Application of the Electronic Differential Analyzer to the Oscillation of Beams, Including Shear and Rotary Inertia by C. E. Howe Professor of Physics, Oberlin College R. M. Howe Instructor n Aeronautical Engineering, University of Michigan L. L. Rauch Associate Professor of Aeronautical Engineering, University of Michigan Aeronautical Research Center Engineering Research Institute University of Michigan UMM- 67 January 1951

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AERONAUTICAL RESEARCH CENTER - UNVERSITY OF MICHIGAN UMM-67 PREFACE This report on the vibrating beam problem represents a continuation of work presented in previous reports9'10 The problem as first proposed for solution by the electronic differential analyzer was presented as one involving the solution of a fourth order differential equation subject to boundary conditions imposed by the restraints, or lack of them, placed on the ends of the beam. In this communication it is demonstrated that, by using the basic first order differential equations from which the fourth order differential equation was derived, the electronic differential analyzer serves to determine the eigen-frequencies and mode shapes for vibrations of both uniform and nonuniform beams with the effects of rotary inertia and of transverse shear force included. Chapters 1 and 2 are devoted to a theoretical analysis of the vibrating beam and the setting up of equations suitable for solution by the computer. Chapters 3 and 4 explain the use of the electronic differential analyzer for obtaining solutions for uniform and non-uniform beams. Chapter 5 describes some experimental techniques involved in the use of the computer. Chapter 6 outlines some proposed work evolving from that which has already been done. In the appendix are given tables and curves for the results obtained for the vibrations of uniform free-free beams. In addition there is a description of a driftstabilized dc amplifier, along with a highly stabilized power supply, While a very brief discussion of the principles of operation of the electronic differential analyzer are given in Chapters 3 and 4, it is assumed that anyone who wishes to do experimental work in this field will thoroughly indoctrinate himself in the construction and use of the computer9'10. The authors express their appreciation for the assistance and advice given on numerous occasions by J. Ormondroyd, Professor of Engineering Mechanics and by C. L. Dolf, Assistant Professor of Mathematics. The work described in this report was supported by a generous grant of funds made available by the former Operating Committee of the Aeronautical Research Center, Engineering Research Institute, University of Michigan. The work was greatly aided by facilities and equipment of the Department of Aeronautical Engineering. ----- -i

AERONAUTICAL RESE.ARCH CENTER- UNIVERSITY OF MICHIGAN UMM-67 TABLE OF CONTENTS Chapter Title Page INTRODUCTION 1 1 THEORETICAL ANALYSIS OF THE VIBRATING BEAM 4 1.1 Basic Equations 4 1.2 Notation 5 1.3 Separation of Variables 6 1.4 Boundary Conditions 8 1.5 Theoretical Analysis of the Vibrations of a Uniform 9 Beam 1.6 Solution for a Free-Free Uniform Beam 11 2 EQUATIONS OF THE VIBRATING BEAM FOR SOLUTION BY AN ELECTRONIC 13 DIFFERENTIAL ANALYZER 2.1 Inexpedience of Using the Fourth Order Differential 13 Equation 2.2 Use of the Basic First Order Differential Equations 14 2.3 Change of Independent Variable 14 2.4 Combination and Simplification of Coefficients 16 2.5 The Equations to be Solved by the Computer 17 2.6 Boundary Conditions 18 3 APPLICATION OF THE ELECTRONIC DIFFERENTIAL ANALYZER TO THE 19 SOLUTION OF THE UNIFORM-BEAM PROBLEM 3.1 Introduction to Operational Amplifiers 19 3.2 Computer Arrangement for Solving the Uniform-Beam 23 Problem 3.3 Method of Obtaining the Correct Solution 24 3.4 Computer Solutions for a Free-Free Beam, Infinitely 29 Long 3.5 Computer Solutions for a Free-Free Beam of Finite 32 Length 4 APPLICATION OF THE ELECTRONIC DIFFERENTIAL ANALYZER TO THE 37 SOLUTION OF NON-UNIFORM BEAM PROBLEMS 4.1 The Equation to be Solved 37 4.2 Methods for Varying the Gain of an Operational 37 Amplifier as a Function of Tim* 4.3 Computer Circuit for Solving the Non-Uniform Beam 45 Problem ii

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-67 TABLE OF CONTENTS (continued) Chapter Title Page 5 COMPONENTS AND EXPERIMENTAL TECHNIQUES 48 5.1 Components 48 5.2 Measurement of Resistances 49 5.3 Measurement of Capacitance 50 5.4 Pulse-Forming Circuit for Checking End Conditions 54 5.5 Use of Time Signals from Radio Station WWV 58 5.6 Proposed Electronic Clock 60 5.7 Technique for Measuring the Length of the Computer 62 Solution 6 PROPOSED WORK 63 6.1 A Complete Set of Frequency and Mode-Shape Curves 63 for All Types of Uniform Beams 6.2 An Experimental Study of the Proper End Conditions 64 for the Clamped End of a Beam 6.3 Comparison of Experimental and Computer Values of 66 Eigen-Frequencies 6.4 Checking the Computer for the Solutions of Non- 66 Uniform Beams 6.5 Validity of Basic Equations for Eigen Vibrations 67 of Wave Lengths Comparable to Transverse Dimensions of Beams 6.6 The Need for a Complete and Comprehensive Bibliography 67 APPENDIX 1 Summary of Normal Mode Frequencies for a Free-Free 69 Beam, Including Transverse Shear and Rotary Inertia Forces APPENDIX 2 Description of the Old DC Amplifier 74 APPENDIX 3 The Drift-Stabilized DC Amplifier 76 APPENDIX 4 Description of the Highly-Stabilized Power Supply 84 APPENDIX 5 Time Constant of the Electronic Integrator 94 BIBLIOGRAPHY 97 DISTRIBUTION 98 --------------------- iii

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 INTRODUCTION The study of flexural vibrations of beams, with particular reference to the determination of eigen-frequencies and mode shapes, is a problem in which considerable interest has been developed. It is the purpose of this report to give a brief theoretical analysis of the problem and to demonstrate the usefulness of an electronic differential analyzer in determining the desired solutions for both uniform and non-uniform beams. The ordinary textbook on vibrations presents a very simplified analysis of the vibrating beam problem. Generally it is assumed that the vibration of the beam takes place in one of the principal planes of flexure of the beam and that the length of the beam is large in comparison with its transverse dimensions. This latter assumption permits the effects of vertical shear force and of rotary inertia to be neglected without much loss in accuracy. However, if the beam being studied no longer has a length large compared with its cross-sectional dimensions, these simple solutions are not sufficiently accurate. Lord Rayleigh derived a correction factor to include the effect of 2 rotary inertia. In 1921 Timoshenko set up a group of differential equations which included the effect of transverse shear force as well as that of rotary inertia. From these equations he derived, for the special case of a beam hinged at both ends, a correction factor 2 which could be used to determine an approximately correct frequency. This method of approximation is not readily applicable to beams with other types of end fastenings, as was pointed out by Goens4. Goens derives exact expressions for the free-free beam with the effects of rotary inertia and vertical shear force included. The roots of these expressions yield the frequencies of vibration if the constants of the beam are known. Actually Goens uses the relations to determine the modulus of elasticity of the material of the bar from experimentally determined eigen-frequencies. More recently Ormondroyd, et al.,' have presented results of theoretical research on the dynamics of a ship's structure with a view toward the determination of the eigen-frequencies and "normal" mode shapes of both uniform and non-uniform beams. In addition studies were made of forced damped vibrations. 1

AERONAUTICAL RESECARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 In this work the effect of rotary inertia is neglected, and for this case orthogonality of the modes of vibration is demonstrated. Hess7 investigates the general problem of transverse vibrations in both uniform and non-uniform beams. After developing the basic equations for the beam and deriving differential equations for mode shape, bending moment, shear force, etc., the author obtains frequency equations and shape factors for beams with different kinds of end restraints. In his study of non-uniform beams the effect of rotary inertia is omitted. For this case it is shown that the modes of vibration are orthogonal. When rotary inertia is not neglected, Dolphl3 has shown that the solutions are not orthogonal, except for a certain type of end fastening. Kruszewski considers the effects of transverse shear and rotary inertia. The following summary of the work is given by the author: "A theoretical analysis of the effect of transverse shear and rotary inertia on the natural frequencies of a uniform beam is presented. Frequency equations are derived for the cases of the cantilever beam, the symmetrically vibrating freefree beam, and the antisymmetrically vibrating free-free beam. Numerical results are given in the form of curves giving the frequencies of the first three modes of the cantilever beam and the first six modes, three symmetrical and three antisymmetrical, of the free-free beam." In all of the above work the solution of the basic equations led to a fourth order differential equation for the mode shape. This equation was solved, subject to the boundary conditions at the ends of the beam, the solutions being equations from which the eigen-frequencies were determined. In most cases no attention was given to mode shapes. Hagelbarger, et al., demonstrated that an electronic differential analyzer can be used for solving the fourth order differential equation with the effects of rotary inertia and transverse shear force included. The eigenfrequencies and mode shapes are determined simultaneously. For higher modes of oscillation of relatively short beams the mode shapes obtained corresponded to oscillations of the center of gravity of the beam. This result was explained by the fact that incorrect boundary conditions had been used. In subsequent work Howel1 showed that satisfactory solutions can be obtained by the electronic differential analyzer, using the correct end conditions 2 -

AERONAUTTICAL RESEARICH CENTER - UNIVERSITY OF MICHIGAN UMM-67 for a free-free beam, when the effects of transverse shear forceeand rotary inertia are included. This particular method of attack does not permit the electronic differential analyzer to operate on the fourth order differential equation to obtain solutions for non-uniform beams. Such a procedure would neglect many derivatives which are important. It is shown in subsequent analysis that satisfactory solutions can be obtained for both uniform and non-uniform beams by having the computer operate on the original basic differential equations. 3 -

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 CHAPTER 1 THEORETICAL ANALYSIS CF THE VIBRATING BEAM 1.1 Basic Equations By applying the laws of dynamics and of elementary strength of materials to a free body diagram, one obtains the five basic equations, o(x) a2y(x't) V+A(x.t) = (1-i) - MAxzt) + Ir(X) + V(x,t)= O, (1-2) ex r at2 2t V(x,t) = - kAG(x) ~ o(x,t), (1-3) X E(x) ay(xt) = (x,t) + t) (1-5) These equations, with changes of notation, are taken from Ormondroyd, et al.,5. Equation (1-1) equates to zero the sum of the transverse forces. Equation (1-2) is the sum of the moments acting on a small element of the beam and includes the effect of rotary inertia. Equations (1-3), (1-4), and (1-5) relate the neutral axis slope to shear and bending. The equivalent of the first two of these equations can be found in textbooks on strength of materials. Equation (1-5) states that the neutral axis slope is the sum of the slopes due to shear and bending and holds true for small deflections only. The derivation of equations (1-1) and (1-2) can be found in Timoshenko's original paper. 4 —--------------..

AERONAUTICAL RESEARCH CENTER- UNIVERSITY OF MICHIGAN UM-67 1.2 Notation x = horizontal distance from left end of beam J = length of beam t = time y(x,t) = transverse deflection of beam at any instant y(x) = maximum transverse deflection of beam, at x M(x,t) = bending moment at any instant M(x) = maximum bending moment, at x Xvx,t) = transverse shear force at any instant V(x) = maximum transverse shear force, at x I = area moment of inertia E = modulus of elasticity, or Young's modulus EI(x) = flexural rigidity, at x EI = a constant value of EI(x) A = cross-sectional area G = modulus of shear, or rigidity modulus k = ratio of average shear stress to stress at neutral axis kAG(x) = shear rigidity, at x kAG = a constant value of kAG(x) /(x) = mass per unit length of beam, at x p = a constant value of p (x) Ir(x) = mass moment of inertia per unit length of beam, at x Ir = a constant value of Ir(x) &(x,t) = neutral axis slope due to shear at any instant O(x) = maximum neutral axis slope due to shear, at x R(x,t) = neutral axis slope due to bending at any instant B(x) = maximum neutral axis slope due to bending, at x (k) (x) = a dimensionless parameter, expressing the effect of rotary \ inertia N =E = a constant value of - (x) *d(x) = a dimensionless variable, reflecting the variation of P (x) with x -------------------- ~~~~5 ----------

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN...... —-----— r UMM-67 r (x) = a dimensionless variable, reflecting the variation of Ir(x) with x ts(x) = a dimensionless variable, reflecting the variation of kAG(x) with x 4f(x) = a dimensionless variable, reflecting the variation of EI(x) with x =(x) = a dimensionless variable, reflecting the variation of ( ) (x) with x El S = -E, a dimensionless parameter kAG = w2,, a dimensionless parameter = a dimensionless independent variable, x = S = the independent variable for the computer, x = L / = a root of equation (1-23) m = a parameter, see equation (1-28) n = a parameter, see equation (1-29) 1.3 Separation of Variables In equations (1-1) to (1-5) there are two independent variables, x and t. The quantities y(x,t), M(x,t), V(x,t), R(x,t) and P(x,t) are functions of both of these variables. EI(x), kAG(x), Ir(x) and p,(x) are constants for a uniform beam and functions of x for a non-uniform beam. To separate the two independent variables x and t it is assumed that the time dependent variables are sinusoidal functions of time. This assumption represents one possible solution with respect to time, though not necessarily the only one. Let y(x,t) = y(x) ejLt, M(x,t) = M(x) ejt, V(x,t) = V(x) ejt, R(x,t) = i(x) ejt, and p(x,t) = p(x) e3jt. -----------------—'~ 6

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-67 Equations (1-1) to (1-5) then become - P(X)W 2y(x) + dV(x- 0, (1-6) - dM( - (x)J(x) + V(x) =, (1-7) dx r V(x) - kAG(x) * o(x), (1-8) dp = M(x) (1-9) dx = )' (1-9) EI(x) dX_)_ <(x) +`(x) (1-10) dx Since the above set of equations are to be applied in general to the study of a non-uniform beam in which the physical characteristics change along the length of the beam, it will be convenient to use the following relations: P(x) =,4d(x), I(x) = I (x), kAG = ( x) kAG, and EI(x) = EI f(x), where y0, Ir, kAG and EI are now constants, independent of distance along the beam, and where the variations of the physical properties of the non-uniform beam are reflected in the *'s. Using the above relations there are obtained from equations (1-6) to (1-10), - wd(X)y (x) dV_ = 0, (1-11) 7

