Vane pump theory for mechanical efficiency

Transcription

1 1269 Vane pump theory for mechanical efficiency Y Inaguma 1 and A Hibi 2 1 Department of Steering Engineering, Toyoda Machine Works Limited, Okazaki, Japan 2 Department of Mechanical Engineering, Toyohashi University of Technology, Toyohashi, Japan The manuscript was received on 2 March 2005 and was accepted after revision for publication on 21 July DOI: / X32002 Abstract: This paper describes the theoretical analysis on the mechanical efficiency of a hydraulicbalanced vane pump and presents a design concept to decide values such as vane thickness and cam lift to improve its mechanical efficiency. In this paper, the friction torque characteristics of the balanced vane pump, especially the friction torque caused by the friction between a cam contour and vane tips are both experimentally and theoretically investigated. Although the friction force increases with vane thickness increase due to increase in the vane force, the mechanical efficiency of the vane pump does not decrease when the cam lift becomes large as the vane thickness is increased. An important parameter 1 defined as the ratio to be obtained by dividing the cam lift by the vane thickness determines essentially the mechanical efficiency of the vane pump. The pumps with the same 1 have the same mechanical efficiencies, even if the cam lift or the vane thickness varies. A larger value of 1 leads the vane pump to obtain a higher mechanical efficiency although the limit of vane strength governs the upper limit for 1. Additionally, reducing the friction coefficient of the vane tip contributes to improvement of the mechanical efficiency. Keywords: hydraulic power system, balanced vane pump, friction torque, mechanical efficiency, cam lift, vane thickness 1 INTRODUCTION A balanced vane pump, now widely used in many hydraulic power systems because of its compactness, lightweight, and low cost, is especially suitable for a hydraulic power source in a power steering system of a vehicle, in which low pressure pulsation and low noise are required. When designing a hydraulic pump including the vane pump, the mechanical efficiency as well as the volumetric efficiency constitutes a key factor in evaluating the pump performance. Friction torque characteristics of various pumps and motors have been already studied and mathematical models of the friction torque have been proposed [1 4]. Authors have also proposed a new mathematical model of the friction torque characteristics for the vane pump because the previously proposed models were not suitable for the vane pump [5]. These mathematical models were constructed conceptually or only to fit measured results in the actual pump torque, but Corresponding author: Department of Steering Engineering, Toyoda Machine Works Limited, 1-18 Miyama, Shinpukuji-cho, Okazaki , Japan. not theoretically. Therefore, the relation between the dimensions of the pump parts and its friction torque in the vane pump has yet to be clarified, and it is generally held that the friction torque in the vane pump has been insufficiently analysed. In this study, the friction torque characteristics, particularly the friction torque caused by the friction between the cam contour and the vane tip are experimentally investigated and theoretically analysed. Further, this investigation tried to clarify the influence of the friction between the cam contour and the vane tip on the mechanical efficiency and to construct the design concept on the relationship between the cam lift and the vane thickness for the vane pump with high mechanical efficiency. 2 EXPERIMENTAL STUDY ON FRICTION TORQUE OF VANE PUMP 2.1 Construction of vane pump Figure 1 shows a cross-sectional view of a balanced vane pump, composed of a cam ring with an elliptic inner bore, a rotor with a series of radially disposed C02805 # IMechE 2005 Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering Science

