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Formulation and optimization of involute spur gear in external gear pump
Mechanism and Machine Theory 117 (2017) 114–132
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Research paper
Formulation and optimization of involute spur gear in external gear pump Xinran Zhao∗, Andrea Vacca School of Mechanical Engineering, Purdue University, Lafayette, IN 47905 USA
a r t i c l e
i n f o
Article history: Received 30 November 2016 Revised 12 May 2017 Accepted 27 June 2017
a b s t r a c t External gear pumps (EGPs) have gained popularity among applications in many fields like fluid power transmissions and systems, automotive, aerospace thanks to their advantage of simplicity, robustness and low cost. Several studies were performed to analyze and innovate the gear profiles of EGPs to achieve better performance, in terms of flow smoothness power to weight ratio. Asymmetric gears represent one of the possible choices. This paper is aimed at developing a methodology of designing asymmetric involute gear, and formulate analytical expression for the instantaneous flowrate and flow non-uniformity given by asymmetric, non-standard involute gear pumps. These analytical expressions are then used within a multi-objective numerical optimization algorithm aimed at minimizing both the flow non-uniformity and the pump size to achieve a specific displacement. The results illustrate the highly constrained nature of the optimization problem, and the relevant impact of certain parameters of the tooth profile. It is shown how gears obtained with the proposed procedure can have significantly higher performance of standard gears, and in particular how the parameters affecting tooth profile asymmetry can further improve the EGP flow irregularity and size. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Among positive displacement machines, external gear pumps (EGPs) are often the preferred solution for many fluid power, injection, lubrication, and fluid transport systems. Key reason for success for EGPs are their low manufacturing cost, compact package, good energy efficiency, high reliability as well tolerance to contamination. Traditionally, the design of the gear profile for an EGP has been taken advantage from the wide literature on gearboxes for power transmission system. However, it is well known that the desirable operating performance of an EGP is in great part given by the fluid dynamics features of the flow through the unit, rather than by the characteristics of the power transmissions between the two matching gears. The force loading of the gears in an EGP essentially given by the fluid pressure, and as documented in [1], for a high pressure unit the radial forces acting on the shaft bearing are significantly higher than the contact forces between the gears. As a consequence, structural requirements typically affecting the gear profile in mechanical power transmissions systems do not always apply in EGPs; and gears with significant modifications are often found in commercial EGP designs. On the other hand, features of the flow as well as fluid compressibility effects often determine the design of an EGP.
∗
Corresponding author. E-mail address: [email protected] (X. Zhao).
http://dx.doi.org/10.1016/j.mechmachtheory.2017.06.020 0094-114X/© 2017 Elsevier Ltd. All rights reserved.
X. Zhao, A. Vacca / Mechanism and Machine Theory 117 (2017) 114–132
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Nomenclature EGP TW CR DC V1 V2 ra rb ruc m
α α rp x z b
ω γ u Q i
θ φ T
ρ e
External Gear Pump Tip width constraint (-) Contact Ratio (-) Displacement Chamber volume of DC1 (mm3 ) volume of DC2 (mm3 ) outer radius of gear (mm) radius of base circle (mm) radius of undercut circle (mm) module of gear (mm) pressure angle of cutting tool (o ) working pressure angle of gear (o ) radius of pitch circle (mm) correction (offset) factor (-) number of teeth (-) whole thickness of gear (mm) shaft speed (rad/s) base pitch (mm) distance from contact point to pitch point (mm) flow rate (mm3 /s) center distance (mm) angular position of gear (o ) angle between inward tangential vector and the vector pointing to the center (o ) Period of theoretical delivery flow (s) Fillet radius of rack cutter (mm) Offset distance (mm)
Flow non-uniformity at the outlet port is considered as one of the most important detrimental features for all positive displacement pump designs. As a matter of fact, low level of outlet flow fluctuation leads to low levels of noise emissions and reduced chances of mechanical vibration in the downstream systems [2,3]. Due to the high number of displacement chambers, the EGP design is particularly prone to provide a low level of outlet flow fluctuation. For this reason, over the last decades, significant research effort was put in studying the relations between gear profile and kinematic flow pulsations as well as in formulating design solutions for lower flow pulsation. In this regard, one of the first relevant contributions known by the authors is the work by Bonacini [4], who provided an analytical expression for the theoretical flow of an EGP, based on the parameter of the involute profile of the gears. The same finding, with alternative analytical methods of derivations, was also reported in the work by Ivantysyn and Ivantysynova [3], and Manring and Kasagaradda [5]. In both works, some considerations about some relevant parameters, such as number of teeth, involute profile pressure angle and correction factor for optimal flow pulsations were also reported. In a recent work [6], Zhou and Vacca proved the equivalency of the analytical expression provided by mentioned works with a numerical control-volume based approach suitable for lumped parameter modeling of the displacing action of an EGP. Several state-of-the-art commercial EGPs have taken advantage of the considerations of the works above mentioned. A significant example is given by the dual flank solution, as described in [7], which consists of gears able to reduce backlash to a minimum value. The feasibility of such solution is proven by several commercial designs available in the market. In other cases, flow non-uniformity is also reduced by adopting helical gear designs as well as unconventional gear profiles [8–11]. The latter solutions, however, penalize some typical advantages of traditional spur gear EGPs such as volumetric efficiency, simplicity and design scalability to different geometric displacement. For this reason, spur gear EGPs with involute type gear still represent the most desirable and cost-effective solution for an EGP. Nevertheless, the literature for spur gear EGPs does not include a general design methodology for formulating optimal profiles for gears obtained through traditional rack cutting processes. This is particularly true for the case of asymmetric involute teeth, which also appears in some commercial solutions [12–15]. With the additional degrees of freedom given by the separate definition of the drive and coast sides of the involute teeth, asymmetric gears could provide additional potentials for reducing the level of flow non-uniformity in an EGP. This is a particular aspect that will be investigated in the present paper. It is important to remark that, similarly to what assumed in the past works mentioned above, all the considerations made in this work are based on theoretical, or kinematic, flow displaced by the EGP. It is well known that actual port flow oscillations are affected also by compressibility effects associated to the pressurization and depressurization of each tooth space volume (TSV), and secondarily on the laminar/turbulent nature of the flow. Several studies focused on the analysis of
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Fig. 1. External gear pump. Lateral and radial gap affecting the energy efficiency.
Fig. 2. Two different configurations in the meshing region (a) single-flank (b) dual-flank contact [6].
the transient pressurization phases of the inter-teeth volumes during the meshing process, and to their reduction through proper design of the recessed machined at the element of the EGPs facing the lateral sides of the gears. Some examples are given by [16–18]. Other studies also addressed the pressurization phases of each TSV, as a function of the actual clearance between tooth tip and internal pump case (radial gap, Fig. 1) [1,5,19]. The leakage flows taking place at both the radial and the lateral gaps in an EGP represent an additional cause of deviation between the actual flow and the theoretical flow. This latter aspect is often studied through CFD approaches, such as in the work by Castilla et al. [20]. Despite all the reasons explained above about the discrepancy between actual and theoretical flow in an EGP, an analysis purely based on theoretical flow still provides the upper limit performance, in terms of minimum flow oscillations, of a given gear profile. If a gear profile is then selected for actual implementation on an EGP, particular care should then be dedicated to all design parameters that plays a role in reducing the actual pump performance with respect to the theoretical one. These parameters, include those affecting radial and lateral leakages flow [1,21], TSV pressure peaks and localized cavitation during the meshing process [1,22], and also the manufacturing tolerances on the actual tooth profile. Optimization techniques such as the one described in [23] were proposed to limit these effects related to fluid properties on the outlet flow oscillations. According to the general idea illustrated in the previous paragraph, this paper describes a procedure for determining the optimal tooth profile for EGPs, spur gear type, including the case of asymmetric teeth. To accomplish this goal, the work first introduces the method used to analytically describe the profile of asymmetric teeth, obtained through a standard rack cut process (Section 2.1). Subsequently, an analytical derivation of the theoretical flow for asymmetric EGPs is provided, considering the drive gear and the pinion with same profile and equal number of teeth. This part extends the findings of the previous work [6], which was limited to the case of symmetric gears (Section 2.3). The main goal of investigating the optimal tooth profile geometry is then accomplished through a numerical optimization procedure, which is detailed in Section 3. This procedure assumes dual-flank operation for the gears, which is already proven to be intrinsically more beneficial for reducing flow non-uniformity up to 75% respect to traditional single-flank technology [6], as shown in Fig. 2. Beside the minimization of flow oscillation, objective of the optimization is also to provide the most compact EGP design in terms of overall volume of the pumping elements (the gear ensemble). An important assumption made throughout this work is given by assuming the driver and the driver gear with equal number of teeth. This assumption reflects the common practice for gear pumps, particularly for high pressure operation. The last section of the paper illustrates the potential of the procedure, showing the effect of the main design parameters of the tooth profile, such as the number of teeth, on the theoretical pump performance.
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Fig. 3. Geometry of a rack-cutter for an asymmetric involute gear.
Fig. 4. Motion of a rack-cutter and decomposition into two sub-steps.
2. Design and theoretical flow rate of asymmetric gears under dual-flank operation The kinematic (or theoretical) flow rate of a positive displacement machine consists on the theoretical fluid displaced by the volume variations of its displacement chambers, under the assumptions of no leakages, negligible fluid compressibility and inviscid flow. This paper aims at finding the best gear profile that can reduce the kinematic flow while minimizing the size of the gears. For this reason, gears working at dual-flank contact (zero-backlash) are considered. For the sake of the generality of the study, non-standard involute gear design is considered, introducing several degrees of freedom in order to include the case of asymmetric teeth, also with profile shift factor. The case of standard gears will result as a particular case of the considered domain of investigation. The definition of the asymmetric profile-shifted for non-standard gear design is provided in Section 2.1, where the design starts from the asymmetric design of the generating rack-cutter. Analytical expressions defining the asymmetric gear profile depending on several design parameters are here derived. The calculation of the center distance for dual-flank operation is then given in Section 2.2. Then Section 2.3 derives the analytical expression of theoretical flow rate for asymmetric non-standard spur gears – never found in the literature known by the authors – by following a rotating control-volume based approach. Related considerations regarding the proper contact and sealing of gears operating in dual-flank contact are discussed in Section 2.4.
2.1. Asymmetric non-standard gear design As this paper is aimed at studying the most general type of involute gears generated by rack-cutter, the undercutting that results by the features of the rack-cutter shape has to be taken into account. In general, undercutting reduces the chance of interference (non-conjugate contact), but it also reduces the contact ratio, and mechanically weakens the gear tooth. While teeth strength considerations are out of the scope of this study, the effect on the contact ratio of gears given by the undercutting is taken careful consideration. A rack cutter of the general shape of Fig. 3 is taken as reference for the generation of gears with involute tooth profile. The shape of cutter is asymmetric, to permit the gear generation of asymmetric gears. The fillet curve at the tip of the cutter is formed by a circular arc with radius ρ d for the drive side, and ρ c for the coast side. The generating (cutting) process of a gear can be viewed as the synchronization of a linear translation of the cutter and a rotational motion of the gear. After a proper coordinate transformation, with a frame of reference fixed at the center of the gear, this process can be treated as a composition of two motions of the rack cutter with respect to the gear: translational and rotational (as shown in Fig. 4). Consequently, the profile of an involute gear can be created with the motion of the straight portion of the cutter, which is
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Fig. 5. Parameters definitions for the involute profile.
given by the parametric equation:
finv (θ ) =
x y
= rb
cos φ sin φ
− sin φ cos φ
sin θ − θ cos θ cos θ + θ sin θ
(1)
where φ is a constant given by:
φ = tan α − α +
π + 4x tan α 2z
(2)
In Eq. (2) α is the geometrical design pressure angle on either drive (α d ) or coast (α c ) side. θ is a parameter whose physical meaning is shown in Fig. 5. For the gear tooth, the involute profile is defined in the interval
θ ∈ 0, ra2 − rb2 /rb
(3)
where the outer radius ra and base radius rb can be expressed as:
ra =
m·z + m · x + ha 2
(4)
rb =
m·z · cos α 2
(5)
The lower limit and upper limit in Eq. (3) stand for the intersection of the involute curve with base circle and addendum circle, respectively (Fig. 5). After defining the involute portion, the next part consists in the definition of the root fillet profile of the tooth. This is assumed to be formed by the circular arc portion of a cutter. Using an approach similar to what proposed by [24], complex analytic function is used to represent the cutter motion on complex plane
w = [ν + (r − e )ϕ + i · (ζ (ν ) + r )] · exp(i · ϕ )
(6)
where e = x · m is the offset distance of the cutter, and r = mz/2 + e is the distance from the pitch line (y = 0 in Fig. 1) to the center of the gear, ν is the horizontal coordinate in Fig. 3, and ζ (ν ) is the circular arc profile in the vertical direction in Fig. 3. The second term in the square brackets of Eq. (6) represents for the translational motion, while the third term is the initial positioning of the gear. The exponential multiplier of Eq. (6) stands for the rotational motion. Notice that when a non-zero correction factor x is applied, the translational velocity does not change; in fact, the only modified term is the positioning of the cutter with respect to the gear. The term ϕ indicates the rotation angle, which can be expressed as a function of ν , i.e. ϕ = ϕ (ν ), by solving following differential equation:
∂ ∂ (Im(w(ν, ϕ ))) (Im(w(ν, ϕ ))) ∂ϕ ∂ = u ∂ ∂ (Re(w(ν, ϕ ))) (Re(w(ν, ϕ ))) ∂ϕ ∂u
(7)
The physical meaning of Eq. (7) is that a point on the cutter can actually cut the profile only when the local slope of its trajectory of motion is the same as the local slope on the cutter, as shown in Fig. 6. The solution can be derived from Eqs. (6) and (7):
ϕ=
−ζ (ζ + e ) − ν r−e
(8)
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Fig. 6. Interpretation of the physical meaning of Eq. (7).
