Paper Number (Assigned by IFPE Staff)

Design of a Variable Displacement Triplex Pump

Shawn Wilhelm, James D. Van de Ven University of Minnesota

ABSTRACT

Conventional variable displacement hydraulic pumps and motors suffer from poor efficiency at low displacements, primarily due to the friction and leakage associated with hydrodynamic bearings, which do not scale with output power. A variable displacement adjustable linkage pump has been developed which can achieve zero displacement and has a constant top dead center position of the pistons, regardless of displacement. This architecture employs roller element bearings in its pin joints, significantly reducing the mechanical losses at low displacements as compared to the hydrodynamic bearings of variable axial piston and bent axis machines. Previous work has described the experimentally validated efficiency model for a low-power single-cylinder pump. In this paper, the optimization and machine design of an 8.5 kW, high pressure, variable displacement, triplex prototype are presented and the potential applications are discussed. It will be shown that this architecture can achieve greater than 90% efficiency across the majority of the operating region. The pump can be applied to a wide range of applications with little compromise, compared with present variable displacement pumps.

INTRODUCTION

According to a 2012 report by Oak Ridge National Labs, the average efficiency of a fluid power system for industrial and mobile hydraulic applications is 50% and 21% respectively. Additionally, these systems consume between 1.5 and 2.4 Quads per year.1) These energy losses result in 544 to 1700 million metric tons (MMT) of CO2 released into the atmosphere at a cost of $16B to $28B. The majority of these losses are due to metering flow control, which throttles flow over a valve to vary the flow rate and pressure of the fluid. There is a clear need for a more efficient flow control method from both an economic and environmental standpoint.

There are three common methods for controlling hydraulic flow: metering valves, speed control of an electric motor with a variable frequency drive, and variable displacement hydraulic pumps. Each of these methods has their own respective tradeoffs and are further discussed here.

Metering valve control uses a valve with a variable orifice to throttle the flow from a fixed displacement pump and is used in both mobile and industrial applications. It is the most popular solution due to the low cost and ease of maintenance. However, it is also the most inefficient

solution due to large amount of energy throttled across the control and relief valves.

Variable frequency drives (VFDs) are used to control the shaft speed of an electric drive motor that is coupled to a fixed displacement hydraulic pump. Due to the low power density of power electronics, their use is restricted to industrial applications where space is available. While being the most efficient solution, VFDs are limited by the high cost associated with high power drives and motors and the reduced motor life.

Variable displacement pumps (VDPs) vary their output by controlling the displaced fluid volume per revolution. The most common VDP is the swash plate type axial piston pump. The displacement is controlled by varying the swash plate angle which in turn varies the piston travel through a stroke, thus resulting in variable flow. Their use is limited to non-corrosive fluids because the working fluid also provides lubrication to the hydrodynamic bearings. They also suffer from poor efficiency at low displacement. If, however, these limitations could be overcome, the variable displacement pump would be a much more attractive hydraulic control solution.

Intrinsic to the design of a swash plate axial piston pump is the hydrostatic bearing between the slipper and the swash plate. This bearing is created by sending pressurized fluid from the cylinder through the piston and spherical bearing to the swash plate. The energy loss associated with this bearing is function of the square of the relative velocity between components and is relatively constant regardless of the volumetric displacement. As a result, the efficiency of these pumps can be high at maximum output, but is significantly reduced at partial displacement.2, 3)

A similar configuration to the axial piston pump is the bent axis type, which circumvents this problem by using spherical joints rather than a swash plate. However, these pumps are not widely used because of their high costs, reliability issues, and their inability support a through-shaft design.4)

Much work has been done on improving the efficiency of existing variable displacement pump architectures.5-7) For the most part, these efforts have resulted in an increase in the maximum efficiency, but they have not significantly improved the performance at low volumetric displacement.

