Original Article

Theoretical investigation of a variable displacement vane-type oil pump

Proc IMechE Part C: J Mechanical Engineering Science 227(3) 592–608 ! IMechE 2012 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0954406212464615 pic.sagepub.com

Dinh Quang Truong1, Kyoung Kwan Ahn1, Nguyen Thanh Trung1 and Jae Shin Lee2

Abstract Variable displacement vane-type oil pumps represent one of the most innovative pump types for industrial applications, especially for engine lubrication systems. This article deals with a mathematical modeling method for theoretical performance investigation of a typical variable displacement vane-type oil pump. This theoretical model is based on the pump geometric design and dynamic analyses. It can be considered as a generation step for a deeper understanding of the pump operation as well as for effectively implementing the pump control mechanisms to satisfy the urgent demands of engine lubrication systems. The developed theoretical pump model is finally illustrated by numerical simulations. Keywords Lubrication, vane pump, variable displacement, flow rate, modeling Date received: 26 June 2012; accepted: 24 September 2012

Introduction The automotive industry is advancing rapidly on a worldwide basis. This growth is directly related to pump development technologies, especially positive displacement pumps due to their reliability and cost effectiveness. A variety of positive displacement pumps (vane pumps, gear pumps, radial piston pumps, axial piston pumps, for example) is applicable in the automotive industry such as lubrication, fuel injection, power steering systems, etc.1–3 Among the applications to this industry, the design requirements for engine lubrication systems, especially for vehicle applications, have been recently oriented to a general performance improvement, coupled with a contemporary reduction of power losses, weights and volumes. To deal with the lubrication purpose in most cases, a fixed displacement lubricating pump driven by a rotating component of the mechanical system is generally designed to operate effectively at a target speed and a maximum operating lubricant temperature. Meanwhile, the lubrication requirements of the machine do not directly correspond to its operating speeds. It results in an oversupply of lubricating oil at most machines. To secure the operational safety in hot idling, these pumps are oversized. Consequently, the low efficiency is obtained in most of operating speeds. A pressure relief valve is then provided to return the surplus lubricating oil back into the pump inlet or a reservoir to avoid over pressure conditions in the mechanical system. As a result, a

significant amount of energy used to pressurize the lubricating oil is exhausted through the relief valve. Oil pumps are regarded as low cost items. Complex designs such as axial-piston pumps are not suitable for this usage. Conventional type oil pumps for passenger car engines are gear pumps.1 External geartype oil pumps are well suited for installation into the oil pan because of their flat design. With this pump type, a continuous change of the delivery volume is possible by an axial shift of one of the two or more pairs of gears. Meanwhile, internal gear oil pumps can be mounted directly onto the crankshaft or into the oil pan. An adjustment of the delivery volume of this pump type can be realized by changing the position of the intake relative to the eccentricity of the rotational gear. However, the disadvantage of these gear pump designs is the large sizes due to their gears and adjustment mechanisms. It is known that vane pump technology is used in a variety of automotive and industrial applications today. Subsequently, a potential trend for machine lubrication is the employment of variable 1

School of Mechanical and Automotive Engineering, University of Ulsan, Ulsan, Korea 2 Material Science and Engineering, University of Ulsan, Ulsan, Korea Corresponding author: Kyoung Kwan Ahn, School of Mechanical and Automotive Engineering, University of Ulsan, San 29, Muger 2dong, Nam-gu, Ulsan, 680-764, Korea. Email: [email protected]

