FURTHER APPLICATION OF THE ELECTRONIC DIFFERENTIAL ANALYZME TO THE OSCILIATION OF BEAMS by Carl E. Howe Umo -47 June 1, 1950 Due to the heavy demand for this report the supply of the first printing from the publisher has been depleted. We trust that the attached ozalid reprint will satisfactorily meet your requirements.

Furthur Application of the Electronic Differential Analyzer to the Oscillation of Beams by Carl E. Howe Engineering Research Institute University of Michigan, Ann Arbor External Memorandum UMM-47 June 1,1950

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF IIICHIGAN -U-iA-47 PREFACE The investigations described in this report were carried out during the summner months of 1949 under the sponsorship of the Research Techniques Group of Project'lizard. This report forms a natural sequel to one aspect of the report UM1I-28 mentioned in the Introduction. The investigations described in this previous report concerning the general utility of the electronic differential analyzer for engineering problems were carried out by the Research Techniques Group principally during the summer months of 1947 and 1948. At the time the work was started it was not generally clear what role the electronic differential analyzer was to play in the field of engineering research, development, and teaching. Since that time it has been demonstrated in many quarters that application of this type of computer is simple and relatively inexpensive and that the range of its application is very wide. Policy changes in Project Wizard have made it impossible to continue basic investigations in the field of structural dynamics beyond those described in this report. In recognition of this, the aeronautical Research Center has made other funds available to continue this work during the summer of 1950. a complete report will be issued. It has been deemed advisable to call the type of computer described in UM&-28 an electronic differential analyzer rather than an electronic analog computer as was done originally. There is another type of electric computer using direct circuit analogs which is better described by the term analog 2 computer. The work described in this report was greatly aided by facilities and equipment of the Department of Aeronautical Engineering and by use of the 1 M. H. Nichols and D. a. Hagelbarger, "A Simple Electronic Differential Analyzer as a Demonstration and Laboratory Aid to Instruction in Engineering," to be published in The Journal of Engineering Education. 2 See, for example, McCann, gJilts, and Locanthi, IRE, 37 954 (1949); McCann and MacNeal, MIE Jour. Appl. Mech., 17 13- (1950) and references thereto. ----------------— ii

AERONAUTICAL RESEARCH CENTER UNIVERSITY OF MICHIGAN UML-47 constant temperature and humidity room of the Departmnent of Chemical and Metallurgical Engineering. L. L. Rauch i-i

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN'. —---—' — UEYL-47 TABLE OF CONTENTS Chapter Title age Introduction I Normal Modes of Oscillations of Uniform Beams 1 A. Normal Modes of Oscillation of a Uniform Beam with Free Ends 7 (1) Neglecting the Effect of Rotary Inertia 8 (2) Including the Effect of Rotary Inertia 16 B. Normal 4Modes of Oscillation of a Uniform Beam with Other End Conditions 28 II Normal Modes of Oscillations of Non-uniform Beams 29 III Proposed Work with Computer 31 IV Changes in the Computer 34 V Proposed Changes and Modifications for the Computer 36 Appendix I 46 Appendix II 47 iv

AERONAUTICAL RESEARCH CENTER- UNIVERSITY OF MICHIGAN 1UMM-47 ILUSTRATIONS Figure No. Title P 1-1 Computer circuit for obtaining normal mode solutions of uniform "floating" beam, excluding the effect of rotary inertia. 9 1-2 First mode solution for uniform "floating" beam excluding the effect of rotary inertia. 11 1-3 Second mode solution for uniform "floating" beam excluding the effect of rotary inertia. 11 1-4 Third mode solution for uniform "floating" beam excluding the effect of rotary inertia. 12 1-5 R vs S for uniform "floating" beam excluding the effect of rotary inertia. 15 1-6 Computer circuit for obtaining normal mode solutions of uniform "floating" beam, including the effect of rotary inertia. 17 1-7 First mode solution for uniform "floating" beam including the effect of rotary inertia. 18 1-8 Second mode solution for uniform "floating" beam including the effect of rotary inertia. 19 1-9 Third mode solution for uniform "floating" beam including the effect of rotary inertia. 20 1-10 R vs 1/S for uniform "floating" leam. 23 4-1 Balancing switch on amplifier chassis. 35 5-1 Partial wiring diagram of an individual amplifier chassis, showing the connections to the automatic balancing relay and the remote initial condition relay. 37 5-2 Circuit for controlling the automatic balancing of individual amplifiers. 39 -- ------ -v, ----- * -—.

*^FONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMI-47 ILLUSRATIONS (continued) IurE b No. Title Page 5-3 Amplifier chassis showing indicator for automatic balancing. 41 vi..,

AERONAUTICAL.RESEARCH CENTER UNIVERSITY OF MICHIGAN UM.47 - nTOQXJCTIOi.. tn a previous reportl on the uses of an electronic differential analyzer one of the problems investigated was that of determining the normal modes of oscillations of beams. In the case of an oscillating beam with free ends it was assumed that the proper end conditions were fulfilled by setting equal to zero, at both ends of the beam, the second and third derivatives of the "displacement" This asumption led to satisfactory solutions when the effects of, shearing force and rotary inertia were neglected. However, when the effects were taken into consideration the solutions obtained by the differential analyzer showed mode shapes corresponding to oscillations of the center of gravity of the beam, a physical impossibility* As explained on page 98 of the previous report this discrepancy was due to the use of incorrect end conditions. "The reason for this discrepancy is that end conditions.... are incorrect. The correct end conditions for a free-free beam are that the bending moment M and shearing force V are zero: at both ends. When shear and rotary inertia are considered, these are no longer proportional to the second and third derivatives respectively.' T* he present report deals primarily with the determination of the normal modes of oscillations of uniform free-free beams, using as end conditions the equating to zero, at both ends, the expressions for bending moment and shearing force. Suggestions for further investigations of oscillating beams are made. / In addition there are described a number of modifications which have been made in the computing and recording equipment. Certain proposals for additional changes have been includedo 1 Hagelbarger, EHowe and Howe, University of Michigan Engineering Research Institute. External Memorandum on Investigation of the Utility of an Electronic Analog Coputer in Engineering Problems (April 1, 1949), Air Force Contract 33S-038-ac-14222 (Project SX-794).. —----- vii...... 1

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Uri-47 It is assumed that the reader is familiar with the fundamental principles and techniques involved in the use of the electronic differential analyzer as described in the previous report to which reference has been made above... -—.._.,- ---- "viii

AERONAUTICAL RESEARCH CENTER UNIVERSITY OF MICHIGAN UI —I47 CHAPTE I NORMAL MODES OF OSCILLATIONS OF UNIFORM BEAMS The equation of motion of a uniform beam may be deduced from a free body diagram by applying the laws of dynamics and elementary strength of mate1 rials. The five basic equations are pft^ + -Qo (1) dt?zsx at2 3x -^ + I'^ + V O (2) V = - kAGM (3) aB M d — M. (4) 9x - El d = _ +L p (5) Notation: x = horizontal distance from left end of beam y = vertical deflection A= neutral axis slope due to shear = neutral axis slope due to bending p= mass per unit length of beam 1 hese equations, with slight changes in notation, are taken from - Ormondroyd, Hess, Htess and Edman; University of Michigan Engineering Research Institute, Second Progress Report (March 31, 1948), Office of Naval Research Contract N5Ori-116 (Univ. of Mich. No. M670-4). Title: Theoretical Research on the Dynamics of a Ship's Structure. --------------------... 1 -..".'''