9.3

Another method to vary the flow output of a fixed displacement pump is to use high speed on/off valves to control the duty ratio effectively pulse-width modulating the output flow. Several works in this area show promise, but they have been hindered by valve speed limitations and transient throttling losses.8-10) A related technique, called digital displacement, involves directing the output flow from individual cylinders of a pump to either the system or reservoir. Studies have shown that this method has potential for high efficiency at partial loads, but it results in significant flow ripple and added complexity due to the large number of actively controlled valves.11-13) Due to the linear relationship between the energy loss and relative velocity of components, pin joints are inherently more efficient than the hydrodynamic planar joints of a swash plate pump at high velocities. Therefore, a pump that purely uses pin joints in a crank-slider linkage has the potential for high efficiency. If a crosshead bearing design is used, the working fluid of the pump can be separated from the mechanism, resulting in a more versatile design. Currently, there are fixed displacement linkage driven pumps on the market based on the four-bar crank-slider mechanism. However, the displacement of this linkage is not easily varied and cannot be done without changing the length of a moving link or moving the pumping cylinder. An example of such as a pump is the hydraulically cushioned variable delivery triplex pump which allows the piston to displace a variable gas spring until a certain force is reached, at which point it will begin to displace working fluid. This high pressure water jet pump can only operate at low speeds on the order of 6 rpm.14) To avoid adjusting a moving member of the mechanism, or the position of the pumping cylinder, a higher order linkage is required. A previous work describes the type synthesis and four possible configurations of a variable displacement six-bar slider linkage which can achieve both zero piston displacement and constant top dead center (TDC) position of the piston regardless of displacement.15) The linkage studied in this paper is of the extended, R1max configuration and is shown in Figure 1.

An input crank, coupler, and rocker constitute the base four bar crank-rocker mechanism. The connecting rod joins the sliding link to the base four bar at a common pin and is the same length as the rocker link. The displacement is varied by changing the position of the ground pivot of the rocker link about point 𝑷 forcing the adjustable ground pivot through the dashed arc. When the adjustable ground pivot is collinear with the axis of slide, the sliding link will exhibit no translation as the crank rotates. As the adjustable ground pivot moves away from the axis of slide, the slider link will translate.

Figure 1 Schematic of Adjustable Linkage Adapted from (16)

The authors previously presented a single cylinder prototype variable displacement linkage pump of the extended, R1max configuration, show in Figure 2. It was constructed to evaluate the concept and experimentally validate the model. The prototype used bronze bushings for bearings. There was strong agreement between model and experimental results. Through an extension of the model, the predicted efficiency of the pump is greater than 90% at partial displacements as low as 10%, if roller element bearings are used instead of bushings.16)

Figure 2 Prototype Variable Displacement Linkage

Reproduced from (16)

The purpose of this paper is to present the design of a three cylinder variable displacement linkage pump (VDLP) which incorporates roller element bearings in order to achieve high efficiency across a large displacement range. First the design specifications are established from application requirements. The second section describes the coupled mechanism and fluid flow optimization and results. In the third section the machine design is discussed and the mechanism is presented. Lastly, conclusions are given.

DESIGN SPECIFICATIONS

The presented mechanism was designed to be incorporated into another project at the University of Minnesota. This application, as well as practical requirements drove the pump task and design specification which are presented below.

TASK SPECIFICATIONS – The application requires a high pressure, variable displacement hydraulic pump that operates at a relatively low fractional displacement for a majority of the operating cycle. As a result, the pump must exhibit high efficiency at low volumetric displacement. To achieve this, roller element bearings are used at the joints. The pump will be run by an AC motor at constant speed and must have feedback control of the displacement to actively control the flow rate.

DESIGN SPECIFICATIONS - The pump must meet the following metrics:

1. Run at synchronous speeds 2. Have an operating pressure of 21MPa 3. Operate at a peak power of 8.5kW 4. Vary displacement from zero to max in < 0.1s 5. Have a pressure drop during intake less than 30kPa

to prevent cavitation 6. Have an operating life of at least 10kHr

OPTIMIZATION

The kinematics of the linkage presented in (15) were previously optimized for power density without regard to pumping behavior. When a piston-cylinder chamber, optimized to reduce energy losses, was added to the linkage, high peak flow rates prevented operating the pump at high speeds due to induced cavitation. If the kinematics and fluid dynamics of the pump are optimized together, the pump performance can be improved significantly. Additionally, there are multiple non-linear objective functions which must be considered to evaluate performance, as well as six design variables to be optimized. In this section, the optimization objectives, design variables, and methods are presented.

