- Email: [email protected]
Dynamic analysis of a low speed high torque hydrostatic drive using steady-state characteristics
Mechanism and Machine Theory 52 (2012) 1–17
Contents lists available at SciVerse ScienceDirect
Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
Dynamic analysis of a low speed high torque hydrostatic drive using steady-state characteristics K. Dasgupta a,⁎, S.K. Mandal b, S. Pan c a b c
Department of Mechanical Engineering and Mining Machinery Engineering, Indian School of Mines, Dhanabad 826004, India Benaras Hindu University, India Department of Electrical Engineering, Indian School of Mines, Dhanabad 826004, India
a r t i c l e
i n f o
Article history: Received 2 July 2010 received in revised form 22 August 2011 accepted 7 December 2011 Available online 31 January 2012 Keywords: Hydrostatic (HST) drive Low-speed-high-torque (LSHT) hydraulic motor Direction control (d.c.) valve Bondgraph modelling Steady state Resistive element Proportional pressure relief valve (PPRV)
a b s t r a c t In this article dynamic analysis of an open-circuit hydrostatic (HST) drive has been carried out to study its performance. A Low Speed High Torque (LSHT) radial piston motor has been considered for the drive. Bondgraph simulation technique is used to model the hydrostatic drive. The relationships of the loss coefficients of the drive with the other system variables, obtained from the steady-state model, are identified through experimental investigation. Using the parametric values, the overall dynamic model of the hydrostatic drive has been validated experimentally. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction An HST drive converts mechanical power of the input drive shaft of a pump to hydrostatic power in a nearly incompressible working fluid and then reconverts it into mechanical power at the output drive shaft of the motor. It is used to convert the rotating mechanical power from one source to another without the use of gears. An LSHT hydrostatic drive is widely used in Heavy Earth Moving Machinery (HEMM) because of its compact design and better low speed characteristics compared to high speed hydrostatic drive with integrated gear unit. It mainly consists of a fixed or variable displacement pump and an LSHT hydraulic motor, either in open loop or in closed loop configuration. Watton [1] has applied the method of characteristics in the time domain dynamic analysis of the hydro-motor drive system when a quasi-analytic method was used in evaluating motor boundary conditions. Chappell [2] has developed a model of radial piston hydraulic motor for evaluation of the factors affecting the motor performance and to study the influence of various internal factors on the performance of the motor. Manring and Luecke [3] had analysed an HST drive that consists of a variable displacement swash-plate controlled pump and a fixed displacement motor. In their study, the system has been linearised and the stability range of the system is presented. However, the torque on the motor shaft is not completely defined. Dasgupta [4] has analysed the quasi-static performance of an open-loop hydrostatic drive using orbit motor, where the torque-speed performance of the motor is investigated using a displacement controlled pump. The open loop dynamic performance of a servo valve controlled axial piston motor transmission system has been studied by Dasgupta et al. [5], where torque loss of the motor has been determined from its steady-state performance. ⁎ Corresponding author. Tel.: + 91 3262296551; fax: + 91 3262296563. E-mail address: [email protected] (K. Dasgupta). 0094-114X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2011.12.004
2
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
The art of model making in system dynamics comprises two stages. The first stage is involved in splitting the system into its very basic components, namely, compliances, inertias, dissipaters and sources. The model may be further refined by identifying more and more elements of the components in the systems. However, one has to stop at one stage either satisfied by the grasp of details for the range of operation or restricted by the limitation of the handling capacity. Putting the pieces back together is the next stage. The work presented in this paper deals with the detail modeling of an open circuit HST drive to study its dynamic performance. Mandal et al. [6,7] have already proposed a reduced model to study the steady-state performance of an open circuit and a closed circuit HST drives with LSHT motor. The said study investigates the effects of the critical parameters of the pumps and the motor on the steady-state performance of the HST drives. In this article a dynamic model of an open-circuit LSHT hydrostatic drive system with pump loading is presented. The determination of the loss coefficients of the system is made from the steady-state analysis of the system [6,7]. With respect to the variation of the inertia and the resistive loads, the model is validated through experiment. The model presented in this article itself guides the design of the experiment and mode of handling of the observations in such a way that values of dependence of the parameters on the operating conditions of the drive are obtained. 2. The physical system Fig. 1 shows the open-circuit hydrostatic drive system considered for the analysis where the speed of the electric motor is controlled by a variable frequency control drive. The pump flow is supplied to the LSHT hydraulic motor through solenoid operated direction control (d.c.) valve. The motor drives loading pump through a step-up gear unit. By pump loading through the proportional pressure relief valve (PPRV) of the loading circuit, the viscous load on the motor shaft is controlled. Fig. 2 shows the multi-piston LSHT radial piston hydraulic motor. The valve integrated with the motor supplies pressurized fluid to the chambers in the expansion mode and also provides a flow path for the fluid being returned from the chambers in the compression mode. There are leakages from the clearances between the pistons and the cylinders. The case drain flow is considered as external leakage resistance (Rem) of the motor. The leakage also occurs from the high-pressure chamber to the lowpressure chamber of the motor; it is termed as inter chamber leakage resistance (Ril) of the motor. These leakage losses are mainly because of the pressure gradient. Due to the support of the pressurized pistons, a tangential force component of the piston pressure force generates rotational motion of the cylinder block connected with the spline shaft. 3. Modelling of the system Dynamics of the open circuit HST drive system and its components are modelled using Bondgraph technique [8]. Such technique is based on the power flow and it is a very useful tool for formulating the state equations of a system and its components. In the development of the dynamic model of the hydrostatic drive system with the loading unit • • • • • •
The fluid inertia is neglected; The mineral based hydraulic oil used is considered to be Newtonian fluid; The resistive and the capacitive effects are lumped wherever appropriate; The effects of pressure on the properties of the fluid are ignored; The dynamics of the gear unit, PPRV and the d.c. valve are not taken into consideration; In the event of the opening the valve ports, flow forces acting on the valve spool are not considered; similar considerations are made for inlet and outlet valve ports of the motor; • The dynamics of the control valves and the loading pump are not considered;
Fig. 1. Open-circuit hydrostatic drive system.
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
3
Fig. 2. Multi-piston LSHT hydraulic motor.
• As the loading pump is placed closed to the PPRV, the effect of the line resistance connecting the pump and the valve is ignored. Similar consideration is made for the line connecting the source pump and the d.c. valve; • The set pressure of the unloading pressure relief valve is considered to be higher than the maximum system pressure, therefore, the flow through the relief valve is not considered; • The leakages other than those from the pumps and the motor are ignored; • Since the temperature is kept almost constant (50 ± 2 °C) during experiment, the effect of temperature change on the properties of the fluid is neglected. Fig. 3 shows the bondgraph model of the open circuit HST drive system. It is described in the following sub-sections. 3.1. The hydrostatic drive unit In Fig. 3, the SF element on 1 junction represents the rotational frequency (ωp) of the pressure compensated pump. The resistance Rls on the same junction represents the drag loss of the pump. Essentially, a hydrostatic unit, pump or motor, transforms hydraulic power to mechanical power or vice-versa. This basic fact is represented by a transformer (TF junction). The TF modulus Dp is the volume displacement rate of the pump with respect to the shaft rotation (i.e., the theoretical pump flow V_ s ¼ ωp Dp Þ. Similarly, the TF element connecting 1 and 0 junctions represents the volume displacement rate of the motor (Dm). A part of the ideal flow of the pump escapes through its clearance as leakage (V_ plkg ). The R element on the 0Pp junction represents the leakage resistance. In the above model, the C elements on the 0 junctions (represent the respective plenum) take into account the flow losses because of the compressibility of the fluid. Similarly, the pressure drop across the valve ports and the hydraulic line are taken into account by the R elements connected with the 1 junctions. The resistances Rvi and Rvo represent the inlet and the outlet ports of the d.c. valve, respectively. The resistance Rln indicates the resistance of the line connecting the d.c. valve and the LSHT motor. Referring to Fig. 2, the valve is an integral part of the motor shaft, through which supply flow passes to the displacement chambers of the LSHT motor. In the model, the resistances Rmv appear on 1 junction indicate the inlet and the outlet ports of the LSHT motor. The internal leakage loss of the motor occurs due to the pressure difference (Pmph – Pmpl) across
Fig. 3. Bondgraph model of the system.