2 1270 Y Inaguma and A Hibi Fig. 1 Cross-sectional view of vane pump vanes, and two-side plates located at both sides of the rotor. The pump is so designed that both suction and delivery ports in the pump are diametrically opposed to provide complete balance of all internal radial forces. In this pump, the side plates have vane backpressure grooves at the sides facing the rotor to introduce the delivery pump pressure to all the bottoms of the vane slot of the rotor. During the pump operation, the vane is always pushed from the bottom by the delivery pressure and rotates with the loads imposed by the vane tip on the cam contour. The rotor is driven by a shaft through a very loose-fitting spline and the rotor and vanes rotate between two-side plates with running clearance. Figure 2 shows notation used in this paper. The vane tip has roundness of radius R v. Figure 2 illustrates two pump chambers, partitioned by two vanes, in which small radius part (u ¼ 0) and large radius part (u ¼ p/2) are located. The pump chamber (cross-hatched area) expands and oil is introduced into the chamber with pump rotation from u ¼ 0to p/2. And the pump rotation from p/2 to p causes the oil to be discharged from the pump chamber. The balanced vane pump will allow one more displacement process to take place between p and 2p, having a double displacement process, with two cycles of suction/discharge per revolution. Calculating the difference between the volume of the pump chamber at large radius part and that at small radius part and considering that the number of pump chambers are z, the pump displacement per revolution V th is given by the following equation n V th ¼ 2zb p o z (R2 2 R2 1 ) w(r 2 R 1 ) (1) cam ring, R 2 the large radius of the cam ring, and w the vane thickness. Table 1 shows dimensions of the test pumps and specifications of the cam lift and the vane thickness. Throughout the tests, each pump tested was assembled by new pump parts. The cam rings were made of ferro-sintered alloy and the inner surfaces of the cam rings were ground with roughness of 1 2 mm Rz. The vanes were finished by the barrel polishing and the roughness of the vane tip was 0.3 mm Rz. The expression of roughness of Rz, average peak-to-valley height, is the average value of the individual peak-to-valley heights in five continuous individual measurement sections. The where z is the number of vanes, b the width of the cam ring and the rotor, R 1 the small radius of the Fig. 2 Notation and pump chamber Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering Science C02805 # IMechE 2005

3 Vane pump theory for mechanical efficiency 1271 Table 1 Dimensions of test pumps Pump no. A-1 A-2 B-1 B-2 B-3 C-1 C-2 Cam ring small radius, R 1 (mm) Cam ring large radius, R 2 (mm) Cam lift, R 2 2 R 1 (mm) Rotor radius, R r (mm) Rotor with, b (mm) Vane thickness, w (mm) Vane tip radius, R v (mm) ¼ (R 2 2 R 1 )/w Pump displacement, V th (cm 3 /revolution) hydraulic fluid used was commercial mineral oil and its density r and viscosity m at 40 8C are 855 kg/m 3 and Pa s, respectively. 2.2 Experimental results of friction torque characteristics The ideal pump driving torque T th without energy loss in the pump is derived from the condition that the power required to drive the pump (the input power) is equal to the hydraulic power output as defined by the following equation T th ¼ V th Dp (2) 2p where Dp is the difference between the delivery pressure and the suction pressure. Adding the friction torque in the vane pump DT to the ideal torque, the actual pump driving torque T is expressed by the following equation T ¼ T th þ DT (3) Figure 3 shows the result of this investigation; DT increases linearly with an increase in Dp and the slope is greater with an increase in vane thickness w. It is worthwhile to notice that all of the lines concentrate on one point at Dp ¼ 0; namely, Y-intercept of all lines are identical. Drawing a line from this point as shown in the figure, DT would be divided into T 0, a constant term independent of Dp, and T n, a term increasing linearly with an increase in Dp. Then, DT can be expressed by the following equation. DT ¼ T 0 þ T n (5) The friction torque in the vane pump is considered to be caused by the following four factors. 1. Friction between the shaft and the oil seal. 2. Friction between the shaft and the bearing. 3. Viscous friction between the rotor and the side plates. 4. Friction between the vane tip and the cam contour. From equations (2) and (3), the following equation can be obtained DT ¼ T V th Dp (4) 2p First, the values of T against Dp for each test pumps were experimentally measured, and DT was calculated by using equation (4). In the experiment, T was measured with a torque meter and Dp was calculated by subtracting the suction pressure, p s, measured at the pump inlet from the delivery pressure, p d, measured at the pump outlet. Three pumps (B-1, B-2, and B-3) with the cam ring of R 2 2 R 1 ¼ 3 mm were used to investigate how three kinds of vane thickness affected the relationships between DT and Dp at N ¼ 2000 r/min and oil temperature of 40 8C. Fig. 3 Relationships between Dp and DT C02805 # IMechE 2005 Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering Science