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UM-47 I = area moment of inertia E = modulus of elasticity A 3 cross sectional area G = modulus of shear k = ratio of average shear stress to shear stress at neutral axis M a bending moment V = vertical shear force I'= mass moment of inertia per unit length of beam a= length of beam t time w= angular frequency of vibration -EI = flexural rigidity kAG = shear rigidity From equations (1) to (5) there may bd obtained 4 /4 4 2 9x- ^kAG C X t kAGCt at 2 2 =EI1^ - (7) V + -=EI^ - | - I - ^ 2 (8) kAG dt C~x \kAG }xt Equation (6) is:t'e differential equation of motion of the uniform beam. Equations (7) qnd (8) are expressions for the bending moment, M, and the vertical shear forde, V. In obtaining a normal mode of oscillation it is assumed that the motion of the beam is simple harmonic. Let: y(x,t) =X(x) e3t, (9) V(x,t). V(x) et, (10) 2

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-47 and M(x,t) - M(x) et. (il) Equations (6), (7) and (8) may now be written di2 d 2 (X- I w,1 dx kAG' dx2A d21 EI f-2X (13) M e EI(13) dx kAG |1 dX (El I' 2 dXZ (14) V EIT "^ ^ ^w "'! (14) ~ I I'L dx3 VkAG dxj kAG.v/ where X, M and V are now functions of x alone. Equation (12) is to be solved by the electronic differential analyzer, subject to boundary conditions imposed by the restraints, or lack of them, placed on the beam. In order to solve the equation by using the computer it is necessary to change the independent variable. The length of the uniform beam is designated byj- so that the range of the solution for the independent variable, x, is OX -i. In solving equation (12) by the electronic analog computer the independent variable, x, is proportional to the time in seconds (as shown on the oscillogram) elapsed from the time of starting the solution. Suppose the range of solution as shown on the oscillogram, is covered in L seconds. Then the independent variable, x, in the above equations should be changed to a new independent variable, 7, according to the relation z x. (15) Then, d L d dx =2 d7, d? J^7' 1 ----- --------------— *5 -----------

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN,, U —4-' UMM-47 2 2 2 L d d2 -2 d72 and, in general, dn Ln dn dx11 In d7 In terms of the new independent variable,, equations (12), (13) and (14) may be written EI L4d4X A EI2 I' 2 d (1 (17) -— L-+-+- L l- I xo (17) p'1 dT4 kAG2 p2 dT2 kAG / 1 L EI d* S X EI I( 2 dX) Vhen - 24 I4. (2319) I ^2 L /oa)W d? kAGP dPi kAG' Let 4 R2. ^ (20) EI SE _ (21) kAt2 and U (22) Then o I I II IIIl, fi, 4i i i i lii 11 >

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-47 Equations (17), (18) and (19) then become L4 d (SL2 L2)X (1 -R2SU X =0 (24) R2 d?4 de = —l;=. [L2 2 dX26 2 Now let L4 C = - (27) D SL2 (28) and F = UL2 (29) Then RBSU. (30) C Equations (24), (25) and (26) may then be written using the further notation that dX X d2X etc dT dr2' CXV + (D + F) X" - (1 - = (31).~i M -u _pf. 2 (CX" + DX) (32) ------—, ---- r _ — ---------

AERONAUTICAL RESEjARCH CENTER - UNIVERSITY OF MICHIGAN UMM-47 and V 2 C"+ (D +F) ] ( 3) These:wpations are now in a form suitable for solution by the compuater. It will be shown later that they can bemused to determine the modes of vibration of a pin-uniform beam by causing the parameters C, D and F to change appropriately. For the solution of a unifonm beam, however, the equations can be put into more useful form., For- a uniform beam I' (34) Equation (22) may then be written u I' fI I I kAGt2 EI |(2. Ae2 Az2 Ak EI AG 2I whence U= - I x = S SN, ( kAG^2 E E where N kG (36) then SL + UL * SL2(1 + N) - D(l + N), (37) F = UL2= SL2N = DN, (38) and DF D N (39)

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN ~- UMM~47......... Equations (31), (32) and (33) become, in terms of the parameters C, D and N, / 2 CX1 + D(1 + N)X" - ( X =0 (40) M e-L (CX" + MD) (41) and L (1 -~ IC )" [] (42) L( [ +D( +N) It should be noted that the new parameter N, as given by equation (36) N. (36) is dependent solely upon the ratio, G/E, of the modulus of shear and the modulus of elasticity and upon the shape of the cross section of the beam, through the constant k. A. Normal Modes of Oscillation of a Uniform Beam with Free Ends For a beam with free ends the bending moment, IM; and the vertical shear force, V, must be zero at each of the ends, i'e., M = V= 0, x = 0,i, or 7'r, L. (43) For this case the equation to be solved is equation (40) CXIV +D(l + N)X - (l X (40) subject to the boundary conditions obtained by setting equations (41) and (42) equal to zero at T= 0, L, CXR + DX. 0 at 7T= 0, L (44) ----------------------! 7; —---------

AERONAUTICAL RESEARCH CENTER-UNIVERSITY OF MICHIGAN UM-47 and CX"' + D(l + N)X' = 0 at r= 0, L. (45) (1) Neglecting the Effect of RotaryInertia Ormondroyd, Hess and Hess have obtained a series of solutions for the first three normal modes of oscillation of a uniform beam. Their solutions include the effects of bending moment and vertical shear force, but exclude the effect of rotary inertia. The exclusion of rotary inertia makes the parameter N equal to zero. Equations (40), (44) and (45) then become CXIV + DX" - X 0 (46) with CX" + DX 0 at 7T O, L. (47) and CX" + DX' = 0 at r= O, L. (48) The computer circuit for obtaining the solution of equation (46) is shown in Figure 1-1. The first six amplifiers represent the equation itself. The last two amplifiers are used to observe the requisite end conditions as expressed by equations (47) and (48). These boundary conditions are met by the proper adjustment of the initial voltages (at time r= 0) V3, V4, V5 and V6 on the feed back capacitors of amplifiers A3, A4, A5 and A6. The switches S3, S4, S5 and S6 are relays with normally closed contacts. The solution is started by energizing all four relays simultaneously. The solution of the equation is obtained by trial and error. The voltage V6 is made a suitable fixed voltage, e.g., 6 or 24 volts. The potentiometer associated with amplifier A is then adjusted until the output from 4 Ormondroyd, Hess and Hess, University of Michigan Engineering Research Institute, Third arogress Report on Dynamics of a Ship's structure (March 1, 1949), Office of Naval Research Contract N5ori - 116 (Univ. of Mich. No. M670-4). ----------------- 8, —---

AERONAUTICAL RESEARCH CENTER- UNIVERSITY OF MICHIGAN l —-l. —-UMII-U47 - I l. —- W V --- i -— y 11, - | - A, -X 1: Am 1' A1 -1'" A AlA i —-'I ~ V v 10 Ay-ca'-ox I;; A, cxr, Figure 1-1. Computer circuit for obtaining normal mode solutions of uniform "floating" beam, excluding the effect of rotary inertia. -— 9."Ae!

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-47 e.| amplifier A7 is zero, thereby satisfying equation (47) at the time T= O. The voltage V5 is then given an arbitrary value and subsequently the voltage V3 is adjusted to such a value that the output of amplifier A8 is zero. Thissatisfies equation (48) at the time r= o. With the initial boundary conditions having been set, a trial solution is obtained by energizing the initial condition relays. The output voltages from amplifiers A7 and A8 are recorded on a suitable recording device. A satisfactory solution is obtained when these output voltages are simultaneously equal to zero. It is highly improbable that a correct solution will be obtained on the first trial. The voltage V5 is now given a new value and the voltage V3 is 0 3 again adjusted for zero output of amplifier A and another trial solution obtained. These trial solutions are continued until the final boundary conditions are satisfied, i.e., until a solution ib obtained where"the outputs from amplifiers A7 and A are simultaneously zero. ^ Figure 1-2 shows a correct solution for a first normal mode of vibration. It should be noted that for the correct solution the curve of equation (47) is tangent to the zero axis at the end of the solution. At the same time the curve of equation (48) must cross the zero axis since it is the derivative of equation (47). Although the record of equation (48) is not necessary for observing the final end conditions it is needed for determining the time L during whioh the solution takes place. As described above, this time L is proportional to the length of the beam being studied and is necessary in interpreting the solution. The procedure, details of which are given later, is to run a series of solutions with appropriate coefficients C and D. In each case the time, L, for the solution is determined. From the value of L, the quantities S and R are obtained py using equations (27) and (28). The several values of Z2 EI R ( 2 are plotted against the corresponding values of S shown in Figure 1-5. Since the values of R and S involve the square of L it is necessary to determine L as accurately as possible. To this end there are recorded on the oscillogram time pulses (in some cases four per second, in others ten per second) obtained from a synchronous motor contactor. The exact value of the time interval between successive pulses is obtained from the instantaneous line 10

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMI-47 C = 1.001 D = l.000, i sec pulses, f = 59.89 Cs8 t-L -- 1 \ -5 - -t — -- v-V - - - "..-. \.:^ ^ ^ ^ \ - \D: \ -\ - \ I igure 1-2. First mode solution for uniform "floating"| beam excluding the effect of rotary inertia. C DCX DXX C- T r I i = 1. 0, _A I i I IA. Figure 1-3. Second mode solution for uniform "floating" beam excluding the effect of rotary inertia.... W11M 1 tH1 riur 1 12 F\irs md solutio fo unfr'fotig