4
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
its high pressure and the low-pressure chambers. It is taken into account by the resistance Ril at the 1 junction. The resistance Rem represents the external leakage flow of the motor. The outlet flow of the motor returns to the sump (Psmp) through the d.c. valve. The 1 junction representing the mechanical part of the load corresponds to the speed of the motor output shaft (ωm). The load inertia (Jl), speed dependent friction load (Rl) and the constant torque loss (Tloss) are represented by the I, R and SE elements, respectively. They are connected with the 1 junction that constitutes the load dynamics. The activated C element on the same junction records the speed of the motor. 3.2. The loading unit The flow supplied by the motor (ωm). The external leak loading pump (0Ppljunction) is proportional to the speed ofthe LSHT age flow of the pump V_ llkg and the flow loss because of the compressibility of the fluid V_ plc at the pump plenum are taken into account by the R and the C elements, respectively, connected with 0 junction. The TF element connecting the 1 and the 0 junctions represents the volume displacement rate of the pump (Dp) multiplied by the step up gear ratio (ng). The pump loading is provided by the proportional pressure relief valve (PPRV). The flow through the valve port V_ rlv depends on the port opening area and the pressure difference (Ppl − Psmp) across it. 4. Describing equations of the system The system equations derived from the bondgraph model are described as follows: 4.1. The hydrostatic drive unit Considering the flow continuity at the pump plenum, the flow supplied to the inlet port of the d.c. valve V_ ¼ V_ −V_ −V_ vi
s
plkg
pp
ð1Þ
where, V_ plkg ¼ P pp =Rip and V_ s ¼ ωp Dp . In the above equation, V_ plkg ; V_ pp , V_ s are the leakage flow of the pump, the compressibility flow loss of the fluid at the pump plenum and the theoretical flow supplied by the pump, respectively. The fluid pressure at the pump plenum is given by P pp ¼ K p V pp :
ð2Þ
The pump torque is expressed as T p ¼ P pp Dp þ wp Rrls :
ð3Þ
In Eq. (3), Rls represents the frictional resistance of the pump shaft. With the compressibility of the fluid, the flow balance at the inlet and the outlet ports of the d.c. valve is given by V_ dci ¼ V_ vi −V_ lni ;
ð4Þ
V_ dco ¼ V_ lno −V_ vo ;
ð5Þ where, the flow through the valve ports are expressed as V_ vi ¼ P pp −P vi =Rvi and V_ vo ¼ P vo −P smp =Rvo . In Eqs. (4) and (5) the inlet and the outlet flows of the hydro-motor are given by V_ lni ¼ ðP vi −P mvi Þ=R ln and V_ lno ¼ ðP mvo −P vo Þ=R ln , respectively. The fluid pressures at the inlet and the outlet ports of the d.c. valve are given by P vi ¼ K v V dci andP vo ¼ K v V dco ; respectively: Considering the flow continuity at the inlet and the outlet ports of the LSHT motor, the flow balance at each port of the motor is given by V_ mi ¼ V_ lni −V_ mvi ;
ð6Þ
V_ mo ¼ V_ lno þ V_ mvo ;
ð7Þ
where, V_ mvi and V_ mvo are the compressibility flow loss of the fluid at the respective plenum of the valve ports. The pressure at the inlet and the outlet ports of the motor are given by P mvi ¼ K mv V mvi andP mvo ¼ K mv V mvo ; respectively:
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
5
The inlet and the outlet flows of the motor depend on the valve port resistances (Rmv) and the pressure difference across it. They are expressed as V_ mi ¼ P mvi −P mph =Rmv and V_ mo ¼ P mpl −P mvo =Rmv ; respectively: With the compressibility of the fluid; the flow balance at the motor chambers is given by: P mph P mph −P mpl V_ mph ¼ −ωm Dm − − þ V_ mi ; Rem Ril
ð8Þ
P mpl P mph −P mpl _ V_ mpl ¼ ωm Dm − þ −V mo ⋅ Rem Ril
ð9Þ
In Eqs. (8) and (9) the high and the low chamber pressures of the motor are expressed as P mph ¼ K mp V mph and P mpl ¼ K mp V mpl ; respectively: The 1 junction in the mechanical load part of the motor comes from the torque balance, which is expressed by the following equation, resulting from the constitutive equation of each element: p_ l ¼ P mph −P mpl Dm −T loss −ωm Rl −P pl ng Dpl ð10Þ where, the first term of the above equation is the torque due to the inertia load Jl, the second term indicates the torque equivalent to the differential pressure across the motor, the third term is the constant torque loss, the fourth term takes into account the torque loss due to viscous friction coefficient Rl and the fifth term represents the load applied on the motor shaft due to the pressure (Ppl) of the loading pump. The motor speed is given by wm ¼ p1 =J 1 :
ð11Þ
4.2. The loading unit Considering the flow continuity at the loading pump plenum, the flow supplied to the proportional pressure relief valve V_ ¼ V_ −V_ −V_ ; rlv
pl
llkg
plc
ð12Þ
where, V_ pl ; V_ llkg and V_ plc are the theoretical flow supplied by the loading pump, the leakage flow of the loading pump, and the compressibility flow loss at the pump plenum, respectively. In Eq. (12) V_ pl ¼ ωm ng Dpl , V_ llkg ¼ P pl =Rpl , V_ rlv ¼ P pl −P smp =Rrlv and the load pressure is given by Ppl = Kpl Vplc. 5. Estimation of the system parameters A practical system discussed in Appendix A was used to validate the dynamic model of the system. Commercially available valves, pressure compensated axial piston pump and the LSHT hydraulic motor were used in the experiment. The values of their resistances were determined from the experimental steady-state performance of the system; the details of which are discussed in Section 6. The other parameters of system components are estimated as follows: • From the geometrical parameters, the total equivalent inertia of the rotating parts connecting the motor and the loading pump is calculated.
Fig. 4. Pressure-flow characteristics of proportional relief valve.
6
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
Fig. 5. Supply flow rate as a function of pressure drop across valve port.
• The values of the bulk stiffness of the fluid at the plenum junctions are obtained based on the volume in the respective cases and the bulk modulus of the fluid used in the experiment (Ka = β / Va). • The volume displacement rate of the pump and the motor are obtained from the manufacturer's product literature [9–11]. • The line resistance (Rln) connecting the pump and the motor is suitably assumed. • The leakage resistance (Rpl) of the loading pump is estimated from its volumetric efficiency [11]. 6. Experimentation, estimation of the resistances of the valves, pump and the motor, verification of the overall dynamic model of the system The experiments were conducted to estimate the resistances of the valves, pump and the motor. From the steady-state pressure-flow characteristics, the load resistance of the PPRV and the port resistances of the d.c. valve are obtained. The resistances of the pump and the motor are determined from the steady-state performance of the HST drive. A brief description of the test set-up along with the experimentation is given in Appendix A. 6.1. Estimation of the load resistance (Rrlv) of the Proportional Pressure Relief Valve (PPRV) Referring to Fig. 1, the resistance of the PPRV of the loading unit provides viscous load on the motor shaft. Varying the area of the orifice of the valve by the command signal (Vcom), the load resistance (Rrlv) is varied. Fig. 4 shows the pressure-flow characteristics of the valve plotted at different Vcom level. From Eq. (B.1) that expresses the variation of the flow through the valve V_ rlv with the pressure drop ( ΔP = Ppl − Psmp) across it, the valve resistance Rrlv is given by Rrlv ¼ ΔP=V_ rlv
ð13Þ
6.2. Estimation of the port resistances of the direction control (d.c.) valve Fig. 5 shows the pressure-flow characteristics of the d.c. valve. From Eqs. (B.2) and (B.3) that express the nature of the above characteristics, the values of the valve port resistances are obtained using the following relations: Rvi ¼ ΔP i =V_ vi
ð14Þ
Fig. 6. Reduced model of hydrostatic unit.
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
7
Fig. 7. Variation of load torque with pressure differential across motor.
Rvo ¼ ΔP o =V_ vo
ð15Þ
In Eqs. (14) and (15), V_ vi and V_ vo are flow through the inlet and outlet valve ports and the pressure drop across the ports are ΔPi = (Ppp − Pvi) and ΔPo = (Pvo − Psmp), respectively. 6.3. Estimation of the resistances of the pump and the motor A reduced model is developed to determine the resistances of the axial piston pump and the LSHT motor from the steady-state performance of the HST drive. Mandal et al. [6] have explained the detail model reduction process in their studies. In the model [6] various losses are lumped into suitable resistive elements. It has fewer adjustable parameters requiring a comparatively few test runs to estimate them. The relationships of the resistances with the operating parameters of the HST drive, obtained through reduced model, are identified through experimental investigation. 6.3.1. Reduced model of the system Fig. 6 shows the reduced model of the open circuit hydrostatic drive. The steady-state performance of the system was tested at various input speed (ωp) of the pump. The arrow shown over SF element represents the modulated rotational frequency (ωp) of the pump. Rmv represents resistances of the valve port of the LSHT motor. The resistance Ril takes into account the inter-chamber leakages that occur because of the differential pressure (Pl) across the motor. Rem on the 0 junction represents the external leakage resistance of the motor. The other parameters of the model are explained in Section 3. From the model shown in Fig. 6, the steady-state load torque provided by the loading circuit discussed in Section 2, is given by T pl ¼ Dm P l –Rl ωm −T loss :
ð16Þ
where, the second term indicates the total torque load on the hydro-motor proportional to the pressure differential across it, the third term represents the torque loss due to fluid viscosity effect assumed to be linearly dependent upon speed and the fourth term is the additional friction losses due to coulomb friction and stiction of the hydro-motor. The following relationships of the resistances with the operating parameters of the HST drive are obtained from the reduced model: Rls ¼
T p −P pp Dp ωp
ð17Þ
Rip ¼ P pp =V_ plkg
ð18Þ
Tl V_ mlkg Dm
ð19Þ
Rem ¼
Table 1 List of the parametric values. Parameter Jl Dpl Kmp Rpl
Value
Parameter 2
6.18 kg m 16 cm3/rev 1012 N/m5 1012 N s/m5
Dm Kpl Kv Rln
Value 3
280 cm /rev 1012 N/m5 1013 N/m5 107 N s/m5
Parameter
Value
Dp Kp Kmv β
28 cm3/rev 1012 N/m5 1013 N/m5 1.5 × 109 N/m2
8
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
Fig. 8. Variation of viscous friction resistance (Rls) of the pump with pump speed ratio (αps).
Rmv ¼
Ril ¼
P pp −ðT l =Dm Þ
α ps ωp max Dp −V_ plkg
Tl _ Dm α ps ωp max Dp −V plkg −V_ mlkg −ωm Dm
ð20Þ
ð21Þ
From the test data of the pump pressure (Ppp), load pressure (Pl), load torque (Tl = Pl Dm), pump speed (ωp), motor speed (ωm) and the leakage flow of the pump and the motor V_ plkg and V_ mlkg Þ, the values of the resistances (Ril, Rmv,, Rls, Rem) are obtained. The empirical relationships that express the variation of the resistances with the pump speed are given in Appendix B. 6.3.2. Estimation of the viscous friction coefficient (Rl) and the constant torque loss (Tloss) of the motor The viscous friction coefficient (Rl) and the constant torque loss (Tloss) of the LSHT motor are estimated experimentally from the steady-state torque characteristics of the motor. Fig. 7 shows the torque (Tl) versus load pressure (Pl) plotted at various speed level. Comparing Eq. (16), the nature of the characteristics is given by Tpl ≈Dm Pl −11:27wm −47:40
ð22Þ
Comparing Eqs. (16) and (22) the values of Rl and Tloss are obtained and they are given by Rl ¼ 11:27 Nm s=rad; T loss ¼ 47:40 Nm: Since the torque characteristics of the hydro-motor shown in Fig. 7 have been determined for speeds greater than 40 rpm, the non-linear stiction term is not considered in the Tloss term that would have been expected at slower speeds. 6.3.3. Estimation of the resistances of the pump and the motor In estimating the resistances of the axial piston pump and the LSHT motor of the hydrostatic drive unit, the test speed of the motor was limited to 30 to 140 rpm to cover its maximum efficiency zone. Using the parametric values given in Table 1 and the test data in Eqs. (17)–(21), the values of the resistances are calculated. Figs. 8–12 show the variations of the resistances with the pump speed ratio (αps = ωp / ωpmax) at various torque (Tl) levels. Eqs. (B.4)–(B.8) express the empirical relationships of the variation of the resistances with αps. Using them, the predicted characteristics of the resistances are obtained for other torque levels. These are compared with the experimental results. One representation is shown in Fig. 13 that compares the predicted and the experimental data of the valve port resistance (Rmv). It is found that there is a close agreement (deviation ± 3%) between the predicted and the experimental data. Therefore, the estimation of resistances given by Eqs. (26)–(30) seems to be acceptable.
Fig. 9. Variation of leakage resistance (Rip) of the pump with the pump speed ratio (αps).
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
9
Fig. 10. Variation of external leakage resistance (Rem) of pump with pump speed ratio (αps).
