Mathematical Analysis of Efficiencies in Hydraulic Pumps for Automatic Transmissions

Transcription

1 TECHNICAL PAPER Mathematical Analysis of Efficiencies in Hydraulic Pumps for Automatic Transmissions N. YOSHIDA Y. INAGUMA This paper deals with a mathematical analysis of pump effi ciencies in an internal ear and a balanced vane pumps. They are commonly used in current automatic transmissions includin a continuously variable transmission. For these pumps, the infl uence of oil temperature as well as pump-operatin pressures and pump speeds is clarifi ed by usin mathematical models takin oil temperature into consideration, which are constructed on the basis of actual measured fl ow and torque data under various pump-operatin conditions. For fuel economy in a vehicle, because the pump should be operated under conditions to obtain hiher pump effi ciencies, it is very important to understand the relationship between the pump effi ciencies and the pump-operatin conditions. This paper reveals that the mathematical models representin the leakae fl ow and friction torque characteristics accurately is helpful to investiate the pump efficiencies as well as the flow and torque performance under various pump-operatin conditions. As a result, the overall effi ciency decreases at extremely low and hih oil temperatures. In the internal ear and the balanced vane pumps tested in this work, the reion with a hiher overall effi ciency lies around an oil temperature of 8. Key Words: Fluid power system, hydraulic pump, pump effi ciencies, mathematical model, operatin condition, oil temperature 1. Introduction In an automatic transmission (AT) and a continuously variable transmission (CVT), an internal ear pump has conventionally been used 1). These days, a balanced vane pump with a fixed-displacement is also used for CVT operated at the maximum pressure of because of its ood volumetric efficiency 2, 3). For an improvement in the efficiency of these transmissions, it is important to reduce enery consumption of the pump 1, 4). Actually, the pump is operated across a broad rane of oil temperatures as well as pressures and pump speeds in the vehicle. Because such pump-operatin conditions affect sinificantly the pump efficiencies, it is very important to understand how the pump efficiencies depend on the pump-operatin conditions. Althouh the volumetric, the mechanical and the overall efficiencies as well as the flow and the pumpdrivin torque characteristics for various pumps for AT and CVT of vehicles have already been investiated 5), they were insufficiently analyzed. Since Wilson constructed a mathematical model for the flow and the torque characteristics in hydraulic pumps and motors 6), Schlösser 7) and Thoma 8) proposed their models based on the Wilson's model. However, their models were conceptual and have an insufficiency to represent accurately the actual characteristics of pump flow and torque. Hibi and Ichikawa 9), then, proposed an improved model for the friction torque characteristics with a non-linear chane aainst pump speed, and the author proposed a model for the friction torque, which was based on the Hibi-Ichikawa model and took oil temperature into consideration 1). In addition, because the model for the pump flow proposed by Wilson, Schlösser or Thoma represented inaccurately the actual pump flow, the author proposed also a model for the leakae flow which took into consideration that the thickness of clearances chaned accordin to pressures 11). And it was revealed that the model represented the flow characteristics accurately in various pumps. These models enable to analyze mathematically the influence of the operatin conditions on the pump efficiencies. In order to improve the efficiency of the transmission for fuel economy, it would be one of the ood methods for that the pump operates under a ood condition of hih pump efficiencies. The effectiveness of calculation usin the mathematical model is validated by comparin the calculated values with the experimental ones estimated from the actually measured pump flow and torque 12). 64 JTEKT ENGINEERING JOURNAL Enlish Edition No. 111E (214)

