A4 Simple Electronic Differential Analyzer as a Demonstration and Laboratory Aid to Instruction in Engineering by M. H. Nichols Associate Professor in Aeronautical Engineering University of Michigan D. W[. Hagelbarger Instructor of Aeronautical Engineering University of Michigan Department of Aeronautical Engineering University of Michigan 1951

Copyright 1951 M. H. Nichols

PREFACE Early in 1947 a small group in the Aeronautical Engineering Department at the University of Michigan began an investigation of the use of an electronic differential analyzer for the solution of engineering problems. Reports covering some of this work have been issued.* More or less parallel with this, a simple electronic differential analyzer has been used in the laboratory of the Engineering Measurements and Servo Mechanism Courses offered by the Department of Aeronautical Engineering to demonstrate simple dynamic systems such as seismic instruments, servos, etc. Inasmuch as the authors have been unable to find in the literature an elementary treatment of the use of the electronic differential analyzer, the following pages have been written to serve as an introduction to more advanced publications in the field and as a preparation for the laboratory work in the above mentioned courses. A short version of this article has been submitted for publication in the Journal of Engineering Education. The authors are indebted to Professor C. E. Howe of Oberlin College, who, while visiting at the University of Michigan, worked out many of the computer connections and ran many records, some of which are included in this article. The authors are also indebted to Professor L. L. Rauch of the University of Michigan for the Appendices and criticism of the manuscript and assistance in the preparation of the manuscript. See reference 3.

TABLE OF CONTENTS Section Page I Introduction............................1 I n Type of Systems Which Can Be Handled by A Simple Electronic Differential Analyzer...................... 1 III Apparatus.................................... 3 IV Use of the High-Gain D.C. Amplifier to Perform Elementary Operations... 7 V Examples 1 One Degree of Freedom Vibration.............. 11 2 Spring and Mass with Dry Friction.............. 18 3 Servomechanisms.............. 18 4 Coupled Systems-The Dynamic Vibration Absorber... 25 5 Boundary Value Problems-The Thin Uniform Beam with Static Uniform Load..... 25 6 Consecutive Chemical Reactions................ 35 VI Precautions in the Use of the Electronic Differential Analyzer Described Here................................. 35 References.................................... 38 Appendix I.................................... 39 Appendix II............................ 40

ILLUSTRATIONS Figure Page 1 Amplifiers on individual chassis in plug-in rack with separate power supplies behind. The plug-in connections are on the individual amplifiers............................. 2 2 Unit consisting of four amplifiers and power supplies. Plug-in connections appear at the front of the unit. The toggle switches on the front side and the knobs between the tubes are for balancing the amplifiers. (See Figure 3)................... 2 3 Balancing circuit for individual amplifiers............. 4 4 Circuit of power supply for +350 volts. The -350 volt supply is the same except the positive end is grounded. The 360 volt battery in this figure and the 200 volt battery in Figure 5 are made up of Eveready #467 battery components............ 5 5 Circuit for -190 volt power supply....................... 6 6 Photograph of low frequency sinusoidal voltage generator..... 7 7 Circuit for low frequency sinusoidal voltage generator....... 8 8 Amplifier connections to perform elementary operations: (1) to multiply by a constant, (2) to sum, and (3) to integrate with respect to time................................. 9 9 General arrangement of amplifiers which obeys equation (1) with positive coefficients. All capacitors and resistors are in units of microfarads and megohms respectively............... 10 10 One degree of freedom mass, spring, and damper system..... 11 11 Amplifier connection which obeys equation (3). All fixed resistors are one megohm and all condensers are one microfarad. The potentiometers indicated are 50,000 ohm wire wound....................................... 13 12 Photograph of the unit of Figure 2 connected according to Figure 11 except that the potentiometers shown in Figure 11 are omitted and the parameters taken care of in the feed back resistors..................................... 14 13 Recording of the steady state response of the system of Figure 10 to a sinusoidal forcing function. The top record is the output voltage and bottom record is the input voltage. The direction of increasing time is left to right..................... 14 14 Steady state amplitude and phase characteristics of the system of the system of Figure 10 for various values of the damping ratio, 5.................................... 15 15 Step function response of the system of Figure 10 for various values of the damping ratio,. Time scale is one division per second..........................^......... 16 16 Alternative amplifier connection for Figure 10. All fixed resistors are one megohm and all condensers are one microfarad. The potentiometers are 50,000 ohms wire wound..... 17

ILLUSTRATIONS Figure Page 17 Amplifier connection for system of Figure 10 with the viscous damper replaced by a dry friction damper................. 19 18 Step function response of system of Figure 10 with dry friction as compared to the response of the same system with viscous friction. The bottom part of the dry friction record is the dry friction force applied through the saturable amplifiers; the overshoot is due to the recorder characteristic........................... 20 19 Alternative circuit to that of Figure 17. All fixed resistors and condensers are one megohm and one microfarad except as labelled. The potentiometers are 50,000 ohms except as labelled........ 21 20 Computer circuit for servomechanism of equation (5). The feed back elements are labelled in units of megohms and in units of microfarads. The 1000 ohm resistor in series with the switch S. is for the protection of the contacts.................... 23 21 Error between the input function and output of a servomechanism for a ramp function input. The upper part of the top recording is the ramp function and the lower part is the output error. The top recording is with viscous output damping, the center recording is with viscous output damping and error rate feedback, and the bottom recording is with viscous output damping, error rate feed back, and error integral feed back, all with critical damping..... 24 22 Computer connection for the damped vibration absorber of equation (7). All feed back resistors and condensers are labelled in units of megohms and microfarads respectively................... 26 23 Response of the damped vibration absorber to a sinusoidal forcing function. The top record is the forcing function, the center record is the deflection of the small mass, and the bottom record is the deflection of the large mass. All the records were taken with the same sensitivity................................. 27 24 Computer connection for the uniform beam of equation (10). All feed back elements are labelled in units of megohms and microfarads.................................... 29 25 Five trial solutions for the deflection of a uniform beam with uniform load hinged at the ends obtained by successive increases in Va. The upper part of each record is y and the lower part is y". The fifth trial gives the proper solution because the y" and y are zero simultaneously................................ 30 26 Recording of y, y', y", and ym for a uniform beam with uniform load hinged at the ends.31 27 Computer connection for the normal modes of a free-free beam. The feed back elements are labelled in units of megohms and microfarads...................................... 33 28 Computer connection for the four component irreversible consecutive reactions described by equations (21).............. 34 29 Recording of the dependence of the variables n of equations (21) on time. From top down are nA, nB, nc, and nD respectively..... 36