Fig. 11. Variation of internal leakage resistance (Ril) of motor with pump speed ratio (αps).
The reasons for these deviations, which are not considered in the present analysis, may be due to minor losses such as variation of the resistances due to the thermal effect, effect of load pressure and speed on the pump displacement rate (Dp), hydrodynamic losses, and compressibility losses of the fluid and so on. Besides these, there may be minor instrumental error. McCandlish and Dorey [13] and Dasgupta et al. [14] have also recognized such variations in their studies on piston type hydrostatic machines.
6.4. Verification of the dynamic model of the open circuit hydrostatic drive The dynamic performance of the open circuit hydrostatic drive system was investigated at constant rotational frequency (ωp) of the pump. The simulation results obtained by solving the system equations given in Section 4 are compared with the experimental data. The derivation of the system equations from the model and their numerical solutions are carried out using the software SYMBOLS 2000 [15]. In simulation studies the parametric values given in Table 1, values of the resistances of the pumps and the motor determined from Eqs. (B.1)–(B.8), the values of Tloss and Rl obtained from Eq. (22) are used. The tests on the dynamic behaviour of the HST drive were conducted at different values of load inertia (Jl) and load resistance (Rrlv). The experiment was repeated several times to examine the repeatability before recording the data in PC through data acquisition system. Figs. 14 and 15 compare the predicted and the experimental transient responses of the pump pressure (Ppp) and the motor speed (ωm) at different values of Jl. As the inertia increases, the speed and the pressure responses become more oscillatory. The maximum overshoot of the speed at the higher inertia (Jl = 0.41 kg m 2) is about 50% over the steady-state motor speed of 100 rpm. It is also clear that with increase in the load inertia, the settling time of the system response increases.
Fig. 12. Variation of valve port resistance (Rvm) of the motor with the pump speed ratio (αps).
10
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
Fig. 13. Comparison of predicted and experimental characteristics of valve resistance Rvm of the motor.
Referring to the experimental test set-up discussed in Appendix A, by changing the command signal (item no. 37) Vcpm, the load resistance Rrlv is controlled. Figs. 16 and 17 compare the transient responses of the pump pressure (Ppp) and the motor speed (ωm) with that of its experimental values at different Vcom values. With the increase in Vcom (that increases Rrlv), the rise time as well as the steady state values of Ppp increases, whereas ωm decreases. The decrease in ωm is due to increase in leakages at higher pressure. The increase in the value of Rrlv that increases the resistive load provides more damping effect and, therefore, the overshoot of ωm and Ppp decrease. In general, various plots shown in Figs. 14–17 indicate that the predicted and the experimental responses are in good agreement. The differences between the simulation and experimental results may be due to the following reasons: • The thermal and hydrodynamic losses of the fluid, local deformation of the internal components of the pump and the motor and the minor leakage loss as well, not considered in the model. • The estimated values of the resistances of the pump and the motor discussed in Section 5 are considered in simulation studies. • Limitations of the experimental test rig and its instruments, lower sampling rate of the data accusation system (DAS). • The fluctuations of the experimental responses at steady state are due to the effects of the flexible hoses, small instrumental errors, as well as minor mechanical misalignment between the loading pump and the hydraulic motor. Further, the fluctuation
Fig.14. Comparison of the predicted and experimental pump pressure at different inertia load.
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
Fig. 15. Comparison of the predicted and experimental motor speed at different inertia load.
Fig. 16. Comparison of the predicted and experimental pump pressure at different load resistance.
11
12
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
Fig. 17. Comparison of the predicted and the experimental motor speed at different viscous load.
of the system's response at low speed range may be attributed to the Coulomb friction / Stiction and unsteady beheviour of the loading pump flow rate as well. Such details are not considered while modeling the hydro-motor. However, the difference between the mean values of the steady state part of the experimental and the predicted results is small. The good correlation between the theoretical and the experimental responses justifies the assumptions made to develop the model. The simulation results would have been more accurate, if the detail dynamics of the pumps, the motor, gear unit and the valves of the overall system are considered. The results of the experimental analysis also indicate that there is a slight time lag in the increase in the pressure and the speed of the motor during the initial rise as compared to the simulation results. One such time lag (tl = 0. 05 s) is shown in Fig. 17. It may be attributed to the dynamics of the direction control valve used in the hydrostatic drive unit, minimum speed required by the loading pump to produce flow and the characteristics of the PPRV at low flow rate. These details of the pump and the valves are not considered in the model. It is seen that with the increase in the load resistance, the rise time of the states of the system increases. The experiments could not be conducted at higher motor speed and load pressure levels due to the limitations of the test unit.
7. Conclusion This paper presents a significant aspect of an integrated study of modeling, analysis, simulation and experimentation of the dynamics of an LSHT open circuit hydrostatic drive. The estimation of the loss coefficients of the pumps and the motor is made through steady-state analysis. The close agreement between the experimental and the simulation results of the dynamic responses of the system validates the proposed model. It shows that the simulated response from a reasonable model when combined with experimental observations renders effective mode of data handling for meaningful estimation of loss coefficients of the systems in terms of dependencies on operating conditions. It is also shown here that the loss coefficients of the pump and the motor that are difficult to estimate in running conditions, are best deduced by considering experimental data with the simulated results and for which the components need not be disassembled and special measurement techniques are not required.
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
13
Justifiably, many plausible parameters like inertia of the working fluid in the conduit, elasticity of the conduit etc. are not incorporated in the detailed model. Still, the model contains several critical parameters that substantially influence the performance of the system with the range of operation considered. The discrepancies found while comparing the theoretical and the test results are due to the dynamics of the gear unit, thermal and hydrodynamic losses of the fluid, and the minor leakage loss as well. Further, it may be noted that the estimation of the loss coefficients is based on the semi-empirical estimation. Consideration of the effects, ignored in the present study, requires further refinement of the model and more rigorous experimentation with better data acquisition system. Due to the limitation of the test rig, the experimental results are obtained within the limited range. However, it may be envisaged that the model developed may be useful for wider range of parameter variation. To ascertain this, further experiments may be undertaken. The dynamic response of the system has been validated with respect to the variation of the inertia and the viscous load. The following conclusions may be derived from the validated model: • With the increase in the inertia load, the speed and the pressure responses become oscillatory with the increase in the maximum overshoot. • With the increase in the load resistances, the steady-state speed decreases and the pressure increases. However, it provides more damping effect to the transient response of the system, therefore settling time of the system's response decreases. Comparing the theoretical and the experimental responses of the system, it may be concluded that the model fairly represents the behaviour of the system and, therefore, it may be useful to the practicing engineers for initial design of the machine, where such a hydrostatic drive is used as a transmission system. Nomenclature C Single port energy storage capacitor element in bondgraph model Dm, Dpl Displacement of the loading pump and the LSHT motor, respectively Dp Displacement of the pressure compensated axial piston pump I Single port energy storage inertial element in bondgraph model SF Single port source of flow element in bondgraph model SE Single port source of effort element in bondgraph model Jl Generalized inertia of the driving shaft with the connected load Ka Generalised bulk stiffness of the fluid Kpp Bulk stiffness of the fluid in the pump column Kmv Bulk stiffness of the fluid in the motor valve port Kmv Bulk stiffness of the fluid in the motor chamber port Kplc Bulk stiffness of the fluid in the loading pump plenum Kv Bulk stiffness of the fluid in the direction control valve port p_ l Torque due to load inertia pl Generalized angular momentum due to load inertia Pmph, Pmpl High and low chamber pressures of the LSHT motor, respectively Pvi, Pvo Pressure at the inlet and the outlet ports of the d.c. valve, respectively Pmvi, Pmvo Pressure at the inlet and the outlet ports of the LSHT motor, respectively Ppl, Psmp Loading pump plenum pressure and the sump pressure, respectively Ppp Plenum pressure of the axial piston pump Rl Resistive load connected with the motor end Rem, Ril External and internal-chamber leakage resistances of the motor, respectively Rrlv Resistance of the proportional pressure relief valve Rv Equivalent port resistance of the d.c. valve in the reduced model Rip, Rpl Leakage resistances of axial piston pump and loading pump, respectively Rln Resistance of the line connecting the valve and the motor ports Rls Viscous resistance of the pump drive shaft Rmvo, Rmvi Resistance of the outlet and the inlet ports of the LSHT motor, respectively Rmv Equivalent resistance of the motor ports in reduced model Tloss Constant torque loss of the motor Tl Steady-state Load torque on the LSHT motor Tp Steady-state Load torque on pump shaft Va Generalised fluid volume at the respective side of the system Flow rate of the axial piston pump and the loading pump, respectively V_ s ; V_ pl V_ plc Volume change rate of fluid at the loading pump plenum Volume change rate of fluid at the high pressure chamber of the motor V_ mph V_ mpl Volume change rate of fluid at the low pressure chamber of the motor V_ pp Volume change rate of fluid at the pressure compensated pump plenum V_ rlv Flow rate that passes through proportional relief valve V_ dci Volume change rate of fluid at the inlet port of the d.c. valve
14
V_ dco V_ plkg V_ mvi V_ mvo V_ mi ; V_ mo V_ lni ; V_ lno ng ωp, ΔP Vcom αps β
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
Volume change rate of fluid at the outlet port of the d.c. valve Leakage flow of the pump Volume change rate of fluid at the inlet port of the motor Volume change rate of fluid at the outlet port of the motor Inlet and outlet flow of the LSHT motor chambers, respectively Inlet and outlet flow from the LSHT motor, respectively Step-up gear ratio ωm Pump and LSHT motor speed, respectively Generalized differential pressure across the valve port Command voltage Pump speed ratio (=ωp / ωpmax) Generalized bulk modulus of the fluid
Acknowledgement The research and development project grant for 2009–2012 from University Grants Commission, Government of India, for carrying out research work on this topic is acknowledged. Appendix A. Experimental test set-up Before validating the dynamic model of the hydrostatic drive system, the resistances of the valves, pump and the motor discussed in Section 5 were determined experimentally. Using them the proposed model was validated. A simplified representation of the experimental test set-up is shown in Fig. 18 for ready reference. The hydro-motor was connected with the direction control valve by flexible hoses; whereas other hydraulic lines are of rigid steel pipes.
Fig. 18. Experimental test set-up.
Table 2 List of components used in the test set-up. Item no.
Item description
Item no.
Item description
1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21.
Power supply with variable speed drive Flow sensor (main pump leakage flow) Unloading pressure relief valve Pressure sensor (supply pressure) Direction control valve Return line filter Pressure sensor (motor inlet pressure) Low speed high torque motor Pressure sensor (motor outlet pressure) Step up gear unit (gear ratio 20:1) Flow sensor (loading pump flow)
2. 4. 6. 8. 10. 12. 14. 16. 18. 20. 22.
