ENGINEERING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR Progress Report DETERMINATION OF BLANK SIZES R.o'o Hess, B, Herzogji oR o F.eaison Project 2569 KELSEY-HAYES WHEEL COMPANY DETROIT, MICHIGAN November 1957

The University of Michigan * Engineering Research Institute TABLE OF CONTENTS Page ABSTRACT iii OBJECTIVE iii INTRODUCTION 1 Statement of the Problem 1 General Introduction to the Various Methods 1 I. NOMOGRAPH METHOD OF CALCULATION MOMENTS OF AREAS 2 Theory 2 Preparation of the Drawing for Measuring Purposes 7 IIo AN ELECTRONIC ANALOG METHOD OF OBTAINING THE FIRST MOMENT OF AN AREA 14 III THE MECHANICAL VOLUMETRIC PLANIMETER 21 IVo OPTICAL METHODS OF PROJECTION AS AIDS TO EVALUATING THE FIRST MOMENT OF AN AREA 23 APPENDIX A-1 ii

The University of Michigan * Engineering Research Institute ABSTRACT At the request of the Kelsey-Hayes Wheel Company of Detroit, members of the staff of the Engineering Research Institute of The University of Michigan investigated methods of obtaining the centroid of an area or the first moment of an area. The sponsors desired such a method to reduce the work load and produce uniformity in determining the size of a metal blank from which piece parts in the form of volumes of revolution are made. Four methods are presented in this reporto Several others were considered but are not reported here since these four present the greatest advantages. Each of the methods require one thing in commono a line drawing of the section of the volume of revolution. The methods areo I. A direct improvement of the "manual" calculating method using ordinary measuring scales and a nomograph in the form of a drafting tool. 2o A fully or semi-automatic electronic analog computing method, using well-known circuitry. 35 A mechanical computing method using well-known linkages. 4o An optical method using principles of projection from or onto curved surfaces to convert the given area into another whose area in turn is the desired quantityo OBJECTIVE The purpose of the work described in this report was to investigate methods of obtaining the centroid of an area or the first moment of an area to reduce the work load and produce uniformity in determining the size of a metal blank from which piece parts in the form of volumes of revolution are made. iii

The University of Michigan * Engineering Research Institute INTRODUCTION STATEMENT OF THE PROBLEM The volume of a surface of revolution made from a flat piece of metal is to be computed. The manufacturing process for the finished product will involve several operations during which the material will be drawn in a somewhat unpredictable manner. In general, however, a finished product drawing is available and it is to its specification that the product is heldo It is assumed here and for the remainder of the report that the manufacturing process is of constant volume. Well-known procedures exist to determine the volume of a surface of revolution by measuring the first moment of the area of the generating cross section about the axis of revolution. The common way of computing the first moment of an area is to subdivide the given area into a number of areas, usually a large number if the area is at all complicated in its shape, for which the centroidal properties are known; then the pertinent dimensions are measured or read from a drawing and entered into a computing table. This is followed by an amount of straightforward work on a desk calculator. This procedure required the operator to have a good knowledge of the subject and also permits a number of arbitrary ways in which different operators will subdivide a given areao These factors, amount of work, knowledge of the subject, and arbitrary procedure, combine to produce different answers to the same problem. The purpose of the procedures presented in this report is to abbreviate the labor involved in the normal calculating method, even to replace it entirely with a method that, with the minimum of operator time or effort, can take a line drawing, operate on it, and present the final result. GENERAL INTRODUCTION TO THE VARIOUS METHODS To obtain the volume V generated by revolving the area A about the nonintersecting x-axis, the following theorem of Pappus or Guldinus is used~ V = 2~ryA, where y is the location of the centroid of the area from the x.axis. ------------------------— 1 —------------

The University of Michigan * Engineering Research Institute Basically. it is necessary to compute the first moment of the area, yA, which need then only be multiplied by the appropriate constants to obtain the volume, weight, or mass, The first moment of the area may be expressed either by integral calculus or by finite summations being written JydA or yA, respectively~ Io NOMOGRAPH METHOD OF CALCULATION MOMENTS OF AREAS* The types of shapes which have provided the stimulus for this report are composed of rectangles, triangles, and portions of circles (in general, annuli)o Furthermore, most complicated areas can, as an approximation, be decomposed into a finite number of combinations of the above-mentioned regular areas. The indicated product and sum can then be performed. However, the calculations involved require that basic data be provided which requires the knowledge and background of, at least, some basic notions from staticso To eliminate the need for knowledge of statics and reduce the number of associated notions to a minimum, a tool that could as far as possible achieve this goal was sought. The element of area that requires more than just measuring a few designated lengths of straight lines is part of a circle. Consequently a nomograph was designed from which necessary numbers could be read while all remaining quantities could be obtained with a scale. All these data will be entered into a complementary calculating sheeto The use of this tool or nomograph will then dictate to a very large extent the subdivision of an area and thus eliminate one of the objections to the conventional methods as described in the previous pageso A diagram of such a tool is shown in Fig. 1.lo The form of the equation for the moment of the area to be used in this method is yA = ZyA THEORY The fundamental notions which led to the form of the nomograph shown in Figo lol will be given in the following paragraphso It is necessary to explain the relation between the conventional notations used for the calculation of moments of areas and those on the nomographo * By Ro Lo Hess 2

B B r 0.7 0.7 02 0207 0.7 56 -6 NOMOGRAPH FOR CONSTANTS USED IN DETERMINATION OF MOMENT OF AREA OF CIRCULAR SECTORS AND ANNULII. 3i FIG. 1.1 7 00.75 i I - 6 0 ~.~~~~~FG