OBJECTIVES – The objective of this optimization was to both improve the pump performance and to control the size of the mechanism. Ultimately, six objective functions were used: maximize efficiency, maximize power density, minimize timing ratio variation, minimize roller bearing load rating, minimize cross head bearing side load, and minimize cylinder pressure drop.

The efficiency of the pump is described by:

𝜂𝑝𝑢𝑚𝑝 =𝑊𝑓𝑙𝑢𝑖𝑑

𝑊𝑠ℎ𝑎𝑓𝑡

(1)

where 𝑊𝑠ℎ𝑎𝑓𝑡 is defined as the sum of the input work per

revolution, found from the kinematic and dynamic analysis and the calculated losses due to coulomb friction, viscous friction, leakage, and fluid compressibility, as defined in (16). The fluid work per revolution, 𝑊𝑓𝑙𝑢𝑖𝑑 , is the product

of the displaced volume and the system pressure. Here, perfect valve operation was assumed to avoid the computational expense of a numerical simulation.

The linkage was designed for a prescribed power output, so the power density is a function of the linkage area and is defined as the area of a box bounding the linkage, parallel to the axis of slide. It should be noted that the linkage area is based on the vector lengths and does not take into account link widths or bearings. It is not a measure of the physical mechanism, but rather a comparison metric between candidates based on the assumption that candidates with similar vector areas will have similar true areas when a linkage is constructed. Minimizing this area maximizes power density and minimizes the mechanism size.

The timing ratio is defined as the ratio of time of the pumping stroke to time of return stroke. If the timing is not equal, the piston will have to move faster for one part of the cycle. In this mechanism, the timing of BDC varies with displacement while TDC remains constant. Timing ratio variation is the maximum difference between 1 and the timing ratio across all displacements.

The load rating of each bearing is determined using a standard bearing life calculation presented in a later section. The bearing life is a function of the relative motion between the bearing and its pin and the applied load. Minimizing this value results in smaller bearings and thus reduced mechanism size and weight.

The adjustable linkage pump design calls for a crosshead bearing design in order to maintain piston cylinder concentricity. The crosshead side load determines the type of bearing which can be used. Minimizing this value, minimizes the bearing size and the friction.

The cylinder pressure drop is defined as the pressure differential along the length of the cylinder during the intake stroke, making the assumption of fully-developed laminar pipe flow. While this assumption is not strictly valid for this situation, this objective qualitatively inserts the influence of piston velocity and diameter on viscous drag. The pressure drop during the intake stroke was minimized to prevent cavitation, which results in partial filling and prevent wear of precision surfaces during collapse of the cavitation bubbles.

DESIGN VARIABLES – There are 6 optimized design variables required to describe the functional operation of the mechanism in addition to the 8 design constants listed in Table 1. The linkage is initially normalized by the length of the input link, 𝑟2, so only two normalized link lengths,

𝑟3𝑁 and 𝑟4𝑁 , and two angles, 𝜃𝑎𝑠 and 𝛾𝑚𝑖𝑛, are required to describe the kinematics as outlined in (15). The bore to stroke ratio (𝑅𝑑𝑠 ), is used to determine the required maximum stroke length according to:

𝑆𝑚𝑎𝑥(𝑅𝑑𝑠) = (4𝑉𝑐𝑦𝑙

𝑅𝑑𝑠2 𝜋

)

13

(2)

where 𝑉𝑐𝑦𝑙, the required displacement per cylinder per

revolution, is be calculated from the design requirements by equation (3).

𝑉𝑐𝑦𝑙 =𝑃𝑤𝑟

(𝑃𝑠𝑦𝑠 − 𝑃𝑡𝑎𝑛𝑘)𝜔𝑠𝑦𝑠𝑁𝑝

(3)

A stretch operator is applied to the dimensionless linkage to scale the normalized stroke to the required stroke, calculated above.

The radial piston to cylinder clearance was set to 10μm, as this is the lowest value we could reasonably achieve using common machining practices. Therefore, the piston length, 𝑙𝑝, is the only independent hydraulic design

variable and is directly related to the viscous friction and leakage losses.