4 1272 Y Inaguma and A Hibi Factors (a) to (c) are considered to be independent of Dp. This fact may be justified by the three following reasonings. First, from the principle of a balanced vane pump, a low pressure at a drain acts on the oil seal for the factor (a). Secondly, the force due to delivery pressure acts symmetrically on the rotor for the factor (b). Finally, the viscous friction due to shearing oil is independent of the pressure for the factor (c). In contrast, the friction torque due to the factor (d) increases with an increase in Dp because the force acting on the cam contour by the vane is proportional to Dp, and that will be described in Fig. 6 later. These reasons may prove that T 0 in equation (5) should constitute the friction torque due to factors (a) to (c) and T n should be caused by the friction between the vane tip and the cam contour of the factor (d). Figure 4 shows the relationship between w and T n, which was arranged from the measured data at Dp ¼ 5.88 MPa in Fig. 3. This figure shows that the straight line joining the measured data passes through the origin and T n is directly proportional to w. From this fact that T n is directly proportional to both w and Dp, it is clear that the friction force between the vane tip and the cam contour is directly proportional to the vane force pushing the cam contour and it resembles the property of Coulomb s friction. As T n is directly proportional to w and Dp, T n is expressed by the following equation T n ¼ kwdp (6) To investigate the dependence of k on the pump speed N, the characteristics of DT against Dp for three kinds of vane thickness were measured at various pump speeds, and Fig. 5 shows the measured result of k. As seen from this figure, k decreases slightly with an increase in N. This fact suggests that the coefficient of friction between the vane tip and the cam contour reduces slightly with an Fig. 5 Relationship between N and k increase in N and the distinctive difference in the vane thickness does not appear. 3 THEORETICAL ANALYSIS ON FRICTION TORQUE AT VANE TIP Figure 6 illustrates the pressure acting on the bottom and the tip of the vane and the vane force F v caused by the pressure difference in the small radius region (0, u, u 1 ), suction (u 1, u, u 2 ), large radius (u 2, u, u 3 ), and delivery (u 3, u, u 4 ). The delivery pressure p d constantly acts on the vane bottom, and p d as much as half the thickness of the vane acts on the vane tip at the small and large radius parts. Further, at the suction part, the suction pressure p s acts on the entire vane tip. On the other side, at the delivery part, p d acts also on the entire vane tip, balancing the radial force on the vane. With the change of F v taking place instantly at the starting point of each process, the changing pattern of F v during one displacement process from u ¼ 0to p is illustrated in Fig. 7. In the case of this test Fig. 4 Relationship between w and T n Fig. 6 Vane force at each part Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering Science C02805 # IMechE 2005

5 Vane pump theory for mechanical efficiency 1273 Fig. 7 Vane force during one displacement process pump, because the angle of suction part, u 2 2 u 1,is equal to that of delivery part, u 4 2 u 3, the average of F v becomes wbdp/2. With the coefficient of friction between the vane tip and the cam contour regarded as l, the friction force of the vane F t is given as follows F t ¼ 1 lwbdp (7) 2 Seeking for a friction torque against a piece of vane by multiplying F t by the average cam radius (R 2 þ R 1 )/2 and considering that the number of vanes is z, the friction torque due to the friction between the vane tip and the cam contour T n can be obtained by the following equation T n ¼ 1 4 zlwb(r 2 þ R 1 )Dp (8) Fig. 8 Coefficient of friction at vane tip Figure 8 shows the measured results of l for the vanes with three kinds of vane thickness by using the experimental apparatus and method for measuring l, which are detailed in Appendix 2. As seen from Fig. 8, l depends on the pump speed N and decreases gradually with an increase in N, namely, the sliding speed of the vane tip. In this test, using the ring with the inner surface of 1 2 mm Rz roughness, l ranges from 0.12 to 0.08 in the pump speed region of r/min and the influence of vane thickness is negligible. Figure 9 shows the comparison between T n obtained by the experiment and T n calculated by equation (8). The calculation is performed by using l ¼ corresponding to the value at N ¼ 2000 r/min in Fig. 8. The value of T n in the experiment almost coincides with the calculated one. Fig. 9 Comparison of T n between calculation and experiment C02805 # IMechE 2005 Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering Science

