Engineering Design Handbook - Servomechanisms, Sections 1-4:

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AMC PAMPHLET DARCOM-P 706-136
AMC PAMPHLET DARCOM-P 706-137
AMC PAMPHLET DARCOM-P 706-138
AMC PAMPHLET DARCOM-P 706-139
ENGINEERING DESIGN HANDBOOK
SERVOMECHANISMS
SECTION 1, THEORY
SECTION 2, MEASUREMENT AND SIGNAL CONVERTERS
SECTION 3, AMPLIFICATION
SECTION 4, POWER ELEMENTS AND SYSTEM DESIGN
HEADQUARTERS, U.S. ARMY MATERIEL COMMAND APRIL 1965
HEADQUARTERS, U.S. ARMY MATERIEL COMMAND AUGUST 1963
AMCP 706-136
AMC PAMPHLET
THIS IS A REPRINT WITHOUT CHANGE OF ORDP 20-136, REDESIGNATED AMCP 706-136
ENGINEERING DESIGN HANDBOOK
SECTION 1, THEORY
HEADQUARTERS, U. S. ARMY MATERIEL COMMAND APRIL 1965
HEADQUARTERS
UNITED STATES ARMY MATERIEL COMMAND
WASHINGTON, D. C. 20315
30 April 1965
AMCP 706- 136, Servomechanisms, Section 1, Theory, forming
part of the Army Materiel Command Engineering Design Handbook
Series, is published for the information and guidance of all concern-
ed.
( AMC RD)
FOR THE COMMANDER:
SELWYN D. SMITH, JR.
Major General, USA
Chief of Staff
OFFICIAL:
A-
STAN EY J. SAWTCKI
Colonel, GS
Chief, Administrative Office
DIS T RIB U TION : Sp e cia1
LIST OF ILLUSTRATIONS
. .
Fig No Title Page
1-1 Elements of a feedback control system .................................. 1-1
1-2 Elements of an open-loop control system ................................ 1-2
1-3 General diagram of a feedback control system ...................... 1-4
1-4 Unity-feedback system main loop ............................................ 1 4
1-6 Basic unity-feedback system .................................................... 1-4
2-1 Speed control of shunt d cm otor .............................................. 2-3
2-2 Transient input functiong .......................................................... 2-6
2-3 Transient responses for system
-d+3x2 -+2d-2+xx = dx [email protected].. .......................................... 2-6
dts dt’ dt dt’
2-4 Straight-line crossing course .................................................... 2-8
3-1 Exponential functions e5 and 1- e+ 3-4
3-2 Cubic chart ................................................ ..................... 3-6
3-3 Cubic chart .................................................................................. 3-7
3-4 Quartic chart .............................................................................. 3-8
3-5 Sketch of the quartic chart .................... ...................... 3-11
3-6 Block diagram manipulation and reduction “rules” .............. 3-19
3-7 Block diagram examples ........ .................................. 3-22
3-8 Signal-flow graph in three variables ....................... 3-25
3-9 Signal-flow graph of order one ................................................ 3-26
3-10 Signal-flow graph of order two ................................................ 3-26
3-1 1 Signal-flow graph showing addition of parallel branches .... 3-26
3-12 Signal-flow graph showing multiplication of cascaded
branches .............................. ................................................... 3-27
3-13 Signal-flow graph showing termination shifted one node
forward .................................................................... 3-27
3-14 Signal-flow graph showing origin shifted one node back-
ward ........................ ........ ........................................ 3-27
3-16 Signal-flow graph showing elimination of a self-loop ............ 3-28
3-16 Signal-flow graph showing reduction of second-order graph 3-28
X
LIST OF ILLUSTRATIONS (cont)
Fig. No. Title Page
3-17 Geometry of general trapezoid for approximating Re
[F(jo)] ...................................................... .......... 3-29
+ + +
3-18 Real-part function for W (8) = 1.4s 0.14/(93 s2 1.4s
+
0.14) ...................................................... 3-31
3-19 Trapezoidal approximation for Re[W(jw)3 ...__.............___3_-.3_2_ _
+
3-20 Impulse response from Floyd's method W(s)= 1.4s 0.14/
+ + +
(93 8 ' 1.49 0.14) ............................................................ 3-32
3-21 Gain-phase loci of constant real part of W = G/(1+ G) 3-33
3-22 Elements of analog computers ..._____..____._.___...__._ _3_-_4_0 ______
3-23 Computer diagram for system of Fig. 3-7A .......................... 3-41
4-1 Single-loop system - block diagram ...................... 4 1
4-2 Locus of s for the Nyquist criterion ........................................ 4-5
+
4-3 LOCUSof [I(I 1)2]-1 for I = jp .......... ............ 4-5
+
4-4 Distortion of locus of [I(I 1)2]-1 ........................................ 4-6
Locus of-(-)1 l+s ..