Truong et al. displacement vane pumps as lubrication oil pumps. To vary the displacement, there are two common approaches which are the use of a linear translating cam ring,4,5 and the use of a pivoting cam ring.6–13 In both cases, each pump generally includes a ring of which the movement is controlled by a mechanism including a return spring. The pump displacement control mechanism is normally supplied with pressurized lubricating oil from the pump output through an orifice. As pressure increases, the ring movement changes its eccentricity with respect to the rotor centerline which in-turn changes the pump displacement. The return spring, which acts to resist the hydraulic force acting on the ring, can be calibrated to achieve the desired pressure regulation characteristics of the pump. By employing this mechanism, over pressure situations in the engine throughout the expected operating range of the system can be avoided. Although this pump series provide improvements in energy efficiency over fixed displacement pumps, they still result in an energy loss. The reason is that the displacement control decision is, directly or indirectly, affected by the pressurized oil rather than by the changing requirements of the lubricating system. Therefore, development of a variable displacement vane-type oil pump model is indispensable and can be considered as a priority task in order to investigate the pump working performance as well as to optimize the pump design structure. Some studies relating to this field have been done to investigate the pump performances.10–13 Giuffrida and Lanzafame10 derived a mathematical model for a fixed displacement balanced vane pump to analyze the theoretical flow rate through the cam shape design and vane thickness. Staley et al.11 carried out a study on a variable displacement vane pump for engine lubrication. Loganathan et al.12 also developed a variable displacement vane pump for automotive applications by simulations and experiments. In another study, Kim et al.13 investigated an electronic control variable displacement lubrication oil pump through a simple mathematical model. Although these studies bring interesting results, a theoretical model for a variable displacement vane-type oil pump based on its geometric design and dynamic analyses was not fully considered. To investigate the dynamic characteristics of vane pumps, Karmel14,15 carried out both the theoretical analysis and parametric study on the pressure distribution inside the variable displacement vane pump as well as the forces and torques applied to the mechanism and pump shaft. In another study, Rundo and Nervegna16 pointed out that the stator ring geometry of variable displacement radial pumps bears on performance characteristics of these units. The type of the stator ring motion (linear or rotational), location of the rotation center, porting plate integral with the casing or with the stator ring, all have remarkable effects on the pump steady state and dynamic performance.

593 From the above analysis, this article presents a feasible way to construct a complete theoretical model for a typical variable displacement vane-type oil pump based on its geometric and dynamic analyses. By using this model, the ideal pump performance characteristics could be easily investigated through numerical simulations. It can be considered as a generation step for a deeper understanding of pump operation as well as for effectively implementing pump displacement control mechanism to satisfy the urgent lubrication demands. It is, therefore, very useful to achieve industry time-to-market goals.

Current lubrication technology Lubrication demand Typical layout of an overall engine lubrication system is shown in Figure 1. The function of this engine lubrication system is to provide sufficiently oil flow to cool and lubricate machine parts such as bearings, wear surfaces, etc. As a result, pressurized oil leaks at various locations throughout the engine. Conversely, a hydraulic system is characterized by providing fluid power in the form of pressure and flow to a component with minimum leakage. Since the engine ties both the lubrication function and hydraulic function together into a common system. The system must operate at higher pressures and resulting flows to satisfy the needs of all components.

Lubrication oil pump design review Fixed displacement oil pump. One of feasible solutions in engine lubrication is the use of fixed displacement vane-type oil pumps. The vane pump is gaining widespread application and acceptance for automotive oil and fuel delivery purposes due to its simplicity and versatility in design and manufacture. Vane pumps provide high volumetric efficiency and smooth pumping action. However, due to the release of excessive oil, there are several drawbacks of fixed displacement vane-type oil pumps as follows: . Oil stress increases through shear effects, which accelerates the aging of the oil . Increase of gas content in engine oil . Hydraulic energy is lost due to the unnecessary delivered oil.

Variable displacement oil pump. The conventional vanetype oil pump with fixed displacement provides the output flow which is approximately linear with the input speed. Subsequently, more flow rates at higher speeds against the engine lubrication demands. Therefore, the main objective for developing a lubrication system is to reduce the flow rate in the high speed region while keeping the flow at idling speeds.

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Proc IMechE Part C: J Mechanical Engineering Science 227(3)

OIL COOLER

MAIN OIL GALLERY

ROCKER SHAFT

TIMING GEARS AND DRIVE

MAIN BEARINGS

CENTER CAMSHAFT

BIG END BEARING

STRAINER

OTHER CAMSHAFT

OIL PUMP

OIL FILTER

RELIEF VALVE

PRESSURIZED LINES OIL SUMP DRAIN/ SUCTION LINES

Figure 1. A typical configuration of an engine lubrication system.