I CX " ~ DX ) Figure 1-4. Third mode solution for-uniform "floating" i7 C =D*1lbeam excluding the effect o i / / f rott in ia, f7/,...,i,,D',, 1,., t Figure 1-4. Third mode solution for uniform "floating" I beam excluding the effect of rotary inertia.:.~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.l 11 4 ~ _.,I_

AERONAUTICAL RESEARCH CENTER-UNIVERSITY OF MICHIGAN U1-47 frequency as Indicated by a Leeds and Northrup recording frequency meter. The value of L in seconds is obtained as follows. By comparative measurements the length of the solution on the oscillogram is expressed as L" in terms of the time pulses. Ihis value of L" is then corrected for line frequency error and expressed as L'. The value of L' is then reduced to a value L in seconds. In Table I is an example of determining one value of L and the corresponding values of R and S. These values of R and S are recorded on the first line of Table II. TABLE I Oscillogram L" (1/4 sec) Frequency Frequency L' No. Mode Uncorrected (cycles/sec) Correction (1/4 sec) 7/15/49/1 1 -6.6 59.89 + 0.03 16.69 7/15/49/2 1 t' 16.70 59.94 + 0.02 16.72 7/15/49/3 1 16.70 59.94 + 0.02 16.72 7/15/49/4 1; 16.68 59.92 + 0.02 16.70 7/15/49/5 1 16.66 59.92 + 0.02 16.68 L' (avg) 16.70 1 L a 16.70 x = 4.175 sec. 1/C = 1.000 L = 17.43 1/D = 1.001 L2 D R = - = 17.4 S = = 0.0573 Slightly different values of the initial voltage, V5, (with V3 adjusted for zero output from amplifier AP, Figure 1) will produce solutions corresponding to higher normal modes. Figures 1-3 and 1-4 are the records of equations (47) and (48) obtained for the second and third normal modes with the 1 1 coefficients C and D of equation (46) being, E = 1.000 and ~ = 1.001. In Table II are listed the pairs of values of R and S obtained for the first three normal modes of vibration for various values of coefficients..............- 1 3

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UM'i-47 TABLE II Lode 1/C 1/D R S 1 1.000 1.001 17.4.0573 1 1.001 1.399 18.8.0381 1 1.001 2.000 20.0.0251 1 1.0dl 4.0C8 21.3.0117 2 1.001 0.3989 27.1.0926 2 1.001 0.5011 31.3.0639 2 1.000 1.001 43.5.0230 2 1.001 1.399 48.1.0149 2 1.001 2.000 52.1.0096 2 1.001 4.008 57.4.0044 3 1.001 0.3989 50.3.0499 3 1.001 0.5011 58.6.0341 3 1.000 1.001 82.8.0121 3 1.001 1.399 91.8.0078 3 1.001 2.000 100.4.0050 C and D. These pairs of values of R and S appear as circles in Figure 1-5. The solid lines are curves obtained by Ormondroyd, Hess and Hess. The curves of Figure 1-5 may be used as follows. From the dimensions of the beam and the properties of the material of which it is made the value of S is calculated. The corresponding value of R is determined, for the first, second or third mode, as desired, and the value of the angular frequency, WA, calculated. If desired the mode shape may be recorded by using the output, X, from amplifier A (Figure l-l). In general it is more satisfactory to take this record at the same time records are made for determining the proper end conditions. The conditions under which the accompanying results were obtained made it impracticable to make more than two recordings simultaneously. Consequently very few oscillogrens are shown for mode shapes. 1 Ibid 14

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-47 120. -\- ORMONDROYD,HESS AND HESS (100~ -______ ~ ELECTRONIC ANALOG COMPUTER El 80.___ _....... 1\ \ I I EI R 60 __' _ 40 _ ___. 20 I —~ I I I FIRST MO..DE 0.0O,.04.06.08.10.12 S Figure 1-5. R vs S for uniform "floating" beam excluding the effect of rotary inertia. -------------------- 35 -------------------

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN U.2.M-47. (2) IncludilS the Effect of Rotary Inertia dhen the effect of rotary inertia is included the equation to be solved is equation (4d), iv OXi D(1 + N)X" - (1 - )X = 0, (40) subject to the boundary conditions CX" + DX = 0, at 7= 0, L, (44) and CX"' + D(1 + N)X' = 0, at 7r= 0, L. (45) In these equations the rotary inertia is introduced by means of the parameter N. Since N = kG/E, and solutions are being obtained for uniform beams this rotary inertia term depends only on the material of the beam and the shape of its cross section. For the solutions given in this report N was arbitrarily given the value 0.25. The computer circuit for obtaining the solution of equation (40) is shown in Figure 1-6. Fundamentally this circuit is the same as that shown in Figure 1-1. The first six amplifiers (A1 to A6) represent the equation itself, while the last two amplifiers (A7 and i8) are used to observe the requisite end conditions as expressed by equations (44) and (45). The circuits differ to the extent that 1 was set equal to zero in the circuit of Figure 1-1. The technique for obtaining a solution from the circuit of Figure 1-6 is the same as that previously described. Arbitrary initial conditions which satisfy equations (44) and (45) are set up and trial solutions made (with different initial conditions) until the final boundary conditions are satisfied. In Figures 1-7, 1-8 and 1-9 are shown correct solutions for the first three normal modes of vibration. It should be noted that for the correct solution the curve of equation (44) is not tangent to the zero axis at the end of the solution, as was the case for the corresponding equation (47) when the effect of rotary inertia was not included. In the present case the criterion for a correct solution is that the curves for equations (44) and (45) cross the zero axis at the same time..'/hen the effect of rotary inertia wos not included a 16

AERONAUTICAL RESEARCH CENTER ~ UNIVERSITY OF MICHIGAN 113 =43 AI —— I'- At X AS - -X. —t/^ —-~ l 1- 1 DCOIX D+N) i Jilli I ~ ~ ~'IV~~~IV AX X A -X AX V_?? -. _ v AI —X D- ) *-DX A GX+D(t+N) Figure 1-6. Computer circuit for obtaining normal mode solutions of uniform "floating" beam, including the effect of rotary inertia. 17

AERONAUTICAL RESEARCH CENTER- UNIVERSITY OF MICHIGAN UM-47' I \THE BRUSH D EVELOPMENT CO. PR.N,= 0,N U 0.S500 A.,.S A S-1.-:.A A A -1/10 G plses f -600 —- - — / —-: —- ------- I i70-:10:1- 1 f I If I IX ITCX" 1.I 1+- t1110I t. fiI THE BRUSH DEVELOPMENT CO. PRINTEQ IN U.S.A. I I7 —-/:t 1 1'/- /I UJ I / / / t 1 r/ /11 t I - ^T -f - 4 * rl 1 l -l Z -;, z Z liL rl- -z- Z 7 70/0 ZX 1 t I E0::0107 T T —f —-li-A-I Figure 1-7. First mode solution for uniform "f ioating* beam includine tie effect of rotary inertia. 1 ------------- ------ ^ 18 _

1 I ^-/ ^-/-/^-N-^N-N^1^^^~~~~~~~~~~~~~~1 0 -CX" DX r %J V\ AM 1 VVV-A A 05 ^ PRINTgp IN U.5.A. CHAR NO. BL99 gm BRI BA>~~~~~~~~~~~~~~~~~~~~~~~~~~ >^ ** 1 t-7 (7f!:/^ tg> ~ ~I::^:i,^-i~i ^^^^ —— ^ ^- ~ ^ ~' - - -- - -i:^:'^- ---- -- -~: \ r~-"~~-_': ~i- |r~"-\-._~ ~"" - - \ —~ """^- ~ ~^.T~~r ~ ~~- i_- ~~e elWC *| ^^fflIffife^^^ V \ A k A\ V V. 7 \ \ \ \ T' or~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~! to 5.? /-/ /:- / / f.:j-+ —}==} -:* rr-/_-/_- /^/.;._ /- /; J ^ _L-J-__:jLjLL. . -. J -JIj. JI I =/ I 7i 4 7Co *c+ z ^~ ^ /t-^^^^ n~C 1.000, D 1 l000,O..25 2'Mi )~k/Pt seI p 0 -CXU DX~~~~~~~~_ - 1. 1 -------- IA —(~~~~~~~~~~~~~~~~~~~3~~~~~~~ C+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C pJ a, c -- ^Ox _-_ DX —. ---- __- - -- ^ —-_ _ ___ ^ __ _''-~i^ j~~~~ ~:: j —L ^i ^.^i.i.J -.- — *-^i lyT r:.. * *.t T ". ~s:. f'..~t ^ T. J:. - 7 10 1 ag pui1 r IC 60 002oesofi CX" I + D(l + )X S1dA~i l b\ \.\ \'.ttl^TOj

