The Shaft Torque of a Tandem Axial- Piston Pump

Transcription

1 The Shaft Torque of a Tandem Axial- Piston Pump oah D. Manring Viral S. Mehta Mechanical and Aerospace Engineering Department, University of Missouri-Columbia, Columbia, MO Frank J. Raab Kevin J. Graf Hydraulic System Research, Technical Center, Caterpillar, Inc., Peoria, IL The objective of this study is to identify the best indexed position of two rotating s within a tandem axial-piston pump for attenuating the torque ripple amplitude that is exerted on the shaft. By attenuating the torque ripple characteristics of the pump, other vibration aspects of the machine are also expected to be reduced. In particular, the objectives of this paper are aimed at reducing the noise that is generated by the pump. This paper begins by considering the theoretical torque ripple that is created by the discrete pumping elements of a single rotating within an axial piston machine. From this analysis, an equation is produced that describes a single pulse for the torque ripple as a function of the average torque and the total number of pistons that are used within the rotating. By superposing another rotating on top of the first, and by indexing the angular position of one rotating relative to the other, a second equation is produced for describing the theoretical torque ripple of a tandem pump design. This equation is also a function of the average shaft torque and the total number of pistons that are used within a single rotating ; however, an additional parameter known as the index angle also appears in this result. This index angle is shown to amplify or attenuate the amplitude of the torque ripple depending upon its value. From these results, it is shown that a proper selection of the index angle can reduce the torque ripple amplitude by as much as 75%. DOI: / Introduction Background. Axial-piston pumps are widely used within hydraulic control systems to convert rotating mechanical power into hydraulic fluid power. As such, these machines are among the most common power sources for hydraulic control systems. Although hydraulic control systems exhibit many advantages over other power transmitting and control devices e.g., power density, continuously variable power transmission, a stiff dynamic response, flexible connections, etc., these control systems are also known for several drawbacks. Among these, one finds the nagging problem of vibration and noise generation. This notorious problem of hydraulic control systems may often be traced back to the Contributed by the Dynamic Systems, Measurement, and Control Division of ASME for publication in the JOURAL OF DYAMIC SYSTEMS, MEASUREMET, AD CO- TROL. Manuscript received September 22, final manuscript received December 7, Review conducted by Perry Y. Li. axial-piston pump, which may produce noise levels as high as 100 db; the equivalent noise level that is experienced when standing near a lawn mower or riding on a subway. In the past, there have been many theories for explaining the problem of hydraulic noise 1 5; however, recent unpublished laboratory experiments have suggested that the noise produced by an axial-piston pump is directly related to the torque ripple that is measured on the pump shaft. Furthermore, these experiments suggest that by reducing the amplitude of the torque ripple, the noise level of the pump will be attenuated accordingly. Thus, current work is being directed toward reducing the amplitude of the shaft torque ripple, which in turn reduces the vibration characteristics of the machine. This paper addresses this issue from a theoretical point of view for a tandem axial-piston pump design. Literature Review. Over the past 25 years, a tremendous effort has been placed on the task of reducing the noise and vibration characteristics of axial-piston pumps. This work has been summarized by Harrison and Edge 1, in which they addressed the use of anti-noise devices that are typically cost prohibitive, the impact and online adjustment of valve plate timing also see 2, the use of independent suction and delivery timing, and the use of check valves for controlling the pressure transition of a single piston within the pump. Pettersson et al. 3 also present many methods for reducing the noise and vibration of an axial-piston pump including the use of a pre-compression filter volume for fixed displacement machines. Another method for attenuating noise and vibration has been suggested by Schutten et al. 4. This method recommends a random variation in the kidney pattern on the cylinder block for the purposes of altering the noise quality by shifting the noise energy from a high frequency to a low frequency. In a recent study, Achten et al. 5 presents the floating cup principle in which a back-to-back configuration of pistons is used to reduce the flow and pressure pulsations of the