TECHNICAL REPORT Calculation of Teoretical Torque and Displacement in an Internal Gear Pump Y. INAGUMA Tis paper describes numerical determination of teoretical torque (ideal torque) and teoretical stroke volume (pump displacement) in an internal gear pump witout crescent. Te design analyzed as been commonly used for an automatic transmission and continuously variable transmission in veicles because of its ig mecanical efficiency. For estimating te pump efficiencies accurately, determining te accurate teoretical torque and teoretical displacement of te pump is important. Tis paper presents a calculation metod tat takes into consideration te contact s for mesing and sealing between te driving and driven gear for te accurate teoretical torque and displacement of an internal gear pump witout crescent suc as a gerotor pump. Key Words: ydraulic power system, ydraulic pump, internal gear pump, teoretical torque, teoretical displacement, calculation metod. Introduction Internal gear pumps used in relatively low pressure conditions ave been widely used for automatic transmissions and engines to supply oil for bot lubrication and ydraulic control. Tey are used because of teir simple structure wit fewer components and easy assembly tan oter options. Currently, tey ave been used in CVT (Continuous Variable Transmission) wit operating pressures over MPa ), ). In te meantime, to improve fuel economy in veicles, auxiliary components including ydraulic pumps are required to increase teir efficiencies. Currently a conventional internal gear pump as a crescent placed between te tips of te driving and driven gears to separate te delivery pressure side and te suction pressure side and to prevent oil leakage from te delivery pressure side. Now tis as been replaced wit an internal gear pump witout crescent. In tis design te sealing is performed by te toot tips of te driving and driven gears. For improving te efficiencies of te pump, it is important to determine accurately te teoretical torque and te displacement volume of te pump. Altoug for te internal gear pump wit crescent including an involute gear pump te calculation metod for teoretical torque and displacement as already been proposed ), it as yet to be clarified for te internal gear pump witout crescent. Tis paper presents te calculation metod for te teoretical torque and te teoretical displacement for internal gear pump witout crescent, by using a similar calculation metod to a balanced vane pump, wic as been previously reported 4).. Nomenclature : Gear widt mesing of driven gear : Force received from driving gear on mesing of driven gear sealing of driven gear mesing of driving gear : Force received from driven gear on mesing of driving gear sealing of driving gear : Pump speed : Delivery pressure : Suction pressure : Difference between delivery and suction pressure (= ) : Radius from center of driven gear to mesing : Radius from center of driven gear to sealing : Radius from center of driving gear to mesing : Radius from center of driving gear to sealing : Teoretical torque : Teoretical displacement per revolution : Number of teet on driving gear 7 JTEKT Engineering Journal Englis Edition No. E ()
u : Pressure angle of toot at mesing : Rotational angle of driving gear Subscript: : Component in radial direction : Component in circumferential direction. Structure of te Pump First, te internal gear pump witout crescent is described below. Te pump is composed of a driving gear (an external toot gear placed inside) and a driven gear (an internal toot gear placed outside). Various toot profiles are in use, among wic te trocoidal profile and te combined profile wit ypocycloidal curves sown in Fig. are well known. Unlike te pump wit crescent to prevent leakage between te tips of te driving and driven gears, sown in Fig., tese pumps witout crescent are designed so te toot tip parts of bot gears come in contact at one to separate suction part from delivery part (in practice a very small clearance is provided to avoid locking of te gears). In contrast to te internal gear pump wit crescent, te internal gear pump witout crescent as a larger volume between te teet of te driving and driven gears. Tis means tat te pump witout crescent can ave a larger teoretical displacement wen te toot widt, te tip diameter of te driving gear and te outer diameter of te driven gear are te same as tose of te pump wit crescent. In oter words, wen te toot widt and te tip diameter of te driving gear are te same, te outer diameter of te driven gear in te pump witout crescent can be smaller wit te same displacement. Especially at ig-speed pump operation, it can decrease te friction torque caused by searing viscous oil on te sliding parts of te gears, improving te mecanical efficiency. Tat is te reason