The University of Michigan * Engineering Research Institute The given area is to be subdivided in a systematic manner and certain references must be established for each of the elements, Fundamental to the whole operation is the identification of the axis of revolution which will hereafter be referred to as the Work Lineo There is then associated with each type of element of area a Base Line, parallel to the Work Line, which is selected appropriately for rectangles, triangles, or sectors in such a manner that it, first, is easily established and, secondly, simplifies the consequent calculations. The Rectangleo. For the rectangle of side lengths D and E shown in Fig. 1o2, the Base Line passes through the lowest corner. The distance C is the vertical intercept between the Base Line and a line drawn parallel to it through the highest corner, Then the area is DE and the centroid is located C/2 above the Base Line. B is the perpendicular distance between the Base Line and the Work Line (for all elements of areas). Finally, the first moment of this area about the Work Line is yA = y(DE) = (B+)DE (1.1) It should be noted that this result holds regardless of the angular orientation of the rectangle. The Triangle oSThe triangle shown in Fig. 1.3 is considered to be composed of two triangles having a common Base Line, The upper triangle having an area 1/2 JL while that of the lower is 1/2 KL. The centroidal distances from the Work Line are for the upper triangle. B + 3 and for the lower one B KH 3 Hence yA = 1/2 JL(B+j) + 1/2 KL(B-K), (1.2) 3 5 which can be more usefully grouped for computational purposesyA = 1/2 L[B(J+K) + l/3(J2K2). (1.3) --------------------- 4 --------------— 1 —

_BASE LINE _WORKING LINE FIG. 1.2 THE RECTANGULAR ELEMENT L BASE LINE co I>, _________ WORKING LINE FIG. 1.3 THE TRIANGULAR ELEMENT / X BASE LINE R 2 RI _c.r o __ WORKING LINE FIG. 1.4 THE ANNULUS SECTOR ELEMENT 5

The University of Michigan * Engineering Research Institute The Annular Sector -For a preliminary discussion of the annular sector consider the one shown in Fig. 1o4, where the lower enclosing radius coincides with the Base Lineo The following quantities are defined with reference to Fig. lo4~ R = radius of outer circle r = radius of inner circle R = radial distance to centroid = half angle subtending the arc in radians G = distance from Base Line to centroid B = distance from Base Line to Work Line A = area of annular sector Then it follows that G = R sin (1.4) And the first moment of the area is 2 sin 2 7A = [B(R2=r2) + S(R3r35) T, (l5) where S= sine (1.6) 3 e and T = (1.7) The sign in Equation (1.5) is chosen plus if the area is as shown in Fig. 1o4, ieo, if the centroid lies above the Base Line, or minus if the centroid lies below the Base Line. The radii of the sector would in general be specified or could be measured from the drawing; the quantities B,C, and D will be obtained from the nomograph shown in Fig. 1.lo C and D, representing functions of the angle e only, are obtained by reading, from the scales} the intercept caused by the enclosing radial lineso In an attempt to avoid, as much as possible, the use of algebraic negative numbers and thus avoiding the decision as to the specific location of the centroid of a sector, i.e,, below or above the axis, the following rule is establishedo Any sector is to be subdivided into component sector so that no one component sector will extend over the horizontal Base Line, and further that it will not subtend an angle greater than 90 degreeso Then to expedite calculations, the above equations can be modified for sectors having an included angle of less than 90 degrees and with the lower enclosing radius not coincident with the Base Line. Such 6

The University of Michigan * Engineering Research Institute sectors are shown in Fig, l15(a) and (b)o The B value will, of course, be constant, but let the radial intercepts of the S and T scale be designated SU and TU, and SL and TL as shown in Fig, 1.5. The first moment of the area then becomes Ay = [B(R-r 2) + S(R -r) Tu - B(R -r2) + SL(R -r ) TL = B(R2-r2) (TU-TL) + (R3-r3) (SUTU-SLTL) M + N, where again, the plus sign is used if the area lies above the Base Line, and the negative sign is used if the area lies below the Base Line. PREPARATION OF THE DRAWING FOR MEASURING PURPOSES A typical section for which the sponsor desires to know the moment of the area is shown in Fig. 1.6. This area must now be subdivided into separate rectangles, triangles, and annuli. Lines are drawn in the drawing denoting the subdivision into these elements, thus establishing the requisite Base Lines as shown in the previous paragraphs and the lines which will make the intercept readings on the S and T scales of the template or nomograph. Dimensions of rectangles and triangles are measured or specified; radii are usually specified on the drawing. All this information can be entered into a computation schedule, a copy of which is attached, which can be followed without any basic knowledge of statics to produce finally the first moment of the area. Details and examples clarifying these procedures are presented in the Instruction Manual for Moments of Areas. This manual, together with the nomograph, is attached to this report. 7

C'S ______ BASE LINE I'__ - _ WORK LINE (a) BASE LINE 8 __ __ _ ___ 61 WORK LINE (b) FIG. 1.5 SECTOR NOT HAVING A RADIUS ON BASE LINE 8

1.06 1.31 0 30'.32N'- 1 -09 22 1o ~:.38 FIG. 1.6 TYPICAL AREA FOR WHICH'A IS TO BE COMPUTED

SUMMARY SHEET ENTER TOTAL I F-. FCNTER HALF OF TOTAL 2 ENTER TOTAL 3 - EN TE R T TAL 4 T F R TOTAL 5 i...-. TnTAL 6 v'(LJi M V E - T yA 628._. J ENTr R O"''. 6

WORK SHEET FOR RECTANGULAR AREAS DATA B C D E C/2 DE B+C2 A COLUMN a b C d e f h O PERATiON | _ | _. -'(b) (c)x(d) (a)+(e) (f)x(g) I I E M TOTAL A