6 1274 Y Inaguma and A Hibi 4 INFLUENCE OF FRICTION AT VANE TIP ON MECHANICAL EFFICIENCY Substituting equation (8) for equation (12), h tc becomes The mechanical efficiency h m is defined as follows h tc ¼ T th T th þ (1=4)zlwb(R 2 þ R 1 )Dp (13) h m ¼ T th T (9) The values of h m are calculated by using the measured pump driving torque T and the calculated ideal torque T th. As a sample, Fig. 10 shows the relationship between Dp and h m for the pump with w ¼ 1.4 mm and cam lift (R 2 2 R 1 ) ¼ 3mm at N ¼ 2000 r/min. In this figure, h m increases with an increase in Dp and then it would approximate to the value of h tc. Further, by substituting equations (3) and (5) for equation (9), the following equation can be obtained h m ¼ T th T th þ T 0 þ T n (10) Then, h tc in the following equation introduces an ultimate value of h m h tc ¼ lim Dp!1 h m ¼ lim Dp!1 T th T th þ T 0 þ T n (11) Because both T th and T n are proportional to Dp and T 0 is independent of Dp, the ultimate value expressed as equation (11) results in the following equation h tc ¼ T th T th þ T n (12) As seen from equations (10) and (12), the friction of the vane tip causes the mechanical efficiency of the vane pump to reduce from 1 to h tc, and then the friction torque T 0 at the rotor sides, the shaft, and the oil seal causes the mechanical efficiency to reduce from h tc to h m. Substituting equation (1) for T th in equation (2) and substituting the result for equation (13), the following equation can be obtained h tc ¼ 1 1 þðlzw=4(r 2 R 1 ){1 (zw=p(r 2 þ R 1 ))}Þ (14) In the case of the test pumps, because zw/ fp(r 2 þ R 1 )g is much smaller than 1, this term can be eliminated. As a result, equation (14) can be expressed with a good accuracy of approximation by the following equation h tc ¼ 1 1 þ (lz=41) where 1 ¼ R 2 R 1 w (15) (16) Equation (15) suggests that the pumps with the same 1 will have the same h tc even if the pump dimensions such as cam lift and vane thickness are changed, and the larger value of 1 as well as the reduction of l and z may increase h tc. Figure 11 shows the relationships between 1 and h tc for z ¼ 10, which is calculated from equation (15) for various l.asseenfromthisfigure,h tc depends greatly on 1 as well as l and increases sharply with an increase in 1, when 1 is less than 1. However, 1 larger than 3 may not increase h tc substantially, resulting in less contribution to improve h tc. As mentioned later, the region of 1 larger than 3 should not be actually used because the vane strength for the bending force at a high pressure depends greatly on 1. To achieve more, 90 per cent of h tc, l should be reduced Fig. 10 Relationship between h m and Dp Fig. 11 Relationships between 1 and h tc Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering Science C02805 # IMechE 2005