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 _dM x Irr(x)W 2() + V(x) = ~ (1-12) V(x) = - kAG4(x) d(x), (1-13) dx) = (1-14) dx EIt+(x)' and dy (x)) + (x) + (1-15) dx In case equations (1-11) to (1-15) are to be used for a uniform beam the variables id, r', s and f may be set equal to unity. 1.4 Boundary Conditions These equations are to be solved subject to the end conditions, or restraints, if any, imposed on the two ends of the beam to be studied. The end of a beam may be in any one of three states; free, hinged, or built-in. The boundary conditions to be met are given in Table I. TABLE I BOUNDARY CONDITIONS End of Beam Boundary Conditions Free M = 0; V = 0 Hinged y =; M = Built-In y =; dy/dx = 0, or y; P'= O In the case of the built-in end there is some difference of opinion as to whether the slope of the neutral axis is zero or whether it has a finite value due to transverse shear force alone. The solution of the differential equations given above for a freely vibrating beam may be accomplished in either of two ways; by a theoretical -8

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 analysis or by a differential analyzer. Both methods are presented here, the theoretical analysis being limited to that for a uniform beam. 1.5 Theoretical Analysis of the Vibrations of a Uniform Beam For a uniform beam d', ry' s and Of in equations (1-11) to (1-15) are set equal to unity. From these modified equations there may be obtained, 2E d'x4 kAG P) 2 - y(x (1-16) | - EId x) *EI M(x) = 2 y (x)] (1-17) kAG e~-(x x) dz+~ -I d+ / \ / dx kAG ic[, W2 dx3 kAG P dx kAG where the bar over the P is omitted because it is of no significance in this part of the work. In the above equations we shall replace the independent variable x (- x 12) by (- 2 1) using the relation x =, where I is the length of the beam. Recognizing that Ir I -A and letting 9\ EI s(1-20) -— 9 ---------

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-67 S EI (1-21) kAGf2 and N =k, (1-22) equations (1-16) to (1-19) may be written, I d4 + S(1 + N) d- (1- A2S2N)y, (1-23) A2 d 5 M =pdi2[+ SY, (1-24) V 1''X2S2N d.' V [ d +N) ] (1-25) and p-(1 - 2sd + (1 + 2s2)3 (1-26) The fourth order differential equation (1-23) is to be solved, subject to the boundary conditions on both ends of the beam as given in Table I. If in equation (1-23) we let y = C e/0 we have 12 4 " + s(1 + N)/,2 (1 - 22N) = 0, (1-27) X2 from which - S(1 + N) + S2(1 + N + - 4S2N /= xA2..2 Since the last term on the left hand side of equation (1-27) is negative for all practicable beams, one of the roots of the equation must be positive and the other one negative. Hence 10

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 /'1 =mX, /2 = - mX, where - S(1 + N) + S(1 - N)2 + 4 X2 2M -= i —---— 2 _, (1-28) and /3 = jnX, /A = - jnX where S(1 + N) + S2(1 N)2 + 42 A2 n - — 2. (1-29) In relations (1-28) and (1-29) only the positive values of the radicals are to be taken. The solution of equation (1-23) may then be written as y = C1 cosh mXg + C2 sinh mg + C3 cos nXk + C, sin nX\. (1-30) In order to obtain the eigen-frequencies for a particular beam it is necessary to substitute the value of y, as given by equation (1-30), and the derivatives of y, into the boundary condition relations for both ends of the beam as given in Table I. This process of substitution results in a set of four homogeneous linear equations in C1, C2, C3 and C4. For solutions other than zero to exist the determinant of the coefficients of C1, C2, C3 and C, must be equal to zero. The expression obtained by setting this determinant equal to zero is an equation the roots of which give all of the eigen-frequencies. 1.6 Solution for a Free-Free Uniform Beam The boundary conditions for a free-free uniform beam are, M = 0 and V = 0 at both ends of the beam. From equations (1-24) and (1-25) we have 1 d - ~ + Sy =0,for 1 1(131) x d2 12 11

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-67 1d S( + (1 N) 0, for - (1-32) x2 dS3 dor2 y2 Substitution of y and its derivatives, as obtained from equation (1-23), into relations (1-31) and (1-32) gives four homogeneous linear equations in C1, C2, C3 and C4. By setting equal to zero the determinant of the coefficients of the C's, there is obtained, tanh 2nX 2 tanh 2 2 It _ 2 m n(n - S) t a m __ 2 m(m + S)= 0 (1-33) The rots of eatn n n. (1-2 The roots of equation (1-33) determine eigen-values of X. From equation (1-20) and the known characteristics of the beam an eigen-frequency, w, can be determined for each eigen-value of \. A very useful family of curves can be made by computing and plotting the eigen-values of X as a function of 1/S for various values of N. Such a set of curves for a free-free uniform beam, obtained from data given by an electronic differential analyzer, is shown in Figure 3-10. ---------------— 12

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN........ —--— r UMv 6-67 CHAPTER 2 EQUATIONS OF THE VIBRATING BEAM FOR SOLUTION BY AN ELECTRONIC DIFFERENTIAL ANALYZER 2.1 Inexpedience of Using the Fourth Order Differential Equation The fourth order differential equation (1-23), derived above, can be solved by an electronic differential analyzer910. For a uniform free-free beam the results obtained are very satisfactory. It was proposed that non-uniform beams be investigated in the same manner, but by representing a non-uniform beam as composed of a discrete number of short uniform beams, each of these short uniform beams having appropriate properties. The coefficients of equation (1-23) and of the corresponding equations which impose the end conditions were to have been changed at appropriate intervals in accordance with the properties of the short uniform beams. In order to investigate the validity of this method, equations similar to (1-16), (1-17) and (1-18) were derived for the bending moment, shear force, and mode shape, taking cognizance of the fact that for a non-uniform beam the coefficients are functions of x. The relations obtained are, M=.EId x [dx V, (2-2) d3S dx dx2 (P kAG r) dx + dx i kAG) p dd kAG) Y 2 kAG 2 d kAGdx +kAG (2-1) M EI + kAG + EI dx (2-2) | dx 1 dV 2 dx (2-3) Examination of the equation for M shows that there should be no difficulty in applying the end condition, M O, at either end of a free-free nonuniform beam, since V is also equal to zero at the same positions. However, the ---- ~ 13

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 equation for V is sufficiently complicated as to make its use in fulfilling boundary conditions exceedingly involved. An insurmountable difficulty arises when one attempts to obtain an explicit equation for the mode shape, y. To obtain this expression it is necessary to take the derivative, with respect to x, of equation (2-1) for V. This process results in an expression involving over a hundred terms, of which very few combine or cancel out. It would be hopeless to attempt to set up a computer to handle this equation. On the other hand, all of these additional terms are neglected, or ignored, in any attempt to apply equation (1-23) to non-uniform beams. 2.2 Use of the Basic First Order Differential Equations Fortunately the actual problem consists not of obtaining the solution of a single fourth order differential equation but of a set of first order differential equations. When these equations are set up on the electronic differential analyzer a number of simplifications result. The end conditions are very readily applied, the technique of obtaining solutions is simplified, and non-uniform beams can be investigated without neglecting any derivatives. 2.3 Change of Independent Variable In order to solve these equations, (1-11) to (1-15), with the differential analyzer it is necessary to change the independent variable. The length of the beam is I so that the range for the independent variable, x, is O C x _<., The independent variable of the computer is time. In solving the equations the independent variable, x, is proportional to the time in seconds which has elapsed since the start of the solution by the computer. If the solution obtained by the computer is completed in L seconds, then the independent variable, x, should be changed to a new independent variable,'r, (0 < r L), according to the relations, x =, and dx L d Equations (1-11) to (1-15) then become, - f d(W Ay(T) + L d = 0, (2-4) 14

AERONAUTICAL RESEARCH CENTER UNIVERSITY OF MICHIGAN UMM-67 -I d(T I_ (T) 2(' + v(T) 0, (2-5) V(T) = - kAGs(T (r), (2-6) L d_(T) M(T) Lf d:(z) = +M~r~ (2-7) I dr EIf<(r)' I dy? = o( + (t. (2-8) In order to place these equations in more suitable form, assume a(7) =i d (9, (2-9) and -(T) = j (T. (2-10) \ Equations (2-4) to (2-8) may then be written, -P W2) y + L dV 0 (2-I1) L dlvZ r 2 caL dV -^ -I^ ^ B+VO, ~(2-12) - rdT - ro,, + V, 0 (2-12) V - kA, (2-13) 14d^'S^^~~ ~(2-1U) j2 dT EIs 2' and dY ^ + ~ ^ (2-15) 15

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 where y, M, V, o, P, d' r' s' and f are functions of r. 2.4 Combination and Simplification of Coefficients The use of the differential analyzer in solving these equations will be made much easier if their coefficients are combined and simplified. To this end we substitute equation (2-13) into equation (2-11) and obtain, after dividing through by kAG/12, tdy + L2d d+ L o= (2-16) This may be written as, El. *^Jg4 + L2 d (4() =0C EI^.f-^ r my + L Cxo -' ~ or Y S12 L dt ( ) ~0' (2-17) where S EI (2-18) kAGL2 and X2 E.* (2-19) Substitution of equations (2-14) and (2-13) into equation (2-12), after division by kAGL/J, results in El L2 kA( r 4 sc = ~ * (2-20) Now.r A kAG 16

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-67 EI+f kG?d kAG, E'n where kG) =kG t 4, = N4, (2-21) Hence 2 2PI4 I -2 El EI o kG Xdfn r kAG t2 ~0~ I. E' rlr kAG - kAG2 kAG EI E or 2 Ir r kAG S2X2+, (2-22) where 4<, +d-f +n (2-23) r s Substitution of relation (2-22) into equation (2-20) gives, after division by SL2, d 2T ffd + SN 2 +F 0+ tK= 0s (2-24) dTUL SL2 2.5 The Equations to be Solved by the Computer The equations to be solved by the computer are then, = ~(+ =, +(2-15) dT Y + S 2 1.L (t cs) = 0 (2-17) a^f^,s' — =0. (2-24) dtk(+f dT' + SN 2 r+ (2 L SL2 17

AERONAUTICAL RESSEARCH CENTER UNIVERSITY OF MICHIGAN UMM-67 2.6 Boundary Conditions Reference to Table I, Section 1.4, indicates that some of the boundary conditions require that M = 0 and V = 0. Now M and V do not appear explicitly in the equations, (2-15), (2-17) and (2-24), to be solved by the computer. However, by reference to equation (2-14) it is seen that M = 0 when do/dT= 0, and equation (2-13) shows that V = 0 when o( = 0. To meet the boundary condition dy/dx = O, we may let dy/dr = 0 or i + 3 = O. The boundary conditions for use with the electronic differential analyzer are given in Table II. TABLE II BOUNDARY CONDITIONS FOR COMPUTER End of Beam Boundary Conditions Free d/dT = 0; 0 = 0 Hinged y = 0; d/dr = 0 Built-In y = 0; dy/dr = + f = O, or yO; =0. 18

AERONAUTICAL RESEARCH CENTER- UNIVERSITY OF MICHIGAN UMM-67 CHAPTER 3 APPLICATION OF THE ELECTRONIC DIFFERENTIAL ANALYZER TO THE SOLUTION OF THE UNIFORM-BEAM PROBLEM 3.1 Introduction to Operational Amplifiers Before proceeding with a description of how the vibrating-beam problem can be solved by means of the electronic differential analyzer, it would seem appropriate to review very briefly the principles behind the operation of such a computer. The basic component of the electronic differential.analyzer is the operational amplifier, which is shown schematically in Figure 3-1. It consists of a dc voltage amplifier of high gain (usually about 40,000), an input impedance Zi' and a feedback impedance Zf (see Appendix 2 for the dc amplifier circuit). L2 - EL ---- ~~1~~~~~ eI e' DC AMPLIFIER Figure 3-1. Operational Amplifier If we eglect the current into the dc amplifier itself (i.e., neglect the current to the rid of the input tube), it follows that i1 = i2. Let us also neglect 19

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 the voltage input e' to the dc amplifier in comparison with the output voltage e2 or the input voltage el to the operational amplifier (e' = 1/40,000 e2). We then have 1 i2 or el e2 Zi Zf from which Zf e2 = -, (3-1) which is the fundamental equation governing the behavior of the operational amplifier. In general Zf/Zi is made the order of magnitude of unity. We shall now consider the scheme by which the amplifier can be used to perform three different types of operations. (a) Multiplication by a constant. If we wish to multiply a certain voltage el by a constant factor k, we need only make Zf/Zi = k. From equation (3-1), then, the output voltage e2 of the operational amplifier will be given by e2 = - k e. (3-2) Thus the required multiplication by a constant has been achieved, except for a reversal of sign. For example, if we wish k to be 10, we may let Zi = 1 megohm resistance, Zf = 10 megohms resistance. If we also desire the sign of e2 to be the same as el, we need only feed e2 through an additional operational amplifier with Z = Zf = 1 megohm..This second operational amplifier merely acts as a sign changer by multiplying any voltage by -1. (b) Addition. In order to add a number of voltages, say ea, eb, and ec, the arrangement shown in Figure 3-2 is used. Here ia + ib + ic = i2, 20

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 + e —-- ~,eb - jII e' DC AMPLIFIER e2 -'e, Figure 3-2. Operational Amplifier Used for Summation and if we neglect e' as small compared with input or output voltages, we have ea eb e 2 r+ b+r= z z +z +z z a b c f or /2Zf Zf Zf e2 = - e + e + e (3-3) Thus the output voltage e2 is the sum of the three input voltages, each multiplied respectively by a constant - Zf/Zn (n = a, b, or c). The operational amplifier can, of course, be used in general to sum any number of input voltages. (c) Integration. If we make the input impedance Z. a resistor and the feedback impedance Zf a capacitor, then the operational amplifier serves as an integrator. 21

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 Referring to Figure 3-3, we see that if we neglect e' and let i =- i2 as before, we have ildr e e -I and i = from which e 1 k e dT (3-4) e2 - RC eld ~'' The output voltage e2 is then the integral with respect to time of the input voltage el (multiplied by a constant factor - 1/RC). L2 R L, e, e' DC AMPLIFIER e2 ~~~~~I~~~~~~~~~~~~~~ Figure 3-3. Operational Amplifier as an Integrator In order to demonstrate how operational amplifiers performing the above three functions can be combined to solve ordinary linear differential equations, we will now set up the amplifier circuits required to solve the uniform beam problem. 22

