Mechanism and Machine Theory
Mechanism and Machine Theory 41 (2006) 262–282
www.elsevier.com/locate/mechmt
Open-loop dynamic performance of a servo-valve controlled motor transmission system with pump loading using steady-state characteristics K. Dasgupta a
a,*
, J. Watton b, S. Pan
a
Department of Mechanical Engineering and Mining Machinery Engineering, Indian School of Mines, Dhanbad 826004, India b Department of Mechanical Engineering and Energy Studies, Cardiff School of Engineering, Cardiff University, Wales, UK Received 16 July 2004; received in revised form 3 May 2005; accepted 1 June 2005 Available online 6 October 2005
Abstract This article presents a study on dynamic performance of a servo-valve controlled axial piston motor used in a transmission system. The modeling of the system has been made using bondgraph simulation technique. The various loss coefficients of the motor have been determined from the steady-state pressure/flow relationships. The simulation results obtained at different operating conditions are validated experimentally. Using the validated model, transient response of the system have been predicted with respect to the variation of some critical parameters of the system. 2005 Published by Elsevier Ltd. Keywords: Bondgraph modelling; Servo-valve controlled hydrostatic transmission; Steady-state performance; Dynamic performance; Loading circuit
*
Corresponding author. Tel.: +91 0326 202487; fax: +91 0326 206372. E-mail address:
[email protected] (K. Dasgupta).
0094-114X/$ - see front matter 2005 Published by Elsevier Ltd. doi:10.1016/j.mechmachtheory.2005.06.005
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Nomenclature Cd discharge coefficient of the servo-valve volume displacement rate of the motor with respect to its shaft rotation Dm volume displacement rate of the pump with respect to its shaft rotation Dp servo-valve amplifier current isv motor plus load inertia Jl servo-valve flow constant kf gain of the servo-valve Ksv generalised effective bulk stiffness of the fluid Kstf effective bulk stiffness of the fluid at the motor plenum Kpm effective bulk stiffness of the fluid at the loading circuit Kpl angular momentum of the motor shaft pl pump plenum pressure Ppp PmiPmo inlet and outlet pressures of the motor boost pressure supply Pbp Psupl supply pressure to the servo-valve sump pressure Ps viscous friction load on the motor Rl cross-port leakage coefficient of the motor Ripl external leakage coefficient of the motor Rlkg load resistance given by the loading circuit Rpl stiction/Coulomb friction torque loss of the motor Tst load torque Tl V generalised fluid volume of the respective side of the system valve dead band vdb V_ 1 ; V_ 2 inlet and outlet flow of the motor V_ mi ; V_ mo compressibility flow loses V_ m mean flow rate of the motor _V lkg ; V_ ilkg external and internal leakage flow of the motor, respectively. V_ pl flow through the loading valve servo-valve displacement xsv n damping coefficient of the servo-valve q density of the fluid load pressure (=Ppp Pbp) DPl b bulk modulus of the fluid speed of the motor output shaft xm natural frequency of the servo-valve xsv C single port energy storage capacitive element in bondgraph model I single port energy storage inertial element in bondgraph model R single port energy dissipative element in bondgraph model SE single port element that indicates the source of effort in bondgraph model
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TF 0 1 •
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Two-port element in bondgraph model, that converts rotational speed to rate of flow and vice-versa common effort junction in bondgraph model common flow junction in bondgraph model it indicates the time derivative of the variable
1. Introduction In a hydrostatic transmission (HST) system the mechanical energy of the input drive shaft of the pump is converted to pressure energy in a nearly incompressible working fluid and then reconverted into mechanical energy at the output drive shaft of the motor. It is used to transmit rotating mechanical power from one source to another without the use of gears. It provides a much softer transmission of power than mechanical gear train. While considering the use of an HST system, the dynamic characteristics of the transmission must be considered whereas with the gear train such consideration is not often necessary. With the available pumps and motors, HST system can provide stepless variation of torque and speed which is one of the most important advantages. Much research work have been accomplished in the area of the performance investigation of HST system with various combinations of pumps and motors [1,2]. A systematic overview of the hydraulic circuits for the servo-hydraulic power drives has been presented by Murrenhoff [3] that may be useful for the designer to find out the best circuit for a given application. Zeiger and Akers [4] have made pioneering work in the area of swash-plate controlled variable displacement pumps and motors. They have developed comprehensive mathematical model for the average torque acting on the swash plate of the pump. Recently Manring and Luecke [5] have analysed a HST system consisting 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. Watton [6] has applied the method of characteristics to determine the transient response of a speed controlled system, which is dominated by the fluid inertia in long hydraulic lines. However, the effects of oil compressibility and load viscous friction are not included in the analysis. The simulation study of the influence of Coulomb friction on the low speed performance of the servo-valve controlled motor has been clarified by Ahmed Abo-Ismail and Wassef [7]. Recently Kim and Lee [8] have analysed the robust control strategy of a variable displacement hydraulic motor assuming a first order differential equation for the servo-valve; where the output torque is adjusted through motor displacement control. The quasi-static performance of an HST system using a low speed high torque motor has been analysed by Dasgupta [9] in recent past, where the torque–speed performance of the motor has been investigated by the displacement control of the pump. This article concerns with the steady-state and dynamic performance investigation of an openloop servo-valve controlled HST system using a fixed displacement axial piston type hydraulic motor with pump loading. Such a motor is usually robust in performance and has many industrial applications where low cost is often more important than operating efficiency. The studies conducted by Watton [10] related to the analysis of hydraulic servo-valve controlled motor system
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Fig. 1. The physical system.