Electric motor Torque and speed sensor Pressure compensated pump Flow sensor (valve inlet flow) Oil cooler Flow sensor (valve outlet flow) Flow sensor (motor leakage flow) Load inertia Torque–speed sensor Pressure sensor (pump pressure) Loading pump (gear pump)
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
15
Table 2 (continued) Item no.
Item description
Item no.
Item description
23. 25. 27. 29. 31. 33. 35. 37.
Proportional pressure relief valve Speed indicator (input speed) Flow indicator (valve inlet flow) Flow indicator (motor leakage flow) Pressure indicator (motor inlet pressure) Speed indicator (motor speed) Pressure indicator (loading pump pressure) Signal input to the Proportional Pressure Relief valve
24. 26. 28. 30. 32. 34. 36. 38.
Torque indicator (input torque) Pressure indicator Flow indicator (valve outlet flow) Pressure indicator (motor outlet pressure) Torque indicator (motor torque) Flow indicator (loading pump flow) Data acquisition system (DAS) Unloading valve
The experiments were conducted over a wide range of speed and torque levels, following a standard procedure [12]. In determining the load resistance (Rrlv) of the proportional pressure relief valve (item no. 23), the flow through the valve port and the pressure drop across it were measured through the sensors (item nos. 20 and 21). The load resistance Rrlv was controlled through command signal Vcom (item no. 37). Similarly, the port resistances (Rvi and Rvo) of the d.c. valve (item no. 9) were obtained from the recorded data of the respective pressure and flow sensors. In order to obtain the viscous friction coefficient (Rl) and the constant torque loss (Tloss) of the hydrostatic drive, the unloading valve (item no. 38) was kept open, so that the pump (item no. 22) driven by the LSHT motor supplies flow at no-load pressure (i.e. Ppl ≈ 0.0). Experiments were conducted at different torque levels (Tl) to estimate the resistances of the axial piston pump and the LSHT motor from the steady-state performance of the HST drive. Constant load torque was maintained by adjusting the set pressure of the valve (item no. 23) through command signal (item no. 37). As such, the transmission considered in the present investigation which is an LSHT drive, the test speed of the motor was limited to 30 to 140 rpm to cover its maximum efficiency zone. The motor speed was controlled by the variable flow supplied by the pressure compensated pump (item no. 6) driven through speed controller (item no. 1). In validating the model of the HST drive discussed in Section 3, the unloading pressure relief valve (item no. 5) was set at high value so as not to fall below the transient response of the system. The tests of the HST drive included step changes in the flow through the d.c. valve, when the motor was initially stationary by the application of its brake and the PPRV was set at a particular position by the command signal (item no. 37). The experiments were conducted in a well-ventilated laboratory and the test rig was equipped with suitable water oil cooler. The oil temperature was maintained at 50 ± 2 °C tokeep constant viscosity with reasonable accuracy. The parameters like load torque (Tl), motor speed (ωm), supply flow rate V_ s , and system pressure (Ppp) were
Table 3 Summary of the major components and instruments used in the test set-up. Components Pressure compensated axial piston pump Make Model Displacement Radial piston low speed high torque motor Make Model Displacement Gear pump Make Model Displacement Proportional pressure relief valve Make Model Maximum set pressure Direction control (d.c.) valve Make Model Maximum flow Torque transducer Make Model Maximum torque range Accuracy Speed sensor Make Model RPM range Accuracy
Parameters Bosch Rexroth, Germany A10VSO28DR/3ZRPPA12N00 28 cm3/rev. Bosch Rexroth, Germany MCR3F280F180Z32B2M 280 cm3/rev. Bosch Rexroth, Germany 1PF2G2-4X/016RA01MB-1 N001 16.2 cm3/rev. Bosch Rexroth, Germany DBEM105X350YG24NK4M 350 bar Bosch Rexroth, Germany 4WE10J 1X/L G24 NZ4 120 lpm Honeywell Sensotec, USA 2100A series data telemetry system 1000 N m Less than ±0.05% full-scale torque Syscon Instruments pvt. Ltd., India ST-60 0–500 Within 0.1%, FSR ± 1 count (continued on next page)
16
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
Table 3 (continued) Components
Parameters
Flow sensor Make Model Flow range Accuracy Pressure sensor Make Model Pressure range Accuracy
Rockwing flowmeter India Pvt. Ltd., India TFM 1015 2–50 lpm ± 0.5% over 10 to 100% flow range Wika, Germany S-10 0–200 bar ≤ 0.25%
measured through suitable sensors and recorded in the respective instruments. All the instruments had their analog output, which were captured in the computer through data acquisition system. Appendix B. Referring to Fig. 4, following empirical relation expresses nature of V_ rlv with ppl and Vcom
For V_ rlv > 0:0 h i h i 4 2 2 2 10 V_ rlv ¼ − 0:0002Vcom –0:002Vcom þ 0:0079 P pl þ 0:0254Vcom þ 0:028Vcom þ 0:6041 Ppl – h i 4:971V2com –1:619Vcom þ 5:1422
ðB:1Þ
where, V_ rlv is the flow in m 3/s, Ppl is the set pressure in bar and Vcom is the command voltage in volt that sets the resistance (Rrlv) of P the relief valve. At a particular value of Vcom, Rrlv ¼ V_ pl rlv Eq. (B.1) as validated by further experiments carried out at other values of Vcom. The deviation was of about ±3%, which is reasonably accepted. Referring to Fig. 5, the following empirical relation expresses the nature of V_ vi and V_ vo with ΔPo and ΔPi, respectively. For V_ vi > 0:0 2 V_ vi ¼ −0:67ΔP i þ 18:42ΔP i þ 8
ðB:2Þ
For V_ vo > 0:0 V_ vo ¼ −0:53ΔP o 2 þ 15:5ΔP o þ 7:2
ðB:3Þ
The resistances of the valve ports are Rvi ¼ ΔP i =V_ vi and Rvo ¼ ΔP o =V_ vo . Referring to Fig. 8, the following empirical relation expresses nature of Rls with αps and Tl for α ps ≥0 2
10 Rls ¼ a2 α ps –b2 α ps þ c2 where a2 ¼ 0:025Tl þ 0:3241; b2 ¼ 0:273Tl þ 3:25 and c2 ¼ 0:814Tl þ 3:15:
ðB:4Þ
Referring to Fig. 9, the following empirical relation expresses the nature of Rip with αps and Tl For α ps ≥0;
10−11 Rip ¼ a1 α ps þ c1
ðB:5Þ
where a1 ¼ 0:0001Tl 2 –0:047Tl þ 4:911 and c1 ¼ 0:0312Tl 2 –8:997Tl þ 671:70: Referring to Fig. 10, the following empirical relation expresses the nature of Rem with αps and Tl For α ps ≥0;
10−11 Rem ¼ a3 α ps þ b3 where a3 ¼ −0:0002Tl þ 0:2438; b3 ¼ −0:0089Tl þ 17:16:
ðB:6Þ
Referring to Fig. 11, the following empirical relation expresses the nature of Ril with αps and Tl For α ps ≥0; 10
−12
Ril ¼ a4 α ps þ b4
where a4 ¼ −10−6 Tl 2 þ 0:0012Tl –0:08; b4 ¼ 8 10−6 Tl 2 –0:01:06Tl þ 10:08:
ðB:7Þ
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
17
Referring to Fig. 12, the following empirical relation expresses the nature of Rvm with αps and Tl For α ps ≥0; 10
−9
Rvm ¼ a5 α ps þ b5
where a5 ¼ −10−7 Tl 2 þ 90 10−5 Tl –0:055; b5 ¼ 5 10−6 Tl 2 –0:007Tl þ 9:886:
ðB:8Þ
References [1] J. Watton, The stability of electro-hydraulic servo-motor systems with transmission lines and non-linear motor friction effects. Part A: system modelling, Journal of Fluid Control 16 (2–3) (1986) 118–136. [2] P.J. Chappel, Modelling of a radial piston hydraulic motor, Proceedings of the Institution of Mechanical Engineers Part I The Journal of Systems and Control Engineering 206 (1986) 171–180. [3] N.D. Manring, G.R. Luecke, Modelling and designing a hydrostatic transmission with a fixed displacement motor, Journal of Dynamic Systems Measurement and Control-Transactions of the ASME 120 (1998) 45–49. [4] K. Dasgupta, Analysis of hydrostatic transmission system using low-speed-high-torque motor, Mechanism and Machine Theory 35 (2000) 1481–1499. [5] K. Dasgupta, J. Watton, S. Pan, Open-loop dynamic performance of a servo-valve controlled motor transmission system with pump loading using steadystate characteristics, Mechanism and Machine Theory 41 (2006) 262–282. [6] S.K. Mandal, K. Dasgupta, S. Pan, A. Chattopadhyay, Theoretical and experimental studies on the steady-state performance of low-speed high-torque hydrostatic drives. Part 1: modelling, Proceedings of the Institution of Mechanical Engineers. Part C 223 (2009) 2663–2674. [7] S.K. Mandal, K. Dasgupta, S. Pan, A. Chattopadhyay, Theoretical and experimental studies on the steady-state performance of low-speed high-torque hydrostatic drives. Part 2: experimental investigation, Proceedings of the Institution of Mechanical Engineers. Part C 223 (2009) 2675–2685. [8] A. Mukherjee, R. Karmakar, Modelling and simulation of engineering systems through bondgraph’, first edition Narora Publishing House, New Delhi, 2000. [9] Bosch Rexroth India Ltd. Product catalogue: RE 15 205/05.93, Hydraulic Pump (pressure compensated fixed displacement type) A10 VS 028 DR 3XRPP A 12 N00. [10] Bosch Rexroth India Ltd. Product catalogue: RE 15 205/05.93, Hydraulic Motor MCR4 3F 280F 180Z 32 B2M. [11] Bosch Rexroth India Ltd. Product catalogue: RE 10028/08.93, Gear Pump, 1PF2G2-4X/016 RA01MB-1N001. [12] BS 4617, Method of testing hydraulic pumps and motors for hydrostatic power transmission, British Standard Institution, London, 1983. [13] D. McCandlish, R.E. Dorey, The mathematical modeling of hydrostatic pumps and motors, Proceedings of the Institution of Mechanical Engineers Part B Journal of Engineering Manufacture 198 (B10) (1984) 165–174. [14] K. Dasgupta, A. Mukherjee, R. Maiti, Estimation of critical system parameters that affect orbit motor performance-combining simulation and experiment, Journal of Engineering for Industry Transactions of the ASME 121 (1999) 300–306. [15] Symbols 2000, High Tech Consultants, STEP, IIT, Kharagpur, India, www.symbols2000.comAvailable from.