wy tis type of gear pump as often been used in te automotive automatic transmissions, including CVTs. In te gear pump, te mesing between te driving and driven gears canges depending on te rotating position (angle) of te driving gear. As sown in Fig., te teet of bot te driving gear and te driven gear are subject to te suction pressure and/or te delivery pressure. Te toot faces mesing wit eac oter receive different pressures separated at te mesing. Also, te toot faces concerned wit sealing receive different pressures separated at te sealing. In addition, for te gear pump witout crescent te sealing is not always on te tip of te toot, as sown in Fig. 4. It canges depending on te rotational angle of te driving gear. Tis difference from te internal gear pump wit crescent makes te accurate calculation of te teoretical torque and te displacement difficult. In te gear pump, te mesing s of te driving gear and te driven gear transfer to te next teet during rotation. Te locus of te mesing as well as te sealing canges in te radial direction. Driving gear Driving gear Fig. Configuration of internal gear pump witout crescent Crescent Driving gear Fig. Configuration of internal gear pump wit crescent O' Mesing O Fig. 4 Tip circle radius and seal radius of driven gear O' o O : Center of driving gear O': Center of driven gear Fig. Pressure acting in internal gear pump JTEKT Engineering Journal Englis Edition No. E () 7
4. Calculation of Teoretical Torque and Displacement 4. Calculation Metod Tere are two calculation metods for te teoretical displacement. One is to calculate geometrically te volume of te space enclosed by te teet of te driving and driven gears. Te oter metod first requires calculation of te teoretical torque of te pump. Ten te displacement of te pump sould be calculated by using te teoretical torque. Tis is based on te principle tat te ydraulic energy given to te oil by te pump operation equals te power required to drive te pump saft, disregarding energy loss. Te former approac used to ave a problem wit accuracy because of te integral of te volume between te teet of te driving and driven gears. In recent years, tanks to te spread of CAD, te calculation as become relatively easier. Neverteless, it still requires drawings of te driving and driven gears for eac minute cange of rotational angle. To avoid tis trouble wit accurate calculation wen calculating te teoretical torque, te latter metod was selected and described below. In tis calculation metod, te torque to rotate te driving gear is considered against te difference between te suction pressure and te delivery pressure. Tese pressures act on te toot face in te direction perpendicular to it. However, for te calculation of te teoretical torque, only te pressure acting in te circumferential direction around te center of rotation must be considered. For te teet witout te mesing or te sealing, te force due to te pressure does not need to be considered. As sown in Fig., te force due to te pressure acts on bot rigt and left faces of te toot evenly. In te case of te internal gear pump, it is necessary only to consider te force caused by te pressure difference on te teet wit te mesing and te sealing. First te force and te torque working on te driven gear are considered. Te relationsip between te pressure and te force working on te driven gear is sown in Fig.. Bot te suction pressure and te delivery pressure act on some of te teet of te driven gear. Te difference between and is denoted as. Te force due to te pressure acts on te toot wit te mesing and te force due to te pressure acts on te sealing in te vicinity of te toot tip in te driven gear. Ten, te driven gear receives te reaction force and te circumferential component against te sum of and are at te mesing from te driving gear. In te actual pump, te teoretical torque to drive te pump depends only on te component of te force in te circumferential direction (). Te values of and are known, and tey can substitute for eac equivalent concentrated force as expressed below: Te radii and from te center of te driven gear O' to te s acting and are expressed by te following equations, respectively. X f f f f 4 f f f7 f 8 f 9 Toot bottom Toot face Toot tips of driven gear Y Y f f f f 4 f f f7 f 8 f 9 X Fig. Pressure and force acting on toot Acting of force due to pressure on te seal side O' Mesing : Tip circle radius Acting of force due to pressure on te mesing side u O': Center of driven gear Fig. Forces acting on driven gear o (u: Pressure angle of toot) 7 JTEKT Engineering Journal Englis Edition No. E ()