pump. All of this work has shown merit in the effort to reduce the noise and vibration characteristics of the axial piston pump, but none of it has specifically addressed the torque ripple that is inherent within axial-piston machinery. As previously noted, this current study is based upon unpublished experimental results that suggest a direct relationship between noise creation and the torque ripple of the input shaft. By attenuating the amplitude of the torque ripple for a tandem pump design, it is believed that the noise generated by this machine will also be attenuated. This paper focuses attention on the torque ripple characteristics of the tandem pump design with an eye toward reducing pump noise and vibration. Pump Description. Figure 1 shows a schematic of a tandem pump design. As shown in this figure, the tandem pump is comprised of two identical rotating s that are designed to pump fluid through a mechanism that will be described shortly. Each rotating in Fig. 1 is used to produce a specified volumetric displacement per revolution of the input shaft. Used together, these two rotating s produce twice the volumetric flow rate as compared to a single rotating. The rotating s in Fig. 1 are connected to each other using a single shaft and a manifold. The shaft is used to turn both rotating s at an angular velocity while the manifold provides a common intake and discharge port for both rotating s. The torque on the input shaft of the tandem pump is shown in Fig. 1 by the symbol T. The flange in Fig. 1 is used to bolt the tandem pump assembly to a prime mover such as an electric motor or an internal combustion engine. Each rotating in Fig. 1 consists of several pistons that are evenly spaced within a circular array about the centerline of the cylinder block. This centerline of the cylinder block is shown in Fig. 1 by the x-axis, which also corresponds to the centerline of the input shaft. The shaft and the cylinder block are connected to each other through the use of a spline that transmits torque and angular displacement from the shaft to the cylinder block. The cylinder block is held tightly against the fixed valve plate using Journal of Dynamic Systems, Measurement, and Control MAY 2007, Vol. 129 / 367 Copyright 2007 by ASME

2 Fig. 1 A schematic of the tandem pump design the compressed force of the cylinder block spring and a less obvious clamping force that results from the fluid pressure within the cylinder block. A thin film of hydraulic fluid separates the valve plate from the cylinder block, which under normal operating conditions forms a hydrodynamic bearing between the two parts. The valve plate provides a bearing surface for the cylinder block and is also used to separate the discharge and intake ports that are more crudely defined by the manifold shown in Fig. 1. ote: the discharge and intake ports are not explicitly shown in this figure. A ball-and-socket joint connects the base of each piston to a slipper, which is held against the thrust surface by a spring loaded retainer. The thrust surface is usually called the swash plate surface and is shown in Fig. 1 by the inclined angle. Hydrostatic and hydrodynamic lubrication mechanisms are used to provide adequate bearing conditions between the slippers and the swash plate. While the valve plate is held in a fixed position, the splined input shaft is used to drive the cylinder block about the x-axis at a constant angular velocity. During this motion, each piston periodically passes over the discharge and intake ports on the valve plate. Furthermore, because the slippers are held against the inclined plane of the swash plate, the pistons also undergo an oscillatory displacement in and out of the cylinder block. As the pistons pass over the intake port, the piston withdraws from the cylinder block and hydraulic fluid is drawn into the piston chamber from the intake port of the pump. As the pistons pass over the discharge port, the piston advances into the cylinder block and hydraulic fluid is pushed out of the piston chamber into the discharge port of the pump. This motion repeats itself for each cylinder block revolution and the basic task of pumping fluid is then accomplished. ote that this behavior is identical for both rotating s shown in Fig. 1 and the flow contributions from each rotating are added together to achieve the net discharge flow for the assembled tandem pump design. Views A-A and B-B of Fig. 1 show a sectional view taken through the cylinder block of rotating s A and B, respectively. This view shows that when the tandem pump is assembled, the discrete pumping elements for each rotating may be indexed slightly relative to each other. This index angle is shown in Fig. 1 by the symbol. In other words, the angular position of the pistons may or may not be coincident with one another depending upon where the shaft