were based on the linearised model that are suitable for the input of small amplitude. In such studies the dynamics of the servo-valve were not included. The present article is aimed at extending the work of Watton [10], that includes the dynamics of the servo-valve and the loading system. The modeling of the system has been performed using bondgraph simulation technique [11]. Using the values of the loss coefficients obtained from the steady-state analysis, the open-loop dynamic behaviour of the system is studied and the simulation results are validated through experiments. The effects of some critical parameters on the dynamic performance of the system have also been studied through simulation. Fig. 1 shows the physical system considered in the present study. A constant pressure source is maintained to the servo-valve which drives a fixed displacement axial piston motor that subsequently drives an identical pump in a loading circuit. The pump loading is controlled through a solenoid operated pressure relief valve.
2. Description of the model The modeling of the system has been performed through Bondgraph simulation technique [11] which is an effective tool for modeling and simulation of physical system. It facilitates the exchange, storage and dissipation of energy among interacting the physical elements efficiently. The bonds of the model portray the paths of the exchange of power within the constraint structure and atomic elements. It is to be noted that all bondgraphs including the present one are lumped element representation. In the development of the dynamic model of the system: • • • • •
A constant pressure source of supply to the inlet port of servo-valve is considered. Fluid inertia is neglected. Fluid properties have Newtonian characteristics. Resistive and capacitive effects are lumped wherever appropriate. Losses of the hydrostatic unit are based on the simple and general theory of positive displacement pumps and motors proposed by Wilson [12].
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Fig. 2. The bondgraph model of the system.
• All springs are assumed to be linear. • Outlet pressure is assumed to be atmospheric. • The detail dynamics of the multi-piston assembly, slipper pads etc. of the pump and the motor are not considered. The bondgraph model of the system shown in Fig. 2 is described in the following subsections. 2.1. Servo-valve with motor circuit From a constant pressure source (Psupl) represented by an SE element, the flow V_ 1 passes through the servo-valve to the inlet port of the motor. Similarly, the outlet flow from the motor V_ 2 returns to the sump (Ps) through the servo-valve. The flow through the servo-valve depends on its port areas, considered as resistances (R1 and R2) that are modulated through the current driving the valve. The 0 junctions representing the pressure Pmi and Pmo correspond to the inlet and outlet ports of the motor, respectively. The equivalent bulk stiffness of the fluid Kpm corresponds to the C elements take into account the compressibility of the fluid ðV_ mi ; V_ mo Þ in each half of the conduit. The external leakage flow V_ lkg of the motor is represented by the resistive elements R connected at the 0 junctions of the motor ports. Similarly, internal leakage flow V_ ilkg also occurs due to the pressure difference across the inlet and outlet ports of the motor which is taken into account by the resistance Ripl at 1 junction. A transformer (TF) element in the bondgraph representation of a hydraulic system transforms the hydraulic power to mechanical power or vice-versa. The TF elements connecting the 1 and 0 junctions represent the volume displacement rate of the motor (Dm). The 1 junction indicating the mechanical part of the load corresponds to the speed of the motor output shaft xm. The load inertia (Jl), speed dependent friction load (Rl) and the constant torque load (Tst) are represented by the I, R and SE elements, respectively. They are connected with the 1 junction that constitute the load dynamics.
