2900-251 -T Report of Project MICHIGAN MECHANICS OF ROTATING PLATES AND PRISMS W. w. C. L.: OWN T. YANG February 1961 INFRARED LABORATORY EIveca e Scarce ai 7ec4d THE UNIVERSITY OF MICHIGAN Ann Arbor, Michigan

NOTICES Sponsorship. The work reported herein was conducted by the Institute of Science and Technology for the U. S. Army Signal Corps under Project MICHIGAN, Contract DA-36-039 SC-78801. Contracts and grants to The University of Michigan for the support of sponsored research by the Institute of Science and Technology are administered through the Office of the Vice-President for Research. Distribution. Initial distribution is indicated at the end of this document. Distribution control of Project MICHIGAN documents has been delegated by the U. S. Army Signal Corps to the office named below. Please address correspondence concerning distribution of reports to: U. S. Army Liaison Group Project MICHIGAN The University of Michigan P. 0. Box 618 Ann Arbor, Michigan ASTIA Availability. Qualified requesters may obtain copies of this document from: Armed Services Technical Information Agency Arlington Hall Station Arlington 12, Virginia Final Disposition. After this document has served its purpose, it may be destroyed. Please do not return it to the Institute of Science and Technology. *II

Institute of Science and Technology The University of Michigan PREFACE Project MICHIGAN is a continuing research and development program for advancing the Army's long-range combat-surveillance and target-acquisition capabilities. The program is carried out by a full-time Institute of Science and Technology staff of specialists in the fields of physics, engineering, mathematics, and psychology, by members of the teaching faculty, by graduate students, and by other research groups and laboratories of The University of Michigan. The emphasis of the Project is upon basic and applied research in radar, infrared, information processing and display, navigation and guidance for aerial platforms, and systems concepts. Particular attention is given to all-weather, long-range, highresolution sensory and location techniques, and to evaluations of systems and equipments both through simulation and by means of laboratory and field tests. Project MICHIGAN was established at The University of Michigan in 1953. It is sponsored by the U. S. Army Combat Surveillance Agency of the U. S. Army Signal Corps. The Project constitutes a major portion of the diversified program of research conducted by the Institute of Science and Technology in order to make available to government and industry the resources of The University of Michigan and to broaden the educational opportunities for students in the scientific and engineering disciplines. Progress and results described in reports are continually reassessed by Project MICHIGAN. Comments and suggestions from readers are invited. Robert L. Hess Technical Director Project MICHIGAN iii

Institute of Science and Technology T h e University of M i c h i g a n Institute of Science and Technology The University of Michigan CONTENTS N otices.................................ii Preface............................... iii List of Figures............................ vi A bstract................................1 1. Introduction.............................1 2. Summary of Results.................... 1 2.1. Deflections and Stresses in Rotating Plates 1 2.1.1. Rotating Elliptical Plates 2 2.1.2. Rotating Circular Plates 3 2. 1. 3. Comparison of Plate Materials on Basis of Strength 4 2.2. Rotating Triangular Prisms 4 3. Rotating Elliptical Plates (Clamped Edges)............... 4 3.1. Deflections in Elliptical Plates 5 3.2. Bending Stresses in Elliptical Plates 7 3. 3. Shearing Stresses in Elliptical Plates 9 3.4. Maximum Total Stress 10 3.5. Sample Calculations 10 4. Rotating Circular Plates (Clamped Edges).............. 11 4.1. Deflections in Circular Plates 12 4. 2. Sample Calculation 12 5. Rotating Elliptical Plates (Simply Supported Edges)........... 13 6. Comparison of Plate Materials from Viewpoint of Rigidity....... 17 6.1. Survey of Materials 17 6. 2. Relationship of Plate Dimensions 18 7. Rotating Triangular Prisms.....................19 7. 1. Deflections in Rotating Triangular Prisms 19 7. 2. Vibrations in Rotating Prisms 24 7. 3. Sample Calculations 25 References.............................. 26 Distribution List........................... 27 V

Institute of Science and Technology The University of Michigan FIGURES 1. Rotating Elliptical Plate.............. 2. Rotating Circular Plate....... 3. Geometry of Elliptical Plate........... 4. Load Distribution on Rotating Elliptical Plate.... 5. Geometry of Circular Plate...... 6. Plate Acted upon by a Bending Moment Only... 7. Location of Moments Applied to an Elliptical Plate 8. Deflections of Simply Supported and Clamped-Edge Elli 9. Diameter-Thickness Relationships for Rotating Plates 10. Geometry of Rotating Prism...... 11. Points Used in Solution of Prism Equations.... 12. Rotating Triangular Prism.............2 3 5 ~ ~ ~~ ~,..6........ 11.........14.........14 )tical Plates.. 18........ 18........ 19........ 23........ 25 p TABLE I. Physical Constants for Plate Materials............... 17 vi