2 In this study, based on the analyzed results, it is clarified how the pump efficiencies depend on the pumpoperation conditions, especially oil temperature. 2. Test pumps In this study, two types of pumps were investiated. Fiure 1 shows a cross-sectional view of pump A, which is an internal ear pump composed of an inner rotor with external ear teeth, an outer rotor with internal ear teeth, a body holdin both the rotors, a cover and a stator shaft supportin the inner rotor via a bush bearin. The dimensions of pump A are presented in Fi. 2. Fiure 3 shows a cross-sectional view of pump B, a balanced vane pump with a fixed-displacement. It is composed of a cam rin with an elliptic inner bore, a rotor with a series of radially disposed vanes, two side plates located on both sides of the rotor and a shaft. The dimensions of pump B are presented in Fi Experimental apparatus Fiure 5 shows a schematic diaram of the experimental system for measurin flow rate out of the pump and pump drivin torque. The test pump was driven by a variable speed electric motor. The delivery pressure reulated with a throttle vale, d, and suction pressure, s, were measured at the outlet and inlet of the pump, respectively. The pressure differential across the pump D was defined by subtractin s from d. The oil temperature h was measured at the pump outlet. The pump flow rate was measured with a ear type flow meter. In this study, by settin pump speed and oil temperature h at a measurin point, and were measured in the process to increase d and to decrease it aain. The mean values Outlet Shaft Oil seal Outer rotor Inner rotor Bush bearin Cover Inlet Stator shaft Body Fi. 1 Cross-sectional view of pump A Side plate Fi. 3 Cross-sectional view of pump B Cam rin Vane Rotor Cover De1 Db D o Di2 De2 Di1 Fi. 2 Dimensions of pump A R 1 R r w R 2 Fi. 4 Dimensions of pump B JTEKT ENGINEERING JOURNAL Enlish Edition No. 111E (214) 65

3 of the two measured and values were adopted as and in this study. The hydraulic fluid is commercial mineral oil for CVT transmission, and its properties are iven in Table 1. Table 1 Properties of oil h q l the ideal flow rate th as follows. D As seen from Fi. 7, D is independent of and depends on oil temperature h (the viscosity of oil l) and D. In the case of pump A, the increasin rate of D becomes reater with an increase in D, whereas D in pump B increases a constant rate aainst D. On the basis of the experimental data, a mathematical model for the leakae flow characteristics will be introduced. The leakae flow D* in the mathematical model for leakae flow proposed by Schlösser 7) is expressed by the followin equation. In this study, because a loss flow Test pump 1 9 =4 min 1 h=4 h=8 h=12 ± ± ± ± ± Fi. 5 Experimental system 4. Experimental results and discussion 4. 1 Flow characteristics and its mathematical model First, relationships between the pressure differential D and the flow rate for various pump speeds and oil temperatures h in tested pumps are shown in Fi. 6. In both the pumps, decreases with an increase in D and the decreasin slope of aainst D becomes reater with risin of oil temperature h. Comparin both the pumps, the chane in aainst D is linear in pump B shown in Fi. 6(b), and is non-linear in pump A shown in Fi. 6(a). To investiate in detail, the leakae flow characteristics for these pumps are effective. The leakae flow D is obtained by subtractin the actual pump flow rate from, 1 6 m 3 /s, 1 6 m 3 /s =3 min 1 =2 min 1 =1 min D, MPa (a) Pump A (internal ear) =4 min 1 =3 min 1 =2 min 1 =1 min 1 h=4 h=8 h= D, MPa (b) Pump B balanced vane Fi. 6 Flow characteristics of test pumps 66 JTEKT ENGINEERING JOURNAL Enlish Edition No. 111E (214)