4-6 . . ..... .. .. .. . . .. .... .. . ... .. ...... ... .... .................. 4-6
s 1-9
('+7
4-6 Distortion of locus of- - __, ,, . . _.. , _.. ..___.., _ __.._ .._._._._. ._ .. .. 4-6
8 1-9
4-7 Root-loci plots ____ .................................................................. 4-10
+ +
4-8 Poles for G(s) = 300K/[s(s lO).(s 301 _._.
49 Asymptotes and real axis behavior for G(s) = 300K/
+ +
[s(s 10) (s 30)] ._._.. ...................................................... 4-15
4-10 Construction for determining imaginary axis crossing G (8)
+ + < <
= 3OOK/[s(s 10) (s 30)] when 0 K ................ 416
00
+ +
4 11 Complete root locus for G(s) = 300K/[s(s 10) (s 3011
<
when O< K 00 ........................................................................ 416
6-1 Single-loop unity-feedback system .......................................... 62
6-2 Bandwidth measures from magnitude of closed-loop fre-
quency response C (j w) /R( j o) .. .. . .. . . .. .. . . .. . .. . . . . . . . .. . . . . . . .. .. . . . . . . . . . 6-2
6-3 Bandwidth measure from phase of closed-loop frequency
response C (jw ) /R( j w ) . .. .. . . .. . .. . . .. .. . . .. . . . .. . . .. . . . . . .. . . . . .. . . .. . .. .. . . .. . . . . . 6-2
6-4 Bandwidth measure from magnitude of error-to-input fre-
quency response E (j o)/ R( j o) .. . . .. . . . . .. . . .. . . . .. . . . . . .. . . . . . . . . . . . . . . . . . . . . 6-3
6-6 Bandwidth measures from open-loop frequency response .... 6 3
xi
LIST OF ILLUSTRATIONS (cont)
Fig. No. Title Page
+
5-6 Direct and inverse polar plots of G (jw) = {jw[ (jw) 0.6
+
jw 1]}4 ......................................... .,_....... _...._.... ._._.._...
,
5-7 Asymptotes and true magnitude curves for the first-order
+
factor (Tjo 1) +1 ............_... ................................................... 5-5
5-8 Asymptotes and true curves for the second-order factor
[(j-)0'+ 2Q*-++]0- ' .......................................................... 5-6
On on
+
5-9 Phase-angle curves for the first-order factor (Tjw 1)* 1 5-7
5-10 Phase-angle curves for the second-order factor
[ ( j- 0 )' + 2 -0 + 13 ............................................................ 5-8
W" On +
c0.2 jw 13
5-11 Magnitude plots for G(jw) = K [ (
jw i ;)+' 0.6 i - + 11 5-9
10
+
(0.2 jo 1)
5-12 Angle plots for G (iw) = K [ ( 5-9
+
iw j $)2+ 0.6 j 10 11
+
(0.2 jw 1)
5-13 Gain-phase plot of G(iw) = 6.5 5-10
[( :)*+
(jw) i 0.6i - +1]
10
5-14 Closed-loop response construction on the G plane . . . . . .. . . . . . . . .. 5-11
5-15 Closed-loop response construction on the G-1 plane . . . . . .. . . . . .. . 5-11
5-16 Contours of constant M in the G plane .................................... 5-13
5-17 Contours of constant phase in the G plane .............................. 5-13
5-18 Contours of constant M and constant cp in the G-1 plane ...... 5-13
5-19 Chart showing symmetry of M-N contours about phase of
180 degrees (Nichols Chart) ... _................. .,_.. ....................... 5-13
5-20 Nichols chart . .._....._._. ,. . ...... __.............__. __.,_. .... ....,_.. ....,,_._..... .. . 5-14
5-21 Nonuni t y -f eed back system . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .. . 5- 15
5-22 System equivalent to the system of Fig. 5-21 .......................... 5-15
5-23 Construction for gain determination on direct (G/K) plane 5-16
5-24 Construction for gain determination on inverse (KG-1)
plane . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . ...... ........ ..... 5-16
5-25 Direct-plane determination of K for Mp = 1.6, G(iw) =
+
K {iw [ (jw)'+ 0.6j0 1]}-1 ._.. ............................................. 5-17
5-26 Inverse-plane determination of K for MI = 1.6, G(iw) =
+ +
K { j ~ [ ( j ~ ) '0 .6i0 1]}-' .................................................. 5-18
5-27 Construction for gain determination on gain-phase plane . . . . 5-19
xii
LIST OF ILLUSTRATIONS (cont)
Fig. No. Title Page
5-28 Gain-phase plane determination of K for MI = 1.5, G(jo)
+
= K (0.2jo 1) ......................................... 5-19
+ + ]
i.[( j;)* 0.6j; 1
5-29 Phase margin and gain margin ................................................ 6-20
5-30 Approximate closed-loop magnitude response of unity-feed-
+
back system, G (jo) = K [jo(jo 1) 3-1 ................................ 5-22
5-31 Loci of characteristic parameters of second-order root ...... 5-24
+
5-32 Gain determination from root locus, G(s) = K (0.28 1)/
+ +
[s(s 1) (0.18 l)] .............................................................. 5-25
6-1 Compensation in a single loop .................................................. 6-1
6 2 Change in open-loop response produced by compensation
illustrating downward motion of MI contour for gain in-