Contents lists available at SciVerse ScienceDirect
Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
Dynamic analysis of a low speed high torque hydrostatic drive using steady-state characteristics K. Dasgupta a,⁎, S.K. Mandal b, S. Pan c a b c
Department of Mechanical Engineering and Mining Machinery Engineering, Indian School of Mines, Dhanabad 826004, India Benaras Hindu University, India Department of Electrical Engineering, Indian School of Mines, Dhanabad 826004, India
a r t i c l e
i n f o
Article history: Received 2 July 2010 received in revised form 22 August 2011 accepted 7 December 2011 Available online 31 January 2012 Keywords: Hydrostatic (HST) drive Low-speed-high-torque (LSHT) hydraulic motor Direction control (d.c.) valve Bondgraph modelling Steady state Resistive element Proportional pressure relief valve (PPRV)
a b s t r a c t In this article dynamic analysis of an open-circuit hydrostatic (HST) drive has been carried out to study its performance. A Low Speed High Torque (LSHT) radial piston motor has been considered for the drive. Bondgraph simulation technique is used to model the hydrostatic drive. The relationships of the loss coefficients of the drive with the other system variables, obtained from the steady-state model, are identified through experimental investigation. Using the parametric values, the overall dynamic model of the hydrostatic drive has been validated experimentally. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction An HST drive converts mechanical power of the input drive shaft of a pump to hydrostatic power in a nearly incompressible working fluid and then reconverts it into mechanical power at the output drive shaft of the motor. It is used to convert the rotating mechanical power from one source to another without the use of gears. An LSHT hydrostatic drive is widely used in Heavy Earth Moving Machinery (HEMM) because of its compact design and better low speed characteristics compared to high speed hydrostatic drive with integrated gear unit. It mainly consists of a fixed or variable displacement pump and an LSHT hydraulic motor, either in open loop or in closed loop configuration. Watton [1] has applied the method of characteristics in the time domain dynamic analysis of the hydro-motor drive system when a quasi-analytic method was used in evaluating motor boundary conditions. Chappell [2] has developed a model of radial piston hydraulic motor for evaluation of the factors affecting the motor performance and to study the influence of various internal factors on the performance of the motor. Manring and Luecke [3] had analysed an HST drive that consists of a variable displacement swash-plate controlled pump and a fixed displacement motor. In their study, the system has been linearised and the stability range of the system is presented. However, the torque on the motor shaft is not completely defined. Dasgupta [4] has analysed the quasi-static performance of an open-loop hydrostatic drive using orbit motor, where the torque-speed performance of the motor is investigated using a displacement controlled pump. The open loop dynamic performance of a servo valve controlled axial piston motor transmission system has been studied by Dasgupta et al. [5], where torque loss of the motor has been determined from its steady-state performance. ⁎ Corresponding author. Tel.: + 91 3262296551; fax: + 91 3262296563. E-mail address: [email protected] (K. Dasgupta). 0094-114X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2011.12.004
2
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
The art of model making in system dynamics comprises two stages. The first stage is involved in splitting the system into its very basic components, namely, compliances, inertias, dissipaters and sources. The model may be further refined by identifying more and more elements of the components in the systems. However, one has to stop at one stage either satisfied by the grasp of details for the range of operation or restricted by the limitation of the handling capacity. Putting the pieces back together is the next stage. The work presented in this paper deals with the detail modeling of an open circuit HST drive to study its dynamic performance. Mandal et al. [6,7] have already proposed a reduced model to study the steady-state performance of an open circuit and a closed circuit HST drives with LSHT motor. The said study investigates the effects of the critical parameters of the pumps and the motor on the steady-state performance of the HST drives. In this article a dynamic model of an open-circuit LSHT hydrostatic drive system with pump loading is presented. The determination of the loss coefficients of the system is made from the steady-state analysis of the system [6,7]. With respect to the variation of the inertia and the resistive loads, the model is validated through experiment. The model presented in this article itself guides the design of the experiment and mode of handling of the observations in such a way that values of dependence of the parameters on the operating conditions of the drive are obtained. 2. The physical system Fig. 1 shows the open-circuit hydrostatic drive system considered for the analysis where the speed of the electric motor is controlled by a variable frequency control drive. The pump flow is supplied to the LSHT hydraulic motor through solenoid operated direction control (d.c.) valve. The motor drives loading pump through a step-up gear unit. By pump loading through the proportional pressure relief valve (PPRV) of the loading circuit, the viscous load on the motor shaft is controlled. Fig. 2 shows the multi-piston LSHT radial piston hydraulic motor. The valve integrated with the motor supplies pressurized fluid to the chambers in the expansion mode and also provides a flow path for the fluid being returned from the chambers in the compression mode. There are leakages from the clearances between the pistons and the cylinders. The case drain flow is considered as external leakage resistance (Rem) of the motor. The leakage also occurs from the high-pressure chamber to the lowpressure chamber of the motor; it is termed as inter chamber leakage resistance (Ril) of the motor. These leakage losses are mainly because of the pressure gradient. Due to the support of the pressurized pistons, a tangential force component of the piston pressure force generates rotational motion of the cylinder block connected with the spline shaft. 3. Modelling of the system Dynamics of the open circuit HST drive system and its components are modelled using Bondgraph technique [8]. Such technique is based on the power flow and it is a very useful tool for formulating the state equations of a system and its components. In the development of the dynamic model of the hydrostatic drive system with the loading unit • • • • • •
The fluid inertia is neglected; The mineral based hydraulic oil used is considered to be Newtonian fluid; The resistive and the capacitive effects are lumped wherever appropriate; The effects of pressure on the properties of the fluid are ignored; The dynamics of the gear unit, PPRV and the d.c. valve are not taken into consideration; In the event of the opening the valve ports, flow forces acting on the valve spool are not considered; similar considerations are made for inlet and outlet valve ports of the motor; • The dynamics of the control valves and the loading pump are not considered;
Fig. 1. Open-circuit hydrostatic drive system.
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
3
Fig. 2. Multi-piston LSHT hydraulic motor.
• As the loading pump is placed closed to the PPRV, the effect of the line resistance connecting the pump and the valve is ignored. Similar consideration is made for the line connecting the source pump and the d.c. valve; • The set pressure of the unloading pressure relief valve is considered to be higher than the maximum system pressure, therefore, the flow through the relief valve is not considered; • The leakages other than those from the pumps and the motor are ignored; • Since the temperature is kept almost constant (50 ± 2 °C) during experiment, the effect of temperature change on the properties of the fluid is neglected. Fig. 3 shows the bondgraph model of the open circuit HST drive system. It is described in the following sub-sections. 3.1. The hydrostatic drive unit In Fig. 3, the SF element on 1 junction represents the rotational frequency (ωp) of the pressure compensated pump. The resistance Rls on the same junction represents the drag loss of the pump. Essentially, a hydrostatic unit, pump or motor, transforms hydraulic power to mechanical power or vice-versa. This basic fact is represented by a transformer (TF junction). The TF modulus Dp is the volume displacement rate of the pump with respect to the shaft rotation (i.e., the theoretical pump flow V_ s ¼ ωp Dp Þ. Similarly, the TF element connecting 1 and 0 junctions represents the volume displacement rate of the motor (Dm). A part of the ideal flow of the pump escapes through its clearance as leakage (V_ plkg ). The R element on the 0Pp junction represents the leakage resistance. In the above model, the C elements on the 0 junctions (represent the respective plenum) take into account the flow losses because of the compressibility of the fluid. Similarly, the pressure drop across the valve ports and the hydraulic line are taken into account by the R elements connected with the 1 junctions. The resistances Rvi and Rvo represent the inlet and the outlet ports of the d.c. valve, respectively. The resistance Rln indicates the resistance of the line connecting the d.c. valve and the LSHT motor. Referring to Fig. 2, the valve is an integral part of the motor shaft, through which supply flow passes to the displacement chambers of the LSHT motor. In the model, the resistances Rmv appear on 1 junction indicate the inlet and the outlet ports of the LSHT motor. The internal leakage loss of the motor occurs due to the pressure difference (Pmph – Pmpl) across
Fig. 3. Bondgraph model of the system.
