- Email: [email protected]
On the relation between vane geometry and theoretical flow ripple in balanced vane pumps
Mechanism and Machine Theory 146 (2020) 103736
Contents lists available at ScienceDirect
Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory
Research paper
On the relation between vane geometry and theoretical flow ripple in balanced vane pumps Mattia Battarra, Ph.D.∗, Emiliano Mucchi University of Ferrara, Engineering Dept. Via G. Saragat, 1 - 44122 Ferrara, Italy
a r t i c l e
i n f o
Article history: Received 5 November 2019 Accepted 29 November 2019
Keywords: Vane pump Cam ring Theoretical flow ripple Design parameter
a b s t r a c t Advancements related to the correlation between pump design parameters and the kinematic flow ripple in balanced vane pumps are addressed in the present work. In particular, the study focuses the attention on the influence of the vane geometry on the oscillations of the flow rate produced by the volume variation of both under-vane pockets and displaced chambers, that is known as one of the most relevant sources of noise in hydraulic systems. The working principle of the machine is detailed and used as starting point to deduce analytical correlations describing both vane kinematics and delivery flow rate ripple. The set of results that can be achieved with the obtained formulation is evaluated by means of a nondimensional parametric study including the two main design parameters defining the vane geometry, i.e. thickness and tip radius. The resulting trends demonstrate that the theoretical delivery flow ripple is closely related to the vane design and the cam ring shape profile. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction In the field of power transmission, balanced vane pumps are widely adopted positive displacement machines characterised by interesting features such as compactness, reliability and high values of the power over weight ratio [1]. These characteristics are particularly appreciated in both automotive and aeronautic applications, where the design of such machines must match a number of constraining requirements related to both fluid-dynamic and Noise, Vibration and Harshness (NVH) performance. As a matter of fact, the operating principle of such pumps is directly responsible for their NVH behavior, which often represents the main drawback to be overcome [2,3]. The pump cyclically displaces oil from the suction side to the delivery port, producing periodically varying outlet flow rate and pressure that excite the pump body and the piping system. The link between fluid-dynamic behavior and NVH characteristics of positive displacement machines has been widely studied in the specialized literature since the ’80s, when researchers started focusing on the determination of the cyclic loads given by delivery pressure and flow rate oscillations [4–7]. The assessment of these dynamic loads has been performed with different approaches, such as zero dimensional models (see for example Refs. [8–11]) and Computational Fluid Dynamic models (as the ones reported in Refs. [12,13]). Within this framework, a peculiar solution is represented by the analytical approach, which allows us to estimate the variable pressure and flow characteristics of the pump on the basis of its kinematics. Despite these methodologies may seem too ∗
Corresponding author. E-mail addresses: [email protected] (M. Battarra), [email protected] (E. Mucchi).
https://doi.org/10.1016/j.mechmachtheory.2019.103736 0094-114X/© 2019 Elsevier Ltd. All rights reserved.
2
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
Nomenclature
β
ˆ·
ω θ EF θ ER θ SF θ SR
ϑ Arot Astat Av B b dv Dth n nd O P p Qth rc rr rv t tv Vdc Vuv z
Superscript indicating derivative with respect to angle Pressure angle between vane tip and stator Symbol indicating a nondimensional quantity Angular velocity in rad/s Angle defining the end of the fall phase in the cam ring profile Angle defining the end of the rise phase in the cam ring profile Angle defining the beginning of the fall phase in the cam ring profile Angle defining the beginning of the rise phase in the cam ring profile Angular coordinate of the reference vane Area of the angular sector covered by the rotor radius between two consecutive vanes Area of the angular sector covered by the cam ring radius between two consecutive vanes Area of the portion of the vane delimiting the displaced chamber Oil Bulk modulus Pump facewidth Vane radial displacement Theoretical volumetric displacement of the pump Angular velocity in rpm Number of displaced chambers and under-vane pockets exposed to the delivery port Pump center and origin of the reference frame Contact point Oil pressure Instantaneous delivery flow rate Cam ring radius Rotor radius Vane tip radius Time Vane thickness Volume of the displaced chamber Volume of the under-vane pocket Number of vanes
generalist and leading to qualitative results, they are often adopted with satisfactory results during the early stages of the design process, when the detailed geometry of the machine is still to be defined and designers need to establish the very first parameters such as number of vanes and aspect ratio. In this context, various scientific works are focused on the determination of analytical formula to predict the delivery flow ripple. Such studies are nowadays performed with the purpose to include the effect provided by an increasing number of geometrical parameters and design solutions. In particular, this approach has been successfully deepened in the field of external gear pumps, where the first formulation proposed by Bonacini in [14] has been fruitfully extended to pumps with helical gears [15], pumps with not-unitary transmission ratio [16] as well as gear pumps with asymmetric tooth profile [17]. However, similar analyses can be found also for other positive displacement machines such as trochoidal pumps [18] and variable displacement vane pumps [19]. In the specific context of balanced vane pumps, the analytical approach still represents a methodology rarely studied. The most representative work pertains to Giuffrida et al. [20], in which the authors defined a dedicated procedure to estimate the theoretical flow ripple in reference to the cam ring shape and the vane thickness. The dissertation is built up on the hypotheses of flat tip vanes with sharp edges and under-vane pockets that contribute to displace oil into the outlet chamber. On the basis of these aspects, the present work has the main purpose to propose an improvement of the methodology provided in [20], by including the effect of the tip radius in the definition of the theoretical flow rate oscillations caused by the pump. The proposed methodology takes advantage of the pump kinematic analysis described by the present Authors in Ref. [21]. A parametric study is adopted to evaluate the capabilities of the method, with a specific focus on the influence of both tip radius and vane thickness on the characteristics of the delivered flow ripple. In particular, the first parameter is shown to cause a progressive deformation of the flow rate waveform produced by the under-vane pockets and an increase of the amplitude of the total delivered flow ripple, which reflects into an amplification of the first order of the vane passage frequency. On the contrary, the latter acts as a tuning parameter to subdivide the delivered flow rate between the undervane pockets and the displaced chambers. In addition, from a NVH point of view, the consequence of the vane thickness increment is observed also on the amplitude of the delivery flow rate waveform, due to the increase of the amplitude of several harmonics of the vane passage frequency. The obtained results underline the lack of knowledge regarding the deep correlation existing between the pump behavior and several design parameters defining its geometry. Within this context, the purpose of the present study is to improve the comprehension of the physics that correlate the pump main geometrical
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
3
Fig. 1. Basic components of a balanced vane pump.
characteristics to its kinematic flow rate ripple. This goal is achieved by presenting an analytical formulation that may find practical applications in real design processes, where the chance to obtain instantaneous indications on the machine behavior are widely appreciated. In this framework, the proposed formulation may provide theoretical strength to design choices that are often based on the experience, in particular when the detailed machine geometry is still to be defined. The following section describes the operating principle of a balanced vane pump and the main components constituting the pump mechanism. Section 3 presents the methodology followed to determine the delivery flow ripple in reference to both vane tip radius and vane thickness. Section 4 reports the results of the parametric study, which is performed on the basis of the nondimensionalization theory in order to provide general results. Eventually, concluding remarks are reported in Section 5. 2. Pump operating principle In the present section, the main components constituting the pump mechanism are described together with the pump operating principle, with a particular focus on the basic design philosophy standing at the basis of this machine. As it will be enlightened in Section 3, this aspect substantially defines the pump theoretical behavior. By taking as reference the schematic representation in Fig. 1, the pump may be described by the interaction between three fundamental items: the external stator, namely the cam ring, the internal rotor and the vanes. Each vane is forced to move on a dedicated track created within the rotor, which moves them from the suction port to the delivery port. During the rotation, the oil is trapped in the chamber limited by two consecutive vanes, namely the displaced chamber, and delivered from the inlet side to the outlet one. Along this process, the peculiar shape of the cam ring guarantees the expansion and subsequent compression of the chambers, which is the phenomenon at the basis of the displacing action. The described working principle leads to the major drawback constituted by flow rate and pressure oscillations, representing a common undesired by-product for all the positive displacement machines. In addition, part of the oil suctioned from the inlet chamber is trapped by the pockets that are formed under the vanes, namely the under-vane pockets. The pumping action of such pockets is naturally caused by the radial motion of the vanes and they provide a not-negligible contribution to the definition of the global displacement of the pump. Inlet and outlet ports related to the under-vane pockets, not reported in Fig. 1, are usually timed with the suction and delivery ports related to the displaced chambers. The overall flow rate oscillation at the outlet port is therefore generated by the synergy between the pumping action of the displaced chambers and the pumping action of the under-vane pockets. The described process is repeated twice per revolution, since the complete pump is constituted by two identical mechanisms, which are symmetrical with respect to the center of the rotor. The overall mechanism is usually packed between two plates providing the axial sealing. The complete pump package includes a number of additional components and features designed to improve the mechanical and fluid-dynamics behavior of the machine; however, the related description is omitted being out of the scope of the present study. On the basis of this operating principle, it may be straightforwardly deduced that the pumping capability of the machine is mainly governed by the shape of the cam ring and its interaction with the sliding vanes. Within this context, the external stator can be considered as a fixed cam, which guides the rotating followers, i.e. the vanes, by means of its internal profile and determines the machine displacement. With reference to Fig. 1, the volume of the displaced chamber, highlighted by yellow stripes, may be defined by:
V1 =
2π b rcmax 2 − rr 2 z 2
(1)
where b is the pump facewidth while rcmax is the maximum value of the cam ring radius, rr is the rotor radius and z is the vane number. Concurrently, the volume of oil brought back from the outlet chamber to the inlet chamber, highlighted by purple stripes in Fig. 1, may be defined by: 2
V2 =
2π b rcmin − rr 2 z 2
(2)
4
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
Fig. 2. Representation of the interaction between vane and cam ring as a cam-follower mechanism including the vectors constituting the closure equation of the linkage.
