suitable approximation methods, it was discovered that the Gaussian approximation method was the closest to the resultant data, so the function equation of the blade contact angle and the rotation angle was fitted by this method as follows. The functional relationship between the contact angle of the arc top profile of the sliding plate and the rotation angle of the sliding plate is described in detail in Appendix B.

The Gaussian approximation method generic function model is shown in Equation (13).

$$f\left(x\right)=a_1·\exp \left(-{\left(\left(x-b_1\right)/c_1\right)}^2\right)$$

(13)

The coefficient (95% confidence level) is as follows.

$$\begin{array}{l}{a}_1=48.11 \left(46.15,50.07\right)\hfill \\ b_1=27.1 \left(26.64,27.57\right)\hfill \\ c_1=13.89 \left(13.27,14.55\right)\hfill \end{array}$$

The results are showed in Equation (14).

$$\alpha =48.11·\exp \left(-{\left(\left(\theta -27.1\right)/13.89\right)}^2\right)$$

(14)

To make it easier to develop the equation of the force on the slider as a function of the slider rotation angle, the relationship between the slider rotation angle θ and the contact angle α of the stator is determined.

As the blade and stator cavity mutual wear situation is serious, fully consider the blade force situation, the blade and stator inner wall between the minimum value of the sliding friction as the optimization objective function. Additionally, the calculation formula of sliding friction is $F_f=f·F_n$, according to Figure 4 force analysis, to obtain the force balance sketch in the direction of the common normal, combined with the actual force situation of the vane pump to obtain the blade by each force, according to the sliding friction formula to find the reverse support force F_n on the wall surface of the blade.

The equation of force balance in the direction of the common normal is shown in Equation (15).

$$\begin{gathered} \begin{array}{l}{F}_n+F_{\eta }·\cos \alpha +\left(F_{n1}+F_k+F_{p3}\right)·\cos \left(90°-\alpha \right)-\left(F_{n2}+F_{p1}\right)·\cos \left(90°-\alpha \right)-\\ \\ \left(F_{p5}+F_k+F_i\right)·\cos \alpha +F_{p4}·\cos \left(\frac{90°-\alpha }{2}\right)+F_{p2}·\cos \left(\frac{90°+\alpha }{2}\right)=0\end{array} \end{gathered}$$

(15)

The collation yields Equation (16).

$$\begin{gathered} \begin{array}{l}{F}_n=\left(F_{p5}+F_k+F_i-F_{\eta }\right)·\cos \alpha +\left(F_{n2}+F_{p1}-F_{p3}-F_{n1}-F_k\right)·\cos \left(90°-\alpha \right)-\\ \\ F_{p4}·\cos \left(\frac{90°-\alpha }{2}\right)-F_{p2}·\cos \left(\frac{90°+\alpha }{2}\right)\end{array} \end{gathered}$$

(16)

The resulting sliding friction relationship between the slider and the stator cavity is shown in Equations (17) and (18).

$$F_f=f·F_n=f·\left[\begin{array}{l}\left(F_{p5}+F_k+F_i-F_{\eta }\right)·\cos \alpha +\left(F_{n2}+F_{p1}-F_{p3}-F_{n1}-F_k\right)·\cos \left(90°-\alpha \right)\\ -F_{p4}·\cos \left(\frac{90°-\alpha }{2}\right)-F_{p2}·\cos \left(\frac{90°+\alpha }{2}\right)\end{array}\right]$$

(17)

$$F_f=f·F_n=f·\left[\begin{array}{l}\left(F_{p5}+k·\left(\mu -\rho \left(\theta \right)+24\right)+m·{\omega }^2\left(\rho \left(\theta \right)-\lambda c\right)-\eta V\left(\theta \right)\left[\frac{A_{vr}}{h_{vr}}-\frac{A_{vs}}{h_{vs}}\right]\right)·\cos \alpha +\cos \left(90°-\alpha \right)·\\ \left(\begin{array}{l}\frac{\xi ·\left[L·\left(f·\cos \alpha -\sin \alpha \right)+\left(L/2\right)·F_k\left(\theta \right)-\left(L_1·\left(f·\cos \alpha -\sin \alpha \right)+\left(L_1-L/2\right)·F_k\left(\theta \right)\right)\right]}{L-L_2}+F_{p1}-F_{p3}-\\ 2m\omega ·V\left(\theta \right)\end{array}\right)\\ -F_{p4}·\cos \left(\frac{90°-\alpha }{2}\right)-F_{p2}·\cos \left(\frac{90°+\alpha }{2}\right)\end{array}\right]$$

(18)

which $A_{vr}=65\ast \left[46-\rho \left(\theta \right)\right]$ and $\xi =\frac{\left(F_p+F_i-F_m\right)\left(L-L_1\right)-L_{1}f{F}_k\left(\theta \right)}{\left[\left(f^2+1\right)L+\left(f^2-1\right)L_1\right]\cos \alpha -2L_{1}f\mathrm{fsin}\alpha }$.