AERONAUTICAL RESEARCH CENTER- UNIVERSITY OF MICHIGAN UMM-67 3.2 Computer Arrangement for Solving the Uniform-Beam Problem (a) The equations to be solved. In chapter 2 we reduced the differential equations of the vibrating beam to the following three simultaneous equations: =x - i B, (2-15) 1L 1d0 (2-17) and,O + CK e~=0. (2-24) dr (f ) L2 SL2 ~ * (2We remember that y is the dependent variable (volts in the case of the computer), and that T is computer time (corresponding to distance along the beam). S and N are dimensionless parameters depending upon the physical characteristics of the beam, and X is the characteristic root of the equations such that the appropriate boundary conditions are satisfied. Bending moment is proportional to d0/d and shear force is proportional to o<. The length of the computer solution is L. For a uniform beam Ad = Of = Or = s = 1, so that equations (2-15), (2-16)x and (2-24) reduce to.t^ ~i + f^P0~ ~(2-15) dT 1L2 d y + l2 dr 0 (3-5) 2 2 ^ + SN v P 1 -- 0( = O (3-6) d + L2 SL2 These equations are of course subject to the end conditions given in Table II at the end of Chapter 2. 23

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-67 (b) The computer circuit. The computer circuit for solving the above three equationsis shown in Figure 3-4. The output functions of each of the operational units are clearly labeled so that the reader should have no trouble in tracing through the circuits to see how the three equations (2-15), (3-5), and (3-6) are set up. Note that the input to each of the integrating units is the derivative of the output of the same unit. By combining several functions in one amplifier, one can simplify the circuit of Figure 3-4 to that shown in Figure 3-5. This arrangement could be further simplified by eliminating A7 and feeding the output of A6 (-n) directly into A4 through a resistor of value L /SNX megohms. Since L /SNX is generally large compared with 1 megohm, it is more convenient to distribute this factor through several resistors. The addition of A7 also allows the feedback resistor X /L4 to be changed in gang with the feedback resistance of Al. 3.3 Method of Obtaining the Correct Solution (a) Initial conditions. The initial conditions utilized on the computer depend upon the type of end fastening for the beam in question. For a "free-free" beam we see from Table II at the end of Chapter 2 that the initial conditions (and also final conditions) are d = O, ( =0 at T= (3-7) dT dt = o, <= O at T= L (3-8) The initial conditions given in (3-7) are imposed by short-circuiting the feedback capacitors of amplifiers A2 and A5 in Figure 3-5 through initialcondition relays. This is equivalent to making the vertical shear force and bending moment zero at the one end of the beam. At the same end the deflection and slope must be finite; these conditions are imposed initially on A3 and A6 through voltages V3 and V6. The solution of the problem is begun by releasing simultaneously the four initial condition relays, two of which were holding dB/dT= 0 and o<= 0, and two of which were holding y = V3 and -6 = V6. 24

2[ > 1[ 0 I _ 2 I t ___ 1 d0( — L.^ J-0< d do ( SX2 dr SL2 d SL2 -T L4 N Al Capacitors areAI m - dT2 d —— vr" 2 L ^ All Resistor Units ore Megohms All Capacitors are I mfd Ground Connections are Omitted for Clarity Figure 3-4. Complete Computer Circuit for Solving the Uniform Beam 1 —-------------------------------------------------------— z

I' A1 O0 22 1 l,~I n - V I z No RS > S X < 1 L 2 d~ o.l ~ ~ ~ ~ ~ A l No AR | A,2R All Capacitors are mfd __1=R H N N All Capacitors are I mfd Figure 3-5. Simplified computer Circuit for Solving the Uniform Beam I___________________________________________________________I z

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 (b) Trial and error method of obtaining the normal-mode solution. The first-trial settings of V3 and V6 were arbitrary, and in general a correct solution will not result. For a correct solution dP/dr must go through zero at precisely the same instant as o( goes through zero. The time elapsed between the start of the solution and the instant when dO/d = o = O is then the length L of the computer solution. A typical first-trial solution is shown in Figure 3-6a, in which the condition d3/dT(L) = i(L) = O is obviously not met. The small pips on the d0/drcurve indicate when o( has passed through zero (see Chapter 5 - Section 4). By varying V3 and holding V6 constant (or vice versa), the ratio V3/V6 can be varied until an exact solution is obtained. The trial solutions leading to an exact first mode solution are shown in Figure 3-6. Usually about half a dozen trial solutions are needed until an exact solution is reached. Higher modes are obtained in exactly the same manner. Second and third mode solutions are shown in Figure 3-7 and Figure 3-8. If we are looking for the solution of a given uniform beam, then we must have chosen originally a suitable computer-solution length L. But the values for the feedback resistor Rv in Al and A7 (Figure 3-5) had to be selected arbitrarily, so that in general the observed computer length L for a correct solution will be different from the L originally chosen. It is then necessary to select a new R7 and rerun the solution, obtaining a new L. By repeating this process several times and by interpolating, we can arrive at the value of R7 which gives a correct solution of the desired length L. The frequency parameter X is then given by X L2. (3-9) On the other hand, if we are interested in solving the problem for a whole family of beams characterized by different values of the parameter S (i.e. having various thickness to length ratios), we may keep R fixed, and calculate X from equation (3-9) and S from the formula 1 - L3 * (3-1) 27

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN U4.M-67 A IL I I l-i' Figure 3-6 Trial arnd Error Mthod for Obtainin I a Correct Solution Figure 3-7. Second-Mode Solution Figure 3-8, Near-Correct Third-Mode for a Uniform, Free-Free Beau Solution for a Uniform, Free-Free Beam _ —--------- 2 _ _ _ -_ _____-_-__ —, I r 1 1X-'MX4WI I I I i =AQ l I,.. IIY I I I IL. I,. -_ IF I E! C tX i tC-XA;l; IF Im I a I 7 7 1 - 1rI /,, ~,.. Z I Ib I I I I I X X X X S n X X w X X T [ r r 1w 1 w T I- - -/1'I l I II I I I 1 1 1\1 1 l NE 28

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN "IU —----— 7 — ~-UMM-67 In this procedure every correct solution of length L represents a certain beam, but we cannot choose the exact 1/S value for that beam beforehand, (c) Method of recording computer solutions. Computer output voltages are recorded by means of a Brush, Model BL-202, two-channel magnetic oscillograph. Since the pen motors require more current than the operational amplifiers can furnish, computer voltages are first fed through Brush, Model BL-913, dc amplifiers, which in turn drive the pens of the magnetic oscillograph. The Brush dc amplifiers were modified as described in a previous reportl~ in order to reduce zero-drift to a negligible value. The frequency response of the oscillograph and modified amplifier together is flat out to 20 cps, which is entirely adequate for the vibrating-beam solutions, except where sharp pulses are also recorded (see Sections 5.4 and 5.5). 3.4 Computer Solutions for a Free-Free Beam, Infinitely Long In order to check the accuracy of the computer it is instructive to solve the vibrating-beam problem for the case in which the length of the beam is very long compared with the thickness. This means that rotary inertia and shear forces are negligible (N = S = 0); such an idealized beam is often termed "infinitely long". The frequency parameter X for the first five modes of an infinitely long free-free beam are given by Timoshenko and can be obtained from the roots of equation (1-33). (a) Comparison of computer solutions with theoretical solutions. In order to solve the problem of the infinitely long beam the connection between A2 and A3 in Figure 3-5 was broken, along with the connection between A? and A5. RX was set equal to unity (1 megohm), so that A= L or ] = L. By using the experimental techniques described in Chapter 5, we obtained the values for shown in Table III. Evidentally the computer solutions are accurate to hundredths of a percent. It is felt that one of the chief limitations on accuracy attainable was the inability to measure the length L of the solution to better than a few milliseconds. In Chapter 5 Section 6 an electronic clock is proposed which would eliminate this inaccuracy in measurement. 29

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-67 TABLE III COMPARISON OF COMPUTER AND THEORETICAL SOLUTIONS FOR A "FREE-FREE" BEAM, INFINITELY LONG 4= L L Mode (Computer) (Theoretical) 1 4.735 sec 4.730 2 7.852 7.853 3 10.997 10.996 4 14.136 14.137 (b) Method of interpolation for higher mode solutions. When obtaining solutions of higher modes one finds that the ratio of V3/V6 in Figure 3-5 becomes extremely critical. A change in these initial voltages of one part in five thousand causes considerable deviation from one solution to the next when a third or fourth mode is being sought. As a result the solutions for the third and fourth modes do not repeat from run to run, and it is extremely fortunate ever to get an exact third or fourt h mode solution. Let us define the length L' of an inexact solutions as the time elapsed from the start of the solution to the instant o< goes through zero near the "end" of the inexact solution. If the value of dP/d'at o = O is plotted as a function of LI, we obtain a curve similar to the one shown in Figure 3-9. From this curve the length L of an exact solution can readily be obtained by interpolation. The L values for the third and fourth modes in Table III were obtained in this manner. It turns out that when one includes shear and rotary inertia effects (N and S finite), the initial conditions for obtaining higher modes become much less critical and solutions for these modes are more easily obtained. The cause for the lack of repetition of solutions for higher modes is believed to be the polarization of the dielectric in the capacitors of the integrators. Thus a solution begun after the capacitors were charged in one direction before being returned to their initial voltages will differ from a solution begun after the capacitors were charged in the opposite direction before being returned to their initial voltages. 30

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-67'p.8-1 THIRD MODE.6 - -04.2,L= I10.997 10.6 10.7 10.8 10.9 11.0 11.1 11.2 11.3 L' IN SECONDS -,4 -6 _ l___ —-- Figure 3-9. Method for Obtaining Exact Solution Length L from Inexact Solutions 31

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 3.5 Computer Solutions for a Free-Free Beam of Finite Length The differential equations presented in Chapter 1 for the vibrating beam include the effects of shear and rotary inertia forces, and hence are representative of a beam whose thickness may be appreciable in comparison with its length. The relative effects of shear force and rotary inertia present in the vibrating beam are represented respectively by the values of the dimensionless parameters S and N (see equations (1-21) and (1-22)). For a long, thin beam, S will be small; for a short, thick beam, S will be large. N, on the other hand, depends only on the ratio of shear modulus G to Young's modulus E for a given shape of beam cross-section. (a) Method of varying parameters. The computer was used to find the frequency parameter X for a considerable range of S and N values in the case of a free-free beam. The circuit shown in Figure 3-5 was used; the resistor R was set equal to unity. The value of Rs was chosen arbitrarily, and N was made 0, 0.1, 0.2, and 0.3 by setting Rn = o, 10, 5, and 3 1/3 megohms. For each value of R and R the n n s length L of the solution of the first three modes was determined. The parameter S for each of these solutions is given by equation (3-10), while X L2 since R =1. x (b) Results of the computer solutions. Figure 3-10 shows the family of curves obtained when X is plotted as a function of 1/S for the four values of N. The actual data obtained, along with more accurate curves, can be seen in Appendix 1. The values chosen for N(0, 0.1, 0.2, 0.3) seemed appropriate for the various types of beams which might be encountered. Variation of X as a function of N is almost linear, so a much wider range in N could be obtained by interpolation from the above values of N without too much loss in accuracy. The parameter 1/S ranges from very high values down to about 14, 35, and 72 in the case of the first, second, and third modes respectively. These lower limits on 1/S correspond to length-thickness ratios of about 2, 3.5, and 5 for a rectangular steel beam; i.e., the length to thickness ratios for the beams represented in Figure 3-10 go down to the region where the wave-length of 32

I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~P 3RD MODE ~ 120 1 m zp 100 ~ ~ ~ ~ ~ ~ - _ - - - - _ _ _ — ^_ _ _- ----- -- _ ___ __ _ E 06 100 t 80 N= 0 3 0~~~ ~ ~ ~_ _ _ _ --- ------ _= 0_ 0^^ - - - - - - - - _ _ —------ -_ _ N= 0. N= 0.2 N 0.3 z0. 2 0 _ _ —--- w - - - - - -------- ------------ _ _ - - o^ 0312ND MODE _1___________ _______________ __ ________ ____ 0 40 15 N 3 - 70 - 100 - 200- 400-700-1000- 2000- 4000 —- -- - - 20 0~~~~~~~~~~~~~~~~~~~~~~~ ~~~0.0. 3 1 0 15 20 30 40 70 100 200 400 700 1000 2000 4000 2~~~~~~~~~~ I/S = KAGI 0 El Figure 3-10. Frequency Parameter A as a Function of N and S for the First Three Modes of a Free-Free Beam ^ 20 N=P ~~~~~~~~~~~~~~~~~d~

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 vibration is almost the same as the thickness of the beam itself. This is the region where one would expect the assumptions involved in deriving the original beam equations (1-1) to (1-5) to break down. (c) Orthogonality of the modes. Dolph13 has shown that when rotary inertia forces are included, the normal mode solutions are not in general orthogonal. The solutions are orthogonal when rotary inertia effects are neglected. Therefore it is of considerable interest to compare the mode shapes obtained with and without rotary inertia effects included. This has been done in Figure 3-11, where three modes are shown, both for N = 0 (no rotary inertia) and N = 0.3. Evidentally there is no great difference in mode shapes. It therefore seems safe to say that a Fourier series of the modes for N = O would not be too bad a representation of an arbitrary function, particularly if we only worry about the first few modes, even though an appreciable rotary inertia effect is present in the beam. (d) Comparison of computer results with theoretical results. The roots of equation (1-33) determine the frequency parameter A. Since this equation is transcendental, the only method of determining the roots is by trial substitution of values of \. However, the computer actually solves for the roots X of equation (1-33), so to check the accuracy of the computer solution it is only necessary to substitute the value of X obtained by the computer into equation (1-33). In general, due to inaccuracies in the computer, the equation will not be satisfied exactly (i.e., the bracketed term yielding the root in question will not be quite zero). By substituting two additional >'s which are slightly different than the original X, one can easily interpolate to find the X for which one of the bracketed terms in equation (1-33) does indeed vanish. This theoretical X is compared with the computer X for a third mode solution in Table IV. Even though we know beforehand almost the exact value of X, the task of computing X from equation (1-33) requires a good many man-hours of work plus the use of a calculating machine. The accuracy of the computer, as exhibited in Table IV, is not as good as that exhibited in Table III. The percentage error in Table IV, however, is doubled since we are comparing \ directly and not, 34

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 FIG. 3-11 ^1ts~ //~ ~FIRST MODE C| | --- N- O=, 1/S =162 I 8: -- N = 0.3,1/S =151 z SECOND MODE /I |z~ \ -— _ N = O, 1/S = 162 Q // | a /-/ \ — N = O, 1/S =151 THIRD MODE N = 0, 1/S = 314 | zi\ - - N = 0, 1/S = 295 a\/ \ Figure 3-11. Comparison of Mode Shapes with and without Rotary Inertia Included 35