where rcmin is the minimum value of the cam ring radius. By considering the contribution of Eqs. (1) and (2) along a complete revolution, pump theoretical volumetric displacement Dth is calculated as:
Dth = 2z (V1 − V2 ) = 2π b rcmax − rcmin
rcmax + rcmin
(3)
The volume reduction affecting the displaced chamber due to the vane thickness is usually compensated by the under-vane pocket contribution. A detailed explanation of Eq. (3) may be found in Ref. [1] pages 343–352, however this brief analysis underlines that the conceptual design of the pump necessarily starts with the determination of the stator internal profile, i.e. the cam ring. By the time the cam ring shape is defined, the designer usually focuses the attention on the vane geometry, that is defined, in its basic form, by the vane thickness and the vane tip radius. Once both vane and cam ring are chosen, the shape of the pockets displacing the oil from the suction chamber to the delivery one is also determined. As a matter of fact, this means that the kinematic flow ripple of the pump is defined and no further geometrical parameters are available to tune this pump characteristic. Based on the depicted scenario, the possibility to correlate both cam ring shape and vane geometry to the theoretical delivery flow rate ripple of the pump is of major relevance during the design process. In this context, an analytical formulation represents a powerful tool to provide theoretical strength to design choices that are often based on the experience. 3. Theoretical flow rate determination The theoretical delivery flow ripple of a positive displacement pump, i.e. the kinematic flow ripple, is represented by the flow rate oscillation at the outlet port calculated on the basis of three main hypotheses. In particular, (i) the pump motion is assumed to be purely kinematic, i.e. dynamic effects such as vane tip detachment and vibration of the components is neglected, (ii) leakage phenomena are not taken into account and (iii) the oil is considered as an incompressible fluid. These hypotheses are always taken as the starting point of all the studies carried out on this subject (see for example Refs. [16,17,20]). Within this context, as also explained in [20], instantaneous delivery flow rate Qth is determined by Eq. (4):
Qth (ϑ ) = −ω
nd k Vdc |ϑk + Vukv |ϑk
(4)
k=1
where nd represents the number of displaced chambers and under-vane pockets exposed to the delivery port, ω is the pump mean velocity, ϑ is the angular position of the reference vane, while Vuv and Vdc are the volume of the under-vane pocket and displaced chamber, respectively. The problem therefore reduces to the determination of the volume variation for both under-vane pockets and displaced chambers along a complete period of the rotation. It is worth specifying that the negative sign in Eq. (4) is mandatory to obtain Qth with a positive mean value. This is due to the fact that, when the pockets are exposed to the outlet port, their volume is necessarily decreasing in order to deliver the fluid. In order to calculate these two contributions, the pump is modeled as a cam-follower mechanism with contact taking place on the inner side of the cam profile. Fig. 2 provides a schematic representation of the geometrical problem, where the cam ring acts as an internal cam fixed to the frame and the vane acts as a rotating follower. Within this framework, by
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
5
Fig. 3. Geometrical representation of the angles described in Eq. (7).
Fig. 4. Calculation of the slope of the cam ring profile at the contact point with respect to its base circle defined by radius rcmin .
considering a circular tip follower, the vector chain constituted by vane radial displacement dv between pump center and circular tip center and the vector between the circular tip center and contact point P must be equal to the vector defining the position of contact point P with respect to the pump center. The described vector chain translates into the following closure equation, written with respect to a reference frame that rotates simultaneously with the rotor:
dv ( ϑ ) + rv e where β
iβ
= rc (P )e
i β +tan−1
rc P rcmin
|
(5)
represents the pressure angle at contact point P, while term rc . |P is the value of the first angular derivative of the
cam ring profile at contact point P. Within this framework, term
r c |P
rcmin .
rcmin
basically represents the angular deviation of the cam
ring profile with respect to its base circle, defined by radius Its geometrical explanation in the cam ring environment is provided in Fig. 3, while Fig. 4 is reported to clarify the mathematical meaning of such a term. As it may be appreciated, term
r c |P rcmin
defines the slope of the cam ring profile in a Cartesian plane, where the x-axis is constituted by the perimeter of
the maximum circumference inscribed inside the cam ring profile, which is actually defined by radius rcmin . For the sake of compactness, the following notation is adopted:
ψ = tan
−1
rc |P rcmin
(6)
6
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
where ψ is the angle associated to the local slope of the cam ring profile. Since the cam ring profile is given by design, Eq. (5) can be expanded in a system of two scalar equations and used to obtain both vane displacement dv and pressure angle β once the tip radius has been defined. A detailed dissertation of the adopted kinematic approach has been proposed by Battarra et al in [21]. It is worth underlining that the vane thickness does not appear in Eq. (5), therefore, in the hypotheses of circular tip, such a parameter does not directly affect the vane motion. This aspect makes the present dissertation different from the methodology based on the flat tip assumption, in which the vane thickness plays a key role in the definition of the vane radial displacement [20]. For the sake of completeness, it has to be underlined that, in order to completely define the vane radial displacement, it is required to obtain also angular position ϑ associated to each value of dv . By taking as reference Fig. 3, the following geometrical relationship can be defined:
ϑ = ϑ (P ) − β − ψ
(7)
where term ϑ(P) represents the angular coordinate associated to contact point P. A graphical explanation of the angular correlation is provided in Fig. 3. Once Eq. (5) is solved, both contact point position and pressure angle are defined, therefore angle ϑ can be straightforwardly calculated. It is worth mentioning that the closure equation approach may be avoided by applying the cosine theorem on the triangle defined by points O, P and the center of the tip circle, which leads to:
dv =
rc 2 + rv 2 − 2rv rc cos ψ
(8)
Eq. (8) is valid throughout the entire revolution and it may be used in place of Eq. (5) to calculate the magnitude of dv . Once the vane radial displacement is obtained, the derivative of the volume related to the under-vane pocket can be calculated by multiplying the cross-sectional area of the pocket with velocity coefficient of the vane radial motion dv :
Vuv = b · tv · dv
(9)
where term tv represents the vane thickness. Here, the term velocity coefficient refers to its kinematic meaning:
dv =
∂ dv 1 ∂ dv = · ∂ϑ ω ∂t
(10)
The computation of dv may be performed both analytically or numerically, depending on the mathematical law adopted to design the cam ring profile.For the sake of clarity, it is worth noticing that the definition in Eq. (10) applies to all the terms with prime superscript. Eq. (9) can be straightforwardly expanded to provide an analytical definition of the derivative of the under-vane volume:
Vuv = b · tv ·
rc · (rc − rv cos ψ ) + rv rc ψ sin ψ
(11)
rc2 + rv2 − 2rv rc cos ψ
It is worth noticing that Eq. (11) is always applicable as long as the mathematical law chosen to describe the cam ring profile admits first and second derivative. in Eq. (4) reWhereas term Vuv has a straightforward definition, the calculation of the contribution given by term Vdc quires, as a first step, to determine the evolution of the displaced chamber volume along a complete pumping period, i.e. half of the revolution. In practice, for each angular position ϑ of the leading vane, volume Vdc may be obtained by the algebraic sum of three terms:
Vdc (ϑ ) = b Astat − Arot −
1 Av (ϑ ) + Av ϑ − 2zπ 2
(12)
where Astat is the area of the angular sector covered by the cam ring profile between two consecutive vanes, Arot is the area of the angular sector covered by the radius of the rotor between two consecutive vanes while Av represents the area of the portion of the vane delimiting the displaced chamber. Fig. 5 provides a graphical explanation of the three contributions defined in Eq. (12). The most relevant term, represented by Astat , may be obtained from the generic integral used to compute the area of an angular sector defined by a variable radius:
Astat (ϑ ) =
1 ϑ 2 [rc (γ )] dγ 2 ϑ − 2π z
(13)
where γ is an auxiliary variable defined within the closed interval ϑ − 2zπ , ϑ . The solution of the integral in Eq. (13) can be obtained both analytically or numerically, depending on the definition of term rc . However, the cam ring profile is commonly described by a piecewise-defined polynomial function, therefore an analytical expression for Vdc (ϑ) can usually be achieved. Known the volume course along half a revolution, the contribution to the delivery flow rate ripple given by the displacement chamber can be directly obtained by differentiating it with respect to the angular position of the leading vane. In addition, since term Arot is a constant value throughout the entire revolution, its practical calculation in Eq. (12) may be neglected. Concurrently, regarding term Av , its derivative reduces to the calculation of the cam ring derivative since:
A v ( ϑ ) = t v [r c ( ϑ ) − r r ]
(14)
For the sake of completeness, it has to be observed that the exact definition of area Av would require to include the vane area constituted by the circular segment defined the by circular vane tip. However, since this term remains constant
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
7
Fig. 5. Geometrical representation of the three contributions adopted to compute the volume of the displace chamber at each angular position of the leading vane.
throughout the entire revolution, it does not concur to the definition of the first derivative of volume Vdc and therefore its computation may be avoided. It is worth noticing that the described procedure allows us to evaluate the theoretical flow rate delivered by a balanced vane pump taking into account the effects produced by both vane thickness tv and tip radius rv . In addition, it is possible to focus the attention on the two sources of flow rate oscillations separately, i.e. the displaced chamber contribution and the under-vane pocket contribution, by simply isolating the two terms in Eq. (4). In particular, the first one is given by: dc Qth (ϑ ) = −ω
nd
k Vdc |ϑk
(15)
k=1
while the second one is determined by: uv Qth (ϑ ) = −ω
nd
Vukv |ϑk
(16)
k=1
Since parameters tv and rv appear both in Eqs. (9) and (13), such terms are expected to influence the displaced chamber contribution and the under-vane pocket one, concurrently. 4. Parametric study The current section is devoted to enlighten the outcomes achievable from the methodology described in Section 3. Moreover, the study assesses the influence of the vane geometry by evaluating the trends imposed to the amplitude of the delivery flow ripple by the variation of different design parameters. In order to perform this study, all the geometrical parameters are defined in nondimensional form (indicated by the ˆ· symbol) by dividing them with respect to the minimum value of the cam ring radius rcmin on the basis of the Buckingham’s Theorem [22]. The pump displacement per facewidth unit may be obtained from Eq. (3) as:
Dth = 2π rcmax − rcmin 2
2
(17)
it is possible to express the specific pump displacement as:
Dˆ th = 2π eˆ2 − 1
(18)
where term eˆ is the ratio between rcmax and rcmin . Based on the Buckingham’s Theorem, the proposed dimensional reduction applies to all the parameters defining the pump geometry. In particular, hereinafter the following dimensionless groups are adopted:
⎡
tv tˆv = speci f ic vane thickness = rmin c rv ⎣rˆv = speci f ic vane tip radius = rmin c rc rˆc = speci f ic cam ring radius = rmin c
(19)
8
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736 Table 1 Main design parameters of the pump design taken as reference. z eˆ ϑSR ϑER ϑSF ϑEF
10
1 + 1/2π
π /10 2π /5 3π /5 9π /10
Fig. 6. Reference cam ring profile in polar coordinates.