79:7K _ _ _~_ ______ L i~~~~~~I~~~~~f.~~~~~~~~.~I.LJ? ^ 7 —^ ^ =e pulses, f ~ 60.T10 1W L- /X - -7-~- - _ _ -- - - HCXH' 4 D(X + ^X' _ 0 I A. -A IL.UP —-rf- L.- 1-%,_A A, —- m V U.B.. ----------— CHA Ic NU- B- AL Iu I Figure 1-9. Third. mo~de solution for uniform "floating" -5'beam including the effect of rotary inertia. v- F!"El O -RM " t -,'I;vettolp"IEW V~S _ f.*56 i L _V_ - --: - 7:~~~~~~~1.Iwo~ ~ ~ ~ ~ ~~~ — A —- V- A'A' k-_1 t-~__- _.~. -f- - -.t_ —~- _~- _ ~ I ) IOPINICf _q T C07. IN V —— 57. Lo"Art I Fq 0. Bt 909 ME WKUSH II~i_- -. I A., T,,Thr lrd ou~~o fruior foai~ bea~m including the effect of rotary inertia.

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN......J UMM-47 correct solution could be obtained by observing the bending moment curve alone, since the vertical shear force curve is proportional to the derivative of the former, Kith the effect of rotary inertia included the vertical shear force equation is not proportional to the derivative of the bending moment equation and hence both curves must be recorded to obtain a correct solution. Figures 1-7, 1-8 and 1-9 include mode shapes for the first three modes. Examination of the mode shape, X, for the third mode solution as shown in Figure 1-9 leaves some doubt as to whether the center of gravity of the curve is on the zero axis as it should be for a correct solution. It was just this situation, but more manifest, as discovered in previous work, that led to a correction in setting up the criteria for the proper boundary conditions. This problem should be investigated thoroughly to see if other modifications in the theory are necessary. In Table III are given the data obtained for solutions including the effect of rotary inertia and the computed values for R, S and 1/S. In order to obtain curves which have a greater spread R is plotted against 1/S on semilogarithmic paper as shown in Figure 1-10. The results in Table III are plotted as triangles in this figure. The other curves represent the results obtained when the effect of rotary inertia is not included in the solution. The large circles represent the results given in Table II, these results being obtained by using as end conditions the equating to zero of both the bending moment, M,, and the vertical shear force, V. The small circles represent the results obtained in previous work where the end conditions were specified as being that the second and third derivatives of the displacement should be equal to zero. The data for these points are given in Tables IV, V, VI and VII. It is interesting to note that points obtained by both methods fall on the same curve for lower values of 1/S with a slight departure for higher values. The curves shown in Figure 1-10 permit the determination of the frequency of vibration of a uniform "floatingn beam of given size, shape and material. It would be of interest to investigate more thoroughly the effect of proper end conditions on the R vs 1/S solutions for normal modes when rotary inertia is neglected, particularly for large values o 1/S, which corresponds to long thin beams. ----— i — "~21 --

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UmiBn;-47 TB3LE III C - 1.000, N = 0.25 L2 L2 S Mode D Avg L L2 R= vc SC L2 1 1.000 3.782 14.30 14.30 0.0699 14.30 1 1.000 3.784 14.32 14.32.0698 14.32 1 0.667 4.052 16.42 16.42.0406 24.63 1 0.500 4.221 17.82 17.82.0280 35.64 1 0.333 4.369 19.09 19.09.0175 57.27 1 0.250 4.466 19.94 19.94.0125 79.78 1 0.200 4.541 20.62 20.62.0097 103.1 2 1.667 4.927 24.27 24.27.0687 16.18 2 1.333 5.573 31.06 31.06.0429 23.30 2 1.000 6.088 37.06 37.06.0270 37.06 2 0.667 6.638 44.06 44.06.0151 66.09 2 0.500 6.941 j8.18 48.18.0104 96.36 2 0.250 7.405 54.83 54.83.00456 219.3 3 1.667 7.264 52.77 52.77.0316 35.18 3 1.333 7.875 62.02 62.02.0215 46.52 3 1.000 8.520 72.59 72.59.0138 72.59 3 0.667 9.282 86.15 86.15.00774 129.2 3 0.500 9.653 93.18 93.18.00537 186.4 -----------— 2 —--------- 22

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN t"UM —47 200 180 4th,MODE ROTARY INERTIA NEGLECTED 4 th MOE R N 0 o INCORRECT END CONDITIONS 160- l lo 0 CORRECT END CONDITIONS ROTARY INERTIA INCLUDED 140 20 40 70 10 200 700 1 N40.25 Figure 1-10. R vs 1/S for uito __ "float bam, 120!2 l -rd MODE i80'. N-0.25- 0"25 80 -00- _______._ _ 100 70 100 200'nd MO DE 60 ~.,. 40 N- 0,25 I St MODE N- 20 I0 20 40 70 I00 200 400 700 I000 2000 406 00 I/S A EI Figure 1-10. R vs 1/S for uniform "floating' beam. -------— fing" beam.

ARONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN TABLB IV First Mode - Rotary Inertia Neglected Incorrect End Conditions 2 2 C D L L Rs _ D 1.000 1.429 3. 900 15.21.21i 10.65 1.000 1.176 4.030 16.24 16.24 13,80 1.000 1.o0 44.a 17.07 17.07 17,07 1Vi000 0.66 t 4.*326 18.71 18.71 28.06.l,000 0Q.500 4.426 19.59 19'.59 f 39.18 1.000 0.33 4.521 20,44 20.44 61.32 1.000 0,200 4*567 20.86 2086 83.43 1.000 0.2004 4.614 21.29 21.29 106.2.QOO i 0.1431 4.641 21,54 21.54 150,,6 4.00.2013 6.625 43.89 21.95 218,1 1.,000 0.0669 4.692 22.02 22.02 328.9 2.000 0.0669 5.606 31.45 22.24 469,9 2.000 0.0503 5,607 31.44 22,23 624.8 1.000 0 4.73 22,37 22.37 oo -2.000; - 0 5.63 31.70 22.41.... —---------------—. g ~~4 -----------

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN U A-47 TABLE V Second Mode - Rotary Inertia Neglected Incorrect End Conditions C D L L2 R L 1.000 2.500 5.192 26.96 26.96 10.78 1.000 2.000 5.588 31.23 31.23 15.61 1.000 1.818 5.700 32.49 32.49 17.87 1.000 1.667 5.838 34.08 34.06 20.45 1.000 1.250 6.263 39.23 39.23 31.38 1.000 1.000 6.541 42.78 42.78 42.78 1.000 0.833 6.736 45.37 45.37 54.40 1.000 0.667 6.947 48.26 48.26 72.40 1.000 0.500 7.177 51.51 51.51 103.0 1.000 0.3333 7.401 54.77 54.77 164.3 1.000 0.2004 7.560 57.15 57.15 285.2 1.000 0.1429 7.670 58.83 58.83 411.2 1.000 0.1005 7.735 59.83 59.83 595.3 1.000 0.0669 7.780 60.53 60.53 905 1.000 0.0503 7.790 60.68 60.68 1206 2.000 0.0503 9.290 86.30 61.02 1715 1.000 0 7.85 61.62 61.62 co 2.000 0 9.33 87.05 61.55 25

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMMl -47 T-BLE VI Third Lode - Rotary Inertia Neglected Incorrect End Conditions 22 2 C D L L2 R L L D 1.000 3.326 6.414 41.17 41.17 12.38 1.000 2.500 7.129 50.82 50.82 20.33 1.000 2.000 7.640 58.37 58.37 29.19 1.000 1.429 8.395 70.48 70.48 49.33 1.000 1.000 9.065 82.17 82.17 82.17 1.000, 0.667 9.635 92.83 92.83 139.2 1.000 0.500 9.920 98.41 98.41 196.8 1.000 0.3333 10.27 105.5 105.5 316.4 1.000 0.2494 10.55 111.2 111.2 555.3 1.000 0.1430 10.71 114.6 114.6 802 1.000 0.1005 10.80 116.6 116.6 1161 1.000 0.0669 10.86 117.8 117.8 1760 1.000 0.0503 10.89 118.6 118.6 2357 2.000 0.0503 12.99 168.7 119.3 3353 1.000 0 11.00 121.00 121.00 ~0~ 0.500 0 9.245 85.47 120.87 or 0.250 0 7.780 60.53 121.06 (No 26

