IFPE Paper 9. A Novel Methodology of Displacement Calculation for the Swash Plate Axial Piston Pump with Angle Cylinder Block Will Guo Danfoss Power Solutions (Zhejiang) ABSTRACT Displacement is the most important and basic performance parameter to be determined during the concept design period of an axial piston pump. However, it is difficult to calculate the displacement of pumps with angle cylinder block bores. To resolve the difficulty in calculation, a novel methodology for displacement calculation is proposed that creates all data without creating a 3D model. The equations for displacement calculation are derived from piston stroke based on the principle that the axial offset from the swash plate center of rotation to the line of piston balls is constant during swash plate movement. The key dimensions and parameters are identified for the components required to determine the pump displacement according to the above method. In addition, the application instructions and procedure for the equations of displacement calculation are stated. This method was used for the design calculation of an axial piston pump of 00cc displacement. With this novel methodology of displacement calculation, it is possible to calculate and analyze pump displacement early in concept design before any 3D model is created, which is very helpful to shorten the product development time. INTRODUCTION Swash plate type axial piston pumps are widely used in different applications throughout the industry, especially in construction machinery. Generally, two kinds of swash plate type axial piston pump designs can be found. These include designs with piston bores in the cylinder block arranged parallel to the shaft axis (named as straight cylinder block bores) and those with an angle to the shaft axis (defined as angle cylinder block bores). Previous studies [, 5] showed that the pump with angle cylinder block bores will have larger geometric displacement, compact structure and can be run at a higher rotational speed in comparison to the pump with straight cylinder block bores. Displacement is the most important and basic design parameter of an axial piston pump, and it needs to be determined early during the concept design phase. The maximum linear travel of the piston (piston stroke) along with piston diameter and number of pistons determines the displacement of the pump. The amount of piston movement in the cylinder bore is determined by the geometry of the cylinder block and swash plate angle. The cylinder block is driven by the shaft and as the block rotates around the shaft, it forces the piston motion due to the swash plate angle. The displacement for the pump with straight cylinder block bore is simple to determine, but it is difficult for the pump with angle cylinder block bore because the piston stroke is not easy to calculate. Ivantysynova [1] gave the equation to calculate the displacement for the pump with angle cylinder block bore based on the swash plate angle, cylinder block bore angle and piston pitch radius of the outer dead point at which the piston is displaced with maximum distance from the cylinder bottom. The piston pitch radius of the outer dead point should be obtained from the 3D model of the pump assembly, which does not exist during the concept design stage. Khalil [] hasn t given the equation to calculate the displacement, but deduced a complex piston stroke equation from the kinematics analysis with some parameters. This includes the pitch diameters of the angle block bore start and end, and the depth of the piston bore, which cannot easily be determined even through 3D modeling of the cylinder block with more [3] accuracy. Yang derived a relatively simple displacement equation with the swash plate angle, cylinder block bore angle and piston pitch radius of outer dead point from the pump kinematics analysis process. However, obtaining the pitch radius is still a problem during the concept design period. To resolve the above difficulty in calculation, this paper will develop a novel displacement calculation method for the pump with angle cylinder bore, which can be easily used during the concept design stage before a 3D model is created. The specific equations will be deduced and the application procedure will be suggested. With this method, the development process of a new pump with an angle cylinder block bore can be simplified and greatly accelerated. THE STRUCTURE AND WORKING PRINCIPLE OF SWASH PLATE TYPE AXIAL PISTON PUMP KIT GROUP The kit group structure with straight and angle cylinder block bore of swash plate type axial piston pump can be seen in Figure 1 and Figure respectively. The straight cylinder block design - The kit group consists of the cylinder block, pistons, slippers, ball guide, block spring, slipper retainer and swash plate.