A SIMPLE ELECTRONIC DIFFERENTIAL ANALYZER AS A DEMONSTRATION AND LABORATORY AID TO INSTRUCTION IN ENGINEERING M. H. NICHOLSl and D. W. HAGELBARGER2 Department of Aeronautical Engineering University of Michigan College of Engineering I. Introduction The demonstration of such things as dynamics of mechanical systems, statistics and dynamics of beams, etc., by mechanical means is often expensive and inconvenient. The demonstration of such idealizations as purely dry or purely viscous friction is difficult, if at all possible, by mechanical apparatus. Many of these demonstrations are more conveniently handled by use of an electronic differential analyzer and, in general, do not lose their instructional value. Also, the use of electronic aids to engineering calculations is rapidly expanding. The purpose of this paper is to describe the use of a simple electronic differential analyzer3 for laboratory instruction and demonstration. Several examples will be worked through completely and numerous additional possibilities will be suggested. Upon sufficient experience, the extension of the computer to the demonstration of many other common engineering problems becomes apparent. A brief description of the apparatus together with references will be included so that it can be duplicated in most college shops. Since the fundamental theory of the type of computer described here is well discussed by Ragazzini, Randall and Russell(l) and by Frost(2), it will not be repeated in this paper.4 II. Type of Systems Which Can Be Handled by a Simple Electronic Differential Analyzer A differential analyzer can be arranged to obey the equations which describe the system to be investigated(1,2). The advantage of the electronic differential analyzer of the type described in this paper is that it can be conveniently and economically arranged to obey to a high degree of accuracy the differential equations describing systems commonly met in many branches of engineering. With only the basic d.c. amplifier to be described in the next section together with simple condensers and resistors, it is easy to handle linear differential equations of the type. m rn-i ^m-a n n-i n-2 Apy+Bp y + Cp y +... = ap +p x + p n x +...'Associate Professor of Aeronautical Engineering,now at California Institute of Technology, Pasadena, California). 2Instructor of Aeronautical Engineering now at Bell Telephone Laboratories, Murray Hill, New Jersey). 3 Another type of electronic computer, usually called an analogue computer, makes use of inductances, capacitances, and resistors, connected circuit wise to simulate the system under study. See, for example, McCann, Wilts, and Locanthi, IRE, 37 954 (1949); McCann and MacNeal, paper No. 49-SA-3 (to be published in the ASME Jour. Appl. Mech., 1949) and references thereto. In the case of continuous media the method of difference equations is used. 4It should be mentioned that Ragazzini et al(l) use only the differentiator type of amplifier connection (see Section IV of this paper) whereas it was later fount that the integrator type of connection is more practical and can be used to accomplish the same end results. This was pointed out to the authors in 1947 byletter from Professor Ragazzini. Frost(2) discusses the amplifier connections for integration.

Figure 1 Amplifiers on individual chassis in plug-in rack with separate power supplies behind. The plug-in connections are on the individual amplifiers. Figure 2 Unit consisting of four am-.... yPa plifiers and power supplies. Plug-in connections appear........at the front of the unit. T toggle switches on the front side and the knobs between the tubes are for balancing the amplifier~. (See Figure 3)

Where the coefficients A, B, C,..., (x, 3, y,..., are constants, p is the differential operator denoting differentiation by the independent variable, y is the dependent variable and x is an arbitrary function of the independent variable. In many cases, x may be considered a forcing function. Since it is considerably more practical to use the amplifier in the connection for integration, it is convenient to divide the equation by p" so that the input function x and its integrals appear: /m*-TTL*" II ^ in.~II~1 - / ~1 -2 (Ap-n + Bp +...)y = (o( p + p + p +...)x (1) It is also possible to handle systems of simultaneous linear differential equations with several dependent variables but with only one independent variable-i.e. coupled systems. In the case of the differential analyzer described here, the independent variable becomes time and the variables x and y are voltages, x being the input voltage and the voltage y the output voltage. By means of more complicated devices such as synchronously driven switches, servo multipliers, etc., more complicated differential equations can be handled(1"23). The description and application of these more complicated computer connections is beyond the scope of this paper. Examples of systems which are described by equation (1) are simple one dimensional linear vibration(4), seismic instruments(4), linear servo systems(5), uniform thin beams(4'^), etc. By use of simple non-linear devices, it is possible to handle such things as dry friction(4), bang-bang controlled servos, etc. III. Apparatus The central part of the computer described here is a high gain d.c. amplifier. The amplifiers used for the results given in this paper are described by Ragazzini, Randall and Russell(l). More recently, Frost(2) has described a similar amplifier which may be more advantageous for demonstration purposes because it provides a higher current output which permits the use of a wider variety of recording instruments. An arrangement of amplifiers which has been proved to be convenient is shown in Figure 1. Here the individual amplifiers plug into a rack. The power supplies are contained in separate cabinets shown on the shelf above the rack. Alternatively, when only a few amplifiers are needed they may all be mounted together with the power supplies on a single chassis as in Figure 2, which is a photograph of a four amplifier unit. Balance controls are provided to balance the amplifier for zero output voltage when zero input voltage is applied-i.e. zero eo when the amplifier is connected as in Figure 8-(1) with zero ei. In order to facilitate balancing of the amplifier, a switch is connected as shown in Figure 3. This makes the balancing possible without disconnecting the unit from the computer circuit. Figures 4 and 5 give the circuit diagrams of the power supplies required(1). The auxiliary condensers used in the feed back connections are Western Electric D161270 luf condensers with polystyrene dielectric. It is also satisfactory to use Condenser Products Co. LAC2C condensers. The auxiliary fixed resistors used are Continental Carbon Co., type x. Any reasonably stable resistor having low temperature coefficient is satisfactory. The variable potentiometer type resistors used in some cases to conveniently vary the parameters of the systems are General Radio, type 314A 50,000 ohm potentiometers. Banana jacks are provided for convenience in plugging in the resistors and condensers. The resistors are mounted on double pole plugs such as the General Radio, type 274M. For the recordings presented in this paper, the Brush Development Company, type BL202 magnetic pen recorder and d.c. amplifier type BL913 were used. 5 Although sometimes inconvenient, slower recorders such as the Esterline Angus Model AW, range 0-1.0 milliampers, can be used by slowing down the time scale of the computer. In this case it would be desirable to use a higher current output amplifier such as that described by Frost(2). Also, photographic type recorders such as are often used in electrical power laboratories can be used but the photographic development process is inconvenient. By speeding up the computer time scale, a repetitive cathode ray tube display can be used. This will generally require a reimposition of initial conditions for each horizontal sweep of the cathode ray 5An opaque type projector (such as that used for projecting the pages of a book) can be used to project the record of the Brush Development Co. magnetic pen type recorder onto a screen as it is made. The recorder is simply placed underneath the projector. 3