Substituting Eq. (8) in the former expression of Eq. (6) gives w = w(ν ), which represents the root fillet profile created by the cutter on complex plane:
w(ν ) = [−ζ (ζ + e ) + i · (ζ + r )] · exp i · −
ζ (ζ + e ) + ν
(9)
r−e
Taking the root fillet profile at the drive side as an example (the coast side can be solved in a similar fashion), on cutter coordinate (Fig. 3), the fillet arc on the drive side of the cutter is written as
ζ ( ν ) = y0 −
ρd2 − (ν − x0 )2
(10)
and consequently its derivative with respect to ν is:
ζ (ν ) =
ν − x0
(11)
ρd2 − (ν − x0 )2
where (x0 , y0 ) is the center of the drive circular fillet arc in cutter coordinate, which can be determined by simple geometric calculation:
x0 =
h b − ρb m·π ρd + + 4 cos αd tan π2 − αd
(12.1)
y0 = −hb + ρb
(12.2)
with the range of ν :
3 ν ∈ ρd cos(π + αd ) + x0 , ρd cos π + x0
(12.3)
2
Sub Eqs. (10) and (11) into Eq. (9), the closed form of the root fillet profile on the drive side can be written as
froot (ν ) =
Where:
β (ν ) =
−(ζ + e ) · ζ cos (β ) + ζ + m2·z + e sin β (ζ + e ) · ζ sin (β ) + ζ + m2·z + e cos β
(ζ + e ) · ζ + ν m·z 2
(13)
3 ν ∈ ρd cos (π + αd ) + x0 , ρd cos π + x0 2
(14)
The intersection between involute profile and root fillet profile can be captured by solving for the minimum value of the equation:
g(θ , ν ) ≡ |finv (θ ) − froot (ν )|
(15)
in range of two variables θ and ν in the intervals given by Eqs. (3) and (12.3), respectively. If there is an intersection between the involute profile and the root fillet profile, the minimum of g should be zero. In the special case where g has a minimum greater than zero, that means there is no intersection; this means that the involute profile does not exist for the given set of geometrical parameters and not in the scope of this paper. This situation can happen when the outer radius specified is unrealistically small. This condition will be used as a constraint in the proposed optimization procedure (Section 3.1). The distance from the intersection point to the gear center is denoted as ruc (Fig. 7 left), because it stands for the actual starting point (undercutting point) of the involute portion of the tooth profile. The undercutting radius is greater or equal to the base circle radius, (i.e. ruc ≥ rb ), and it is written as
ruc = |finv (θ ∗ )| = |froot (ν ∗ )|
(16)
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Fig. 7. Real starting point of involute i.e. working profile (left) and root fillet profile given by the analytical expressions compared to a cutting simulation (right).
Fig. 8. Tooth thickness on gear pitch circle.
where θ ∗ and ν ∗ are the solutions corresponding to the minimum value of g(θ , ν ) given by Eq. (15). In the following part of the paper, ruc, d and ruc, c will represent the distance between the start point of the involute profile and the gear center, respectively for the drive and the coast sides.
2.2. Dual-flank interaxis distance To achieve dual-flank (no-backlash) contact, the working center distance for the general case of asymmetric non-standard gears has to be calculated. As it is well known, for x = 0 the working pressure angle is different from the design pressure angle, i.e. a = α . The case of same geometry for the drive gear and driven gear is considered. Thus, as shown in Fig. 6, for the case of corrected tooth profile (x = 0), the pitch circle results to be different from the centrode circle. The radius of centrode circle is r p = m · z/2. This value is defined regardless of x, as the profile correction changes only the position of rack-cutter, without affecting the translational velocity of the cutter. In this case, the pitch circle stands for the circle at which the tooth circumferential width equals to the gap width. On the centrode circle, the width of tooth (or of the gap) is equal to the linear length of the cutter on the centrode line:
t p = mπ /2 + xm(tan αd + tan αc )
(17)
w p = mπ /2 − xm(tan αd + tan αc )
(18)
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Fig. 9. The DC volume curve and the different shapes that each DC assumes at minimum volume, maximum volume, and intermediate volume conditions [6].
Geometrically, the involute function inv(x ) ≡ tan(x ) − x accounts for the angle shift from the centrode circle to the pitch circle (see Fig. 8). In particular, the angular interval on the pitch circle for tooth and gap is given by:
t p tp = − inv(α c ) − inv(αc ) − inv(α d ) − inv(αd ) r p rp =
π + 2x(tan αd + tan αc ) z
w p wp = + inv(α c ) − inv(αc ) + inv(α d ) − inv(αd ) r p rp =
π − 2x(tan αd + tan αc ) z
Equating t p /r p to wp /r p gives
inv(αd ) + inv(αc ) =
2x z
− inv(α c ) − inv(αc ) − inv(α d ) − inv(αd )
+ inv(α c ) − inv(αc ) + inv(α d ) − inv(αd )
(19)
(20)
+ 1 (tan αd + tan αc ) − αc − αd
(21.1)
It can also be observed that the pitch circle radius is the same for both drive and coast sides, therefore:
cos α d cos αd = cos α c cos αc
(21.2)
These two equations with respect to two variables αd and αc can be solved by numerical root-finding method such as Newton Iteration. The solution of Eqs. (21.1) and (21.2) can be used to calculate the pitch circle, which corresponds to half of the dual-flank interaxis distance:
r p =
1 m · z · cos αd i = 2 df 2 cos α d
(22)
2.3. Theoretical flow rate The kinematic flow rate can be determined by first introducing the concept of a displacement chamber (DC) for an EGP. A DC is a moving control volume, which relates to the tooth space between adjacent teeth of a gear, and rotates as the gears rotate. In an EGP, there are 2z DCs, as shown in Fig. 9. With the meshing process realized by the gears, each DC first reduces then increases its volume, realizing the displacement of the fluid from the inlet to the outlet. Considering the inlet and outlet environments, the total internal fluid volume bounded by the gears and the housing of can be divided into 2z + 2. For every angular position of the gears, the sum of their volume is always equal to a constant value. The theoretical (kinematic) flow rate for an EGP can be formulated based on DC volume conservations, with the algorithm already presented by the authors in [6]. Overall, a formula expressing the theoretical flow rate for the case of symmetric gears was already found by several authors [3,5,6,25]. Those expressions apply also to the case of asymmetric gears, but only if they operate in single-flank contact conditions. However, a formula that expresses the theoretical flow rate in case of asymmetric dual-flank gear with profile correction is not available literature. For this reason, in the following part of this section this particular case is studied using a method similar to differential volume approach used in [6]. For an EGP, unless different timing strategies are purposely introduced (like for the case of variable timing in a EGP introduced by the authors in [26]), it is reasonable to assume that an increase of each DC volume always contributes to
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Fig. 10. Differential volume of a DC at dual-flank contact (a); distance of two contact points with the gear centers, denoted by l1 , l2 , l3 and l4 , respectively.
fluid delivery, and similarly a decrease of each DC volume yields to suction of fluid from the inlet. This implies that only the DCs with an increasing volume contribute to the fluid delivery. According to Zhao and Vacca [6], with this assumption the instantaneous flow rate is given by
Qout = −
dV = b · ω · ra2 − r 2p − u2 dt
(23)
Eq. (23) is valid for all cases, including dual-flank or single-flank, symmetric gear or asymmetric gear. The value of u is the distance from the pitch point to the contact point on the low-pressure side which is delimiting the boundary of all DCs that are decreasing volume. As also observed in [6], u has a definition that depends on the particular gear profile. It is therefore important to find the position corresponding to the minimum DC volume. After the DC reaches that position, it will start increasing its volume, and therefore it will not be counted as part of volumes contributing to the outlet flow. As shown in Fig. 10(a), during an infinitesimal rotation of the gears four sub-differential volumes, from V1 to V4 , can be found with a first order accuracy as
V1 =
1 2 2 (l − rr1 )d θ 2 1
(24.1)
V2 =
1 2 (r − l22 )dθ 2 a2
(24.2)
V3 =
1 2 (r − l42 )dθ 2 a2
(24.3)
V4 =
1 2 2 (l − rr1 )d θ 2 3
(24.4)
Therefore, the differential volume dV can be formulated as:
dV = −V1 + V2 − V3 + V4 1 2 = −l1 + l32 − l22 + l42 dθ 2
(25)
In order to make DC volume V to be the minimum, it must have dV = 0. It yields
−l12 + l32 − l22 + l42 = 0
(26)
Using the geometric identity shown by Fig. 10(b)
l12 + l22 = 2r 2p + 2d12
l32 + l42 = 2r 2p + 2d22
(27)
By combining Eqs. (26) and (27), it yields d1 = d2 . This value can be denoted as d∗ , i.e.
d ∗ ≡ d1 = d2
(28)
This result indicates that a DC in dual-flank contact condition reaches its minimum volume when two contact points confining it have the same distance to the pitch point. It further indicates that, for a contact point moving away from pitch point - either on drive side or coast side of a tooth - once its distance from the pitch point reaches d∗ , the DC will begin to increase its volume and it will not be considered to find u in Eq. (17) any more. At this point, another contact point, on the
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123
Fig. 11. Reference Position (a) and the position with minimum DC2 volume (b).
other line of action (drive or coast), with also has a distance d∗ from the pitch point but moving toward it, will start to give the correct value of u. Therefore, the value of d∗ gives the range of variation of u in Eq. (23), which needs to be determined. For this purpose, a reference position is selected, where the coast contact point coincides with pitch point, as shown in Fig. 11. At this position, two DCs are symmetric with respect to the coast contact point (i.e. pitch point). Being the line of action also symmetric with respect to the pitch point, for a gear with contact ratio greater than one two contact points on drive side of the gears have to exist, and they will have the same distance to the pitch point, which is equal to the half of the base pitch γ d :
γd = m · π · cos αd
(29)
If the time for the reference position of Fig. 11 is denoted as t0 , considering a motion of the contact point on the line of action, the minimum volume position for DC1 can be found by solving the equation d1 (t ) = d2 (t ). With the knowledge on the initial position and the speed of movement of each point on respective line of action:
d1 (t0 ) = γd /2 =
vd =
1 mπ cos ad 2
d2 (t0 ) = 0
1 dd1 = − mz cos αd · ω dt 2
vc =
(30.1)
dd2 1 = mz cos αc · ω dt 2
(30.2)
by solving the problem
d1 (t0 + t ) = d2 (t0 + t )
(31.1)
It gives
t=
π cos αd z · ω cos αd + cos αc
(31.2)
Therefore
d∗ = d1 (t0 + t ) = d2 (t0 + t ) =
π 2
m
cos αd · cos αc cos αd + cos αc
(32)
Notice that d∗ is also the solution of d∗ = d1 (t0 − t ) = −d2 (t0 − t ), which corresponds to the minimum-volume position of DC2. The sign convention is given in Fig. 11 with proper arrows along the lines of action. It is therefore clear that the value of u in Eq. (23) changes periodically from −d∗ to d∗ . Due to the fact that the lower boundary of control volume with decreasing volume is delimited by drive contact point and coast contact point alternatively, for which u(t) has different expression, too. For simplicity, t0 , which is the time instant when coast contact point coincides with pitch point, is set to be 0: This, for the coast side:
uc (t ) =
1 mz cos αc · ω (t − T ) 2
for t ∈ −
d∗
vc
+ T,
d∗
vc
+T
(33.1)
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Fig. 12. Kinematic flow rate calculated for a 12-tooth gear with b = 32 mm, shaft speed of 10 0 0 rpm, ra = 18.5 mm, r p = 16 mm, x = 0, pressure angle of 15° and 45° for coast and drive side of gear, respectively.