AERONAUTICAL RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-47 TABLE VII Fourth Mode - Rotary Inertia Neglected incorrect End Conditions 22 2 2L2 1 L C *D L L 1 L S D 1.000 4.000 7.397 54.72 54.72 13.68 1.000 3.326 7.987 63.79 63.8 19.18 1.000 2.853 8.455 71.49 71.5 25.06 1.000 2.222 9.280 86.12 86.1 38.75 1.000 2.000 9.591 91.99 92.0 46.00 1.000 1.667 10.15 103.0 103.0 61.8 1.000 1.250 10.96 120.1 120.1 96.1 1.000 1.000 11.48 131.8 131.8 131.8 1.000 0.7133 12.14 147.3 147.3 206.5 1.000 0.5000 12.70 161.3 161.3 322.6 1.000 0.2857 13.28 176.4 176.4 617.3 1.000 0.2000 13.51 182*5 182.5 913 1.000 0.1176 13.78 189.9 189.9 1614 1.000 0.0717 13.94 194.2 194.2 2709 1.000 0.0503 13.96 194.9 194.9 3873 1.000 0 14.10 198.8 198.8 0.5000 0 11.87 i 140.9 199.2 ~~ 27

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN......- -.....- UM —47 B. Normal Modes of Oscillation of a Uniform Beam with Other End Conditions t In a previous report solutions for normal modes were obtained (1) for a beam clamped on both ends, (2) for a beam hinged on both ends, and (5) for a cantilever beam. The solutions obtained for these beams were made with the assumption that the effect of shear forces and rotary inertia could be neglected. The end conditions included the equating to zero the second and third derivatives of the deflection. In order to obtain completely accurate solutions the effects of shear forces and rotary inertia should be included as well as the correct end conditions of setting equal to zero the bending moment, A, as expressed by equation (44) and the shear force, V, as expressed by equation (45). Hagelbarger, Howe and Howe, University of Michigan Engineering Research Institute, External Memorandum on Investigation of the Utility of an Electronic Analog Computer in Engineering Problems (April 1, 1949), Air Force Contract 133-038-ac-14222 (ProJect L3X-794). 28 -

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN -U1~-47...... CHAPTiR II NORiMAL MODES OF OSCIL4TIONS OF 1CON;-UNIFOR BEAMS In obtaining normal modes of oscillation of non-uniform beams it is convenient to consider the beam to be divided along its length into an equal number of parts. For each of these parts there is determined, or approximated, the average properties of that section of the beam. The beam is then considered to be made up of a discrete number of uniform parts. For each of these parts the equation to be solved is eXiv + D(1* N)Xw - (1 - -)X. (40) For each section of the beam the coefficients C, D and N take on different values. Resistors corresponding to the successive values of these coefficients can be introduced into the computing circuit by means of stepping relays, or their equivalent, as described in the previous report. From equations (20) and (27), (21) and (28), and (36) it can be seen that the expressions for these coefficients are c 1 L.C=EI ( 2' kAl These relations involve, in addition to the physical prdperties of the section Of the beam, the ratio L/(and the frequency. The ratio L (the length of time of the solution) to 1 (the length of the beam) is arbitrarily and conveniently chosen. The values of D and N can then be determined uniquely for each section of the beamrand appropriate computer resistances found. 29 - | ----—. —

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMl-47 The value of C can not be determined uniquely since it involves the unknown quantity.i However, since 1/w enters as a common constant factor for all the sections of the beam, the resistances involving C can be set up so that the part of C that changes from section to section will be handled by the step2 ping relays and the part involving 1/c. will constitute other computer resistances to be changed from trial solution to trial solution. For example, the changing part of C could be put in the feedback resistance of an amplifier and the 1/J in the input resistance. The details of the aost satisfactory computer circuit have not yet been worked out. In general the computer would follow the plan of the circuit shown in Figure 1-6, with such modifications as would make a minimum the number of variable resistors required and make the solutions obtainable with a reasonable number of circuit changes from trial solution to trial solution. One very evident change is to arrange the two amplifiers (corresponding to A, and A8 in Figure 1-6), for obtaining the proper end conditions, so that but one variable resistance is required for each amplifier. In order to obtain a correct solution not only must the initial voltages on the integrating capacitors be adjusted so that the proper end conditions are observed, but also the variable part of C, i.e., 1/o2, must be adjusted so the solution is obtained in exactly the right amount of time, L. 30

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN 1 1~47 CHAR IIIIl PROPOSED WORK WITH COMPUTER In view of information gained by the use of the computer while doing the experimental work included in this report, the following additional work is suggested. 1. The investigation of whether the mode shapes obtained with the revised end conditions are in agreement with theory. 2. The determination of the effect of the revised end conditions on the R vs 1/S curves. 3. The solutions for normal modes of vibration of uniform beams other than the free-free beam. 4. The solutions for normal modes of non-uniform beams. 5. The solution of the Bessel equation with variable resistances only in the feed back circuits of the amplifiers. In the original solutions for normal modes of vibration of uniform beams it was discovered that the mode shapes obtained in some cases were such as to indicate a vibration of the center of gravity of the bean, a physical impossibility. As a consequence, the boundary conditions at Witber end of the beam were changed from, setting equal to zero the second and third derivations of the displacement, to, equating to zero the expressions for the bending moment, M and the vertical shear force, V. In connection with the present work it Was found that these new end conditions improved the situation but left some doubt concerning whether the mode shapes obtained are in perfect agremnent with the physical case. For small amplitudes of vibration of the bar the area under the mode shape curve, X, should be equal to zero for the center of gravity of the curve to remain on the axis. Consequently, the output of an amplifier, integrating the value of the displacement, X, for the duration of the solution, should give the 31 l

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN,Ub.; L-47 required information. In case the value of this integral, at the end of the solution, is not equal to zero the matter of boundary conditions should be investigated further. In Figure 1-10 are plotted curves of R vs 1/S on semi-logarithmic paper. For the cases where rotary inertia is neglected there are plotted data obtained by using (1) the original end conditions (second and third derivatives of the displacement equal to zero) and (2) the revised end conditions (M and V equal to zero). Since the points representing this data fall almost exactly on the same curves, it is suggested that further investigation be made to determine what effect, if any, the end conditions have on determining the frequency of vibration of the beam. As implied earlier in the report it would be of interest to make studies of uniform beams other than the free-free beam. This would include the clamped-clamped, hinged-hinged and cantilever beams. The correct end conditions should be used with particular attention to how these end conditions affect the determination of the frequency. Mode shapes should be examined to see if their centers of gravity remain on the zero axis. Since the ultimate objective of the project involving the study of vibrating beams by means of the electronic differential analyzer is to determine the normal modes of vibration of non-uniform structures, such as ships and aircraft, it is suggested that considerable attention be given to the development of a computing circuit which will handle this problem satisfactorily. In connection with the solution of Bessel's equation as given in the previous report it was pointed out that the step approximations of 1/x and 1/x are not equivalent to the reciprocals of the step approximations of x and x, respectively. Because of the difficulty involved in obtaining step approximations of the reciprocal of a'-variable in the neighborhood of zero, it is suggested that the equation be used in the form 2 2,x X2 + x Z + (x2 - n2)y = 0 d2 dx In this form the step approximations for x and x2 can be used satisfactorily. Some of the computer curves, such as those for J,/q and J./ differed 1 loc. cit.....-.-.............-..-. 32 -----------------------

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN.........:....I UM11-47 materially from the theoretical curves. It would be- of interest to determine how the proposed method of solution affects the disagreement. 33