As a result, less power consumption can be obtained as depicted in Figure 2.12 One advanced solution for the lubrication demand is a variable displacement vane-type oil pump. It can vary the flow through the adjustment of the eccentricity (or pump displacement). To vary the pump eccentricity, there are two common approaches including the uses of a linear translating cam ring and of a pivoting cam ring.

Variable displacement vane pump model design Variable displacement vane-type oil pump In this study, a variable displacement vane-type oil pump made by MyungHwa Co. Ltd.13 as displayed in Figure 3 was used for the investigation. As seen in this figure, the pump mainly consists of a housing, a slotted rotor, a main circular-shaped ring, a pivot pin, sliding vanes, a control chamber with an orifice, a compression spring, an oil inlet (suction port), and an oil outlet (deliver port). The vanes are positioned into the rotor slots while the rotor is placed eccentrically inside the main ring. During the pump operation, the vanes slide along the rotor slots and the vane tip-edges always contact with the ring inner surface due to their centrifugal effects. Subsequently, it performs pumping chambers between succeeding vanes to carry oil from the inlet to the outlet. An increase in volumes forming by the pumping chambers sucks the oil at atmospheric pressure from the oil sump through suction pipe. These chamber volumes continue to be increased during the first half of each rotation. In the second haft, the oil is then carried to the other side where the pumping

chamber volumes are reduced. Finally, the oil is squeezed out of the pump through the delivery pipe. For varying the pump displacement, rotation of the main ring around the pivot pin is controlled by the pressurized oil inside the pump, the centrifugal force effects and the return spring.

Variable displacement vane-type oil pump model Vane movement analysis. In order to investigate the theoretical flow rate of the van-type oil pump, the geometric analysis of a generic vane ith is carried out for a cross section of the pump as shown in Figure 4. First of all, let consider that Or and Os are in turn the centers of the rotor and the main ring inner contour of which their radii are Rr and Rs, respectively. The eccentricity between the rotor and the ring inner surface is represented by ec. The vane with thickness tv and radius Rv at its tip curve (center point is Ovi) can slide along the slot on the rotor during its rotation. At the initial state, each vane is at the end of the corresponding slot, which is defined by radius Rrv. Point Bi is selected as the intersection point between the tip arc and center line of the vane. Generally, considering that the rotor rotates with a constant velocity ! (corresponding to the pump speed is n). At a current angular position of the rotor represented by , position of the vane ith can be defined as i ¼  

2 ði  1Þ, i ¼ 1, . . . , N N

ð1Þ

Figure 4 shows that due to the centrifugal effect, the vane contacts with the inner surface of the main

595

Power

Truong et al.

Conventional oil pump– Power Consumption

Desired Power Consumption Actual variable displacement oil pump – Power Consumption

Engine Speed

Figure 2. Oil pump with ideal power consumption.

ring at point Ai at the current time. Point Ai is far from point Bi, which is presented by angle i . To determine the contact point Ai, it is necessary to find the distance Or Ai  i and the corresponding angle i . Considering small triangles OrOviAi and OrOviOs, following relations can be obtained qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 i ¼ R2v þ Or Ovi þ 2Rv Or Ovi cos i , i ¼ 1, . . . , N ð2Þ i ¼ acos

2i þ e2c  R2s 2i ec



ec sin i i ¼ asin Rs  Rv Or Ovi ¼

ð3Þ

 ð4Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2c þ ðRs  Rv Þ2 þ2ec ðRs  Rv Þ cosði þ i Þ ð5Þ