INPUT TO AMPLIFIER ", 7.5 MEG. SWITCH 120 K 0~1~ 0~i(O (-pOO COUPLING TERMINALS OUTPUT TERMINALS INPUT TERMINALS TO AMPLIFIER Figure 3 Balancing circuit for individual amplifiers. 4

THORDARSON L1 TYPE T- 19R30 6B4 15 hy 15 hy 25K. -K, < \ 10.5 meg 115V ^ ^ ___'^ ^^ ^^ ___ ^ ^^6SJ7 uC C 0 8 mfd 8 mfd 8 mfd 50 K Th — 350 20 K I meg,' " 6.3 V 5Y3GT 25 K F GND. Figure 4 Circuit of power supply for +350 volts. The -350 volt supply is the same except the positive end is grounded. The 360 volt battery in this figure and the 200 volt battery in Figure 5 are made up of Eveready #467 battery components.

115 V STANCOR TYPE P-6011 6B4G ^115V ^ Pi V6SJ7 V 8 mfd 8mfd 8mfd 0meg 5Y3Gv^T TT HEATERS OF 25K 15 hy 15hy Figure 5 Circuit for -190 volt power supply.

beam. Sufficiently high speed relays (such as the "Millisec" relay, manufactured by Stevens-Arnold, Inc. or any other relay with a sufficiently light weight armature) or electronic switches can be used for imposing the initial conditions and are triggered in synchronism with the start of each horizontal sweep of the cathode ray beam. Due care must be taken to keep the 60-cycle hum level sufficiently low. For handling systems with sinusoidal forcing functions, it is satisfactory to use some form of motor driven signal generator which can be operated at frequencies of one cycle per second or less. A device which has been found convenient is shown in Figure 6. It consists of a oltzer CabotTypeRW Figure 6 ponent occurring in the rectified output from the one phase of the three phase windings in series with the such as gear-ing, etc. are obvious.6 Figure 8 shows how the amplifier can be used to (6) multiply by a constant, (2) sum and (3) integrate each with a high degree of accuracy if the gain of the amplifier is sufficiently hlgh.v The derivations are given in the appendices. voAlso, it is pofsible to use three d.c. amplifiers connected as in Figure 16 except with no net rate feed back. The difficulty with this method has been that there are aeways small amounts of feed back via the power upply etc., which requires a careful adjustment of inserted feed back in order to stabilize the amplitude. This adjustment requires ppreciable time particularly at low frequencies.

HOLTZER- CABOT GENERAL ELECTRIC A.C. SYNCHRONOUS MOTOR SELSYN GENERATOR TYPE RWC 2505 MODEL 2JIFI ovA~10 V.A.G. I PA 30 TO 300 C.RS. WINDINGS FROM \ HEWLETT- PACKARD OSCILLATOR - - - m -~ ^ >^" ~' ^^ ^ ~ 1 /^ ^^r^ ^ ^^ONE PHASE OF THREE PHASE WINDING DIODES ARE TRANS. 2mf TYPE IN34 FORMER ( 0.25mf Figr IOK 50K Figure 7 Circuit for low frequency sinusoidal voltage generator.

0= RO 0i -R( ei A 2eo0 II 1 e0eo=RR eR (3) to i R withi2 r c t Rim eo RO(e/R += -' ei A eo 0o -- - — o (3) e ~ ~- RiC J-e^ dt eo R. Co Figure 8 Amplifier connections to perform elementary operations: (1) to multiply by a constant, (2) to sum, and (3) to integrate with respect to time.

x I I ~ 1 I j ~1VVV I I ~ 1 A, A^ P yx A y x 3~~~~~~~~~~ I I I/A ~l I ~ II.~~A V A A^ 4 A6 mwn ^m-ni / P y P y m-n2 P Figure 9 General arrangement of amplifiers which obeys equation (1) with positive coefficients. All capacitors and resistors are in units of microfarads and megohms respectively.

