The article presents some considerations on the effect of the assumed mathematical models of the pump efficiency and the hydraulic engine, exerted upon the static and dynamic properties of a hydrostatic transmission. For this purpose some simulation tests of the transmission described have been carried out with two models: one - simplified, containing efficiency constants, and the other - an extended one with various efficiency values

Jêdrzykiewicz  Zenon

Pluta Janusz

Stojek Jerzy

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 Acta  Montanistica  Slovaca     Ročník 2 (1997), 4, 373-380   Research on the properties of a hydrostatic transmission for different efficiency models of its elements    Zenon Jędrzykiewicz 1, Janusz Pluta 1, Jerzy Stojek 1   The article presents some considerations on the effect of the assumed mathematical models of the pump efficiency and the hydraulic engine, exerted upon the static and dynamic properties of a hydrostatic transmission. For this purpose some simulation tests of the transmission described have been carried out with two models: one - simplified, containing efficiency constants, and the other - an extended one with various efficiency values.  Key  words: hydrostatic transmission, efficiency, mathematical model, simulation tests.   Introduction    Modern design works entail still better and better knowledge of the phenomena that may occur in real systems. In the case of hydrostatic transmissions, an essential problem may be the va-riability of efficiency coefficients of both the pump and the hydraulic engine. Usually, while modelling  a transmission it is proposed that the indicated efficiency values are constant, assuming average levels, in most cases as high as in the catalogues of typical parts. It seems that such an assumption is not justified and the efficiency values should be treated as variable. Preliminary simulation tests (Jedrzykiewicz et al., 1995) indicate that such an approach is not without foundations. This article is aimed at showing some simulation tests of a transmission described with two models - one is simplified and with constant efficiency values; the other is extended and with variable efficiency values. In the simulation tests a modern MATLAB-SIMULINK packet (Matlab, 1992), (Simulink, 1992) has been applied. This packet is commonly used in research and technical calculations.  Assumptions    Mathematical modelling and simulation tests of real systems are usually carried out after having assumed a set of simplifying proposals. Such assumptions should arise  out of the construction of the systems and regard the degree of being familiarised with the phenomena under modelling.  It could seem that the more complex a mathematical model is, the better it is - as it describes   the object under investigation in a more exact and precise way; thus it is more reliable. On the other hand, the more complex a model is, the more difficult are its formulation, modification and analysis. All these factors lead to two diametrically opposed postulates: •  models should be little complicated, thus soluble in an easy and fast way, •  models should exactly description work of real systems. In the case of a hydrostatic transmission, usually the following set of assumptions is taken up: 1.  The transmission works in an open circuit of fluid. 2. The properties of the transmission do not depend upon the direction of revolutions, therefore a one-way operation of the pump and the hydraulic engine can be assumed. 3.  The rotational velocity of the engine driving the pump is constant. 4.  The transmission  is at the thermal equilibrium, therefore changes in the operational parameters do not appear as a temperature function. 5.  Fluid stream pulsation’s, arising out of the construction of the pump and the hydraulic engine, are negligibly small. 6. The transmission is compact, thus energy losses in the duct connecting the pump and the hydraulic engine are negligibly small.                                                        