4
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
its high pressure and the low-pressure chambers. It is taken into account by the resistance Ril at the 1 junction. The resistance Rem represents the external leakage flow of the motor. The outlet flow of the motor returns to the sump (Psmp) through the d.c. valve. The 1 junction representing the mechanical part of the load corresponds to the speed of the motor output shaft (ωm). The load inertia (Jl), speed dependent friction load (Rl) and the constant torque loss (Tloss) are represented by the I, R and SE elements, respectively. They are connected with the 1 junction that constitutes the load dynamics. The activated C element on the same junction records the speed of the motor. 3.2. The loading unit The flow supplied by the motor (ωm). The external leak loading pump (0Ppljunction) is proportional to the speed ofthe LSHT age flow of the pump V_ llkg and the flow loss because of the compressibility of the fluid V_ plc at the pump plenum are taken into account by the R and the C elements, respectively, connected with 0 junction. The TF element connecting the 1 and the 0 junctions represents the volume displacement rate of the pump (Dp) multiplied by the step up gear ratio (ng). The pump loading is provided by the proportional pressure relief valve (PPRV). The flow through the valve port V_ rlv depends on the port opening area and the pressure difference (Ppl − Psmp) across it. 4. Describing equations of the system The system equations derived from the bondgraph model are described as follows: 4.1. The hydrostatic drive unit Considering the flow continuity at the pump plenum, the flow supplied to the inlet port of the d.c. valve V_ ¼ V_ −V_ −V_ vi
s
plkg
pp
ð1Þ
where, V_ plkg ¼ P pp =Rip and V_ s ¼ ωp Dp . In the above equation, V_ plkg ; V_ pp , V_ s are the leakage flow of the pump, the compressibility flow loss of the fluid at the pump plenum and the theoretical flow supplied by the pump, respectively. The fluid pressure at the pump plenum is given by P pp ¼ K p V pp :
ð2Þ
The pump torque is expressed as T p ¼ P pp Dp þ wp Rrls :
ð3Þ
In Eq. (3), Rls represents the frictional resistance of the pump shaft. With the compressibility of the fluid, the flow balance at the inlet and the outlet ports of the d.c. valve is given by V_ dci ¼ V_ vi −V_ lni ;
ð4Þ
V_ dco ¼ V_ lno −V_ vo ;
ð5Þ where, the flow through the valve ports are expressed as V_ vi ¼ P pp −P vi =Rvi and V_ vo ¼ P vo −P smp =Rvo . In Eqs. (4) and (5) the inlet and the outlet flows of the hydro-motor are given by V_ lni ¼ ðP vi −P mvi Þ=R ln and V_ lno ¼ ðP mvo −P vo Þ=R ln , respectively. The fluid pressures at the inlet and the outlet ports of the d.c. valve are given by P vi ¼ K v V dci andP vo ¼ K v V dco ; respectively: Considering the flow continuity at the inlet and the outlet ports of the LSHT motor, the flow balance at each port of the motor is given by V_ mi ¼ V_ lni −V_ mvi ;
ð6Þ
V_ mo ¼ V_ lno þ V_ mvo ;
ð7Þ
where, V_ mvi and V_ mvo are the compressibility flow loss of the fluid at the respective plenum of the valve ports. The pressure at the inlet and the outlet ports of the motor are given by P mvi ¼ K mv V mvi andP mvo ¼ K mv V mvo ; respectively:
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
5
The inlet and the outlet flows of the motor depend on the valve port resistances (Rmv) and the pressure difference across it. They are expressed as V_ mi ¼ P mvi −P mph =Rmv and V_ mo ¼ P mpl −P mvo =Rmv ; respectively: With the compressibility of the fluid; the flow balance at the motor chambers is given by: P mph P mph −P mpl V_ mph ¼ −ωm Dm − − þ V_ mi ; Rem Ril
ð8Þ
P mpl P mph −P mpl _ V_ mpl ¼ ωm Dm − þ −V mo ⋅ Rem Ril
ð9Þ
In Eqs. (8) and (9) the high and the low chamber pressures of the motor are expressed as P mph ¼ K mp V mph and P mpl ¼ K mp V mpl ; respectively: The 1 junction in the mechanical load part of the motor comes from the torque balance, which is expressed by the following equation, resulting from the constitutive equation of each element: p_ l ¼ P mph −P mpl Dm −T loss −ωm Rl −P pl ng Dpl ð10Þ where, the first term of the above equation is the torque due to the inertia load Jl, the second term indicates the torque equivalent to the differential pressure across the motor, the third term is the constant torque loss, the fourth term takes into account the torque loss due to viscous friction coefficient Rl and the fifth term represents the load applied on the motor shaft due to the pressure (Ppl) of the loading pump. The motor speed is given by wm ¼ p1 =J 1 :
ð11Þ
4.2. The loading unit Considering the flow continuity at the loading pump plenum, the flow supplied to the proportional pressure relief valve V_ ¼ V_ −V_ −V_ ; rlv
pl
llkg
plc
ð12Þ
where, V_ pl ; V_ llkg and V_ plc are the theoretical flow supplied by the loading pump, the leakage flow of the loading pump, and the compressibility flow loss at the pump plenum, respectively. In Eq. (12) V_ pl ¼ ωm ng Dpl , V_ llkg ¼ P pl =Rpl , V_ rlv ¼ P pl −P smp =Rrlv and the load pressure is given by Ppl = Kpl Vplc. 5. Estimation of the system parameters A practical system discussed in Appendix A was used to validate the dynamic model of the system. Commercially available valves, pressure compensated axial piston pump and the LSHT hydraulic motor were used in the experiment. The values of their resistances were determined from the experimental steady-state performance of the system; the details of which are discussed in Section 6. The other parameters of system components are estimated as follows: • From the geometrical parameters, the total equivalent inertia of the rotating parts connecting the motor and the loading pump is calculated.
Fig. 4. Pressure-flow characteristics of proportional relief valve.
6
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
Fig. 5. Supply flow rate as a function of pressure drop across valve port.
• The values of the bulk stiffness of the fluid at the plenum junctions are obtained based on the volume in the respective cases and the bulk modulus of the fluid used in the experiment (Ka = β / Va). • The volume displacement rate of the pump and the motor are obtained from the manufacturer's product literature [9–11]. • The line resistance (Rln) connecting the pump and the motor is suitably assumed. • The leakage resistance (Rpl) of the loading pump is estimated from its volumetric efficiency [11]. 6. Experimentation, estimation of the resistances of the valves, pump and the motor, verification of the overall dynamic model of the system The experiments were conducted to estimate the resistances of the valves, pump and the motor. From the steady-state pressure-flow characteristics, the load resistance of the PPRV and the port resistances of the d.c. valve are obtained. The resistances of the pump and the motor are determined from the steady-state performance of the HST drive. A brief description of the test set-up along with the experimentation is given in Appendix A. 6.1. Estimation of the load resistance (Rrlv) of the Proportional Pressure Relief Valve (PPRV) Referring to Fig. 1, the resistance of the PPRV of the loading unit provides viscous load on the motor shaft. Varying the area of the orifice of the valve by the command signal (Vcom), the load resistance (Rrlv) is varied. Fig. 4 shows the pressure-flow characteristics of the valve plotted at different Vcom level. From Eq. (B.1) that expresses the variation of the flow through the valve V_ rlv with the pressure drop ( ΔP = Ppl − Psmp) across it, the valve resistance Rrlv is given by Rrlv ¼ ΔP=V_ rlv
ð13Þ
6.2. Estimation of the port resistances of the direction control (d.c.) valve Fig. 5 shows the pressure-flow characteristics of the d.c. valve. From Eqs. (B.2) and (B.3) that express the nature of the above characteristics, the values of the valve port resistances are obtained using the following relations: Rvi ¼ ΔP i =V_ vi
ð14Þ
Fig. 6. Reduced model of hydrostatic unit.
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
7
Fig. 7. Variation of load torque with pressure differential across motor.
Rvo ¼ ΔP o =V_ vo
ð15Þ
In Eqs. (14) and (15), V_ vi and V_ vo are flow through the inlet and outlet valve ports and the pressure drop across the ports are ΔPi = (Ppp − Pvi) and ΔPo = (Pvo − Psmp), respectively. 6.3. Estimation of the resistances of the pump and the motor A reduced model is developed to determine the resistances of the axial piston pump and the LSHT motor from the steady-state performance of the HST drive. Mandal et al. [6] have explained the detail model reduction process in their studies. In the model [6] various losses are lumped into suitable resistive elements. It has fewer adjustable parameters requiring a comparatively few test runs to estimate them. The relationships of the resistances with the operating parameters of the HST drive, obtained through reduced model, are identified through experimental investigation. 6.3.1. Reduced model of the system Fig. 6 shows the reduced model of the open circuit hydrostatic drive. The steady-state performance of the system was tested at various input speed (ωp) of the pump. The arrow shown over SF element represents the modulated rotational frequency (ωp) of the pump. Rmv represents resistances of the valve port of the LSHT motor. The resistance Ril takes into account the inter-chamber leakages that occur because of the differential pressure (Pl) across the motor. Rem on the 0 junction represents the external leakage resistance of the motor. The other parameters of the model are explained in Section 3. From the model shown in Fig. 6, the steady-state load torque provided by the loading circuit discussed in Section 2, is given by T pl ¼ Dm P l –Rl ωm −T loss :
ð16Þ
where, the second term indicates the total torque load on the hydro-motor proportional to the pressure differential across it, the third term represents the torque loss due to fluid viscosity effect assumed to be linearly dependent upon speed and the fourth term is the additional friction losses due to coulomb friction and stiction of the hydro-motor. The following relationships of the resistances with the operating parameters of the HST drive are obtained from the reduced model: Rls ¼
T p −P pp Dp ωp
ð17Þ
Rip ¼ P pp =V_ plkg
ð18Þ
Tl V_ mlkg Dm
ð19Þ
Rem ¼
Table 1 List of the parametric values. Parameter Jl Dpl Kmp Rpl
Value
Parameter 2
6.18 kg m 16 cm3/rev 1012 N/m5 1012 N s/m5
Dm Kpl Kv Rln
Value 3
280 cm /rev 1012 N/m5 1013 N/m5 107 N s/m5
Parameter
Value
Dp Kp Kmv β
28 cm3/rev 1012 N/m5 1013 N/m5 1.5 × 109 N/m2
8
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
Fig. 8. Variation of viscous friction resistance (Rls) of the pump with pump speed ratio (αps).
Rmv ¼
Ril ¼
P pp −ðT l =Dm Þ
α ps ωp max Dp −V_ plkg
Tl _ Dm α ps ωp max Dp −V plkg −V_ mlkg −ωm Dm
ð20Þ
ð21Þ
From the test data of the pump pressure (Ppp), load pressure (Pl), load torque (Tl = Pl Dm), pump speed (ωp), motor speed (ωm) and the leakage flow of the pump and the motor V_ plkg and V_ mlkg Þ, the values of the resistances (Ril, Rmv,, Rls, Rem) are obtained. The empirical relationships that express the variation of the resistances with the pump speed are given in Appendix B. 6.3.2. Estimation of the viscous friction coefficient (Rl) and the constant torque loss (Tloss) of the motor The viscous friction coefficient (Rl) and the constant torque loss (Tloss) of the LSHT motor are estimated experimentally from the steady-state torque characteristics of the motor. Fig. 7 shows the torque (Tl) versus load pressure (Pl) plotted at various speed level. Comparing Eq. (16), the nature of the characteristics is given by Tpl ≈Dm Pl −11:27wm −47:40
ð22Þ
Comparing Eqs. (16) and (22) the values of Rl and Tloss are obtained and they are given by Rl ¼ 11:27 Nm s=rad; T loss ¼ 47:40 Nm: Since the torque characteristics of the hydro-motor shown in Fig. 7 have been determined for speeds greater than 40 rpm, the non-linear stiction term is not considered in the Tloss term that would have been expected at slower speeds. 6.3.3. Estimation of the resistances of the pump and the motor In estimating the resistances of the axial piston pump and the LSHT motor of the hydrostatic drive unit, the test speed of the motor was limited to 30 to 140 rpm to cover its maximum efficiency zone. Using the parametric values given in Table 1 and the test data in Eqs. (17)–(21), the values of the resistances are calculated. Figs. 8–12 show the variations of the resistances with the pump speed ratio (αps = ωp / ωpmax) at various torque (Tl) levels. Eqs. (B.4)–(B.8) express the empirical relationships of the variation of the resistances with αps. Using them, the predicted characteristics of the resistances are obtained for other torque levels. These are compared with the experimental results. One representation is shown in Fig. 13 that compares the predicted and the experimental data of the valve port resistance (Rmv). It is found that there is a close agreement (deviation ± 3%) between the predicted and the experimental data. Therefore, the estimation of resistances given by Eqs. (26)–(30) seems to be acceptable.
Fig. 9. Variation of leakage resistance (Rip) of the pump with the pump speed ratio (αps).
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
9
Fig. 10. Variation of external leakage resistance (Rem) of pump with pump speed ratio (αps).
Fig. 11. Variation of internal leakage resistance (Ril) of motor with pump speed ratio (αps).