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2.2. Loading circuit The flow supplied to the loading pump plenum Ppp (indicated at the 0 junction) is proportional to the speed of the motor (xm). The SE element represents the boost pressure supply (Pbp) provided to the loading circuit. The TF element connecting the 1 and 0 junctions represents the volume displacement rate of the pump (Dp). The relief valve in the loading circuit provides load to the pump. The flow through the valve port V_ pl depends on its port opening area and the pressure difference (Ppp Pbp) across it. The C element corresponds to the bulk stiffness of the fluid Kpp on the 0 junction takes into account the compressibility flow loss at the pump plenum.
3. Describing equations of the system The system equations derived from the bondgraph model are described as follows: 3.1. Servo-valve with motor circuit The resistances of the servo-valve ports (R1 and R2) are modulated depending on the current input to the servo-valve and the pressure difference across the ports. As these resistances are in conductive causalities, they are considered as modulated flow sources (V_ 1 and V_ 2 ) and expressed as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3:1Þ V_ 1 ¼ k f xsv ðP supl P mi Þ pffiffiffiffiffiffiffiffi V_ 2 ¼ k f xsv P mo ðconsidering P s ¼ 0Þ ð3:2Þ pffiffiffiffiffiffiffiffi In the above equations k f ¼ C d pd s 1=q is the valve flow constant and xsv is the spool displacement that depends on the current input to the servo-valve. The servo-valve is of rectangular ports, symmetrical spool design with critically lapped lands. On the basis of the manufacturerÕs specification and the measurement conducted by Watton [13], the valve dynamics is represented by an equivalent second order differential equation, €xsv þ 2nxnsv x_ sv þ x2nsv xsv ¼ x2nsv k sv isv
ð3:3Þ
In Eq. (3.3) nsv and xnsv are the damping coefficient and the undamped natural frequency of the valve, whereas ksv and isv are the gain and the current driving the servo-valve respectively. In Fig. 2, Pmi and Pmo depicted at the 0 junctions are the pressure at the inlet and outlet port of the motor. With the compressibility of the fluid, the flow balance in each port of the motor is given by V_ mi ¼ V_ 1 V_ lkg1 V_ ilkg Dm xm V_ mo ¼ V_ ilkg þ Dm xm V_ lkg2 V_ 2
ð3:4Þ ð3:5Þ
where V_ ilkg is the cross-port leakage flow of the motor, whereas V_ lkg1 and V_ lkg2 are the external leakage flow from the inlet and the outlet port of the motor, respectively.
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In Eqs. (3.4) and (3.5) the pressure at the inlet and the outlet port of the motor are expressed as P mi ¼ K pm V mi
and P mo ¼ K pm V mo ; 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 ðP mi P mo ÞDm T st xm Rl Dp ðP pp P bp Þ
ð3:6Þ
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 due to coulomb friction and stiction that mainly dominates at the low speed range of the motor, the fourth term is the torque loss due to viscous friction coefficient Rl and the fifth term is the torque applied on the motor shaft due to the differential pressure across the loading pump (Ppp Pbp). The motor speed is given by xm ¼ pl =J l
ð3:7Þ
3.2. Loading circuit Considering the flow continuity at the loading pump plenum, the flow returns to the sump through the relief valve port is given by V_ pl ¼ xm Dp V_ pp
ð3:8Þ
where V_ pp is the compressibility flow loss at the pump plenum. The load pressure is expressed as Ppp = KppVpp. In calculating the pressure at the respective 0 junction of the model, the effective bulk stiffness of the fluid is Kstf = b/V, where b is the bulk modulus of the fluid and V is the volume of the fluid column. 3.3. Steady-state equations of the system While formulating the steady-state equations of the system, the time derivative of the system equations derived from the model (Eqs. (3.4)–(3.8)) are made equal to zero. The compressibility losses of the fluid ðV_ mi ; V_ mo Þ and the load inertia effect on the output torque of the motor shaft ðp_ l Þ are neglected. The steady-state flow equations derived from Eqs. (3.4) and (3.5) are: Inlet flow to the motor V_ 1 ¼ Dm xm þ V_ ilkg þ V_ lkg1