On the basis of this method, all the angles need to be reported in radians, which represent specific quantities by definition. The adopted approach allows to deduce general results that are common to a family of pumps with the same specific displacement defined by the value associated to eˆ and same cam ring shape. Table 1 reports the main parameters defining the machine geometry while the cam ring profile is reported in Fig. 6 in polar coordinates. Parameter eˆ has been chosen equal to 1 + 1/2π in order to obtain Dˆ th = 1. Rise phase, i.e. profile geometry between ϑSR and ϑER , and fall phase, i.e. profile geometry between ϑSF and ϑEF , have been obtained by means of a 5th order polynomial law:
rˆc (ϑ ) =
5
ak ϑ k
(20)
k=0
where ak is the kth coefficient to be determined. The coefficients are computed by imposing the continuity of the law, together with its first two derivatives, at the boundary of the reference angular interval. This law is a commonly adopted mathematical law in cam design, since it guarantees the smoothness of the radial displacement, velocity and acceleration of the vane, which actually behaves as a proper follower [23]. It is worth noticing that this choice does not affect the generality of the dissertation that would lead to similar conclusions independently on the mathematical law adopted to define the cam ring profile. As underlined in Ref. [21], the mathematical law describing rise and fall phase affects the smoothness of the vane radial motion, however it does not alter the main kinematic characteristics of the vane-cam ring mechanism. For this reason, different mathematical laws would lead to different waveforms of the flow rate ripple, but they would not affect how the vane tip radius and the vane thickness affect such waveforms. This aspect is further highlighted in Section 4.4, where the parametric study is carried out with respect to multiple cam ring profiles, characterized by the same theoretical displacement and different mathematical laws describing rise and fall phases. The following subsections detail the results of the parametric study, which is carried out by varying the vane tip radius and the vane thickness separately and then simultaneously. Within this framework, it is worth noticing that tˆv and rˆv are not independent quantities and they must respect the following mathematical constrains at each angular position of the reference vane:
0 ≤ tˆv ≤ 2 tˆv 2
≤ rˆv ≤
√
tˆv2 rˆc cos ψ −tˆv rˆc sin |ψ | 2 tˆv2 −4rˆc2 sin ψ
4rˆc2 −tˆv2
(21)
The complete theoretical dissertation regarding this subject may be found in Ref. [21]. For the scope of the present work, it is sufficient to state that maximum and minimum allowable values for both vane thickness tˆv and tip radius rˆv are calculated on the basis of Eq. (21) in order to guarantee the admissibility of the vane geometry.
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
9
Fig. 7. Effect of the vane tip radius increment on the theoretical flow ripple referring to displaced chambers (a) and under-vane pockets (b).
Fig. 8. Effect of the vane tip radius increment on the theoretical flow ripple globally delivered by the pump.
4.1. Vane tip radius influence The effect produced by the vane tip radius is now evaluated by considering the pump design parameters defined in Table 1 with a fixed value of the specific vane thickness equal to 0.1. Within this framework, the delivery flow ripple pertaining to the displaced chambers and the one referring to the under-vane pockets is estimated (Fig. 7a and b, respectively) for different values of the vane tip radius from 0.05 to 0.35. For the sake of generality, specific flow rate oscillation is obtained by dividing the delivered flow rate by the working speed and the pump facewidth. As it can be observed, both displaced chamber and under-vane pocket ripples are dominated by the periodic compression of the volumes exposed to the delivery chamber. In this scenario, the influence of the tip radius increment on the displaced chamber contribution, despite present, is substantially inconsequential. On the contrary, such an increment produces a not-negligible effect on the undervane pocket contribution by increasing the amplitude of the oscillation and varying its timing. The phenomenon has similar drawbacks on the total theoretical flow ripple, shown in Fig. 8, in which the amplitude of the oscillation follows the same trend of the under-vane pocket contribution, as expected. The reason for this behavior has to be found in the consequences that the tip radius variation produces on the machine kinematics. As observed in [21], if rˆv increases, the vane radial motion diverges from the cam ring profile producing the phase-shift in Fig. 7b. On the other hand, the flow rate pertaining to the displaced chambers is mainly governed by the cam ring profile, while modifications on the vane kinematics exclusively affect term Av in Eq. (12).
10
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
Fig. 9. Amplitude of the first five orders of the VPF in reference to the vane tip radius.
The delivery flow rate ripple can be considered as an indicator of the pressure ripple loading the pump since it has been widely demonstrated that the pockets of positive displacement machines tend to respect the following continuity equation:
∂p B = Q ∂t V
(22)
where B is the oil Bulk modulus. On the basis of this consideration, from a NVH point of view, the analysis of the delivery flow ripple may be further deepened by evaluating it in the frequency domain. Within this framework, Fig. 9 reports the amplitude of the first five orders of the Vane Passage Frequency (VPF) calculated from the waveforms in Fig. 8. The VPF represents the fundamental frequency of the pump and it has the following definition:
V PF = z
ω
2π
(23)
As it can be observed from Fig. 9, the vane tip radius strongly affects the amplitude of the first order, while the remaining harmonics show negligible variations.
4.2. Vane thickness influence In analogy to the previous subsection, the analysis is now focused on evaluating the effect of the vane thickness by considering the pump design parameters defined in Table 1 with a fixed value of the vane tip radius equal to 0.2. The chosen value for rv allows us to span an interval for the vane thickness that goes from 0.1 to 0.4. The results of the vane thickness increment on the two flow ripple contributions are reported in Fig. 10. As it can be observed, this parameter has a relevant effect on both flow rate ripples by influencing their amplitude and mean value, simultaneously. In particular, the mean value of the under-vane pocket flow ripple tends to be increased by the vane thickness increment while the behavior is the opposite for the displaced chamber flow ripple. As a matter of fact, the vane thickness mainly acts as a tuning parameter of the ratio between the mean flow rate provided by the displaced chambers and the one provided by the under-vane pockets. The amplitude of the oscillation, on the contrary, tends to increase in both cases, and this effect is clearly recognized also by focusing the attention on the total theoretical flow ripple that is shown in Fig. 11. It is worth noticing that, despite the outlet flow rate is differently distributed between under-vane pockets and displaced chambers depending on the value of tˆv , the mean total delivery flow rate remains unaltered. Fig. 11 also reveals a secondary drawback related to the vane thickness increment that tends to be masked when the attention is focused on the two contributions separately. As tˆv increases, the influence of term Av in Eq. (12) becomes more and more relevant, producing peaks in the global flow rate ripple with higher frequency content with respect to the main flow rate pulsation. Such a phenomenon is generated in the displaced chamber flow rate ripple, despite slightly noticeable (see Fig. 10), and it results amplified as the two contributions are summed up. With the purpose to further investigate the link between the vane thickness and the theoretical flow rate ripple, the waveforms reported in Fig. 11 are now analyzed in the frequency domain. Fig. 12, in particular, depicts the amplitude of the first five orders of the VPF in correlation with the analyzed range of vane thickness values. The detected behavior is considerably different with respect to the one observed in the previous subsection, since in this case the geometric parameter is affecting all the considered harmonics. In particular, the vane thickness increase causes a less severe increment of the first order compared to the one generated by the tip radius, while it produces a not-negligible increase of the amplitude of the higher orders. This phenomenon is a direct consequence of the behavior observed in Fig. 10, where the increase of tˆv is shown to cause the appearance of peaks with higher frequency content with respect to the main flow rate pulsation.
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
11
Fig. 10. Effect of the vane thickness increment on the theoretical flow ripple referring to displaced chambers (a) and under-vane pockets (b).
Fig. 11. Effect of the vane thickness increment on the theoretical flow ripple globally delivered by the pump.
4.3. Tip radius and thickness interaction Sections 4.1 and 4.2 described the influence of a single parameter on the theoretical flow ripple, demonstrating that tip radius and vane thickness produce different effects on the flow rate oscillation. In order to provide a global overview of their influence, the purpose of the present subsection is to analyze the delivery flow rate waveform with respect to the simultaneous variation of rˆv and tˆv . Within this framework, the globally delivered flow ripple has been calculated for the entire admissible domain for the couple (rˆv , tˆv ). In order to evaluate the result of each geometrical configuration, the obtained waveforms are replaced by the rating parameter ARMS , which is calculated as the Root Mean Square (RMS) value of the first five harmonics of the VPF:
ARMS =
5
Ak 2
(24)
k=1
where Ak is the amplitude of the kth harmonics of the VPF. The rating parameter, as also reported in various works (see for example Refs. [10,24,25]), is strictly correlated to the energy content of the waveform. The result of the described analysis is reported in Fig. 13, where the dashed lines represent the boundaries of the geometry admissible domain. The rating parameter is displayed in logarithmic scale for the ease of the representation and the chart is limited to tˆv < 0.8, since higher values of vane thickness are not feasible for real applications. As it may be observed, the RMS value tends to decrease as both the tip radius and the vane thickness decrease, reaching zero for both rˆv and tˆv equal to zero. In agreement with the results of the analyses in the previous subsections, as a general rule, the amplitude of the flow ripple oscillations tends to decrease as both vane thickness and tip radius are decreased. In addition,
12
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
Fig. 12. Amplitude of the first five orders of the VPF in reference to the vane thickness.
Fig. 13. Contour plot of the RMS value of the first five harmonics of the VPF referred to Qˆth , obtained within the admissible domain for rˆv and tˆv .
it is worth noticing that rˆv produces different behaviors depending on the value associated to tˆv . For value of tˆv between 0 and 0.1, the isolines are almost vertical, meaning that the increase of rˆv produces negligible effects on the amplitude of the delivery flow ripple. On the contrary, as the vane thickness is increased, the isolines tend to flatten themselves, meaning that the influence of rˆv and tˆv is comparable in magnitude. 4.4. Cam ring profile influence The purpose of the present subsection is to extend the results obtained in Section 4.3 to different cam ring profiles in order to ensure the general applicability of the proposed method and evaluate the influence of the mathematical law adopted to describe rise and fall phases. Within this framework, six different cam ring profiles have been defined, each of them based on the same design parameters reported in Table 1. Fig. 14 reports a detailed view of the rise phase profile of the six cam rings. In agreement with the parameters in Table 1, rise and fall profiles are symmetrical. As it may be noticed, four profiles are obtained from the polynomial law with order from two to seven, while the last two profiles are the cycloidal law and the double harmonic law. Detailed information regarding the mathematical formulation behind these laws and their application to real scenarios is reported in Ref. [23], while Ref. [21] provides explanation on how they affect the vane radial motion. For the sake of clarity, it is worth noticing that the 2nd order polynomial law seems to follow a higher divergence rate from the linearity than the 3rd order polynomial law. However, this is due to fact that, while a single polynomial is sufficient to describe the rise phase with the 3rd order law, the 2nd order law requires two distinct polynomial in order to guarantee the smoothness of the profile.