While for the drive side:
1 1 ud (t ) = − mπ cos αd + mz cos αd · ω (t − T ) for t ∈ 2 2
d∗
vc
+ T,
d∗
vc
+
2d ∗
vd
+T
(33.2)
where the period is given by
T = 2N d ∗
1
+
vd
1
vc
N∈Z
(33.3)
Combining Eqs. (23), (32) and (33.1), the theoretical flow rate for asymmetric involute spur gear pump working at dualflank condition can be expressed by a piecewise function:
⎧
1 2 ⎪ 2 2 ⎪ b · ω · r − r − mz cos α · ω ( t − T ) c ⎪ a p ⎪ ⎪ 2 ⎪ ⎪ π cos αd π cos αd ⎪ ⎨ for t ∈ − + T, +T zω (cos αd + cos αc ) zω (cos αd +cos αc ) Qout (t ) =
1 2 1 ⎪ 2 2 ⎪ b · ω · ra − r p − − mz cos αd + mz cos αd · ω (t − T ) ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ π cos αd π cos αd + 2π cos αc ⎪ ⎩ for t ∈ + T, +T zω (cos αd + cos αc ) zω (cos αd + cos αc )
(34)
This formula captures the important feature that the theoretical flow ripple given by asymmetric gears is also asymmetric. Fig. 12 shows the theoretical flow given by a reference gear with highly asymmetric gear. The peak-to-peak difference (magnitude of the fluctuation) and mean flow rate are written as:
Qmean = b · ω ·
δ=
bω 4
πm
cos αd ·cos αc (π m cos αd +cos αc )
2
ra2
− r p2 −
cos αd · cos αc cos αd + cos αc
12
(35)
2 (36)
2.4. Contact ratio The starting point of the involute portion of the profile, i.e. the intersection between the root fillet profile and the involute profile, is important for determining the contact ratio. This point is also important for determining the dual flank operation, which is guaranteed by the sealing through a contact point on the coast side. For involute gears, proper contact conditions can be realized only on the involute profile. The contact ratio (CR) can be expressed as the ratio between the length of the line of action with involute-involute contact and the base pitch on drive side γ d given by Eq. (37).
CR = Where
min L1,drive , L2,drive
(37)
γd
L1,drive = 2 r p sin α p −
2 2 ruc − rbd ,d
L2,drive = 2r p cos(π /2 + αd ) +
(2rp cos(π /2 + αd ))2 − 4(rp2 − ra2 )
(38.1) (38.2)
As shown by Fig. 13, L1, drive represents the double of the length of segment OB, and L2, drive represents double of the length of segment OA. Point B is a point on the drive-side of the line of action with distance rbd, real to the gear center. Point B is closer to the pitch point O than to the point C, which is the tangential point between the base circle and the line of action. Point A is the intersection between the addendum circle of the mate gear and the line of action.
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125
Fig. 13. Schematic of the segment on the line of action representing the occurrence of the contact point. Point O is the pitch point. Point A, B, C are the intersection of the line of action with the outer circle of the mating gear, the real base circle, and the base circle, respectively. The actual contact length on the pressure line is the smaller one between OA (i.e. half of L2 ) and OB (i.e. half of L1 ). Table 1 Normalization of variables. ha = ha /m hb = hb /m d = ρd /m ρ ρc = ρc /m
ˆ tip = wtip /m w ra = ra /m = z/2 + ha + x r = r /m = z/2 − h − x r
r
b
rp = r p /m = z/2 αd rp = r p /m = 2z cos cos αd ruc,d = ruc,d /m r uc,c = ruc,c /m rp = r p /m = z/2 δˆ = δ /(m2 bω ) Q˙ mean = Q˙ mean /(m2 bω )
To accomplish dual-flank gear contact, the contact on the coast side is also needed to be considered, which is related to the backlash of the gears. Only when the contact point exists on the coast side to separate the trapped volume in the meshing zone into two disconnected displacement chambers, the condition of dual-flank operation can be satisfied from fluid dynamic point of view. The detailed explanation can be found in [6]. Therefore, in a similar fashion, another contact ratio can be defined for the coast. 3. Optimization algorithm This section formalizes the optimization problem, in terms of design variable, constrains and objective functions, utilized to study the optimal gear profile. All the derived quantities from previous Section 2 are used but in normalized form. Variables with length scale in radial direction and axial direction are normalized by module m and length of the gear b, respectively, while volume is normalized by m2 b, and flowrate is normalized by m2 bω. The normalized variables are shown in Table 1. 3.1. Design variables and constraints The geometry of an involute asymmetric gear can be fully described by combination of 8 non-dimensional design variables, 6 of them are describing the geometry of the cutter, which are shown in Fig. 3. The remaining two are given by the
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X. Zhao, A. Vacca / Mechanism and Machine Theory 117 (2017) 114–132 Table 2 Design variables and the ranges of their variation. Parameter
Range
Number of teeth z [-] Correction factor x [-] Normalized addendum ha [-] Normalized deddendum hb [-]
12–19 −1.0–2.0 0.0–3.0 0.0–3.0
Parameter Drive Coast Drive Coast
Range
pressure angle α d [ ] pressure angle α c [o ] fillet radius ρc [-] d [-] fillet radius ρ o
0–30 0–30 0–0.5 0–0.5
number of teeth, z, and the profile correction factor, x. Among eight design variables, the number of teeth z is an integer, for which objective functions and constraints are not continuous functions. For this reason, and also to better highlight the effect of the number of teeth on the optimization results, optimization algorithm is conducted by studying each number of teeth as separated case. The eight parameters after scaling are given in Table 2 together with their range of variation considered in this study. In order to exclude designs not corresponding to gears that can properly work in dual-flank conditions, 13 constraints in total are considered. As for the scope of this paper, all the constraints we used are purely geometrical, and they do not account for these structural aspects. Constraints (C.1) to (C.5) are related to the cutter geometry, and they ensure that a given combination of parameters forms a closed cutter profile. In particular, constraints (C.1) and (C.2) state that the addendum/deddendum should be large relative to the fillet radius. To simplify the problem, it is assumed that the center of the root fillet arc is in between the top land and the root land of the cutter:
d ha + hb > ρ
(C.1)
c ha + hb > ρ
(C.2)
Constraints (C.3) and (C.4) give the fact that the addendum/deddendum cannot exceed the position of the intersection of two straight lines of the cutter which have slopes equal to the geometric pressure angles:
hb <
π /2 tan(αd ) + tan(αc )
(C.3)
ha <
π /2 tan(αd ) + tan(αc )
(C.4)
Constraint (C.5) expresses the requirement that the tip of the cutter should be flat, i.e. the calculated position of the center for the coast root fillet arc should be on the right, or coincide with that of drive fillet arc (based on the coordinate configuration of Fig. 3):
d c ρ −hb + ρ ρc π + − − ≤ cos(αd ) cos(αc ) 2 tan π2 − αd tan π2 − αc d hb − ρ
(C.5)
Constraint (C.6) states that the addendum radius has to be greater than either the drive base circle radius or the coast base circle radius, so that the involute profile exists for both drive and coast side:
z z z + x + ha > max cos αd , cos αc 2 2 2
(C.6)
For gears operating in dual-flank contact, switching the features (pressure angle and fillet radius) among the drive side or the coast side does not change the performance of the unit in terms of fluid displacement. Therefore, in order to reduce the degree of freedom of design space, the drive pressure angle is arbitrarily assumed to be larger than the coast pressure angle i.e. (C.7). Also, small pressure angle on the drive side generally gives smaller contact ratios, so (C.7) will promote fulfilling the contact ratio constraint. Asymmetric tooth can have different pressure angles α d and α c on drive and coast flanges. It is shown in Eqs. (35) and (36) that exchanging the value of α d and α c will not change the kinematic displacement and fluctuation. However, higher pressure angle on the drive side α d will be favorable for increasing the contact ratio, which makes it easier to fulfill the contact ratio requirement, i.e. to make a design feasible. Therefore, the Constraint (C.7) is used:
αd ≥ αc
(C.7)
(C.8) related to the minimum width of gear tooth tip. Pointed teeth (corresponding to zero width) are normally avoided for reasons that include manufacturing feasibility, proper sealing of the DCs and reduction of the effects of gear wear on the EGP performance. In this work, this constraint is written in non-dimensional form:
ˆ tip ≡ w
wtip ≥ TW m
For a general representation, TW should be greater than 0, i.e. TW > 0.
(C.8)
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Fig. 14. Virtual bounding box for external gear pump used for volume calculation.
The calculation for the contact ratio is given in Section 2, which needs to be greater than unity of ensure enough contact on drive:
CR > 1
(C.9)
Interference between the gears needs is further check as last step. There are two types of interferences that can most commonly happen which can be described by closed equations. The first type of interference is when the tip of one gear is touching the root land of the other mating gear. For this case, the interaxis distance needs to be greater than the sum of addendum and deddendum radii. In general, a root clearance croot is specified such that 2r p ≥ rr + ra + croot . In this work, c root = croot /m ≥ 0 is used. Another type of interference occurs when the center distance is small enough that contact happens out of the working profile. To avoid this under dual-flank contact conditions, the pitch radius should be larger than either ruc, d or ruc, c . Summarizing the two cases, an interference constraint can be written in the form of (C.10)
2rˆP ≥ max rr + ra + c root , 2rˆuc,d , 2rˆuc,c
(C.10)
An additional – and trivial – consideration is that, in order to mesh, two gears needs to have a pitch radius smaller than the addendum radius, i.e. r p < ra , substitute in Eqs. (4) and (22).
z cos αd z < + ha + x 2 cos α d 2
(C.11)
The last constraint (C.12) states that the intersection between involute profile and root fillet profile must exist on both drive and coast sides, as given by Section 2.3. This can be written as:
|g(θ ∗ , ν ∗ )| < · m
(C.12)
where a common value for is 1 is used in the present work. Among 13 constraints mentioned above, (C.1)–(C.7) are explicit constraints, which can be expressed explicitly by design variables, while (C.8)–(C.12) they are implicit constraints, which are based upon intermediate calculation results. × 10−4
3.2. Objective functions In this work, the performance of a given design is measured by the flow non-uniformity, which needs to be minimized in order to obtain an EGP design with low flow pulsation. This can be quantified by the objective function OF1, the ratio between peak-to-peak value in the delivery flow and the mean flow rate, which has to be minimized:
OF1 =
δˆ Qˆmean
cos α · cos α 2 c d π cos αd + cos α =
cos α · cos α 2 1 c d π 4rˆa2 − 4rˆp2 − 3 cos αd + cos αc
(41)
In addition, compactness is of great importance in an EGP, since it relates to the power to weight ratio of the unit. Therefore, it is desirable to minimize the overall size of the pump required to realize a certain target unit displacement. For each set of design parameters, the total volume occupied by the gear set is estimated as the volume of the minimum bounding box for two mating spur gears (Fig. 14):
Vpump = 2ra (2r p + 2ra )b
(42)
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In normalized form:
Vˆpump = Vpump /(m2 b) = 4rˆa (rˆp + rˆa )
(43)
In first approximation, this volume relates to the overall size of the EGP. Obviously, the overall size of the EGP will be higher to include a casing containing the journal bearings of the gears. In this work, the ratio between pump volume and the specific displacement (mean flowrate per angular velocity) is defined as the objective function to be minimized, which stands for the minimum volume required for achieving a certain displacement:
OF2 =
4rˆa (rˆp + rˆa ) Vˆpump = cos αd ·cos αc 2 1 Qˆmean rˆa2 + rˆp2 − 12 π cos α +cos αc
(44)
d
3.3. Results and discussions The optimization algorithm described in the previous sections was implemented and executed with the commercial software modeFRONTIER®. The highly-constrained nature of the problem makes difficult to generate an initial set of design parameters corresponding to feasible designs. Also the high number of non-linear constraints makes the design space highly separated. Therefore, in order to solve this global optimization problem, an initial coordinate search is necessary to create a suitable set of initial designs. In this work, for each number of teeth, 10 0,0 0 0 initial random designs are generated by the SOBOL algorithm, in such a way that these designs automatically satisfy all the 7 explicit constraints of the problem, (C.1– C.7 in Section 3.2). Then feasible designs from these initial random designs are found checking the remaining constraints. These feasible designs are then used as Design of Environment (DOE) for the main optimization workflow. Typically, despite a low feasible rate, from 10 to 100 feasible designs can be found from the initial 10 0,0 0 0 random designs. The presence of two objective functions, a suitable scheme for multi-objective optimization needs to be utilized. NSGA-II algorithm [27] in modeFRONTIER® was selected and executed for each number of teeth. For any given number of teeth, a Pareto front can be found with more than 10 0 0 design evaluations. In the design selection process, linear Multiple Criteria Decision Maker (MCDM) is selected with equal weights flow non-uniformity (OF1) and volume (OF2). The optimum designs for different number of teeth conditions are reported in Table 3, for TW = 0. The plots of the gears with the optimum design in Table 3 for 12-tooth and 16-tooth conditions are shown in Fig. 15.