In order to use the amplifiers as they are connected in Figure 8, it is necessary, for stability reasons, that the output voltage be opposite in polarity from the voltage at the input terminals of the amplifier-i.e., the amplifier must have a negative gain. It is also possible to differentiate(1) but this is generally not practical because of the resultant amplification of stray 60-cycle pick up and the instability resulting from the phase-shift characteristic of the amplifiers at high frequencies. A general arrangement of the amplifiers which obeys equation (1) with positive coefficients is shown in Figure 9.7 In particular cases there are alternative connections which sometimes make possible the elimination of one or more amplifiers, but Figure 9 can be used as a guide for setting up the problem. The following examples will illustrate the arrangement for common problems and will show how initial conditions are handled. V. Examples 1. One degree of freedom vibration. FIXED SUPPORT x(t)3i I/ MASS Figure 10 One degree of freedom mass, spring, and damper system. Consider the spring, mass, and viscous damper system represented by Figure 10. Let a force x(t) act on the mass m. The differential equation of motion of the mass is my + cy + ky = x(t) (2) 7In the case of negative coefficients, the appropriate connections should be changed from bus (1) to bus (2) or vice versa. 11

where the dots represent differentiation with respect to time. In case the frequencies occurring in the problem are not in the range which are convenient to record or observe with the apparatus at hand, a change in independent variable can be made. Let Tbe the computer time. Make the substitution = t where r is a constant. For convenience in varying the parameters, it is desirable to divide the equation by a constant Q such that 2 1 and k <l Equation (2) can then be written m r2i c k 1 Q y Q k Q Q C- (3) where now the dots represent differentiation with respect to T. A circuit which obeys equation (3) is given in Figure 11 which is a special case of Figure 9.8 The parameters of equation (3) are varied by means of potentiometers at the appropriate amplifier outputs. Figure 12 is a photograph of the four amplifier unit connected according to Figure 11 except that the potentiometers are omitted and the parameters taken care of in the feed back resistors. The sinusoidal voltage generator is shown connected to the input. The steady state response(4) to a sinusoidal forcing function can be obtained by inserting a sinusoidal voltage at the input terminals (x terminals) and observing the output amplitude and phase at the output terminals (y terminals) after the transient has died out. Figure 13 is a recording of the steady state response and the forcing function taken at a frequency of 5.7 radians/sec. The natural undamped frequency on of the system is 5.0 radians/sec and the damping ratio(4), A, is 0.5. Figure 14 is a plot of the amplitude and phase characteristic as a function of the forcing frequency co expressed as a ratio to the natural undamped frequency on. For various values of S the output amplitude is plotted as a ratio to the amplitude of the forcing function-i.e., the ratio of output amplitude Y to the input amplitude X is plotted. The points are the results from the computer and the solid lines are the calculated values. In order to obtain transient solutions resulting from a step function input, a constant voltage is applied at the x terminals and after the transient has died out or been shorted out by switches, the voltage is suddenly removed (i.e. a voltage step function is applied) and the response is recorded. Figure 15 shows a series of step functions (transient) responses taken with on = 3.2 radians/sec for several values of 5. An alternative computer connection for this problem which requires only three amplifiers is shown in Figure 16. By an analysis similar to the one applied to the circuit of Figure 10, it is easy to show that this circuit also obeys equation (3). 10 8That this circuit obeys equation (3) can be seen by applying Kirchhoff's law to the currents at the junction "0". Since the input resistance of the amplifier is very large, the input current to the amplifier can be neglected and since the gain of the amplifier is very large, the junction 0 can be considered essentially at ground potential(^ ). Kirchhoff's law states that the sum of the currents into a junction must be zero. Since all feed-back resistors are one megohm, the relation is just equation (3). This is analogous to equating all the forces acting on the mass of Figure 10, including the inertial force, to zero. 9it should be pointed out that the feed-back impedances are essentially in parallel with the output of the potentiometer. Normally this makes only a small error if neglected but in cases in which high accuracy is required, this should be taken into account or the potentiometers should be dispensed with and the parameters taken care of in the feed-back elements. 10In some applications in which simplicity and economy of equipment are desired, a combination of analogue computer techniques and differential analyzer techniques can be used to advantage. For example, Professor L. L. Rauch, of the University of Michigan, has pointed out to us that by combining the two techniques, equation 2 can be handled accurately by only one amplifier provided that there is some positive damping. The accompanying figure shows a circuit which obeys equation 2 and which uses only one amplifier. It is true that -x(t) must be inserted instead of x(t) but inasmuch as x(t) is the forcing function, it is usually convenient to "reverse" the connections to provide a voltage -x(t). By applying Kirchhoff's law to the currents at the nodes A and B (neglecting as before the current into the amplifier 12

AAAAAAAAA~rI Ad~~4~ r ~ rlYIAA~IA I A- kY Q Q Q x^ ^1 X Al 2 A3 A JUNCTION " 10^ ^.l Mr^y c -ry Y Figure 11 Amplifier connection which obeys equation (3). All fixed resistors are one megohm and all condensers are one microfarad. The potentiometers indicated are 50,000 ohm wire wound.

':.... ":'..'.'i"''...'..' ".'".:.'. "''.:.".......:......... " " "'':.:i::~l -...................z..'.:....:..:::":':"......... Figure 12 Photograph of the unit of Figure 2 connected according to Figure 1 1 except that the potentiometers shown in Figure 11 are omitted and the parameters taken care of in the feed back resistor s. IL ~ \ t- i f-\_IL:I Fgr 1 3.14

~'o.tj O utdudtp aqj jO snleTA SfOTJ.1eA JOJ 01 aOn121A jo uiwlsAs 9tp jo snTasI S etJ x'e:qo as9e5d pue apnjidwuie a9^13s Ap-eapq 0 0~~~~~~~~0, U"/ " ~ ~ 03um ~~~~~~~~Z - 0 09 > CA r ~ 1 ~ \ ~ 1~-T ~ 1 ~ 1 ~ 1~-aiQ01: g z O'' m oeli -- 0* O'l cn x' O~~~~~~~~~~~' ozI Z~~~~~~~~~~~~~' =1D ~~~~~= m___l_^ l ____________ __!__ ___________ 2 ____^ ^ - gi^^^ m 01^-J ^ ^ ^ ^ 1:^ -) X ~~_~~o_ ~ ~ ~ A~- ~ ^~ ~ 3^I - \ ^r 09 1~~~~~~~~~~~~

.= 0.52 f= 1.0 3-= 1.7 Figure 15 Step function response of the system of Figure 10 for various values of the damping ratio, 3. Time scale is one division per second. 16

are one microfarad. The potentiometers are 50,000 ohms wire wound Y -y y Figure 16 Alternative amplifier connection for Figure 10. All fixed resistors are one megohm and all condensers are one microfarad. The potentiometers are 50,000 ohms wire wound.

