Chapter - 4. Variable Displacement Axial-Piston Pump - Design and Analysis

Transcription

1 Chapter - 4. Variale Displaement Axial-Piston Pump - Design and Analysis 4.1 Overview In a pressure-ompensated design of a variale displaement axial piston pump as shown in Fig. 4.1, the inlination of the swash plate is adjusted y utilizing the pressure at the delivery kidney port to produe a ontrol torque for having a relative swiveling motion etween the plate and the head of eah piston. A numer of splines diretly onnet the arrel and the shaft passing through a hole at the enter of the swash plate. Thus, the swash plate remains stationary as long as the disharge demand from the pump remains onstant. During the variation of delivery pressure over a ertain range, a torque produed y the ontrol arrangement of a spool valve, ylinders and springs auses a swiveling motion of the swash plate therey hanging its inlination and the pump disharge. Zaki and Baz [43] developed a model apturing the dynamis of a pressureompensated axial piston pump and analyzed the effet of variation of some design parameters at a fixed operating ondition. For minimizing the settling time, steadystate error and maximum pressure overshoot for different loading onditions, Baz [44] reommended the use of a high natural-frequeny stroking arrangement for the ontrol. In oth the analyses, a very simple onfiguration for the pump was assumed that together with the physial data of a speifi pump provided the oeffiients of linear transfer-funtion model of the dynamis. Akers and Lin [8] performed an optimal design analysis, onsidering a single stage eletrohydrauli servovalve along with ylinders for the ontrol of the swash angle of the pump. 51

2 Variale Displaement Axial-Piston Pump - Design and Analysis Figure 4.1: Shemati of a swash-plate axial-piston pump with pressure ompensator. 5

3 Variale Displaement Axial-Piston Pump - Design and Analysis Zeiger and Akers [] developed a more detailed model for the dynamis of the fixed-displaement swash-plate pump. Besides analyzing the effets of different design variations and toleranes, the steady-state estimate of the ontrol torque was provided. Of ourse, the knowledge of this torque is essential for designing the ontrol system. Shoenau et al. [0] arried out the dynami analysis of a pump, having a swash plate together with a simple swiveling arrangement involving a hydrauli ylinder and a spring. The variation of the swash angle with time in response to a given delivery pressure signal was suessfully ompared with experimental result. Manring [45, 46] onsidered an arrangement of a ontrol and ias ylinders for swiveling the swash plate over a pair of radle earings. He otained the fores on the swash plate from eah of these memers. In the former analysis [45], the axes of atuation pistons were assumed to e oriented in a transverse diretion with respet to the arrel axis. In the latter arrangement [46] that alls for larger pump diameter to aommodate the atuation pistons, all the axes were taken as parallel to eah other. A properly designed axial arrangement results in nearly equal loading on oth the radles that ontrasts the grossly unequal load sharing in the former arrangement. A transversely atuated swash plate not only has low earing life due to unequal loading, ut also suffers from ontainment limitation at low delivery pressure and high rotational speed. In the design and analysis presented over the remaining setions, existing models [47] have een omined and modified for an axially atuated swash plate. Modeling of the swiveling dynamis has een presented in terms of the omplete freeody diagrams of the relevant assemly as well as in terms of the individual pistons. A design optimization of the ompensator has een aomplished y onsidering a mean torque estimated over a revolution of the arrel from the predited variation of pressures within the arrel ores, assuming to e not influened y the ompensator dynamis. The model developed in Chapter has een applied here to otain the pressure variation. A valve plate design with optimized pre-ompression angle ahieved in Chapter 3 has een used. Following the stand-alone optimization, a oupled analysis of the swash-plate swiveling and arrel-pressure dynamis has een 53