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
13
Fig. 14. Detail of the rise phase cam ring profile obtained from six different mathematical laws, adopting the same values of ϑSR and ϑER .
Fig. 15. Contour plot of the RMS value of the first five harmonics of the VPF referred to Qˆth , obtained with six different mathematical laws for the cam ring profile design.
14
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
According to Eq. (3), the six cam ring profiles guarantee the same pump theoretical displacement since they reach the same maximum value of rcmax , even though the vane radial motion during rise and fall phases does not coincide. As a direct consequence, each cam ring profile is expected to produce a delivery flow ripple with a characteristic waveform, which is basically governed by the derivative of both under-vane pocket volume Vuv and displaced chamber volume Vdc . In order to provide a general overview regarding the link between cam ring profile and delivery flow ripple, each cam ring shown in Fig. 14 has been used to compute the same analysis reported in Section 4.3. The results are depicted in Fig. 15, showing the RMS value of the amplitude of the first five harmonics of the VPF referred to Qˆth . The magnitude of the contour plots refers to the same logarithmic scale adopted in Fig. 13. As a preliminary consideration, it may be noticed that the 5th order polynomial law and the cycloidal law, which give rise to cam ring profiles that are almost superimposed in Fig. 14, consequently provide similar amplitude fields (Fig. 15c and e). The same applies to the 7th order polynomial law and the double harmonic law, despite the presence of major discrepancies both in the cam ring shape and the amplitude field. From a practical perspective, it is worth underlying that the overall amplitude of the delivery flow ripple tends to increase as the mathematical law diverges from the linear trend. This behavior is a direct consequence of the fact that higher order polynomial laws involve higher values of the associated derivatives. Based on Eqs. (15) and (16), higher values of the volume derivatives produce higher amplitude waveforms of the delivery flow rate as detrimental effect. It is interesting to notice that this result is in contrast with the basic design guidelines of cam-follower mechanisms, where high order polynomial laws are usually advised since they guarantee the smoothness of high order derivative of the follower radial motion. 5. Concluding remarks The present work addresses the correlation between the main vane design parameters and the theoretical flow ripple in balanced vane pumps. The analysis, based on the hypotheses of no leakages and incompressible fluid, is carried out by means of an analytical procedure, which starts from the determination of the vane radial motion in reference to the exact position of the contact point. Within this context, the method takes into account the influence of the vane geometry, represented by the two parameters that define it in practice: the vane tip radius and the vane thickness. The radial motion of the vanes is then used to calculate the volume variation of the under-vane pockets and the displaced chambers along a complete pumping period. Finally, the described method allows us to calculate the contribution to the delivery flow ripple given by the displaced chambers and the under-vane pockets and analyze them separately. The described methodology has been then applied on a realistic pump geometry, in order to assess the potential outcomes and address the influence of the vane shape on the theoretical flow ripple. The analysis has shown that the vane tip radius mainly affects the under-vane pocket contribution: as the tip radius increases, the amplitude of the flow ripple increases as well and its timing is varied. On the other hand, the influence on the flow ripple caused by the pumping action of the displaced chambers is negligible. As a final outcome, the amplitude of the total delivery flow ripple is increased. The phenomenon is observed also in the order domain, underlining that such a behavior is linked to the increment of the amplitude of the first VPF order. The method has been also adopted to enlighten the flow ripple behavior in reference to the vane thickness. The results have shown that the vane thickness acts as a tuning parameter of the ratio between the mean flow rate provided by the displaced chambers and the one provided by the under-vane pockets. In addition, such a parameter is linked to the NVH behavior of the pump since the amplitude of the flow fluctuations related to both the under-vane pockets and the displaced chambers tends to increase in accordance with the vane thickness increment. Moreover, by focusing the attention on the delivery flow ripple in the order domain, the thickness increase is shown to consistently alter the produced waveforms by influencing a large number of harmonics of the VPF. In order to provide a global overview of the interaction between the delivery flow ripple and the pump geometry, the analysis has been repeated by varying both vane thickness and tip radius simultaneously. The results demonstrated that, as a general rule, the amplitude of the flow ripple oscillations tends to decrease as both vane thickness and tip radius are decreased. In addition, the influence of the tip radius is shown to become progressively more relevant as the thickness increases, while it is almost negligible for low-thickness vane layouts. With the purpose to further investigate the link between the pump geometry and its kinematic flow ripple, this latter analysis has been then repeated for six different cam ring profiles, designed to provide the same theoretical displacement with different mathematical laws describing rise and fall phases. The results demonstrate that the overall amplitude of the delivery flow ripple tends to increase as the mathematical law diverges from the linear trend and therefore low order polynomial laws should be preferred. References [1] J. Ivantysyn, M. Ivantysynova, Hydrostatic Pumps and Motors: Principles, Design, Performance, Modelling, Analysis, Control and Testing, first english ed., Tech Books International, New Delhi, India, 2003. [2] M. Battarra, E. Mucchi, A method for variable pressure load estimation in spur and helical gear pumps, Mech. Syst. Signal Process. 76–77 (2016) 265–282, doi:10.1016/j.ymssp.2016.02.020. [3] M. Battarra, E. Mucchi, Incipient cavitation detection in external gear pumps by means of vibro-acoustic measurements, Measurement 129 (June 2018) 51–61, doi:10.1016/j.measurement.2018.07.013. [4] K. Foster, R. Taylor, M. Bidhendi, Computer prediction of cyclic excitation sources for an external gear pump, Proc. Inst. Mech.Eng. Part B 199 (3) (1985) 175–180, doi:10.1017/CBO9781107415324.004.
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
15
[5] T. Costopoulos, A. Kanarachos, E. Pantazis, Reduction of delivery fluctuation and optimum tooth profile of spur gear rotary pumps, Mech. Mach. Theory 23 (2) (1988) 141–146. [6] E. Mucchi, G. Dalpiaz, A. Fernandez Del Rincon, Elastodynamic analysis of a gear pump. Part I: pressure distribution and gear eccentricity, Mech. Syst. Signal Process. 24 (7) (2010) 2160–2179, doi:10.1016/j.ymssp.2010.02.003. [7] E. Mucchi, G. Dalpiaz, A. Rivola, Elastodynamic analysis of a gear pump. Part II: meshing phenomena and simulation results, Mech. Syst. Signal Process. 24 (7) (2010) 2180–2197, doi:10.1016/j.ymssp.2010.02.004. [8] J.J. Watton, K.L. Watkins-Franklin, The transient pressure characteristic of a positive displacement vane pump, Proc. Inst. Mech.Eng. Part A 204 (4) (1990) 269–275, doi:10.1243/PIME_PROC_1990_204_036_02. [9] B.L. Jones, D.N. Johnston, D.K. Longmore, Simulation of suction flow ripple in power steering pumps, in: SAE Technical Paper, 1998, pp. 89–95, doi:10. 4271/982023. [10] M. Buzzoni, M. Battarra, E. Mucchi, G. Dalpiaz, Motion analysis of a linear vibratory feeder: dynamic modeling and experimental verification, Mech. Mach. Theory 114 (2017) 98–110, doi:10.1016/j.mechmachtheory.2017.04.006. [11] M. Battarra, E. Mucchi, G. Dalpiaz, A model for the estimation of pressure ripple in tandem gear pumps, in: IDETC/CIE 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 2015. V010T11A018; 9 pages; doi: 10.1115/DETC2015-46338. [12] E. Frosina, A. Senatore, D. Buono, K.A. Stelson, F. Wang, B. Mohanty, M.J. Gust, Vane pump power split transmission: three dimensional computational fluid dynamics modeling, ASME/BATH 2015 Symposium on Fluid Power and Motion Control, 2015. V001T01A011 doi: 10.1115/FPMC2015-9518. [13] M. Rundo, Massimo, Models for flow rate simulation in gear pumps: a review, Energies 10 (9) (2017) 1261, doi:10.3390/en10091261. [14] C. Bonacini, Sulla portata delle pompe a ingranaggi, L’ingegnere 9 (1961). [15] C. Bonacini, Sulle pompe a ingranaggi a dentatura elicoidale, Tecnica Italiana 3 (1965). [16] N.D. Manring, S.B. Kasaragadda, The theoretical flow ripple of an external gear pump, J. Dyn. Syst. Meas. Control 125 (3) (2003) 396–404, doi:10.1115/ 1.1592193. [17] X. Zhao, A. Vacca, Numerical analysis of theoretical flow in external gear machines, Mech. Mach. Theory 108 (October 2016) (2017) 41–56, doi:10.1016/ j.mechmachtheory.2016.10.010. [18] G. Mimmi, P. Pennacchi, Involute gear pumps versus lobe pumps : a comparison, J. Mech. Des. 119 (December) (1997) 458–465. [19] A.M. Karmel, A study of the internal forces in a variable-displacement vane-Pump–Part I: a theoretical analysis, J. Fluids Eng. 108 (2) (1986) 227. [20] A. Giuffrida, R. Lanzafame, Cam shape and theoretical flow rate in balanced vane pumps, Mech. Mach. Theory 40 (3) (2005) 353–369, doi:10.1016/j. mechmachtheory.20 04.07.0 08. [21] M. Battarra, A. Blum, E. Mucchi, Kinematics of a balanced vane pump with circular tip vanes, Mech. Mach. Theory 137 (2019) 355–373, doi:10.1016/j. mechmachtheory.2019.03.034. [22] E.S. Taylor, Dimensional Analysis for Engineers, Oxford University Press, 1974. [23] R.L. Norton, Cam Design and Manufacturing Handbook, second ed., 2009, doi:10.1037/023990. [24] A. Kahraman, R. Singh, Non-linear dynamics of a spur gear pair, J. Sound Vib. 142 (1) (1990) 49–75, doi:10.1016/0022- 460X(90)90582- K. [25] A. Kahraman, G.W. Blankenship, Experiments on nonlinear dynamic behavior of an oscillator with clearance and periodically time-varying parameters, J. Appl. Mech. 64 (1) (1997) 217–226.