WORK S HEET FCR TRIANGULAR AREAS DATA B 2 2JK) d+ ~DATA B J K L J+K J2 K2 2- K2 B(J+K) (J-)K2) P 27A COLUMN b c d e f 9 h i j k OPERATION -.... (c)(d) (b)x(b) (c)x(c) (f)-(g) (a)x(e) -(h) (i)(j)(d)x(k) ITEM ~~~~~~~~~~~~..._:,. _ _ TOTAL 2 = HALF OF TOTAL 2 =

WORK SHEET FOR AREAS OF CIRCULAR SECTORS AND ANNULII DATA B R r Su T SL T, RL FT r2 r3 (RZr2)TU-T_ 8(R~r2^ M (R~r3) SUTU SJ^ 5^-5^ N +N -N COLUMN a b C d e f g h I j k Imen o Ps t u v OPERATION - - - -- - - - (b)x(b) (b)x(h) (c)x(c) (clx(j)(h)-(j) (e)-(g) (o ) l(m)x(n)(i) k) (d)x(e) (f)x(g)j(q )-( ITEM TOTAL ~3 = TOTAL 4 = TOTAL 5 NOTE: USE COt-LUMN "V" FOR AREAS ABOVE WORKLINE UIE COLUMN "%WII FOR AREAS BELOW WORKLNEF

The University of Michigan * Engineering Research Institute IIo AN ELECTRONIC ANALOG METHOD OF OBTAINING THE FIRST MOMENT OF AN AREA* The electronic analog computer permits the computation of the moment of the area by evaluating the integral Ay = ydA, where means integration around the whole region, area, concerned. This may be more suitably written (see Fig. 2.1) in the following two ways: Ay = jf y2dx = fxydy. Consider for the purpose of this report that the "Analog Computer" consists of a series of "black boxes" which will perform the required operationso** The simplest of these operations are addition, subtractionmultiplication by a constant, and integration. Other operations, which present more difficulty, are squaring, square root, multiplication of two dependent variables etc. It should be understood at this point that the independent variable is time and all other variables will be voltages. Generally the other components, resistors and capacitors, may be plugged into the operational amplifier, which is the core of the analog computer, to produce a circuit having the desired properties. Some examples of such circuits are shown in Fig. 2.2, where eo refers to the output voltage and ei to the inputl voltage, It should be noted that, in general, a sign change is involved in these operations; see, for example, the first three operations in Fig. 2.2o Consider now the evaluation of the first moment of the area Ay = 1 y2dx. Let the curve be drawn out on a sheet of paper which will be placed in an X-Y plotter. This device is, essentially, a two-axis voltmeter. The horizontal axis (x axis) is to be driven by a voltage proportional to the independent variable which is x in the problem and time for the computer, i.e., time on the computer corresponds to x. An operator will then turn a potentiometer in such a manner that the recording point of the X-Y plotter will follow the y coordinate of the designated curve. The output voltage on this potentiometer will then be proportional to y, which is then squared and integrated. Thus, in outline form the desired integral has been evaluated. Figure 2.3 shows the block diagram for this operation. This diagram omits such items as scale factors, time constants inherent in certain operations, and excursion limits of the various devices. As long as these are constants they will not affect the general direction of the solution. The figure then represents, essentially, a flow diagramo XIBy Bo Herzogo For a detailed discussion of the analog computers, the reader is referred to any one of the several books published on the subjecto An excellent introductory text is a book by Co Lo Johnson, Analog Computer Techniques, McGrawHill Book Coo, June, 19560 14

y — t-dx SCHEME I: Ay - 2.ydx y dYyf I -'4 X -' SCHEME E: Ay =f y xdx FIG. 2.1 15

TRANSFER FUNCTION CIRCUIT DIAGRAM BLOCK DIAGRAM MULTIPLICATION BY A CONSTANT Rf R. | e l A R R R= e e0 ei eo -I eil - nf ei2 Rjfe-3 R e -- -D 0 mil R2 I Iei eio r I I | v RiI R1 e~Rf ADDITATION 8 o-, -..-ei e- ei2 - e. o-,, e3 Vei3 INTEGRATION C MULTIPLICATION kI FIG. 2.2

V= CONSTANT 5. Y g Y2 | — Y^drev Y2dX X-Y PLOTTER Y _ -k f 4X ^"r t o FIG. 2.3 BLOCK DIAGRAM FOR SCHEME I

.k fI Y___ ___ X-Y PLOTTER L' — MULTI - XY PLIER i-^Y / y' - l co V- CONSTANT -- L —- X IXYdYI XYdr FIG. 2.4 BLOCK DIAGRAM FOR SCHEME:I

The University of Michigan * Engineering Research Institute In a similar manner the block diagram, Fig. 2 4, is obtained for the operationo Ay = $xydyo Here the variable y is made proportional to T and the operator adjusts the potentiometer setting to be proportional to Xo The X-Y plotter then serves as a means of tracking the given curve, It is possible to obtain a curve-following device which will automatically track the curve and thus eliminate the need for an operator except to start the deviceo The first method is better suited for an area extended essentially parallel to the x axis, while the second method will best serve an area extended along the y axis. There are several reasons for these preferenceso First, it will take some amount of practice for an operator to become expert at adjusting the potentiometer setting to follow the curve. It is, of course, possible to have the voltage driving the independent variable axis increase slowly, thus permitting more time for the operator to make his adjustments. However, if the operation is extended over too long a time, errors introduced by drift of the electronic equipment may become significant. These methods were tried for some elementary shapes. Equipment which was on hand in laboratories of the Engineering Mechanics Department was used. It should be remarked here that such equipment is mostly of the general utility type and not designed for this particular problem only. Consequently, some of the components were too sensitive for this purpose, while others could use some modification to be more accurate or stable for this particular use. Several of the components in themselves are rated as having tolerances of 1%o Thus, from a combination of these we would expect perhaps an accuracy within several percent. A servo-driven potentiometer was used as a multiplying and squaring unit to complement the X-Y plotter and the general analog computer. By Scheme I the first moment of the area of the triangle shown in Fig. 2o5 was obtained. The calculated result was 67.5 in3, while the computer obtained a value of 61o2, which is about 2% in error. The first moment of the area of the rectangle of Fig. 2.6 was obtained by Scheme IIE The calculated value was 96 in3s, while that obtained by the computer was 95 in.3, in error about 1%. Under the conditions previously stated, these results are much better than were expected and it remains only to state that with equipment designed for the particular purpose better results should be possible. 19