ð3:9Þ
Outlet flow from the motor V_ 2 ¼ Dm xm þ V_ ilkg V_ lkg2
ð3:10Þ
Subtracting Eq. (3.10) from Eq. (3.9) the combined flow losses of the motor V_ 1 V_ 2 ¼ V_ lkg1 þ V_ lkg2 ¼ ðP mi þ P mo Þ=Rlkg P supl =Rlkg
ð3:11Þ
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Hence, the losses are linear in form, since the sum of the pressure (as very small pressure loses across the servo-valve ports) P mi þ P mo P supl
ð3:12Þ
The mean flow rate may be obtained from Eqs. (3.9) and (3.10) V_ m ¼ Dm xm þ ðP mi P mo Þ=Ripl þ ðP mi P mo Þ=2Rlkg
ð3:13Þ
where V_ m ¼ ðV_ 1 þ V_ 2 Þ=2. Therefore from Eq. (3.13) the cross-port leakage term f(Pmi, Pmo, xm) may be expressed as V_ ilkg ¼ V_ m ðP mi P mo Þ=2Rlkg Dm xm
ð3:14Þ
where V_ ilkg ¼ ðP mi P mo Þ=Ripl . Ignoring the inertia torque loss (i.e. p_ l ¼ 0) in Eq. (3.6), the load torque Tl may be expressed as T l ¼ ðP mi P mo ÞDm xm Rl T st
ð3:15Þ
where Tl = (Ppp Pbp)Dp (as Dp = Dm). Similarly ignoring the compressibility flow loss at the pump plenum (i.e. V_ pp ¼ 0Þ of the loading circuit, from Eq. (3.8), the flow through the relief valve is given by V_ pl ¼ xp Dp
ð3:16Þ
(as the motor and the pump are directly coupled together xm = xp). The load resistance Rpl is expressed as Rpl ¼
P pp P bp DP l ¼ V_ pl V_ pl
ð3:17Þ
4. Experimental investigation The values of the leakage coefficients, the viscous friction coefficients and the constant torque loss (Tst) of the motor were obtained experimentally from the steady-state performance of the practical system shown in Fig. 3. Commercially available axial piston pump, motor, servo-valve and other accessories were used in the experiment. The major items of the test set-up are as follows: Power pack (7.5 kW capacity) Servo-valve Pressure transducer Flow transducer Torque and speed sensor (capacity—0–50 N m and 0–5000 rpm) Relief valve Data acquisition system
Vickers System International/UK Dowty type—4551/UK Kistler piezo-resistive element Kracht/VC0.2 Sensor Technology Ltd./UK Oil gear/UK—No. 8 Canal system/UK
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Fig. 3. Experimental test set-up.
A stable source of supply from a power pack was set at 100 bar by adjusting the pressure relief valve (item no. 3). By controlling the relief valve of the loading circuit (item no. 14), the motor (item no. 5) was tested at different load torque (Tl). Using suitable sensors and instruments, the speed and the torque of the motor (xm and Tl), the supply pressure (Psupl), the inlet and the outlet pressures of the motor (Pmi and Pmo) the inlet and the outlet flow of the servo-valve (V_ 1 and V_ 2 ) and the load pressure DPl were measured. After setting the valve (item no. 14) at a particular pressure, by switching off/on the valve (item no. 12), the step changes in the load resistance Rpl of the loading circuit is made. The current input given to the servo-valve was controlled through suitable amplifier unit. The viscosity of the fluid was kept constant by maintaining the temperature of the oil constant at 40 ± 2 C. Experiment was repeated several times to examine repeatability before recording the data in PC through data acquisition system. 5. Results and discussion 5.1. Steady-state performance of the system The steady-state performance of the system was investigated at constant supply pressure Psupl = 100 bar. Figs. 4 and 5 show the flow and torque characteristics of the motor, where the
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Fig. 4. Pressure–flow characteristics of the motor.