AERONAUTICAL RESEARCH CENTER- UNIVERSITY OF MICHIGAN.... ——.... UMLd-47. CHAPTER IV CHANGES IN THE C0EPUTER For the work associated with this report several changes and additions were made to the computer. These include an improvement in the technique of balancing individual amplifiers, a more exact method of measuring time intervals on the records, and a plug-in mounting for a number of amplifiers. Balancing of Individual Amplifiers In order to facilitate the balancing of individual amplifiers there was mounted on the chassis of each amplifier a double pole double throw switch. When this switch is closed in one direction (for balancing) the input of the amplifier is connected through a resistor RB (approximately 160,000 ohms) to ground and simultaneously a feedback resistor Ri (approximately 10 megohms) is connected into the circuit (Figure 4-1). Since the values of Ri and Rf are selected to give large amplification, balancing is readily accomplished by adjusting one or both of the balancing controls of the amplifier until the output voltage as read on a DC vacuum tube voltmeter is reduced to zero. then the switch is closed to the other position the input and feedback impedances used for computing are properly connected. This procedure permits the operator to check the DC balance of the amplifier without disconnecting any of the impedances used in the computing pperation. Time Measurements The results obtained in the experimental work depend on the square of L, the time, as measured on the oscillograph records, for a solution to be obtained. In order to obtain the desired accuracy in these measurements, a penmarker was arranged to record time pulses on the oscillograph paper. These time pulses were obtained from a one revolution per second synchronous motor with a regular polygon cam operating a microswitch. For some of the records the time pulses were four per second; for others, ten per second. Simultaneously with the taking of the record there was observed and recorded the power line frequency. -, I. I —-------------....... I 34 -.~~~~~

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN.,I -------- UMIU47v-47 ein eoutz Figure 4-1. Balancing switch on amplifier chassis. as indicated by a Leeds and Northrup frequency recorder. From these observations time measurements could be made to an accuracy of about one-tenth-of one percents Plug-In Mountin In order to arrange the amplifiers so that the computer should occupy a minimum amount of space, a plug-in mounting chassis was made to accommodate ten amplifiers. This chassis consisted of two shelves arranged step-wise, one above the other, each shelf supporting five amplifiers. The female power supply receptacles were mounted in vertical risers and spaced so that adjacent amplifiers were separated about one-eigth inch. All Meads to these receptacles were shielded. In addition to giving a compact set-up, this arrangement dispenses with the clumsy power cable leading to each amplifier. 35

ERONATICAL RESEARCH CENTER -UNIVERSITY OP MICHIGAN ^ —-------— UaI-47 ----—. CHAPTER V PROPOSED CHANGES AND KODCIFIOATINS FOR THE CCPFUTZEt iDurng the work with the computer it became evident that a number of modifications and additions should be considered. These proposed changes include automatio balancing of. individual ampliiers, the application of initil condiioins trin- a remote position, reconstruction of the basic amplifiers, different tyes of recording equipment, improvement in the making of accurate time measure, Itits, 0etc. These suggestions should be interpreted in the light of the work to be done with the computer. Some of them would be important for certain types of work and not for others. atomatic Balancing of Individual Amplifiers Since the individual amplifiers of the computer frequently require | balancing it is suggested that some method of automatic balancing be considered. In the method here proposed the individual amplifiers are balanced consecutively. By means of auxiliary equipment each amplifier is automatically converted into Ia high gain DC amplifier, the output of which is reduced to zero by a suitable control mechanism. Each amplifier is caused to be balanced in turn by a stepping relay. In Figure 5-1 are shown several additions to the individual anmplifier chassis. One is a DPDT relay used for automatic balancing. In its norm&l, unenergized position this relay connects the input impedance Zi and the feedback itpedance Zf to the amplifier for computing. When the relay is energized (by remote control) the amplifier is converted into a high gain DC amplifier using as input and feedback impedances the resistors Ri and Rf* When the balancing process is completed these two resistors are disconnected from the computer, except for a ground connection, so that any voltage picked up on the long balancing output lead is not introduced into the computer. It should be noted that during the process of balancing the output terminal at e0 remains connected to the feedback impedance Z 80so that the initial

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN -----------— U Uiam-47 REMOTE INITIAL CONDITIONS REMOTE OUTPUT FOR CONTROL REMOTE * ( C BALANCING I^~ RELAY i,. mCOMPUTER OUTPUT INITIAL CONDITION RELAY REMOTE COM BR-CONTROL Z7 AUTOMATI Rf BALANCI — RELAY COM_ W —71T3 L Zi ei R; a I AMPLIFIER | Figure 5-1. Partial wiring diagram of an individual amplifier chassis, showing the connections to the automatic balancing relay and the remote initial condition relay, --------------------—. 37_.. —------------

AERONAUTICAL RESEARCH CENTER UNIVERSITY OF MICHIGAN UMM-47 condition, if any, imposed on Zf is maintained at the output terminals. In Figure 5-2 are shown some of the details of the auxiliary automatic balancing equipment. The stepping relay has three banks of contacts, each bank having connections to the individual amplifiers. The terminals of bank I, labeled BA1, BA2, etc., are connected to the balancing output leads of the several amplifiers, which are connected consecutively to the input terminals of the special amplifier for balancing. The terminals of bank II, labeled BM1, BM2, etc., are connected to leads carrying power to small, reversible DC motors, one on each amplifier chassis for operating a balancing control. The terminals of bank III, labeled BR1, BR2, etc., are connected to leads which energize, at the proper time, the automatic balance relay on each amplifier chassis. The stepping relay is stepped forward periodically by a contactor operated by a motor of suitable speed. As shown, the circuit provides for continuous automatic balancing. Provision could be made for one cycle of balancing, for push-button balance of any one amplifier, for automatic "homing" of the stepping relay when fewer than the maximum number of amplifiers are being used, etc. As described above, during the process of balancing the output voltage of each amplifier is connected to the input of the special amplifier for balancing. This amplifier has input and feedback impedances, Bi and Bf, of such values as to give suitable gain to the amplifier. The output of this amplifier is fed into a load consisting of two resistors, R1 and R2, in series. These two resistors are given such values that the output voltage is limited for relatively large input voltages and that voltages of suitable magnitude are applied to the input of the controller. If necessary for the proper performance of the controller, attention should be paid to its input impedance. bhe controller as shown in the circuit is indicated as performing a simple FORNARD-OFF-REVERSE operation, actuating either one of the two relays, electrically interlocked, to turn the balancing motors in the necessary direction. This method of balancing, if unmodified, would probably result in sluggish action (if the balancing motors are geared way down) or in excessive hunting (if the motors operate the balancing controls very rapidly). There are several methods by which this condition could be corrected. One is to use a controller which energizes the balancing motors for a fraction of a given time cycle, this fraction of time to be proportional to the error in balancing; This method is used. —- 3 1 -.-,,.-...38

o~ 2=- ~ ~ I I I H Q Q Qt ft fQ Q Q 0 AMPLIFIER | FOR | |^ | | tBALANCING 0 l 6 - B, f- R' CONTROLLER L(D Bi RI - 0 — 0 <R2: COM I I- w BM3.(._ - _ BM2 I BMIO -- - BM I = I C) BMIO — -0 IL j _.C) BM9 o 0 —- 0 BM7 0 —----— O — I |BM6 0 BM5 0 BM40 -- ro(0.9... 0_, 0 =-~ — a D /f - -1 I -I,,,. Y0)0 Fi:,,. ct r -, on t u,-,,, B-l o, Fig. -2 Circuit for Controlling the Automa"tc Balancing of Indivoial An-liL'iers. I-._____________________________________Iz

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UM-147 by the Leeds and Northrup Company in their Micromax frequency controller. Another method would be to use a Speedomax (Leeds and Northrup) controller with a potentiometer or rheostat mounted on it in such a way as to cause the speed of the balancing motors to be proportional to the error in balance. A still further method would be to use electromagnetic damping on the balancing motors. For this method of balancing it is suggested that the balancing motors be small 24 V DC permanent magnet motors of which there are many available from war surplus. Each motor should be geared down to operate one of the two DC amplifier balance controls. This control could advantageously be of the helipot type. The details of these motor operated controls are not described here, but in Figure 5-3 is shown a dial on the amplifier chassis to indicate the position of the automatic balancing control. Reference to Figure 5-2 shows that the special amplifier for balancing is itself balanced automatically in its turn. This amplifier needs no balancing relay. The chassis relays for automatic balancing and the small DC motors have one lead in common. The battery furnishing power for the balancing motors is floating so that either one of its terminals is connected as required to the common line. If desired two batteries could be used, one with the positive lead connected to the common line and the other with the negative lead connected. This method of balancing would probably result in some economy of equipment as compared with some other methods. However, since the amplifiers are balanced one after the other it entails some loss of time. How serious this loss would be depends upon the types of problems being solved as well as the frequency with which the balancing needs to be done. A much more efficient method would be to have an automatic balancing mechanism for each individual amplifier. Such a system might consist of a twophase AC motor driving the balance control, this motor having one phase of its supply voltage furnished by a DC to AC converter-servo-amplifier such as is used in the Speedomax (Leeds and Northrup) of in the Electronik (Brown Instrument Co.) recorders. This method would permit the balancing of all amplifiers simultaneously in a few seconds. A compromise between these two methods would be to use individual twophase motors on each chassis with a single DC to AC servo-amplifier, the automatic balancing to be done consecutively but rapidly. oec.-D^endix I -. 40