E x a m p 1 e 2. Spring and Mass with Dry Friction This is a non-linear problem and can not be handled with the circuit of Figure 9, but can be easily handled by making use of the saturation characteristic of the amplifier. Let the viscous damper in Figure 10 be replaced by a dry friction damper. A computer connection which will handle this case is given in Figure 17. Here the input to amplifier A,, which is connected to operate at very high gain, is the velocity of the mass. Because of the saturation characteristic of this amplifier, a very small deviation of the input voltage from zero will give a constant output voltage either positive or negative depending on whether the input is negative or positive. However, because of the nature of the amplifier, the saturated voltage on the positive side is smaller than that on the negative side. In order to equalize the positive and negative voltages, the output of amplifier A1 is fed into amplifier A2 the output of which is equalized. An appropriate fraction of this output is then fed in at the input terminals (x terminals) to simulate a constant dry friction force which changes sign each time the velocity changes sign. (See reference 4, page 407 or reference 6, page 30.) For convenience in obtaining a step function, the battery across the feed-back condenser of A6 is included in Figure 17. When switch S. is closed the mass is displaced and when the switch is opened the mass is released. The response of this system to a step function is given in Figure 18. Shown in Figure 18 are the dry friction response, the corresponding dry friction force, and a viscous friction step function response for comparison. The steady state response to a sinusoidal forcing function(4) can be obtained in the same was as in example 1. Actually an unequalized voltage such as occurs in the output of amplifier A. in Figure 17 does not invalidate the result inasmuch as the effect of the inequality may be considered as a constant displacement of the system. By making use of Figure 16, it can be seen that the problem can also be handled by the circuit of Figure 19. Combinations of dry and viscous friction can be handled by combining the circuits of Figure 11 and 17. E x a m p le 3. Servomechanisms From reference 5, page 177, the fundamental differential equation for a servomechanism with viscous output damping, error rate damping, and integral control may be written: J dt2 + (L+F) d + K No + N Godt = L dti + K + N i dt (4) where Gi = input angle Go = output angle K = output torque per unit error angle F = friction torque per unit output rate J = output moment of inertia L = torque per unit error rate N = torque per unit time integral of error t = time (Footnote continued) and considering node B essen- tially at ground potential, which is legitimate if the 2 gain of the amplifier is high 2)) it can be shown that RI R, R3 B m = C1C2RR~2, c = (R1R2 + R1Rs + R2R3) C2/R1, AV ~ -x(t ) A k + Ps/R, A when m, c, and k are defined by equation (2). This leaves any two circuit parameters arbitrary; these two parameters may be chosen in such a way as to __ - result in convenient values for all the parameters. 18

30 MEG I 50 K 10K VVV —' - 1^20Kl SIN g 111 1'i II I-I —Figure 17 Amplifier connection for system of Figure 10 with the viscous damper replaced by a dry friction damper.

Figure 18 Step function response of system of Figure 10 with dry friction as compared to the response of the same system with viscous friction. The bottom part of the dry friction record is the dry friction force applied through the saturable amplifiers; the overshoot is due to the recorder characteristic. 20

30 MEG. 50K A l,~ A2.,'a A 25K ie -- - - -^ 4-I I t Figure 19 Alternative circuit to that of Figure 17. All fixed resistors and condensers are one megohm or one one microfarad except as labelled. The potentiometers are 50,000 ohms except as labelled.

and = i - o where 0 = error angle. Since this is of the form of equation (1) the circuit of Figure 9 can be used. However, for the illustration which follows, it will be more convenient to use a modified set-up as will be clear from the following. For this example the response of a servosystem to a ramp input function-i.e., the input function is a linear function of time starting at zero at time zero-will be investigated. Many ways of generating such a ramp function will be obvious, but a convenient one consisting of a single amplifier connected as an integrator is used for this example. For, to generate a ramp function, it is only necessary to integrate a constant voltage-i.e. to integrate a velocity step function. It is desirable to record the error between the input and output-i.e., 0o - 0i. It is convenient, for this purpose to rewrite equation (4) to give dt + (L+F) + K0 - L d- Ki - N 0 dt = It is then easy to show that the circuit of Figure 20 is governed by equation (5) for the case of a velocity step function input. In this case it is necessary to provide the switch S2 which is opened at time t = 0 so that a small unbalance in amplifier # 1 does not result in a slow build-up in the voltage across the condenser. Switch S2 is opened and switch S, is closed simultaneously at t = 0. Figure 21 is made up of recordings for three different combinations of parameters. In each case the input ramp friction Gi(t) is recorded and the error between the input function Gi(t) and the output 0o(t) is also recorded. The latter is recorded with ten times the sensitivity of the former. The three cases are for Case I J =, K = 1, 91 = 0.1, F = 1, L = 0, N = 0; Case II J = K = 1, 0.1, F =, L =, N = 0; Case mI J = 1, K = 1, 1 =0.1, F = L, L =, N =. Since in each case, F + L = 1, and JK = 1, the damping is always critical. It is interesting to note that the addition of rate feed back halves the steady state output error and that the further addition of integral control reduces the steady state output error to zero. To illustrate the accuracy of the computer and recording system, the problem on page 75 reference (5) was run through on the computer. The results are compared in Table I. In order to handle this problem conveniently with the apparatus used for this paper it was necessary to change variables as illustrated in example 1. In this Q = 40 x 10-6 and T = 10 t. TABLE I Z t 0i - go fi - go Radians Measured Radians From Reference 5 0 0 0 0 0.5 0.05 0.044 0.041 1.0 0.10 0.072 0.073 1.5 0.15 0.087 0.087 2.0 0.20 0.094 0.094 2.5 0.25 0.097 0.097 3.0 0.30 0.099 0.099 ~~ Oc_ 0.100 0.100 22

IK S ~St I-~ I Ai,' I,. _,! _, A A iAA 5K\ A lj A^ iA. A3 | A^ dt I/K I\ L +F) I/K N I I _'- i I Go -8o Figure 20 Computer circuit for servomechanism of equation (5). The feed back elements are labelled in units of megohms and in units of microfarads. The 1000 ohm resistor in series with the switch S1 is for the protection of the contacts.