crease ................................................ ................................ 6-2
6-3 Universal lag functions ....................................................
6-4 Lagcompensation procedure .................................................... 6-4
6-5 Universal lead functions .................... ... .... 6-4
6-6 Leadcompensation procedure .................................................. 6-5
6-7 Magnitude curves for lagcompensation procedure employ-
ing phase margin ........................................................................ 6-8
6-8 Phase curves for lagcompensation procedure employing
phaqe margin .................................................................... 6-8
6 9 Magnitude curves for leadcompensation procedure employ-
ing phase margin ............................ ............................ 6-9
6-10 Phase curves for leadcompensation procedure employing
phase margin ........................................ .............................. 6-9
611 General feedback-compensation configuration ...................... 6-11
6-12 Cascade equivalent of feedbalk-compensation configuration 6-11
6-13 Open-minor-loop asymptote for feedback compensation pro-
cedure .......................................................................................... 6-13
614 Closed-minor-loop magnitude for feedback compensation
procedure ............ ................................................................ 6-14
6-15 Closed-minor-loop phase angle for feedback compensation
procedure ................................................................. ........ 6-14
616 D-C compensation networks ...................................................... 6-17
617 Equivalent circuit elements for carrier-frequency networks 6-18
xiii
LIST OF ILLUSTRATIONS (cont)
Fig. No. Title Page
618 Parallel-T notch filter ................................................ 6-19
619 Inertia damper ........................................................................ 6-20
6-20 Block diagram of inertia damper (motor damping neg-
iigible) ............................................................................ 6-20
6-21 Hydro-mechanical compensation network .............................. 6-20
6-22 Schematic diagram of a pneumatic controller ...... 621
623 Controller block diagram .......................................................... 6-21
6-24 Schematic diagram of a proportional controller .................... 6-22
6-26 Schematic diagram of a proportional plus integral con-
.................................................................................... 6-22
6-26 Schematic diagram of proportional plus derivative con-
troller ............................................................................. 6-23
7-1 Normalized curves yielding time for 10-percent transient
response corresponding to combinations of various time
constants .......... ...................................................... 7-2
7-2 Normalized curves yielding time for 40-percent transient
response corresponding to combinations of various time
constants .................................... ..................................... 7-2
7-3 Normalized curves yielding time for 80-percent transient
response corresponding to combinations of various time
constants ..................................... ............................................ 7-3
7-4 Normalized curves yielding time-interval ratios of the tran-
sient response corresponding to combinations of various time
constants ............................................ .................................. 7-3
7-6 Normalized curves yielding the time- interval between 10-
and 80-percent response of the transient corresponding to
combinations of various time constants ................. ...... 7-4
7-6 Rectangular approximation to step response .......................... 7-6
7-7 Rectangular approximation to impulse response .................... 7-6
7-8 Triangular approximation to time function ............................ 7-6
7-9 Triangle function .............................................. ...................... 7-6
7-10 Partial derivatives for Linvill's procedure ................... 7-7
7-1 1 Dimensionless transient error-response curves of a second-
order servomechanism to a unit-ramp input ............................ 7-17
7-12 Transient error-response curves of a second-order servo-
mechanism to a unilistep input ................................................ 7-18
xiv
LIST OF ILLUSTRATIONS (ant)
Fig. No. Title Page
7-13 Transient output-response curves of a second-order servo-
mechanism to a unibstep input .... ................ 7-18
7-14 Peak magnitude M, versus damping ratio ( for a second-
order servomechanism .............................................................. 7-18
