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.
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
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.
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:
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:
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.