spline connection happens to place each rotating. As it turns out, this angular difference between the two rotating s shown in View A-A and B-B of Fig. 1 is very important in determining the magnitude and frequency of the torque ripple that is generated on the pump shaft. With this being the case, this study is aimed at determining the best value for that may be used to attenuate the amplitude of the net torque ripple that is exerted on the input shaft by rotating s A and B. By attenuating this amplitude, the vibrational characteristics of the tandem pump will also be reduced, thereby providing a machine that operates more smoothly with less noise generation. Objectives. The objective of this study is to identify the best indexed position of two rotating s within a tandem axialpiston pump for attenuating the torque ripple amplitude that is exerted on the shaft. By attenuating the torque ripple characteristics of the pump, other vibrational aspects of the machine are also expected to be reduced. In particular, the objectives of this paper are aimed at reducing the noise that is generated by the pump. This paper begins by considering the theoretical torque ripple that is created by the discrete pumping elements of a single rotating within an axial-piston machines. From this analysis, an equation is produced that describes a single pulse for the torque ripple as a function of the average torque and the total number of pistons that are used within the rotating. By superposing another rotating on top of the first, and by indexing the angular position of one rotating relative to the other, a second equation is produced for describing the theoretical torque ripple of a tandem pump design. This equation is also a function of the average shaft torque and the total number of pistons that are used within a single rotating ; however, an additional parameter known as the index angle also appears in this result. This index angle is shown to amplify or attenuate the amplitude of the torque ripple depending upon its value. From these results, it is shown that a proper selection of the index angle can reduce the torque ripple amplitude by as much as 75%. Analysis One Rotating Group. The analysis for the instantaneous torque that is exerted on the shaft of an axial-piston pump by a single rotating has been presented in previous literature 6. Figure 2 shows a schematic of forces that generate this torque, where F n is the side load exerted on the cylinder block of the rotating by the nth piston and d n is the instantaneous location of this force away from the centerline of the shaft. Using these quantities, it may be shown that the torque exerted on the shaft by the nth piston is given by T n =F n d n. In Fig. 2, r is the piston pitch radius, n is the angular position of the nth piston relative to bottom dead center, A p is the cross-sectional area of a single piston, is the swash plate angle for the rotating, and P n is the instantaneous pressure within the nth piston chamber. From Fig. 2, it may be shown that the net torque exerted on the pump shaft by a single rotating is given by T = F n d n = A p r tan P n sin n 1 where is the total number of pistons within the rotating. Equation 1 has neglected the linear momentum of the piston because it can be shown that the net torque effect of piston inertia is identically zero for a rotating that utilizes evenly spaced pistons while rotating at a constant angular velocity 6. Ifthe ideal pressure profile shown in Fig. 2 is assumed, then the pressure within the nth piston chamber may be written as P n = P d 0 n 2 P i n 2 where P d and P i are the discharge and intake pressures of the pump, respectively. Using this result with Eq. 1, the instantaneous torque exerted on the shaft by a single rotating is given by 368 / Vol. 129, MAY 2007 Transactions of the ASME

3 Fig. 3 Schematic of the net torque ripple produced by a single rotating Fig. 2 A schematic of the forces that exert torque on the input shaft of a single rotating n T = A p r tanp d sin n + P i n=n+1 sin n 3 where n is the total number of pistons instantaneously pressurized by the discharge port of the pump. For a rotating with an odd number of pistons, n=±1/2. For a rotating with an even number of pistons, n=/2. If the pistons are evenly spaced within the cylinder block, the following expression may be written to describe the angular location of the nth piston n = n 1 4 where 1 is the angular position of piston number 1. Using Eq. 4, it may be shown that b sin n = csc 1 a + b a + b 2 sin n=a sin 1 + where a and b are dummy variables to indicate the limits of the summation operator. Substituting this general expression into Eq. 3, the following expression for the instantaneous torque exerted on the pump shaft by a single rotating may be produced where 5 T = T csccos 1 for A p r tanp d P i T = 8 Equation 