Table 1 - Design Constants of Optimization

Design Constant

Value Description

𝑷𝒔𝒚𝒔 21𝑀𝑃𝑎 The hydraulic system pressure

𝑷𝒕𝒂𝒏𝒌 101𝑘𝑃𝑎 Pressure of the hydraulic reservoir

𝝁𝒌 0.005 Friction Coefficient of Roller bearings

𝑷𝒘𝒓 8.5𝑘𝑊 Maximum Power Rating

𝑵𝒑 3 Number of Pistons

𝝎𝒔𝒚𝒔 60𝐻𝑧 Input Shaft Speed

𝒉 10𝜇𝑚 Radial Piston-Cylinder Clearance

𝜼𝒅 . 065𝑃𝑎

𝑠 Dynamic Viscosity of Oil

METHODS - Due to the large number of objective functions, design variables, and the coupled kinematic and fluid dynamics of this problem, a robust global optimization technique was required. The fast non-dominated sorting genetic algorithm (NSGA-II) has been shown to be effective in solving non-linear multi-objective problems and was selected to optimize the pump.17) The algorithm was completed with 10,000 individuals per generation for 100 generations with a mutation probability of 0.9 for a total computation time of 1hr 24 min on a single core of a 2.7GHz AMD processor.

The output of a multi-objective optimization is not a single optimized solution, but rather an N-dimensional optimal surface where N is the number of objective functions. The candidate solutions vary in optimality relative to each of the objective functions meaning that any one solution will have performance tradeoffs relative to the various objectives. This solution space is known as the Pareto front.

Figure 3 shows the six-dimensional Pareto front projected on a two-dimensional plane to show the design trade-off

between linkage area and bearing load. Each point represents a candidate mechanism plotted by its objective values. When the linkage area decreases towards zero, there is an exponential increase in bearing load rating. As the bearing load rating approaches it minimum, the size of the linkage becomes greater. Tradeoffs exist between each of the objective functions and it is the job of the designer to select the “best” solution for the given application.

Figure 3 Two Dimensional Projection of Pareto-Front

Representing Design Trade-offs of Optimal Solutions

As seen in Figure 3, there can be a large number of Pareto-optimal candidate solutions. In this problem, there are on the order of 2000. In order to reduce the number of solutions, a set of threshold values are applied to act as a filter. These values were chosen as to disqualify candidates which were not viable. Table 2 shows the thresholds used for each objective. These values where developed to reduce the list of candidate solutions to those which would perform reasonably well from both a performance and practical standpoint.

Table 2 - User Defined Threshold Values and the Objective Function Values of the Selected Solution

Objective Threshold Solution Values

Efficiency at 50% Displacement (min)

92% 94.7%

Linkage Area (max) 45 c𝑚2 42 𝑐𝑚2

Timing Ratio Variation (max) . 07 . 055

Bearing Load Rating (max) 32 𝑘𝑁 25.4𝑘𝑁

Crosshead Side Load (max) 2 𝑘𝑁 0.85 𝑘𝑁

Pressure Drop (max) 7𝑘𝑃𝑎 (~1𝑝𝑠𝑖) 6.6 𝑘𝑃𝑎

After filtering the initial large solution space, there are generally still multiple candidate solutions. These solutions must then be evaluated further to select a single candidate. For this application, the timing ratio variation is the most important constraint due to its coupling with shaking forces and peak flow rates. Therefore, the solution with the smallest timing ratio variation was selected.

RESULTS –Table 2 shows the objective function values for the selected solution. The reduction in timing ratio variation is demonstrated by Figure 4 where the performance of the optimized linkage is compared the linkage presented in (16) and a perfect timing ratio variation of zero.

Figure 4 Timing Ratio Variation

The predicted efficiency of the mechanism is compared to that of a generic axial piston pump in Figure 5. An average efficiency of 94% is expected between 10 % and 100% displacement. Figure 6 shows the energy loss contributions due to coulomb friction, viscous friction, and Leakage. The coulomb friction has been reduced by an order of magnitude compared to the plane bearing friction of (16)

Figure 5 Predicted Efficiency of VDLP as Compared

to an Equivalent Axial Piston Pump

Figure 6 Loss Contributions Showing

Additionally, a contour plot of the efficiency, seen in Figure 7, shows a predicted efficiency of greater than 90% for the majority of operating conditions.