4 D, 1 6 m 3 /s h min 1 3 min 1 1 min 1 3 min 1 1 min 1 3 min 1 D Exp. D Cal. (h=12) Cal. (h=8) pump-operatin conditions, such as pressures, pump speeds and oil temperatures for various types of pumps. The model used is as follows: D D l D q l D D q D, 1 6 m 3 /s due to cavitaion is not taken into account, this effect was removed from the Schlösser s model D h D l 1 min 1 3 min 1 1 min 1 3 min 1 1 min 1 3 min 1 D, MPa (a) Pump A (internal ear) D Exp D, MPa D q Removin the second term proportional to D 1/2, the equation becomes the model of Wilson 6) or Thoma 8). In the model proposed by Wilson 6) and that by Thoma 7), it is considered that the leakae flow D is proportional to the pressure differential D. This conception cannot explain the non-linear D D characteristics in pump A. The Schlösser's model 7) added the component of the orifice flow leakae proportional to the square root of D to the Wilson's model, but cannot also explain the D D characteristics of pump A, in which the increasin rate of the leakae flow aainst D becomes reater and reater with an increase in D. Therefore, the author proposed a new model for the leakae flow D, in which the thickness of the clearance causin the leakae chaned accordin to the pressure differential D. As a result, the model was able to represent the D D characteristics under various Cal. (h=4) D Cal. (h=12) Cal. (h=8) Cal. (h=4) Fi. 7 Leakae flow characteristics in test pumps where a to d are pump constants accordin to the exponent of D, l is the viscosity of oil and q is the density of oil. The derivation of equation (3) is described in the reference 11). The terms on the riht-hand side of equation (3) mean from the left as follows: the first term dependent on D is the viscous flow leakae throuh the initial clearance in parts like parallel disks or plates, the second term dependent on D 2 is the chaned viscous flow leakae due to the chane in the clearance in parts like the parallel disks or plates, the third term dependent on D 1/2 is the orifice flow leakae throuh the initial clearance, and the fourth term dependent on D 3/2 is the chaned orifice flow leakae due to the chane in the clearance. The first and second terms have an inverse of l, and the third and fourth terms have that of q and are independent of l. In the case of c = d =, equation (3) becomes the Schlösser' model, and furthermore by settin b = it results in the Wilson's model. All the constants of a to d for the leakae flow in Table 2 were estimated from the actual D D characteristics in both the test pumps. Substitutin the values in Table 2 for a to d in equation (3), D aainst D can be calculated for each oil temperature. The calculated leakae flow usin the mathematical model is expressed as D* with comparison tod experimented. Fiure 7 shows the comparisons between D experimented and D * calculated for three oil temperatures. As seen from this fiure, D* accurately arees with D and the mathematical model can well represent also non-linear D D characteristics in pump A. Table 2 Values of pump constants for leakae flow in test pumps JTEKT ENGINEERING JOURNAL Enlish Edition No. 111E (214) 67

5 4. 2 Torque characteristics and its mathematical model Next, the characteristics of the pump drivin torque in pumps A and B were investiated and their results are shown in Fi. 8. The conditions of D=1, 2 and are chosen because the transmissions are mostly operated nearby at, the maximum operatin pressure of AT is and that of CVT is. Also the torque characteristics and the pump efficiencies are investiated and evaluated at these pressure differentials. In contrast to the flow characteristics, the torque characteristics have a complex confiuration affected by pump-operatin conditions such as the pump speed, the operatin pressure differential D and oil temperature h. In order to understand the torque characteristics in detail, the relationship between and the friction torque D should be investiated. The friction torque D is obtained by subtractin the ideal torque th from the pump drivin torque, as expressed by the followin equation. D Fiure 9 shows the relationships between and D for three kinds of D and h in pumps A and B. Althouh D varies complicatedly accordin to, D and h, D increases with an increase in D in all cases of h. When D is low, D has a nearly monotonous increase with an increase in. In contrast, D decreases first and then increases with an increase in when D becomes hiher for all cases of h. This chane appears remarkably with increasin h. Pump B has the characteristic similar to that of pump A. The mathematical model for friction torque was oriinally proposed by Wilson 6) throuh his excellent concept that the friction torque consists of three components: (1) a component dependent on the friction proportional to the pressure, (2) a component dependent on the viscous friction due to oil shearin concernin the pump speed and the viscosity of fluid, and (3) a component independent of pump-operatin conditions. The friction torque of his model is expressed as D* by the followin equation. D D l x In equation (5), f, n and c are constants independent of D, x(=2p/6) and oil temperature h. However, this model cannot simulate the non-linear characteristics between and D shown in Fi. 9. Schlösser proposed his model 7) addin a component proportional to the product of the mass of oil and the square of the pump speed to the Wilson' model, but it cannot also represent such characteristics. Hibi and Ichikawa 8) proposed a new model, in which the pump constant f in the Wilson' model was treated as a function of the pump anular velocity x. Althouh their model can represent the, Nm, Nm h=4 h=8 h=12 Exp. Cal. Exp. Cal. Exp. Cal , min 1 (a) Pump A (internal ear) h=4 h=8 h=12 D = D = D =5 MPa D = D = D = Exp. Cal. Exp. Cal. Exp. Cal , min 1 Fi. 8 Pump drivin torque characteristics in test pumps 68 JTEKT ENGINEERING JOURNAL Enlish Edition No. 111E (214)