Contents lists available at ScienceDirect
Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory
Research paper
On the relation between vane geometry and theoretical flow ripple in balanced vane pumps Mattia Battarra, Ph.D.∗, Emiliano Mucchi University of Ferrara, Engineering Dept. Via G. Saragat, 1 - 44122 Ferrara, Italy
a r t i c l e
i n f o
Article history: Received 5 November 2019 Accepted 29 November 2019
Keywords: Vane pump Cam ring Theoretical flow ripple Design parameter
a b s t r a c t Advancements related to the correlation between pump design parameters and the kinematic flow ripple in balanced vane pumps are addressed in the present work. In particular, the study focuses the attention on the influence of the vane geometry on the oscillations of the flow rate produced by the volume variation of both under-vane pockets and displaced chambers, that is known as one of the most relevant sources of noise in hydraulic systems. The working principle of the machine is detailed and used as starting point to deduce analytical correlations describing both vane kinematics and delivery flow rate ripple. The set of results that can be achieved with the obtained formulation is evaluated by means of a nondimensional parametric study including the two main design parameters defining the vane geometry, i.e. thickness and tip radius. The resulting trends demonstrate that the theoretical delivery flow ripple is closely related to the vane design and the cam ring shape profile. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction In the field of power transmission, balanced vane pumps are widely adopted positive displacement machines characterised by interesting features such as compactness, reliability and high values of the power over weight ratio [1]. These characteristics are particularly appreciated in both automotive and aeronautic applications, where the design of such machines must match a number of constraining requirements related to both fluid-dynamic and Noise, Vibration and Harshness (NVH) performance. As a matter of fact, the operating principle of such pumps is directly responsible for their NVH behavior, which often represents the main drawback to be overcome [2,3]. The pump cyclically displaces oil from the suction side to the delivery port, producing periodically varying outlet flow rate and pressure that excite the pump body and the piping system. The link between fluid-dynamic behavior and NVH characteristics of positive displacement machines has been widely studied in the specialized literature since the ’80s, when researchers started focusing on the determination of the cyclic loads given by delivery pressure and flow rate oscillations [4–7]. The assessment of these dynamic loads has been performed with different approaches, such as zero dimensional models (see for example Refs. [8–11]) and Computational Fluid Dynamic models (as the ones reported in Refs. [12,13]). Within this framework, a peculiar solution is represented by the analytical approach, which allows us to estimate the variable pressure and flow characteristics of the pump on the basis of its kinematics. Despite these methodologies may seem too ∗
Corresponding author. E-mail addresses: [email protected] (M. Battarra), [email protected] (E. Mucchi).
https://doi.org/10.1016/j.mechmachtheory.2019.103736 0094-114X/© 2019 Elsevier Ltd. All rights reserved.
2
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
Nomenclature
β
ˆ·
ω θ EF θ ER θ SF θ SR
ϑ Arot Astat Av B b dv Dth n nd O P p Qth rc rr rv t tv Vdc Vuv z
Superscript indicating derivative with respect to angle Pressure angle between vane tip and stator Symbol indicating a nondimensional quantity Angular velocity in rad/s Angle defining the end of the fall phase in the cam ring profile Angle defining the end of the rise phase in the cam ring profile Angle defining the beginning of the fall phase in the cam ring profile Angle defining the beginning of the rise phase in the cam ring profile Angular coordinate of the reference vane Area of the angular sector covered by the rotor radius between two consecutive vanes Area of the angular sector covered by the cam ring radius between two consecutive vanes Area of the portion of the vane delimiting the displaced chamber Oil Bulk modulus Pump facewidth Vane radial displacement Theoretical volumetric displacement of the pump Angular velocity in rpm Number of displaced chambers and under-vane pockets exposed to the delivery port Pump center and origin of the reference frame Contact point Oil pressure Instantaneous delivery flow rate Cam ring radius Rotor radius Vane tip radius Time Vane thickness Volume of the displaced chamber Volume of the under-vane pocket Number of vanes
generalist and leading to qualitative results, they are often adopted with satisfactory results during the early stages of the design process, when the detailed geometry of the machine is still to be defined and designers need to establish the very first parameters such as number of vanes and aspect ratio. In this context, various scientific works are focused on the determination of analytical formula to predict the delivery flow ripple. Such studies are nowadays performed with the purpose to include the effect provided by an increasing number of geometrical parameters and design solutions. In particular, this approach has been successfully deepened in the field of external gear pumps, where the first formulation proposed by Bonacini in [14] has been fruitfully extended to pumps with helical gears [15], pumps with not-unitary transmission ratio [16] as well as gear pumps with asymmetric tooth profile [17]. However, similar analyses can be found also for other positive displacement machines such as trochoidal pumps [18] and variable displacement vane pumps [19]. In the specific context of balanced vane pumps, the analytical approach still represents a methodology rarely studied. The most representative work pertains to Giuffrida et al. [20], in which the authors defined a dedicated procedure to estimate the theoretical flow ripple in reference to the cam ring shape and the vane thickness. The dissertation is built up on the hypotheses of flat tip vanes with sharp edges and under-vane pockets that contribute to displace oil into the outlet chamber. On the basis of these aspects, the present work has the main purpose to propose an improvement of the methodology provided in [20], by including the effect of the tip radius in the definition of the theoretical flow rate oscillations caused by the pump. The proposed methodology takes advantage of the pump kinematic analysis described by the present Authors in Ref. [21]. A parametric study is adopted to evaluate the capabilities of the method, with a specific focus on the influence of both tip radius and vane thickness on the characteristics of the delivered flow ripple. In particular, the first parameter is shown to cause a progressive deformation of the flow rate waveform produced by the under-vane pockets and an increase of the amplitude of the total delivered flow ripple, which reflects into an amplification of the first order of the vane passage frequency. On the contrary, the latter acts as a tuning parameter to subdivide the delivered flow rate between the undervane pockets and the displaced chambers. In addition, from a NVH point of view, the consequence of the vane thickness increment is observed also on the amplitude of the delivery flow rate waveform, due to the increase of the amplitude of several harmonics of the vane passage frequency. The obtained results underline the lack of knowledge regarding the deep correlation existing between the pump behavior and several design parameters defining its geometry. Within this context, the purpose of the present study is to improve the comprehension of the physics that correlate the pump main geometrical
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
3
Fig. 1. Basic components of a balanced vane pump.
characteristics to its kinematic flow rate ripple. This goal is achieved by presenting an analytical formulation that may find practical applications in real design processes, where the chance to obtain instantaneous indications on the machine behavior are widely appreciated. In this framework, the proposed formulation may provide theoretical strength to design choices that are often based on the experience, in particular when the detailed machine geometry is still to be defined. The following section describes the operating principle of a balanced vane pump and the main components constituting the pump mechanism. Section 3 presents the methodology followed to determine the delivery flow ripple in reference to both vane tip radius and vane thickness. Section 4 reports the results of the parametric study, which is performed on the basis of the nondimensionalization theory in order to provide general results. Eventually, concluding remarks are reported in Section 5. 2. Pump operating principle In the present section, the main components constituting the pump mechanism are described together with the pump operating principle, with a particular focus on the basic design philosophy standing at the basis of this machine. As it will be enlightened in Section 3, this aspect substantially defines the pump theoretical behavior. By taking as reference the schematic representation in Fig. 1, the pump may be described by the interaction between three fundamental items: the external stator, namely the cam ring, the internal rotor and the vanes. Each vane is forced to move on a dedicated track created within the rotor, which moves them from the suction port to the delivery port. During the rotation, the oil is trapped in the chamber limited by two consecutive vanes, namely the displaced chamber, and delivered from the inlet side to the outlet one. Along this process, the peculiar shape of the cam ring guarantees the expansion and subsequent compression of the chambers, which is the phenomenon at the basis of the displacing action. The described working principle leads to the major drawback constituted by flow rate and pressure oscillations, representing a common undesired by-product for all the positive displacement machines. In addition, part of the oil suctioned from the inlet chamber is trapped by the pockets that are formed under the vanes, namely the under-vane pockets. The pumping action of such pockets is naturally caused by the radial motion of the vanes and they provide a not-negligible contribution to the definition of the global displacement of the pump. Inlet and outlet ports related to the under-vane pockets, not reported in Fig. 1, are usually timed with the suction and delivery ports related to the displaced chambers. The overall flow rate oscillation at the outlet port is therefore generated by the synergy between the pumping action of the displaced chambers and the pumping action of the under-vane pockets. The described process is repeated twice per revolution, since the complete pump is constituted by two identical mechanisms, which are symmetrical with respect to the center of the rotor. The overall mechanism is usually packed between two plates providing the axial sealing. The complete pump package includes a number of additional components and features designed to improve the mechanical and fluid-dynamics behavior of the machine; however, the related description is omitted being out of the scope of the present study. On the basis of this operating principle, it may be straightforwardly deduced that the pumping capability of the machine is mainly governed by the shape of the cam ring and its interaction with the sliding vanes. Within this context, the external stator can be considered as a fixed cam, which guides the rotating followers, i.e. the vanes, by means of its internal profile and determines the machine displacement. With reference to Fig. 1, the volume of the displaced chamber, highlighted by yellow stripes, may be defined by:
V1 =
2π b rcmax 2 − rr 2 z 2
(1)
where b is the pump facewidth while rcmax is the maximum value of the cam ring radius, rr is the rotor radius and z is the vane number. Concurrently, the volume of oil brought back from the outlet chamber to the inlet chamber, highlighted by purple stripes in Fig. 1, may be defined by: 2
V2 =
2π b rcmin − rr 2 z 2
(2)
4
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
Fig. 2. Representation of the interaction between vane and cam ring as a cam-follower mechanism including the vectors constituting the closure equation of the linkage.