Fig. 5. Pressure–torque characteristics of the motor.
symbols indicate the test data and the lines give the model fit. It appears that the flow differences are constant, and hence independent of speed and pressure differential. Therefore, from Eq. (3.11), the external flow losses V_ 1 V_ 2 ¼ P supl =Rlkg With sufficient accuracy, it therefore follows that Rlkg 1:5 1012 N s=m5
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From Fig. 4 it is also clear that the mean flow rate ðV_ m ¼ ðV_ 1 þ V_ 2 Þ=2Þ varies linearly with the pressure differential (Pmi Pmo); then it is assumed that the cross-port leakage flow V_ ilkg characteristics also has linear form as obtained from Eq. (3.14). From the characteristics shown in Fig. 4, the value of Ripl is estimated as Ripl ¼ 3:2 1012 N s=m5 From Fig. 5 the steady-state torque characteristics of the motor may be represented by the following equation: T l ¼ Dm ðP mi P mo Þ 0:01146xm 2:5
ð5:1Þ
where Tl is in N m and xm is in rad/s. Comparing Eqs. (3.15) and (5.1) the viscous friction coefficient (Rl) and the coulomb friction/ stiction load (Tst) on the motor shaft are obtained as Rl ¼ 0:01146 N m s=rad and T st ¼ 2:5 N m 5.2. Dynamic performance of the system Using the test set-up shown in Fig. 3, the dynamic performance of the system was investigated at medium flow rate passes through the loading valve (item no. 14). Considering the linear pressure–flow characteristics of the valve, the load resistance (Rpl) is expressed as Rpl ¼
P pp P bp Dp xm
ð5:2Þ
The additional data required are the load inertia (Jl), the motor and pump displacement flow rate (Dm and Dp), the servo-valve flow constant (kf) and the servo-valve gain ksv. These are obtained from the earlier study conducted by Watton [10] and given by
Fig. 6. Characteristics of current input given to servo-valve.
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J l ¼ 0:014 kg m2 ;
Dm ¼ Dp ¼ 1:68 106 m3 =rad;
K pm ¼ K pl ¼ 1 1014 N=m5
kf ¼
273
1:04 108 m3 =s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; xsv pressuredrop
and k sv ¼ 0:03 mm=mA
To validate the dynamic response of the test motor, the system Eqs. (3.3)–(3.8) are solved numerically using the software Symbols 2000 [14] and the simulation results are compared with their experimental counterpart under similar conditions. The time responses of the system are obtained with respect to the step changes in the load resistance Rpl and the square wave current input given to the servo-valve. Because of the limitations of the current driver in the experimental set-up, the actual current steps (3–8 mA) shown in Fig. 6 were not ideal. There were fluctuations of the current input over its steady-state values.
Fig. 7. (a) Comparison of the experimental and the simulation results of the pressure (Pmi) with respect to the step increase in the load resistance (Rpl). (b) Comparison of the experimental and the simulation results of the motor speed (xm) with respect to the step increase in the load resistance (Rpl).
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In Figs. 7 and 8, the comparisons are made between the present simulated response and the measured transient response of the system when the motor is initially running at a constant speed. The pressure and speed characteristics of the motor shown in Fig. 7 are obtained with respect to the step increase in the load resistance Rpl from the value of 7.5 · 109–6 · 1010 N s/m5; whereas, the characteristics shown in Fig. 8 are obtained with respect to the step decrease in the load resistance Rpl from the value of 5 · 1011–8 · 1010 N s/m5. In both the cases, it is found that the experimental and the simulated response are very similar; both showing approximately the same rise time of 50–100 ms. However, the Coulomb friction/stiction effect could not be realized as would be expected at much lower motor speed. Due to the resistive load applied to the motor through the loading circuit and the low inertial load connected to the motor output shaft, there is almost a negligible pressure overshoot observed in the experimental responses; however, there are fluctuations of the system pressure as well as the motor speed over their steady-state values.
Fig. 8. (a) Comparison of the experimental and the simulation results of the pressure (Pmi) with respect to the step decrease in the load resistance (Rpl). (b) Comparison of the experimental and the simulation results of the motor speed (xm) with respect to the step decrease in the load resistance (Rpl).
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The performance of the system is also investigated based on the square wave current input (Fig. 6) given to the servo-valve and at a load resistance Rpl = 5 · 109 N s/m5. These are shown in Fig. 9. The experimental results show that there are pressure and speed fluctuations over the steadystate values. The deviations between the experimental and the simulation results at the low speed operation may be due to the non-linear pressure–flow characteristics of the resistance Rpl of the loading valve at low flow range. This aspect is not considered while modeling the system. The minor fluctuations of the input current given to the servo-valve over its steady-state value (Fig. 6) also cause the fluctuations of the systemÕs response. However, the similarity between the nature of the predicted and experimental responses of the system match with each other validates the model.