6 D, Nm h=4 h=8 h=12 h=4 h=8 h=12 D = D = D = D Exp. D Cal. D Exp. D Cal. D Exp. D Cal. (a) Pump A (internal ear) D Exp. D Cal. D Exp. D Cal. D Exp. D Cal., min 1 D = D = D = non-linear characteristics between and D, it was insufficient in consideration of oil temperature. Therefore, the author proposed an improved model addin a factor reardin oil temperature to the component dependent on D and another factor independent of the viscosity of oil to the component dependent on x in the model of Hibi and Ichikawa 1). D a xx l x h h h h D where coefficients in equation (6) such as, x, a, h, h,, and are pump constants for friction torque characteristics independent of pump-operatin conditions of x(=2p/6), h, D and l. The values of these constants are determined from the experimental data of and presented in Table 3 for two pumps tested in this study. The pump drivin torque is calculated as * by addin the ideal torque th = th D/(2x) to D* in equation (6). The chanes in * and D* calculated usin equation (6) and the values in Table 3 are indicated with various lines, respectively. Comparin D* with D in Fi. 9, the mathematical model can not exactly represent the chanes in D in a reion of low at a hih oil temperature for pump A. However, it can be seen that for actual pumps A can be well estimated by usin the mathematical model under various pump-operatin conditions, as seen from Fi. 8. D, Nm , min 1 Fi. 9 Friction torque characteristics in test pumps 5. Influence of oil temperature on pump efficiencies 5. 1 Volumetric efficiency In this study, the volumetric efficiency denoted usin the actual flow rate v and that denoted usin the leakae flow calculated from equation (3) v * are expressed as follows. D D Fiure 1 shows the relationships between the pump speed and the volumetric efficiency for three D of 1, 2 and at oil temperature of 8. In Fi. 1, the volumetric efficiency of pump B is hiher than that of pump A and v * aree well with v in both the pumps. Table 3 Values of pump constants for friction torque in test pumps x a h h JTEKT ENGINEERING JOURNAL Enlish Edition No. 111E (214) 69

7 Oil temp. h=8 D Cal. Exp..4.2 Oil temp. h=8 D Cal. Exp , min 1 (a) Pump A (internal ear) min 1 (a) Pump A (internal ear) Oil temp. h=8 D Cal. Exp..4.2 Oil temp. h=8 D Cal. Exp , min min 1 Fi. 1 Chanes in volumetric efficiency in test pumps 5. 2 Mechanical efficiency Similarly to the volumetric efficiency, the mechanical efficiency denoted usin the actual pump drivin torque m and that denoted usin the friction torque calculated from equation (6) m * are expressed as follows. D D Fiure 11 shows the chanes in the mechanical efficiency aainst the pump speed for aforementioned D at h=8. For the mechanical efficiency, m * usin D* calculated from equation (6) arees well with m obtained from the actual pump drivin torque. Fi. 11 Chanes in mechanical efficiency in test pumps 5. 3 Overall efficiency Finally, two overall efficiencies and * are denoted as the product of v and m obtained from measured and and that of v * and m * calculated usin the mathematical models for D* and D*, respectively. Fiure 12 shows the chanes in the overall efficiency aainst the pump speed for three kinds of D at h=8. In the case of pump A shown in Fi. 12(a), the chane in the overall efficiency aainst as well as D is lare. In contrast, pump B has a small chane in the overall efficiency aainst and D, as shown in Fi.12(b). In both the pumps, * arees well with throuh all the measured conditions. Fiure 13 shows the influence of oil temperature h on the overall efficiency for various kinds of and D. Both the pumps have similar chanes in the overall efficiency. In the rane of h from 2 and 14, with decreasin, the overall efficiency increases at a lower D of and decreases at a hiher D of. In this fiure, the areement of * with is excellent. 7 JTEKT ENGINEERING JOURNAL Enlish Edition No. 111E (214)