The reasons for these deviations, which are not considered in the present analysis, may be due to minor losses such as variation of the resistances due to the thermal effect, effect of load pressure and speed on the pump displacement rate (Dp), hydrodynamic losses, and compressibility losses of the fluid and so on. Besides these, there may be minor instrumental error. McCandlish and Dorey [13] and Dasgupta et al. [14] have also recognized such variations in their studies on piston type hydrostatic machines.
6.4. Verification of the dynamic model of the open circuit hydrostatic drive The dynamic performance of the open circuit hydrostatic drive system was investigated at constant rotational frequency (ωp) of the pump. The simulation results obtained by solving the system equations given in Section 4 are compared with the experimental data. The derivation of the system equations from the model and their numerical solutions are carried out using the software SYMBOLS 2000 [15]. In simulation studies the parametric values given in Table 1, values of the resistances of the pumps and the motor determined from Eqs. (B.1)–(B.8), the values of Tloss and Rl obtained from Eq. (22) are used. The tests on the dynamic behaviour of the HST drive were conducted at different values of load inertia (Jl) and load resistance (Rrlv). The experiment was repeated several times to examine the repeatability before recording the data in PC through data acquisition system. Figs. 14 and 15 compare the predicted and the experimental transient responses of the pump pressure (Ppp) and the motor speed (ωm) at different values of Jl. As the inertia increases, the speed and the pressure responses become more oscillatory. The maximum overshoot of the speed at the higher inertia (Jl = 0.41 kg m 2) is about 50% over the steady-state motor speed of 100 rpm. It is also clear that with increase in the load inertia, the settling time of the system response increases.
Fig. 12. Variation of valve port resistance (Rvm) of the motor with the pump speed ratio (αps).
10
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
Fig. 13. Comparison of predicted and experimental characteristics of valve resistance Rvm of the motor.
Referring to the experimental test set-up discussed in Appendix A, by changing the command signal (item no. 37) Vcpm, the load resistance Rrlv is controlled. Figs. 16 and 17 compare the transient responses of the pump pressure (Ppp) and the motor speed (ωm) with that of its experimental values at different Vcom values. With the increase in Vcom (that increases Rrlv), the rise time as well as the steady state values of Ppp increases, whereas ωm decreases. The decrease in ωm is due to increase in leakages at higher pressure. The increase in the value of Rrlv that increases the resistive load provides more damping effect and, therefore, the overshoot of ωm and Ppp decrease. In general, various plots shown in Figs. 14–17 indicate that the predicted and the experimental responses are in good agreement. The differences between the simulation and experimental results may be due to the following reasons: • The thermal and hydrodynamic losses of the fluid, local deformation of the internal components of the pump and the motor and the minor leakage loss as well, not considered in the model. • The estimated values of the resistances of the pump and the motor discussed in Section 5 are considered in simulation studies. • Limitations of the experimental test rig and its instruments, lower sampling rate of the data accusation system (DAS). • The fluctuations of the experimental responses at steady state are due to the effects of the flexible hoses, small instrumental errors, as well as minor mechanical misalignment between the loading pump and the hydraulic motor. Further, the fluctuation
Fig.14. Comparison of the predicted and experimental pump pressure at different inertia load.
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
Fig. 15. Comparison of the predicted and experimental motor speed at different inertia load.
Fig. 16. Comparison of the predicted and experimental pump pressure at different load resistance.
11
12
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
Fig. 17. Comparison of the predicted and the experimental motor speed at different viscous load.
of the system's response at low speed range may be attributed to the Coulomb friction / Stiction and unsteady beheviour of the loading pump flow rate as well. Such details are not considered while modeling the hydro-motor. However, the difference between the mean values of the steady state part of the experimental and the predicted results is small. The good correlation between the theoretical and the experimental responses justifies the assumptions made to develop the model. The simulation results would have been more accurate, if the detail dynamics of the pumps, the motor, gear unit and the valves of the overall system are considered. The results of the experimental analysis also indicate that there is a slight time lag in the increase in the pressure and the speed of the motor during the initial rise as compared to the simulation results. One such time lag (tl = 0. 05 s) is shown in Fig. 17. It may be attributed to the dynamics of the direction control valve used in the hydrostatic drive unit, minimum speed required by the loading pump to produce flow and the characteristics of the PPRV at low flow rate. These details of the pump and the valves are not considered in the model. It is seen that with the increase in the load resistance, the rise time of the states of the system increases. The experiments could not be conducted at higher motor speed and load pressure levels due to the limitations of the test unit.
7. Conclusion This paper presents a significant aspect of an integrated study of modeling, analysis, simulation and experimentation of the dynamics of an LSHT open circuit hydrostatic drive. The estimation of the loss coefficients of the pumps and the motor is made through steady-state analysis. The close agreement between the experimental and the simulation results of the dynamic responses of the system validates the proposed model. It shows that the simulated response from a reasonable model when combined with experimental observations renders effective mode of data handling for meaningful estimation of loss coefficients of the systems in terms of dependencies on operating conditions. It is also shown here that the loss coefficients of the pump and the motor that are difficult to estimate in running conditions, are best deduced by considering experimental data with the simulated results and for which the components need not be disassembled and special measurement techniques are not required.
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
13
Justifiably, many plausible parameters like inertia of the working fluid in the conduit, elasticity of the conduit etc. are not incorporated in the detailed model. Still, the model contains several critical parameters that substantially influence the performance of the system with the range of operation considered. The discrepancies found while comparing the theoretical and the test results are due to the dynamics of the gear unit, thermal and hydrodynamic losses of the fluid, and the minor leakage loss as well. Further, it may be noted that the estimation of the loss coefficients is based on the semi-empirical estimation. Consideration of the effects, ignored in the present study, requires further refinement of the model and more rigorous experimentation with better data acquisition system. Due to the limitation of the test rig, the experimental results are obtained within the limited range. However, it may be envisaged that the model developed may be useful for wider range of parameter variation. To ascertain this, further experiments may be undertaken. The dynamic response of the system has been validated with respect to the variation of the inertia and the viscous load. The following conclusions may be derived from the validated model: • With the increase in the inertia load, the speed and the pressure responses become oscillatory with the increase in the maximum overshoot. • With the increase in the load resistances, the steady-state speed decreases and the pressure increases. However, it provides more damping effect to the transient response of the system, therefore settling time of the system's response decreases. Comparing the theoretical and the experimental responses of the system, it may be concluded that the model fairly represents the behaviour of the system and, therefore, it may be useful to the practicing engineers for initial design of the machine, where such a hydrostatic drive is used as a transmission system. Nomenclature C Single port energy storage capacitor element in bondgraph model Dm, Dpl Displacement of the loading pump and the LSHT motor, respectively Dp Displacement of the pressure compensated axial piston pump I Single port energy storage inertial element in bondgraph model SF Single port source of flow element in bondgraph model SE Single port source of effort element in bondgraph model Jl Generalized inertia of the driving shaft with the connected load Ka Generalised bulk stiffness of the fluid Kpp Bulk stiffness of the fluid in the pump column Kmv Bulk stiffness of the fluid in the motor valve port Kmv Bulk stiffness of the fluid in the motor chamber port Kplc Bulk stiffness of the fluid in the loading pump plenum Kv Bulk stiffness of the fluid in the direction control valve port p_ l Torque due to load inertia pl Generalized angular momentum due to load inertia Pmph, Pmpl High and low chamber pressures of the LSHT motor, respectively Pvi, Pvo Pressure at the inlet and the outlet ports of the d.c. valve, respectively Pmvi, Pmvo Pressure at the inlet and the outlet ports of the LSHT motor, respectively Ppl, Psmp Loading pump plenum pressure and the sump pressure, respectively Ppp Plenum pressure of the axial piston pump Rl Resistive load connected with the motor end Rem, Ril External and internal-chamber leakage resistances of the motor, respectively Rrlv Resistance of the proportional pressure relief valve Rv Equivalent port resistance of the d.c. valve in the reduced model Rip, Rpl Leakage resistances of axial piston pump and loading pump, respectively Rln Resistance of the line connecting the valve and the motor ports Rls Viscous resistance of the pump drive shaft Rmvo, Rmvi Resistance of the outlet and the inlet ports of the LSHT motor, respectively Rmv Equivalent resistance of the motor ports in reduced model Tloss Constant torque loss of the motor Tl Steady-state Load torque on the LSHT motor Tp Steady-state Load torque on pump shaft Va Generalised fluid volume at the respective side of the system Flow rate of the axial piston pump and the loading pump, respectively V_ s ; V_ pl V_ plc Volume change rate of fluid at the loading pump plenum Volume change rate of fluid at the high pressure chamber of the motor V_ mph V_ mpl Volume change rate of fluid at the low pressure chamber of the motor V_ pp Volume change rate of fluid at the pressure compensated pump plenum V_ rlv Flow rate that passes through proportional relief valve V_ dci Volume change rate of fluid at the inlet port of the d.c. valve
14
V_ dco V_ plkg V_ mvi V_ mvo V_ mi ; V_ mo V_ lni ; V_ lno ng ωp, ΔP Vcom αps β
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
Volume change rate of fluid at the outlet port of the d.c. valve Leakage flow of the pump Volume change rate of fluid at the inlet port of the motor Volume change rate of fluid at the outlet port of the motor Inlet and outlet flow of the LSHT motor chambers, respectively Inlet and outlet flow from the LSHT motor, respectively Step-up gear ratio ωm Pump and LSHT motor speed, respectively Generalized differential pressure across the valve port Command voltage Pump speed ratio (=ωp / ωpmax) Generalized bulk modulus of the fluid
Acknowledgement The research and development project grant for 2009–2012 from University Grants Commission, Government of India, for carrying out research work on this topic is acknowledged. Appendix A. Experimental test set-up Before validating the dynamic model of the hydrostatic drive system, the resistances of the valves, pump and the motor discussed in Section 5 were determined experimentally. Using them the proposed model was validated. A simplified representation of the experimental test set-up is shown in Fig. 18 for ready reference. The hydro-motor was connected with the direction control valve by flexible hoses; whereas other hydraulic lines are of rigid steel pipes.
Fig. 18. Experimental test set-up.
Table 2 List of components used in the test set-up. Item no.
Item description
Item no.
Item description
1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21.
Power supply with variable speed drive Flow sensor (main pump leakage flow) Unloading pressure relief valve Pressure sensor (supply pressure) Direction control valve Return line filter Pressure sensor (motor inlet pressure) Low speed high torque motor Pressure sensor (motor outlet pressure) Step up gear unit (gear ratio 20:1) Flow sensor (loading pump flow)
2. 4. 6. 8. 10. 12. 14. 16. 18. 20. 22.