where rcmin is the minimum value of the cam ring radius. By considering the contribution of Eqs. (1) and (2) along a complete revolution, pump theoretical volumetric displacement Dth is calculated as:
Dth = 2z (V1 − V2 ) = 2π b rcmax − rcmin
rcmax + rcmin
(3)
The volume reduction affecting the displaced chamber due to the vane thickness is usually compensated by the under-vane pocket contribution. A detailed explanation of Eq. (3) may be found in Ref. [1] pages 343–352, however this brief analysis underlines that the conceptual design of the pump necessarily starts with the determination of the stator internal profile, i.e. the cam ring. By the time the cam ring shape is defined, the designer usually focuses the attention on the vane geometry, that is defined, in its basic form, by the vane thickness and the vane tip radius. Once both vane and cam ring are chosen, the shape of the pockets displacing the oil from the suction chamber to the delivery one is also determined. As a matter of fact, this means that the kinematic flow ripple of the pump is defined and no further geometrical parameters are available to tune this pump characteristic. Based on the depicted scenario, the possibility to correlate both cam ring shape and vane geometry to the theoretical delivery flow rate ripple of the pump is of major relevance during the design process. In this context, an analytical formulation represents a powerful tool to provide theoretical strength to design choices that are often based on the experience. 3. Theoretical flow rate determination The theoretical delivery flow ripple of a positive displacement pump, i.e. the kinematic flow ripple, is represented by the flow rate oscillation at the outlet port calculated on the basis of three main hypotheses. In particular, (i) the pump motion is assumed to be purely kinematic, i.e. dynamic effects such as vane tip detachment and vibration of the components is neglected, (ii) leakage phenomena are not taken into account and (iii) the oil is considered as an incompressible fluid. These hypotheses are always taken as the starting point of all the studies carried out on this subject (see for example Refs. [16,17,20]). Within this context, as also explained in [20], instantaneous delivery flow rate Qth is determined by Eq. (4):
Qth (ϑ ) = −ω
nd k Vdc |ϑk + Vukv |ϑk
(4)
k=1
where nd represents the number of displaced chambers and under-vane pockets exposed to the delivery port, ω is the pump mean velocity, ϑ is the angular position of the reference vane, while Vuv and Vdc are the volume of the under-vane pocket and displaced chamber, respectively. The problem therefore reduces to the determination of the volume variation for both under-vane pockets and displaced chambers along a complete period of the rotation. It is worth specifying that the negative sign in Eq. (4) is mandatory to obtain Qth with a positive mean value. This is due to the fact that, when the pockets are exposed to the outlet port, their volume is necessarily decreasing in order to deliver the fluid. In order to calculate these two contributions, the pump is modeled as a cam-follower mechanism with contact taking place on the inner side of the cam profile. Fig. 2 provides a schematic representation of the geometrical problem, where the cam ring acts as an internal cam fixed to the frame and the vane acts as a rotating follower. Within this framework, by
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
5
Fig. 3. Geometrical representation of the angles described in Eq. (7).
Fig. 4. Calculation of the slope of the cam ring profile at the contact point with respect to its base circle defined by radius rcmin .
considering a circular tip follower, the vector chain constituted by vane radial displacement dv between pump center and circular tip center and the vector between the circular tip center and contact point P must be equal to the vector defining the position of contact point P with respect to the pump center. The described vector chain translates into the following closure equation, written with respect to a reference frame that rotates simultaneously with the rotor:
dv ( ϑ ) + rv e where β
iβ
= rc (P )e
i β +tan−1
rc P rcmin
|
(5)
represents the pressure angle at contact point P, while term rc . |P is the value of the first angular derivative of the
cam ring profile at contact point P. Within this framework, term
r c |P
rcmin .
rcmin
basically represents the angular deviation of the cam
ring profile with respect to its base circle, defined by radius Its geometrical explanation in the cam ring environment is provided in Fig. 3, while Fig. 4 is reported to clarify the mathematical meaning of such a term. As it may be appreciated, term
r c |P rcmin
defines the slope of the cam ring profile in a Cartesian plane, where the x-axis is constituted by the perimeter of
the maximum circumference inscribed inside the cam ring profile, which is actually defined by radius rcmin . For the sake of compactness, the following notation is adopted:
ψ = tan
−1
rc |P rcmin
(6)
6
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
where ψ is the angle associated to the local slope of the cam ring profile. Since the cam ring profile is given by design, Eq. (5) can be expanded in a system of two scalar equations and used to obtain both vane displacement dv and pressure angle β once the tip radius has been defined. A detailed dissertation of the adopted kinematic approach has been proposed by Battarra et al in [21]. It is worth underlining that the vane thickness does not appear in Eq. (5), therefore, in the hypotheses of circular tip, such a parameter does not directly affect the vane motion. This aspect makes the present dissertation different from the methodology based on the flat tip assumption, in which the vane thickness plays a key role in the definition of the vane radial displacement [20]. For the sake of completeness, it has to be underlined that, in order to completely define the vane radial displacement, it is required to obtain also angular position ϑ associated to each value of dv . By taking as reference Fig. 3, the following geometrical relationship can be defined:
ϑ = ϑ (P ) − β − ψ
(7)
where term ϑ(P) represents the angular coordinate associated to contact point P. A graphical explanation of the angular correlation is provided in Fig. 3. Once Eq. (5) is solved, both contact point position and pressure angle are defined, therefore angle ϑ can be straightforwardly calculated. It is worth mentioning that the closure equation approach may be avoided by applying the cosine theorem on the triangle defined by points O, P and the center of the tip circle, which leads to:
dv =
rc 2 + rv 2 − 2rv rc cos ψ
(8)
Eq. (8) is valid throughout the entire revolution and it may be used in place of Eq. (5) to calculate the magnitude of dv . Once the vane radial displacement is obtained, the derivative of the volume related to the under-vane pocket can be calculated by multiplying the cross-sectional area of the pocket with velocity coefficient of the vane radial motion dv :
Vuv = b · tv · dv
(9)
where term tv represents the vane thickness. Here, the term velocity coefficient refers to its kinematic meaning:
dv =
∂ dv 1 ∂ dv = · ∂ϑ ω ∂t
(10)
The computation of dv may be performed both analytically or numerically, depending on the mathematical law adopted to design the cam ring profile.For the sake of clarity, it is worth noticing that the definition in Eq. (10) applies to all the terms with prime superscript. Eq. (9) can be straightforwardly expanded to provide an analytical definition of the derivative of the under-vane volume:
Vuv = b · tv ·
rc · (rc − rv cos ψ ) + rv rc ψ sin ψ
(11)
rc2 + rv2 − 2rv rc cos ψ
It is worth noticing that Eq. (11) is always applicable as long as the mathematical law chosen to describe the cam ring profile admits first and second derivative. in Eq. (4) reWhereas term Vuv has a straightforward definition, the calculation of the contribution given by term Vdc quires, as a first step, to determine the evolution of the displaced chamber volume along a complete pumping period, i.e. half of the revolution. In practice, for each angular position ϑ of the leading vane, volume Vdc may be obtained by the algebraic sum of three terms:
Vdc (ϑ ) = b Astat − Arot −
1 Av (ϑ ) + Av ϑ − 2zπ 2
(12)
where Astat is the area of the angular sector covered by the cam ring profile between two consecutive vanes, Arot is the area of the angular sector covered by the radius of the rotor between two consecutive vanes while Av represents the area of the portion of the vane delimiting the displaced chamber. Fig. 5 provides a graphical explanation of the three contributions defined in Eq. (12). The most relevant term, represented by Astat , may be obtained from the generic integral used to compute the area of an angular sector defined by a variable radius:
Astat (ϑ ) =
1 ϑ 2 [rc (γ )] dγ 2 ϑ − 2π z
(13)
where γ is an auxiliary variable defined within the closed interval ϑ − 2zπ , ϑ . The solution of the integral in Eq. (13) can be obtained both analytically or numerically, depending on the definition of term rc . However, the cam ring profile is commonly described by a piecewise-defined polynomial function, therefore an analytical expression for Vdc (ϑ) can usually be achieved. Known the volume course along half a revolution, the contribution to the delivery flow rate ripple given by the displacement chamber can be directly obtained by differentiating it with respect to the angular position of the leading vane. In addition, since term Arot is a constant value throughout the entire revolution, its practical calculation in Eq. (12) may be neglected. Concurrently, regarding term Av , its derivative reduces to the calculation of the cam ring derivative since:
A v ( ϑ ) = t v [r c ( ϑ ) − r r ]
(14)
For the sake of completeness, it has to be observed that the exact definition of area Av would require to include the vane area constituted by the circular segment defined the by circular vane tip. However, since this term remains constant
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
7
Fig. 5. Geometrical representation of the three contributions adopted to compute the volume of the displace chamber at each angular position of the leading vane.
throughout the entire revolution, it does not concur to the definition of the first derivative of volume Vdc and therefore its computation may be avoided. It is worth noticing that the described procedure allows us to evaluate the theoretical flow rate delivered by a balanced vane pump taking into account the effects produced by both vane thickness tv and tip radius rv . In addition, it is possible to focus the attention on the two sources of flow rate oscillations separately, i.e. the displaced chamber contribution and the under-vane pocket contribution, by simply isolating the two terms in Eq. (4). In particular, the first one is given by: dc Qth (ϑ ) = −ω
nd
k Vdc |ϑk
(15)
k=1
while the second one is determined by: uv Qth (ϑ ) = −ω
nd
Vukv |ϑk
(16)
k=1
Since parameters tv and rv appear both in Eqs. (9) and (13), such terms are expected to influence the displaced chamber contribution and the under-vane pocket one, concurrently. 4. Parametric study The current section is devoted to enlighten the outcomes achievable from the methodology described in Section 3. Moreover, the study assesses the influence of the vane geometry by evaluating the trends imposed to the amplitude of the delivery flow ripple by the variation of different design parameters. In order to perform this study, all the geometrical parameters are defined in nondimensional form (indicated by the ˆ· symbol) by dividing them with respect to the minimum value of the cam ring radius rcmin on the basis of the Buckingham’s Theorem [22]. The pump displacement per facewidth unit may be obtained from Eq. (3) as:
Dth = 2π rcmax − rcmin 2
2
(17)
it is possible to express the specific pump displacement as:
Dˆ th = 2π eˆ2 − 1
(18)
where term eˆ is the ratio between rcmax and rcmin . Based on the Buckingham’s Theorem, the proposed dimensional reduction applies to all the parameters defining the pump geometry. In particular, hereinafter the following dimensionless groups are adopted:
⎡
tv tˆv = speci f ic vane thickness = rmin c rv ⎣rˆv = speci f ic vane tip radius = rmin c rc rˆc = speci f ic cam ring radius = rmin c
(19)
8
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736 Table 1 Main design parameters of the pump design taken as reference. z eˆ ϑSR ϑER ϑSF ϑEF
10
1 + 1/2π
π /10 2π /5 3π /5 9π /10
Fig. 6. Reference cam ring profile in polar coordinates.