Figure 7 Contour Plot Showing Predicted

Performance of Pump at Various Operating Conditions

MACHINE DESIGN

With an optimized analytical solution found, the detailed design of the pump is now considered. There are four main components required to make a working reciprocating piston pump: The mechanism to drive reciprocation, the piston-cylinder chamber, the check-valve manifold, and the pump body. An inline three cylinder configuration was selected to simplify manifold design, but other multi-cylinder configurations are possible. Here the design of the driving mechanism and piston-cylinder chamber are discussed.

LINKAGE ARRANGEMENT – This six-bar mechanism consists of 5 moving links with 6 unique joints, which require roller bearings, and a control link. The links are numbered 1-6 with the assignments given in Table 3. The joints are labeled as 𝐶𝑖𝑗 where 𝑖 and 𝑗 denote the links

which share the joint, with 𝐶𝑖𝑗 being equal to 𝐶𝑗𝑖.

0.8

0.8

0.8

0.9

0.9

0.9

0.9

0.9

0.9

5

0.9

5

0.9

5

0.9

50

.95

Syste

m P

ressure

(MP

a)

Displacement (%)

Model Efficiency at 60 Hz

0.2 0.4 0.6 0.8 1

5

10

15

20

25

30

35

Table 3 - Link Assignments

Link Name Assignment Shared Joints

Ground Link* 1 𝐶12, 𝐶14, 𝐶16

Input Link 2 𝐶12, 𝐶32

Coupler 3 𝐶32, 𝐶34, 𝐶35

Rocker 4 𝐶41, 𝐶34

Connecting Rod 5 𝐶35, 𝐶65

Piston 6 𝐶65, 𝐶16 *Includes the Control Link, Input Shaft, and Slider Axis

In order to reduce the number of bearings required and the axial length of the pump, duplicate links were avoided when possible. As a result, most of the pin joints are loaded in single shear. Using a common crankshaft dictates that 𝐶12 , the joint between the input link and coupler link, be duplicated. Additionally, the piston center was prescribed to pass through the central plane of the connecting rod as is common practice.

To reduce moments, links with shared joints were located adjacent to each other. As such, it is beneficial to for the control link to be adjacent to the rocker link, as this joint, 𝐶14, constitutes the movable ground pivot. Because link 3 has a common joint with both links 4 and 5, it should be located between the two.

As a result of these constraints, there are 2 unique linkage arrangements which are depicted in Figure 8. Here, the various links are labeled by their assignment, and represented by two dimensional top view cross sections with holes present at pin locations. Because arrangement “b” resulted in interference between the control link (1) and the coupler (3), arrangement “a” was selected for the design.

Figure 8 Unique Linkage Arrangements

BEARING SELECTION – A kinematic and dynamic analysis was completed to determine the pin forces on each of the joints, in addition to the relative rotations of the associated links. Using this information, a dynamic load rating for each bearing was determined. From this information, the bearings were selected and their link locations determined.

Figure 9 shows a schematic of two representative links who share a common joint and are rotating relative to

each other. FAB is the joint load and ω is the angular velocity of the links. In order to determine their relative

rotation, θAB, the angular velocity difference between the two links is integrated over a cycle as shown in equation (4). From these values and the desired life rating, a bearing load rating was determined.

𝜃𝐴𝐵 = ∫ 𝑎𝑏𝑠(𝜔𝐴 − 𝜔𝐵)𝑑𝑡

(4)

Figure 9 - Relative Angular Velocity of Two Links

with a Shared Joint

The bearing life calculation is based off the L10 designation which means that 90% of bearings will survive for the prescribed life. The bearing life is determined by:

𝐿10 =𝜃𝐴𝐵

2𝜋𝜔𝑠𝑦𝑠

3600𝑠

𝐻𝑟𝐿ℎ (5)

where 𝜔𝑠𝑦𝑠 is the system frequency, and 𝐿ℎ is the desired

life in hours. The required dynamic load rating 𝐶 and equivalent load P are found by:18)

𝐶 = √𝐿10

310 𝑃 (6)

𝑃 =2

3max(𝐹𝐴𝐵) (7)

As stated in the design specifications, the design life is 10kHr. The load rating is calculated at 25% increments of displacement and the maximum is used for bearing selection. Table 4 gives the required load ratings for the associated joints.