Case II _ —— _Case IL__ I I Case III Figure 21 Error between the input function and output of a servomechanism for a ramp function input. The upper part of the top recording is the ramp function and the lower part is the output error. The top recording is with viscous output damping, the center recording is with viscous output damping and error rate feed back, and the bottom recording is with viscous output damping, error rate feed back, and error integral feed back, all with critical damping. 24

It should be pointed out here that the differential equation for a seismic instrument(4) can be written in a form identical to equation (5) in which only error and rate control are applied. To show this, let x(t) be the coordinate of the support of the seismic instrument and y(t) be the coordinate of the seismically supported mass of the instrument, both relative to a fixed axis. Then y(t) - x(t) is the reading on the seismic instrument scale. The differential equation of motion of the seismically supported mass is my + c( - ) + k(y - x) = 0 (6) Thus if F = N= 0 in equation (5) the two equations are identical and the instrument reading (y-x) can be taken off amplifier A3 in Figure 20. The steady state response to a sinusoidal vibration can be run off in a manner similar to that of example I and the response to a velocity step function-i.e. a ramp input function-can be run off as in the servo mechanism case. One dimensional vibration isolation problems (see reference 4 page 85) can be handled in a similar way. Such things as dry friction and bang-bang control can be handled by the saturation of the amplifiers as discussed in example 2. E x ample 4. Coupled Systems-The Dynamic Vibration Absorber In order to handle linear coupled systems with one independent variable, the computer may be set up for each dependent variable in accordance with Figure 9. Then the necessary cross coupling connections are inserted. As an example of coupled systems the problem of the damped vibration absorber will be treated. From reference (4) page 115, the differential equations of the system with a sinusoidal forcing function are: mly, + c2 (Y1-Y2) + kL2 (Y1-Y2) + kly1 = PO sinwt m2Y2 + cn (Y2-Y1) + km (Y2-YL) = 0 (7) A computer connection for these equations is given in Figure 22. By way of illustration the damped dynamic vibration absorber discussed in the example in reference (4) page 128 will be simulated with km = 8.4 lb/in. and cm = 0.054 lb. sec./in. In order to keep the voltage levels of all the amplifiers in a reasonable range, it turns out that with these values, it is desirable to change the independent variable to a new time scale T such that T = 25 t and in order to use reasonable feed back elements, to divide the equation through by 100. The equation then becomes: P0 1.94yj + 0.162 (Y,-Y2) + 1.01 (y,-y2) + 12.2y 100 sinwt 0.194Y2 + 0.162 (Y2-Y,) + 1.01 (Y2-y1) = 0 where the dots denote differentiation with respect to -Tand all displacements are in feet. Figure 23 shows the response of this system to a forcing function of 53 radians per second with equal sensitivity in each channel of the recorder. The ratio of the amplitude of yL to Po from Figure 23 is 2.9/8.0 = 0.36. Since this response was obtained with one unit of Po/100-i.e. with 100 units of Pothe true deflection of the main mass is 0.36/100 = 0.0036 feet which is in agreement with the result of reference (4) page 128. Example 5. Boundary Value Problems-The Thin Uniform Beam with Static Uniform Load One of the fields in which the differential analyzer is of particular practical use is the solution of boundary value problems. As an illustration, the thin uniform beam with static uniform loading and with both ends hinged will be treated. From reference 4, page 179, the differential equation for this case can be written. 25

0.0755 I 5.1 0.515,, 1 ASPo l A 2. 5, IS~~~~~~~~~~~~AA8,o5,,,t6 y''y Figure 22 Computer connection for the damped vibration absorber of equation (7). All feed back resistors and condensers are labelled in units of megohms and microfarads respectively.

Figure 23 Response of the damped vibration absorber to a sinusoidal forcing function. The top record is the forcing function, the center record is the deflection of the small mass, and the bottom record is the deflection of the large mass. All the records were taken with the same sensitivity. 27

EIl t =W (8) where W is the load per unit length and for the examples here is considered constant. The end conditions are y(0) = y(L) = 2 dx2j x=L (9) A convenient method of getting a solution which satisfies (9) is to arrange a computer to obey equation (8) in which the independent variable becomes time. The conditions y(0) = y) = = 0 are imposed by means of switches and the ratio of y'(0) to y'" (0) is varied until at some later time T, the conditions y(T) = y"(T) = 0 are satisfied. This then is a solution for the case of both ends hinged. As a special case of Figure 9, a circuit for equation (8) is given in Figure 24 where initial condition switches Si, S2, S3 and S4 are opened simultaneously at t = 0. This may be done by a multiple pole switch or by means of relays. The voltage Va is then varied until a solution which satisfies the end conditions at some later time T is obtained. Figure 25 is a recording of y and y" for five trial solutions obtained by successive settings of Va. Figure 26 is a recording of y, y', y" and y' for the correct solution. T Since the independent variable on the computer is time, let t = - x where L is the actual length of the beam. Then equation (8) becomes d4y W L4 V (10) dt - EI T4V Once T is determined, it is possible to determine the loading of the beam which corresponds to the solution obtained; i.e. W T4 V (11) EI L4 ( Corresponding to any point t in the computer solution, the bending moment is given by M= EI 2 t (12) The shear force corresponding to any point, t, on the computer solution is given by Q = EI T3 d (13) Since the problem is linear, the displacement, bending moment, and shear force can be determined for any loading by simply multiplying the corresponding quantities from the computer solution by the ratio of the desired loading to the loading computed by equation (11). A convenient way to determine the values of y, y', y", and ym from the computer solution is to apply the voltage V directly to each channel of the recorder and record the number of units deflection R in each channel. Then the values of y, y', y", and ym are the corresponding deflections divided by the Rv for the respective channel. For example, for the recordings of Figure 26 the unit input voltage V gives 6.0 small units deflection when connected directly to the y channel of the recorder. The maximum deflection y of the beam read from the figure is approximately 14 small units. Therefore, the maximum deflection y of the beam is 14/6.0 = 2.3 units for one unit of V. The computer length T of the beam is read from the records of Figure 26 as 3.66 seconds. Therefore, from equation (11) the load, W, per unit length corresponding to the solution is W = (3.66)4 EL - 180L (14) 28

S2 Vb S3 s4 S ld 3 I S Va _V_ s-Al A2 A3 A _0 _ _),,, —'. - -y Figure 24 Computer connection for the uniform beam of equation (10). All feed back elements are labelled in units of megohms and microfarads.