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 TABLE IV COMPARISON OF COMPUTER AND THEORETICAL SOLUTIONS FOR A FREE-FREE BEAM, INCLUDING SHEAR AND ROTARY INERTIA FORCES Third Mode 1/S N \(Computer) X(Theoretical) 551 0 111.43 111.32 537 0.2 108.56 108.32 which is actually what the computer finds (L = 5). The error can probably be attributed to a small increase in the time-constant of the integrating units between the time they were calibrated and the time the solution was run off. Additional errors are involved in measuring the length L of the exact solution. It is probably safe to say that the X's of Figure 3-10 are good to 0.2%. By incorporating better measuring and calibration techniques, we feel that this accuracy could be improved by a factor of 10 (see Chapter 5, in particular Section 5.6). Accuracies of 0.01% or even 0.1% are obviously of no great engineering significance in this vibrating-beam work. Usually the physical quantities describing the beams themselves are not known to this accuracy, and most engineers would be very happy with 1% accuracy in normal-mode frequency predictions. The purpose behind these measurements of high accuracy is really twofold; (1) to check the accuracies attainable with the electronic differential analyzer, and (2) to stimulate further investigation of lateral vibrations of actual beams in the laboratory in order to check the theory given in Chapter 1. 36

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 CHAPTER 4 APPLICATION OF THE ELECTRONIC DIFFERENTIAL ANALYZER TO THE SOLUTION OF NON-UNIFORM BEAM PROBLEMS 4.1 The Equation to be Solved In Chapter 2 we found, after separation of variables, the following equations for the non-uniform beam: = O( + (2-15) dS S + 2 - 5 -(d-=0 (2-17) dt + SN L2X tr SL 1 0 (2-24) where the notation has been defined in Section 1.2. These equations are subject to the boundary conditions given in Table II. Note that d, 4)' It' and 4r are all functions of the independent variable T. Since o(,, and dO/dt are dependent variables, it is necessary to vary somehow the gain of the operational amplifiers as a function of T (computer time) in order to obtain soc, l/od d/dT (QX), +f do/d1 and ~r respectively. In other words, we have the problem of solving differential equations with coefficients which are varying functions of the independent variable. Several of the methods for accomplishing this are described in the next section. 4.2 Methods for Vaying the Gain of an Operational Amplifier as a Function of Time (a) Two methods of approach. In Section 3.1 we saw that the output voltage e2 of an operational amplifier is given by 37

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 Zf e2 - el (3-1) where el is the input voltage, and Zf and Zi are feedback and input impedances respectively. Our problem is to find an arrangement whereby e2 =- f(T) el, (4-1) where f(T) is the variable coefficient. Evidentally equation (4-2) will be realized providing we can make Zf f = f((). (4-2) But this can be accomplished if we let Zi be a fixed resistor Ri and make Zf a resistor Rf which varies with time according to the relation Rf( = Rif(T). (4-3) Likewise, if we want e2 = l/f(T) el, we may let Zf be a fixed resistor Rf and Zi be a resistance R. which varies with time in accordance with Ri(1) = Rff(r). (4-4) Thus one scheme of changing the operational amplifier gain as a function of time T is to vary the feedback or input resistance as a function of time. A second scheme for varying the gain of the operational amplifier is indicated in Figure 4-1. Here the impedances Z. and Zf are fixed; the resistance r between the tap on the potentiometer R and ground is made to vary with time in accordance with f(7) = r/R, where f(1' is the function f(T) multiplied by a constant factor K so that the maximum value of?f() is unity. Thus we let l f() = K f(T), (4-5) 38

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UNM-67 - — II - e ie2 (T)e, _- Rl Figure 4-1. A Method for Obtaining Variable Gain and if we set K = Zi/Zf, then e2 = - f(T) el, which is the relationship desired. In general the voltage e2 in Figure 4-1 will be fed into an additional operational amplifier having input resistance Ri. Unless R is very much smaller 1 than RI, an appreciable error in e2 results. In order to eliminate this error we must arrange r(T) such that rR.' R r+R ) f (T). (4-6) Thus we see that there are two general methods for varying the gain of an operational amplifier; (1) by varying feedback or input resistors, and (2) by varying the tap position of a potential divider across the output. (1) has the advantage of allowing division as well as multiplication by a function f(T). (2) has the disadvantage of introducing an error unless the correction factor given in equation (4-6) is incorporated. Let us consider several methods for varying resistors as a function of time. 39

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN.... —---..UMM-67 (b) Cam operated variable resistances. Angular rotation of an output shaft can be made any desired function of the angle of rotation of an input shaft by driving the output shaft through a properly-shaped cam turned by the input shaft. The output shaft is then connected to the sliding-contact shaft of a linear potentiometer, so that the resistance of the potentiometer will vary with input shaft angle by the desired functional relationship. If the input shaft is turned at constant speed by a synchronous motor, the resistance of the potentiometer will vary properly with time. This scheme can be used to vary the gain of an operational amplifier as a function of time by either of the two methods described in (a). Its accuracy is limited by the precision of the cam and connecting linkage as well as the linearity of the potentiometer. The proper cutting of cams may also involve a costly or time-consuming procedure. (c) Non-linear potentiometers. It is possible to obtain a potentiometer wound so that the resistance is any desired function of the angle of rotation of the sliding-contact shaft. If this shaft is driven at constant speed, then the resistance of the non-linear potentiometer will vary with the desired function of time. Again cost and lack of flexibility may be serious disadvantages to this method. (d) Stepping-relay method. One method for varying resistance as a function of time which has been used with considerable success by the authors of this report is the socalled stepping-relay method. In this scheme the resistance, instead of being varied continuously, is changed in discrete steps after equal time intervals. For example, a linear function of resistance with time would be replaced by the staircase function shown in Figure 4-2. Note that at the end of each step the integral of the step-function (the area under the curve) is the same as the integral of the continuous function. The circuit for obtaining the resistance steps of Figure 4-2 is shown in Figure 4-3. 40

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UTM-67 LINEAR FUNCTION STEP APPROXIMATION / / / R/ 7g W s 4 5 a:: w /!, AR/2 7 I TIME T Figure 4-2. Step-Method of Approximating a Linear Function aR/2 AR AR AR AR AR AR AR o 0) 1 8 STEPPING RELAY Figure 4-3. Circuit for Obtaining Staircase Resistance Function 41

AERONAUTICAL RES EARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 At T = 0 the contact is in the position shown in the figure. After time AT has elapsed the contact moves to position (2), after additional time AT to position (3), etc. The contact is moved around by the stepping relay mechanism, which is energized every AT7 seconds once a problem solution has begun. The AT pulses to drive the stepping relay are obtained through a synchronous contactor driven by a synchronous motor. For complete details of the entire stepping-relay mechanism, the reader is referred to a previous report by the authors. The step-method is most effective when used to vary directly the input or feedback resistors of the operational amplifier, as described in Section 4.la. The input or feedback circuit is never opened during the switching operations because bridging-type relay contacts are employed. The extension of this step-approximation scheme to arbitrary continuous functions is obvious. The whole scheme has proved highly satisfactory, and surprising accuracy is obtained even when the steps are made comparatively large. Bessel's equation and Legendie's equation have both been solved with good results on the electronic differential analyzer using the stepping-relay method of varying coefficients.9 A total of forty steps were employed in this work. Several advantages are apparent for the stepping-relay scheme. First of all, it is easy to build up any desired function of time. One need only plug appropriate resistors into a panel provided for this purpose. Then too, accuracy of resistance is limited not by the percentage accuracy of full-scale resistance, but only by the percentage accuracy of the resistor in question. Thus 0.1% accuracy can be maintained in going from an operational amplifier gain of 1/50 to 50. This cannot be realized with the methods of continuously varying the resistance described in Sections 4.lb or 4.1c. A third advantage to the steppingrelay scheme is that the total resistance utilized can be of the order of megohms, which is convenient when that resistance is being used as input or feedback impedance in the operational units. Many times the data for a given engineering problem involving variable coefficients is presented in the form of discontinuous or stepped functions. The stepping relay method would seem ideally suited for transforming this type of data into the computer circuit. One disadvantage to the stepping-relay method is that it introduces discontinuous functions. However, in most computer circuits one or more 42

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 integrators follow the amplifiers employing the stepping relays, so that the discontinuous functions get smoothed out fairly well. If one desires to synthesize an input or forcing function using the step-method, a smoothly-varying function can be obtained by setting up the derivative of the desired function with resistance steps and by then running the resulting stepped function through an integrating unit. The stepping-relay method of varying amplifier gain is utilized in the computer to solve the non-uniform beam problem. (e) Binary digital method. An alternative circuit arrangement for varying resistance by steps is shown in Figure 4-4. Here any desired resistance is obtained by selecting the proper series combination of the resistors at the top of the figure. A particular resistor is added to the total series resistance R if the relay across this resistor is open. When the relay is closed, this resistor is short-circuited and makes no contribution to R. If we desired R to be 41 ohms, we would close relays R5, R3 and R2 and leave relays R6, R4, and R1 open. R would then be equal to Ix 32 + Ox 16 + 1 x 8 + 0 x 4 + x 2 + 1x 1 = 41 ohms. In the binary system which is employed here the number 41 is then represented as 101001. The position which each of the relays R1 to R6 will assume is controlled by the position of the corresponding toggle switches in one of the rows of toggle switches shown in Figure 4-4. Thus for Step 1 the switches are set in positions up, down, up, down, down, up, going from left to right. This represents the number 41 and makes R = 41 ohms. The position of the 6-gang stepping relay determines which row of toggle switches is connected to the resistor-controlling relays R1 to R6. During the first step, the top row of toggle switches controls the total resistance R; during the second step, the second row of toggle switches controls R, etc. Any step function of resistance varying with time can be set up merely by positioning the toggle switches properly to represent the function at each step. The same set of resistors and resistor-controlling relays is used over and over again. If we wish 0.01% accuracy of maximum R, 13 resistors instead of 6 will be required (213 = 8192 = 10). This means of course that we need 13 resistorcontrolling relays, 13 toggle switches in each row, and a 13 gang stepping relay, 43

> I- -R,- 0R\ 32. 16.n. 4. 2n_ I n. 0 f6 R5 R4 E 3 2 1 -6 IC -; 6 GANGI STEPPINRELAY (4 STEPS) ___. 20 3 82T E3 I T T t T T.' T I? T F STEPI,- ~ ~BANII STEP2 BANK OF --- - - -- - - - - - -- - - ~-~ -- ~ ~ ~ ~TOGGLE STEP 3. SWITCHES STEP4 * oV-e Liue44 inr iia chm o ayngCefcet0 Figure 4-4. Binary Digital Scheme for Varying Coefficients

AERONAUTICAL RESEIARCH CENTER UNIVERSITY OF MICHIGAN UMM-67 The total number of steps possible is determined merely by the number of steps available on the stepping relay. Naturally there must also be as many rows of toggle switches as there are steps. It would probably be desirable to add an additional digit to change the sign of a function. This could be done by adding an extra gang and accompanying relay. A sign-changing operational amplifier would be connected into or out of the circuit, depending on the sign of the variable coefficient. The initial cost of this binary digital circuit would probably be somewhat higher than the stepping-relay circuit used at present (Figure 4-3), but the increased versatility would soon pay off in decreased man hours of operation. Any function could be set up in a matter of a very few minutes - merely the time required to position the toggle switches. Furthermore, the digital scheme requires only one set of 13 precision resistors for 0.01% accuracy, whereas the arrangement of Figure 4-3 necessitates a vast supply of different resistors which could in themselves involve considerable expense. 4.3 Computer Circuit for Solvin the Non-Uniform Beam Problem The equations to be solved for the non-uniform beam are repeated again for convenience. WT 0~ + P (2-15) dT 2ds S K (dT0 (2-17) dTf d + SN 2 r SL2 s (2-24) The computer circuit for solving the above equations is shown in Figure 4-5. The variable functions d, A', 1 f and Ir are approximated by the step-method described in the previous section. No initial condition circuits have been indicated. These will be dictated by the type of end-conditions appropriate for the beam in question (see Table II at the end of Chapter 2). 45

0 2_ L4 x M SLn s 4 ^ I I R -- W \ / -- I 1 I ----- s 2 P I --------- A AA C it r oc 3 A2 ---- s^ ^ ---- SL2 r X2 ~ A's~'A5 f4 a 6 3 A L? L2~~~~~~~~~~~~~~ N~~~~~~~~~~~~~~~ All Resistor Units are Megohms All Capacitors are lmfd Ground Connections are Omitted for Clarity SL2 C All Resistor Units are Megohms, ^ All Capacitors are Imfd I Ground Connections are Omitted for Clarity Clj Figure 4-5. Computer Circuit for Solving the Non-Uniform Beam C z

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 The techniques for solving the uniform beam are again used to obtain the solution of the non-uniform beam. When a starting button is pressed, the initial condition relays are released and the stepping relays begin changing the appropriate resistors. One of the initial voltages is varied until a solution with the correct end conditions is obtained. In general this solution will have a length different from the assumed length L for the computer solution. RX must then be varied until a correct solution of length L is obtained (or a length close enough to L to allow interpolation of R ). The frequency parameter X is then given by X = L2 /T^ (3-9) where R is the exact feedback resistance for Al and A7 which gives a correct solution of exact length L. For beams of reasonable length to width ratios the value of the feedback resistor of Al in Figure 4-4 will be much more critical than that of A7. Hence RXin equation (3-9) should be the value of the feedback resistor of Al, although ideally this resistor should be equal to the feedback resistor of A7. For solving the non-uniform beam problem it is quite important to have a good standard-frequency source available for driving the stepping relays (see Section 5.6). This is necessary not only to insure good accuracy but also to allow solutions to repeat when solving for the critical higher modes. The time scale of the computer is fixed; so, therefore, must be the time scale of the stepping-relays. As was the case for the uniform-beam computer circuit, here again care should be taken to maintain the gains of amplifiers in the main loop (A1-A2-A 4 A5-A6-A3-A1) close to unity. This means that one should make an intelligent guess for > beforehand, and choose L so that R X is about 1 megohm. The electronic differential analyzer has been used to solve several 9,11 non-uniform vibrating beam problems to date with good success, 9 although shear and rotary inertia forces were not included properly in either case. It is in the solution of the non-uniform beam problem that the computer should have its greatest utility, for here is a realm in which mathematical solutions that include shear and rotary inertia forces are virtually impossible by any hand methods, 47

AERONAUTICALL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM —67 CHAPTER 5 COMPONENTS AND EXPERIMENTAL TECHNIQUES 5.1 Components (a) Power supply. The dc power supply for the direct current amplifiers furnishes the following voltages: +300, -190, and -350. The absolute values of these voltages are extremely close to the values given and are held constant to t 1.5 millivolts by continuous automatic checking against a standard cell. The ac ripple is approximately 1 millivolt. The details of the power supply are given in Appendix 4. (b) Direct current amplifiers. The dc amplifiers used in this work are those described previously9'0, modified for use with the power supply voltages given above (see Appendix 2). Improved amplifiers with a continuous automatic balancing feature are in the process of construction. These are described in Appendix 3. (c) Resistors, feedback and input. For most of the work the feedback and input resistors were Continental "Nobeloy X" type. This type of resistor has a low voltage coefficient and a low noise characteristic. The temperature coefficient is less than 0.05 percent per degree Centigrade negative. There is some change of resistance with age so that for very accurate work frequent calibration is necessary. The humidity characteristics are fair, but not perfect. A ten megohm resistor is temporarily changed in value about ten percent when breathed upon by a good slow "ah" with the open mouth very close to the resistor. A coat of ceresin wax removes most of this difficulty. For the most accurate work it is suggested that precision wire-wound resistors be used. These can be obtained with a temperature coefficient as low as + 0.002 percent per degree Centigrade. 48