Subsequently, the vane lift (lvi) can be obtained as

Figure 5 displays a volume variation analysis for the chamber under the considered vane ith corresponding to a small rotational angle of the rotor d. During the operation, the lift of vane ith at any angle can be easily obtained as described in the previous section. Subsequently, the volume derivative of the chamber under vane ith, dVuv ði Þ=d, is computed as dVuv ði Þ dlv ði Þ dlvi ¼ b  tv   b  tv  ð7Þ d d d where b is the ring depth. Next, the volume variation for a chamber between two consecutive vanes (ith and (i þ 1)th), occupying respectively the angular positions i and iþ1  i þ 2=N, is considered and analyzed in Figure 6. The volumes of oil entered into and pumped out of this chamber corresponding to a small rotational angle of the rotor d can be seen in Figure 6(b). Subsequently, the volume variation for this chamber can be computed dVbv ði , iþ1 Þ ¼ dVbv in ði , iþ1 Þ  dVbv

out ði , iþ1 Þ

ð6Þ

ð8Þ

Based on equations from (1) to (6), the movement of vane ith is defined at any working point.

From equation (8), it is important to determine the in and out volumes of this chamber. As depicted in Figure 6(a), at the current time when the rotor is considered at angle , the two vanes ith and (i þ 1)th contact with the ring inner surface at points Ai0 and A(i þ 1)0, respectively. After the small rotation d of the rotor, these vanes contact with the ring inner surface in turn at two new points Ai1 and A(i þ 1)1, which

lvi ¼ Or Bi  Rr ¼ Or Ovi þ Rv  Rr

Theoretical pump flow rate analysis. Generally, a pump presenting N vanes is rigorously characterized by N pumping chambers between consecutive vanes and N pumping chambers under vanes, then, all the pumping chambers come to 2N.

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x6 2

(a)

3

4

1

5

Outside view of the pump 8

(b)

X8 10

9

11

7 12

13 6

14

15

16

Internal structure of the pump 1

Base

6

Main-covered coat

11

Sub ring

2

Tapped holes

7

Located pin

12

Pivot pin

3

Sub-covered coat

8

Main ring

13

Deliver port

4

Drive shaft

9

Rotor

14

Orifice

5

Suction port

10

Vanes

15

Oil seals

16

Spring

Figure 3. Configuration of the researched vane-type oil pump.

are different from the previous ones due to the eccentricity between the rotor and the main ring. The angle analysis for these contact points is clearly shown in Figure 6(a). Consequently, it causes both of the input and output volumes to be reduced small amounts, which are presented by part Si2 in Figure 6(b). Based on this figure and a simple calculation, the volumes of oil entered into and pumped out of the chamber between two consecutive vanes can be in turn derived as in equations (9) and (10) (represented as part Si1 in Figure 6(b)) dVbv

 1  2 2 in ði , iþ1 Þ ¼ b iþ1 diþ1  iþ1 diþ1 2

ð9Þ

dVbv

out ði , iþ1 Þ

 1  ¼ b 2i di  2i di 2

ð10Þ

where the factors , d can be obtained from the previous section; factor d can be derived as di ¼ di  d

ð11Þ

From equations (8) to (11), the volume derivative of the chamber between vanes is finally obtained as  dVbv ði , iþ1 Þ b  2 ¼ iþ1  2i d 2

ð12Þ

Truong et al.

597

γi Ai

Bi

Main Ring

Rv ρi Vane i th

Ovi

ω tv Sliding Slot

Rrv ec Or

βi αi

Rr

Os

Rs

Rotor

Figure 4. Geometric analysis of a generic vane.

dlvi

ω

Main ring rotation analysis. The rotation of the main ring is largely depended on following forces:

Ai0

. Force caused by pressurized oil inside the main ring . Force caused by pressurized oil outside the main ring . Force caused by centrifugal effects of vanes and oil volumes between vanes . Force of the compression spring

Ai1 Ai0 tv

dlvi

dVuv

dα d βi

Rr

Figure 5. Analysis of volume variation for a chamber under a generic vane.