7777I I A t -Y~y Figure 25 Five trial solutions for the deflection of a uniform beam with uniform load hinged at the ends obtained by successive increase in Va. The upper part of each record is y and the lower part is y". The fifth trial gives the proper solution because the y" and y are zero simultaneously. 30

Figure 26 Recording of y, y', y", and y"' for a uniform beam with uniform load hinged at the ends. 31

Suppose that the problem to be solved is for a beam with L = 1, EI = 1, and W = 1. From equation (14), the load per unit length of the solution of Figure 26 is 180. Hence the value of Ymax for W = 1 is 2.3/180 =.013 units. From reference 7, page 121, the maximum deflection of a hinged-hinged beam is given by (\5 WL4 Ymax = (384] E = 0.013 (15) which is in good agreement with the computer result. By use of equation (12) and the y" record of Figure 26 the maximum bending moment can be calculated as follows. For the y" channel, the input voltage V gives 5.6 small units deflection when connected directly to the recorder. From Figure 26, 1.5 y"max = 15 small divisions on the record. Therefore, y"max = 15/(1.5x5.6) = 1.8. Thus max = E ) ym = (3.66)2 x 1.8 = 24.(16) But as before this corresponds to W = 180. Hence for W = 1 24 Ma 1 = 0.13. (17) From reference 7, page 121 for this case WL2 Mmax 8 0.13. (18) The shear force Q can be computed in a like manner by use of equation (13). In order to handle different types of support of the beam it is only necessary to change the end conditions-otherwise the computer remains the same. For example for the beam clamped at each end, y(0) = y'(0) = y(T) = y'(T) = 0 and the ratio of y"(0) to ym(0) is varied to get a solution. It should be pointed out that it is always possible to adjust the input voltage V, each time adjusting the ratio of the two unspecified initial voltages until the end conditions are satisfied, so that the computer length T is numerically equal to the actual length L (or to any specified value). In this case the ratio T/L is numerically equal to unity. The normal modes of vibration of the thin uniform beam can be obtained in a similar fashion as follows. From reference 6, page 332 the differential equation of the space dependent part of the motion, Y(x), is a2 d4 p2 4 - Y(x) = 0 (1 9) where a2 and p2 are constants. A computer arrangement for equation (19) for a free-free beam is given in Figure 27. The end conditions are Y"(0) = Y"(L) = Ym(0) = Y"'(L) = 0. The ratio of Y(0) to Y'(0) is varied by adjusting Va until the end conditions are satisfied at some later time T. Various settings of Va which satisfy the end conditions will give the various modes. The setting of Va for the higher modes is very sensitive. The static deflection and normal modes for thick uniform beams and for non-uniform beams and loads can be handled by the electronic differential analyzer3), but the circuits and equipment are more complicated. Nevertheless, the method can be used to considerable advantage in cases where these solutions must otherwise be worked out by numerical methods.(3) 32

Va Vb I J I- I I I. - I I A2 A3 | A4 -Y 1 1 Figure 27 Computer connection for the normal modes of a free-free beam. The feed back elements are labelled in units of megohms and microfarads.

{V SI.8/K 2 / S3 I./K 1K1 1 /K2 I/K3 NA -N4 NI/ Figure 28 Computer connection for the four componreaction for the four component irreveribe consecutive reacby equations described (21)

Examp e 6. Consecutive Chemical Reactions Another very simple problem illustrating a system of linear differential equations is that of consecutive chemical reactions.(8) Consider the unimolecular case A —B — >C —D (20) The differential equations for this reaction are, dnA dt - k"nA dnB dt = klnA - k2nB (21) dn0 = k2nB - ksnC dnD dt = kn, where the constants are characteristic of the respective reactions and the quantities n are the number of moles of the respective substances present. These equations also describe the behavior of a series of radio active disintegrations in which the mother substance decays into daughter products, the final product having a decay rate negligibly small compared to the others.(9) If only the mother substance A is present initially, then at time t = 0; nA = nAO nB = nC = nD = It is easy to show by application of Kirchhoff's law that the circuit of Figure 28 obeys equations (21). The voltage Va is set to provide the initial condition Va = nAO. The switches, S, are opened simultaneously at t = 0. As an example consider the case in which(9) kI = 3.79 x 10-3/sec. kg = 4.31 x 10 /sec. kg = 5.86 x 10-4/sec. k4 = 0, practically In order to have convenient values of the feed back elements the time scale is changed so that T= t/600 This has the effect of multiplying all k's by 600 so that the feed back resistors used in the first three amplifiers are, in order, 0.44 megs., 3.86 megs., and 2.84 megs. Figure 29 gives a computer record of the dependence of the variables n on time. The case of reversible consecutive reactions can be handled in a similar manner. VI. Precautions in the Use of the Electronic Analogue Computer Described Here Computer time may be thought of in units of the input resistance times the feed back condenser (RC products) of the integrators. If for a particular problem, the RC products of the individual integrators are all changed by the same factor (and the time scale of the forcing function changed by the same factor) the only result is to change the time scale of all the voltages. This time scale change is often convenient in order to suit the response of the recording or observing instrument. 35

Figure 29 Recording of the dependence of the variables n of equations 21 on time. From top down are nA, ng, nC, and nD respectively. 36