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 There would be considerable advantage in housing the computing equipment in a constant temperature and constant humidity room. However, this procedure would probably be very expensive, since considerable power is dissipated. One exception to the use of low temperature coefficient resistors occurs in the case of resistors associated with feedback capacitors. (d) Capacitors, feedback. The feedback capacitors used in the integrating circuits are Western Electric D161270 Condensers, 1 microfarad, with polystyrene dielectric. The characteristics of these capacitors are not known by us but it is assumed that they are not far different from those of the Plasticon laboratory-grade condensers with polyethylene dielectric manufactured by the Condenser Products Company. These Plasticon condensers have a negative temperature coefficient of 0.04-0.05 percent per degree Centigrade. When an amplifier is used as an integrator it is important that the feedback capacitance, C, and the associated input resistance, R, be such that the product RC remains constant. If the temperature coefficients of both R and C are negative the product RC will be temperature sensitive. Such was found to be the case. Fairly satisfactory results were obtained by using Akrohm precision wire-wound resistors with a positive temperature coefficient of approximately 0.017 percent per degree Centigrade. Using this type of resistor with a Western Electric condenser gave an RC product that remained constant to within 0.05 percent for a temperature change of 5 C. Ideally R and C should have equal and opposite temperature coefficients, as well as the other properties demanded by accurate computing. The Condenser Products Company make condensers with temperature coefficients from +800 to -800 parts per million per degree Centigrade. The suitability of these condensers should be investigated. 5.2 Measurement of Resistances Most of the resistors used have a tolerance of ~ 1 percent and must be calibrated if better than one percent accuracy is desired. The ordinary box bridge, or test set, was not found suitable, particularly for measuring resistances of one megohm and higher. The bridge arrangement shown in Figure 5-1 49

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-67 proved to be satisfactory both in accuracy and in speed of measurement. All of the equipment, with the exception of the galvanometer and recorder, is mounted on a small polystyrene panel (1/4" thick). The bridge proper consists of two "ratio" resistors, R1 and R2, the unknown resistor, Rx, and a decade box, R. General Radio (or equivalent) jacks, with standard 3/4" separation, are provided for plugging in the resistors and the galvanometer and recorder connections. The resistors R1 and R2 are General Radio Type 500 with plug type terminals. Minimum requirements are two 10,000 ohm standards and one each of 1,000, 100 and 10 ohms. A decade box similar to General Radio Type 602-L (111,100 ohms total, in steps of 10 ohms) makes a suitable variable resistor at Rs. B2 is a battery of suitable voltage (45 volts) and B is a shorting plug which may be removed to insert either a protective resistance or an additional battery in series with B2. The bridge may be balanced either by the galvanometer G or by a Speedomax recorder (0 - 10 mv), the recorder being used merely as an indicator. The dry cell B1 and the associated network permits a five millivolt bias voltage to be put on the recorder to place its zero point for bridge balancing at the middle of the scale. The use of the recorder as a null indicator gives speed to the measurements and good sensitivity. With R1 = 10,000 ohms and R2 = 100 ohms a sensitivity of well over one-tenth of one percent exists in measuring a ten megohm resistor. Resistances as high as 30 megohms have been measured. For most measurements an accuracy of the order of one-tenth of one percent is to be expected. In using the bridge care must be taken not exceed the current rating of any of the resistors. If R1 were made permanently 10,000 ohms and a fixed 10,000 ohm resistor were connected in series with R (and added to the value of Rs) the bridge would be protected against overload up to B2 = 90 volts. 5,3 Measurement of Capacitance A very suitable method for determining the capacity of condensers involves measurements of the period of oscillation of the computer circuit shown in Figure 5-2. At least three capacitors are needed. In the circuit shown C1 and C2 are two of the 1 mfd condensers to be measured. R1 and R2 are one megohm 50O

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 R IT B R2I d 10 K J lOOn- 25 K Rn^D2/ v Rs I —---- O-IOMV RECORDER WHEATSTONE BRIDGE CIRCUIT FOR MEASURING- HIGH RESISTANCES Figure 5-1. SI 82 Rf c| C2 Ri RI R2 Al A2 A3 OSCILLATOR CIRCUIT FOR CALIBRATING CAPACITORS Figure 5-2 51

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 resistors which have been carefully calibrated. Ri and Rf are a pair of accurately matched resistors of equal value. The circuit is an undamped harmonic oscillator representing the equation y + RR2C1C2Y = 0, for which the frequency is given by /02 1 The circuit is put into operation by simultaneously opening switches S1 and S2, and the output is recorded on a Brush oscillograph. Timing pulses from an accurate source are recorded on the same oscillograph. The timing pulses should be accurate to at least 0.01 percent. In the work associated with this report seconds pulses from WWV were used. From the oscillograph records the period of oscillation is determined. This procedure is repeated for condensers C1 and C3, and then for C2 and C3. On the assumption that resistors R1 and R2 differ but very little from one megohm and that the capacitors differ but little from one microfarad, we write R1 = 1 + rl, C1 = 1 + C1, R2 = 1 + r2 C2 = 1 + c2, C3 1+ c3. Also let T 1 T= 1 + e = 1 Then for the first oscillator set up, using C1 and C2, 2 1 + 12 1C2 52

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-67 = j(1 + rl)(1 + r2)(1 + cl)(l + c2) 1 + r + r2 cI + c2 rl + r2 + C1 + C2 - 1 + —.-. 2 Hence r + r2 + C1 + 2 12 2 Similarly, rl + r2 + c + c3 13 2 and r1 + c + 23=' 2 Solving these last three equations for cl, c2, and c3, we obtain I +% 9 2r- + r c3 = 6 12 2 C = 12 + 13 23 2 CE r + r 2 2 = 12 +-23 ~13 2 C E + E E rl + r2 C3 + 23 12 2 From the observed deviations, 612, E13 and 123, of'from unity and the known values of r1 and r2, the values of cl, c2, and c3 and hence C1, C2 and 03 can be determined. 53

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN u2i-67 Having measured the capacitances in this manner it is possible to select resistors to obtain a desired RC product to within 0.01 percent. In selecting these resistors cognizance must be made of the fact that the temperature at the computer may be several degrees higher than room temperature. 5.4 Pulse-Forming Circuit for Checking End Conditions Frequently occasions arise which make it necessary to determine accurately when a function becomes zero, or when two functions become zero simultaneously. Ordinarily a precise observation is difficult because of such effects as recorder pen lag, error in determining the exact point of crossing the zero axis, etc. A method which aids in making more precise measurements is shown in Figure 5-3. Assume that a function F1 crosses the zero axis with a finite slope. Amplifier A1 increases the amplitude by a factor of 30. The 0.05 megohm input resistor to amplifier A2 acts as a load on amplifier Al, preventing, due to saturation effects, a large voltage from being developed across the output of Al. However, the rapid passage through zero of the output voltage is a faithful reproduction of the function F1 as it passes through zero. An additional gain of 200 is obtained through amplifier A2, the output of which is a saturation voltage at all times except for the extremely abrupt passage through zero. The differentiating CR circuit produces a sharp pulse at the exact instant of the passage through zero of the output voltage. This pulse can be recorded directly, or mixed in amplifier A3 with another function F2. Using the output of A3 it is possible to observe on the same record the passage through zero of each of the two functions, F1 and F2. Figure 5-4 shows a sinusoidal input function, F1. The output of amplifier Al is shown in Figure 5-5 and that of A2 in Figure 5-6. Figures 5-7 and 5-8 show the pulse recorded at paper speeds of 1 division per second and 25 divisions per second, respectively. Figure 5-9 shows the output of amplifier A3 when the same function is fed into amplifiers A1 and A3. It can be seen that the pulse occurs slightly before the function, as recorded, crosses the zero axis. This effect is caused by pen lag in recording the sinusoidal function. On the same record are shown seconds time pulses as received from WWV. In this particular recording of the 54

30 10 I0.00125 m _fd ~~~~~F,~~~~~~~~~~~~~~~ II INPUT - t Fl PULSE m ALL RESISTORS IN MEGOHMS 0 F2 | ~A3 |FUNCTION + PULSE CIRCUIT FOR FORMING PULSES AS FUNCTION GOES THROUGH ZERO 0 Figure 5-3 I__________I______11z

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN. —-------- ~~UMM-67 I~~~~~~~~~~~~~ ~~~~~~~ A;_rLI I; I It I I I. I'I FigI. 1. I IIIa I,, Al I IL I I: I'.L I - = —a',_,' __- ~. A __-W-_ i\l I Figure 5-4. Sinusoidal Input Function, F1 Figure 5-56. Output of Amplifier A1,,, ---—,,- _! i, l'1 —-— r —-------- S I I I I I "'/ I'' r ~I IL I II II7__!1 1 IF1 A" 1 I F rl I I1r F 1 r l-'IL~~~~ I f., 1\1 1 11 1 11 I I 11 1\1 1 =-ill IIL I IF 1.. 1 ~ I,' J,. {i,. I..... I, IF, i... Figure 5-5. Out:put, of Amp:Lifier A2 IF~5 I I / I I ~~~~~~~~~~~~~~~~~~~~I I IF I I I / I / I'I 1~~~~~~~~~~~~~1 I = -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~7 =k F~~.gr.5...4. Snsia Inu,F.nct.o.. F e~~~~~~~~~~~~~~~~~~~~~~i I I I T I I I IL I I I1 I ILI 11 5 1 01 1 1 Figre 5-5. Output of Amanlifier A |~~~~~~~~~~~~~~~~~~~~~~~~~~~~o - Il - 56~~~~~~~~~~~~~~~~~~~~~~~a

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN, —-----,, [eU4MM-67 Figure 5-7. Pulse Recorded at a Speed of 5 div/sec Figure 5-8. Pulse Recorded at a Speed of 25 div/sec 7 7 I I I CHART NO. BL 909 I/'' I 1'. Figure 5-97. Sinesoidal Function with Superimposed HZero Figure 58.Pulses, and Seconds Pat a ses from W25 div/sec 57 57

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 time signals the noise level was rather high, but the time pulses are readily distinguishable. 5.5 Use of Time Signals from Radio Station WWV Very accurate timing pulses are needed for several reasons: (1) to aid in the measurement of capacitances, (2) to determine L, the exact time taken by the computer to complete its solution, and (3) to operate stepping relays for changing computer resistors when studying non-uniform beams. By far the most satisfactory method of obtaining accurate timing pulses is to use an oscillating quartz crystal frequency standard or equivalent, as described in a previous re10 port. Such a standard should have an accuracy of at least 0.01 percent under all operating conditions. Since a standard frequency source was not available for the present work, seconds time signals from Radio Station XWV were used instead. WWV transmits continuously, day and night, standard radio frequencies of 2.5, 5, 10, 15, 20, 25, 30 and 35 megacycles per second with an accuracy of one part in 50,000,00OO Each of these carrier frequencies is modulated by three audio frequencies, 1, 440 and 600 cycles per second. A 1,000 cycles per second pulse of 0.005 second duration may be heard as a faint tick every second, except the 59th second of each minute. The time interval marked by the pulse every second is accurate to one microsecond. The two audio frequencies of 440 and 600 cycles per second are transmitted alternately for four minutes out of each five minute interval. The audio frequency is interrupted at precisely one minute before each hour and each five minutes thereafter (59th minute; 4 minutes past the hour; 9 minutes past the hour, etc.); the alternate audio frequency is resumed in each case after an interval of precisely one minute. This one minute interval is used to announce Eastern Standard Time and to give certain other information. In order to receive the seconds-pulses, exclusive of other modulation frequencies, the output of a radio receiver tuned to one of WWV's carrier signals is passed through a 1,000 cycle filter. The output of the filter is rectified and the resulting DC pulse used to place time markers on the oscillograph record. Figures 5-lOa and 5-lOb show the circuit arrangements used. In Figure 5-lOa a General Radio band pass filter Type 830R is employed. This filter permits ---------------- ---------------

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 IN 34A 0.006,I/d 5000n.TO 500^. lo 5000.nr, — RECORDER FI LTER AMPLIFIER RADIO FILTER FL-8-A 30K 0.1 MEG. 0.005 f/4d (a) 2.4 K 500o 5000 aL IN 34A'S 1000", FILTER GR TYPE 830R 0.014d 39K IOOK "0.01614F TO RECORDER AMPLIFIER (b) Figure 5-10. Circuits for Obtaining Seconds Pulses from WWV 59

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN l..... —---- UMM-67 the use of a full wave bridge rectifier as shown. In Figure 5-lOb the filter is a surplus property Radio Filter FL-8-A. Since this filter has a terminal common to both input and output, a half wave rectifier is used. The networks associated with the filter outputs were not scientifically designed, but merely selected by trial and error to give satisfactory pulses. The time pulses obtained are shown in Figure 5-9. These time pulses were obtained and used during the latter part of August and the first part of September (1950). At night and during the forenoon suitable signals could usually be picked up on 5 or 10 megacycles; sometimes one carrier frequency was better than the other. After about 2:30 P.M. it was difficult to receive satisfactory signals. For this and other reasons it would be highly desirable to have a local standard frequency source. 5.6 Proposed Electronic Clock In many problems it is necessary to determine with a high degree of accuracy the length (in time) of a solution obtained by the computer. Thus far the procedure has been to record simultaneously the solution of the problem and accurate time pulses. The time of solution is then determined by measuring (in centimeters) the length of solution shown on the oscillogram and a nearly equal length (in centimeters) covered by an integral number of time pulses. From these two measurements the length (in time) is computed. Considerable labor could be saved and better accuracy obtained by using an electronic clock for measuring the time it takes for the computer to obtain the solution. This electronic clock would consist of a local standard frequency source and an accurate electronic counting device. The standard frequency source should furnish pulses at the rate of 1,000 per second. The counting device would consist of a set of decade scaling units to indicate elapsed time to the nearest 0.001 second. The Decascale Unit SC-11 of Tracerlab Inc., would serve very well for the basic counting units. The clock would be started by a pulse at the time the solution is begun and stopped by a pulse at the end of the solution. This latter pulse could be obtained in the manner described in Section 5.4 above. The clock would indicate the length (in time) of the solution without calculation or measurements of length. Figure 5-11 shows a block diagram of the proposed electronic clock. 60