4 Variale Displaement Axial-Piston Pump - Design and Analysis arried out. From the oupled analysis, the final ompensator design has een reommended. 4. System Desription and Mathematial Model Figure 4.1 depits a swash-plate pump with pressure ompensation. The valve plate A ontains the sution port B and delivery port C for external pipe onnetions. Ports B and C are internally onneted with two large plate kidneys D and E respetively. The rotation of the arrel F with angular speed auses eah of the nine ores with axes laeled as 1 to 9 to move alternatively over these plate kidneys along the pith irle of radius R p. A arrel ore H enters eah plate kidney from the side having a silening groove, namely G1 and G. For proper interfaing with the plate kidneys, eah arrel ore towards the plate side has a small kidney ore of axial length l, kidney angle and angle for the semiirular ends. Through the part of eah ore with irular ross setion, a arrel piston I moves from BDC to TDC and then TDC to BDC, while rotating essentially over D and E respetively. The figure shows the valve plate to have ridge angle etween the plate kidneys and pre-ompression angle. Figure 4.: Shemati of the pressure-ompensating spool valve. With the rotation of eah arrel ore, the piston in it reiproates with stroke length that depends on the swash angle at a stationary position of the swash plate J. Figure 4.1 shows the swash plate supported at the maximum angle 0 y the slippers K and two radle earings L1 and L. When the delivery pressure P d exeeds the 54

5 Variale Displaement Axial-Piston Pump - Design and Analysis ut-in limit, the spool inside the valve M moves direting flow to the ontrol ylinder N that auses the swash plate to rotate against the fore exerted y the springs inside the ias piston O and ehind the ush P. Aout the ommon enter of L1, L and P, the rotation of the swash plate together with the slipper retainer Q takes plae. As a result, the flow Q L through the needle valve R hanges. The spool valve M, shown in Figures 4.1 and 4., has a ore ontaining a three-land spool of mass m sp and land diameter d sp, a spring of stiffness k sp loated at the left end hamer of the ore and a numer of orifies on the ore wall. There are two orifies of diameters d osl and d osr respetively at the spring-side left and right end hamers of the ore, another orifie metered y the entral land of the spool and two more fixed-opening orifies at the hamers etween the entral land and the two end lands of the spool. While the two rightmost orifies supply oil to the ore from the delivery port at pressure P d, the two leftmost orifies arry the oil to a reservoir at pressure P r. Up to the ut-in level of pressure seated on the right fae of the ore due to a pre-ompression sp P d, the spool remains in the spring. During this phase of pump operation, the spool displaement equations an e written as x for ( Pd Pr )( d sp /4) ksp sp, (4.1) sp 0 and msp xsp ksp ( xsp sp ) {( Pd Pr ) 4 dsp x 4 4 sp (1/ dosr 1/ dosl )/(Cd )}( dsp / 4) Fd for d r sp ( P P )( d /4) k sp sp, (4.) where is the oil density, damping due to the transient flow fores [48] is F d C d is the disharge oeffiient of an orifie and the l Q l ( Q Q )}, (4.3) { d sp r sp where l d and l r are the distanes etween the nearest points in the metered orifie and its adjaent orifies at the right and left respetively, Q sp and Q are the flow 55

6 Variale Displaement Axial-Piston Pump - Design and Analysis rates through this right orifie and the ylinder N respetively. For a irular metering port of radius r P, initial overlap d delivery line and underlap o towards the hamer side onneted with the u r towards the hamer side onneted with the return line, these flow rates an e expressed with referene to Fig. 4. as Q Q sp sp d rp os sin ( P P )/ C, (4.4) d d rp d d d m os sin ( P P )/ Q C, (4.5) r r r m r with and os 1 max( x o,0)/ r, (4.6a) d sp d p os min {1 ( u x )/ r }, 1, (4.6) r r sp p and P m is the average pressure in the ylinder N. Figure 4.3: Complete free-ody diagrams of the pistons. 56

7 Variale Displaement Axial-Piston Pump - Design and Analysis Figure 4.3 ontains the free ody diagrams of the pistons in ylinders N and O along with the piston-slipper pair within the arrel ore I equal to 1. Of ourse, the aelerations of these pistons remain linked suh that all these omponents remain in ontat all the while. Hene following the rotation of the swash plate aout the swiveling axis z (perpendiular to the plane of the paper) as shown in Fig. 4.4, the ias piston displaement and the swash angle an e expressed in terms of ontrol piston displaement x as x, (4.7a) x and 0 x / l, (4.7) where l is the distane of the axes of the ores of the ontrol and ias ylinders from the arrel axis. Inside the ontrol ylinder ore of length l 0, there is a piston of diameter d and radial learane r. During piston movement, the learane gets squeezed towards the arrel side and piston gets tilted. The tilt leads to an angle etween the axes of the piston and its ore, reduing the angle etween the piston axis and the swash plate. Of ourse due to the small radial learane, the tilt angle remains very small. The tilt and squeeze provide a variation of pressure, oth in the radial diretion r and tangential diretion at the up inlet and along the axial and tangential diretions in the annular leakage path. In terms of periodi amplitude P a, the pressure variation at the up inlet, shown as setion XX in Figure 4.3, an e written as P P (r/ d ) P sin. (4.8a) m a Assuming the onsequent leakage through the learane to e governed y Poiseuille flow [31], it an e determined y onsidering elemental flows parallel to the piston axis and integrating these around the piston periphery. This yields the leakage as Q 3 lk ( r m r 0 d )( P P )/{3 ( l x )}, (4.8) 57