Table 3 Optimization results for selected number of teeth with constraint wtip /m ≥ 0. No. of teeth
12
13
14
15
16
17
18
19
Addendum ha Deddendum hb Drive Pressure Angle [deg] Coast Pressure Angle [deg] d Drive Fillet Radius ρ Coast Fillet Radius ρc Correction Factor x Contact Ratio ˆ tip Tooth tip width w OF1 Non-uniformity grade: [%] OF2 Normalized volume [-]
1.3617 1.5147 26.115 22.411 0.0798 0.0699 −0.2117 1.0817 0.0 2.8838 20.8013
1.4762 1.5771 19.777 17.134 0.4850 0.2225 −0.0215 1.0 0 0 0.0 2.6256 21.7377
1.5040 1.6180 20.933 17.992 0.3262 0.2770 −0.0998 1.0 0 0 0.0 2.3894 22.4247
1.5346 1.6542 20.845 17.868 0.1905 0.2483 −0.1197 1.0 0 0 0.0 2.1973 23.2583
1.5755 1.6649 19.54 17.97 0.101 0.453 −0.1246 1.0099 0 2.0268 23.993
1.5641 1.7346 21.861 19.625 0.0963 0.2712 −0.2371 1.037 0.0 1.8801 24.7453
1.5981 1.8106 21.598 19.243 0.2415 0.0632 −0.2589 1.0 0 0 0.0 1.7445 25.3673
1.6711 1.8016 18.699 16.708 0.2318 0.3259 −0.1440 1.0 0 0 0.0 1.6393 26.3182
Fig. 15. Geometry of gears with optimized geometry for different number of teeth (a) 12-tooth (b) 16-tooth with constraint wtip /m ≥ 0.
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Fig. 16. Increased number of teeth of gear pump (from 10 to 15) tends to decrease the flow non-uniformity but it increases the pump size.
Table 3 shows the design parameters and the main numerical results, which depicts the trend of the optimal design at convergence of the optimization with wtip /m ≥ 0. The table shows that an increased number of teeth gives lower flow non-uniformity; however, in reality high number of teeth is not always desirable, as it increases the pump size. Therefore, there is a trade-off between volume and flow non-uniformity; but the trend for the optimal desings show how pumps with higher volume-specific displacement tends to give higher flow fluctuation. Also, it can be noticed that the optimization algorithm finds optimal values for the OFs by reaching constrains bounds, namely, the minimum contact ratio and the minimum tip width. Therefore, these constrains behave as important limiting factors. As reflected by the data in Tables 2 and 3, the designs with optimum performance in flow non-uniformity and pump size always reached a limiting bound in contact ratio or tip-width. The results also shows that the profile correction factor x gives the optimum performance when it has a value very close to zero. However, this also depends on the value assumed for the TW as constraint. Also, the optimum designs have addendum values close to 1.5 (increased with number of teeth from 1.4 to 1.7). Both facts are in contrast to the conclusion given by several references, such as [3], which recommend higher profile correction x and outer radius to reduce flow pulsation. This is due to the fact that in this past mentioned work, constraints such contact ratio, tip width are not taken into account. It can be interesting to observe how the optimal designs always give asymmetry in the flank pressure angles, as shown in Table 3. This emerges from the fact that the contact ratio on the coast side is always easier to satisfy than the drive-side contact ratio constraint. This gives room to further reduce the coast pressure angle to minimize the backlash interval to turn improve the flow non-uniformity. At last, the fillet radius shows less effect than other parameters in this optimization. In reality root fillet radius affects the compressibility effects in the meshing process mentioned in the introduction, since it changes the trapped volume between teeth, as well as the connection to the trapped volume. However, this is out of the scope of this paper, where root fillet radius serves only as constraints for tooth size, contact ratio, etc. The optimal parameters shown in Table 3 results from a numerical optimization, therefore they are close to the actual optimum but they not represent the exact analytical optimum solution. Moreover, the trend of the optimal designs of Table 3 can depends on the value assumed for the constraints. Changes in one or more constraints, results in a different set of optimal designs. For example, Fig. 16 represents the effect of the TW (toot tip width constraint value in (C.8)) on the Pareto front found by the optimization algorithm. The figure graphically shows how higher number of teeth gives smaller flow non-uniformity, but larger pump size. Additionally, the figure shows how the change in the value of TW gives significant influence on the optimal performance of both objective functions. Fig. 17 shows the details of the local Pareto frontiers found by the optimization algorithm. The Pareto front near the global optimal point is quite narrow, which is given by the highly constrained nature of this optimization problem. With the optimal parameters for different number of teeth being shown, the decision of the proper number of teeth to be used is often up to specific designs and customer requirements. In general, typical consideration will cover the following aspects: higher number of teeth will give not only larger pump size and lower power density, but also higher-frequency noises. Also the actual operation conditions and pump designs, and associated natural modes and harmonics are also important. In order to quantify the advantages given by the designs found in this optimization, the results of Table 2 are compared with those of a standard spur gear used as reference by Litvin and Fuentes in [28] (ha = 1.00, hb = 1.25, pressure angle α =20° without profile offset, working in dual-flank condition). As shown in Fig. 18, asymmetric gear profile optimization is able to give 1/3 additional decrease in flow non-uniformity compared to standard full-depth gear. Fig. 18 also compares the results from the optimization algorithm for the case of asymmetric design with the case of symmetric tooth profile with T W = 0 (dual flank condition for both cases). This permits to quantify the advantage given by the use of asymmetric teeth. As shown in figure, asymmetric designs (evident by the different pressure angles at drive/coast sides) give improvem‘ents around 5% to flow non-uniformity (OF1) and about 3% of pump size (OF2). Therefore, compared
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Fig. 17. Optimizatioin for 18-tooth gear pump with TW = 0 (1) Pareto Frontier between pump volume required for 30 cc-rev displacement and flow nonuniformity given by the pump given by the multi-objective optimization for (2) distribution of feasible designs.
Fig. 18. Flow non-uniformity given by optimized gear geometry, and comparison to standard full-depth gears working at single-flank and dual-flank contact conditions.
to the influence given by other parameters defining the tooth profile, advantages due to profile asymmetry are not of a great impact on the objective functions. The optimization results for symmetric tooth with T W = 0 for different number of teeth is shown in Table 4. 4. Conclusions This paper has presented an original approach to analytically study the fluid displacing action realized by external gear pumps (EGPs) with asymmetric tooth profiles. The generation of the profile as well as physical constrains that guarantee correct meshing of the gears are presented. The study particularly concentrates to the derivation of equations that expresses the theoretical performance of a given design, in terms of flow uniformity and pump size. These expressions form the basis
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131
Table 4 Optimization results with symmetric tooth design and constraint wtip /m ≥ 0. No. of teeth Addendum ha Deddendum h
b
Pressure Angle [deg] d Fillet Radius ρ Correction Factor x Contact Ratio ˆ tip Tooth Tip Width w OF1 Non-uniformity grade: [%] OF2 Normalized volume [-]
12
13
14
15
16
17
18
19
1.3276 1.4629 25.527 0.1376 −0.2274 1.0 0 0 0.0 2.9156 21.1764
1.3662 1.4833 25.140 0.1326 −0.2668 1.0 0 0 0.0 2.6346 21.8550
1.3907 1.5445 25.182 0.0845 −0.313 1.0 0 0 0.0 2.4029 22.6153
1.4525 1.5749 23.39 0.1375 −0.2668 1.015 0.0 2.2119 23.4886
1.4494 1.6412 24.637 0.0510 −0.3738 1.0 0 0 0.0 2.0311 24.0625
1.5724 1.6615 19.697 0.2706 −0.15667 1.0 0 0 0.0 1.9004 25.1516
15.469 17.212 22.340 0.0871 −0.32528 1.0 0 0 0.0 1.7584 25.6668
1.5877 1.7496 21.468 0.0599 −0.30784 1.0 0 0 0.0 1.6462 26.4690
of a multi-objective numerical optimization algorithm formulated to find the best EGP design for a given number of teeth. The optimization scheme uses a NSGA-II algorithm, implemented within the commercial software modeFRONTIER®. The optimization study focuses on the case of dual-flank operation, which is already proven to be an efficient solution for reducing flow oscillations. Optimization results show that the number of teeth has primary influence on the EGP performance: a higher number of teeth reduces outlet flow oscillations but increases the overall volume of the unit. The resulting optimal profiles are strongly affected by physical constraints such as the minimum contact ratio on drive and coast side, minimum tip width, etc. These constraints are also the reason why certain parameters such as profile correction factor cannot be used in practice to increase pump performance, as often indicated in the known literature. The results also shown that asymmetric designs for the drive/coast sides of a tooth can give advantages in terms of flow non-uniformity (5%) and EGP size (3%) with respect to the case of optimal symmetric gears. Much larger improvements can be noticed by comparing a standard gear profile with one optimized with the proposed procedure. This confirms the importance of using methods such as the one described in this work to determine the optimal shape of the gears for EGPs. Acknowledgment The authors would like to thank Esteco for the use of the software modeFrontier. References [1] A. Vacca, M. Guidetti, Modelling and experimental validation of external spur gear machines for fluid power applications, Simul. Modell. Pract. Theory 19 (9) (2011) 2007–2031. [2] T. Opperwall, A. Vacca, A combined FEM/BEM model and experimental investigation into the effects of fluid-borne noise sources on the air-borne noise generated by hydraulic pumps and motors, Proc. Inst. Mech. Eng. Part C 228 (3) (2014) 457–471. [3] J. Ivantysyn, M. Ivantysynova, Hydrostatic Pumps and Motors, Akademia Books International, New Delhi, India, 2001. [4] C. Bonacini, Sulla Portata delle Pompe ad Ingranaggi, L’Ingegnere, Anno, Milan, Italy, 1961 (in Italian). [5] N.D. Manring, S.B. Kasaragadda, The theoretical flow ripple of an external gear pump, J. Dyn. Syst. Meas. Control Trans. ASME 125 (3) (2003) 396–404. [6] X.R. Zhao, A. Vacca, Numerical analysis of theoretical flow in external gear machines, Mech. Mach. Theory 108 (2017) 41–56. [7] S. Negrini, A gear pump designed for noise abatement and flow ripple reduction, in: Proceedings of the national conference on fluid power, National Fluid Power Association, 1996. [8] M. Lätzel, D. Schwuchow, An innovative external gear pump for low noise applications, in: Proceedings of 8th International Fluid Power Conference, Dresden, Germany, 2012. [9] Y. Zhou, S.H. Hao, M.H. Hao, Design and performance analysis of a circular-arc gear pump operating at high pressure and high speed, Proc. Inst. Mech. Eng. Part C 230 (2) (2016) 189–205. [10] C.K. Chen, S.C. Yang, Geometric modelling for cylindrical and helical gear pumps with circular are teeth, Proc. Inst. Mech. Eng. Part C 214 (4) (20 0 0) 599–607. [11] S.M. Luo, Y. Wu, J. Wang, The generation principle and mathematical models of a novel cosine gear drive, Mech. Mach. Theory 43 (12) (2008) 1543–1556. [12] M.A. Morselli, “Geared hydraulic machine and relative gear wheel”, US Patent No. 20150330387 A1. Nov 19, 2015. 2015. [13] Y. Kita, J. Hida, “Zahnradpumpe oder -motor” Patent No. DE2242269 A1. Mar 8, 1973. 1973. [14] D. Schwuchow, “Hydraulische Zahnradmaschine” Patent No. DE10003735 A1. Nov 16, 2000. 20 0 0. [15] K. Koganei, “Zahnradpumpe” Patent No. DE112005003637 B4. Nov 28, 2013. 2013. [16] M. Borghi, et al., The Influence of Cavitation and Aeration on Gear Pumps and Motors Meshing Volumes Pressures, 47713, 2006, pp. 47–56. [17] M. Eaton, P.S. Keogh, K.A. Edge, The modelling, prediction, and experimental evaluation of gear pump meshing pressures with particular reference to aero-engine fuel pumps, Proc. Inst. Mech. Eng. Part I 220 (I5) (2006) 365–379. [18] P. Casoli, A. Vacca, G. Berta, Optimization of relevant design parameters of external gear pumps, in: Proceedings of the JFPS International Symposium on Fluid Power, The Japan Fluid Power System Society, 2008. [19] E. Mucchi, G. D’Elia, G. Dalpiaz, Simulation of the running in process in external gear pumps and experimental verification, Meccanica 47 (3) (2012) 621–637. [20] R. Castilla, et al., Numerical simulation of turbulent flow in the suction chamber of a gearpump using deforming mesh and mesh replacement, Int. J. Mech. Sci. 52 (10) (2010) 1334–1342. [21] A. Vacca, S. Dhar, T. Opperwall, A coupled lumped parameter and CFD approach for modeling external gear machines, SICFP2011–The Twelfth Scandinavian International Conference on Fluid Power, 2011. [22] J.J. Zhou, A. Vacca, P. Casoli, A novel approach for predicting the operation of external gear pumps under cavitating conditions, Simul. Modell. Pract. Theory 45 (2014) 35–49. [23] R.S. Devendran, A. Vacca, Optimal design of gear pumps for exhaust gas aftertreatment applications, Simul. Modell. Pract. Theory 38 (2013) 1–19. [24] B. Laczik, P. Zentay, R. Horvath, A new approach for designing gear profiles using closed complex equations, Acta Polytechnica Hungarica 11 (6) (2014) 159–172.
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[25] K.J. Huang, W.C. Lian, Kinematic flowrate characteristics of external spur gear pumps using an exact closed solution, Mech. Mach. Theory 44 (6) (2009) 1121–1131. [26] R.S. Devendran, An Innovative Working Concept For Variable Delivery Flow External Gear Machine, 2015. [27] K. Deb, et al., A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Trans. Evol. Comput. 6 (2) (2002) 182–197. [28] F.L. Litvin, A. Fuentes, Gear Geometry and Applied Theory, second ed., Cambridge University Press, New York, 2004, p. 800. xvi.