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN Nm M-67 STANDARD FREQUENCY SOURCE PULSE FROM COMPUTER INDICATES BEGINNING OF SOLUTION AND STARTS CLOCK _ COUNTING CIRCUIT (ELAPSED TIME PULSE FROM COMPUTER INDICATES INDICATED END OFSOLUTION AND STOPS CLOCK BY LIGHTS) Figure 5-11. Block Diagram for Proposed Electronic Clock CHART NO. BL 909 THE BRUSH DEVELOPMENT CO. PRINTED IN U.S.A. Figure 5-12, Computer Solution for Second Mode, Showing Superimposed I —---- \t /\- I\- \I — \ I\ x- — = —-\\ \\ - I_ ----—' —--- CH- NO - -... BL..... CO... U. S.A "Zero" Pulses and Seconds Pulses from WWV 61

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 5.7 Technique for Measuring the Length of the Computer Solution In order to obtain the frequency parameter X of the vibrating beam it is necessary to measure the length L of the computer solution (see Sections 3.2 and 3.3). L is defined as the elapsed time from the beginning of the solution to the instant when the proper end conditions are met. For the free-free beam the initial and end (or final) conditions are that do/dT = O and that of = 0 (i.e., that shear and bending moment are zero). The function dO/dTis recorded from the computer, along with pulses occurring at ( = O (using the technique described in Section 5.4). One of the initial voltages is varied until a solution satisfying the proper end conditions is obtained. In order to determine more accurately when the correct end conditions have been realized, the d/dT record is "blown-up" by a factor of ten in the region where it is near zero in value, i.e., at the end of a correct solution. The dp/dT record is also blown-up by a factor of ten at the beginning of a solution. A correct solution obtained in this manner for a second mode is shown in Figure 5-12. The pulses superimposed on the dO/dT curve occur at the instant Co goes through zero. The length of the solution L is just the time between initial and final pulse. This time is obtained by comparison with the WWV seconds-pulses on the upper channel. 62

AERONAUTICAL RESEARCH CENTER — UNIVERSITY OF MICHIGAN UMM-67 CHAPTER 6PROPOSED WORK In addition to continuing the investigation of the general utility of the electronic differential analyzer for obtaining solutions to theoretical and physical problems, there are a number of specific lines of investigation suggested by the work done thus far, including the following: (1) the publication of a complete set of frequency and mode-shape curves for all types of uniform beams, (2) an experimental study of the proper end conditions for a cantilever beam, (3) the comparison of experimentally determined eigen-frequencies for vibrating uniform beams with the eigen-frequencies obtained with the computer, (4) a check on the accuracy obtained in using the electronic computer to determine solutions for non-uniform beams, (5) the degree of validity of the basic equations of Chapter 1 for eigen-vibrations having wave-lengths comparable to the transverse dimensions of the beam, and (6) a complete and comprehensive bibliography of the work which has been done on the problem of vibrating beams. 6.1 A Complete Set of Frequenc and Mode-Sha Curves for all Tyes of Uniform Beams In this report we have included data and families of curves giving the relation between frequency parameter X and shear and rotary inertia parameters S and N for the first three "normal modes" of vibration of free-free beams. A definite contribution would be made by obtaining similar data and curves for all types of uniform beams. Including the one already done, these would be: (1) Free-Free Beam, (2) Clamped-Free Beam, (3) Hinged-Free Beam, (4) Clamped-Clamped Beam, 63

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-67 (5) Clamped-Hinged Beam, and (6) Hinged-Hinged Beam. Although some of the above types of beams are more important than others, it is suggested, for the sake of completeness, that data and curves for all types be obtained. In determining the data for the free-free-beam curves as given in this report the values for rotary inertia parameter N (0.0, 0.1, 0.2 and 0.3) were selected somewhat arbitrarily. It is desirable to determine values of N for beams of various materials and forms of cross-section. The cross-sectional forms should include cylinders, hollow cylinders, rectangles, I shapes, U shapes, etc. From the values of N found in this way one can determine the most suitable values to be included in the data for the families of frequency curves. Publication of these data and curves, together with typical mode shape curves, is desirable. 6.2 An Experimental Study of the Proper End Conditions for the Clamped End of a Beam There seems to be some question concerning the correct expressions for the type of restraint placed on a beam end when it is "clamped". When the effects of rotary inertia and transverse shear force are neglected, the boundary conditions at the built-in end are assumed to be y = 0, and dy/dx = 0. When the effects of rotary inertia and transverse shear force are included, several 7,8 authors8 have given the boundary conditions as y = 0 and v = 0, where 3 is the neutral axis slope due to bending moment. This latter condition means that dy/dx, the slope of the neutral axis, has a finite value at the built-in end of the beam. This condition seems to contradict the usual definition of a clamped beam. However, in general there will be distortion in the built-in portion of the beam so that the assumption made in deriving our basic equations of Chapter 1, namely that the planes of flexure remain parallel, will no longer be valid. In order to solve the problem rigorously we must consider carefully not only the elastic properties of the beam but also of the supporting wall. To apply the equations of Chapter 1 with either end condition (y = 8 = 0 or y = dy/dx = 0) is an oversimplification. The problem is not merely academic, since preliminary results with the electronic computer indicate a difference of about 20 percent in eigen frequencies of the first mode, for a 1/S value of 11, depending upon which end condition is used. 64

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 13 One interesting sidelight to this problem is that Dolph has shown the f = 0 end condition is the only one which leads to orthogonal solutions when the effect of rotary inertia is included. A direct experimental approach to the problem is to construct a small uniform cantilever beam and experimentally to observe its eigen-frequencies. These values should then be compared with those obtained by calculations using the two proposed sets of end conditions. To compute the eigen-frequencies from theoretical considerations it is necessary to know the values of the modulus of elasticity, E, and the modulus of rigidity, G. It is suggested that these moduli be determined dynamically, following the general procedure given by Goens4. Attention is called to the fact that static and dynamic determinations of G differ by approximately one percent. Little difficulty should be experienced in exciting transverse and longitudinal vibrations in a steel bar by an audio oscillator with an electromagnetic drive. The excitation of torsional vibrations cannot be obtained as easily. One method worth considering is the use of a fluctuating, as opposed to a rotating, magnetic field. This fluctuating field could be obtained by means of a four pole stator, similar to that used in a small two phase motor. One pair of opposing poles should be excited by direct current, the other pair by alternating current. One would expect the round rod, acting as a rotor, to receive oscillatory torque impulses of the same frequency as the alternating current applied to the stator, and that relatively large torsional oscillations would be set up in the rod when the correct frequency is applied. In setting up the experimental cantilever beam care must be taken to imbed the beam tightly in the "wall" at all points. The mass into which the beam is imbedded should be so large that no measurable vibration of the wall mass takes place. A somewhat less direct method, but one which might possibly give more enlightenment on the conditions which exist in the built-in portion of the beam, involves the use of the electronic computer for solving non-uniform beams. Consider the cantilever beam as consisting of a uniform rod of relatively small diameter imbedded in another uniform beam of much larger diameter. By obtaining a series of solutions for greater and greater sizes of this large beam it is possible to approach the ideal case of having the smaller beam clamped in a rigid 65

AERONAUTICAL RESEARCH CENTETR -UNIVERSITY OF MICHIGAN UMM-67 massive wall. From the observed curves of y, dy/dx, oc, and v there should be some indication of the conditions which exist when the mass of the "wall" becomes infinite. 6.3 Comparison of Exxperimental and Computer Values of Eigen-Frequencies Goens4 has made some very precise experimental measurements on vibrations of free-free cylindrical steel bars and on an aluminum bar of rectangular cross-section. In most cases the necessary data is given to four significant figures. The comparison of his results with those obtained by the electronic differential analyzer would serve as a good check on the validity of the equations. Alternately, the check could be made with vibrating beams set up and measured in the laboratory. Dolphl3 has shown that when the effect of rotary inertia is included, the solutions are in general not orthogonal. Dolph further argues that when rotary inertia effects are included, equations (1-1) to (1-5) are fourth order in both time and displacement along the beam, resulting in four boundary and four initial conditions. But the standing wave hypothesis involved in separating the time variable from equations (1-1) to (1-5) allows only two initial conditions in time to be satisfied. Thus the validity of equations (1-1) to (1-5) might seriously be questioned. Good laboratory checks on beam frequencies, where rotary inertia effects are quite appreciable, ought to throw considerable light on this whole question. 6.4 Checkin the Computer for the Solutions of Non-Uniform Beams It is assumed that a non-uniform beam may be represented approximately by a finite number of equal length uniform beams having appropriate dimensions and properties. It is suggested that small non-uniform beams be constructed and that the experimentally determined eigen-frequencies be compared with those obtained by using the electronic computer. At least one of the beams should consist of a series of equal length uniform beam increments. In this case the computer solution would be made without approximations and a direct check could be obtained between frequencies determined by experiment and by the computer. In addition a beam should be constructed of continuously varying size, preferably with symmetry about the center of the beam. For solution by the 66

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 computer this beam would be approximated by a finite number of uniform beam increments. Frequency comparisons would indicate the accuracy with which the computer gives the frequency when operating on the approximated equations. Another check on the computer would be to use the approximation data available for the ship APA87. This ship is 400 feet long and is considered divided into 20 equal-length sections for each of which are given the necessary properties and dimensions. This ship problem has been solved by several other methods. 6.5 Validity f Basic Equations for Eigen-Vibrations of Wave-Lengths Comparable to Transverse Dimensions of Beams For higher and higher modes of vibration of a beam the corresponding wave-lengths become smaller and smaller. The question arises as to what happens when the wave-lengths become comparable in size to the transverse dimensions of the beam. Goens attacks the problem analytically and shows that above a certain critical frequency limit a change in the type of oscillation should develop. He states that Giebe and Scheibe, working with prismatic quartz bars of a few millimeters thickness, obtained limiting frequencies which, from data unpublished at the time of writing his paper, seemed to be somewhat smaller than the above mentioned critical frequency. Goens raises the question as to whether this limiting frequency has any physical meaning, even under the assumption that the basic equations give at least a qualitative description of the vibration process under such extreme conditions. It would be of interest to determine to what extent the basic equations used in the present approach to the beam problem are valid when the wave-lengths become comparable in size to the transverse dimensions of the beam. 6.6 The Need for a Complete and Comprehensive Bibliography Originally there was no thought of making the study of vibrating beams a major project associated with the computer. The problem was presented to the group working with the electronic differential analyzer as the solution of a specific fourth order differential equation subject to certain boundary conditions for a free-free beam. The problem was welcomed as an opportunity to show what the computer could do. 67

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 The computer, itself, demonstrated that the assumed end conditions were in error. Furthermore it was soon discovered that the use of the fourth order equation for non-uniform beams neglected many derivatives which are important. However, after some investigation it was found that by using the basic relations from which the fourth order equation was derived the computer could obtain solutions without neglecting any derivatives. If, before attempting to solve the fourth order differential equation by the computer, adequate reference had been made to easily available literature, it is probable that neither of these problems would have arisen. Subsequently, enough reference has been made to various articles to find that there is available a considerable amount of information and data which would be helpful in continuing work on the general problem of vibrating beams. Among the vast amount of work which has been done on vibrating quartz crystals there is considerable information directly applicable to the present problem. If work is continued on vibrating beams, it is suggested that a complete bibliography, with abstracts, be compiled. 68

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN IUM4-67 APPENDIX 1 SUMMARY OF NORMAL MODE FREQUENCIES FOR A FREE-FREE BEAM, INCLUDING TRANSVERSE SHEAR AND ROTARY INERTIA FORCES Notation (see Chapter 1) EI EI S= 2 kG E \o = \ for S = N = O (i.e., for an infinitely-long beam) L = length of computer solution in seconds 69

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 FIRST MODE "FREE-FREE" BEAM Xo = 22.373 = 0.0446967 1 N X- L2 1 XlOO 1 L2 1 X ~2 N = L2 1x 100 2 N =L2 S x 100 SL2 b SL2 O 29.55 0.0 22.203 656.1 99.240 3.982 0.0 20.867 83.09 93.268 0.1 22.099 653.0 98.775 0.1 20.412 81.28 91.234 0.2 22.034 651.1 98.484 0.2 19.909 79.28 88.986 0.3 21.930 648.0 98.019 0.3 19.430 77.37 86.845 19.83 0.0 22.090 438.0 98.735 2.977 0.0 20.421 60.79 91.275 0.1 21.977 435.8 98.229 0.1 19.803 58.95 88.512 0.2 21.837 433.0 97.604 0.2 19.193 57.14 85.786 0.3 21.697 430.3 96.978 0.3 18.602 55.38 83.144 14.92 0.0 21.977 327.9 98.229 1.990 0.0 19.483 38.77 87.082 0.1 21.809 325.4 97.479 0.1 18.697 37.21 83.569 0.2 21.678 323.4 96.893 0.2 17.893 35.61 79.975 0.3 21.511 320.9 96.147 0.3 17.148 34.12 76.645 9.92 0.0 21.772 216.0 97.313 1.502 0.0 18.636 27.99 83.296 0.1 21.520 213.5 96.187 0.1 17.707 26.60 79.144 0.2 21.326 211.6 95.320 0.2 16.810 25.25 75.135 0.3 21.114 209.5 94.372 0.3 15.928 23.92 71.192 6.958 0.0 21.548 149.9 96.312 0.9963 0.0 16.991 16.928 75.944 0.1 21.197 147.5 94.743 0.1 15.952 15.893 71.300 0.2 20.867 145.2 93.268 0.2 14.900 14.845 66.598 0.3 20.593 143.3 92.043 0.3 13.883 13.832 62.052 4.946 0.0 21.197 104.84 94.743 0.1 20.739 102.58 92.696 0.2 20.349 100.64 90.953 0.3 19.927 98.56 89.067. —----------------- 70