6 describes a single torque pulse that is exerted on the pump shaft for a limited range of angular positions for piston number 1; i.e., By restarting Eq. 6 each time these limits are exceeded, the schematic of Fig. 3 may be produced to illustrate the pulsating torque that is exerted on the shaft by a single rotating. In this figure, the pulse period and amplitude are shown by the dimensions 2 and T, respectively. Two Rotating Groups. In practical tandem pump designs, two identical rotating s are typically used on a single shaft to generate twice as much pump flow as compared to the flow produced by a single rotating. In this case, both rotating s exert a torque on the pump shaft that is governed by Eq. 6. The only difference between the two rotating s used in a tandem pump design is the angular position of piston one 1 that may be indexed relative to the other rotating by an amount. As shown in Fig. 1, this angular index results from a difference in the assembled positions of each rotating on the shaft. In other words, the spline connection between the cylinder block and the rotating shaft may not be aligned perfectly so as to position piston 1 in the same place for each rotating. To account for this potential angular difference, the following equations are written to describe the torque exerted on the pump shaft by each rotating T A = T csccos 1 for T B = T csccos 1 for In this equation, the subscripts A and B refers to rotating s A and B, respectively, and and T are given in Eqs. 7 and 8, respectively. By adding these two results, the following torque pulsation is described for the tandem pump design T =2T csccos/2cos/2 + 1 for where 0 for an examination of the largest pulsation of the torque ripple. The results of this equation will be discussed in the following section. = 2 odd number of pistons even number of pistons and the average shaft torque is given by 7 Results and Discussion The objective of this paper is to identify the index angle that produces the smallest amplitude of the torque ripple. As shown in Fig. 3, the maximum torque exerted on the shaft will occur at the middle of the pulse which, for the tandem pump design, is given by 1 = /2 see Eq. 10. This maximum torque value is Journal of Dynamic Systems, Measurement, and Control MAY 2007, Vol. 129 / 369

4 Table 1 Index angles to be used to achieve the smallest torque amplitude for a tandem pump deg Fig. 4 Shaft torque characteristics of a tandem pump utilizing nine pistons for each rotating T max =2T csccos/2 11 Also shown in Fig. 3, the minimum torque exerted on the shaft will occur at the ends of the pulse. These ends are identified when 1 =0 or when 1 =2 for the tandem pump design. Evaluating Eq. 10 at one of these points produces the following result for the minimum torque T min =2T csccos/2cos /2 12 Subtracting Eq. 12 from Eq. 11 produces the following result for the torque ripple amplitude of the tandem pump T =2T csccos/21 cos /2 13 where T is schematically shown in Fig. 3. To illustrate the torque result of Eq. 10, the quantity T/2T is plotted in Fig. 4 for a tandem pump utilizing 9 pistons for each rotating. As shown in this figure, the torque amplitude is a maximum when =0 and is a minimum when =. The reader will recall that is bounded between 0 and for the examination of the largest pulsation of the tandem pump torque ripple. In other words, we have analytically identified the index angles that maximize and minimize the amplitude of the torque ripple. These analytical results are summarized as follows T max =2T csc1 cos when =0 T min =2T csccos/21 cos/2 when = 14 To illustrate the difference between these two results, Fig. 5 shows a bar-chart comparison of the maximum and minimum torque Fig. 5 Torque amplitude for tandem pumps using a specified number of pistons for each rotating amplitudes for tandem pumps utilizing rotating s with a specified number of pistons in each. As shown in this figure, tandem pumps that use an even number of pistons within each rotating exhibit the largest torque amplitude, while tandem pumps that use an odd number of pistons in each rotating exhibit the lowest torque amplitude. This is a well-known feature of axial-piston pumps and is the primary reason for using an odd number of pistons within most rotating designs. From Eq. 14, the percent difference between the maximum and minimum torque amplitude may be calculated as T max T min T max 100 % = 2 + cos/2 4 cos 2 /4 100 % 75% 15 In other words, by selecting the proper index angle for the assembly of the two rotating s, i.e., setting =, the torque amplitude may be reduced by 75% from its largest possible value. As the reader can see from Figs. 4 and 5, this reduction is significant. To achieve this smallest torque amplitude for the tandem pump, the index angles shown in Table 1 should be used for tandem pumps utilizing rotating s