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Research paper
Formulation and optimization of involute spur gear in external gear pump Xinran Zhao∗, Andrea Vacca School of Mechanical Engineering, Purdue University, Lafayette, IN 47905 USA
a r t i c l e
i n f o
Article history: Received 30 November 2016 Revised 12 May 2017 Accepted 27 June 2017
a b s t r a c t External gear pumps (EGPs) have gained popularity among applications in many fields like fluid power transmissions and systems, automotive, aerospace thanks to their advantage of simplicity, robustness and low cost. Several studies were performed to analyze and innovate the gear profiles of EGPs to achieve better performance, in terms of flow smoothness power to weight ratio. Asymmetric gears represent one of the possible choices. This paper is aimed at developing a methodology of designing asymmetric involute gear, and formulate analytical expression for the instantaneous flowrate and flow non-uniformity given by asymmetric, non-standard involute gear pumps. These analytical expressions are then used within a multi-objective numerical optimization algorithm aimed at minimizing both the flow non-uniformity and the pump size to achieve a specific displacement. The results illustrate the highly constrained nature of the optimization problem, and the relevant impact of certain parameters of the tooth profile. It is shown how gears obtained with the proposed procedure can have significantly higher performance of standard gears, and in particular how the parameters affecting tooth profile asymmetry can further improve the EGP flow irregularity and size. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Among positive displacement machines, external gear pumps (EGPs) are often the preferred solution for many fluid power, injection, lubrication, and fluid transport systems. Key reason for success for EGPs are their low manufacturing cost, compact package, good energy efficiency, high reliability as well tolerance to contamination. Traditionally, the design of the gear profile for an EGP has been taken advantage from the wide literature on gearboxes for power transmission system. However, it is well known that the desirable operating performance of an EGP is in great part given by the fluid dynamics features of the flow through the unit, rather than by the characteristics of the power transmissions between the two matching gears. The force loading of the gears in an EGP essentially given by the fluid pressure, and as documented in [1], for a high pressure unit the radial forces acting on the shaft bearing are significantly higher than the contact forces between the gears. As a consequence, structural requirements typically affecting the gear profile in mechanical power transmissions systems do not always apply in EGPs; and gears with significant modifications are often found in commercial EGP designs. On the other hand, features of the flow as well as fluid compressibility effects often determine the design of an EGP.
∗
Corresponding author. E-mail address: [email protected] (X. Zhao).
http://dx.doi.org/10.1016/j.mechmachtheory.2017.06.020 0094-114X/© 2017 Elsevier Ltd. All rights reserved.
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Nomenclature EGP TW CR DC V1 V2 ra rb ruc m
α α rp x z b
ω γ u Q i
θ φ T
ρ e
External Gear Pump Tip width constraint (-) Contact Ratio (-) Displacement Chamber volume of DC1 (mm3 ) volume of DC2 (mm3 ) outer radius of gear (mm) radius of base circle (mm) radius of undercut circle (mm) module of gear (mm) pressure angle of cutting tool (o ) working pressure angle of gear (o ) radius of pitch circle (mm) correction (offset) factor (-) number of teeth (-) whole thickness of gear (mm) shaft speed (rad/s) base pitch (mm) distance from contact point to pitch point (mm) flow rate (mm3 /s) center distance (mm) angular position of gear (o ) angle between inward tangential vector and the vector pointing to the center (o ) Period of theoretical delivery flow (s) Fillet radius of rack cutter (mm) Offset distance (mm)
Flow non-uniformity at the outlet port is considered as one of the most important detrimental features for all positive displacement pump designs. As a matter of fact, low level of outlet flow fluctuation leads to low levels of noise emissions and reduced chances of mechanical vibration in the downstream systems [2,3]. Due to the high number of displacement chambers, the EGP design is particularly prone to provide a low level of outlet flow fluctuation. For this reason, over the last decades, significant research effort was put in studying the relations between gear profile and kinematic flow pulsations as well as in formulating design solutions for lower flow pulsation. In this regard, one of the first relevant contributions known by the authors is the work by Bonacini [4], who provided an analytical expression for the theoretical flow of an EGP, based on the parameter of the involute profile of the gears. The same finding, with alternative analytical methods of derivations, was also reported in the work by Ivantysyn and Ivantysynova [3], and Manring and Kasagaradda [5]. In both works, some considerations about some relevant parameters, such as number of teeth, involute profile pressure angle and correction factor for optimal flow pulsations were also reported. In a recent work [6], Zhou and Vacca proved the equivalency of the analytical expression provided by mentioned works with a numerical control-volume based approach suitable for lumped parameter modeling of the displacing action of an EGP. Several state-of-the-art commercial EGPs have taken advantage of the considerations of the works above mentioned. A significant example is given by the dual flank solution, as described in [7], which consists of gears able to reduce backlash to a minimum value. The feasibility of such solution is proven by several commercial designs available in the market. In other cases, flow non-uniformity is also reduced by adopting helical gear designs as well as unconventional gear profiles [8–11]. The latter solutions, however, penalize some typical advantages of traditional spur gear EGPs such as volumetric efficiency, simplicity and design scalability to different geometric displacement. For this reason, spur gear EGPs with involute type gear still represent the most desirable and cost-effective solution for an EGP. Nevertheless, the literature for spur gear EGPs does not include a general design methodology for formulating optimal profiles for gears obtained through traditional rack cutting processes. This is particularly true for the case of asymmetric involute teeth, which also appears in some commercial solutions [12–15]. With the additional degrees of freedom given by the separate definition of the drive and coast sides of the involute teeth, asymmetric gears could provide additional potentials for reducing the level of flow non-uniformity in an EGP. This is a particular aspect that will be investigated in the present paper. It is important to remark that, similarly to what assumed in the past works mentioned above, all the considerations made in this work are based on theoretical, or kinematic, flow displaced by the EGP. It is well known that actual port flow oscillations are affected also by compressibility effects associated to the pressurization and depressurization of each tooth space volume (TSV), and secondarily on the laminar/turbulent nature of the flow. Several studies focused on the analysis of
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Fig. 1. External gear pump. Lateral and radial gap affecting the energy efficiency.
Fig. 2. Two different configurations in the meshing region (a) single-flank (b) dual-flank contact [6].
the transient pressurization phases of the inter-teeth volumes during the meshing process, and to their reduction through proper design of the recessed machined at the element of the EGPs facing the lateral sides of the gears. Some examples are given by [16–18]. Other studies also addressed the pressurization phases of each TSV, as a function of the actual clearance between tooth tip and internal pump case (radial gap, Fig. 1) [1,5,19]. The leakage flows taking place at both the radial and the lateral gaps in an EGP represent an additional cause of deviation between the actual flow and the theoretical flow. This latter aspect is often studied through CFD approaches, such as in the work by Castilla et al. [20]. Despite all the reasons explained above about the discrepancy between actual and theoretical flow in an EGP, an analysis purely based on theoretical flow still provides the upper limit performance, in terms of minimum flow oscillations, of a given gear profile. If a gear profile is then selected for actual implementation on an EGP, particular care should then be dedicated to all design parameters that plays a role in reducing the actual pump performance with respect to the theoretical one. These parameters, include those affecting radial and lateral leakages flow [1,21], TSV pressure peaks and localized cavitation during the meshing process [1,22], and also the manufacturing tolerances on the actual tooth profile. Optimization techniques such as the one described in [23] were proposed to limit these effects related to fluid properties on the outlet flow oscillations. According to the general idea illustrated in the previous paragraph, this paper describes a procedure for determining the optimal tooth profile for EGPs, spur gear type, including the case of asymmetric teeth. To accomplish this goal, the work first introduces the method used to analytically describe the profile of asymmetric teeth, obtained through a standard rack cut process (Section 2.1). Subsequently, an analytical derivation of the theoretical flow for asymmetric EGPs is provided, considering the drive gear and the pinion with same profile and equal number of teeth. This part extends the findings of the previous work [6], which was limited to the case of symmetric gears (Section 2.3). The main goal of investigating the optimal tooth profile geometry is then accomplished through a numerical optimization procedure, which is detailed in Section 3. This procedure assumes dual-flank operation for the gears, which is already proven to be intrinsically more beneficial for reducing flow non-uniformity up to 75% respect to traditional single-flank technology [6], as shown in Fig. 2. Beside the minimization of flow oscillation, objective of the optimization is also to provide the most compact EGP design in terms of overall volume of the pumping elements (the gear ensemble). An important assumption made throughout this work is given by assuming the driver and the driver gear with equal number of teeth. This assumption reflects the common practice for gear pumps, particularly for high pressure operation. The last section of the paper illustrates the potential of the procedure, showing the effect of the main design parameters of the tooth profile, such as the number of teeth, on the theoretical pump performance.
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Fig. 3. Geometry of a rack-cutter for an asymmetric involute gear.
Fig. 4. Motion of a rack-cutter and decomposition into two sub-steps.
2. Design and theoretical flow rate of asymmetric gears under dual-flank operation The kinematic (or theoretical) flow rate of a positive displacement machine consists on the theoretical fluid displaced by the volume variations of its displacement chambers, under the assumptions of no leakages, negligible fluid compressibility and inviscid flow. This paper aims at finding the best gear profile that can reduce the kinematic flow while minimizing the size of the gears. For this reason, gears working at dual-flank contact (zero-backlash) are considered. For the sake of the generality of the study, non-standard involute gear design is considered, introducing several degrees of freedom in order to include the case of asymmetric teeth, also with profile shift factor. The case of standard gears will result as a particular case of the considered domain of investigation. The definition of the asymmetric profile-shifted for non-standard gear design is provided in Section 2.1, where the design starts from the asymmetric design of the generating rack-cutter. Analytical expressions defining the asymmetric gear profile depending on several design parameters are here derived. The calculation of the center distance for dual-flank operation is then given in Section 2.2. Then Section 2.3 derives the analytical expression of theoretical flow rate for asymmetric non-standard spur gears – never found in the literature known by the authors – by following a rotating control-volume based approach. Related considerations regarding the proper contact and sealing of gears operating in dual-flank contact are discussed in Section 2.4.
2.1. Asymmetric non-standard gear design As this paper is aimed at studying the most general type of involute gears generated by rack-cutter, the undercutting that results by the features of the rack-cutter shape has to be taken into account. In general, undercutting reduces the chance of interference (non-conjugate contact), but it also reduces the contact ratio, and mechanically weakens the gear tooth. While teeth strength considerations are out of the scope of this study, the effect on the contact ratio of gears given by the undercutting is taken careful consideration. A rack cutter of the general shape of Fig. 3 is taken as reference for the generation of gears with involute tooth profile. The shape of cutter is asymmetric, to permit the gear generation of asymmetric gears. The fillet curve at the tip of the cutter is formed by a circular arc with radius ρ d for the drive side, and ρ c for the coast side. The generating (cutting) process of a gear can be viewed as the synchronization of a linear translation of the cutter and a rotational motion of the gear. After a proper coordinate transformation, with a frame of reference fixed at the center of the gear, this process can be treated as a composition of two motions of the rack cutter with respect to the gear: translational and rotational (as shown in Fig. 4). Consequently, the profile of an involute gear can be created with the motion of the straight portion of the cutter, which is
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Fig. 5. Parameters definitions for the involute profile.
given by the parametric equation:
finv (θ ) =
x y
= rb
cos φ sin φ
− sin φ cos φ
sin θ − θ cos θ cos θ + θ sin θ
(1)
where φ is a constant given by:
φ = tan α − α +
π + 4x tan α 2z
(2)
In Eq. (2) α is the geometrical design pressure angle on either drive (α d ) or coast (α c ) side. θ is a parameter whose physical meaning is shown in Fig. 5. For the gear tooth, the involute profile is defined in the interval
θ ∈ 0, ra2 − rb2 /rb
(3)
where the outer radius ra and base radius rb can be expressed as:
ra =
m·z + m · x + ha 2
(4)
rb =
m·z · cos α 2
(5)
The lower limit and upper limit in Eq. (3) stand for the intersection of the involute curve with base circle and addendum circle, respectively (Fig. 5). After defining the involute portion, the next part consists in the definition of the root fillet profile of the tooth. This is assumed to be formed by the circular arc portion of a cutter. Using an approach similar to what proposed by [24], complex analytic function is used to represent the cutter motion on complex plane
w = [ν + (r − e )ϕ + i · (ζ (ν ) + r )] · exp(i · ϕ )
(6)
where e = x · m is the offset distance of the cutter, and r = mz/2 + e is the distance from the pitch line (y = 0 in Fig. 1) to the center of the gear, ν is the horizontal coordinate in Fig. 3, and ζ (ν ) is the circular arc profile in the vertical direction in Fig. 3. The second term in the square brackets of Eq. (6) represents for the translational motion, while the third term is the initial positioning of the gear. The exponential multiplier of Eq. (6) stands for the rotational motion. Notice that when a non-zero correction factor x is applied, the translational velocity does not change; in fact, the only modified term is the positioning of the cutter with respect to the gear. The term ϕ indicates the rotation angle, which can be expressed as a function of ν , i.e. ϕ = ϕ (ν ), by solving following differential equation:
∂ ∂ (Im(w(ν, ϕ ))) (Im(w(ν, ϕ ))) ∂ϕ ∂ = u ∂ ∂ (Re(w(ν, ϕ ))) (Re(w(ν, ϕ ))) ∂ϕ ∂u
(7)
The physical meaning of Eq. (7) is that a point on the cutter can actually cut the profile only when the local slope of its trajectory of motion is the same as the local slope on the cutter, as shown in Fig. 6. The solution can be derived from Eqs. (6) and (7):
ϕ=
−ζ (ζ + e ) − ν r−e
(8)
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Fig. 6. Interpretation of the physical meaning of Eq. (7).