7-15 Overshoot variation with ( .......... ................ 7-19
7-16 Typical open-loop asymptote function .................................... 7-21
7-17 Sketches showing nomenclature used to describe various
characteristics of servomechanism performance .................... 7-22
7-18 Comparison of steady-state frequency response character-
istics and transient response following a step function of
input as a function of ol/w, .................................... 7-24
7-19 Sketch of open-loop asymptote function .................................. 7-42
8-1 Configuration for ISE minimization ............................ 8-3
8-2 Step-function responses of the optimum unit-numerator
transfer systems, second to eighth orders ..... .......... 8-4
8-3 Frequency responses of the optimum unit-numerator trans-
fer systems ....................................... .......................... 8-5
8-4 Step-f unction responses of the optimum zero-velocity-error
systems, second to sixth orders ................................................ 8-6
8-6 Configuration for MSE minimization ..... .................... 8-6
8-6 Configuration for MSE minimization ...................................... 8-8
9-1 Sampleddata system ................................................ . 9-1
9-2 Model of sampleddata system .................................................. 9-2
9-3 Train of unit impulses which represents the carrier A (t) .... 9-3
9-4 Action of sampledclamper ................................................ 9-3
9-5 Simplified picture of sampleddata system .............................. 9-3
9-6 Operation of sampling switch ................................ 9-3
9-7 Relations between s plane and z plane .................................... 9-6
9-8 Sampling a smoothed sampled signal ......................... 9-6
9-9 Sampling a filtered continuous signal ................ .............. 9-6
9-10 A sampleddata feedback system .................................... 9-6
9-11 Re.lati.ons between s and G*(s lication of Nyquist
criterion .................................... ................................. 9-7
9-12 Comparison between discrete and continuous compensation 9-9
LIST OF ILLUSTRATIONS (cont)
. .
Fig No Title Page
9-13 Sampled-time function .............................................................. 9-11
9-14 Step response of sampleddata system ...................................... 9-11
9-15 Determination of c (t) between sampling instants by samp-
ling at nn rad/sec ...................................................................... 9-12
9-16 Determination of c (t) between sampling instants through
the use of an artificial delay ...................................................... 9-12
10-1 Nonlinear feedback control system .......................................... 10-2
10-2 Dimensionless representation of contactor characteristics .... 10-2
10-3 Plot of the describing function N .... ....................... 10-3
10-4 Nonlinear characteristics with dead zone and saturation ...... 10-4
10-5 Describing function for dead zone .......................................... 10-4
10-6 Describing function for-s aturation .......................................... 10-4
10-7 Describing function for saturation and dead zone ................ 10-4
10-8 Hysteresis nonlinearity ............................................................. 10-5
10-9 Describing function for hysteresis-type nonlinear element .... 10-5
10-10 Simplified nonlinear system ...................................................... 10-5
10-11 Stability determination with describing function .................. 10-5
10-12 Contactor servomechanism study ............................................ 10-7
10-13 Contactor servomechanism ........................................................ 10-7
10-14 Degree of stability variation with input amplitude for con-
tactor servomechanism ............................................................ 10-7
10-15 Phase portrait of linear second-order system with { = 0.5 .... 10-9
10-16 Portrait in the vicinity of a stable node .................................. 10-10
10-17 Portrait in the vicinity of an unstable node .......................... 10-10
10-18 Portrait in the vicinity of a stable focus .................... 10-10
10-19 Portrait in the vicinity of an unstable focus ................
10-20 Portrait in the vicinity of a center ................................
10-21 Portrait in the neighborhood of a saddle point ...................... lO-lX
10-22 Types of singularities ................................................................ 10-12
10-23 Portrait with soft self-excitation ............................................ 10-12
10-24 Portrait with hard self-excitation ............................................ 10-12