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Hydraulic Servo 

 

Pump Displacement Position Controller for a Hydrostatic Transmission

 

Exerpts from Chapter 9 of User’s Manual – Hemmer Example 13-PD    

 

Introduction: The pump displacement or swashplate position control is a hydraulic position control servo driving a load consisting of an inertia and a spring located between the inertia and the frame. The motor is a hydraulic cylinder controlled by a two stage servo valve. The linear motion of the motor is converted to angular motion at the load by the linkage shown in Figure 9-1. The conversion ratio of the linkage is radius, R, in./ rad. or lb.in./ lb. There are actually two sets of cylinders, links, and springs but only one set is shown to simplify the illustration of the principle of operation. Block diagrams are shown in Figure 9-2. 

                                       

 Appendix A contains an analysis where typical, conventional servo compensation is used. The amplifier gain, loop gain, and stiffness are less than that of the Hemmer design, described herein, by a factor of 19.6, the static error is greater by a factor of 18.4, and the bandwidth is less by a factor of  22.6.   

 

This servo is used in the hydrostatic transmission of Example 13 which is used in another Hemmer compensated servo to control the tension of a winch. When conventional servo compensation is used, the peak error due to a change in speed exceeds that of the Hemmer compensated system by a factor of 43 when each system is subjected to the same disturbance. When the disturbance is slowed down by a factor of 40, the system with conventional compensation has the same peak error as that of the Hemmer system. The advantage of  using Hemmer compensation, where many of the drive train lags are cancelled by leads in the servoamplifier, is to tolerate much larger or faster disturbances; The servo is much more robust.

 

 

Figure 9-1 – Example of a drive containing a spring between the load inertia and a fixed point.

 

Gx = Kt Ga Gv            Gu1 = Gu/(GiGx) 

G = Gx Gpt / Kg          C = (R - Gu1 Tu) Gcl

Gcl = Gi G Go /(1+GH)    If Gi = H/Go, Gcl approximates 1 when GH >>1.

 

 

Figure 9-2 – Block Diagrams: Position Control Servo for Case 2, spring between the load Inertia and a fixed or moving point and where angular load position is fed back.  It conforms to Examples 1 and PD-13 where the outboard end of the spring is fixed. 

 

 

           TABLE 9-1: HEMMER PROGRAM DT: Drive Train Analysis

 

        DATA FILE: V-DT-13PD.TXT              TIME: 08:17:10              DATE: 02-07-2005

 

        The load is driven by a hydraulic cylinder or rectilinear hydraulic

        motor controlled by a 2 stage servovalve.

 

            TABLE 9-1-1: CHARACTERISTICS OF SERVOVALVE, MOTOR, FLUID, AND PLUMBING

 

        SYMBOL  VALUE         UNITS                            DESCRIPTION

 

         I.D.   0.5           inches                   inside diameter of high pressure oil line

         L      8             inches                   length of high pressure oil line

         SumL   180           -----                    sum of fitting L/D for above

 

         S.G.   0.87          -----                    specific gravity of oil

         VIS    38            centistokes              kinematic viscosity of oil

         B      250000        psi                      bulk modulus of elasticity of oil

         Td     0.000289      sec.                     sonic time delay in oil

         Wd     302.          rad./sec.                W where Td causes -5 deg shift

 

         V      54.5          cu.in.                   supply side oil volume

         R      196.4         psi/(cu.in./sec)         supply side linear resistance

         C      0.000218      cu.in./psi               supply side capacitance

         Te     0.0428        sec.                     time constant, RC

 

         Ps     200           psid                     supply to return pressure drop

         P1     200           psid                     servovalve dP at Qsat flow

         Isat   50            ma                       servovalve max. current

         Qsat   12.5          cu.in./sec.              servovalve max. flow at P1 & Isat or Esat

         Ilin   30            ma                     servovalve max. current for linear operation

         Qlin   10            cu.in./sec.              servovalve flow at P1 & Ilin or Elin

         Kpp    130.95        psi/ma                   servovalve pressure gain

         Kqq    0.3333        (cu.in./sec)/ma          servovalve flow gain

 

         A      8.92          sq.in.                   area of motor

         Ntm    0.9           -----                    motor torque or force efficiency

         Ntmo   0.9           -----                    same as above at zero speed

         Nvm    0.99          -----                    volumetric efficiency of motor

         Mm     0.0399        lb.sec.^2/in.            motor inertia

         Dm     28415.12      lb./in./sec              motor internal damping factor

         KvKt   1051.23       lb./ma                   servovalve & motor gain, dF/di

         Kv     0.0333        (cips/psi^.5)/ma         servovalve gain, dg/di

         Kt     31536.9600    lb./(cips/psi^.5)        motor gain, dF/dg

         Tna    0.03          sec.                     servovalve time constant

         Tnb    0.0037        sec.                     servovalve time constant

         Kp     10.00         psi^0.5                  servovalve pressure constant

         RL     19.64         psi^0.5 / cips           servovalve leakage resistance

         QL     0.72          cu.in./sec.              servovalve internal leakage at P1

         Glim   1.250         cips /(psi^0.5)          servovalve conductance limit

 

 

        NOTE: Kv is the static gain of the servovalve from the electric

              signal input to the valve conductance and Kt is the static

              gain from the above conductance to force or torque exerted

              by the motor at stall.