Substituting Eq. (8) in the former expression of Eq. (6) gives w = w(ν ), which represents the root fillet profile created by the cutter on complex plane:
w(ν ) = [−ζ (ζ + e ) + i · (ζ + r )] · exp i · −
ζ (ζ + e ) + ν
(9)
r−e
Taking the root fillet profile at the drive side as an example (the coast side can be solved in a similar fashion), on cutter coordinate (Fig. 3), the fillet arc on the drive side of the cutter is written as
ζ ( ν ) = y0 −
ρd2 − (ν − x0 )2
(10)
and consequently its derivative with respect to ν is:
ζ (ν ) =
ν − x0
(11)
ρd2 − (ν − x0 )2
where (x0 , y0 ) is the center of the drive circular fillet arc in cutter coordinate, which can be determined by simple geometric calculation:
x0 =
h b − ρb m·π ρd + + 4 cos αd tan π2 − αd
(12.1)
y0 = −hb + ρb
(12.2)
with the range of ν :
3 ν ∈ ρd cos(π + αd ) + x0 , ρd cos π + x0
(12.3)
2
Sub Eqs. (10) and (11) into Eq. (9), the closed form of the root fillet profile on the drive side can be written as
froot (ν ) =
Where:
β (ν ) =
−(ζ + e ) · ζ cos (β ) + ζ + m2·z + e sin β (ζ + e ) · ζ sin (β ) + ζ + m2·z + e cos β
(ζ + e ) · ζ + ν m·z 2
(13)
3 ν ∈ ρd cos (π + αd ) + x0 , ρd cos π + x0 2
(14)
The intersection between involute profile and root fillet profile can be captured by solving for the minimum value of the equation:
g(θ , ν ) ≡ |finv (θ ) − froot (ν )|
(15)
in range of two variables θ and ν in the intervals given by Eqs. (3) and (12.3), respectively. If there is an intersection between the involute profile and the root fillet profile, the minimum of g should be zero. In the special case where g has a minimum greater than zero, that means there is no intersection; this means that the involute profile does not exist for the given set of geometrical parameters and not in the scope of this paper. This situation can happen when the outer radius specified is unrealistically small. This condition will be used as a constraint in the proposed optimization procedure (Section 3.1). The distance from the intersection point to the gear center is denoted as ruc (Fig. 7 left), because it stands for the actual starting point (undercutting point) of the involute portion of the tooth profile. The undercutting radius is greater or equal to the base circle radius, (i.e. ruc ≥ rb ), and it is written as
ruc = |finv (θ ∗ )| = |froot (ν ∗ )|
(16)
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Fig. 7. Real starting point of involute i.e. working profile (left) and root fillet profile given by the analytical expressions compared to a cutting simulation (right).
Fig. 8. Tooth thickness on gear pitch circle.
where θ ∗ and ν ∗ are the solutions corresponding to the minimum value of g(θ , ν ) given by Eq. (15). In the following part of the paper, ruc, d and ruc, c will represent the distance between the start point of the involute profile and the gear center, respectively for the drive and the coast sides.
2.2. Dual-flank interaxis distance To achieve dual-flank (no-backlash) contact, the working center distance for the general case of asymmetric non-standard gears has to be calculated. As it is well known, for x = 0 the working pressure angle is different from the design pressure angle, i.e. a = α . The case of same geometry for the drive gear and driven gear is considered. Thus, as shown in Fig. 6, for the case of corrected tooth profile (x = 0), the pitch circle results to be different from the centrode circle. The radius of centrode circle is r p = m · z/2. This value is defined regardless of x, as the profile correction changes only the position of rack-cutter, without affecting the translational velocity of the cutter. In this case, the pitch circle stands for the circle at which the tooth circumferential width equals to the gap width. On the centrode circle, the width of tooth (or of the gap) is equal to the linear length of the cutter on the centrode line:
t p = mπ /2 + xm(tan αd + tan αc )
(17)
w p = mπ /2 − xm(tan αd + tan αc )
(18)
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Fig. 9. The DC volume curve and the different shapes that each DC assumes at minimum volume, maximum volume, and intermediate volume conditions [6].
Geometrically, the involute function inv(x ) ≡ tan(x ) − x accounts for the angle shift from the centrode circle to the pitch circle (see Fig. 8). In particular, the angular interval on the pitch circle for tooth and gap is given by:
t p tp = − inv(α c ) − inv(αc ) − inv(α d ) − inv(αd ) r p rp =
π + 2x(tan αd + tan αc ) z
w p wp = + inv(α c ) − inv(αc ) + inv(α d ) − inv(αd ) r p rp =
π − 2x(tan αd + tan αc ) z
Equating t p /r p to wp /r p gives
inv(αd ) + inv(αc ) =
2x z
− inv(α c ) − inv(αc ) − inv(α d ) − inv(αd )
+ inv(α c ) − inv(αc ) + inv(α d ) − inv(αd )
(19)
(20)
+ 1 (tan αd + tan αc ) − αc − αd
(21.1)
It can also be observed that the pitch circle radius is the same for both drive and coast sides, therefore:
cos α d cos αd = cos α c cos αc
(21.2)
These two equations with respect to two variables αd and αc can be solved by numerical root-finding method such as Newton Iteration. The solution of Eqs. (21.1) and (21.2) can be used to calculate the pitch circle, which corresponds to half of the dual-flank interaxis distance:
r p =
1 m · z · cos αd i = 2 df 2 cos α d
(22)
2.3. Theoretical flow rate The kinematic flow rate can be determined by first introducing the concept of a displacement chamber (DC) for an EGP. A DC is a moving control volume, which relates to the tooth space between adjacent teeth of a gear, and rotates as the gears rotate. In an EGP, there are 2z DCs, as shown in Fig. 9. With the meshing process realized by the gears, each DC first reduces then increases its volume, realizing the displacement of the fluid from the inlet to the outlet. Considering the inlet and outlet environments, the total internal fluid volume bounded by the gears and the housing of can be divided into 2z + 2. For every angular position of the gears, the sum of their volume is always equal to a constant value. The theoretical (kinematic) flow rate for an EGP can be formulated based on DC volume conservations, with the algorithm already presented by the authors in [6]. Overall, a formula expressing the theoretical flow rate for the case of symmetric gears was already found by several authors [3,5,6,25]. Those expressions apply also to the case of asymmetric gears, but only if they operate in single-flank contact conditions. However, a formula that expresses the theoretical flow rate in case of asymmetric dual-flank gear with profile correction is not available literature. For this reason, in the following part of this section this particular case is studied using a method similar to differential volume approach used in [6]. For an EGP, unless different timing strategies are purposely introduced (like for the case of variable timing in a EGP introduced by the authors in [26]), it is reasonable to assume that an increase of each DC volume always contributes to
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Fig. 10. Differential volume of a DC at dual-flank contact (a); distance of two contact points with the gear centers, denoted by l1 , l2 , l3 and l4 , respectively.
fluid delivery, and similarly a decrease of each DC volume yields to suction of fluid from the inlet. This implies that only the DCs with an increasing volume contribute to the fluid delivery. According to Zhao and Vacca [6], with this assumption the instantaneous flow rate is given by
Qout = −
dV = b · ω · ra2 − r 2p − u2 dt
(23)
Eq. (23) is valid for all cases, including dual-flank or single-flank, symmetric gear or asymmetric gear. The value of u is the distance from the pitch point to the contact point on the low-pressure side which is delimiting the boundary of all DCs that are decreasing volume. As also observed in [6], u has a definition that depends on the particular gear profile. It is therefore important to find the position corresponding to the minimum DC volume. After the DC reaches that position, it will start increasing its volume, and therefore it will not be counted as part of volumes contributing to the outlet flow. As shown in Fig. 10(a), during an infinitesimal rotation of the gears four sub-differential volumes, from V1 to V4 , can be found with a first order accuracy as
V1 =
1 2 2 (l − rr1 )d θ 2 1
(24.1)
V2 =
1 2 (r − l22 )dθ 2 a2
(24.2)
V3 =
1 2 (r − l42 )dθ 2 a2
(24.3)
V4 =
1 2 2 (l − rr1 )d θ 2 3
(24.4)
Therefore, the differential volume dV can be formulated as:
dV = −V1 + V2 − V3 + V4 1 2 = −l1 + l32 − l22 + l42 dθ 2
(25)
In order to make DC volume V to be the minimum, it must have dV = 0. It yields
−l12 + l32 − l22 + l42 = 0
(26)
Using the geometric identity shown by Fig. 10(b)
l12 + l22 = 2r 2p + 2d12
l32 + l42 = 2r 2p + 2d22
(27)
By combining Eqs. (26) and (27), it yields d1 = d2 . This value can be denoted as d∗ , i.e.
d ∗ ≡ d1 = d2
(28)
This result indicates that a DC in dual-flank contact condition reaches its minimum volume when two contact points confining it have the same distance to the pitch point. It further indicates that, for a contact point moving away from pitch point - either on drive side or coast side of a tooth - once its distance from the pitch point reaches d∗ , the DC will begin to increase its volume and it will not be considered to find u in Eq. (17) any more. At this point, another contact point, on the
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Fig. 11. Reference Position (a) and the position with minimum DC2 volume (b).
other line of action (drive or coast), with also has a distance d∗ from the pitch point but moving toward it, will start to give the correct value of u. Therefore, the value of d∗ gives the range of variation of u in Eq. (23), which needs to be determined. For this purpose, a reference position is selected, where the coast contact point coincides with pitch point, as shown in Fig. 11. At this position, two DCs are symmetric with respect to the coast contact point (i.e. pitch point). Being the line of action also symmetric with respect to the pitch point, for a gear with contact ratio greater than one two contact points on drive side of the gears have to exist, and they will have the same distance to the pitch point, which is equal to the half of the base pitch γ d :
γd = m · π · cos αd
(29)
If the time for the reference position of Fig. 11 is denoted as t0 , considering a motion of the contact point on the line of action, the minimum volume position for DC1 can be found by solving the equation d1 (t ) = d2 (t ). With the knowledge on the initial position and the speed of movement of each point on respective line of action:
d1 (t0 ) = γd /2 =
vd =
1 mπ cos ad 2
d2 (t0 ) = 0
1 dd1 = − mz cos αd · ω dt 2
vc =
(30.1)
dd2 1 = mz cos αc · ω dt 2
(30.2)
by solving the problem
d1 (t0 + t ) = d2 (t0 + t )
(31.1)
It gives
t=
π cos αd z · ω cos αd + cos αc
(31.2)
Therefore
d∗ = d1 (t0 + t ) = d2 (t0 + t ) =
π 2
m
cos αd · cos αc cos αd + cos αc
(32)
Notice that d∗ is also the solution of d∗ = d1 (t0 − t ) = −d2 (t0 − t ), which corresponds to the minimum-volume position of DC2. The sign convention is given in Fig. 11 with proper arrows along the lines of action. It is therefore clear that the value of u in Eq. (23) changes periodically from −d∗ to d∗ . Due to the fact that the lower boundary of control volume with decreasing volume is delimited by drive contact point and coast contact point alternatively, for which u(t) has different expression, too. For simplicity, t0 , which is the time instant when coast contact point coincides with pitch point, is set to be 0: This, for the coast side:
uc (t ) =
1 mz cos αc · ω (t − T ) 2
for t ∈ −
d∗
vc
+ T,
d∗
vc
+T
(33.1)
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Fig. 12. Kinematic flow rate calculated for a 12-tooth gear with b = 32 mm, shaft speed of 10 0 0 rpm, ra = 18.5 mm, r p = 16 mm, x = 0, pressure angle of 15° and 45° for coast and drive side of gear, respectively.