y IC) a I - x 15 FIG. 2.5 y 4 FIG. 2.6 20

The University of Michigan * Engineering Research Institute IIIo THE MECHANICAL VOLUMETRIC PLANIMETER* The design utilizes a conventional planimeter wheel mounted upon a linkage which produces the integration of the differential of the first moment of the area 2 ydA. 2 Thus finally the moment of the area, when multiplied by 2xc, which equals the volume of the body of revolution, is read out on the planimeter dial. There are similar instruments commercially available,** but these are quite general in their utility. It is believed that the design presented here will be more specifically suitable. The design is based upon the following expressions which are derived from the line diagram of Fig. 3.1. A print of the curve in question is placed on the board on which the instrument is mounted. The pointer A on the arm AB of length L is used to trace around the outline of the curve. The arm BC is mounted on GH so that their intersection is always perpendicularo GH is permitted to travel in the x-direction through two bushings, The planimeter wheel D is mounted on arm CED. Member EF is pinned to CED at E so that distance CE equals distance BC. EF is constrained to slide through a bushing at B so that EF always remains perpendicular to ABo Let the angle between AB and GH on the x axis be o; the angle the plane of the wheel makes with the x-direction is 2&. It follows that the center of the wheel will travel a distance dx cos 29 when the pointer A moves a distance dx. Then if the planimeter wheel has a radius r, it will turn through the angle do when the pointer moves a distance dx so that d = dxcos2G = dx (1-2sin2G)= dx 2sin2 r r r r Thus = dx sin2dx r r But d x = 0, r and sine = y *By Jo Ro Pearson. **Keuffel and Esser Coo, Hoboken, N, J., Catalog Item 4270. Eugene Dietzgen Coo, Chicago, Illo, Catalog Item 1813A. 21

xdx WHEEL-dx RADIUS = r /> dx cos 28 FIG. 3.1 SCHEMATIC OF MECHANICAL VOLUMETRIC PLANIMETER

The University of Michigan * Engineering Research Institute Then = dx = 2dx L r rL2 2 Finally = yA rL2 The desired volume is 2iCrL2 V = 2yA = It is apparent, then, that a scale factor or characteristic constant of the instrument exists which, by selecting appropriate values for r and L, will permit designing to a particular sensitivity. Consider, for example, the following values: r = 0.500 in. L = 20 in. The volume/revolution then becomes 2i (v) = rL = 1975 in. / revolution This constant will, of course, be modified by manufacturing errors and the proper constant must be found by calibrating the instrument on an area for which the moment can be easily computed, IV. OPTICAL METHODS OF PROJECTION AS AIDS TO EVALUATING THE FIRST MOMENT OF AN AREA* An optical method using projection techniques offers some distinct advantages since only a few calculations are involved based upon ordinary planimeter measurements of areas. A film negative of the original drawing is placed in a frame and its image projected on positive paper or on another piece of film. The negative of the original drawing may be placed in a frame of predetermined shape such that the image cast on the positive paper is the first moment of the area. Alternatively, the positive paper could be placed in a curved frame and have the image from a flat negative cast upon it with the same result. A *By R. L. Hess and Bo Herzogo 23

The University of Michigan * Engineering Research Institute third method using a point source of light moving along a curved path can accomplish the desired effect by casting the image from a flat film on a flat positive paper. These methods are outlined in the following paragraphs. Consider first the method using a curved surface to hold the positive paper A photographic negative of the original drawing is placed on a frame in the xy plane (see Fig. 4.1). The positive paper is placed on the surface z = f(x,y)o Parallel light is used in the z direction to project the image of the area, on the drawing, on the film. The film is then developed, laid out on a board, and an ordinary planimeter is used to evaluate the resulting area which will be proportional to the moment of the area. In choosing the surface z = f(x,y) consider the integral for the first moment of the area in the form A^ = J dx. (4.1) Therefore, it is required that a projection be made which will be proportional (directly, preferably) to x along the x-direction and proportional to y2 in the orthogonal direction. This may be accomplished if we define a coordinate s measured along the curve in Fig. 4.2, which is a section parallel to y-z plane. This coordinate will be perpendicular to x and s-x will be conventional Cartesian system when the film is laid out flat. The quantity s must be of the form s = ky2 + k, (4.2) where k and k are arbitrary constants, The equation for z which satisfies this condition is z = k y2-1- loge(2ky + k y -) + C, (4 where C is an arbitrary constant. The above equation is restricted by 4ky2> 1. (4.4) The selection of k, kI, and C is governed by design requirements and scale factors. However, Eq. (4.4) places some definite restrictions which can be avoided by considering the following equation, where n is an arbitrary constant. nA + yA = (n+y)dA, (4.5) 24

Z f(x,y) / | PLANE OF ORIGINAL PLANE OF AREA // PROPORTIONAL TO MOMENT OF AREA\ ~FfG. 4. ~IG..r ~~z FIG. 4.1 z Z,) I. (,z) FIG. 4.2 25