10,-|9) W S0, BAL. RELAY PWR. m ~ g 1^ l? -^" pr3 ~BAL. MOTOR PWR. L 0 I Z V ~ \: RELAY a MOTOR GOM.? _.N 5~ OA \ X / - CHASSIS GND. 0 I.. 0. _ /.16 V. FIL. =- ^ |. -:'""!~;-350 VOLTS S g: -190 VOLTS,.o_"'-+ S 3350 VOLTS P g., \ INITIAL CONDITIONS t 0 fr |0^ () < 4 w cO j OUTPUT FOR BALANCING Z o m gO'- INIT. CON. RELAY PWR. 0~*, | (h) 03 r 3: 7 Fl COMPUTER OUTPUT 03. -4 ~,~~~~~~~~~~~~~~~~

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMIg-47 Initial Conditions In order to keep the computing assembly more compact and convenient for operation it is proposed that an initial condition relay be mounted permanently on each chassis, the connections to it being as indicated in Figure 5-1. This scheme permits thy initial condition voltage to be applied to the feedback capacitor from a remote position. It would be convenient to have the remote initial condition terminals for the several amplifiers mounted on a panel. On the same panel, or on an adjacent one, there should be available a suitable number of variable voltage sources for the initial condition voltages. Connections would be made by patch cords. In the cases where the initial condition voltage is zeo, a shorting resistor of low value could be placed across the remote initil condition terminals. The coils of all of the initial condition relays would be connected in parallel for simultaneous operation by a single starting switch. Jhen these relays are energized the remote initial condition voltage supplies are completely disconnected from the computing system, as can be seen in Figure 5-1. In case there is a situation where one initial condition voltage depends on another one, it could easily be arranged to have this relationship adjusted automatically by means of a servomechanism similar to that used in the automatic balancing of the amplifiers, This provision would have been particularly useful in the solution of the problem on the determination of the modes of vibration of oscillating beeas where two such interrelations between initial conditions occur. Amplifiers In doing the experimental work contributing to the results summarized in the first part of this report considerable difficulty was experiencedwith the basic DC amplifiersa Without apparent cause an anplifier would change so that it could no longer be balanced with the two balancing controls provided. The trouble was found to be caused by large changes in the Values of some of the 1 megohm and 2 megohm resistors in the amplifier circuit. There seemed to be some correlation rith the hot, humid weather prevailing at the time. Consequently, the computer was moved to a room with constant temnerature (70~ F) and constant humidity (50%). The mortality rate of the amplifiers changed from one or two a................- ---- - -42 -..

kERONAUTICAL R FSEARCH CENTER- UNIVERSITY OF MICHIGAN ------- ----—:~ UIxi-47 day to almost none. In the six weeks during which the computer was used in this roos only oei pmplifier had.to be repaired. Jbhile it was never determined justwhat caused the resistors to change values, the fact that the difficulty disappeared as soon as the amplifiers were put into a constant temperature and constant humidity atmosphere suggests high humidity as a large contributing factor. In order to avoid difficulties of this: kind it is suggested that the ordinary carbon resistors used in thQ amplifiers be replaced by a more stable type, such as the Continental X-type.- Alternately, o:rn- ~addition the resistors could be covered with a layer of suitable wax to prevent the absorption of moisture. The use of the more stable type of resistor is recommended.' On the other hand, if the trouble experienced was really due to atmospheric conditions it would be desirable to investigate the effect of these conditions on other electronic equipment. The results of such an investigation might indicate the desirability of housing the computing equipment in a room in which the atmosphere is maintained at suitable temperature and humidity. Since the amplifiers used in this work were constructed other DC anplitiers with more satisfactory charcteristics have been developed. This general field should be investigated with the idea of adopting new basic amplifiers for usea:i: the computer.. Recording Equiment Ihe recording equipment used in the experimental work consisted of a 2 Brush, Model BL-202, double channel magnetic oscillograph and two Brush, Model B-L-913, DC amplifiers designed to work into the magnetic oscillograph. Considerable difficulty was experiencedwith zero drift of these amplifiers, so much S0 that the work was made tedious and much time consumed in taking unusable records. The Brush magnetic oscillograph has the advantage of speed in recording, being useful from DC up to frequencies above 60 cycles per second. For New and improved basic DC amplifiers are now under construction and should be -available by June. 2:Brush Development Company, Cleveland, Ohio....-. eB- --------- 43 —. —--— 3

AERONAUTICAL RESEARCH CENTER- UNIVERSITY OF MICHIGAN i erate accuracy it is suggested that this oscillograph be used with a more atisfactory input amplifier.l'-[. W'here more accuracy is demanded, and speed is not so important, it It suggested that either Speedomax (Leeds and Northrup Company) or Electronik (Brown Instrument Company) recording potentiometers be tried. Both of these instruments have wide chart scales, ten and eleven inches, respectively, and record with an accuracy of the order of one-fifth of one percent. At least tone of' tem (Speedomax) can be obtained with chart speeds up to six inches per minute Because of the high input impedance of the recording potentiometer no intermediate amplifier is needed between the computer and the recorder. The Speedotax recorder with a range of 0-10 millivolts has an effective input imped acane, when off balance, of the order of magnitude of 7500 ohms. As a consequence it could be'driven' from a voltage divider connected directly across an output of the computer. These recording potentiometers- have the adisadvantage of being, primarily, DC recorders with the result that they can satisfactorily record only very low frequency alternating voltages. Consequently their use would require the slowing down of the computer solution to such frequencies as the recorders would handle satisfactorily. For general computer work it would be highly desirable to have both the high speed magnetic recorders with suitable amplifiers and the more accurate recording potentiometers. Time.Yeasuremenis In using the computer time is the independent variable and if maximru use is to be nde of the result obtained, the length (in time) of a solution should be known to a high degree of accuracy. While the method for obtaining accurate time ilses on the computer records as described above under Computer Changes is practicable, there is an alternative method of more convenience and accuracy. Several manufacturers produce secondary frequency standards of high accuracy. The Hewlitt-Packard Company's Models 1OOA and 100lOB produce frequencies See.4ppendix II. -i' —- 44 -i-... i, —.....

AERONAUTICAL RESEARCH CENTER- UNIVERSITY OF MICHIGAN t..... -- UMLE-47 --- of 100, 10, 1 and 0.1 kilocycles per second with accuracies of + 0.01% and 0.001% respectively. A supplementary frequency divider.is available to obtain a frequency of ten cycles per second. The General radio Company offers for sale Type 1100-A primary and secondary frequency standards with a frequency drift of 7 8 the order of one part in 10 or 10 per day. These standards give frequencies of 100, 10, 1 and 0.1 kilocycles per second. The General Radio Company uses multivibrators for dividing the frequencies. No 100 to 10 cycles per second multivibrator is offered for sale by General Radio, but this multivibrator could be constructed easily. One disadvantage of multivibrator frequency division is, if any failure, or loss of A synchronism, occurs in the system of frequency division, one or more of the multivibrators will become free running and give inaccurate frequencies. However, for well designed circuits the probability of this occurrence is small. The Hewlitt-Paokasd Company uses a regenerative modulation frequency dividing circuit i lits instrument. This circuit has the characteristic that it is non-regenerative in the sense that it canenot function without a suitable input frequency asd hence can not operate free running. For records taken at high speed the tenth-second time pulses could be recorded directly on the recording paper by using a suitable pen. For slower speeds one-second pulses would probably be desirable. These could be obtained by another multivibrator or by any other suitable frequency dividing device. If desired, a'synchronous motor could be driven by a power amplifier, using 50 or 100 cycles per second obtained from the standard frequency source. Gears and cams connected to the motor could be used to obtain any time pulses desired. In any event the time pulses used would be so accurate that no corrections would be needed. In case variable input or feedback resistances are to be obtained by stepping relays, or their equivalent, pulses from the standard source should be used for operating these relays. 45