Figure 1 The structure of a swash plate type pump with straight cylinder block Figure 1 illustrates the schematic drawing of a swash plate type pump with straight cylinder block design. The cylinder bores are distributed equally in the cylinder block in a circular array arrangement, which is parallel to the shaft axis. The cylinder block is rotated by means of the pump shaft driving, and is held tightly against the stationary valve plate by the effect of delivery pressure and compressed cylinder spring. A ball and socket joint connects the spherical end of each piston to a slipper, which is always kept in contact with the swash plate by means of the slipper retainer, ball guide and the compressed cylinder spring. During cylinder block rotation, each piston passed periodically over the discharge and intake port on the valve plate. Since the slippers are held against the inclined plane of swash plate, each piston undergoes a simple linear reciprocating motion in and out of cylinder block. As the cylinder block rotates, the volume of each cylinder bore in front of its respective piston will increase or decrease. When it passes over the intake port, its respective piston is retracted and hydraulic oil is drawn into the cylinder bore. When it passes over the discharge port, its piston advances into the cylinder and displaces the hydraulic oil out of the cylinder bore. The angle cylinder block design - Figure illustrates an improved design of swash plate pump design, in which pistons are arranged in a cylinder block with the bores at a small angle to the shaft axis. In this type of pump, the piston axis is inclined to the pump shaft axis. An axial component of the centrifugal force acting on each piston is generated and pushes the piston towards the swash plate. This reduces the piston inertia force effect, which normally tends to detach the pistons from the slipper away from the swash plate. Besides, the angle cylinder bore design can reduce the outer diameter of the valve plate. Figure The structure of a swash plate type pump with angle cylinder block With this design, the pump can be rotated at a high speed, thereby increasing pump specific power. The axial offset - Variable displacement axial piston means that the displacement can be changed as needed. During the displacement change process, the swash plate will tilt to some degree around the swash plate rotating center. From the structure in Figure, the line of piston balls is parallel to the swash plate running surface; the axial offset is defined as the distance between the center of the swash plate rotation and the line of piston balls. The axial offset is constant for a given design during the displacement change process. Normally, the axial offset is optimized for pump response. The axial offset produces a similar effect on both open and closed circuit pumps, however, the specific axial offset need differs. For open circuit pumps, axial offset is typically used to improve recovery to full stroke, but can also aid in return to neutral for closed circuit pumps. Axial offset needs and requirements will not be addressed in detail for the purposes of this paper. A NOVEL DISPLACEMENT CALCULATION METHODOLOGY PUMP DISPLACEMENT - The maximum linear travel of the piston (piston stroke) along with piston diameter and number of pistons determines the displacement of the pump defined as the volume of fluid displaced per one revolution. As defined above, the geometric displacement q of an axial piston pump is calculated as follows when the number of pistons is given by z, the piston diameter by d and piston stroke by s: q d zs (1)
PUMP WITH STRAIGHT CYLINDER BORE DESIGN - In Figure 3, line AB denotes the plane defined by the piston ball centers with the swash plate at zero degrees. This plane is parallel to the swash plate surface. Line CD denotes the plane defined by the piston ball centers when the swash plate is at angle ɣ. In the case of design without axial offset, line AB and CD will intersect at point O, thus point O defines the swash plate pivot point (i.e. center of rotation). In the right triangle ΔCEF, Thus, in the right triangle ΔDEF, 90 () 90 (5) EF DF s cos( ) (6) If the piston pitch radius of outer dead point is given by r, in the right triangle ΔCEF Figure 3 Conceptual schematic of swash plate type axial piston pump If the piston pitch radius is defined by r, in right triangle ΔAOC, then the piston stroke s can be defined as follows: s AC r tan () The geometric displacement q of an axial piston pump with straight cylinder block bore is then given by: q d zs d z r tan (3) PUMP WITH ANGLE CYLINDER BORE - Figure shows that the piston ball centers are defined by line AB when the swash plate is at the zero degree position. If the swash plate tilts to the maximum angle ɣ position, all the piston ball centers are on the line CD. Lines DH and CF are parallel to AB, and line DG is perpendicular to line CF. Line FE is perpendicular to line CD at E. O is the intersection of