On the basis of this method, all the angles need to be reported in radians, which represent specific quantities by definition. The adopted approach allows to deduce general results that are common to a family of pumps with the same specific displacement defined by the value associated to eˆ and same cam ring shape. Table 1 reports the main parameters defining the machine geometry while the cam ring profile is reported in Fig. 6 in polar coordinates. Parameter eˆ has been chosen equal to 1 + 1/2π in order to obtain Dˆ th = 1. Rise phase, i.e. profile geometry between ϑSR and ϑER , and fall phase, i.e. profile geometry between ϑSF and ϑEF , have been obtained by means of a 5th order polynomial law:
rˆc (ϑ ) =
5
ak ϑ k
(20)
k=0
where ak is the kth coefficient to be determined. The coefficients are computed by imposing the continuity of the law, together with its first two derivatives, at the boundary of the reference angular interval. This law is a commonly adopted mathematical law in cam design, since it guarantees the smoothness of the radial displacement, velocity and acceleration of the vane, which actually behaves as a proper follower [23]. It is worth noticing that this choice does not affect the generality of the dissertation that would lead to similar conclusions independently on the mathematical law adopted to define the cam ring profile. As underlined in Ref. [21], the mathematical law describing rise and fall phase affects the smoothness of the vane radial motion, however it does not alter the main kinematic characteristics of the vane-cam ring mechanism. For this reason, different mathematical laws would lead to different waveforms of the flow rate ripple, but they would not affect how the vane tip radius and the vane thickness affect such waveforms. This aspect is further highlighted in Section 4.4, where the parametric study is carried out with respect to multiple cam ring profiles, characterized by the same theoretical displacement and different mathematical laws describing rise and fall phases. The following subsections detail the results of the parametric study, which is carried out by varying the vane tip radius and the vane thickness separately and then simultaneously. Within this framework, it is worth noticing that tˆv and rˆv are not independent quantities and they must respect the following mathematical constrains at each angular position of the reference vane:
0 ≤ tˆv ≤ 2 tˆv 2
≤ rˆv ≤
√
tˆv2 rˆc cos ψ −tˆv rˆc sin |ψ | 2 tˆv2 −4rˆc2 sin ψ
4rˆc2 −tˆv2
(21)
The complete theoretical dissertation regarding this subject may be found in Ref. [21]. For the scope of the present work, it is sufficient to state that maximum and minimum allowable values for both vane thickness tˆv and tip radius rˆv are calculated on the basis of Eq. (21) in order to guarantee the admissibility of the vane geometry.
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
9
Fig. 7. Effect of the vane tip radius increment on the theoretical flow ripple referring to displaced chambers (a) and under-vane pockets (b).
Fig. 8. Effect of the vane tip radius increment on the theoretical flow ripple globally delivered by the pump.
4.1. Vane tip radius influence The effect produced by the vane tip radius is now evaluated by considering the pump design parameters defined in Table 1 with a fixed value of the specific vane thickness equal to 0.1. Within this framework, the delivery flow ripple pertaining to the displaced chambers and the one referring to the under-vane pockets is estimated (Fig. 7a and b, respectively) for different values of the vane tip radius from 0.05 to 0.35. For the sake of generality, specific flow rate oscillation is obtained by dividing the delivered flow rate by the working speed and the pump facewidth. As it can be observed, both displaced chamber and under-vane pocket ripples are dominated by the periodic compression of the volumes exposed to the delivery chamber. In this scenario, the influence of the tip radius increment on the displaced chamber contribution, despite present, is substantially inconsequential. On the contrary, such an increment produces a not-negligible effect on the undervane pocket contribution by increasing the amplitude of the oscillation and varying its timing. The phenomenon has similar drawbacks on the total theoretical flow ripple, shown in Fig. 8, in which the amplitude of the oscillation follows the same trend of the under-vane pocket contribution, as expected. The reason for this behavior has to be found in the consequences that the tip radius variation produces on the machine kinematics. As observed in [21], if rˆv increases, the vane radial motion diverges from the cam ring profile producing the phase-shift in Fig. 7b. On the other hand, the flow rate pertaining to the displaced chambers is mainly governed by the cam ring profile, while modifications on the vane kinematics exclusively affect term Av in Eq. (12).
10
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
Fig. 9. Amplitude of the first five orders of the VPF in reference to the vane tip radius.
The delivery flow rate ripple can be considered as an indicator of the pressure ripple loading the pump since it has been widely demonstrated that the pockets of positive displacement machines tend to respect the following continuity equation:
∂p B = Q ∂t V
(22)
where B is the oil Bulk modulus. On the basis of this consideration, from a NVH point of view, the analysis of the delivery flow ripple may be further deepened by evaluating it in the frequency domain. Within this framework, Fig. 9 reports the amplitude of the first five orders of the Vane Passage Frequency (VPF) calculated from the waveforms in Fig. 8. The VPF represents the fundamental frequency of the pump and it has the following definition:
V PF = z
ω
2π
(23)
As it can be observed from Fig. 9, the vane tip radius strongly affects the amplitude of the first order, while the remaining harmonics show negligible variations.
4.2. Vane thickness influence In analogy to the previous subsection, the analysis is now focused on evaluating the effect of the vane thickness by considering the pump design parameters defined in Table 1 with a fixed value of the vane tip radius equal to 0.2. The chosen value for rv allows us to span an interval for the vane thickness that goes from 0.1 to 0.4. The results of the vane thickness increment on the two flow ripple contributions are reported in Fig. 10. As it can be observed, this parameter has a relevant effect on both flow rate ripples by influencing their amplitude and mean value, simultaneously. In particular, the mean value of the under-vane pocket flow ripple tends to be increased by the vane thickness increment while the behavior is the opposite for the displaced chamber flow ripple. As a matter of fact, the vane thickness mainly acts as a tuning parameter of the ratio between the mean flow rate provided by the displaced chambers and the one provided by the under-vane pockets. The amplitude of the oscillation, on the contrary, tends to increase in both cases, and this effect is clearly recognized also by focusing the attention on the total theoretical flow ripple that is shown in Fig. 11. It is worth noticing that, despite the outlet flow rate is differently distributed between under-vane pockets and displaced chambers depending on the value of tˆv , the mean total delivery flow rate remains unaltered. Fig. 11 also reveals a secondary drawback related to the vane thickness increment that tends to be masked when the attention is focused on the two contributions separately. As tˆv increases, the influence of term Av in Eq. (12) becomes more and more relevant, producing peaks in the global flow rate ripple with higher frequency content with respect to the main flow rate pulsation. Such a phenomenon is generated in the displaced chamber flow rate ripple, despite slightly noticeable (see Fig. 10), and it results amplified as the two contributions are summed up. With the purpose to further investigate the link between the vane thickness and the theoretical flow rate ripple, the waveforms reported in Fig. 11 are now analyzed in the frequency domain. Fig. 12, in particular, depicts the amplitude of the first five orders of the VPF in correlation with the analyzed range of vane thickness values. The detected behavior is considerably different with respect to the one observed in the previous subsection, since in this case the geometric parameter is affecting all the considered harmonics. In particular, the vane thickness increase causes a less severe increment of the first order compared to the one generated by the tip radius, while it produces a not-negligible increase of the amplitude of the higher orders. This phenomenon is a direct consequence of the behavior observed in Fig. 10, where the increase of tˆv is shown to cause the appearance of peaks with higher frequency content with respect to the main flow rate pulsation.
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
11
Fig. 10. Effect of the vane thickness increment on the theoretical flow ripple referring to displaced chambers (a) and under-vane pockets (b).
Fig. 11. Effect of the vane thickness increment on the theoretical flow ripple globally delivered by the pump.
4.3. Tip radius and thickness interaction Sections 4.1 and 4.2 described the influence of a single parameter on the theoretical flow ripple, demonstrating that tip radius and vane thickness produce different effects on the flow rate oscillation. In order to provide a global overview of their influence, the purpose of the present subsection is to analyze the delivery flow rate waveform with respect to the simultaneous variation of rˆv and tˆv . Within this framework, the globally delivered flow ripple has been calculated for the entire admissible domain for the couple (rˆv , tˆv ). In order to evaluate the result of each geometrical configuration, the obtained waveforms are replaced by the rating parameter ARMS , which is calculated as the Root Mean Square (RMS) value of the first five harmonics of the VPF:
ARMS =
5
Ak 2
(24)
k=1
where Ak is the amplitude of the kth harmonics of the VPF. The rating parameter, as also reported in various works (see for example Refs. [10,24,25]), is strictly correlated to the energy content of the waveform. The result of the described analysis is reported in Fig. 13, where the dashed lines represent the boundaries of the geometry admissible domain. The rating parameter is displayed in logarithmic scale for the ease of the representation and the chart is limited to tˆv < 0.8, since higher values of vane thickness are not feasible for real applications. As it may be observed, the RMS value tends to decrease as both the tip radius and the vane thickness decrease, reaching zero for both rˆv and tˆv equal to zero. In agreement with the results of the analyses in the previous subsections, as a general rule, the amplitude of the flow ripple oscillations tends to decrease as both vane thickness and tip radius are decreased. In addition,
12
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
Fig. 12. Amplitude of the first five orders of the VPF in reference to the vane thickness.
Fig. 13. Contour plot of the RMS value of the first five harmonics of the VPF referred to Qˆth , obtained within the admissible domain for rˆv and tˆv .
it is worth noticing that rˆv produces different behaviors depending on the value associated to tˆv . For value of tˆv between 0 and 0.1, the isolines are almost vertical, meaning that the increase of rˆv produces negligible effects on the amplitude of the delivery flow ripple. On the contrary, as the vane thickness is increased, the isolines tend to flatten themselves, meaning that the influence of rˆv and tˆv is comparable in magnitude. 4.4. Cam ring profile influence The purpose of the present subsection is to extend the results obtained in Section 4.3 to different cam ring profiles in order to ensure the general applicability of the proposed method and evaluate the influence of the mathematical law adopted to describe rise and fall phases. Within this framework, six different cam ring profiles have been defined, each of them based on the same design parameters reported in Table 1. Fig. 14 reports a detailed view of the rise phase profile of the six cam rings. In agreement with the parameters in Table 1, rise and fall profiles are symmetrical. As it may be noticed, four profiles are obtained from the polynomial law with order from two to seven, while the last two profiles are the cycloidal law and the double harmonic law. Detailed information regarding the mathematical formulation behind these laws and their application to real scenarios is reported in Ref. [23], while Ref. [21] provides explanation on how they affect the vane radial motion. For the sake of clarity, it is worth noticing that the 2nd order polynomial law seems to follow a higher divergence rate from the linearity than the 3rd order polynomial law. However, this is due to fact that, while a single polynomial is sufficient to describe the rise phase with the 3rd order law, the 2nd order law requires two distinct polynomial in order to guarantee the smoothness of the profile.