with a specified number of pistons. The explanation for these results is as follows. For each rotating used in the tandem pump design, the frequency of the torque ripple is equal to the piston pass frequency for a rotating that utilizes an even number of pistons. For a rotating that utilizes an odd number of pistons, the frequency of the torque ripple is equal to two times the piston pass frequency. To minimize the amplitude of the net torque ripple for two rotating s used on a tandem pump shaft, the minimum torque value for one of the rotating s must coincide with the maximum torque value for the other rotating. This means that the torque ripple for one of the rotating s must be shifted or indexed by one-fourth of the piston pass frequency for a pump utilizing an odd number of pistons for each rotating ; or, the torque ripple must be shifted by one-half the piston pass frequency for a pump utilizing an even number of pistons for each rotating. These shifted dimensions are shown in Table 1 by the angular dimension, which is a physical index that may be designed into the assembly process of the tandem pump for the purposes of generating a torque ripple with the lowest possible amplitude. Conclusion The results and discussion of this paper support the following conclusions: 1. Tandem axial-piston pumps produce an oscillatory torque ripple that is known to contribute to the noise and vibration characteristics of the machine. 2. The torque ripple for the tandem pump is generally less for a pump that utilizes an odd number of pistons for each rotating as compared to a pump that uses an even number of pistons for each rotating. 3. In order to minimize the amplitude of the tandem pump torque ripple, one rotating must be indexed relative to the other rotating by the angular dimension =. See Eq. 7 for the definition of and Table 1 for tabulated values. 4 By assembling the pump so that the index angle = i.e., according to the previous conclusion, the torque ripple am- 370 / Vol. 129, MAY 2007 Transactions of the ASME

5 plitude may be reduced by 75% from is maximum possible value, which exists when the index angle is zero. Acknowledgment This project was proposed and sponsored by engineers at Caterpillar, Inc. who are devoted to the continuous improvement of off-highway equipment. The authors would like to give a note of special thanks to B. J. Landsberger who is now a research faculty member at the University of evada Las Vegas for his initiation of this work and for his participation in discussions that have proven to be helpful. omenclature A p cross-sectional area of a single piston d n normal distance of the nth piston away from the centerline of the pump shaft F n side load produced on the cylinder block by the nth piston total number of pistons within a single rotating n dummy variable for identifying the nth piston n number of pistons within a single rotating pressurized by the discharge port P d pump discharge pressure P i pump intake pressure P n fluid pressure within the nth piston chamber r piston pitch radius T net instantaneous shaft torque T average shaft torque produced by a single rotating T A instantaneous shaft torque produced by rotating A References T B instantaneous shaft torque produced by rotating B T max maximum torque generated within the pump torque ripple T min minimum torque generated with the pump torque ripple T n torque produced by the nth piston swash plate angle T torque ripple amplitude T max -T min index angle between rotating s number defined in Eq. 7 n angular position of the nth piston 1 angular position of piston number 1 angular velocity of the pump shaft 1 Harrison, K. A., and Edge, K. A., 2000, Reduction of Axial Piston Pump Pressure Ripple, Proc. Inst. Mech. Eng.: Part I: J. Syst. Cont. Eng., 2141, pp Edge, K. A., and Darling, J., 1989, The Pumping Dynamics of Swash Plate Piston Pumps, ASME J. Dyn. Syst., Meas., Control, 111, pp Pettersson, M., Weddfelt, K., and Palmberg, J.-O., 1991, Methods of Reducing Flow Ripple From Fluid Power Piston Pumps A Theoretical Approach, SAE International Off-Highway and Powerplant Congress, pp. 1 10, Milwaukee, USA. 4 Schutten, H. P., Wakefield, D. M., Zimmerer, D. R., and Malaney, D. W., 1994, oise Reduction at the Second Order Frequency, US Patent o. 5,358, Achten, P. A. J., Van den Brink, T. L., Paardenkooper, T., Platzer, T., Potma, H. W., Schellekens, M. P. A., and Vael, G. E. M., 2004, Design and Testing of an Axial Piston Pump Based on the Floating Cup Principle, Proceedings SICFP 03, The Eighth Scandinavian International Conference on Fluid Power, Tampere University of Technology, Vol. 2, pp Manring,. D., 1998, The Torque on the Input Shaft of an Axial-Piston Swash-Plate Type Hydrostatic Pump, ASME J. Dyn. Syst., Meas., Control, 120, pp Journal of Dynamic Systems, Measurement, and Control MAY 2007, Vol. 129 / 371