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 SECOND MODE "FREE-FREE" BEAM o = 61.6696 1 = 0.01621544 N2 1 2x100 12 2 N \ =L2 -x100 2 N X= L2 1 x 100 SL2 o SL O 29.55 0.0 60.934 1800.6 98.807 3.982 0.0 56.190 223.7 91.114 0.1 60.746 1795.0 98.502 0.1 55.145 219.6 89.420 0.2 60.575 1790.0 98.225 0.2 54.081 215.4 87.694 0.3 60.482 1787.2 98.074 0.3 52.984 211.0 85.915 19.83 0.0 60.528 1200.2 98.148 2.977 0.0 54.361 161.83 88.148 0.1 60.187 1193.5 97.595 0.1 53.086 158.04 86.081 0.2 60.063 1191.0 97.394 0.2 51.926 154.58 84.200 0.3 59.939 1188.6 97.193 0.3 50.652 150.79 82.134 14.92 0.0 60.125 897.1 97.495 1.990 0.0 51.094 101.68 82.851 0.1 59.815 892.4 96.992 0.1 49.421 98.35 80.138 0.2 59.537 888.3 96.541 0.2 47.734 94.99 77.402 0.3 59.213 883.5 96.016 0.3 46.050 91.64 74.672 9.92 0.0 59.367 588.9 96.266 1.502 0.0 48.066 72.20 77.941 0.1 58.890 584.2 95.492 0.1 46.158 69.33 74.847 0.2 58.446 579.8 94.772 0.2 44.209 66.40 71.686 0.3 57.912 574.5 93.906 0.3 42.250 63.46 68.510 6.958 0.0 58.446 406.7 94.772 0.9963 0.0 42.576 42.42 69.038 0.1 57.821 402.3 93.759 0.1 40.284 40.13 65.322 0.2 57.199 398.0 92.750 0.2 37.896 37.76 61.450 0.3 56.475 393.0 91.576 0.3 35.355 35.22 57.329 4.946 0.0 57.184 282.8 92.726 0.1 56.355 278.7 91.382 0.2 55.413 274.1 89.854 0.3 54.597 270.0 88.531 71

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN -UMM-67 THIRD MODE "FREE-FREE" BEAM 0 = 120.890 = 0.008271983 12 ~ ~ ~ -1012 1 X2 2 N = L2 x 100 N L 1 xlOO SL N O SL\ 29.55 0.0 119.40 3528 98.767 3.982 0.0 109.29 435.2 90.404 0.1 119.18 3522 98.585 0.1 107.45 427.9 88.882 0.2 118.81 3511 98.279 0.2 105.78 421.2 87.501 0.3 118.61 3505 98.113 0.3 104.04 414.3 86.o61 19.83 0.0 118.77 2355 98.246 2.977 0.0 105.33 313.6 87.128 0.1 118.16 2343 97.741 0.1 103.33 307.6 85.474 0.2 117.81 2336 97.452 0.2 101.24 301.4 83.475 0.3 117.57 2331 97.253 0.3 99.14 295.1 82.008 14.92 0.0 117.74 1756.7 97.394 1.977 0.0 98.66 195.05 81.611 0.1 117.25 1749.4 96.989 0.1 95.90 189.59 79.328 0.2 116.62 1740.0 96.467 0.2 93.20 184.26 77.094 0.3 116.25 1734.4 96.161 0.3 90.44 178.80 74.811 9.92 0.0 116.23 1153.0 96.145 1.502 0.0 92.37 138.74 76.408 0.1 115.30 1143.8 95.375 0.1 89.36 134.22 73.918 0.2 114.40 1134.8 94.631 0.2 86.27 129.58 71.362 0.3 113.53 1126.2 93.911 0.3. 83.14 124.88 68.773 6.958 0.0 114.08 793.8 94.366 0.9963 0.0 81.07 80.77 67.060 0.1 113.06 786.7 93.523 0.1 77.67 77.38 64.248 0.2 111.98 779.2 92.629 0.2 74.11 73.84 61.303 0.3 110.97 772.1 91.794 0.3 70.34 70.08 58.185 4.946 0.0 111.43 551.1 92.174 0.1 109.96 543.9 90.958 0.2 108.56 536.9 89.800 0.3 107.02 529.3 88.526 72

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN rn-MM47 O 0 L 0 o O 0O 0 3oo - co -( o~Ic~~o........................... *,.< o oo o o o -:.........*- -*. -- -*-. —-- - f 0 -:::-..,,.,..-,: C t 0 tO 0 —- tOMi- - -i oi - HIE t ~~ ~ O 1, - t — (0 (0 a37 1 i. -.. -4 -. ^ u o + o 4 -- - -j1 M!"I II i Nil li I Iiii ill!lit 7 + +El! H I fe ^ ^^ 1 il L. S o t g -H4+`1W 7 tt3tti -H+- 8 1 WIWmtrZI t $If gW ze I ~ i Ill I I I tt j | ffi ffi ffi tit-lRl- l 1+1 1 W1 1 1 If-F FIT IN ------ ~ ~r c~~~~~~~lc ~ ~ ~~~l_~ U0.............. -- _To T _r F............O L O O O d d d d, d d~~~~~~~~~~~~~~~~~~~~~ —------ — +H+ H H+I — ----- -----

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN " —----, ~ ~ ~~ UMM-67 APPENDIX 2 DESCRIPTION OF THE OLD DC AMPLIFIER The work covered in this report was carried on using the dc amplifier shown in Figure A2-1. This circuit is similar to one presented by Ragazzini 12 et al and has been used with very satisfactory results in the electronic 9,10 differential analyzer 910 It has an overall dc gain of about 40,000, which is completely adequate for the problems solved thus far. This amplifier is liable to drift off of balance over extended periods of time. The amplifier chassis is pictured in Figure A2-2. Balance potentiometer controls are provided, along with jacks for plugging in feedback or input impedances. A more complete description of these amplifiers is given in a previous report.9 OPERATE BALANCE *K * TO INPUT JACK ON CHASSIS ------ 250K -V75 MMFD - 74 11 i i.5 1MEG- 3.9 MEG ME^0D 6SL7 SL7 --- SL7 INPUT 500KK - - I800K OUTPUT 3.3 K 2 800K 13.3 K MEG E X= A FIXED RESISTANCE ADDED AS NECESSARY o TO INDIVIDUAL AMPLIFER TO MAKE DC BALANCING- POSSIBLE 500 K J~_ -~Id A4 6~ 6 = —- 6V(OC) -350 -190 +300 G ND. Figure A2-1. Old DC - Amplifier Circuit -- -----— 74

AI4RIUONAUTICAt RESEElARGCH CENTI& R- UNIVER:,SITY OF MICH-IGAN -____ Bu1 67...UM-67 Figure A2-2. Operational Amplifier Chassis 75

AE!Wt!ONAUTICAL REvS AU..Ae C CENTER - U NIVERSITY OFi MICHIIGAN Uv- UM-67 APPENI)X 3 THEi DRIFT-STABILIZED DC AMPLIFIER A3-1 Importance of jow-Drift Operation The dc amplifier described in Appendix 2 and used for the work covered in this report is liable to drift off of balance over extended periods of time. This drift may be due to very slight changes in power supply voltages or in amplifier components. In general the drift is quite small during the several seconds required for a computer solution, particularly when the feedback is enough so that the operational amplifier is operated at a gain of about unity. However, it only takes a zero-drift of the order of a millivolt to affect noticeably the behavior of higher-mode solutions (see Section 3.4b). For many types of problems it is desirable to run solutions for several minutes at a time (this may be necessary, for example, if a recorder of low frequency response is used). It may even be necessary to stop the solution midway through a problem, perhaps to read the computer voltage outputs with a potentiometer. Low-drift do amplifier operation is an obvious requirement if either of the above procedures are employed, A3-2 General DescrtipJt of the Drift-Stabilized DC Amlifier An ingenious method for practically eliminating do drift was worked out by RCA and Leeds and Northrup, and is shown schematically in Figure A3-l. The arrangement consists of a conventional do amplifier of gain A and having input and feedback impedances Zi and Zf respectively. The do amplifier actually has two isolated inputs, at P and P'. The net input voltage is the sum of the two voltages appearing at these points. The potential at the point P is fed through a 60 cycle vibrator, chopped into ac, amplified, and reconverted to do. This amplified dc signal is fed into the do amplifier at 1. From here it gets amplified through the do amplifier and fed back through Z2 to P., which is driven back to zero potential (relative to ground). The function of the additional drift-free loop, therefore, is to maintain zero potential at the junction point P. If ideally met, this condition 76

AE.RONAUTICAl.'RES —-EISARCII CENT"ER -UNIVERSI1TY OF MICHIGAN UI-67 INPUT |P ----— f - OUTPUT D. 0. AMPLIFIER:............................................:.'.:4:.. LOW PASS LOW PASS FILTER FILTER CONVERTOR AO,.C. PS.E 44 " VI~B~RATOR ~......SENSITIVE [.... VIBRATVOR 4:AMPLIFIER RECTI Figure A3-1. Drift-Stabilized DC Amplifier guarantees that e2 = Zf/Zi el, where eI is the input voltage, e2 is the output voltage. Thus the amplifier is kept in balance, since if e0 = 0, then e2 = 0 It can be shown that the effective gain of the de amplifier and ac loop system is A + AG, where A is the gain of the dc amplifier and G is the gain of the auxiliary, drift-free loop. The frequency response of the auxiliary loop begins to decrease at around 0.1 cps, so that this drift-stabiling loop can only correct for slow drifts. At higher frequencies for which G -> 0, the net gain of the system is still A, while for low frequencies, where G >>l, the net gain is A(G + 1). Thus for the actual amplifier described here, where A 1 40,000 and G = 2000, the effective gain of the system at low frequencies (<O.1 cps) is about 80,000,000. A3-3 CircSit Descpion of the DriftStabilzed D Amplifier The actual circuit employed for the drift-stabiliszed d amplifier is shown in Figure A3-2. This circuit is essentially similar to one developed by the Rand Corporation for their electronic differential analyzer. The input proper and error-signal input are fed into the de amplifier through cathode followers (each half of a 6SU7). Three stages of amplification follow using a 5691, 12SJ7, and 1631 tube respectively. The 6bU7 and 5691 tubes 77

____________________ 3X T___ _........ I*vf! i.:o=V'~=~:3 V K __ ___ 3AUXC2 yOXS ( ^ 2|'| &5 ~~~~~~~~~~~~~~~O} 3 -2 l7 1/2s 67 7 500V 5 693 I 3:??2 G? 2G0Xo | A5?'I, ~ ~: =2 5,' 22____.SS._..... I 2X2?2^ WZSW^ 1| ^( 1 | ^^ I — - ^^ Rw v -I Oi.... I r I - t e _ 1 200? i |)S I5 I | <| |R SS$? -T -.'? A Fige A3-'2. Cirit for Drif-Stbizd DOC Aplifier. 7 ___ Al ________________________ -__f 600 69 -- 3I S9X I Z2 0R Ni6CATOR aE7ES SV63:?iN STr~0?K z ) G J IASS(25 7 ^: g j jl; X o ^ 6317 X! 3 $ 702 56 91'2!? A 5692fl$2 I6E 2?2O5XASCZ OEA6ACSa:S?:C3 CAN S3?0XaAZ^, LIrg 22-r Fignre~i A3-20 Cic M o rf-tblie CApiir.

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AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-67 are operated at reduced filament voltages of 4.2 and 5.25 volts respectively. This prevents the flow of appreciable current to the grids. A Leeds and Northrup Model 3338-9 special vibrator is used to chop the error signal into ac. A 5693 tube and a 6SL7 tube are employed in the ac amplifier, and the same L & N vibrator reconverts the ac signal to the amplified dc error signal. The error signal itself can be read directly off the errorindicating meter. It is desirable in normal operation to have the amplifier balanced with the manual control so that a minimum of error-signal is required to keep the system in balance. Relays are provided in order that initial conditions can be imposed on the operational amplifier; a solution can also be stopped and held fixed at any time. The amplifier panel is shown in Figure A3-3. Jacks are provided on the face of the panel to receive General Radio plugs with standard 3/4" separation, so that input or feedback resistors can readily be plugged into the unit. Space is available back of the chassis to plug in a feedback capacitor when the unit is to be used as an integrator. A switch is provided to select manual or automatic balance operation. The manual balance control operates a ten-turn heliopot. Error-signal can be read directly off of the meter on the panel provided for this purpose. The signal-ground connections for all amplifiers should be connected to the power ground at just one point. This is necessary to prevent circulating ground currents from introducing voltage errors in the operational amplifiers. The +300, +100, -190, and -350 volt dc power for the amplifier is supplied by the highly-stabilized power supply described in Appendix 4. All filaments are supplied by direct current. This is necessary to reduce 60 cycle ripple level, particularly in the ac loop, where any 60 cycle pickup is amplified and reconverted to dc error signal. In the final dc amplifiers we plan to replace all 12SJ7 and 6SL7 tubes by 5693 and 5691 tubes respectively in order to prolong tube life. A3-4 Performance Data on the DC Amplifier The drift-stabilizing loop can work only when a feedback impedance is present across the dc amplifier. Therefore it is necessary to measure the 80

100o I I I | I 0 1 - I \\ [ | FREQUENCY RESPONSE OF DC AMPLIFIER MANUAL BALANCE 60 j > AMPLIFIER WIDE OPEN 60 0 0 100 1000 1 OKC 100KC 1c 10M C I, FREQUENCY Fge 4FeecRsneuefthiOe ~~~~~~~ 2~~~~~~~~~~~01 I

AERONAUTICAL RESEARCH CENTER UNIVERSITY OF MICHIGAN UMM-67 wide-open gain of the amplifier with the balance control on manual, i.e., with the stabilizing loop disconnected. Amplifier gain as a function of frequency is shown in Figure A3-4. Note that the Bode criteria for stability is met; the gain does not fall off faster than 12 db per octave. No combination of input or feedback impedance can cause the amplifier to go into oscillation. Output saturation voltage as a function of load resistance is shown in Figure A3-5. The automatic balance feature works extremely well; the stabilizing loop holds the output voltage within 50 microvolts of balance for an operational gain of unity. When the amplifier was set up as an integrator with a time constant of one second and with the input short-circuited, an initial output of 100 volts held constant to better than 0.1% for over one hour. Because of the rapid decrease in the stabilizing-loop gain for frequencies above about 0.1 cps, recovery of the amplifier from a saturation condition is very slow when operated on automatic balance. For this reason the stabilizing loop cannot be used when the amplifier is employed in the zeropulsing circuit described in Section 5.4. At the time of writing of this report only one of the drift-stabilized amplifiers had been constructed. Performance was deemed highly satisfactory, and nine more are presently under construction. Eventually we hope to have twenty or even thirty of these amplifiers aervailable for the electronic differential analyzer. -------------------— 82 —---------

0O 1 DRIFT STABILIZED DC AMPLIFIER OUTPUT SATURATION VOLTAGE FIG. A-3-5 + 250 ~ VERSUS + t~ LOAD RESISTANCE L POSITIVE SATURATION +~200 VOLTAGE /1~~~~+100 q -—. NEGATIVE SATURATION -... ——. —-—..-. —-—. —— ^ —. —- l o. VOLTAGE | | S Figure A3-5 Saturation Voltage as a Function of Load / / I I r~~:-I I I z /. ~ - x/- /'- ----- - - -----------— ~-~50' 10052 1K 10K 1OOK 1000K C1O LOAD RESISTANCE t 0 Figure A3-5. Saturation Voltage as a Function of Load