Fig. 9. (a) Comparison of the experimental and the simulation results of the pressure (Pmi) with respect to the square wave current input to the servo-valve. (b) Comparison of the experimental and the simulation results of the motor speed (xm) with respect to the square wave current input to the servo-valve.
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Fig. 10. Effect of the cross-port leakage resistance on the transient response of the motor speed (xm).
Fig. 11. (a) Effect of the variation of the bulk stiffness of the fluid on the transient response of the motor speed (xm). (b) Effect of the variation of the bulk stiffness of the fluid on the transient response of the system pressure (Pmi).
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5.3. Perturbations of some parameters on system’s performance Using the validated model of the system, the simulation is also carried out to investigate the effects of some parameters on the dynamic performance of the system. 5.3.1. Effect of cross-port leakage resistance (Rilkg) The effect of the cross-port leakage resistance on the dynamic performance of the system is shown in Fig. 10. It is observed that the motor performance deteriorates with the decrease in Ripl, that is with the increase in the clearance between the port plate and the piston–cylinder assembly as expected. Similar results would have been obtained with the variation of Rlkg. It also shows that the motor leakage has little effect on the systemÕs damping characteristics at high rotational speed.
Fig. 12. (a) Effect of the variation of the load inertia on the transient response of the system pressure (Pmi). (b) Effect of the variation of the load inertia on the transient response of the motor speed (xm).
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5.3.2. Effect of bulk stiffness of the fluid (Kstf) Sometimes the hydraulic lines in the experimental set-up are quite flexible. In such case, the effective bulk stiffness of the fluid in the line reduces. Fig. 11 indicates the transient responses of the pressure and the motor speed due to the variation of the effective bulk stiffness of the fluid between the motor and the pump plenum. The maximum overshoot of the systemÕs response increases with the increase in the ratio of the bulk stiffness of the fluid. It shows that with the increase in the ratio of Kpm and Kpp, the setting time of the systemÕs response increases, however, the delay time of the response decreases. It is consistent with the findings of Dasgupta et al. [15] that the dominant parameters of the systemÕs performance are the leakage of the system, the compressibility of the fluid and the applied load characteristics. 5.3.3. Effect of load inertia (Jl) The effect of load inertia on the dynamics of the system are shown in Fig. 12. As the inertia increases, the speed and the pressure responses become more oscillatory with an increase in the magnitude of the maximum overshoot over the steady-state operating speed of 50 rad/s. The
Fig. 13. (a) Effect of the variation of the load resistance on the transient response of the system pressure (Pmi). (b) Effect of the variation of the load resistance on the transient response of the motor speed (xm).
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speed characteristics presented in Fig. 12b indicate that the delay time, defined at 50% of the final steady-state speed, increased when the load inertia decreases from Jl = 0.003 kg m2 to 0.001 kg m2. The corresponding maximum overshoot of the speed at the higher inertia is about 40%. It is also clear that with the increase in the load inertia the settling time of the system response increases. 5.3.4. Effect of load resistance (Rpl) With the increase in the load resistance (Rpl) controlled through the relief valve of the loading circuit, the steady-state system pressure increases and the speed decreases. As it is a resistive type load (Rpl), with the increase in the load resistance Rpl, it provides more damping effect and therefore, the overshoot of the speed and the pressure decrease. Fig. 13a and b shows the transient responses of the system pressure (Pmi) and the motor speed (xm) due to the variation of the load resistance Rpl. 5.3.5. Influence of valve dead band (vdb) The influence of valve dead band (vdb) on the system response is exhibited in Fig. 14. While modeling, it is considered as the displacement of the valve spindle (xsv) till the port opens that is equivalent to the current input (isv) to the servo-valve. The results are obtained with the dead band of vdb = 0.001 A and, without dead band, vdb = 0.0 A. Comparing both the results, which are obtained at an operating speed of about 85 rad/s, it indicates that the valve dead band has a delay characteristics and it also has the influence on the maximum overshoot of the response. With the increase in the valve dead band from 0.0 to 0.001 A, the starting time (ts), the delay time (td) and the rise time (tr) increases from 0.01 to 0.03 s, 0.04 s to 0.052 s and 0.06 s to 0.07 s respectively. 5.3.6. Effect of combined ramp and step input Many applications require the servo-drive to accelerate a load to a constant velocity such as tape drive and cutting tools. Fig. 15 shows the speed characteristics of the motor with respect to the ramp current input given to the servo-valve.