Finally, the theoretical flow rate of the vane pump can be computed based on equations (7) and (12) as follows10 N 1 X dVuv ði Þ 1 Qth ðÞ ¼ ! ½signðdVuv ði ÞÞ  1 þ ! 2 i¼1 d 2



N 1 X i¼1

dVbv ði , iþ1Þ ½signðdVbv ði , iþ1 ÞÞ  1 d

1 dVbv þ ! ðN , 1 Þ½signðdVbv ðN , 1 ÞÞ  1 2 d ð13Þ

(a) Force due to pressurized oil inside the main ring. Figure 7 displays the distribution of working pressure inside the main ring. In this research pump type, the eccentricity between the rotor and the ring is maximum (ec_max ¼ Rs–Rr) at the initial state of the main ring. In this case, the rotor contacts with the inner contour of the ring at point C10 and the center point of the ring inner contour is at point Os0. Two coordinate systems have been used, which are a Cartesian coordinate system positioning at the rotor center (OrXrYr) and a polar coordinate system. With the polar coordinate system, the pole is located at center point of the ring inner contour and the axis points to a point on the ring at which the ring contour is closest to the rotor. Hence, this pole axis is along the line Os0 C10 and coincides with the Yr axis at the initial state. During the operation, the eccentricity between the rotor and the ring makes the ring inner contour similar to a cam contour if considering Or is the center point. Based on the pump working principle, at the initial position of the ring, the profile of pressure distribution on the ring inner surface can be divided into three regions as (Figure 7(a)). . The first region within arc C10 C20 (where the arc radius presents a positive gradient (the minimum

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Proc IMechE Part C: J Mechanical Engineering Science 227(3)

dθ1

(a)

A11

A10

B1

dα

β11

α11

ω

β10

α10

dθ 2

A21

dα

α 21 β 21

A20 B2

α 20 β 20 Or

ec

Os Rotational analysis of two consecutive vanes

A11

A10

ω

dlvi

(b)

Ai1 A

B1

Ai0

i0

Out-volume dVbv _ out(α 1 ,α 2 )

Si2 In-volume dVbv _ in(α 1 ,α 2 )

A21

Si1

dlvi

ω

Rr

A20 B2

ec

Or

dθ i dα d βi

Os

Rr

In/out-volume analysis of a chamber between two consecutive vanes

Figure 6. Analysis of volume variation for a chamber between two consecutive vanes.

value (Rr) at C10)). This region relates to the suction zone. Consequently, the pressure is almost the same as the minimum pressure Pmin (the tank pressure) during the rotation angle of rotor r 2 ½0, r10  ðr ¼  þ Þ. The effect of this pressure region on the ring is represented by an angle range s 2 ½0, s10   ½0, C20 . . The second region within arc C20 C30 (where the arc radius increases or decreases very slowly (near the maximum value (Rs þ ec_max)). This region gives the pre-compression for the oil chambers. The pressure increases from Pmin (the tank pressure) to Pmax (the outlet pressure), corresponding to the rotation angle of rotor r 2 ½r10 , r10 þ r20 . The effect of this pressure region on the ring is represented by an angle range s 2 ½s10 , s10 þ s20   ½C20 , C30  . The third region within arc C30 C10 (where the arc radius decreases with the same gradient in the first region). This region relates to the delivery zone. The pressure is then almost the same as the

maximum pressure Pmax (the outlet pressure), corresponding to the rotation angle of rotor r 2 ½r10 þ r20 , r10 þ r20 þ r30   ½r10 þ r20 , 2. The effect of this pressure region on the ring is represented by an angle range s 2 ½s10 þ s20 , 2  ½C30 , 2. For a small rotation of the ring, d’, around the pivot pin Op, the center point of the ring inner contour is moved from point Os0 to point Os1. Based on a geometric analysis, the three regions of pressurized oil are re-positioned as displayed in Figure 7(b) in which points C10, C20, C30 are moved to points C11, C21, C31, respectively. The axis of the polar coordinate system is also rotated to a new position along to line Os1 C11 . It is clear that the initial coordinate of the center point Os (Os0) is easily determined based on the coordinate of the fixed pivot point Op as  Op Os0 ¼ Rrot ð14Þ ffHOp Os0 ¼ ’Os0

Truong et al.