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UIM-67 APPENDIX 4 DESCRIPTION CF THE HIGHLY-STABILIZED POWER SUPPLY A4-1 Power Suppl Requirements for the Computer The high-gain dc amplifiers used in the electronic differential analyzer require well regulated dc power supplies, since the output balance of these amplifiers will be highly sensitive to any changes in the plate-voltage supplies of the vacuum tubes. Even with the considerable feedback employed in the operational amplifiers themselves, the output balance is still sensitive to fluctuations in the power-supply voltage. The solution of vibrating-beam problems depends in a very critical manner on the initial starting voltages and hence on the initial balance of the amplifiers. Amplifier balance is also very sensitive when the pulsing techniques described in Section 5.4 are employed (where the overall operational amplifier gain used is 6000). Thus the importance of well-regulated power supplies is apparent. The dc amplifiers used for the work described in this report (see Appendix 2) require power supply voltages of +300, -190, and -350 volts. The new amplifiers now being put into operation (see Appendix 3) require +100 volts in addition to the above voltages. A4-2 General Description of the Highly-Stabilized Power Supply A well known scheme for obtaining a highly-regulated dc power-supply voltage is shown in the block diagram of Figure A4-1. The output voltage from the rectifier and filter is regulated in the usual way by means of loop (a), in which a VR tube is used as the reference voltage. This loop is able to provide regulation over a wide range of frequencies (for example it eliminates most of the 60 cycle ripple) but in general will be subject to slow drifts. Loop (b) provides an additional feedback which takes care of any slow drifts. The output voltage in this loop is compared with a standard cell reference. Any dc error signal is chopped into ac by means of a vibrator, amplified, and reconverted to dc with the phase sensitive rectifier. This amplified dc error signal is finally fed back into the regulator amplifier. 84

FILTERTU A MPLI FIER | COMPARISON STAN DARD COMPARISON' CELL REFEREN CE cn tlt LOW PASS REFERENC FILTER T PHASESiE A C CONVERTOR SENSITIVE AMPLIFIER VIBRATOR RECTIFIERE Figure A4-1. Block Diagram of Power Supply [ |P~~~~~~~~~~~~~

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN Ji UM-67 The (a) loop therefore serves the purpose of stabilizing the output voltage against fast or sudden fluctuations while the (b) loop prevents the output voltage from drifting away from the voltage set by the standard cell reference. The frequency response of the (b) loop is such that it cannot compensate for fluctuations faster than about 0.1 cps. A4-3 Components and Circuits Used in the Power Supply The 60 cycle 115 volt input to the power supply is regulated by a Sorenson Model 2000S Regulator. The rectifier and filter circuits are shown in Figures A4-2, A4-3, A4-4, and A4-5, while the regulator circuits are given in Figures A4-6 and A4-7. The current capacities of the various voltage supplies are sufficient to operate 20 computer amplifiers under the most adverse load conditions and 30 computer amplifiers under normal load conditions, About 1/350th of the -350 volt output is tapped off with a precision potential divider and compared with the standard cell voltage (see Figure A4-6). The other voltage outputs (-190, +100, and +300) are compared with the -350 volt supply by means of potential divider arrangements (Figures A4-6 and A4-7). Since only a small fraction of the -350 volt output is used for comparison with the standard cell, the gain of the ac amplifier in the -3,50 volt loop needs to be considerably higher than that of the ac amplifiers for the -190, +100, and +300 volt loops, where a considerable fraction of the voltage is compared with the -350 volt supply. The 12AU7 and 6AU6 tubes in Figures A4-6 and A4-7 are part of the ac amplifiers. The 6AL5 tubes are used in the phase sensitive rectifier, the output voltage of which can be observed directly by means of an error-indicating meter. Two Leeds and Northrup Model 3881 standard vibrators are used to chop the dc error signal. The panel housing for the highly-stabilized power supply is shown in Figure A4-8. Current and voltage outputs for each of the four supplies can be read directly off of panel meters. The error signal of the (b) loop for each of the supplies can also be read directly from panel meters. Full scale on the meters represents an error signal of 25 volts. If it becomes desirable to add current capacity to the power supplies, it is only necessary to increase the capacity of the rectifier, filter, and control circuits. The regulator circuits can remain unchanged. 86

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 The output voltages of this supply remain constant to the order of one millivolt under normal conditions of operation. 60 cycle ripple in the outputs is also the order of one millivolt. 10 HY 200 MA 10 HY 200 MA 5R4-GY F5(1.6) F6(1.6) 95K 2OW -190V 12.6.3V.3 V'A 315-05P9AMS L J L l l l, 10 T8, T9, T10: SURPLUS TRANSFORMERS l| AN 3102-20-5P AN-3102-1OSL-4P AN-3102-18-3S. -- AN 3106-20-5S AN-3106-1OSL-4S AN-3106-18-3P 1I.6:,J r -; --- r "-,-, —- r9 2 TO 117 V 60 RELAY IN. TO P6, P18, P19 TO -190 V TO FILAMENTS TO FILAMENTS TO FILAMENTS OF -190 V 0.4 AMPS CHASSIS 0.4 AMPS OF DC AMPLI. OF DC AMPLI. FUTURE DC AMPLI. Figure A4-2. -190 Volt Power Supply and Filaments Part I 0.6 Amps DC 87 —---------

10 HY 200 MA c TO -190 V.6 AMP CHASSIS r 5R4-OY OIL 1 J5 P5 TO REGULATOR CHASSIS I, r C -190 V 1.0 AMP T712V9ACTGND L a b C - J 4iB P o AN 3102-20-40 T~ 6.3 7. 5A AIVAN CE20- _____ I _1 __ 1 B A i b a —-.- -, r 2 MFD 1 00 10 2 0A 2 00a ba HPO1~~~~~ l ADVANC E. F|-22 a. T: SOVERLOAD RELAY ADVANCE 750B I a> 964R'__0______________________ __________ M 0 60RELA 1C RESET SW |ITCH | Figure 11-6.3V 1.2A CT -190 VV l 117w 60-d FI | | Part II 0v1; Amp V DC A g 8~~ ~ ~~~00.~~~ 0 i d( **I ADVANCE 1 a a PANEL 1V T MAYN ITCH IN0 Iz~~~2- I l11 6.3V 1.2A CTG -190 V'g0-6 —" 46.3V.9A a o, T,: SURPLUS TRANSFORMERS 6.3V 14A CT G ND ~-' ~ (1.2A) J PA A +100 v +300 V 5 T: TANCOR T 21P10 T T2, T3: STANCOR P 6134 3 (1.-25A) J AN 4A -— A -+300 V Ti: STANCOR P 6309 /b J3 Lia — b --— d G- J AN 3102-20-40+ TO +300 V AND +100 V OF JUNCTION BOX Figure A4-3, -190 Volt Power Supply and Filaments Part II 0.4 Amps DC

OVERLOAD RELAY RESET SWITCH, -350 V SUPPLY j —-- ______-__________ z; OVERLOAD RELAY RESET SWITCH, +100 V SUPPLY r 0 10.'2A).A' 10 5R4-GY 010 0 F11 (.1A) - c —---—, - < (- nTrrrr'- )- - -- - -^ i ----- 15 HY 55 MA ~~~~~~~15 ~~~~1000 V OIL 1(EACH){ (Jr *1 __ _____ t~~~ F12 (1A) 5R4-GY _Ti-,' 16 I -~r +100 V LINE I F13 (.1A) --'- NP - H IIii' -------- I= = 1' - l I CA 15HY250 MA 350V LINE I I T_17 I -I, L x J - - T~?O~ S B-, ~~DPDT OVERLOAD RELAY WITH 6V RESET COIL MPG_ 1000^~ V__ \ ^GUARDIAN X-300 ER f 4-20 MFD 1000 V OIL 400V~~~~~~~~~~~ ------------------— i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 ci: ~~~~~~~~~~~~~~~~~IDPST MAIN SWITCH -— AN 3102 —20 —5P11 30 117 V 60 _15 j... —--- DIAL LGHT \ \ DP3T MHII SWITCH P ~ ~ ~ ~ ~ ~ ~ ~ ~ -_____-. ^ r — J [\ l7 *~ no02.2o.5P pr ^_____117 y!! f 0 1P5 r | AN 3106-20-5S TOGGLE SW. CONTROLS' 15 | | 1 AM 3106-2015S. ALL B" VOLTAGES J19 Pl9 I TO 117 V 60- MAINS F i-i Figure A4-4. +100 Volt 30 Ma and -350 Volt 200 Ma Power Supplies z_2

z CO P.~~~~~~~P 0 z H c) t-4 12 2 MFD ss sa F7 (GA) ~~~~~~~5R4-GY 54G TO REGULATOCHSI Fe (2A) 110 V PANEL LIGHT T31 1 II __ l "R-Y(r~ i^ (In i lir Fr ^ ADVANCE ELECTRIC O_ < _ ____-10 HY 500 MATYPE 750B ~~~~0 FI~~~~~~ \. (1/2A1)~~ <EACH COND. 20 MFD 9$)~'~~~~~~~~~~~~~~~~~~;: —--------------------------— 1000 V OIL ODPST OVERLOAD RELAY 0, _ _ 11_____WITH 110 V RESET COIL I s,______^"! ~~ ~ ~ ~~~~~~~r —? 0 1 I — g g -"" -- zi L'^ 01 V AE IlHT3_TT ",1 I_____-________________________I z 0 ~ ~ ~ ~ ~ ~~~~o ADVANCE ELECTRIC P~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ff to HY 500 MA u TYPE 750B~~~18 c P1 RESET COIL SWITCH i a J1 I.J AN 3102-20-5P AN 3106-20-5S 5 Z,~~~~~~~ 12 & T1, STANCOR P-5000 \0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1 g ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~2: h-l 0 C) 1 Figure A4-5. +300 Volt 780 Ma Power Supply z

CD~~~~~~~~~~~~~~~~~~~~ II e ^ I O. l;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...... o t H. K___________________ _ ~.....~,.,...., ~.,.. ~.. _,,...... Os~ ~ ~ ~ ~~~~~~E t I I -*1-1-^......-....~, —_,.._^^^^^^^^^.~.~~*g~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...... N m >~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0^ \30 o A i G ~~~~~~~~~~~~~~t0 NO 0" i..................................... #. 4 #4.#..........4........ 4Z< 44~~~~~~~~~~~~~~~~~~~~~~~~~.................... 4....4 / ~ 4'4 A: ~ ~ 0 H 1H __________________.....I o 4

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-67 APPENDIX 5 TIME CONSTANT OF THE ELECTRONIC INTEGRATOR Rf ri I I eI Figure A5-1. The DC amplifier has a gain of -u so that e = -/ve. (A5-1) The total current entering the input junction must be zero e -e e.-e + (e + (A5-2) Rf + o Ri Using (A5-1) to eliminate e in (A5-2) gives O O(1 ) C (1 ( ) - --- (A5-3) --- o | (l +/) + C 00(1 +,) + (A5-3) e e0 r * 3.+.) I1.. 94

AERONAUTICAL RESEARCH CENTER- UNIVERSITY OF MICHIGAN UMM-67 Integration furnishes the result t.t e C (-1 i e dt 11 t1e dt. (A5-5) RiC (1 P)' ICf^ / Ri (1 +f ) 0o It is seen that a finite value of/ reduces slightly the value of the coefficient before the integral of e. as compared with the simplified derivation which assumes e = o. More important is the coefficient 1 /RiC (1 + ) 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 > t1 and e = e for t > tl. Then equation (A5-4) can be solved to give e = exp (- ) (A5-6) where 1 + 1 1 (A5-7) R R + f /Ri (1 + ) Thus we see that when e. = o the integrator time constant is equivalent to two leakage resistors paralleled across the condenser: the first is the actual resistance Rf and the second, due to the finite gain of the amplifier, is /Ri (1 + l/~/). When R = oo and RiC = 1 the time constant is closely equal to/r. The general solution of (A5-4) for e0 as a linear time operation on ei is 95

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-67 where 1 RC 1' f Ri (1/ )J and k is determined by the initial value of e0 when the integrator is started at t = 0. Condensers with C = 106 farads are available with Rf > 1012 ohms. 96

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UM1-67 BIBLIOGRAPHY 1. Lord Rayleigh, Theory of Sound, paragraph 186. 2. Timoshenko, S.P., On the Correction for Shear of the Differential Equation for Transverse Vibrations of Prismatic Bars, London Philosophical Magazine, Series 6, Volume 41, p. 744. 3. Timoshenko, S.P., Vibration Problems in Engineering, Second Edition, D. Van Nostrand Co., Inc. I, 11 11 4. Goens, E., Uber die Bestimmung der Elastizitatsmoduls von Staben mit Hilfe von Biegungsschwingungen, Annalen der Physik, Series 5, Volume 11, pp. 649-678. 5. Ormondroyd, Hess, Hess and Edman, Theoretical Research on the Dynamics of a Ships Structure, 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). 6. Ormondroyd, Hess and Hess, Theoretical Research on the Dynamics of a Ship's Structure, University of Michigan Engineering Research Institute, Third Progress Report (March 1, 1949), Office of Naval Research Contract N50ri-116, (Univ. of Mich. No. M670-4). 7. Hess, Robert L., Theoretical Vibration of Beams, Ph.D. Thesis, University of Michigan, 1949. 8. Kruszewski, Edwin T., Effect of Transverse Shear and Rotary Inertia on the Natural Frequency of a Uniform Beam, National Advisory Committee for Aeronautics, Technical Note 1909. 9. Hagelbarger, Howe and Howe, Investigation of the Utility of an Electronic Analog Computer in Engineering Problems, External Memorandum UMM-28 (April 1, 1949), University of Michigan Engineering Research Institute, Air Force Contract W33-038-ac-14222 (Project MX-794). 10. Howe, Carl E., Further Application of the Electronic Differential Analyzer to the Oscillation of Beams, External Memorandum UMM-47 (June 1, 1950), University of Michigan Research Institute, Air Force Contract W33-038ac-14222 (Project MX-794). 11. Erickson and Parrett, The Use of an Electronic Analog Computer in the Determination of the Normal Modes of Lateral Vibration of Non-Uniform Beams, Masters Thesis, Department of Aeronautical Engineering, University of Michigan (June, 1949). 12. Ragazzini, Randall, and Russell; Proc. IRE, 35, 444 (1947). 13* Dolph, C.L., Normal Modes of Oscillation of Beams, External Memorandum UMM-79 (Februay, 1951), University of Michigan Research Institute. 97

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-67 DISTRIBUTION Distribution of this report is made in accordance with ANAF-GM Mailing List No. 15, dated 15 October 1950, to include Part A, Part B, and Part C. _____________________ 98,

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