8 Oil temp. h=8 D Cal. Exp , min 1 (a) Pump A (internal ea) Oil temp. h=8 D Cal. Exp , min 1 Fi. 12 Chanes in overall efficiency in test pumps A hydraulic pump for a vehicle operates under various conditions of, D and h. Therefore, it would be important to understand the actual overall efficiency of the pump. Then, the overall efficiency * calculated usin the mathematical models for the leakae flow and the friction torque was plotted on the h plane for three kinds of D. Fiure 14 shows the result that the pumps have an area of hiher overall efficiency as a whole at h nearby 8. The increase in prevents the overall efficiency from oin down when h becomes hiher. In addition, pump operation at a hiher prevents the overall efficiency from fallin down at a hih oil temperature. 6. Conclusions In this study, the influence of oil temperature on various pump efficiencies was analyzed by usin mathematical models for the leakae flow characteristic and the friction torque characteristics in an internal ear and a balanced vane pumps used for an automatic transmission in a vehicle. As a result, the followin conclusions were drawn. Usin the mathematical models, it could be possible to survey the pump operatin conditions with hiher efficiencies. Because the viscosity of oil decreased sharply with an increase in oil temperature, the volumetric efficiency fell down remarkably in the rane of hih oil temperature. In contrast, the mechanical efficiency fell down remarkably in the reion of a low oil temperature because D = Cal. Exp. 1 min 1 2 min 1 3 min 1 4 min D = Cal. Exp. 1 min 1 2 min 1 3 min 1 4 min h, h, h, (a) Pump A (internal ear) D =.4 Cal. Exp. 1 min min 1 3 min 1 4 min D = Cal. Exp. 1 min 1 2 min 1 3 min 1 4 min D = Cal. Exp. 1 min 1 2 min 1 3 min 1 4 min h, h, h,.4.2 D = Cal. Exp. 1 min 1 2 min 1 3 min 1 4 min Fi. 13 Chanes in overall efficiency aainst oil temperature JTEKT ENGINEERING JOURNAL Enlish Edition No. 111E (214) 71