In te same way, only te component of direction sould be considered for te reaction force from te driving gear. Denoting tat te pressure angle of te toot face at te mesing is u, cosu means. Disregarding friction and oter loss, te moment equilibrium equation around te center of te driven gear O' is expressed by te following equation from tese tree forces and te radii. Te equation can be canged to te following equation to determine : By substituting equations troug in equation : Next, te force and te torque to rotate te driving gear are considered, as sown in Fig. 7. Just like te teet of te driven gear, te forces due to te pressure and act on te teet wit te mesing and te sealing in te driving gear respectively loaded. Also, te reaction force acts against te aforementioned force on te mesing of te driven gear and te driving gear. Only te components of force in te direction are actually producing te torque to drive te pump. Te forces generated by te ydraulic pressure and are te known values. Also, as in te case of te driven gear, tey can replace te equivalent concentrated forces as expressed below: Te radii and from te center of te driving gear O to te s acting and are expressed as follows. Te force acting on te mesing from te driven gear as te same magnitude as te aforementioned force, toug te direction is opposite. Terefore, te direction component of can be expressed as follows: Disregarding te friction and oter losses, te torque to rotate te driving gear against te moments due to and is defined as te teoretical torque. Tis torque can be obtained from te moment around te center of te driving gear O, as expressed by te following equation. Substituting equations troug in equation, In te above equations, te values of and are fixed. Because te coordinates of te mesing and te sealing vary according to te rotational angle of te driving gear,,,,,,,, as well as,,,,, are functions of. However, tey are not expressed as te functions of for simplicity of expression. Wile equation defines te teoretical torque at eac angle, te averaged teoretical torque * can be calculated by te following equation. O Center of driving gear Fig. 7 Forces acting on driving gear Mesing o p p Following te calculation of te teoretical torque, te calculation of te teoretical displacement volume can be performed. Here, te teoretical displacement is related to te teoretical torque as follows. Te amount of te work required to drive te pump for one rotation is * p wen te averaged teoretical torque is *. Consequently, because te pressure energy of is given to te oil of volume *, te energy given to te oil is *. Disregarding te loss, p * and * must be equal. Ten, te following equation is obtained ). p JTEKT Engineering Journal Englis Edition No. E () 7
Equation defines te relationsip between te averaged teoretical torque and te teoretical displacement in one revolution. Te oil volume delivered from te pump troug rotation of (radians) from te rotational angle, (), could be obtained by substituting te averaged teoretical torque * in equation for (). Tus, te momentary fluctuation of teoretical displacement would be obtained by equation Wit = p/ in equation, te fluctuation of te delivered oil per degree can be calculated. For te internal gear pump wit crescent sown in Fig., te sealing always exists on te tip circle radius, and te teoretical torque is expressed as follows: Equation replaces in te first term and in te second term of te rigt side of te equation wit and respectively. Since > and < stand in te pump, te value of te teoretical torque obtained from equation is larger tan tat obtained from equation, and ence te teoretical displacement becomes larger too. Table Dimensions of pump 4 7 O' 8 7 4 4 4 Mesing 7 O O' O 8 7 = Mesing O : Center of driving gear O' : Center of driven gear 4 4 7 Mesing 8 7 O' O 9 8 = Fig. 8 Transfer of contact s = 4. Calculation Results Te calculation results based on te assumption of = MPa will now be explained. Table sows te dimensions of te test pump used for te calculation and te experiment. In tis internal gear pump, te driving gear rotates for one toot pitc, wile te teet involved in mesing and sealing in te driving and driven gears cange in tree patterns. Tese patterns are sown in Fig. 8, were te gears rotate clockwise, and eac toot of bot te gears is numbered. Initially, No. driving gear toot meses wit No. driven gear toot, wile No. driving gear toot and No. 7 driven gear toot ave te sealing. Wen te rotational angle of te driving gear exceeds, te mesing teet of bot te driving and te driven gears transfer to No., wereas te sealing is still formed by No. driving gear toot and No. 7 driven gear toot. Wen te rotation progress furter, te teet wit te sealing of te driving and driven gears cange to te next teet. Table summarizes tese results. Table Pattern of number of teet wit contact p Depending on te transfer of te mesing and sealing teet, bot te forces acting on te teet and te moment arms cange significantly. Figure 9 sows canges of te magnitudes of te forces, and acting on te driven gear teet. Figure sows te canges of te magnitudes of te forces and acting on te driving gear teet. At te rotating angle =, te forces and as well as ( is te same as ) cange suddenly because te mesing moves from te toot bottom to te next toot tip of te driving gear (or from te toot tip to te toot bottom of te driven gear) wit te transfer of te mesing teet. Te increase and decrease in te forces and of te driven gear are opposite to te of te driving gear. 74 JTEKT Engineering Journal Englis Edition No. E ()