It is desirable to operate each amplifier at as high an output level as possible but still on the linear part of the amplifier characteristic (this is analogous to choosing an ammeter on which the current to be measured is at least half scale or better) in order to minimize effects of drifts, hum pickup, etc. This can be done in part by using suitable values of the input voltage. In some problems in which the parameters are such that certain integrator stages would operate at a low level, a change in independent variable will help (see examples 1, 3, 4, and 7). In this case the change is not made in the RC products of the individual integrators but in the connections to the other amplifiers. Another similar method is to change the voltage level of a given integrator by changing its RC product (see Figure 8). As an example, consider an amplifier connection for the problem of Figure 10 derived directly from Figure 9. For this case only amplifiers A2, A4, A6 and A. are used-i.e. ( = = 5... = 0 and D = E... = 0. Let the feed back condenser and input resistor of As be Cs and R8 and let C6 and R1 apply to A6. Then the equation for the currents into junction "J" is A B Rs6R8Cs Y + R8C8 + cy = ax (22) The output level of A8, for example, could be raised by decreasing R8C8. Then B can be adjusted to keep B/RICs constant, if desired. If desired, the term A/R6RsC6C8 can be left constant by adjustment of A, R8, or C8, etc. It is also possible to use voltage dividers at the output of a given amplifier in order that the amplifier may be operated at nearly its maximum output which is then divided down to the proper level for feeding to the other amplifiers. (The voltage divider divides the undesired voltage by the same amount as the desired.) It is desirable to operate the amplifiers at as low a net gain as possible consistent with the high output level requirement. It is, of course, necessary to avoid saturation of any of the amplifiers-i.e. exceeding the linear voltage output range of the amplifiers except when special non-linear effects (dry friction, for example) are desired. If the differential equation which the differential analyzer is to obey represents a stable system, then it is always necessary to include an odd number of amplifiers in each feed back loop. The reason for this is that each amplifier changes the sign of the input voltage. At first sight it might appear that the differential analyzer described here could be used for linear algebraic equations in which case only resistances would appear in the feed back connections. However, if more than one amplifier is used in such a loop, parasitic oscillations are apt to occur because of the phase shifts in the amplifiers at high frequencies which are due to tube capacities, etc. However, if an amplifier connected as an integrator is inserted into the loop, the high frequencies are sufficiently degenerated to overcome this tendency. The amplifiers described in references (1) and (2) require frequent d.c. balancing. Means for automatic balancing have been developed but involve additional complications. 37

REFERENCES 1. Ragazzini, Randall and Russell, IRE, 35 344 (1947). 2. Frost, Electronics 21, 116, July (1948), see also Korn, Electronics 21 122, April (1948); Macnee, IRE 37 1315 (1949). 3. a. University of Michigan, External Memorandum #UMM-28 (April 1, 1949) Investigation of the Utility of an Electronic Analaogue Computer in Engineering Problems USAF Project MX-794 (Contract W33-038-ac-14222) by Hagelbarger, Howe and Howe. b. University of Michigan, External Memorandum #UMM-47 (June 1, 1950) Further Application of the Electronic Differential Analyzer to the Oscillation of Beams USAF Project MX-794 (Contract W33-038-ac-14222) by C. E. Howe. c. University of Michigan, External Memorandum #UMM-67 (October 1, 1950) Application of the Electronic Differential Analyzer to the Oscillation of Beams Including Shear and Rotary Inertia by Howe, Howe and Rauch. 4. Den Hartog, Mechanical Vibrations, (Second Edition 1940) McGraw-Hill. 5. Lauer, Lesnick, and Matson, Servomechanism Fundamentals (1947) McGraw-Hill. 6. Timoshenko, Vibration Problems in Engineering (Second Edition, 1937) Van Nostrand. 7. Hudson, The Engineers' Manual. 8. Hougen and Watson, Chemical Process Principles, Part III (1947), Wiley, Page 829. 9. Rasetti, Element of Nuclear Physics (1936) Prentice-Hall, Page 28. 38

APPENDIX I Amplifier as a Constant Multiplier Let us consider an amplifier with gain -p connected as in Figure 8(1). We have eo = -e (Al-1) where e is the voltage at the amplifier input. The current entering the input of the amplifier is zero so that e -e ei - e + = 0. (Al-2) Ro Ri Using (Al-1) to eliminate e in (Al-2) gives iRo e= t Ri e (A1-3) R R i i+~ ~ As the first fraction in the right member approaches unity giving the expression in Figure 8(1). 39

APPENDIX II Amplifier as an Integrator Let us consider an amplifier with gain - } connected as in Figure 8(3) with an additional resistance Ro in parallel with the capacitance Co as a leakage resistance. We have e= - pe (A2-1) where e is the voltage at the amplifier input. The current entering the input of the amplifier is zero so that e e ee e0- e - e + Co(eo - e) + 0. (A2-2) Using (A2-1) to eliminate e in (A2-2) gives eo ei eo Ro (+! )C o0( + + + AR= 0 (A2-3) e;C r[1\ 2,. 1 (A2-4) R.C 1+o +jARi +-= Integration furnishes the result t t e eR -dt + i O L ~ dt. (A2-5) RiCo( + C }Ri1 + 0o to It is seen that a finite value of i reduces slightly the value of the coefficient before the integral of ei as compared with the simplified derivation which assumes e = o. More important is the coefficient pRiCo ( +-) introduced before the integral of eo. This second integral acts as a time constant of exponential decay for the integrator. This can be seen by assuming ei = o for t->t and eo = e for t t,. Then equation (A2-4) can be solved to give eo = o exp (- (A2-6) where 1 1+ 11(A2-7) R-Ro (1 1' Thus we see that when e i = o the integrator time constant is equivalent to two leakage resistors paralleled across the condenser: the first is the actual leakage resistance Ro and the second, due to the 40

finite gain of the amplifier, is Ri(l + "). When Ro = o and RiCo = 1 the time constant is closely equal to p. The general solution of (A2-4) for e as the result of a linear time operation on ei is eo = k exp. (-bt) - a exp (-bt) exp (bt) ei dt (A2-8) o where R~c- l + ~ - ~ b Co iR = R + + and k is determined by the initial value of eo when the integrator is started at t = 0. Condensers with C = 10-6 farads are available with Ro 1012 ohms. 41

UNIVERSITY OF MICHIGAN 111111111 0E48 5507111111111111111111 3 9015 03483 5507