While for the drive side:
1 1 ud (t ) = − mπ cos αd + mz cos αd · ω (t − T ) for t ∈ 2 2
d∗
vc
+ T,
d∗
vc
+
2d ∗
vd
+T
(33.2)
where the period is given by
T = 2N d ∗
1
+
vd
1
vc
N∈Z
(33.3)
Combining Eqs. (23), (32) and (33.1), the theoretical flow rate for asymmetric involute spur gear pump working at dualflank condition can be expressed by a piecewise function:
⎧
1 2 ⎪ 2 2 ⎪ b · ω · r − r − mz cos α · ω ( t − T ) c ⎪ a p ⎪ ⎪ 2 ⎪ ⎪ π cos αd π cos αd ⎪ ⎨ for t ∈ − + T, +T zω (cos αd + cos αc ) zω (cos αd +cos αc ) Qout (t ) =
1 2 1 ⎪ 2 2 ⎪ b · ω · ra − r p − − mz cos αd + mz cos αd · ω (t − T ) ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ π cos αd π cos αd + 2π cos αc ⎪ ⎩ for t ∈ + T, +T zω (cos αd + cos αc ) zω (cos αd + cos αc )
(34)
This formula captures the important feature that the theoretical flow ripple given by asymmetric gears is also asymmetric. Fig. 12 shows the theoretical flow given by a reference gear with highly asymmetric gear. The peak-to-peak difference (magnitude of the fluctuation) and mean flow rate are written as:
Qmean = b · ω ·
δ=
bω 4
πm
cos αd ·cos αc (π m cos αd +cos αc )
2
ra2
− r p2 −
cos αd · cos αc cos αd + cos αc
12
(35)
2 (36)
2.4. Contact ratio The starting point of the involute portion of the profile, i.e. the intersection between the root fillet profile and the involute profile, is important for determining the contact ratio. This point is also important for determining the dual flank operation, which is guaranteed by the sealing through a contact point on the coast side. For involute gears, proper contact conditions can be realized only on the involute profile. The contact ratio (CR) can be expressed as the ratio between the length of the line of action with involute-involute contact and the base pitch on drive side γ d given by Eq. (37).
CR = Where
min L1,drive , L2,drive
(37)
γd
L1,drive = 2 r p sin α p −
2 2 ruc − rbd ,d
L2,drive = 2r p cos(π /2 + αd ) +
(2rp cos(π /2 + αd ))2 − 4(rp2 − ra2 )
(38.1) (38.2)
As shown by Fig. 13, L1, drive represents the double of the length of segment OB, and L2, drive represents double of the length of segment OA. Point B is a point on the drive-side of the line of action with distance rbd, real to the gear center. Point B is closer to the pitch point O than to the point C, which is the tangential point between the base circle and the line of action. Point A is the intersection between the addendum circle of the mate gear and the line of action.
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Fig. 13. Schematic of the segment on the line of action representing the occurrence of the contact point. Point O is the pitch point. Point A, B, C are the intersection of the line of action with the outer circle of the mating gear, the real base circle, and the base circle, respectively. The actual contact length on the pressure line is the smaller one between OA (i.e. half of L2 ) and OB (i.e. half of L1 ). Table 1 Normalization of variables. ha = ha /m hb = hb /m d = ρd /m ρ ρc = ρc /m
ˆ tip = wtip /m w ra = ra /m = z/2 + ha + x r = r /m = z/2 − h − x r
r
b
rp = r p /m = z/2 αd rp = r p /m = 2z cos cos αd ruc,d = ruc,d /m r uc,c = ruc,c /m rp = r p /m = z/2 δˆ = δ /(m2 bω ) Q˙ mean = Q˙ mean /(m2 bω )
To accomplish dual-flank gear contact, the contact on the coast side is also needed to be considered, which is related to the backlash of the gears. Only when the contact point exists on the coast side to separate the trapped volume in the meshing zone into two disconnected displacement chambers, the condition of dual-flank operation can be satisfied from fluid dynamic point of view. The detailed explanation can be found in [6]. Therefore, in a similar fashion, another contact ratio can be defined for the coast. 3. Optimization algorithm This section formalizes the optimization problem, in terms of design variable, constrains and objective functions, utilized to study the optimal gear profile. All the derived quantities from previous Section 2 are used but in normalized form. Variables with length scale in radial direction and axial direction are normalized by module m and length of the gear b, respectively, while volume is normalized by m2 b, and flowrate is normalized by m2 bω. The normalized variables are shown in Table 1. 3.1. Design variables and constraints The geometry of an involute asymmetric gear can be fully described by combination of 8 non-dimensional design variables, 6 of them are describing the geometry of the cutter, which are shown in Fig. 3. The remaining two are given by the
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Range
Number of teeth z [-] Correction factor x [-] Normalized addendum ha [-] Normalized deddendum hb [-]
12–19 −1.0–2.0 0.0–3.0 0.0–3.0
Parameter Drive Coast Drive Coast
Range
pressure angle α d [ ] pressure angle α c [o ] fillet radius ρc [-] d [-] fillet radius ρ o
0–30 0–30 0–0.5 0–0.5
number of teeth, z, and the profile correction factor, x. Among eight design variables, the number of teeth z is an integer, for which objective functions and constraints are not continuous functions. For this reason, and also to better highlight the effect of the number of teeth on the optimization results, optimization algorithm is conducted by studying each number of teeth as separated case. The eight parameters after scaling are given in Table 2 together with their range of variation considered in this study. In order to exclude designs not corresponding to gears that can properly work in dual-flank conditions, 13 constraints in total are considered. As for the scope of this paper, all the constraints we used are purely geometrical, and they do not account for these structural aspects. Constraints (C.1) to (C.5) are related to the cutter geometry, and they ensure that a given combination of parameters forms a closed cutter profile. In particular, constraints (C.1) and (C.2) state that the addendum/deddendum should be large relative to the fillet radius. To simplify the problem, it is assumed that the center of the root fillet arc is in between the top land and the root land of the cutter:
d ha + hb > ρ
(C.1)
c ha + hb > ρ
(C.2)
Constraints (C.3) and (C.4) give the fact that the addendum/deddendum cannot exceed the position of the intersection of two straight lines of the cutter which have slopes equal to the geometric pressure angles:
hb <
π /2 tan(αd ) + tan(αc )
(C.3)
ha <
π /2 tan(αd ) + tan(αc )
(C.4)
Constraint (C.5) expresses the requirement that the tip of the cutter should be flat, i.e. the calculated position of the center for the coast root fillet arc should be on the right, or coincide with that of drive fillet arc (based on the coordinate configuration of Fig. 3):
d c ρ −hb + ρ ρc π + − − ≤ cos(αd ) cos(αc ) 2 tan π2 − αd tan π2 − αc d hb − ρ
(C.5)
Constraint (C.6) states that the addendum radius has to be greater than either the drive base circle radius or the coast base circle radius, so that the involute profile exists for both drive and coast side:
z z z + x + ha > max cos αd , cos αc 2 2 2
(C.6)
For gears operating in dual-flank contact, switching the features (pressure angle and fillet radius) among the drive side or the coast side does not change the performance of the unit in terms of fluid displacement. Therefore, in order to reduce the degree of freedom of design space, the drive pressure angle is arbitrarily assumed to be larger than the coast pressure angle i.e. (C.7). Also, small pressure angle on the drive side generally gives smaller contact ratios, so (C.7) will promote fulfilling the contact ratio constraint. Asymmetric tooth can have different pressure angles α d and α c on drive and coast flanges. It is shown in Eqs. (35) and (36) that exchanging the value of α d and α c will not change the kinematic displacement and fluctuation. However, higher pressure angle on the drive side α d will be favorable for increasing the contact ratio, which makes it easier to fulfill the contact ratio requirement, i.e. to make a design feasible. Therefore, the Constraint (C.7) is used:
αd ≥ αc
(C.7)
(C.8) related to the minimum width of gear tooth tip. Pointed teeth (corresponding to zero width) are normally avoided for reasons that include manufacturing feasibility, proper sealing of the DCs and reduction of the effects of gear wear on the EGP performance. In this work, this constraint is written in non-dimensional form:
ˆ tip ≡ w
wtip ≥ TW m
For a general representation, TW should be greater than 0, i.e. TW > 0.
(C.8)
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Fig. 14. Virtual bounding box for external gear pump used for volume calculation.
The calculation for the contact ratio is given in Section 2, which needs to be greater than unity of ensure enough contact on drive:
CR > 1
(C.9)
Interference between the gears needs is further check as last step. There are two types of interferences that can most commonly happen which can be described by closed equations. The first type of interference is when the tip of one gear is touching the root land of the other mating gear. For this case, the interaxis distance needs to be greater than the sum of addendum and deddendum radii. In general, a root clearance croot is specified such that 2r p ≥ rr + ra + croot . In this work, c root = croot /m ≥ 0 is used. Another type of interference occurs when the center distance is small enough that contact happens out of the working profile. To avoid this under dual-flank contact conditions, the pitch radius should be larger than either ruc, d or ruc, c . Summarizing the two cases, an interference constraint can be written in the form of (C.10)
2rˆP ≥ max rr + ra + c root , 2rˆuc,d , 2rˆuc,c
(C.10)
An additional – and trivial – consideration is that, in order to mesh, two gears needs to have a pitch radius smaller than the addendum radius, i.e. r p < ra , substitute in Eqs. (4) and (22).
z cos αd z < + ha + x 2 cos α d 2
(C.11)
The last constraint (C.12) states that the intersection between involute profile and root fillet profile must exist on both drive and coast sides, as given by Section 2.3. This can be written as:
|g(θ ∗ , ν ∗ )| < · m
(C.12)
where a common value for is 1 is used in the present work. Among 13 constraints mentioned above, (C.1)–(C.7) are explicit constraints, which can be expressed explicitly by design variables, while (C.8)–(C.12) they are implicit constraints, which are based upon intermediate calculation results. × 10−4
3.2. Objective functions In this work, the performance of a given design is measured by the flow non-uniformity, which needs to be minimized in order to obtain an EGP design with low flow pulsation. This can be quantified by the objective function OF1, the ratio between peak-to-peak value in the delivery flow and the mean flow rate, which has to be minimized:
OF1 =
δˆ Qˆmean
cos α · cos α 2 c d π cos αd + cos α =
cos α · cos α 2 1 c d π 4rˆa2 − 4rˆp2 − 3 cos αd + cos αc
(41)
In addition, compactness is of great importance in an EGP, since it relates to the power to weight ratio of the unit. Therefore, it is desirable to minimize the overall size of the pump required to realize a certain target unit displacement. For each set of design parameters, the total volume occupied by the gear set is estimated as the volume of the minimum bounding box for two mating spur gears (Fig. 14):
Vpump = 2ra (2r p + 2ra )b
(42)
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In normalized form:
Vˆpump = Vpump /(m2 b) = 4rˆa (rˆp + rˆa )
(43)
In first approximation, this volume relates to the overall size of the EGP. Obviously, the overall size of the EGP will be higher to include a casing containing the journal bearings of the gears. In this work, the ratio between pump volume and the specific displacement (mean flowrate per angular velocity) is defined as the objective function to be minimized, which stands for the minimum volume required for achieving a certain displacement:
OF2 =
4rˆa (rˆp + rˆa ) Vˆpump = cos αd ·cos αc 2 1 Qˆmean rˆa2 + rˆp2 − 12 π cos α +cos αc
(44)
d
3.3. Results and discussions The optimization algorithm described in the previous sections was implemented and executed with the commercial software modeFRONTIER®. The highly-constrained nature of the problem makes difficult to generate an initial set of design parameters corresponding to feasible designs. Also the high number of non-linear constraints makes the design space highly separated. Therefore, in order to solve this global optimization problem, an initial coordinate search is necessary to create a suitable set of initial designs. In this work, for each number of teeth, 10 0,0 0 0 initial random designs are generated by the SOBOL algorithm, in such a way that these designs automatically satisfy all the 7 explicit constraints of the problem, (C.1– C.7 in Section 3.2). Then feasible designs from these initial random designs are found checking the remaining constraints. These feasible designs are then used as Design of Environment (DOE) for the main optimization workflow. Typically, despite a low feasible rate, from 10 to 100 feasible designs can be found from the initial 10 0,0 0 0 random designs. The presence of two objective functions, a suitable scheme for multi-objective optimization needs to be utilized. NSGA-II algorithm [27] in modeFRONTIER® was selected and executed for each number of teeth. For any given number of teeth, a Pareto front can be found with more than 10 0 0 design evaluations. In the design selection process, linear Multiple Criteria Decision Maker (MCDM) is selected with equal weights flow non-uniformity (OF1) and volume (OF2). The optimum designs for different number of teeth conditions are reported in Table 3, for TW = 0. The plots of the gears with the optimum design in Table 3 for 12-tooth and 16-tooth conditions are shown in Fig. 15.
Table 3 Optimization results for selected number of teeth with constraint wtip /m ≥ 0. No. of teeth
12
13
14
15
16
17
18
19
Addendum ha Deddendum hb Drive Pressure Angle [deg] Coast Pressure Angle [deg] d Drive Fillet Radius ρ Coast Fillet Radius ρc Correction Factor x Contact Ratio ˆ tip Tooth tip width w OF1 Non-uniformity grade: [%] OF2 Normalized volume [-]
1.3617 1.5147 26.115 22.411 0.0798 0.0699 −0.2117 1.0817 0.0 2.8838 20.8013
1.4762 1.5771 19.777 17.134 0.4850 0.2225 −0.0215 1.0 0 0 0.0 2.6256 21.7377
1.5040 1.6180 20.933 17.992 0.3262 0.2770 −0.0998 1.0 0 0 0.0 2.3894 22.4247
1.5346 1.6542 20.845 17.868 0.1905 0.2483 −0.1197 1.0 0 0 0.0 2.1973 23.2583
1.5755 1.6649 19.54 17.97 0.101 0.453 −0.1246 1.0099 0 2.0268 23.993
1.5641 1.7346 21.861 19.625 0.0963 0.2712 −0.2371 1.037 0.0 1.8801 24.7453
1.5981 1.8106 21.598 19.243 0.2415 0.0632 −0.2589 1.0 0 0 0.0 1.7445 25.3673
1.6711 1.8016 18.699 16.708 0.2318 0.3259 −0.1440 1.0 0 0 0.0 1.6393 26.3182
Fig. 15. Geometry of gears with optimized geometry for different number of teeth (a) 12-tooth (b) 16-tooth with constraint wtip /m ≥ 0.