9 the viscous friction to shear oil increased sharply. In addition, even in the rane of hih oil temperature, it did not increase because the friction torque dependent on the pressure differential sinificantly increased. The overall efficiency obtained as the product of the volumetric and the mechanical efficiencies had a ood condition in the vicinity of 8, and increasin or decreasin oil temperature from the vicinity worsened the overall efficiency. Also operatin the pumps at a hiher speed could maintain a hih overall efficiency for an increase in oil temperature. a : pump constant independent of pump-operatin condition (m 3 ) b : pump constant independent of pump-operatin condition (m 3 /Pa=m 3 /N) c : pump constant independent of pump-operatin condition (m 2 ) d : pump constant independent of pump-operatin condition (m 2 /Pa=m 4 /N) fo : pump constant independent of pump operatin condition ( ) : pump constant independent of pump operatin condition ( ) h : pump constant independent of pump operatin condition (Pa s) h : pump constant independent of pump operatin condition ( ) : pump speed (r/min) : delivery pressure (MPa) : suction pressure (MPa) : actual pump flow rate (m 3 /s) : ideal pump flow rate (= /6) (m 3 /s) : drivin torque of pump (Nm) : friction torque independent of D and (Nm) : ideal torque of pump (= D) (Nm) : theoretical pump displacement per radian (cm 3 /rev.) a : pump constant independent of pump operatin condition ( ) D : pressure differential across pump (= ) (Pa) D : total leakae flow (= th ) (m 3 /s) D : total friction torque (= th ) (Nm) : overall efficiency (= m + v ) ( ) m : mechanical efficiency (= th /) ( ) v : volumetric efficiency (=( th D)/ th ) ( ) h : oil temperature () l : viscosity of oil (Pa s) x : anular velocity of pump (=2p/6) (rad/s) x : pump constant independent of pump operatin condition (rad/s) q : density of oil (k/m 3 ), min 1 D = = h,, min = =.7 2 (a) Pump A (internal ear) h, D =, min h, D = D = D = =.8 D = , min =.75, min , min h, 1 5 = h, h, Fi. 14 Overall efficiency map in test pumps 72 JTEKT ENGINEERING JOURNAL Enlish Edition No. 111E (214)

10 References 1) Kanada, T., Iijima, Y., Yasue, H. and Takahashi, T.: Toyota's New Generation Super ECT (U-14E) Four- Speed Automatic Transaxle, SAE Paper (1999). 2) Ozeki, T. and Umeyama, M.: Development of Toyota's Transaxle for Mini-van Hybrid Vehicles, SAE Paper (22). 3) Shimizu, K., Waki, H., Saito, T., Sawayama, M., Nishiyama, H., Kuroda, S. and Oohori, T.: Development of a New-Generation CVT with Medium Torque Capacity for Front-Drive Cars, SAE Paper (26). 4) Ide, T.: Effect of Belt Loss and Oil Pump Loss on the Fuel Economy of a Vehicle with a Metal V-Belt CVT, Proceedins of Seoul 2 FISITA World Automotive Conress, paper no. FA2A131 (2) ) Kluer, M., A., Fussner D., R. and Roethler, B.: A Performance Comparison of Various Automatic Transmission Pumpin Systems, SAE Paper (1996). 6) Wilson, W., E.: Rotary-Pump Theory, Trans. ASME, vol. 68, no. 4 (1946) ) Schlösser, W., M., J.: Ein mathematisches Modell für Verdränerpumpen und -motoren, Oelhydraulik und pneumatik, vol. 5, no. 4 (1961) ) Thoma, J., U.: Mathematische Modelle und die effective Leistun hydrostatischer Maschinen und Getriebe, ölhydraulik und pneumatic, vol. 14, no. 6 (197) ) Hibi, A. and Ichikawa, T.: Mathematical Model of the Torque Characteristics for Hydraulic Motors, Bull. JSME, vol. 2, no. 143 (1977) ) Inauma, Y.: Oil temperature influence on friction torque characteristics in hydraulic pumps, Proc. Inst Mech En, Part C: J. Mechanical Enineerin Science, vol. 226, no. 9 (212) ) Inauma, Y.: A practical approach for analysis of leakae flow characteristics in hydraulic pumps, Proc. Inst Mech En, Part C: J. Mechanical Enineerin Science, vol. 227, no. 5 (213) ) Inauma, Y. and Yoshida, N.: Mathematical Analysis of Influence of Oil Temperature on Efficiencies in Hydraulic Pumps for Automatic transmissions, SAE Paper (213). N. YOSHIDA * Y. INAGUMA ** * Hydraulic System Enineerin Dept., Automotive Systems Business Headquarters ** Hydraulic System Enineerin Dept., Automotive Systems Business Headquarters, Doctor of Enineerin JTEKT ENGINEERING JOURNAL Enlish Edition No. 111E (214) 73