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
13
Fig. 14. Detail of the rise phase cam ring profile obtained from six different mathematical laws, adopting the same values of ϑSR and ϑER .
Fig. 15. Contour plot of the RMS value of the first five harmonics of the VPF referred to Qˆth , obtained with six different mathematical laws for the cam ring profile design.
14
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
According to Eq. (3), the six cam ring profiles guarantee the same pump theoretical displacement since they reach the same maximum value of rcmax , even though the vane radial motion during rise and fall phases does not coincide. As a direct consequence, each cam ring profile is expected to produce a delivery flow ripple with a characteristic waveform, which is basically governed by the derivative of both under-vane pocket volume Vuv and displaced chamber volume Vdc . In order to provide a general overview regarding the link between cam ring profile and delivery flow ripple, each cam ring shown in Fig. 14 has been used to compute the same analysis reported in Section 4.3. The results are depicted in Fig. 15, showing the RMS value of the amplitude of the first five harmonics of the VPF referred to Qˆth . The magnitude of the contour plots refers to the same logarithmic scale adopted in Fig. 13. As a preliminary consideration, it may be noticed that the 5th order polynomial law and the cycloidal law, which give rise to cam ring profiles that are almost superimposed in Fig. 14, consequently provide similar amplitude fields (Fig. 15c and e). The same applies to the 7th order polynomial law and the double harmonic law, despite the presence of major discrepancies both in the cam ring shape and the amplitude field. From a practical perspective, it is worth underlying that the overall amplitude of the delivery flow ripple tends to increase as the mathematical law diverges from the linear trend. This behavior is a direct consequence of the fact that higher order polynomial laws involve higher values of the associated derivatives. Based on Eqs. (15) and (16), higher values of the volume derivatives produce higher amplitude waveforms of the delivery flow rate as detrimental effect. It is interesting to notice that this result is in contrast with the basic design guidelines of cam-follower mechanisms, where high order polynomial laws are usually advised since they guarantee the smoothness of high order derivative of the follower radial motion. 5. Concluding remarks The present work addresses the correlation between the main vane design parameters and the theoretical flow ripple in balanced vane pumps. The analysis, based on the hypotheses of no leakages and incompressible fluid, is carried out by means of an analytical procedure, which starts from the determination of the vane radial motion in reference to the exact position of the contact point. Within this context, the method takes into account the influence of the vane geometry, represented by the two parameters that define it in practice: the vane tip radius and the vane thickness. The radial motion of the vanes is then used to calculate the volume variation of the under-vane pockets and the displaced chambers along a complete pumping period. Finally, the described method allows us to calculate the contribution to the delivery flow ripple given by the displaced chambers and the under-vane pockets and analyze them separately. The described methodology has been then applied on a realistic pump geometry, in order to assess the potential outcomes and address the influence of the vane shape on the theoretical flow ripple. The analysis has shown that the vane tip radius mainly affects the under-vane pocket contribution: as the tip radius increases, the amplitude of the flow ripple increases as well and its timing is varied. On the other hand, the influence on the flow ripple caused by the pumping action of the displaced chambers is negligible. As a final outcome, the amplitude of the total delivery flow ripple is increased. The phenomenon is observed also in the order domain, underlining that such a behavior is linked to the increment of the amplitude of the first VPF order. The method has been also adopted to enlighten the flow ripple behavior in reference to the vane thickness. The results have shown that the vane thickness acts as a tuning parameter of the ratio between the mean flow rate provided by the displaced chambers and the one provided by the under-vane pockets. In addition, such a parameter is linked to the NVH behavior of the pump since the amplitude of the flow fluctuations related to both the under-vane pockets and the displaced chambers tends to increase in accordance with the vane thickness increment. Moreover, by focusing the attention on the delivery flow ripple in the order domain, the thickness increase is shown to consistently alter the produced waveforms by influencing a large number of harmonics of the VPF. In order to provide a global overview of the interaction between the delivery flow ripple and the pump geometry, the analysis has been repeated by varying both vane thickness and tip radius simultaneously. The results demonstrated that, as a general rule, the amplitude of the flow ripple oscillations tends to decrease as both vane thickness and tip radius are decreased. In addition, the influence of the tip radius is shown to become progressively more relevant as the thickness increases, while it is almost negligible for low-thickness vane layouts. With the purpose to further investigate the link between the pump geometry and its kinematic flow ripple, this latter analysis has been then repeated for six different cam ring profiles, designed to provide the same theoretical displacement with different mathematical laws describing rise and fall phases. The results demonstrate that the overall amplitude of the delivery flow ripple tends to increase as the mathematical law diverges from the linear trend and therefore low order polynomial laws should be preferred. References [1] J. Ivantysyn, M. Ivantysynova, Hydrostatic Pumps and Motors: Principles, Design, Performance, Modelling, Analysis, Control and Testing, first english ed., Tech Books International, New Delhi, India, 2003. [2] M. Battarra, E. Mucchi, A method for variable pressure load estimation in spur and helical gear pumps, Mech. Syst. Signal Process. 76–77 (2016) 265–282, doi:10.1016/j.ymssp.2016.02.020. [3] M. Battarra, E. Mucchi, Incipient cavitation detection in external gear pumps by means of vibro-acoustic measurements, Measurement 129 (June 2018) 51–61, doi:10.1016/j.measurement.2018.07.013. [4] K. Foster, R. Taylor, M. Bidhendi, Computer prediction of cyclic excitation sources for an external gear pump, Proc. Inst. Mech.Eng. Part B 199 (3) (1985) 175–180, doi:10.1017/CBO9781107415324.004.
M. Battarra and E. Mucchi / Mechanism and Machine Theory 146 (2020) 103736
15
[5] T. Costopoulos, A. Kanarachos, E. Pantazis, Reduction of delivery fluctuation and optimum tooth profile of spur gear rotary pumps, Mech. Mach. Theory 23 (2) (1988) 141–146. [6] E. Mucchi, G. Dalpiaz, A. Fernandez Del Rincon, Elastodynamic analysis of a gear pump. Part I: pressure distribution and gear eccentricity, Mech. Syst. Signal Process. 24 (7) (2010) 2160–2179, doi:10.1016/j.ymssp.2010.02.003. [7] E. Mucchi, G. Dalpiaz, A. Rivola, Elastodynamic analysis of a gear pump. Part II: meshing phenomena and simulation results, Mech. Syst. Signal Process. 24 (7) (2010) 2180–2197, doi:10.1016/j.ymssp.2010.02.004. [8] J.J. Watton, K.L. Watkins-Franklin, The transient pressure characteristic of a positive displacement vane pump, Proc. Inst. Mech.Eng. Part A 204 (4) (1990) 269–275, doi:10.1243/PIME_PROC_1990_204_036_02. [9] B.L. Jones, D.N. Johnston, D.K. Longmore, Simulation of suction flow ripple in power steering pumps, in: SAE Technical Paper, 1998, pp. 89–95, doi:10. 4271/982023. [10] M. Buzzoni, M. Battarra, E. Mucchi, G. Dalpiaz, Motion analysis of a linear vibratory feeder: dynamic modeling and experimental verification, Mech. Mach. Theory 114 (2017) 98–110, doi:10.1016/j.mechmachtheory.2017.04.006. [11] M. Battarra, E. Mucchi, G. Dalpiaz, A model for the estimation of pressure ripple in tandem gear pumps, in: IDETC/CIE 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 2015. V010T11A018; 9 pages; doi: 10.1115/DETC2015-46338. [12] E. Frosina, A. Senatore, D. Buono, K.A. Stelson, F. Wang, B. Mohanty, M.J. Gust, Vane pump power split transmission: three dimensional computational fluid dynamics modeling, ASME/BATH 2015 Symposium on Fluid Power and Motion Control, 2015. V001T01A011 doi: 10.1115/FPMC2015-9518. [13] M. Rundo, Massimo, Models for flow rate simulation in gear pumps: a review, Energies 10 (9) (2017) 1261, doi:10.3390/en10091261. [14] C. Bonacini, Sulla portata delle pompe a ingranaggi, L’ingegnere 9 (1961). [15] C. Bonacini, Sulle pompe a ingranaggi a dentatura elicoidale, Tecnica Italiana 3 (1965). [16] N.D. Manring, S.B. Kasaragadda, The theoretical flow ripple of an external gear pump, J. Dyn. Syst. Meas. Control 125 (3) (2003) 396–404, doi:10.1115/ 1.1592193. [17] X. Zhao, A. Vacca, Numerical analysis of theoretical flow in external gear machines, Mech. Mach. Theory 108 (October 2016) (2017) 41–56, doi:10.1016/ j.mechmachtheory.2016.10.010. [18] G. Mimmi, P. Pennacchi, Involute gear pumps versus lobe pumps : a comparison, J. Mech. Des. 119 (December) (1997) 458–465. [19] A.M. Karmel, A study of the internal forces in a variable-displacement vane-Pump–Part I: a theoretical analysis, J. Fluids Eng. 108 (2) (1986) 227. [20] A. Giuffrida, R. Lanzafame, Cam shape and theoretical flow rate in balanced vane pumps, Mech. Mach. Theory 40 (3) (2005) 353–369, doi:10.1016/j. mechmachtheory.20 04.07.0 08. [21] M. Battarra, A. Blum, E. Mucchi, Kinematics of a balanced vane pump with circular tip vanes, Mech. Mach. Theory 137 (2019) 355–373, doi:10.1016/j. mechmachtheory.2019.03.034. [22] E.S. Taylor, Dimensional Analysis for Engineers, Oxford University Press, 1974. [23] R.L. Norton, Cam Design and Manufacturing Handbook, second ed., 2009, doi:10.1037/023990. [24] A. Kahraman, R. Singh, Non-linear dynamics of a spur gear pair, J. Sound Vib. 142 (1) (1990) 49–75, doi:10.1016/0022- 460X(90)90582- K. [25] A. Kahraman, G.W. Blankenship, Experiments on nonlinear dynamic behavior of an oscillator with clearance and periodically time-varying parameters, J. Appl. Mech. 64 (1) (1997) 217–226.