7 Vane pump theory for mechanical efficiency 1275 Fig. 12 Relationships between h tc and Dp to about 0.1. Thus, reducing l to less than 0.05 could bring the h tc close to 95 per cent. 5 COMPARISON OF h tc BETWEEN EXPERIMENT AND CALCULATION Figure 12 shows the comparison of the h tc obtained by the experiment with that calculated by equation (15). The experimental values of h tc in Fig. 12 were calculated by equation (12) after estimating T n as shown in Fig. 3. The value of l used for the calculation is 0.095, with the value estimated at N ¼ 2000 r/min in Fig. 8. As all of the pumps shown in Fig. 12 have the same cam lift of 3 mm, a pump with a thinner w produces higher h tc because of less friction force of the vane tip. For all the pumps shown in Table 1, the experimental values of h tc were investigated using the same procedure as shown in Fig. 12. The values of h tc in Table 2 are the average of h tc, ranging from 1 to 5 MPa of Dp. Table 2 shows h tc at four pump speeds and the values of 1 for each test pump. Figure 13 shows the comparison of the experimental results of h tc with the calculated ones. The values of l used for the calculation were and 0.1. Figure 13 shows that a pump with a larger 1 produces a higher h tc and pumps with the same 1 result in almost the same h tc despite variation of the vane thickness and the cam lift. Strictly speaking, the experimental points are plotted almost along the curve of the calculated values for l ¼ at N ¼ 1000 r/min and close to that for l ¼ 0.1 at N ¼ 3000 r/min. Therefore, it may be verified that the decrease of l improves h tc at a higher N. It is well known that the friction torque tends to decrease with machine parts running-in after machines such as hydraulic pumps and motors have been operated for a longer period of time. Figure 13 shows the data of h tc for the pumps that were assembled with new pump parts. In contrast, Fig. 14 shows the data of h tc for the pumps operated for 500 h under a condition of N ¼ 1500 r/min, Dp ¼ 3 MPa, and oil temperature of 40 8C. All the pumps shown in Fig. 14 produce the values of h tc higher than those of the new pumps and these values correspond to the 1 2 h tc curve calculated with less than 0.1 of l. Some pumps have h tc for l less than The aforementioned result may justify the reasoning that the running-in effect has smoothed the surface at sliding parts in the pump, contributing to cause the coefficient of friction to be reduced extremely low. 6 LIMIT OF 1 TO BE DETERMINED BY VANE STRENGTH Figure 15 shows a vane posture at a large radius part between u 2 and u 3. In this region, a vane separates the suction area and the delivery area. The delivery pressure p d acts on one side of the vane part Table 2 Estimated h tc from experimental data Pump speed N Pump speed (r/min) Pump no. 1 h tc N (r/min) Pump no. 1 h tc 1000 A A A A B B B B B B C C C C A A A A B B B B B B C C C C C02805 # IMechE 2005 Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering Science

8 1276 Y Inaguma and A Hibi Fig. 13 Relationships between 1 and h tc (oil temperature ¼ 40 8C) protruded from the rotor and the suction pressure p s acts on the other side. To calculate the vane strength, the principle of the cantilever can be applied. At the large radius part, the largest bending moment M max and the largest tensile stress s max acts on the point of slot edge in the vane. Because the length of the vane creating the pressure difference Dp (¼p d 2 p s ) could be almost denoted as R 2 2 R 1, the bending moment M max is given by the following equation M max ¼ 1 2 b(r 2 R 1 ) 2 Dp (17) Fig. 14 Relationships between 1 and h tc after 500 h (oil temperature ¼ 40 8C) Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering Science C02805 # IMechE 2005