all axes of the cylinder bore. EF CF sin r sin (7) By substituting Eq. (7) in Eq. (6), we can get the following equation for piston stroke s: r sin cos( ) s (8) The geometric displacement q of an axial piston pump with angle cylinder block bore is given by following expression d zs d r sin z cos( ) q (9) With axial offset - Using the orthogonal coordinates system xyz, as shown in Figure 5. Its origin, O coincides with the point of intersection of the angle cylinder bore axes, the x-axis coincides with the pump driving shaft axis, z-axis is parallel to the axis around which the swash plate is swinging. The piston ball centers are on the line AB when the swash plate is at the zero degree position. If the swash plate tilts to the maximum angle ɣ position, then all of the piston ball centers are on the line CD. Line CF is parallel to AB. Point K is the rotating center of the swash plate. Line KM is perpendicular to line CD. Figure Conceptual schematic of the swash plate type axial piston pump with angle cylinder bore During the cylinder block rotation, the piston will move from outer dead point C to inner dead point D which will be a piston stroke. Thus, the piston stroke will be equal to the length of HC or DF. Figure 5 Working principle of the swash plate type axial piston pump with axial offset In the triangle ΔOFC,
Since the y-intercept is equal to 0, the equation of the straight line OC in slope-intercept form can be written as following Where x and y are variable. y x tan( ) (10) Similarly, the equation of the straight line CD in slopeintercept form is given by y x tan( 90 ) c (11) Where x, y are variable and c is a constant (i.e. y- intercept). Based on the working principle of the axial piston pump, the perpendicular distance from the swash plate center of rotation K to the projected line CD of the piston balls is constant. Therefore, we can calculate the distance of the swash plate rotating center ( ) to line CD with x, y equation 1 based on the standard distance equation of the point to line: k xk tan(90 ) yk c h tan (90 ) ( 1) k (1) Where, h is a constant and should be determined during the pump concept design. From equation (1), the value of constant c can be calculated. Substituting equation (1) in (11), and together with equation (10), the intersectional coordinate of line OC and CD will be resolved as x c, y c. And it is obvious that y c r (13) Thus, we can use the Eqs. (9) to (13) to calculate the displacement of the pump with angle cylinder block bore with the parameters z, d, ɣ, β, x k, y k and h. Without axial offset - For the design without axial offset from the swash plate center of rotation to the projected line CD of the piston balls, the swash plate will rotate at the point O1 as depicted in Figure 6. The piston ball centers are on the line AB when the swash plate is at the zero degree position. When the the swash plate rotates to the maximum angle ɣ position, all the piston ball centers are on line CD. Figure6 Working principle of the swash plate type axial piston pump without axial offset Using the orthogonal coordinates system xyz, its origin, O coincides with the intersection of the angle cylinder block bore axes, the x-axis coincides with the pump driving shaft axis, and the z-axis is parallel to the axis around which the swash plate is rotating. Without an axial offset, Eq. (1) will be equal to zero. In this case, Eq. (11) of line CD can be given based on its slope and the coordinate (x o1,y o1 ) of point O1 by ( o1 o1 y y ) tan(90 )( x x ) (1) Where x o1 is the x coordinate of point O1 and it can be taken from the specific design. By substituting Eq. (10) in Eq. (1), the coordinate of the intersectional point C will be resolved x c, y c. Thus, we can use the Eqs. (9), (10) and (1) to calculate the displacement of the pump with angle cylinder block bore with the parameters z, d, ɣ, β and x o1. A PROCEDURE OF PUMP DISPLACEMENT DESIGN The displacement should be the first primary performance parameter needed to be determined in any new pump development. According to the above analysis, the displacement is a function of the piston number, piston diameter, swash plate angle, cylinder bore angle, the coordinate of the swash plate rotating center and the axial offset from swash plate rotating center to the line of the piston balls. When a new pump is designed, the displacement can be determined by the following procedure. 1. Piston number. A pump with an even number of pistons produces a significantly larger flow ripple than a pump which is designed with an odd number of pistons. In fact, the majority of today s pumps are designed with a total of nine or seven pistons.. Maximum swash plate angle. For the variable axial piston pump, the swash plate angle (ɣ) governs the pump displacement. If ɣ is zero, the pump will not displace any fluid. Maximum displacement is achieved when ɣ is at the maximum value, usually in a range of 15 to 0 degrees for swash plate type pumps with common used material today.