PLANE OF AREA PROPORTIONAL TO MOMENT OF AREA PLANE OF 5 >^ >^ \ ORIGINAL s I I // I FIG. 4. 3 26

y PLATE WITH DRAWING PATH OF POINT FILM LIGHT SOURCE SHEET OF LIGHT z.FIG. 4.4 POINT LIGHT SOURCE <h h' FIG. 4.5 27

The University of Michigan * Engineering Research Institute which is equivalent to transferring the axiso The above method can now be applied to the evaluation of the integral of Eq, (4~5). However, to obtain the first moment of the area now requires measuring both the original area and the projected area. This information combined with the knowledge of the constant n will allow the evaluation of the first moment of the area from the following equation: yA = A' - nA, (4.6) where Al is the magnitude of the projected area, i.e., A' = (n + y)dA. (4.7) Similarly the negative of the original drawing may be placed on a surface and have its image cast on a flat plane (see Fig. 4.3). Here the y coordinates appear along the surface. In this case it is required that y2 = m + m (4,8) where m and ml are arbitrary constants. The resulting equation for the surface is z = ~loge(ms + m) - s + D,(4o9) where D is an arbitrary constanto For both of the above methods parallel light is required. This is most satisfactorally accomplished by using a double slit source and restricting it to a sheet in the y-z or s-z plane and having it traverse along the x axiso A third method consists of a point source producing a sheet of light normal to the drawing and traveling along a curve in the y-z plane. The negative of the drawing is mounted in a flat plane parallel to a piece of positive paper also mounted flato The light source travels along a curved path to satisfy the necessary transformation. A sketch of this arrangement is shown in Fig. 4.4. The basis for this operation is the relation of Eq. (4.5), repeated for convenience: nA + yA = (n + y)dA (4.5) In Fig. 4.5 a section perpendicular to the y axis is shown. Let h represent the x intercept between the two boundaries of the area. Then he is the projection of h onto the receiving plane located a distance a from the x-y plane. It follows then that 28

The University of Michigan * Engineering Research Institute h - =z+a h (4.10) z where z is the distance of the point light source from the x-y plane. Let h9 = (n + y)h. (4.11) Rewriting the right-hand side of Eq. (4.5) and substituting Eq. (4.10) yields Y2 Y2 (n + y)dA y (n + y)hdy = h hdy, (4.12) where yL and Y2 are the limits of integration. The result represented in the form of the last term of Eq. (4.12) may be obtained by the use of an ordinary planimeter, Equation (4.11) provides, when combined with Eqo (4.10) and simplified, the equation of the path of the source of light, ioe., a n+y- (4113) The constant n is provided here to prevent the denominator of (4,13) becoming zeroo Other modifications of the ideas outlined in the above paragraphs are possible. These methods were presented to explain some of the details of a projecting systemo The advantages of such a system are that once the mechanical and optical components have been carefully built, the operator's only efforts will be developing the photographs and measuring the pertinent area with a planimetero 29

APPENDIX

The University of Michigan * Engineering Research Institute INSTRUCTION MANUAL FOR FIRST MOMENT OF AN AREA A-2

- The University of Michigan * Engineering Research Institute INTRODUCTION The method described in this manual for determination of the first moment of an area from a drawing of the cross section utilizes ordinary drafting procedures and a specially designed nomograph. The nomograph is shown in Fig. 1. A-3

The University of Michigan * Engineering Research Institute Procedure The given area is subdivided into rectangles, triangles, sectors of a circle or annuluso The Work Line is selected - usually this is the center line of the object of which this area is the half cross section - and for each element of area a Base Line is selected according to the procedure outlined in the following paragraphso The first step is to take the line drawing and subdivide it into the rectangles, triangles, etco, by drawing dividing lines on the drawing, ioe, lines defining the edge of a rectangle where it joins a triangle or a sector of a circle, etco Secondly, the Base Lines for each element of area are drawn as follows 1l Rectangles (Figo 2)o-The Base Line is drawn through the lowest corner of the rectangle and parallel to the Work Line. For a rectangle having sides parallel to the Work Line the lower side becomes the Base Lineo 2o Triangles (Figo 3)0-The Base Line is drawn parallel to the Work Line through the vertex of the triangle that divides it into two triangles having a common side on the Base Line. Triangles having a side parallel to the Work Line will have that side as the Base Line. 35 Sectors of Circles or Annuli (Figso 4 and 5)o-All sectors to be used in the computation must have subtending angles of ninety degrees or less and lie in a single quadranto All sectors shown in Fig. 5 satisfy these requirementso Many sectors will be encountered that do not qualify for these requirements and these must be further dividedo Figure 4 presents such situations and shows the method of subdivisiono Note that each of the resulting sectors is of the type to be found in Figo 5o After subdivision into the proper areas, each resulting element of area shall be given a number to identify it. In FigSo 2, 3, and 5 various dimensions have been indicatedo These are the only dimensions required for the calculations. Three work sheets, one each for rectangles, triangles, and circular elements, have been prepared. Into these enter the dimensions indicated in Figs. 2, 3, and 5 and obtained by measurement from the given drawingo In many cases the drawing will be dimensioned and thus some of this information may be read directly from it In any case, for rectangles and triangles an ordinary scale only is required, while for the circular elements the "Nomograph for Constants used in Determination of Moment of Area of Circular Sectors and Annulii"t will provide all dimensions except the inside and outside radius, whic may be obtained with a scale or from given dimensionso I —-------------------- ^ 4 —---------— 1

The University of Michigan * Engineering Research Institute This nonograph is placed on the drawing so that the center of the circular scale coincides with the center of the circular elements and the 0-0 (zero-zero) line coincides with the Base Lineo The intercepts on the S and T scale will be read as outlined in Figo 5 and the distance B from the Base Line to the Work Line is read from the B scaleo Having entered all the dimensions into the appropriate Work Sheets, the calculation may be started and carried out according to the order as given in the entry "Operation"' The indicated totals are obtained and entered into the Summary Sheet where the indicated operations will yield the final resulto A-5