9 Vane pump theory for mechanical efficiency 1277 Fig. 15 With the section modulus of the vane defined as bw 2 /6, the largest tensile stress s max is defined as s max ¼ 31 2 Dp (18) The s max at the maximum rated pressure Dp R must be lower than the allowable bending stress of the vane material s al as expressed in the following equation 31 2 Dp R, s al (19) As a result, 1 should satisfy the following condition rffiffiffiffiffiffiffiffiffiffiffi s al 1, 3Dp R Vane posture and load condition at large radius part (20) As shown in equation (20), the upper limit of 1- value is determined by the strength limit of the vane material. When the strength of the vane material is identical, reducing of the maximum rated pressure will cause the upper limit of 1 to be increased. 7 CONCLUSIONS In this study, the friction torque characteristics of the vane pump, especially the friction torque characteristic between the vane tip and the cam contour and its influence on the mechanical efficiency were experimentally and theoretically investigated. As a result, the following conclusions are obtained. The friction force of the vane tip against the cam contour is proportional to the vane thickness and also to the pressure difference between pump outlet and inlet. In other words, the friction force is proportional to the vane force, which is proportional to the vane thickness and the pressure difference. This behaviour is similar to Coulomb s law. In addition, the coefficient of friction decreases as the sliding speed of the vane is increased. Although the load at the contact point of the vane tip on the surface of the cam contour increases with an increase in vane thickness, enlarging the cam lift in proportion to the vane thickness can prevent the mechanical efficiency from going down. The pumps produce the same mechanical efficiency when the parameter 1 defined as the ratio calculated by dividing the cam lift by the vane thickness remains the same, even if the cam lift or the vane thickness is varied. The mechanical efficiency becomes higher with an increase in 1, although the upper limit for 1 must be considered to exist because of the limit of vane strength. To design the vane pump with high mechanical efficiency, it is necessary to reduce the coefficient of friction on the sliding surface as well as to adopt a larger value of 1 within the allowable strength of the vane material. REFERENCES 1 Wilson, W. E. Rotary pump theory. Trans. ASME, 1946, 68(4), Wilson, W. E. Performance criteria for positivedisplacement pumps and fluid motors. Trans. ASME, 1949, 71(2), Schlösser, W. M. J. Ein mathematisches Modell für Verdrängerpumpen und -motoren. Oelhydraulik und pneumatik, 1961, 5(4), Hibi, A. and Ichikawa, T. Mathematical model of the torque characteristics for hydraulic motors. Bull. JSME, 1977, 20(143), Inaguma, Y., Watanabe, K., Kato, H., and Hibi, A. Energy saving and reduction of oil temperature rising in hydraulic power steering system. SAE paper APPENDIX 1 Notation b width of cam ring, rotor, and vane F v pushing force of vane k coefficient for T n defined by equation (6) N pump speed p d delivery pressure p s suction pressure R 1 small radius of cam ring R 2 large radius of cam ring T driving torque of pump T 0 friction torque independent of Dp T n friction torque of vanes T th ideal torque of pump V th pump displacement w vane thickness z number of vanes C02805 # IMechE 2005 Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering Science

10 1278 Y Inaguma and A Hibi Dp pressure difference between delivery- and suction-pressure (p d 2 p s ) DT total friction torque (¼T 2 T th ) 1 parameter defined as (R 2 2 R 1 )/w h m mechanical efficiency (¼T th /T) h tc ultimate value of h m defined by equation (11) l coefficient of friction APPENDIX 2 The coefficient of friction between the vane tip and the cam contour was experimentally measured by using a ring. The inner surface of the ring was ground with the same roughness as that of the cam rings for the test pump. The inner radius of the ring is the same as the small radius of cam ring. In this test, three kinds of vane thickness were used to investigate the influence of the vane thickness on the coefficient of friction. Figure 16 shows the specifications of the test parts. Figure 17 shows a schematic diagram of the experimental apparatus and hydraulic circuit of the test for measuring the coefficient of friction. The apparatus is an actual pump equipped with a ring instead of a cam ring and the side plates without ports connected with pump delivery. Oil delivered from a feed pump was fed to the vane backpressure groove of the side plate in the test apparatus. The vanes in the test apparatus were lifted to touch their tips to the inner surface of the ring by the pressure regulated by a relief valve. All the vane tips ran on the inner surface of the ring under a pressure as low as zero. The shaft of the rotor in the test apparatus was driven by an electric motor via a torque meter. Without the cam lift, the apparatus did not work as a pump to deliver oil, namely, T th expressed by equation (2) is zero. Therefore, the driving torque of this apparatus constitutes the total friction torque of the shaft, Fig. 16 Fig. 17 Dimensions of test ring and vanes Experimental apparatus and circuit rotor, and vanes. As described in Fig. 3, the friction torque of ten vanes, T n10, could be measured by subtracting the friction torque of the shaft and rotor (corresponding to Y-intercept in Fig. 3) from the driving torque. The coefficient of friction at the vane tip l could be calculated by the following equation. l ¼ T n10 =(zbwr 1 Dp) (21) where z is the vane number, b the rotor width, w the vane thickness, R 1 the inner radius of the ring, and Dp the pressure difference between the inlet and outlet of the apparatus. Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering Science C02805 # IMechE 2005