1 Department of Process Control, University of Mining and Metallurgy, Cracow, PL    (Received September 29, 1997)   373   Jedrzykiewicz et al.: Research on the properties of a hydrostatic transmission for different efficiency models of its elemets 7.  Natural volumetric and mechanical losses in both the pump and the hydraulic engine are described with  either constant or variable efficiency coefficients. 8. The high pressure fluid volume consists of the volume of the fluid in the pump, in the hydraulic engine and in the duct connecting the pump with the engine. 9.  Fluid bulk module is constant. 10.  The safety valve opening pressure exceeds  the operational pressures of the transmission.  A mathematical transmission model    Basing upon the said simplifying assumptions, the calculating diagram for the transmission, as presented in Fig. 1, has been accepted. QpQQ szQ hMoIh, BhVspsp = 0pb= 0 pshαp Mh, ωhp Fig. 1 The mathematical model of the transmission presented in Fig. 1 consists of three equations: flow-rates balance, equilibrium of loads and efficiency ones. Their general form is given below: z s h p Q Q Q Q + + =     (1)  o B I h M M M M + + =     (2)  mh vh mp vp o η η η η η =     (3) where: Qp is pump delivery, Qh is hydraulic engine absorptivity, Qs is flow rate resulting from fluid compressibility, Qz is flow rate through the safety valve, Mh is torque generated by the hydraulic engine, MI is torque resulting from  the inertia of moving parts, MB is torque resulting from viscous friction in moving parts, Mo is torque resulting from technological load, ηo  is general efficiency of the transmission, ηvp is pump's volumetric efficiency, ηmp is pump's mechanico-hydraulic efficiency, ηvh is engine's volumetric efficiency and ηmh is engine's mechanico-hydraulic efficiency. Respective terms of the equations (1) - (3), basing upon (Jedrzykiewicz, 1981), (Jedrzykiewicz, 1996), (Stryczek, 1995) are shown below: vp qp p p K Q η α =     (4) where: Qp - pump delivery [m3/s], αp - deflection angle of either a pump disk or casing [o], Kqp - pump's efficiency coefficient [m3/(o⋅s)], ηvp -  pump's volumetric efficiency coefficient [-].   374 Acta  Montanistica  Slovaca     Ročník 2 (1997), 4, 373-380 vhh qhhKQηω=     (5) where:  Qh - engine absorptivity [m3/s], Kqh - absorptivity coefficient [m3], ωh - angular velocity of the engine shaft [1/s], ηvh - volumetric efficiency coefficient [-]. DpEVQsss =     (6) where: Qs - flow rate related to compressibility [m3/s], Vs - fluid volume, subject to pressure effect [m3], Es - fluid bulk module [Pa], p - fluid pressure [Pa], D - differentiating operator.                                                   ) ( b zb z p p K Q − =        for    p > pb    (7)                                                                             for    p ≤ p 0 = z Q b    (8) where: Qz - flow rate through a valve [m3/s], Kzb - average slope of the valve static characteristics [m5/(N⋅s)], p - system operational pressure [Pa], pb - valve opening pressure - set while in operation [Pa]. mh h mh h p K M η ∆ =     (9) where: Mh -  rotational moment generated by the engine [N⋅m], Kmh - moment coefficient [m3], ∆ph pressure fall in the engine [Pa], the value being ∆ph = p - psh = p, ηmh  - coefficient of engine's mechanico-hydraulic efficiency [-]. MI D Ihh = ω   (10)  h h B B M ω =   (11) where: Ih - moment of inertia of the engine and machine parts, reduced upon the engine shaft [N⋅m⋅s 2], Bh - resistance coefficient of viscous friction in the engine and machine parts, reduced upon the engine shaft [N⋅m⋅s]. () () () () + +∆−+ +∆− =np p ep ppvpnp p ep ppvp v vp npAnpAε ε ε ε ε εε η ρ µ 1   (12) where: ∆pp - pressure fall in the pump [Pa], its value being ∆pp = p - psp = p, np - rotational speed of the pump driving shaft [rev/s], Avpµ - fluid  loss coefficient during laminar flow [m2⋅rev /(N⋅s)], determined from experimental data, Avpρ - fluid loss coefficient in turbulent flow [m⋅rev/(N0.5⋅s)], determined as well from experimental data, εp - pump delivery setting coefficient [-], (εp = 1 for a constant delivery  pump and 0 ≤ εp  ≤ 1 for a variable delivery pump), εep - a very small positive number e.g. 10e-10 [-] enabling calculations to be continued when εp = 0, εnp - a very small positive number, e.g. 10e-10 [rev/s] that enables to keep on calculating when np = 0, εv - corrective factor depending upon the expression in   375   Jedrzykiewicz et al.: Research on the properties of a hydrostatic transmission for different efficiency models of its elemets square brackets (ε = 1 where the expression is not  negative, εv = 0 where the expression is negative - it results from the fact that the efficiency cannot be negative while considering energy losses). pp pp pmppp ppmp mpfmppnApnA Aεεεηρ µ + ∆++ ∆+ +=3 2 211     (13)where Ampf - loss coefficient of mechanical friction [-] determined from experimental data, Ampµ - loss coefficient of viscous friction [N⋅s/(rev⋅m2)] determined from  experimental data, Ampρ - hydraulic loss coefficient in turbulent flow [N⋅s2/(rev2⋅m2)] determined also from experimental data, εpp a very small  positive number e.g. 10e-10 [Pa] enabling  the calculations to be carried on when ∆pp = 0. () () () () nh h eh hhvhnh h eh hhvhvhnpBnpBε ε ε ε ε εηρ µ + +∆++ +∆+=11     (14) where: nh - rotational speed  of the hydraulic engine shaft [rev/s], Bvhµ - fluid loss coefficient in laminar flow [m2⋅rev/(N⋅s)] determined experimentally, Bvhρ - fluid loss coefficient in turbulent flow [m⋅rev/(N0.5⋅s)], also determined experimentally, εh - engine absorptivity setting factor [-],  (εh = 1 for constant absorptivity engines, 0 ≤ εh ≤ 1 for variable absorptivity engines), εeh and εnh - very small positive numbers per analogiam  to the abovesaid. ηεεεεµρ mh m mhf mhhhp hmhhhhp hBBnpBnp=−−+−+122 3∆∆  (15) where: Bmhf - loss coefficient of mechanical friction [-] determined from experimental data, Bmhµ - loss coefficient of viscous friction [N⋅s/(m2⋅rev)] determined from  experimental data, Bmhρ - hydraulic loss coefficient in turbulent flow  [N⋅s2/(m2⋅rev2)] determined as well from experimental data, εph - a very little positive number analogous to the abovesaid, εm - corrective factor of the same meaning as for the pump.   Under the simulation research programme of the transmission, two versions of its mathematical model have been considered: Model 1. Efficiency values of the pump and hydraulic engine in formulae (3), (4), (5) and (9) assume     constant values, as in catalogue specifications. Model 2. Efficiency values of the pump and hydraulic engine in formulae (3), (4), (5) and  (9)  are va- riable, according to formulae (12) - (15).  Block diagrams of the transmission    By using the graphical editor of the MATLAB-SIMULINK packet, basing upon equations (1) -  (15 detailed block diagrams and a general), diagram have been made (shown in Figs 2, 3 and 4).  Simulation tests    Basing upon the catalogue and experimental data-sheets, the following coefficients of the transmission models have been found: Aα=16 [o], Am=250 [N⋅m], Avpµ=9.4860e-8 [m2⋅rev/(N⋅s)], Avpρ=-7.3539e-5 [m⋅rev/(N0.5⋅s)], Ampf=-1.4151e-3 [-], Ampµ=-9.1763e+6 [N⋅s/(m2⋅rev)], Ampρ=3.7036e+5 [N⋅s2/(m2⋅rev2)], Bh=12 [N⋅s⋅m], Bvhµ=4.8705e-9 [m2⋅rev/(N⋅s)], Bvhρ=1.4612e-5 [m⋅rev/(N0.5⋅s)],    376 Acta  Montanistica  Slovaca     Ročník 2 (1997), 4, 373-380 QhemhηmhFluidp-+-PumpQpevpηvp1αpHydraulicmotorHydraulicmotorMhLoad Mux 2ω h ω hevhηvh3Mo0.004Scale 3ValveQz2pb1e-6Scale 2Mux0.187Scale 11pModel 1 Fig. 2 1 αpPumpQpehεh ηvhQh0.125Scale 11e-6Scale 20.004Scale 3LoadωhFluidpHydr.motorHydr.motorHydr.motorHydr.motorMh3Moηmh-+-ehεhValveQz2pbPumpηvpepεpnpnhnpPumpηmp2 Mux1ωhpMux*0.1592epεpnpnp0.0125 ηoScale 5Mux 30.1Scale 4Hydr.motorModel 2Fig. 3    377   Jedrzykiewicz et al.: Research on the properties of a hydrostatic transmission for different efficiency models of its elemets HydraulicHydraulic Hydraulic Hydraulic HydraulictransmissionModel 1DataModel 2MuxMux30e+6pbαpMopωhηoHydraulictransmission Fig.  4   Bmhf=1.0817e-2 [-], Bmhµ=2.7200e+5 [N⋅s/(m2⋅rev)], Bmhρ=-1.1345e+4 [N⋅s2/(m2⋅rev2)], Es=1.65e+9 [Pa], Ih=0.04 [N⋅m⋅s2], Kqp=2.6882e-5 [m3/(o⋅s)], Kqh=3.979e-5 [m3], Kmh=3.979e-5 [m3], Kzb=0.20e-9 [m5/(N⋅s)], pb=30.0e+6 [Pa], tα=0.01 [s], tm=0.05 [s], Vs=1.4145e-4 [m3],  ηvp=0.97 [-], ηmp=0.80 [-], ηvh=0.93 [-], ηmh=0.95 [-]. The tests were performed  for a jump-like change in the pump  delivery αp=Aα⋅1(t-tα) and a jump-like  change in the motor load Mo=Am⋅1(t-tm). The tests results are shown in Figs 5, 6 and 7. Fig. 5 shows  changes in the angular velocity of a hydraulic engine, Fig. 6 - pressure variations in the press duct and Fig. 7 -  changes in the general efficiency of the transmission - model 2.  0 0.02 0.04 0.06 0.08 0.10246810121416Time [s]ωh [1/s]η=constη=var η=varη=constMo=250[Nm]αp=16 [o] Fig. 5    378 Acta  Montanistica  Slovaca     Ročník 2 (1997), 4, 373-380 0 0.02 0.04 0.06 0.08 0.102468101214Time [s]αp=16 [o]Mo=250 [Nm]p [MPa]η=var η=constη=var Fig. 6 0 0.02 0.04 0.06 0.08 0.100.10.20.30.40.50.60.70.8Time [s]ηo [-]αp=16 [o]Mo=250 [Nm]model 1model 2 Fig. 7  Comparison of the results    It can be deduced form the presented simulation tests that the application of model 2 instead of model 1 can lead to essential  quantitative differences in the rotational speed, pressure and general efficiency. Time courses of the first two speeds indicate lower over-controlling, smaller control times and higher stationary values. For a numerical presentation of the differences abovementioned, the following relative error has been defined: [%] 1002 mod1 mod 2 modelel elWW W −= δ   (16)   379   Jedrzykiewicz et al.: Research on the properties of a hydrostatic transmission for different efficiency models of its elemets   380where: Wmodel2, Wmodel1 - physical properties obtained, respectively, from model 2 and 1. Table 1 shows relative errors for the stationary operational states of the transmission, whereas Table 2 presents the maximum values of relative errors for non-stationary states.   Table 1. Stationary states Physical quantity Operation of unloaded transmission  δ [%] Operation of loaded transmission δ [%] ωh  6.8 1.9 p  13.6 1.1 ηo  -28.9 4.5    Table 2. Non-stationary states Physical quantity Operation of unloaded transmission δmax [%] Operation of loaded transmission δmax [%] ωh  16.2 9.4 p  61.8 8.3 ηo  -93.0 6.8  Summary    Simulation tests, made on hydrostatic transmissions, indicate the necessity of having a mathematical model adapted to  concrete design circumstances: •  Model 1 can be used exclusively in the situations during an operation under a load, close to the nominal one, only stationary states are essential; i.e., when the device operates for long time intervals under a constant load and unchanged control, •  Model 2 - can supply significant design tips when: - during a larger part of an operational cycle the operation of a transmission occurs under a slight load against the nominal one, - while operating under a load close to the nominal one, non-stationary states are significant; they occur  relatively often under variable loads and controlling courses. As it can be seen, model 2 is exceptionally useful in case of analysing the operation of transmissions designed for  automated devices.    The presented research was carried out within the framework of the Department of Process Control of, Contract no 11.130.174.    References  Jędrzykiewicz Z.: Elementy i układy hydrauliczne. Skrypt Uczelniany AGH, nr 849. Kraków, 1981. Jędrzykiewicz Z.: Modele matematyczne sprawności pomp i silników hydraulicznych. Hydraulika i Pneumatyka, nr 3/1996. Jędrzykiewicz Z., Pluta J.: Symulacja pracy przekładni hydrostatycznej w środowisku Matlaba&Simulinka. Materiały I Krajowej Konferencji Użytkowników MATLAB-a. Kraków, 1995. Matlab. Reference Guide. The MathWorks, Inc., 1992. Simulink. User’s Guide. The MathWorks, Inc., 1992. Stryczek S.: Napęd hydrostatyczny. Tom I. Elementy. WNT Warszawa, 1995. 

Research on the properties of a hydrostatic transmission for different efficiency models of its elements

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