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Fig. 16. Increased number of teeth of gear pump (from 10 to 15) tends to decrease the flow non-uniformity but it increases the pump size.
Table 3 shows the design parameters and the main numerical results, which depicts the trend of the optimal design at convergence of the optimization with wtip /m ≥ 0. The table shows that an increased number of teeth gives lower flow non-uniformity; however, in reality high number of teeth is not always desirable, as it increases the pump size. Therefore, there is a trade-off between volume and flow non-uniformity; but the trend for the optimal desings show how pumps with higher volume-specific displacement tends to give higher flow fluctuation. Also, it can be noticed that the optimization algorithm finds optimal values for the OFs by reaching constrains bounds, namely, the minimum contact ratio and the minimum tip width. Therefore, these constrains behave as important limiting factors. As reflected by the data in Tables 2 and 3, the designs with optimum performance in flow non-uniformity and pump size always reached a limiting bound in contact ratio or tip-width. The results also shows that the profile correction factor x gives the optimum performance when it has a value very close to zero. However, this also depends on the value assumed for the TW as constraint. Also, the optimum designs have addendum values close to 1.5 (increased with number of teeth from 1.4 to 1.7). Both facts are in contrast to the conclusion given by several references, such as [3], which recommend higher profile correction x and outer radius to reduce flow pulsation. This is due to the fact that in this past mentioned work, constraints such contact ratio, tip width are not taken into account. It can be interesting to observe how the optimal designs always give asymmetry in the flank pressure angles, as shown in Table 3. This emerges from the fact that the contact ratio on the coast side is always easier to satisfy than the drive-side contact ratio constraint. This gives room to further reduce the coast pressure angle to minimize the backlash interval to turn improve the flow non-uniformity. At last, the fillet radius shows less effect than other parameters in this optimization. In reality root fillet radius affects the compressibility effects in the meshing process mentioned in the introduction, since it changes the trapped volume between teeth, as well as the connection to the trapped volume. However, this is out of the scope of this paper, where root fillet radius serves only as constraints for tooth size, contact ratio, etc. The optimal parameters shown in Table 3 results from a numerical optimization, therefore they are close to the actual optimum but they not represent the exact analytical optimum solution. Moreover, the trend of the optimal designs of Table 3 can depends on the value assumed for the constraints. Changes in one or more constraints, results in a different set of optimal designs. For example, Fig. 16 represents the effect of the TW (toot tip width constraint value in (C.8)) on the Pareto front found by the optimization algorithm. The figure graphically shows how higher number of teeth gives smaller flow non-uniformity, but larger pump size. Additionally, the figure shows how the change in the value of TW gives significant influence on the optimal performance of both objective functions. Fig. 17 shows the details of the local Pareto frontiers found by the optimization algorithm. The Pareto front near the global optimal point is quite narrow, which is given by the highly constrained nature of this optimization problem. With the optimal parameters for different number of teeth being shown, the decision of the proper number of teeth to be used is often up to specific designs and customer requirements. In general, typical consideration will cover the following aspects: higher number of teeth will give not only larger pump size and lower power density, but also higher-frequency noises. Also the actual operation conditions and pump designs, and associated natural modes and harmonics are also important. In order to quantify the advantages given by the designs found in this optimization, the results of Table 2 are compared with those of a standard spur gear used as reference by Litvin and Fuentes in [28] (ha = 1.00, hb = 1.25, pressure angle α =20° without profile offset, working in dual-flank condition). As shown in Fig. 18, asymmetric gear profile optimization is able to give 1/3 additional decrease in flow non-uniformity compared to standard full-depth gear. Fig. 18 also compares the results from the optimization algorithm for the case of asymmetric design with the case of symmetric tooth profile with T W = 0 (dual flank condition for both cases). This permits to quantify the advantage given by the use of asymmetric teeth. As shown in figure, asymmetric designs (evident by the different pressure angles at drive/coast sides) give improvem‘ents around 5% to flow non-uniformity (OF1) and about 3% of pump size (OF2). Therefore, compared
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Fig. 17. Optimizatioin for 18-tooth gear pump with TW = 0 (1) Pareto Frontier between pump volume required for 30 cc-rev displacement and flow nonuniformity given by the pump given by the multi-objective optimization for (2) distribution of feasible designs.
Fig. 18. Flow non-uniformity given by optimized gear geometry, and comparison to standard full-depth gears working at single-flank and dual-flank contact conditions.
to the influence given by other parameters defining the tooth profile, advantages due to profile asymmetry are not of a great impact on the objective functions. The optimization results for symmetric tooth with T W = 0 for different number of teeth is shown in Table 4. 4. Conclusions This paper has presented an original approach to analytically study the fluid displacing action realized by external gear pumps (EGPs) with asymmetric tooth profiles. The generation of the profile as well as physical constrains that guarantee correct meshing of the gears are presented. The study particularly concentrates to the derivation of equations that expresses the theoretical performance of a given design, in terms of flow uniformity and pump size. These expressions form the basis
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Table 4 Optimization results with symmetric tooth design and constraint wtip /m ≥ 0. No. of teeth Addendum ha Deddendum h
b
Pressure Angle [deg] d Fillet Radius ρ Correction Factor x Contact Ratio ˆ tip Tooth Tip Width w OF1 Non-uniformity grade: [%] OF2 Normalized volume [-]
12
13
14
15
16
17
18
19
1.3276 1.4629 25.527 0.1376 −0.2274 1.0 0 0 0.0 2.9156 21.1764
1.3662 1.4833 25.140 0.1326 −0.2668 1.0 0 0 0.0 2.6346 21.8550
1.3907 1.5445 25.182 0.0845 −0.313 1.0 0 0 0.0 2.4029 22.6153
1.4525 1.5749 23.39 0.1375 −0.2668 1.015 0.0 2.2119 23.4886
1.4494 1.6412 24.637 0.0510 −0.3738 1.0 0 0 0.0 2.0311 24.0625
1.5724 1.6615 19.697 0.2706 −0.15667 1.0 0 0 0.0 1.9004 25.1516
15.469 17.212 22.340 0.0871 −0.32528 1.0 0 0 0.0 1.7584 25.6668
1.5877 1.7496 21.468 0.0599 −0.30784 1.0 0 0 0.0 1.6462 26.4690
of a multi-objective numerical optimization algorithm formulated to find the best EGP design for a given number of teeth. The optimization scheme uses a NSGA-II algorithm, implemented within the commercial software modeFRONTIER®. The optimization study focuses on the case of dual-flank operation, which is already proven to be an efficient solution for reducing flow oscillations. Optimization results show that the number of teeth has primary influence on the EGP performance: a higher number of teeth reduces outlet flow oscillations but increases the overall volume of the unit. The resulting optimal profiles are strongly affected by physical constraints such as the minimum contact ratio on drive and coast side, minimum tip width, etc. These constraints are also the reason why certain parameters such as profile correction factor cannot be used in practice to increase pump performance, as often indicated in the known literature. The results also shown that asymmetric designs for the drive/coast sides of a tooth can give advantages in terms of flow non-uniformity (5%) and EGP size (3%) with respect to the case of optimal symmetric gears. Much larger improvements can be noticed by comparing a standard gear profile with one optimized with the proposed procedure. This confirms the importance of using methods such as the one described in this work to determine the optimal shape of the gears for EGPs. Acknowledgment The authors would like to thank Esteco for the use of the software modeFrontier. References [1] A. Vacca, M. Guidetti, Modelling and experimental validation of external spur gear machines for fluid power applications, Simul. Modell. Pract. Theory 19 (9) (2011) 2007–2031. [2] T. Opperwall, A. Vacca, A combined FEM/BEM model and experimental investigation into the effects of fluid-borne noise sources on the air-borne noise generated by hydraulic pumps and motors, Proc. Inst. Mech. Eng. Part C 228 (3) (2014) 457–471. [3] J. Ivantysyn, M. Ivantysynova, Hydrostatic Pumps and Motors, Akademia Books International, New Delhi, India, 2001. [4] C. Bonacini, Sulla Portata delle Pompe ad Ingranaggi, L’Ingegnere, Anno, Milan, Italy, 1961 (in Italian). [5] N.D. Manring, S.B. Kasaragadda, The theoretical flow ripple of an external gear pump, J. Dyn. Syst. Meas. Control Trans. ASME 125 (3) (2003) 396–404. [6] X.R. Zhao, A. Vacca, Numerical analysis of theoretical flow in external gear machines, Mech. Mach. Theory 108 (2017) 41–56. [7] S. Negrini, A gear pump designed for noise abatement and flow ripple reduction, in: Proceedings of the national conference on fluid power, National Fluid Power Association, 1996. [8] M. Lätzel, D. Schwuchow, An innovative external gear pump for low noise applications, in: Proceedings of 8th International Fluid Power Conference, Dresden, Germany, 2012. [9] Y. Zhou, S.H. Hao, M.H. Hao, Design and performance analysis of a circular-arc gear pump operating at high pressure and high speed, Proc. Inst. Mech. Eng. Part C 230 (2) (2016) 189–205. [10] C.K. Chen, S.C. Yang, Geometric modelling for cylindrical and helical gear pumps with circular are teeth, Proc. Inst. Mech. Eng. Part C 214 (4) (20 0 0) 599–607. [11] S.M. Luo, Y. Wu, J. Wang, The generation principle and mathematical models of a novel cosine gear drive, Mech. Mach. Theory 43 (12) (2008) 1543–1556. [12] M.A. Morselli, “Geared hydraulic machine and relative gear wheel”, US Patent No. 20150330387 A1. Nov 19, 2015. 2015. [13] Y. Kita, J. Hida, “Zahnradpumpe oder -motor” Patent No. DE2242269 A1. Mar 8, 1973. 1973. [14] D. Schwuchow, “Hydraulische Zahnradmaschine” Patent No. DE10003735 A1. Nov 16, 2000. 20 0 0. [15] K. Koganei, “Zahnradpumpe” Patent No. DE112005003637 B4. Nov 28, 2013. 2013. [16] M. Borghi, et al., The Influence of Cavitation and Aeration on Gear Pumps and Motors Meshing Volumes Pressures, 47713, 2006, pp. 47–56. [17] M. Eaton, P.S. Keogh, K.A. Edge, The modelling, prediction, and experimental evaluation of gear pump meshing pressures with particular reference to aero-engine fuel pumps, Proc. Inst. Mech. Eng. Part I 220 (I5) (2006) 365–379. [18] P. Casoli, A. Vacca, G. Berta, Optimization of relevant design parameters of external gear pumps, in: Proceedings of the JFPS International Symposium on Fluid Power, The Japan Fluid Power System Society, 2008. [19] E. Mucchi, G. D’Elia, G. Dalpiaz, Simulation of the running in process in external gear pumps and experimental verification, Meccanica 47 (3) (2012) 621–637. [20] R. Castilla, et al., Numerical simulation of turbulent flow in the suction chamber of a gearpump using deforming mesh and mesh replacement, Int. J. Mech. Sci. 52 (10) (2010) 1334–1342. [21] A. Vacca, S. Dhar, T. Opperwall, A coupled lumped parameter and CFD approach for modeling external gear machines, SICFP2011–The Twelfth Scandinavian International Conference on Fluid Power, 2011. [22] J.J. Zhou, A. Vacca, P. Casoli, A novel approach for predicting the operation of external gear pumps under cavitating conditions, Simul. Modell. Pract. Theory 45 (2014) 35–49. [23] R.S. Devendran, A. Vacca, Optimal design of gear pumps for exhaust gas aftertreatment applications, Simul. Modell. Pract. Theory 38 (2013) 1–19. [24] B. Laczik, P. Zentay, R. Horvath, A new approach for designing gear profiles using closed complex equations, Acta Polytechnica Hungarica 11 (6) (2014) 159–172.
132
X. Zhao, A. Vacca / Mechanism and Machine Theory 117 (2017) 114–132
[25] K.J. Huang, W.C. Lian, Kinematic flowrate characteristics of external spur gear pumps using an exact closed solution, Mech. Mach. Theory 44 (6) (2009) 1121–1131. [26] R.S. Devendran, An Innovative Working Concept For Variable Delivery Flow External Gear Machine, 2015. [27] K. Deb, et al., A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Trans. Evol. Comput. 6 (2) (2002) 182–197. [28] F.L. Litvin, A. Fuentes, Gear Geometry and Applied Theory, second ed., Cambridge University Press, New York, 2004, p. 800. xvi.