Fig. 14. Effect of the valve dead band on the transient response of the motor speed (xm).
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Fig. 15. The speed response with respect to the ramp input to servo-valve.
With the increase in the current input to the servo-valve, the response tends to become oscillatory. To satisfy the desired performance specifications, proper proportional–integral control action may be incorporated for such a drive.
6. Conclusion This paper presents a significant aspect of an integrated study of modeling, analysis and simulation of the dynamics of a servo-valve controlled axial piston motor with pump loading. The results presented here are the estimation of some parameters, which significantly influence the response of the motor and the behaviour of the plant with respect to various types load input. The close agreement between the experimental and the simulation results validates the proposed model. It is shown that the simulated response from a model when combined with the experimental observations reduces effective mode of data handling for meaningful estimation of the system parameters like Ripl and Rlkg in terms of dependencies on the operating conditions. Therefore, the proposed model can be used to predict the performance and to provide insights for improving the design of the system. Justifiably, many plausible parameters like inertia of the working fluid in the conduit, elasticity of the conduit etc. are not incorporated in the details of the modeling. Still, the model contains several critical parameters those substantially influence the performance of the system within the range of operation considered. The discrepancies found while comparing the theoretical and the test results of the motor are mainly due to the dynamics of the multi-piston machine, fluctuation of the current input to the servo-valve over its steady-state value, dead band of the servo-valve and the instrumental error as well. Consideration of the effects ignored in the study requires further refinement of the model and more rigorous experimentation with better data acquisition system. Due to the limitations of the test-rig, the experimental results are obtained within a limited range. However, it is envisaged that the model developed may be useful for wider range of parameter variation. To ascertain this, further experiments may be undertaken.
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Based on the validated model, the dynamic response of the system has been evaluated with respect to the changing parameters of the system. The following conclusions may be derived from the simulation results: (a) The systemÕs response can be improved by controlling leakage resistances (Ripl and Rlkg). (b) With the increase in the effective bulk stiffness of the fluid, the peak rise pressure increases, however, the delay time decreases. (c) With the increase in the inertia load, the speed and pressure responses become more oscillatory with the considerable increase in the maximum overshoot. (d) 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. (e) The valve dead band has a delay characteristics and it also increases the maximum overshoot of the response. (f) With the combined ramp and step input given to the servo-valve, the response becomes more oscillatory with the increase in the speed. To achieve the desired performance of the system with respect to the various loading conditions, close-loop control system may be designed with suitable controller like P, PI and PID; that may be further scope of studies. Further refinement of the model may be made incorporating more detail analysis of the servo-valve as proposed by Anderson and Li [16].
Acknowledgments The first author gratefully acknowledges the fellowship and facilities provided to him by the Indian National Science Academy, India, and the Royal Society, UK, and kind cooperation by the University staff during his 3 months work in the Cardiff School of Engineering, UK. The research and Development Project Grant for 2003–2006 from MHRD, Government of India, for carrying out the research work on this topic is also acknowledged. References [1] H.E. Merritt, Hydraulic Control Systems, Wiley, New York, 1967. [2] J.U. Thoma, Mathematical models and effective performance of hydrostatic machines and transmissions, Hydraulic Pneumatic Power November (1969) 642–651. [3] H. Murrenhoff, Systematic approach to the control of hydrostatic drives, Journal of Systems and Control, ImechE, Part-I 213 (15) (1969) 333–347. [4] G. Zeiger, A. Akers, Dynamic analysis of an axial piston pump swashplate control, Journal of Systems and Control, ImechE, Part-I 200 (C1) (1986) 49–58. [5] N.D. Manring, G.R. Luecke, Modelling and designing a hydrostatic transmission with a fixed-displacement motor, ASME Journal of Dynamic Systems, Measurement, and Control 120 (1998) 45–49. [6] J. Watton, The stability of electro-hydraulic servomotor systems with transmission lines and non-linear motor friction effects. Part A: System modeling, The Journal of Fluid Control Including Fluidics Quarterly 16 (2–3) (1986) 118–136.
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