Tis is a typical feature of te gear pumps including te external gear pump. On bot te driving and driven gears, te forces and acting on te teet wit te sealing, are relatively small but tey cannot be negligible. (), cm /deg..4... Average, N, N 8 4 4 8 4 8, deg Fig. 9 Cange of forces acting on driven gear 8 4 4 8 4 8, deg Fig. Cange of forces acting on driving gear Te teoretical torque to drive te internal gear pump is te sum of te driving torque for te driven gear (te first term of rigt side in equation ) and te driving gear (te second term of rigt side in equation ). Te calculated results of, and te teoretical torque ( in Fig. ) to drive te pump are sown in Fig.. Altoug and vary greatly along te rotational angle of te driving gear, tey cancel out te fluctuations of eac oter, and te total of tem as as little fluctuation. Also, from tese results, te fluctuation of te teoretical delivery flow rate remains very small, as sown in Fig.. 4 8 4 8 Rotational angle of driving gear, deg. Fig. Cange of pump displacement per degree As te calculated results for te test pump, te values of te averaged teoretical torque at = MPa and te teoretical displacement were 4.9 Nm and.4 cm /rev., respectively. Incidentally, te difference between te calculated result by equation and tat by equation amounted to about %.. Experimental Measurement of Displacement. Experiment Equipment and Measuring Metod Wit te experimental apparatus sown in Fig., te actual value of te displacement for te test pump wit te dimensions given in Table was measured. Te simple experimental apparatus consists of te test pump driven by a DC motor via pulleys and belt. Te delivery pressure was measured wit a Bourdon's tube pressure gauge equipped at te pump outlet. Te delivery flow from te test pump was measured wit a flow meter and oil temperature was measured at te pump outlet wit a termistor termometer. Te oil used was commercial mineral oil and its density and viscosity at 8 were 8 kg/m and.8 Pa s, respectively. Te test pump was operated at various pump speeds, and delivery pressures, and te volumetric flow rate delivered from te test pump was measured. 4 = + (Teoretical pump driving torque) (Driving torque for driven gear), Nm Cange of mesing teet (Driving torque for driving gear) 4 8 4 8, deg. Test pump DC motor Torque meter Trottle valve Fig. Test circuit Fig. Torques for driving eac gear and pump driving torque JTEKT Engineering Journal Englis Edition No. E () 7
Tis time, te teoretical displacement was calculated in te following procedure. Te volumetric flow rate was measured at various delivery pressures for tree pump speeds. From te results measured at various for eac, te delivery flow rate at = was ten extrapolated. Dividing by te pump speed, te pump displacement was obtained.. Experimental Results Te measured delivery flow rates for tree pump speeds wit canging delivery pressure are plotted in Fig. 4. In Fig. te values of te pump displacement are sown. Tey were obtained by dividing te delivery flow rate at = by for eac. Te experimental result of agrees well wit te calculated one., L/min Condition: Oil temperature 8 = min =8min =min. Conclusions In tis study, te teoretical torque and te teoretical displacement of te internal gear pump witout crescent were investigated. As a result, te following results were obtained: Te calculation metod for te teoretical displacement of te pump as been constructed, based on calculation of te teoretical pump driving torque from te forces acting on te gears. Te teoretical displacement value obtained by using tis calculation metod agrees well wit te value obtained by te experiment. References ) N. Emoto and A. Ito: Toyoda Koki Gio (Toyoda Koki Tecnical Review), 4, () 4. ) S. Takita: Toyooki Gio (Toyooki Engineering Review), (). ) T. Icikawa: Haguruma Ponpu, Nikkan Kogyo Sinbunsa (9). 4) Y. Inaguma: Toyoda Koki Gio (Toyoda Koki Tecnical Review),, (99). ) T. Icikawa and A. Hibi: Yuatsu Kogaku, Asakura-soten (979) 9....., MPa Fig. 4 Real flow of pump Condition: Oil temperature 8 / (= ) cm /rev 4.4 Fig., min Vt estimated from measured flow Y. INAGUMA * * Engineering Department, Driveline System Operations Headquarters, PD 7 JTEKT Engineering Journal Englis Edition No. E ()