 

 

                      TABLE 9-1-2: GENERAL CHARACTERISTICS

 

        SYMBOL  VALUE         UNITS                            DESCRIPTION

 

         JL     0.843         lb.in.sec^2              load inertia or mass

         KL     4888.4        lb.in./rad.              spring rate

         Kgb    5.58          in./rad.                 gear ratio of converter (b)

         Ngb    1             -----                    efficiency of converter (b)

         Kg     5.58          in./rad.                 overall conversion ratio

         Ng     1.000         -----                    overall conversion efficiency

 

         Vss    0             rad./sec.                steady state load speed 

         Plim   0.314         rad.                     +/- load travel limit

         Ylim   1.752         in.                      +/- load travel limit

         Qss    0.0           cu.in./sec.              steady state flow rate

         Plss   0.0           psid                     steady state line loss

         TLmax  8959.2        lb.in.                   max. load at 200 psi

         Qmax   12.5          cu.in./sec.              flow rate at no load

         PLmax  7.6           psid                     line loss at no load

         Vmax   0.249         rad./sec.                no load speed at load shaft

 

         Vs     0.244         rad./sec.                <= load slew rate

         Acl    769.9         rad./sec./sec.           => cont. acceleration limit

 

 

                  TABLE 9-1-3: TRANSFER FUNCTIONS

 

        Case 2: Spring between the load inertia and a fixed or movable point.

 

        TL = JL x A + DL x V + KL x P + TLs  (reference)

 

        Gu = Ku ( Te S + 1 )

 

        Where: Ku = 0.1792 lb.in./lb.in.

               Te = 0.0428 sec.

 

        Gb = 1/Kgb

 

        Where: 1/Kgb = 0.179 rad./in.

 

                               Kpt

        Gpt = ---------------------------------------------

               ( T1 S + 1 ) ( T5^2 S^2 + 2 Z5 T5 S + 1 )

 

        Where:    K = 157.000 lb./in.

                  D = 28415.120 lb./(in./sec.)

                  M = 0.067 lb./(in./sec./sec.)

 

                Kpt = 1/K = 0.0063694 in./lb.

 

                 T1 = 181.031400 sec.

                 T5 = 0.000318 sec.

                 Z5 = 0.0037

 

        Guf = Kuf S ( Tuf^2 S^2 + 2 Zuf Tuf S + 1 )

 

        Where: Kuf = 158556.4 lb.in./(in./sec.)

               Tuf = 0.0003 sec.

               Zuf = 0.0037

 

 

       TABLE 9-2: HEMMER PROGRAM FR: Frequency Response

 

     DATA FILE: V-FR-PD-13.TXT         TIME: 10:33:12       DATE: 02-07-2005

 

     The loop gain is 477275.00 sec.^0

     The phase margin is 46.81 deg.

     The unity gain crossover frequency is 2638.26 rad./sec.

 

     The resonant peak for the closed loop system is 2.497 or

     7.95 db. This occurs at a resonant frequency of 3162.28 rad./sec.

                    

 

GH and A are the magnitude & phase shift, respectively, for the open  loop.  M and B are the magnitude & phase shift, respectively, for the closed loop.  Mu is the magnitude of the disturbance response. 

 

              Figure 9-3: Open & Closed Loop Frequency Responses   for

                                 Frequencies from 100 to 100,000  rad./sec.

 

 

      TABLE 9-5:  HEMMER PROGRAM SE: Servo Error Analysis

 

     DATA FILE: V-SE-PD-13.TXT              TIME: 10:42:54              DATE: 02-09-2005

 

     This is a 1 loop, Type 0 servo with a # 4 drive that controls position.

    There is a spring between the final inertia & a fixed or movable point.

 

      The steady state error, excluding the feedback transducer error,

    is 0.001195 deg. or .0066 % of the 18 deg. range.

      This includes a proportional error of 0.000038 deg. at command,

    r = 18 deg..

 

    The following error is infinite.

 

    The output stiffness is 40,721,040 lb.in./deg..

Fig. 9-4 :  Transient Responses to a Step Command  where Slewing does not Occur   

          

     The maximum position is 1.592 deg., occurring at 0.0006 + delay = .00089

     sec. The peak overshoot is 59.19 percent. The sonic time delay is

     .00029 sec., as shown.

 

  Figure 9-5: Transient Responses to a Step Command where Slewing  Occurs

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Exerpts from Chapter 11 of the User’s Manual:

 

 

 Figure 11-17: Servoamplifier for Example PD-13

 

    The servoamplifier for Example PD-13 has the following transfer function:

 

              Ka ( Ta S + 1 ) ( Tb S +1 ) ( Tg 2 S 2 + 2 Zg Tg S +1 )

Ga =  ----------------------------------------------------------------------- 

                                      ( 1E-06 S + 1 ) 4

 

where: Ka =  13,299 ma / vdc 

 

 Ta = Tnb  = .03 sec.    

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    

 Tb = Tnb  = .0037 sec

                         

 Tg =  .00032 sec.        

 Zg =  2 x .0037 = .0074             

 

The ( Tg 2 S 2 + 2 Zg Tg S +1 ) double lead is handled by a notch filter plus two single leads, where they all have the same time constant. 

 

 

The Hemmer Engineering Corp. will bid, on requst, for the design and fabrication of  analog servoamplifiers, such as the schematic of Figure 11-17 above and its  3” x 5” printed circuit shown below.

 

 

Figure 11-23: Printed Circuit for the Servoamplifier for the Pump Displacement Controller for Example PD13.