B -'r B.0.1 40.0. 4 0 -0: 07 0.7 5.1 5.5 -1 / ).1 0.2 0 2 7_- 0.2 NOMOGRAPH FOR CONSTANTS USED IN DETERMINATION OF MOMENT OF AREA OF CIRCULAR SECTORS AND ANNULII. 0.3 / 0.$ FIG. I A-6 _p I 05 A-6

_ __ __ _BASE LINE WORK LINE t_ E 9 __ __.,_._.__...,....ASE.LINE. - ----------—.ORK a,....... T.... BASE LINE /' _? *WORK LINE II u, — s e /'I BASE LINE FIG. 2 DIMENSIONING FOR RECTANGLES A-7

/_'S., E LINE..... _............ WoRK =iNE I.'.\. Y I BASE LINE t,TRK -::- ~ WORK LINE I _ V-I r A I t-0 BASE LINE _ \......WORK LINE FIG. 3 WiMrN S O iN iG FOR T' IANGLF S A-8

GIVEN ELEMENT SUB- ELEMENTS..1. + + I Xu —~,- +,+ 4 — FIG 4. DETAILS OF DIVISION OF CIRCULAR ELEMENTS A-9

i- oB —- --- 6 SL=O Su = 06.424 J S=O.IIG SU0 04.325 6 0 T.-O TU = 0.785 T TL -0178 TU = 0.5-8 OF MOMENT OF AREA OF CIRCULAR SECTORS AND ANNULII. OF MOMENT OF AREA OF CIRCULAR SECTORS AND ANNULII. S -- -S S.S 1:, o,? o 1 6- S= 6 S 01 Su 0.424 TL=0 T 0 0.2752 TL -. T 0.8 5 NOMOGRAPH FOR CONSTANTS USED IN DETERMINATION NOMOGRAPH FOR CONSTANTS USED IN DETERMINATION OF MOMENT OF AREA OF CIRCULAR SECTORS AND ANNULII OF MOMENT OF AREA OF CIRCULAR SECTORS AND ANNULII. 0. 0.6 FIG. 5 DIMENSIONS FROM NOMOGRAPH FOR VARIOUS SECTOR T T.ELEMENTS18 T NOMOGRAPH FOR CONSTANTS USED IN DETERMINATION NOMOGRAPH FOR CONSTANTS USED IN DETERMINATION 0.7 0.7 oz C~. 0 0 0 10.5 Q0 ~0- 7 07 070 \ 070 0 SL: 0oSu 0.: 0. 165 Su = 0.4 T 0 T: 0.474 TL:0.252 Tu 0.785 NOMOGRAPH FOR CONSTANTS USED IN DETERMINATION NOMOGRAPH FOR CONSTANTS USED IN DETERMINATION OF MOMENT OF AREA OF CIRCULAR SECTORS AND ANNULII OF MOMENT OF AREA OF CIRCULAR SECTORS AND ANNULII. ELEMENTS

0.7 __ 07 0 0'5' 5 5 r 6 SL= 0L SU 0.424. 6 S)=01 SU=O 688 6 L 6 1 TL = 0 3 T 0.7 85o TL0331 TU 40.685 NOMOURAPH FOR CONSTANTS USED IN DETERMINATION NOMOGRAPH FOR CONSTANTS USED IN DETERMINATION OF MOMENT OF AREA OF CIRCULAR SECTORS AND ANNULII. OF MOMENT OF AREA OF CIRCULAR SECTORS AND ANNULII. 02 _. ol8 - 2 2 2 0 5 1 5-5 FIG. 5 DIMENSIONS FROM NOMOGRAPH FOR VARIOUS SECTOR ELEMENTS A-10 0 0 0 SL:0: 0 0 0..Su = 0.8 06 ~ ~'l''"r~~rrmTTT"' 0 6 0-6 6 — 6 TL = 0 Tu = 0.795 T: 0 331 TU- 0.685 7 __- T -? NOMOGRAPH FOR CONSTANTS USED IN DETERMINATION NOMOGRAPH FOR CONSTANTS USED IN DETERMINATION OF MOMENT OF AREA OF CIRCULAR SECTORS AND ANNULII. OF MOMENT OF AREA OF CIRCULAR SECTORS AND ANNULII. o. 0,6 07 0.7 6 0.6 o0. 4 0. 0.5 5 S 0/ 07./0 % o o. o 0 0 O —o 3 ~ 4I....... 4 4 04 - 0.211 - 0.424 11 6 6 L 0 6 L = 0 T 0.330 T 0.78 T 0 T 0.465 NOMOGRAPH FOR CONSTANTS USED IN DETERMINATION NOMOGRAPH FOR CONSTANTS USED IN DETERMINATION OF MOMENT OF AREA OF CIRCULAR SECTORS AND ANNULII. OF MOMENT OF AREA OF CIRCULAR SECTORS AND ANNULII. A-10

The University of Michigan * Engineering Research Institute Examples lo Consider the area shown in Figo 60 This can be divided as ten separate elements to qualify for the computation, The Work Line and all the Base Lines are shown in Figo 7o Data are entered into the work sheetso 2o Consider a typical area for the cross section of a wheel spider which is reproduced in Fig. 80 A complete subdivision of this area, with subdivision lines and Base Lines identified, is reproduced in Fig. 9. Data are entered into the work sheets and the computation carried to completiono A-11

1.25. _,_30,# _ _ 45 0 10 /3 \ 450 FG / / ~ / X s 5 7 5 1350'-. 450 FIG. 6 SAMPLE AREA I\-12

----- -- -— _- BASE LINE -__ _ WORK LINE. 25 __30 o_ —-._ I| | 45~ Cl R- C-2 5 R-2 C-A -6 \ C - 4" 1350 1 \ 450 C45 FIG. 7 SAMPLE AREA WITH WORK LINE AND BASE LINE SHOWN A-13