Table 4 - Bearing Load Ratings

Joint 𝐶12 𝐶32 𝐶34 𝐶41 𝐶35 𝐶65

Load Rating 𝑘𝑁

25 25 13 16.8 18.5 18.2

For each of the 6 joints, a bearing is to be mounted in one of the associated links. The bearings for joints 𝐶32 and 𝐶65 were mounted in links 3 and 5 respectively as to reduce the moving mass of the material required to support the

bearing and thus shaking forces. Additionally, joint 𝐶12 is the crank shaft bearing and was to the crank case. Joints 𝐶34 and 𝐶35 share a common pin and therefore the bearings for these pins are concentric. To prevent large pin moments, it is better for the common pin to be mounted in link 3, which is in between the other links. Therefore, the bearings for joints 𝐶34 and 𝐶35 were located in links 4 and 5 respectively. Joint 𝐶41 is the movable ground pivot and the location of the associated bearing was not as readily deduced as with the other joints. Ultimately, the bearing was mounted in link 4 for convenience of design and assembly. With the required load ratings, and locations found, feasible bearings were selected. The bearing OD was constrained to be less than 95% of the link length in which it is mounted. The optimal bearing had the smallest width while meeting the other constraints. In order to reduce manufacturing costs, the number of hardened precision surfaces was minimized by selecting bearings which have inner and outer races. Therefore caged roller and drawn cup needle rollers were not considered. Additionally, full complement bearings were avoided due to their higher friction and wear characteristics.

MECHANISM DESIGN – The moving links, were designed to accommodate their bearings and the applied load with a design life of 1e9 cycles. Their dimensions were first estimated with a two-force member assumption and evaluated considering axial load, bearing, and tear-out failure.

The pin dimensions were restricted by the inner diameter of the selected bearings. A Von Misses combined bending and shear stress calculation was used to calculate an equivalent stress to determine if the pins would be able to survive the required life. The moment load was reduced whenever possible by minimizing the gap between the applied load and the furthest point from the pin.

A split input crank-shaft is required to accommodate a bearing between the input link and the coupler. Flats are machined into the crank pin and input shaft to create eccentric centers when mated. After the crank pin is installed inside of its roller bearing, it is clamped to the input shaft to create a rigid structure. A counter mass is built into the clamp to balance the crankshaft and crank-pin bearings. Figure 10 below shows the assembled crankshaft for clarification.

Figure 10 Split Crank-Shaft with Counter Mass Clamps

The linkage was then modeled using commercial FEA software where stresses can be more accurately evaluated and dimensions are further refined to reduce weight. Because the majority of link dimensions are driven by factors other than stress, aluminum was used whenever mass could be reduced without affecting the integrity of the part.

To conserve space the piston cylinder chamber serves both as the pumping chamber and the crosshead bearing. Since the working fluid is hydraulic oil, leakage is permitted to drain to the linkage housing. If a different working fluid is to be used, the mechanism is designed to accommodate a separate cross-head bearing and leakage flow path with a modular cylinder housing and manifold assembly.

Figure 11 shows a CAD rendering of the triplex pump design. Some of the bearings and mounting frame have been excluded for clarity. The input crank is built into the crankshaft as described previously and the other links are labeled. The adjustment cylinder is used to actuate the central control link which is coupled to the other two control links with a common pin. Doing so synchronizes the displacement control of the three pistons.

Figure 11 CAD Rendering of Triplex VDLP

CONCLUSIONS

A design methodology for a variable displacement triplex pump has been presented which includes optimization and machine design. An 8.5kW triplex check-valve pump was designed with links which incorporate roller bearings to improve efficiency. The experimentally validated model predicts an efficiency greater than 90% for a majority of the operating conditions. Such a pump could have a large impact on the hydraulics industry vastly reducing energy consumption carbon emissions. In the future, a functioning prototype will be manufactured to evaluate the performance of this new architecture

While this design was selected for its timing ratio and efficiency characteristics, the multi-objective solution space gives opportunities to design a pump that excels in other areas as well, depending on application requirements and allowable tradeoffs. For example, mechanism size could be minimized further to increase power density for mobile applications. The linkage based platform creates much design flexibility allowing a pump to be easily optimized for a large variety of applications and shows great promise as an efficient and versatile architecture.

ACKNOWLEDGMENTS

This work is supported by the National Science Foundation under grant number EFRI-1038294.

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