599

(a)

Yr ≡ X s

C10

ω

α r/ s ⊕

ψ r0

H

α r10

⊕

ψ s0

α s10 Or

Pmin

Cα s

2π − α s

Os0

α s30

P (α s ) MOp Op

ϕ C0 ϕ Os0 α r30

Xr

Pmax

α s20 α r20

Fspr

xspr C0

Pmin C20

Pmax C30

Pressure distribution at initial position of the main ring

Yr

Xs

(b)

C11

ω

C10

α r/ s ⊕ ψ s1 Os1

Or

dϕ

Rrot

Op Xr

Os0

C21 C20

C31 C30

Pressure distribution after a small rotation of the main ring

Figure 7. Analysis of pressure distribution inside the main ring.

Consequently, trajectory of the point Os when the ring is rotated around the pivot point (Op) is also determined as a rotating vector of which the length is Rrot and the angle is ’Os0  d’ . Therefore, the three pressure regions can be completely determined. Let define the angle difference between the pole axis and the center line Op Ost at the current time t as st . Then the total moment acting on the ring caused by the pressurized oil inside the ring to make it rotate around the pivot pin can be generally computed as Z C2t X MOp oil inside ¼ Rrot sinð st þ s ÞPmin bRs ds 0 Z C3t þ Rrot sinð st þ s Þ C2t s  C2t þ Pmin bRs ds  ðPmax  Pmin Þ C3t  C2t Z 2 þ Rrot sinð st þ s ÞPmax bRs ds ð15Þ C3t

(b) Force due to pressurized oil outside the main ring (through the control orifice).

The pressurized oil not only affects on the ring inner surface but also acts on the ring outer surface through the control orifice (Figure 3). Considering the oil chamber outside the ring (or called the outside chamber), relationship between the input and output flow rates during a small rotation of the ring, d’, can be expressed as Vout dPout Qinflow  Qoutflow ¼ ð16Þ oil dt where Qinflow and Qoutflow are input and output flow rates of this outside chamber, respectively; oil is the bulk modulus; Vout and dPout are the volume and pressure derivation of this chamber, respectively. From the pump design, the volume of the chamber outside the ring can be measured accurately with respect to the ring rotation angle. The flow entering to this chamber through the orifice is computed by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Qinflow ¼ Cor Aor ðPmax  Pout Þ ð17Þ oil where Cor , Aor , and oil are in turn flow coefficient, area of the orifice, and mass density of oil, respectively.

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Proc IMechE Part C: J Mechanical Engineering Science 227(3)

Due to the shape of the ring, the distribution of longitudinal velocity as well as the distribution of pressure acting on the ring outer surface can be divided into 4 areas corresponding to 4 curved surfaces with different radii as shown in Figure 8. Subsequently, the output flow of the outside chamber is derived Z 1b   Qoutflow ¼  Op O1 d’ sin  bR1 d  Z 1a3b    Op O3 d’ sin  bR3 d  Z 3a2b   þ Op O2 d’ sin  bR2 d  Z 2a4b   þ Op O4 d’ sin  bR4 d ð18Þ

evaluate these effects, force generated by an oil volume between two consecutive vanes is firstly analyzed. As discussed in the section ‘Theoretical pump flow rate analysis,’ the volume of chamber between each two consecutive vanes at the present time t is described in Figure 9. As shown in this figure, the boundary of the oil chamber is the polygon AitA(i þ t)tMN. The centrifugal force of this volume is presented by a vector Foil_cen(i) positioning at the chamber mass center Gi. Hence, it is necessary to determine the coordinate of Gi. Let divide the chamber volume AitA(i þ t)tMN into a rectangular block Ait Ait MN and a triangular block Ait Ait Aðiþ1Þt , and consider that the oil density distributes uniformly in all the chamber volume. Therefore, the center mass of this oil chamber can be represented by Gi W1 SAit Ait Aðiþ1Þt ¼ ð20Þ ¼ SAit Ait MN Gi Q