Electric motor Torque and speed sensor Pressure compensated pump Flow sensor (valve inlet flow) Oil cooler Flow sensor (valve outlet flow) Flow sensor (motor leakage flow) Load inertia Torque–speed sensor Pressure sensor (pump pressure) Loading pump (gear pump)
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
15
Table 2 (continued) Item no.
Item description
Item no.
Item description
23. 25. 27. 29. 31. 33. 35. 37.
Proportional pressure relief valve Speed indicator (input speed) Flow indicator (valve inlet flow) Flow indicator (motor leakage flow) Pressure indicator (motor inlet pressure) Speed indicator (motor speed) Pressure indicator (loading pump pressure) Signal input to the Proportional Pressure Relief valve
24. 26. 28. 30. 32. 34. 36. 38.
Torque indicator (input torque) Pressure indicator Flow indicator (valve outlet flow) Pressure indicator (motor outlet pressure) Torque indicator (motor torque) Flow indicator (loading pump flow) Data acquisition system (DAS) Unloading valve
The experiments were conducted over a wide range of speed and torque levels, following a standard procedure [12]. In determining the load resistance (Rrlv) of the proportional pressure relief valve (item no. 23), the flow through the valve port and the pressure drop across it were measured through the sensors (item nos. 20 and 21). The load resistance Rrlv was controlled through command signal Vcom (item no. 37). Similarly, the port resistances (Rvi and Rvo) of the d.c. valve (item no. 9) were obtained from the recorded data of the respective pressure and flow sensors. In order to obtain the viscous friction coefficient (Rl) and the constant torque loss (Tloss) of the hydrostatic drive, the unloading valve (item no. 38) was kept open, so that the pump (item no. 22) driven by the LSHT motor supplies flow at no-load pressure (i.e. Ppl ≈ 0.0). Experiments were conducted at different torque levels (Tl) to estimate the resistances of the axial piston pump and the LSHT motor from the steady-state performance of the HST drive. Constant load torque was maintained by adjusting the set pressure of the valve (item no. 23) through command signal (item no. 37). As such, the transmission considered in the present investigation which is an LSHT drive, the test speed of the motor was limited to 30 to 140 rpm to cover its maximum efficiency zone. The motor speed was controlled by the variable flow supplied by the pressure compensated pump (item no. 6) driven through speed controller (item no. 1). In validating the model of the HST drive discussed in Section 3, the unloading pressure relief valve (item no. 5) was set at high value so as not to fall below the transient response of the system. The tests of the HST drive included step changes in the flow through the d.c. valve, when the motor was initially stationary by the application of its brake and the PPRV was set at a particular position by the command signal (item no. 37). The experiments were conducted in a well-ventilated laboratory and the test rig was equipped with suitable water oil cooler. The oil temperature was maintained at 50 ± 2 °C tokeep constant viscosity with reasonable accuracy. The parameters like load torque (Tl), motor speed (ωm), supply flow rate V_ s , and system pressure (Ppp) were
Table 3 Summary of the major components and instruments used in the test set-up. Components Pressure compensated axial piston pump Make Model Displacement Radial piston low speed high torque motor Make Model Displacement Gear pump Make Model Displacement Proportional pressure relief valve Make Model Maximum set pressure Direction control (d.c.) valve Make Model Maximum flow Torque transducer Make Model Maximum torque range Accuracy Speed sensor Make Model RPM range Accuracy
Parameters Bosch Rexroth, Germany A10VSO28DR/3ZRPPA12N00 28 cm3/rev. Bosch Rexroth, Germany MCR3F280F180Z32B2M 280 cm3/rev. Bosch Rexroth, Germany 1PF2G2-4X/016RA01MB-1 N001 16.2 cm3/rev. Bosch Rexroth, Germany DBEM105X350YG24NK4M 350 bar Bosch Rexroth, Germany 4WE10J 1X/L G24 NZ4 120 lpm Honeywell Sensotec, USA 2100A series data telemetry system 1000 N m Less than ±0.05% full-scale torque Syscon Instruments pvt. Ltd., India ST-60 0–500 Within 0.1%, FSR ± 1 count (continued on next page)
16
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
Table 3 (continued) Components
Parameters
Flow sensor Make Model Flow range Accuracy Pressure sensor Make Model Pressure range Accuracy
Rockwing flowmeter India Pvt. Ltd., India TFM 1015 2–50 lpm ± 0.5% over 10 to 100% flow range Wika, Germany S-10 0–200 bar ≤ 0.25%
measured through suitable sensors and recorded in the respective instruments. All the instruments had their analog output, which were captured in the computer through data acquisition system. Appendix B. Referring to Fig. 4, following empirical relation expresses nature of V_ rlv with ppl and Vcom
For V_ rlv > 0:0 h i h i 4 2 2 2 10 V_ rlv ¼ − 0:0002Vcom –0:002Vcom þ 0:0079 P pl þ 0:0254Vcom þ 0:028Vcom þ 0:6041 Ppl – h i 4:971V2com –1:619Vcom þ 5:1422
ðB:1Þ
where, V_ rlv is the flow in m 3/s, Ppl is the set pressure in bar and Vcom is the command voltage in volt that sets the resistance (Rrlv) of P the relief valve. At a particular value of Vcom, Rrlv ¼ V_ pl rlv Eq. (B.1) as validated by further experiments carried out at other values of Vcom. The deviation was of about ±3%, which is reasonably accepted. Referring to Fig. 5, the following empirical relation expresses the nature of V_ vi and V_ vo with ΔPo and ΔPi, respectively. For V_ vi > 0:0 2 V_ vi ¼ −0:67ΔP i þ 18:42ΔP i þ 8
ðB:2Þ
For V_ vo > 0:0 V_ vo ¼ −0:53ΔP o 2 þ 15:5ΔP o þ 7:2
ðB:3Þ
The resistances of the valve ports are Rvi ¼ ΔP i =V_ vi and Rvo ¼ ΔP o =V_ vo . Referring to Fig. 8, the following empirical relation expresses nature of Rls with αps and Tl for α ps ≥0 2
10 Rls ¼ a2 α ps –b2 α ps þ c2 where a2 ¼ 0:025Tl þ 0:3241; b2 ¼ 0:273Tl þ 3:25 and c2 ¼ 0:814Tl þ 3:15:
ðB:4Þ
Referring to Fig. 9, the following empirical relation expresses the nature of Rip with αps and Tl For α ps ≥0;
10−11 Rip ¼ a1 α ps þ c1
ðB:5Þ
where a1 ¼ 0:0001Tl 2 –0:047Tl þ 4:911 and c1 ¼ 0:0312Tl 2 –8:997Tl þ 671:70: Referring to Fig. 10, the following empirical relation expresses the nature of Rem with αps and Tl For α ps ≥0;
10−11 Rem ¼ a3 α ps þ b3 where a3 ¼ −0:0002Tl þ 0:2438; b3 ¼ −0:0089Tl þ 17:16:
ðB:6Þ
Referring to Fig. 11, the following empirical relation expresses the nature of Ril with αps and Tl For α ps ≥0; 10
−12
Ril ¼ a4 α ps þ b4
where a4 ¼ −10−6 Tl 2 þ 0:0012Tl –0:08; b4 ¼ 8 10−6 Tl 2 –0:01:06Tl þ 10:08:
ðB:7Þ
K. Dasgupta et al. / Mechanism and Machine Theory 52 (2012) 1–17
17
Referring to Fig. 12, the following empirical relation expresses the nature of Rvm with αps and Tl For α ps ≥0; 10
−9
Rvm ¼ a5 α ps þ b5
where a5 ¼ −10−7 Tl 2 þ 90 10−5 Tl –0:055; b5 ¼ 5 10−6 Tl 2 –0:007Tl þ 9:886:
ðB:8Þ
References [1] J. Watton, The stability of electro-hydraulic servo-motor systems with transmission lines and non-linear motor friction effects. Part A: system modelling, Journal of Fluid Control 16 (2–3) (1986) 118–136. [2] P.J. Chappel, Modelling of a radial piston hydraulic motor, Proceedings of the Institution of Mechanical Engineers Part I The Journal of Systems and Control Engineering 206 (1986) 171–180. [3] N.D. Manring, G.R. Luecke, Modelling and designing a hydrostatic transmission with a fixed displacement motor, Journal of Dynamic Systems Measurement and Control-Transactions of the ASME 120 (1998) 45–49. [4] K. Dasgupta, Analysis of hydrostatic transmission system using low-speed-high-torque motor, Mechanism and Machine Theory 35 (2000) 1481–1499. [5] K. Dasgupta, J. Watton, S. Pan, Open-loop dynamic performance of a servo-valve controlled motor transmission system with pump loading using steadystate characteristics, Mechanism and Machine Theory 41 (2006) 262–282. [6] S.K. Mandal, K. Dasgupta, S. Pan, A. Chattopadhyay, Theoretical and experimental studies on the steady-state performance of low-speed high-torque hydrostatic drives. Part 1: modelling, Proceedings of the Institution of Mechanical Engineers. Part C 223 (2009) 2663–2674. [7] S.K. Mandal, K. Dasgupta, S. Pan, A. Chattopadhyay, Theoretical and experimental studies on the steady-state performance of low-speed high-torque hydrostatic drives. Part 2: experimental investigation, Proceedings of the Institution of Mechanical Engineers. Part C 223 (2009) 2675–2685. [8] A. Mukherjee, R. Karmakar, Modelling and simulation of engineering systems through bondgraph’, first edition Narora Publishing House, New Delhi, 2000. [9] Bosch Rexroth India Ltd. Product catalogue: RE 15 205/05.93, Hydraulic Pump (pressure compensated fixed displacement type) A10 VS 028 DR 3XRPP A 12 N00. [10] Bosch Rexroth India Ltd. Product catalogue: RE 15 205/05.93, Hydraulic Motor MCR4 3F 280F 180Z 32 B2M. [11] Bosch Rexroth India Ltd. Product catalogue: RE 10028/08.93, Gear Pump, 1PF2G2-4X/016 RA01MB-1N001. [12] BS 4617, Method of testing hydraulic pumps and motors for hydrostatic power transmission, British Standard Institution, London, 1983. [13] D. McCandlish, R.E. Dorey, The mathematical modeling of hydrostatic pumps and motors, Proceedings of the Institution of Mechanical Engineers Part B Journal of Engineering Manufacture 198 (B10) (1984) 165–174. [14] K. Dasgupta, A. Mukherjee, R. Maiti, Estimation of critical system parameters that affect orbit motor performance-combining simulation and experiment, Journal of Engineering for Industry Transactions of the ASME 121 (1999) 300–306. [15] Symbols 2000, High Tech Consultants, STEP, IIT, Kharagpur, India, www.symbols2000.comAvailable from.