8 Variale Displaement Axial-Piston Pump - Design and Analysis where is the oil visosity and l 0 is the initial length of the ontrol piston within its ore. The visous frition due to the leakage opposing the piston veloity x an e modeled [0, 31] as f d x ( l 0 x )/. (4.8) r The net fore due to pressure on the ontrol piston, the visous frition and the reation at the point of ontat etween the swash plate and the piston gives rise to an aeleration of the piston of mass m. Of ourse, the pressure of oil having ulk modulus rises inside the ore due to aumulation within the piston up of volume V, partially relieved y the piston displaement. The total ontat reation along the piston axis at an inlination ) with the swash plate has a omponent F normal ( to the swash plate and a stati frition stati frition is given y F along the plate, where the oeffiient of tan( ). (4.8d) Finally, the oupled dynamis of pressure and displaement of the ontrol piston is modeled as m/ Q Qlk ( /4) d ( V x. d /4) P x, (4.8e) and m r m x ( P P )( d /4) f 1 F. (4.8f) Similar to Equations (4.8a), it is possile to express the variation of pressure at the entry to the piston up of the ias ylinder in terms of a spatial mean pressure P m and angular amplitude P a within the ias ylinder as, P P (r/ d ) P sin. (4.9a) m a Between the ore of diameter d and the ias piston, the leakage and the visous frition ating on the ias piston an e expressed in view of Equation (4.7a) as Q lk 3 ( r m r 0 d )( P P )/{3 ( l x )}, (4.9) 58

9 Variale Displaement Axial-Piston Pump - Design and Analysis and f d x ( l 0 x )/, (4.9) r where the initial engaged length and radial learane are l 0 and r respetively. Similarly in terms of the angle etween the axes of the ias piston and its ore, the oeffiient of stati frition on the ias piston an e represented as tan( ). (4.9d) Figure 4.4: Free-ody diagram of the swivelling dynamis of the swash plate. Both the pistons in ylinders N and O are sujeted to fores due pressures given respetively y Equations (4.8a) and (4.9a) and fritions given respetively y Equations (4.8) and (4.9). As shown in Figure 4.4, the total fore on the ontrol piston an e written as F p r ( P P ) d / 4 f. (4.10a) Figure 4.1 shows a ompression spring inside the ylinder O. At this position this spring of stiffness k has a pre-ompression. Thus, orresponding to the position shown in Figure 4.3, the spring fore on the piston is given y F s k ( x ). (4.10) 59

10 Variale Displaement Axial-Piston Pump - Design and Analysis Figure 4.4 shows the total fore on the ias piston F p due to the spring, pressure and frition as F p s r F ( P P )( d /4) f. (4.10) Besides an orifie of diameter d ol at the left end of the ore, there is another orifie of diameter d or in the urved wall of the ias piston at an axial distane l or from the eginning of the ore. While the first orifie has a onstant area, the seond orifie gets losed eyond a ertain displaement. Thus, a relatively steeper rise in the iasylinder pressure takes plae thereafter, providing a ushioning effet against impating etween the piston and the ore wall. The flow rates through oth the orifies an e expressed together as Q o ol or C [ d /4 ( d /4){( ) os sin }] ( P P )/, (4.11a) d where 0 for x lor, (4.11) for or or x l d, (4.11) r and os 1 ( x lor) / dor for lor x lor dor. (4.11d) Finally, the oupled dynamis of the pressure inside the ias piston of up volume V and the displaement of the ias piston of mass m an e written as ( V x. d /4) P / ( d /4 ) x Q Q, (4.1a) m lk o and m x 1 F F f. (4.1) p Consequent to an inrease of delivery pressure eyond the ut-in limit, the displaement of the spool in the valve M sets in the oupled motion of the ontrol and ias pistons together with the swash plate with mass moment of inertia I s. Figure 4.4 shows the free ody diagram of the oupled system. Besides the fores from ontrol and ias pistons given y Equations (4.10a) and (4.10), eah arrel piston exerts a net fore due to pressure and frition given y 60