4a

By replacing equations (17) and (18) into equation (16), the working pressure in the outside chamber, Pout , is obtained. This pressurized oil acts directly on the main ring as analyzed in Figure 8(b). The total moment acting on the ring outer surface caused by this pressurized oil to make it rotate around the pivot pin can be obtained as Z 1b X MOp oil outside ¼ Op O1 sin Pout bR1 d 1a Z 3b þ Op O3 : sin Pout bR3 d 3a Z 2b  Op O2 sin Pout bR2 d 2a Z 4b  Op O4 sin Pout bR4 d ð19Þ

where Wl and Q are the mass centers of the Ait Ait MN and Ait Ait Aðiþ1Þt blocks, respectively; and 8 1 > > ¼  Ait Ait  Aðiþ1Þt Ait S  > < Ait Ait Aðiþ1Þt 2 1 ¼ ði  iþ1 Þðiþ1  i Þ > > 2 > : SAit Ait MN ¼ MN  Ait N ¼ ði  iþ1 Þði  Rr Þ ð21Þ By using simple geometric calculation for the chamber analysis as in Figure 9, the following relations can be obtained 8 i þ iþ1  2Rr 3 þ 2 i  Rr > >   þ >  :UM ¼ 3 þ 3 2 ð22Þ

4a

(c) Force due to centrifugal effects of vanes and oil volumes between vanes. It is clear that the oil chambers as well as the vanes generate centrifugal forces acting on the ring. To

(a)

(b)

MOp

dϕ

VO2 O2

R1

μ2

R2

μ3

VO4

J1

O2

μ1

J2

VO3

Op

Op

VO1

O4 J5

J1

μ 2 μ2a

R1

R2

μ3 R3

R4 Motion analysis of the main ring

μ 4b

μ4

μ1b

μ3a Pout

J3

μ3b O3 J4

Pout

J2

O1

J3 R3

μ2b

O4 J5

Pout O3 J4 μ

μ4

R4

4a

Pout

Pressure analysis at the outside chamber

Figure 8. Analysis of pressure distribution outside the main ring.

μ1 O1 Control Orifice, Aor

Truong et al.

601

Foil_cen(mi)

ω

Vane ith

A(i+1)t D

Ait

A*it

Q

ω

αr/s

Rr

⊕M O p

Gi

r i –Rr

r i+1 –Rr

Xs

Vane i+1th

W1

W2

Or

SU V

ψ r (t ) ≡ψ r

Os1

W3

N

Op

dϕ M

Os0

R ( mi )

α r ( mi ) Rr

β i − β i+1

Mass mi

8 > Or Gi ¼ Rr þ G iU > >  

!

! <

ff Or Xs , Or Gi  r ðGi Þ  r Ai Aðiþ1Þ MN > 3 þ 2 i  iþ1 > > : ¼ iþ1 þ  þ 3 þ 3 2

δ ( mi )

Fcen

Figure 9. Analysis of centrifugal force generated by an oil chamber volume between two consecutive vanes.

From equation (22), the coordinate of Gi is finally calculated as

Mi

Figure 10. Analysis of centrifugal force effect on the ring rotation.

figure, the total moment acting on the ring caused by the centrifugal forces of N vanes and 2N oil chambers are derived as ð23Þ

X

MOp

cen

¼

N X

Fcen ðmi Þ  cosð ðmi ÞÞ  Op Mi

i¼1

Centrifugal force of an object of mass mi travelling in a circle of radius R(mi) around the rotor center can be computed as Fcen ðmi Þ ¼ mi !2 Rðmi Þ

ð24Þ

where mi and R(mi) are defined as in equations (25), (26) and (27) corresponding to an oil chamber between two consecutive vanes, an oil chamber under each vane and a vane.

mi ¼ tv  ðRr  Rrv þ lvi  hv Þ  b  oil Rðmi Þ ¼ Rrv þ ðRr  Rrv þ lvi  hv Þ=2



O p Mi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ¼ Op Or þ R2 ðmi Þ  2Op Or Rðmi Þ cosðr ðmi Þ þ r Þ ð29Þ

ðmi Þ ¼

ðfor an oil chamber between two 

Here Fcen ðmi Þ is obtained from equations (24) to (27); Op Mi and ðmi Þ are obtained from (29) and (30), respectively.

"

8