The remaining values of each parameter are shown in

Table 1. Table of parameter values.

Ab (mm2)	Avs (mm2)	Avr1 (mm2)	Avr2 (mm2)	η (mm2/s)	hvr (mm)	hvs (mm)
340.58	80.03	423.54	910	30	0.03	0.30

.

The comprehensive values above can be obtained and will be the slide and the stator cavity of the minimum value of the sliding friction of the objective function relationship, as shown in Equation (19).

$$\min F_f=f·F_n=f·\left[\begin{array}{l}\left(F_{p5}+k·\left(\mu -\rho \left(\theta \right)+24\right)+m·{\omega }^2\left(\rho \left(\theta \right)-\lambda c\right)-\eta V\left(\theta \right)\left[\frac{A_{vr}}{h_{vr}}-\frac{A_{vs}}{h_{vs}}\right]\right)·\cos \alpha +\cos \left(90°-\alpha \right)·\\ \left(\begin{array}{l}\frac{\xi ·\left[L·\left(f·\cos \alpha -\sin \alpha \right)+\left(L/2\right)·F_k\left(\theta \right)-\left(L_1·\left(f·\cos \alpha -\sin \alpha \right)+\left(L_1-L/2\right)·F_k\left(\theta \right)\right)\right]}{L-L_2}+F_{p1}-F_{p3}-\\ 2m\omega ·V\left(\theta \right)\end{array}\right)\\ -F_{p4}·\cos \left(\frac{90°-\alpha }{2}\right)-F_{p2}·\cos \left(\frac{90°+\alpha }{2}\right)\end{array}\right]$$

(19)

3. Stator Inner Cavity Curve Length and Radius Optimization

Eberhart and Kennedy proposed that the particle swarm optimization (PSO) algorithm consists of a series of iterative forms shown in Equations (20) and (21), iterating the velocity magnitude and direction of travel at times until the termination condition is reached [22].

$$V_{i,j}\left(t+1\right)=\omega V_{i,j}\left(t\right)+C_{1}R \mathrm{and} \left[P_{i,j}-x_{i,j}\left(t\right)\right]+C_{2}R \mathrm{and} \left[P_{i,j}-x_{i,j}\left(t\right)\right]$$

(20)

$$x_{i,j}\left(t+1\right)=x_{i,j}\left(t\right)+V_{i,j}\left(t+1\right) \left(i,j = 1, 2, \dots , d\right)$$

(21)

where ω is the inertia weight factor, C₁ and C₂ are learning elements with values in the range between 0 and 2, and rand and Rand are two mutually independent random functions following U(0,1). C₁ is the particle’s distance from its optimal position step, C₂ is the particle’s distance from the global optimal position step, and C₁ = C₂ = 1.49445 is taken in the requested objective function.

Constraints on the optimization of the stator length and radius parameters. The objective function of minimizing the sliding friction between the slider and the stator cavity is solved by the intelligent particle swarm algorithm, and some constraints still need to be set to make the solution converge more easily and find the optimal solution according to the design requirements and size limitations [23]. The constraints are shown in

Table 2. Constraints.

a	ω (rad/s)	ƒ	α	r (mm)	R (mm)	R/r (R ≠ r)
50°	100	0.13	[0, 90°]	[32, 34]	[32, 40]	(1, 1.2603]

.

We solve the objective function with constraints according to the particle swarm algorithm.

(1)

When R − r = 1, the minimum value of the objective function is 4.8298 × 10⁶, at which time R = 33, r = 32, θ = 26.65°.

(2)

When R − r = 2, the minimum value of the objective function is 4.4148 × 10⁶, when R = 34, r = 32, θ = 24.56°.

(3)

When R − r = 3, the minimum value of the objective function is 4.0240 × 10⁶, when R = 35, r = 32, θ = 23.71°.

(4)

When R − r = 4, the minimum value of the objective function is 3.6583 × 10⁶, when R = 36, r = 32, θ = 22.90°.

(5)

When R − r = 5, the minimum value of the objective function is 3.3217 × 10⁶, when R = 37, r = 32, θ = 23.13°.

(6)

When R − r = 6, the minimum value of the objective function is 3.4313 × 10⁶, when R = 38, r = 32, θ = 22.14°.

The objective function value of the sliding friction between the slide and the stator cavity is minimized for R = 37, r = 32 and θ = 23.13°, as calculated by the particle swarm algorithm analysis. The simulation results are shown in Figure 5 and Figure 6.

The optimized results R = 37, r = 32 are brought into the octant curve simulation program. The comparison results before and after the optimization of the stator cavity transition curve can be obtained by the simulation program, as shown in Figure 7.

4. Results

Based on the stator model before and after optimization, the ADAMS dynamics simulation of the virtual prototype before and after stator optimization is carried out to predict the effect of the optimized stator on pump kinematics and dynamics performance.

We created the simulation model in ADAMS/View, set up the working environment and set up the gravity, coordinate system, etc., as shown in Figure 8.