11 Variale Displaement Axial-Piston Pump - Design and Analysis F pi pi p P. d / 4 f, (4.13a) pi where f d y ( l y ) / h, (4.13) pi p i i i d p is the piston diameter, P pi is the average pressure inside the ore of the i th arrel piston and the piston displaement has een shown in Fig..1 in Chapter. Figure 4.4 also shows the reations on the swash plate from the radle earings L1 and L and the reation from the ush P on the slipper retainer Q. These fore-earing surfaes have een given spherial shapes suh that all the reation fores pass through point o on the z axis. Due to the rolling nature of these ontats, the onsequent fritions are negligile in omparison to the visous fritions. Hene, taking moments of all the fores aout z axis, the equation of the swiveling dynamis an e derived with referene to Figure 4.4 as {( I s / l) ( m m ) l} x max [0, F R p p F p 9 ] l [( F i 1 pi m p x pi ) os{ t / ( i 1) / n}], (4.14) where the zero within the max funtion represents the pre-swiveling stati phase and m p is the mass of eah arrel piston. In pratial designs, the maximum swash angle is kept within 18 0 to 0 0 so that the availale stati frition prevents slippage etween any ontating pair. The existing torque models [0, 31, 49] have een developed y negleting the stati frition and onsidering the ontat fore to e direted along the normal to the swash plate. However, it should e kept in mind that the net reation fore for eah ontating pair should always e direted along the piston axis, irrespetive of whether stati frition has een onsidered or negleted. In view of the small radial learane around eah piston, this axis always remains almost parallel to the arrel axis. Now with referene to Figure 4.3, y otaining the expression for the total reation at eah ontating pair and taking their moments aout the swiveling axis, an expression idential to that given y Equation (4.14) an e found out. 61

12 Variale Displaement Axial-Piston Pump - Design and Analysis The visous frition on the i th arrel ore represented y Equation (4.13) has an assoiated leakage Q li. Other leakages Q lsi and Q li through the arrel-piston apillary and etween the valve plate and the arrel have een shown respetively in Figures 4.3 and 4.1. These equations have een derived in Chapter Figure 4.5: Geometri details of varying flow areas and delivery pressure. A numer of other disharges have also een shown in Figures 4.3 and 4.5 at the entries and exits of the ontrol volumes of grooves G1 and G. At the entries, these are respetively respetively flow Q 1 and Q 1 vi shown for i equal to 1 and Q. At the exits, these are Q vi shown for i equal to 5. For the arrel kidney entrane, the Q svi has een shown for i equal to 5. Through a needle valve, the disharge Q L from the delivery port goes ak to the reservoir. A general representation for these flows has een already shown with referene to Tale.1 and Tale. in the Chapter. Figure 4.5 gives the variations of areas indiated in the Tale.1, over a omplete yle interfaing with arrel kidney 1 and silening groove G1 or delivery plate kidney along with arrel kidney 5 and silening groove G or sution plate 6