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN -~ Ul47 ----—? -4 APPENDIX I CLOSED-LOOP CONTkIUOUS DRIFT COLPENSATION For future work it is planned to use DC computer amplifiers equipped with an additional drift-free slow response feedback loop to compensate slow drifts in the basic DC computer amplifier. A slow response drift-free DC ampll-:fier with an amplification of several hundred is connected with its input to the input junction of the high-gain computer amplifier. The output of th drift-free amplifier is connected to the other input of the computer amplttiferi: which was normally used for manual balancing. The polarity is such that it the7 input junction of the computer amplifier is not at zero potential then the drift-free amplifier applies a voltage to the balancing input of the computer amplifier resulting in an output voltage which acting through the feedback impedance returns the input junction to zero. The slow response drift-free DC amplifier is of the type which uses a synchronous mechanical vibrator switch to modulate the input signal on a 60-cycle supressed carrier. The sidebands are then amplified by an AC amplirier and demodulated back to a relatively large DC signal by another synchronous vibrator switch. The effective drift at the input of such a slow response amplifier can be made negligibly small. Ihe application of this closed-loop continuous drift compensation scheme in no way changes the high frequency response of the computer amplifier, L.L.R. I The condition for ideal operation of the computer amplifier (input grid current neglected) is that the potential of the input junction remains eXactly. at zero. -—. -....... 4,6 - - 11 i...'-'

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN pr —---------- llDX47 --- APPENDIX II MODIFICATION OF THE BRUSH1 DC AMPLIFIER BL-913 FOR USE DIITH AN ELECTRONIC DIFERENIAL ANALYZER by L. L. Rauch and R. H. Dougherty The problem was to make a simple modification in the Brush D.C. Amplifier Model BL-913 in order to reduce excessive zero-polt drift. The Brush Company developed the amplifier primarily for use with their Magnetic Direct Inking Oscil lgraphs such as Lodel BL-202. The amplifier and recorder together form a convenient means of recording the solutions of an electronic differential analyzer, however, the zero-point drift of the anplifiers ovdr periods of several minutes was found to be many times greater than the maximum computer errors. Since the weak link in the computer accuracy was the Brush i paplifiers there were two alternatives, either design new driving amplifiers for;-'' the recorders, or modify the Brush instruments. For economy and convenience. modification of the Brush instruments offered the best solution. The electronic differential analyzer never requires a full scale recorder sensitivity greater than approximately one volt. The unmodified Brush equipment has approximately fifty times this sensitivity and the weak point is that the Brush amplifier sensitivity control is a voltage divider at the input. Thus the amplifier operates at full gain for all sensitivity adjustments and no reduction in the, zero-point drift is obtained when operating at reduced sensitivity. Most of the voltage drift comes from the first stage and rather than attempt to reduce the equJiralent voltage drift at the input grid it appears more practical, in view of the lower sensitivity requirements of the computers 1 Brush Development Company, Cleveland, Ohio 4.7

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN [ —----------- UMLM17 ----- *.. ^'*go.ll to override the drift by operating at a larger equivalent input signal voltage. This is easily accomplished by an inverse feedback loop from the output of the amplifier to the input stage which reduces the voltage gain by a factor of 50 from the original 1000. Herewith is presented the determination of the feedback loop, the design of a D.C. calibraion circuit and the characteristics of the modified. Brush D. C. Amplifier and Recorder. 4 SUMMARY The modifications for the feedback loop are simple and inexpensive, two resistors and a condenser are required (Figure 1). If the original characteristics of the amplifier are desired again it is only necessary to short Rf and open R and the stabilizing condenser. The characteristics of the modified amplifier are as follows: 1. The drift is less than the width of the pen line over hours of operation as compared to as much as 5 nmn on an unmodified anplifier over periods of several minutes. 2. The amplifier - recorder gain versus frequency response is flat to 25 cps (Figure 5) compared to 80 cps on an unmodified amplifier. This reduced bandwidth is more than enough for the computer application so no compensation was included to ext4nd it. 3. The maximum voltage gain of the amplifier is 20, compared with approximately 1000 originally. 4. The balance control has a range of 5 nr each way from center compared to more than full scale each way on the unmodified amplifier. A D.C. calibration voltage is more convenient than the 60 cycle A.C. employed originally because the A.C. required critical adjustments and had to be readjusted for the frequency response of different recorders. The D.C. calibration circuit is designed to put plus or minus one volt D.C. on the input: 48

AERONAUTICALe RESEARCH CENTER UNIVERSITY OFP MICHIGAI -J0&47 -,-i gr ii^(iigure 2). The original milliameter and 50 C potentiometer (aB46).at i used. The extras required are a four pole - double throw switch (C-4 8888) which is off unless depressed, one resistor (R ) and a 6 volt battery. The P | milliameter is recalibrated to read 1 volt at half scale (.375 ma) and is a constant checke on the calibration circuit. R-46 is adiusted for a ha seae reading on the millimeter.: To calibrate the amplifier remove any input voltage, turn the attenU 0 ator switch to "calibrate", throw the calibration switch to plus or minus ione volt, observing that the meter is reading half scale, and vary the gain contro for desired output. EE FEECBACK LOOP To stabilize the amplifier against oscillation when employing 0 tt feedback loop the.01 mf condenser shunting R-14 was necessary. Bode's, "Minimal Phase Shift Criterion" indicated that oscillation would occur up application of 34 db of feedback (/#= 50) because of the 27 db per odetav - slope shown ii Figure 4; by Bode's criterion the slope must not exceed 12 db per octave. The shunting condenser acts as part! of an attenuation network to give a maximum slope of 8 db per octave in the first 34 db of attenuation without feedback. The feedback loop was designed to leave the anplifier gain appro imately 20, or a little less than one volt input forfull scale deflection on the recorder. The values of R and Rf were determined as follows: db gain at 20 cps without feedback (Figure 4) = 60 Voltage gain without feedback t 1000 = Or Voltage gain with feedback. 20 A |Froa negative feedback relationship, A." 5.049 Aiao A' R+R ~ "f l-E 49

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN..-........... UMWI-47 R was chosen as 5000 ohms so R. 250 ohms. E I CALIBRATION CIRCUIT The D.C. calibration circuit for plus or minus one volt input is made up of circuit mponents already in the amplifier except for a 6 volt battery, a double th w - four pole switch and a resistor. Ihen the switch is depressed R is placed in parallel with R-40 and the series network of p R-45 and R-56 so that one volt will appear across the combination at *375 ma.The 50K potentiometer (R-46) is adjusted for a half scale reading onthe - millianmeter which is.375 ma, thus the meter is a check on the condition f' the calibration circuit. The calibration switch does not open the input lead, thus it is necessary to disconnect any input voltage when calibrating. There is pace for a 6 volt "A" battery on top of the chassis; the battery was wrapped with asbestos paper to reduce voltage variation due to the heat from the tubes. The attenuation switch was relabeled as shown in Figure 3. --—: —-: —. —,: 50 - -'' —- - - L-: —50

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN -ML-47....... V-5 V-7 STABILIZING CONDENSER Tr'\ /Rf2.I I I c-13~1>.03 INPT I I: _ oeV_ II'- 4 ^I - Mvi. T INPUT /^5 0-13;^?LJ___ 5000 Rf ww ^25 FEEDBACK LOOP Figure 1. odification or Brush empliier BL-913. 51

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OP MICHIGAN -------—. - -TUMM-47 ON OFF ON 2 3 |O 6 SWITCH'719 * LCUTLER-HAMMER- 8888 811I 7 4g 1 Iv V.-., 0..: 2 I\ 3 __ 0I 1 I R-46 I 50K I =6V _ Figure [ I -10 MEG. 3633 9 MEG I MEG 90K 9K 900 100 R-56 CALIBRATE - Figure 2 - Calibration Circuit (+) or (-) IV D-C Input. b001 1 o00 Figure 3 - Recalibration of Attenuator Control 52

h~ V ^U~~~~~~NMIODIFIED AMPLIFIER~ 00 27db /OCTAVE'1 0 Ill, 0~~ ~~~~~~~~~~~~~~~~~~~~~~~!I II I IL ~* 2 40 __ --- -- -- _____.. ^ \ 88db /OCTAVE___ _" C.,, 0 MODIFIED AMPOUFIER 1~~~~~~~~~~~~~~~~~~~1 ~~~~~~~~~~~~~~:::2: 2S.0 20 100 1000 10W000' o P.0 1000~~~~~~~~~~~~~0,0 FREQUENCY C.RS.

0 24 I I I I I I I I I FREQUENCY RESPONSE OF 2BL-913 AMPLIFIER AND BL-202 RECORDER 12 ='______ I - - oL t~l ~ —w-. zX 0 10w 1 FREQUENCY C.RESPON S.E OF gur 5 reqency reponse of BL-915 AMPLlifier AND BL-202 Recorder Q0 <~~~~~~~~~~~~~~~I aN