3. Angle of the cylinder block bore. The angle of the cylinder block β can be theoretically chosen up to 5. In actual pump designs, it is usually much smaller, normally less than 10.. Axial offset. Normally, for open circuit pumps the offset from swash plate rotating center to the line of the piston balls is optimized for pump response. The offset is usually less than 10 mm in actual existing pump designs. 5. Piston diameter and coordinate of the swash plate rotating center. These two parameters depend on each other to some extent since the coordinate of the swash plate rotating center will determine the cylinder block size with angle β which should have enough room to accommodate the pistons. One of them should be defined based on experience. The other will be calculated according to the above equations. Normally, the piston diameter will be defined by manufacturing to utilize existing tooling. Together these parameters define the displacement of an axial piston pump. In the actual development process, it should be an iterative process to optimize displacement considering the geometry constraints, and performance and endurance requirement of the whole pump. CASE STUDY The described process in this paper was used to design an axial piston pump with angle cylinder block bore. The main parameters are listed below in Table 1. Table 1 Parameters determination Parameters Values Piston Number, z 9 Swash Plate Angle, ɣ, degree 16.5 Angle of Cylinder Bore, β, 5 degree Axial offset, h, mm 7.5 swash plate type axial piston pump with angle cylinder bore. 3. A displacement design procedure was suggested using the novel methodology.. An example pump with angle cylinder bore structure was calculated for 00cc based on this novel methodology calculation procedure. REFERENCES 1. Ivantysyn J. and Ivantysynova, M., Hydrostatic Pumps and Motors, Academic Books International, 1st ed., New Delhi, 001, pp 113-10.. Khalil M., Performance Investigation of The Swash Plate Axial Piston Pumps With Conical Cylinder Blocks, Ph.D Thesis, Concordia University, 003, pp 1-6. 3. Yang Fengyu, Hu Min, Ma Dejiang etc, Analysis of Kinematic And Dynamic Characteristics of Piston In Cam-type Axial Piston Pump With Inclined Piston, Journal of Lanzhou University of Technology, Vol. 37 No., April 011, pp 5-56.. Ivantysyn, R., Computational Design of Swash Plate Type Axial Piston Pumps, Masters Thesis, Purdue University, 011, pp 1-1. 5. Manring N. D., The Discharge Flow Ripple of an Axial-Piston Swash-Plate Type Hydrostatic Pump, Journal of Dynamic Systems, Measurement, and Control, Vol. 1, June 000, pp 63-68. CONTACT Will Guo is an Engineering Team Leader for Danfoss Power Solutions. He graduated from Mechanical Engineering with his Bachelors in Henan University of Science of Technology. His Masters in Mechanical Engineering is from Guangxi University and he also has Ph.D in Mechanical Engineering from Zhejiang University. Will specializes in developing the swash plate axial piston pumps and motors for the past years. Will may be contacted at wguo@danfoss.com Piston Diameter, d, mm 8 Coordinate of Swash Plate 700 Rotating Center, x k, mm According to Eqs. (9) to (13) and above parameters, the displacement can be calculated as 00cc. CONCLUSION 1. Analyzing the working principle of an axial piston pump shows that the distance from the rotating center of the swash plate to the line of the piston balls is constant during pump operation.. A novel displacement calculation was proposed and the specific equations have been derived for the