13 Variale Displaement Axial-Piston Pump - Design and Analysis kidney. Tale.3 in Chapter lists the various dimensions used for alulating these areas. Eah of grooves in the figure is a tetrahedral wedge. The view of a groove from the arrel side is a triangle ade or hkl of height h 1 and ase triangle opq with onstant flow area triangles with a ommon edge sloping with 1 de. In addition to a A, eah groove is ounded y two slanting s. The disharges have een taken as positive, if these are in the diretions shown in the figure. Hene, in grooves G1 and G, the flows respetively towards the plate kidney and the ore are positive. Through the onstant areas A, these flows have een shown as Q 1 and respetively. Through the variale areas A 1v1 equal to de and A v5 as iklji, shown for the arrel ores 1 and 5 respetively, the orresponding disharges are Q 1v1 and Q v5. The varying areas dv1 Q A and A sv5 respetively etween the arrel kidney 1 and the delivery plate kidney and etween the arrel kidney 5 and the sution plate kidney an e identified in the figure as equal to dfged and kmnlk respetively. Figure 4.5 also shows the variation of the delivery pressure otained y employing the model developed in Chapter ignoring the oupled effet of the ompensator dynamis. Equations for arrel piston displaement, flow and pressure gradients are shown in Chapter that provides the model of the onneted dynamis of pressure in the flow loop. This set of equations has een solved for a fixed swash angle and needle valve opening so as to otain the mean torque on the swash plate for one omplete revolution of the arrel. The mean torques otained for different pairs of the fixed values have een used as the input for otaining an optimum design of the pressure ompensator. 4.3 Optimisation of pressure ompensator Using the model presented aove, an optimized design study for the pressure ompensated pump has een taken up. This has een done y speifying a step hange in the opening of the load orifie from the steady operation of the pump at the ut-in ondition to the ut-off ondition. Correspondingly, the model predits transient variations during whih the swash angle adjusts to the hanged flow demand and all other variales ome ak to the ut-in states within a speified time T. In 63

14 Variale Displaement Axial-Piston Pump - Design and Analysis defining the performane index to e minimized, four variales j with j=1 to 4 have een taken respetively as the swash angle, the spatial mean pressure P m within the ias ylinder, the spatial mean pressure and the spool displaement P m within the ontrol ylinder x sp. Using susripts s and n to eah j for the final steady state and the nondimensionalizing value respetively, the performane index to e minimized is defined as I I 4 0 j 1 I j, (4.15a) with and I I 5 0 [ sgn( min P Pv Prsv ) sgn( Pmax max )] 10 T j [ 0 j js jn / a, (4.15) {( )/ } dt] T for j=1 to 4, (4.15) where the sgn funtion returns 1 for positive argument inluding 0 and -1 for negative argument. Eah integral in equation (4.15) attains the minimum for the fastest approah to the steady state with minimum overshoots and osillations. However, neither over-pressurization nor avitation in the ias ylinder should our during the proess. This is ahieved through the non-zero large penalization [4] ontriuted y the sgn funtion in equation (4.15), if either of the two onditions is violated. The penalization fator involving the sgn funtion returns a zero, if the two onditions are simultaneously satisfied. One violation orresponds to the failure in keeping the maximum value of the ias ylinder pressure presried limit of pressure P m max elow a P max. The other violation ours, if the minimum ias ylinder P m min falls elow a reserve level P rsv that makes avitation a serious onern. Of ourse avitation ours, if the pressure falls elow oil vapor pressure at the prevailing oil temperature. It may e noted that the vapor pressure P v is usually speified in asolute terms in ontrast to speifying all other pressures with respet to the amient pressure P a. Of ourse, the aeptane of all the optimized sizes is sujet to the investigation of the oupled dynamis of the swiveling of the swash plate and pressures inside the ores of the arrel. 64

15 Variale Displaement Axial-Piston Pump - Design and Analysis The mathematial model desried through Equations (4.1) to (4.14) along with Equations (.1) to (.15) has een implemented in Matla/Simulink in a similar manner mentioned in Chapter. The solver and simulation time step have een kept idential with the simulation model for fixed displaement pump. 4.4 Results and Disussion Steady State Response of Pressure Compensator Figure 4.6: Response of pressure ompensator at steady state ondition The formulation desried aove an e utilized for arriving at the steady-state preditions using Equation (.4) for load flow and y dropping the time derivative terms in the Equations (4.) and (4.14) for spool displaement and swash plate angle respetively. Tale.3 in Chapter lists all the system speifiations other than that for the ompensator that have een used to otain the steady-state results for different fixed pairs of swash angle and needle valve opening. Figure 4.6 shows the predited variations of the displaement of the ompensator spool and the pump disharge against the delivery pressure along with the orresponding imposed variations of the swash angle s and the needle valve opening perentage given y pod (0 s )/19. (4.16) 65