SUMMAH,' SFHEET* ENTE R TO TR- i) L ENTER HALF OF TOTAL 2... E N T E R 7 TA L F r 7 D 58 ENTER OT, L....7 8.... ENTER TOTAL 4,.2 ENTER TOTAL 5 ___ -1.9 TC TAi 6 L 15 JOL JME =2 -ry b.28 x _18-._ =..__ __1- INj ENTER TOTAL 6'Refer to Fig 7. A -14

WORK SHEET FOR RECTANGULAR AREAS DATA B C D E C/2 DE B+/2 YA COLUMN b c d e f g h OPERATION - - - -(b) (c)x(d) (o)+(e) (f)x(g) ITEM.R-l _ 4.65.50.50 1.25.25.625 2 4.9o 5.06 __ 3 _. oo R-2 2.47 2.17.84 2.00 1.09 1.68 3.56 5.28 R-3 0.82 1.18.84 1.50.59 1.26 1.41 1.79 TOTAL I 10.8'-. -T:-: = 10,..

WORK SHEET FCR TRIANGULAR AREAS B2J 2 DATA i B J K L J+K J 2 IJ2 K2 B(J+K) K P COLUMN a b c d e f g h i j k OPERATION.. (c)t-(b) (b)x(b) (c)x(c) (f)-(g) (a)x(e) ITEM T-l 5.15: O ~50.50.50 O 25 2.,...................... ^..:15.........- 0....*5,.50 _ o... 2....... >.._.,..57 _.j.o..'........ ~74 T-2 4...~ 65.1. 4......4... o.54.019.160 -.141.... 1 -.4.2-.46 1._47L ^,.........:....,....L. ^ ^ ~..... ~:A TOTA L 2 = 2.225 NAL~F OF TOTAL 2 = 1.111 a-16

WORK SHEET FOR AREAS OF CIRCULAR SECTORS AND ANNULII QATA B H r Su Tu S_ TL R2 R' r2 r~ (R r2) Tu-Ti_ B(R2r2): M (R3- r3) Sy- SJ^ SJT,-ST. N +N -N COLUMN a b C d e fi j k I mn-q s t u v w OPERATION (b)x(b)(b)(h) (c)x(c) (c t (u) -(u) ITEM c-l.A 6.. _ ^ _...T _0..~ 24 _ 785,.., Q, _ _ o^ 1.._, 25.. __._. ^_.12 5- - _Q_-_. __ ___.._.-0 _._,._, 2L__1 _ ~, _Z85 __ ^.1.l6_.._,,51_0..,.1;25:- I.-^ -._.t _ _,. _.b.~. -.__-,^5 -IQ~. ^ -_.U2_ C-2 4.65.6...526.52Q_,oao__.l22 _,.. 5 _,.216 o 0.5. 67.66^.2166o el.O9 2186 5o4o o26 214o _ C-3 2.25 ( 1.40..*5T5_,. _-268 I _Q _ _ J-. S_6,,.4... __l..5 _.768 _.,67_._26.984.2.55__ __ 1.19.01Z.1Z_ c-4 _2,25. _. 4o_ 57. 424.7585 _ _q_ _ _ 1.56.._.2.74_,50.~__Ll.$0, l~. 6l._. _. 785 _ 5.67 _2,88 _ _2.55~_-_. *JJ'J.....0__.1._ _.213_?. 1.849_.__ ___.849 c-4 2.25 i.4o I87 4 44.1 10. c-6 2.60 1. 4q.575.380.664.1'74 207 1.96 2 74__.6j_...l^ ^190_.... -7A —. —. 397 4.2 1.68 _2.5_,.52 1.047. TOTAL? =7.562 TOTAL 4 =.199 TOTAL 5 =-1.586~ NCTL: USE C7LUMN "V" FOFi AREaS ABOVE WOSKLlNE UE C(LUMN "N" FOR Ai'EAI, ELO \ WORELLNE A-1U7

I L'-~.31 o 30.32R -.12.57 ^0 lhn Ir-~~ ^ 09n 6 FIG. 8 TYPICAL WHEEL SPIDER CROSS SECTION -"p iA.2 2R.,5' I FIGs 8 TYPICAL WHEEL SPIDER CROSS SECTION A-18

1.06 1.31 o - --- ----- ^ 30' R-.-32.12 C-~ - 4 /^V^ 50^ R-6 430 R-3 C-7 i 7' R-4 _. 3; C-8 A-19 -09', C I10 I 0 —'1 R22.... I 45 FIG, 9 WHEEL SPIDER CROSS SECTION SUBDIVIDED A-19

SUMMARY SHEET* ENTER TOTAL I 1.740 ENTER HALF OF TOTAL 2.-___...~ ENTER TOTAL 3 1.846 ENTER TOTAL 4 _ 0.078 ENTER TOTAL 5 -0.026 TOTAL 6 3.658 VOLUME 2TTA = 6.28x 368 22.85 IN3 ENTER TOTAL 6 *Refer to Fig. 9. A-20

WORK'MIEET FOR RECTANGULAR AREAS DATA b C c/ B- / A COLU/Nt L c J e f 9 h OWI. ATI rL _ —- — (L ) (,c) ~ x(J~ ()t- e) (f ) x,) ITM R-1 _ _.828_ 0.14 0.14 1.06 0.0 0.148 _.924 0877_ _ R-2_ __5. 4_...0.4 0.504. 0o. 14. 152 0..._. Q 5.522 _0. 2' F -3 3.*45 0 o._o 1__01 I.87 o.365 0.122 5.815 0.465 P-4 3.0o4 o.16 o.1 6 0.14 0.08 0.022 3.12 0.069 _R-5_ 1.82 0. 18 0.18 o.14 0__.09_. 02? _ 1.~ _0. o48.-6 _1.59 0.15 0.0 0.14 o.0 O.013.665 0.022 __.- _ ___ _l~~~0 _.1,. o_. _.022_ -._____ __ -__ _ _ ___- __._ _ __t __. _ TLTAI.. -. 1.740 A-21 i - 1740 A-21