Spring force is applied between the blade and the rotor blade slot, friction force between the edge and the inner wall of the rotor blade slot. Solid-to-solid contact force is applied between the blade and the stator cavity and a force acting on the bottom and both sides of the edge, applying a given speed to the spindle, setting the simulated spindle speed to 300 rpm.

Sliding vane pump simulation results are compared and analyzed as follows.

(1)

Spindle torque curve before and after stator cavity length to radius ratio optimization.

Comparing Figure 9 and Figure 10, it can be seen from the simulation curve that there is a prominent peak in the shaft torque before the optimization of the central stator cavity. This analysis is put forward because when the difference of R − r is significant, a curve section with large curvature will be generated at the transition curve. Therefore, an ample torque is developed in this curve section when the main shaft rotates. The optimized simulation results show that the maximum value of the spindle torque is controlled within 5800 N·mm. Compared with the maximum torque peak of 16,000 N·mm before optimization, there is no prominent peak of the spindle torque in the whole rotation cycle. The optimization of the length and radius of the stator cavity has a good effect on reducing the spindle torque.

(2)

Slider acceleration curves before and after optimizing stator cavity length to radius ratio.

As shown in Figure 11 and Figure 12, the simulation results before the optimization of the stator cavity show that the acceleration of the slide has a sudden change of the great peak; the reason is that when the difference of R − r is significant, it will produce a curve section with large curvature at the transition curve. Therefore, when the slide slides on the stator transition curved surface, it will significantly impact this curve surface, resulting in severe wear and vibration. The simulation results of stator cavity optimization show that the maximum value of slide acceleration is controlled within 3.0 × 10⁵ mm/s² compared with the peak value of 5.0 × 10⁶ mm/s² before optimization; there is no significant peak value of slide acceleration in the whole rotation cycle, which shows that the optimization of the length and radius of stator cavity has a more significant effect on reducing the slide acceleration impact.

(3)

Friction curves between the slider and rotor slider slot before and after optimizing the stator cavity length and short radius ratio.

As shown in Figure 13 and Figure 14, the simulation results before optimizing the stator cavity show that the friction between the slide and the rotor slide slot changed abruptly to a great peak. The reason is that when the difference of R − r is significant, a curve section with large curvature will be generated at the transition curve. Therefore, when the slider slides on the stator transition curved surface, it will generate a large reaction force on the slider arc side, which will increase the pressure between the slider and the rotor slider slot wall, and the sliding friction will increase when the slider slides radially, resulting in deeper scratches. From the stator cavity optimization simulation results, it can be seen that the friction force between the slider and the rotor slider slot is controlled within 0.0032 N. Compared with the peak friction force of 50 N between the slider and the rotor slider slot before optimization, there is no large peak friction force between the slider and the rotor slider slot during the whole rotation cycle.

(4)

Slider–stator contact force curve before and after optimizing the stator cavity length to short radius ratio.

As shown in Figure 15 and Figure 16, the simulation results before stator cavity optimization show that the contact force between the slide and the stator has a sudden and extreme peak because when the difference of R − r is significant, a curve section with large curvature will be generated at the transition curve. As a result, when the slide slides on the stator transition curved surface, it will produce an excellent acceleration impact on this curved surface. Then, the slide and the stator will have a great elastic deformation, resulting in a sudden change in the contact force and causing a severe wear phenomenon. The simulation results of the stator cavity optimization show that the maximum value of the contact force between the slider and the stator is controlled within 21 N. Compared with the peak value of the contact force between the slider and the stator before optimization, which reaches 75 N, the peak value of the contact force between the slider and the stator is significantly reduced during the whole rotation cycle, which indicates that the optimization of the length and radius of the stator cavity has a beneficial effect on reducing the contact force between the slider and the stator.

From the ADAMS simulation results, it can be seen that the spindle torque, slider acceleration, blade and rotor friction, and blade and stator contact force numerical magnitude are reduced before and after stator optimization, indicating that the optimization of the length and radius of the stator inner cavity has a beneficial effect on lowering the blade and stator contact force.

5. Conclusions

With algorithm optimization and dynamics simulation of the transition curve of the stator cavity of the vane pump, the following research results were obtained.

• The explicit expression of the blade contact angle as a function of the blade rotation angle was obtained by fitting the values of the sampling points. The analysis showed that the blade contact angle was maximum in the middle of the transition curve and that the Gaussian approximation method had a good approximation effect.
• The optimization model of the long–short radius ratio using an intelligent particle swarm algorithm. According to the force balance equation, the expression of the sliding friction between the blade and the stator was derived. To minimize the sliding friction force on the edge, the optimized solution of the long–short radius ratio of the inner cavity of the stator was calculated.
• The dynamics of the vane pump was simulated using Adams separately, and the main shaft torque, vane acceleration, vane and rotor friction, and vane and stator contact force before and after optimization were compared and analyzed. The results showed that the optimized stator performance was better.