16 Variale Displaement Axial-Piston Pump - Design and Analysis The aove equation has een otained from the oservation of the variation of the swash angle s with needle valve opening perentage pod shown in Fig A linear variation is apparent with 100% needle valve opening orresponds to 0 o swash angle and 4% needle valve opening orresponds to 1 o swash angle.` The speifiations of the ompensator for whih the alulations have een done are given in Tale 4.1. These have een ompiled from the earlier analyses due to Kavanagh [50] and Wu [51]. The swash angles of 0 o and 1 o respetively with full and 4% needle valve openings have een found to produe the ut-in and ut-off limits of delivery pressures P d1 and P d respetively equal to 7.MPa and 7.5MPa. It has already een mentioned in Setion 4.3 that equations (4.15a) to (4.15) have een used to evaluate the performane index in the proess of designing the ompensator. The swash angle of 1 o at the ut-off limit and the orresponding predition of 157 m for the ompensator spool have een taken respetively as and 4n in equation (4.15). By hoosing n and 3n in equation (4.15) as 0.1MPa and 0.7MPa, y and large similar order of value for I 1 to I 4 have een ensured. 1n Tale 4.1: Parameter values used in the simulation of the Compensator [50, 51] Symol Value Symol Value Symol Value d m 0 l m m sp kg d m 0 l m o d d sp m l d m u r 0 m 0 m I s kg-m l r m V m k N/m m kg V m k sp N/m m kg m l m m p kg sp m 66

17 Variale Displaement Axial-Piston Pump - Design and Analysis 4.4. Dynami Response of Pressure Compensator Figures 4.7 to 4.11 show a numer of dynami variations orresponding to a step hange in the load orifie opening from the ut-in limit to the ut-off limit for the values listed in Tale.3 in Chapter, Tale 4.1 and the physial dimensions of Sets 1 to 4 of Tale 4.. Besides a numer of dimensions, Tale 4. also onsists of the orresponding performane index and its omponents for eah set. While the integrals in equation (4.15) have een alulated from the predited variations over 0.15s from the imposition of the step hange, the figures depit the results for a span of 0.04s from the onset of the step hange of the opening. This has een done for the ease of distinguishing the urves in eah figure. A thorough study of Tale 4. would make it lear that among the physial dimensions onsidered, the size of the ak orifie diameter d ol in the ias ylinder, the spring-side orifie diameter d osl along with the radius r p of the metering orifie in the spool valve are ritial in arriving at an optimum design. Tale 4.: Optimization results Sl r r d ol d or l or d osl d osr r p I 0 I 1 I I 3 I 4 I No ( m) ( m) (mm) (mm) (mm) (mm) (mm) (mm) E E

18 Variale Displaement Axial-Piston Pump - Design and Analysis Figure 4.7: Swiveling dynamis of swash plate etween extreme limits. Figure 4.8: Dynamis of ias piston hamer pressure during swiveling. It is evident from Fig. 4.7 that the swiveling movement of the swash plate from initial to final steady angles is the slowest orresponding to Set 1 in Tale 4.. This is due to hoosing a low value of d ol leading to very steep rise of pressure in the ias ylinder with the retration of the piston into its ore. The rise is apparent from the orresponding urve in Fig. 4.8 and the predited high penalizing value of I 0 in Tale 4.. The penalizing value learly indiates oth over-pressurization and 68

19 Variale Displaement Axial-Piston Pump - Design and Analysis loss of avitation margin for this set. However for the sake of larity of presentation, the maximum pressure limit in Fig. 4.8 has een shown as 3MPa instead of the overpressurization limit P max set equal to 10MPa. In spite of the onsequent rise of pressure in the ontrol ylinder seen in Fig. 4.9, the driving torque for swiveling the swash plate remains low that makes the swiveling slow. Figure 4.9: Dynamis of Control piston hamer pressure during swiveling. Between Sets 1 and, the notale hange in Tale 4. is an inrease in from 0.5mm to 1mm. Fig. 4.7 learly depits a faster approah to the final steady swash angle for the latter set and Figs. 4.8 and 4.9 show muh lower rise in ylinder pressures. This is oserved in spite of a redution in the radial learane around the ias piston from 90 m in Set 1 to 10 m in Set. Of ourse, oth the akside orifie and the radial learane provide paths for the oil to leave the ias ylinder to make room for the retrating piston. As a result of the faster redution of swash angle in response to the sudden redution in the load area, the initial rise in the delivery port pressure shown in Fig is learly lower for Set. This explains the lower initial displaement oserved in Fig for the ompensator spool that is set in motion y the inreased delivery pressure eyond a limit. d ol 69