WORK SHEET FOR AREAS OF CIRCULAR SETRANANL1 DATA B R r Sy T SL R R3 r2 r (R -r2 - R r) M (R r3 ST S Su Si N +NN COLUMN a b cd e f i kimoqtv OPERATION - --- (b)x(b) (b)x(h) (c)xk~c) (c)x(j) (h)-()(e -g (ax l (m x )(i (k d x )(fx )(q (s (px t +( )(u I TE M C-1 5.68 0.32 O. 18 0. 425 0.785 0 0 0.102 0.033 0.052 0.00o6 0.070C 075 058 052 007 0.5_0 _ 054 009 009 _ C -2 5.37 0.26 0.12 0.248 0. 596 0 0 0.0o68 0.0o18 0.0o14 0. 002 0.05 0.9.9 015 006 _.1 O_1.02 _ _,._.0*.2 C-5 ~4.54 1.26 1.12 0.262 0. 418 0.126 0.190 1.588 2.001 1.254 1. 4o4 0-33.2.5 _ ^5.9.1 0.2 _.8 _q^5l _^5_. _ C -4 4.625 0.50 0.36 0.125_ 0.187_ 0_ 0 0.250 0.125 0.150 0.0o47 0.120C ~ 8_055 _014 008 _^.2.2.0 _002 _ c-5 ~4.625_ o.50 o_ 3^6_ 0.374_ o.65o0_j0_ _o. o2r, 0.125 o~150_ o. o47 0.120 o__0 0.5 _.6 0.08 ^ A -,.. -0-^ -0^ c-6 3.90 0.59 0.25__q.372_ 0_641_ 0. 24o_ 0376_ 0.152 0.059 O. o63 0.0o16 0.0o89c.6.4 ^9 0 4 0g 000.4.0.0 _C-7 _ 5.1.9_ 0_.52_ O _q^8 0.241 _0.3579 0__ _0. 0.270 0. l4o_ 0.144, 0.055 0.126.7.0 0.5 (8_ _00 1.,0 _.9._.0 _ ^p _ _ _ c-8 3.03 0.23 0.09 o. 184 0.285 0 0 0.053 0.012 0.008 o. ool 0.0o45.8.-6 009 ool o02,,.oo5 0_o ____. C.9 2. 0 p.5 7 0. 6l q0.l85_ 0.285. 0 0 0. 565 0. 422 0.572 0.227 0.191 028 _04 _ 0. 6 0_15 __pi _0 _,005_.,.1 _0_1_ _ _ q -10 2.50 0.75 o_.6l_ o.l85_ 0.282 0 0 0.563 0.422 _0.372 _0.227 0.191 022 048..5 _015 _002 0 _,5 _,^ _._._001 C -11 1. 98 0.23 0.09 0.182 0.282 0 0 0.05'-5 0. 012 0.008 0.001 0.0o45.8.8.2.1.5.5 *0_,0_0_,.__ C -12 1.81 0.22 0.15 0.250 0.595 0_ 0_ 0.048 0.011l 0.017 0.002 0.051 055 0063.02 O_0.9,,.9,0 ____,_ 001 C -13 1.76 0.22 0.13 0. 425 _0.785 0.250 0.394 0.0o48 0.011 0.017 0.002 0.051 051 005 0.2.0.54.9.5.002002 NOTE: USE COLUMN "v" FOR AREAS ABOVE WORKLINE ---------

The University of Michigan * Engineering Research Institute COMMENTS REGARDING AN IDEAL METHOD FOR OBTAINING THE FIRST MOMENT OF AN AREA* All the methods for obtaining the first moment of an area presented in this report have one feature in common: the aim to abbreviate the work involvedo The accuracy of any of these methods depends, even at the outset, upon the accuracy of the line drawing of the cross sectiono Therefore, the most attention must be given to the drawing. Consider Fig. 8, which is a reproduction of a typical drawingo A few minutes' study will reveal that dimensionally several of the elements of area are not completely specified. For example, see elements marked 4, 5, 6, 7, 11, 12, and 13 which lack one or more data. What are the data required? For rectangles, the side dimensions and location dimensions will suffice. Location of the rectangle may, of course, be specified in many ways, e.go, locate any two corners. For the circular elements shown in Figo 4, the following data are required: internal and external radii, horizontal and vertical location of center, and the angle the enclosing radii make, for example, with the horizontal axis (these can provide the subtended angle of the sector). For a section to be consistent dimensionally, all this information must be available in one form or anothero However, were it actually provided, then a calculating procedure could be designed that would yield any desired degree of accuracyo Such a procedure could be programmed for a desk calculator or a punch-card system for a high-speed digital computero The conclusion to be drawn is that a modification of drafting room procedure is necessary to provide sufficient data for a satisfactory calculating method and at the same time provide a dimensionally consistent drawing. t is further advsable to make the informaton on a drawing available in a specified form to facilitate the consequent procedures, A drawing has other purposes than to provide information for the calculation of the first moment of an area and might consequently be cluttered with dimensions which would needlessly confuse manufacturing efforts. However, an adjunct work drawing will be sufficient for the present purpose since the information is used infrequently once the results have been obtained. Any other method must be expected to provide varied results unless the very same drawing and exactly similar procedures are utilized to obtain the first moment of an area. *By R. Lo Hess and Bo Herzog. A-23