20 Variale Displaement Axial-Piston Pump - Design and Analysis Figure 4.10: Dynamis of delivery kidney port pressure during swiveling. Figure 4.11: Dynamis of spool displaement of ompensator during swiveling. The dynamis of spool motion exhiited in Fig for Sets 3 and 4 are quite different. This differene an e explained y the redution of d osl from Set 3 to Set 4. Indeed the redued d osl provides inreased damping to the spool motion. As a result, the oserved osillation in Set 4 is apparent only during the terminal phase of the spool motion. In ontrast, for Set 3 the osillation, though with reduing amplitudes, is evident throughout the time span shown in Fig In turn the 70

21 Variale Displaement Axial-Piston Pump - Design and Analysis omparatively lower osillation of spool in Set 4 is responsile for the lower osillations in the ontrol ylinder pressure and the swash angle, as oserved in Figs. 4.9 and 4.7 respetively. For Set 3, the ias ylinder pressure in Fig. 4.8 an e seen to fall elow the avitation mark that an e explained on the ground of spool osillation. It is quite lear from Fig.4.11 that during the osillating phase, temporary losure of the delivery-side metering orifie in the spool valve together with the opening of the drain-side metering orifie has een predited. Of ourse, it would ause retration of the piston in the ontrol ylinder. This in turn would make the piston inside the ias ylinder to extend out that ould indeed lead to avitation in the ias ylinder. Nearly similar pressure variations within the ias ylinder presented in Fig. 4.8 for Sets and 4 an e due to the ompeting effets in the onneted system. Between Sets 3 and 5 in Tale 4., the redution of only d osl from 1mm to 0.5mm is seen to redue the performane index drastially. Of ourse, this redution takes plae due to the inreased spool damping that eliminates the hane of avitation, as explained in the last paragraph. The results for Set 5 and 13 learly show that inrease in the radial learanes in oth the ontrol and ias ylinders y almost one-order auses not only a minor inrease in the overall performane index, ut also insignifiant hanges in eah omponent. An inrease only in the metering orifie radius r p of the ompensator spool valve from Set 5 and Set 4 show a redution in the performane index from to Of ourse, a larger rate of opening of the metering orifie assoiated with larger r p supplies flow to the ontrol ylinder at a higher rate, whih in turn makes the swash plate swiveling faster. This explains the lower value for Set 4 in omparison to Set 5 in Tale 4. for the integral I 1 assoiated with swash angle movement. The tale reveals the design values of Set 4 to the most optimum among the ases studied. 71

22 Variale Displaement Axial-Piston Pump - Design and Analysis 4.5 Summary Flow and motion dynamis of the different omponents of an axially atuating pressure-ompensated in-line axial-piston swash-plate pump have een modeled. Motions of the spool in the pressure-ompensating valve and an assemly of all the pistons together with the swash plate have een analyzed. Free-ody diagrams onsidering the pressure fores and frition on the arrel pistons, the ontrol piston and the ias piston have een used for desriing the motion dynamis. Effets of oil inertia and ompressiility along with various leakages have een onsidered in modelling the flow. The main flow from the reservoir to the reservoir has een modeled in terms separate ontrol volumes for the sution and delivery ports, ores in the rotating arrel and the silening grooves that periodially estalish the flow onnetivity etween a arrel ore and the respetive kidney port. A load orifie in the delivery line has een onsidered for setting a desired steady disharge from the pump. Implementing the omined model in Matla-Simulink platform, the dynamis have een solved orresponding to a step hange in the load orifie area to study the swash plate motion from the ut-in limit angle to the ut-off limit angle. Different sets of design parameters have een used to perform the dynami studies. It has een found that the orifie diameters in the spring-side and at the metering port of the spool valve and in the akside of the ias ylinder have ritial role in the design that should work within the onstraints of speified ut-in and ut-off limit pressures. Besides minimizing the transient osillations in the swash plate, the ompensator spool and the pressures within the ontrol and ias ylinders, avoidane of avitation and over-pressurization in the ias ylinder has een ensured in arriving at the optimum design. 7