MECHANICS OF ROTATING PLATES AND PRISMS ABSTRACT This is the report of an analysis of rotating plates and prisms from a theoretical viewpoint for the purpose of setting limits on the size and rotational speed of such bodies for scanner applications. The analysis was carried completely through for a fused-quartz elliptical plate rotating about an axis, in the plane which contains a normal to the plate and the major axis, at a 450 angle to the major axis, and for a fused-quartz prism of triangular cross section rotating about an axis through its centroid, normal to the cross section. The analysis indicated that the stresses and displacements are quite large for a 0. 5-inch-thick plate with a 9-inch major axis and a 9/V2-inch minor axis, but negligible for a 6-inch prism whose cross section is an equilateral triangle of 3-inch altitude. The foregoing applies for rotational frequencies of approximately 100 cps, which is a typical operating speed for an infrared scanner. 1 INTRODUCTION Modern scanning techniques (infrared and other) require the use of mirrors which rotate at extremely high velocities. The high-resolution requirements of the optical systems used for such applications cause some concern about the effects of forces upon the shapes of the mirror surfaces during the process of rotation. The problems which are the subject of this report lend themselves well to solution by stress functions, which satisfy the equations for equilibrium of the body, continuity of deflection, and certain applicable boundary conditions. The solutions may be used in numerous situations where one is concerned with the mechanics of rotating plates or prisms. The specific purpose of this report is to give the deflections and stresses for two particular scanning mirrors. However, every attempt has been made to make the solutions general and of a form convenient for application to other situations. 2 SUMMARY OF RESULTS 2. 1. DEFLECTIONS AND STRESSES IN ROTATING PLATES Several problems were solved in an attempt to obtain a comparison of rotating plates of similar shapes. Circular and elliptical plates were of conern in this case, and hence are those used in the calculations. 1

Institute of Science and Technology The University of Michigan Institute of Science and Technology The University of Michigan 2. 1. 1. ROTATING ELLIPTICAL PLATES. The plate to be considered as a numerical example for this analysis is elliptical in shape (Figure 1). It is 0. 5 inch thick and has a major axis of 9 inches and a minor axis of 9/V2 inches. The center of rotation is the center of the ellipse, and the axis rotation makes an angle of 450 with the major axis and 900 with the minor axis. The speed of rotation is assumed to be 5000 rpm. The plate is constructed of fused quartz, although only the numerical results depend upon the type of material. y (0", 9/12") (9?", 0o") X FIGURE 1. ROTATING ELLIPTICAL PLATE The magnitudes of the stresses and deflections in a rotating plate depend in large measure on the method of supporting the plate. Two extremes may be considered: (1) the edge simply supported, or (2) the edge rigidly clamped. In the general case, something intermediate would be encountered. In no event would one consider a thin plate inclined to the axis of rotation rotating at high speed with no edge support, since the bending stresses would either shatter the plate or at least warp it hopelessly out of shape. For a plate with a clamped edge, the results are: (a) Maximum deflection: 0. 223 x 10 inch (b) Maximum bending stress: 406 psi along major axis 350 psi along minor axis (c) Maximum shearing stress: 187 psi (d) Maximum total tensile stress: 753.3 psi 2

Institute of Science and Technology The University of Michigan -3 For a simply supported plate, the maximum deflection is 1.1 x 10 inch. No stresses were calculated for this case. In the general case, the deflection would fall somewhere between those given above, which are possibly 5 to 25 times the desirable limit for a high-resolution system. The deflection can be held to within a desirable tolerance by decreasing the area of the plate or by increasing its thickness, or both, as discussed in Section 6. 2. The maximum total tensile stress in the rotating elliptical plate under consideration is 753.3 psi, which is much less than the 7110-psi design limit of fused quartz. The practicality of using crown glass in such an application is marginal since the design limit of glass is generally assumed to be 1000 psi. However, the deflection curves are given for crown glass as well as for quartz in Figure 8. 2. 1. 2. ROTATING CIRCULAR PLATES. An analysis of a circular plate (Figure 2), of dimensions comparable to the elliptical plate described in Section 2. 1. 1, was made for the sake of comparing results. The plate is 0. 5 inch thick and has a radius of 4. 5 inches. The center of rotation is the center of the plate, and the axis of rotation is at an angle of inclination of 450 to the surface of the plate. The rotational speed is 5000 rpm. The plate is constructed of fused quartz, although, as in the elliptical case, only the numerical results depend upon the material. FIGURE 2. ROTATIG C R FIGURE 2. ROTATING CIRCULAR PLATE 3

Institute of Science and Technology The University of Michigan If the plate has a clamped edge, the maximum deflection of the surface is 0. 363 x 103 inch. This is the same magnitude as the deflection of the elliptical plate. 2.1.3. COMPARISON OF PLATE MATERIALS ON BASIS OF STRENGTH. A survey of the available materials for use in rotating plates showed that preference should be given as follows: Clamped-Edge Case Simply Supported Case (1) Aluminum (1 Quartz Fused Fused Quartz ( Molybdenum (2) Molybdenum (2) Aluminum (3) Steel (3) Steel (4) Magnesium (4) Magnesium (5) Light Borate Crown (5) Light Borate Crown The difference between fused quartz and crown glass is not great enough to warrant giving serious consideration to the selection of a material on the basis of deflection alone, since in each case the crown glass deflects only 1. 5 times as much as the fused quartz. The complete analysis of these materials is discussed in Section 6.1. 2.2. ROTATING TRIANGULAR PRISMS A 6-inch-long fused-quartz prism, with a cross section in the form of an equilateral triangle 3 inches in altitude, is rotated at 100 rpm about the centroid of its cross section (Figure 10). The maximum deflection of the face of the prism, in a radial direction from the axis of rotation to the corners of the prism, is 12. 6 iiin. The natural frequency of vibration of the prism is 8. 17 kcs if the ends are simply supported and 18. 8 kcs if the ends are clamped. 3 ROTATING ELLIPTICAL PLATES (CLAMPED EDGES) A plate of elliptical shape is rotated about an axis passing through its center, the axis of rotation making a 450 angle with the major axis of the ellipse, as shown in Figure 3. The co2 2 ordinate axes are such that the edge of the ellipse is described by the equation (x/a) + (y/b) = 1. The constants which describe the plate are 4

Institute of Science and Technology The University of Michigan w = deflection of plate (inches) = angular velocity of plate (radians/second) q = load intensity (pounds/inch2) D = flexural rigidity of plate (inch-pounds) E = Young's modulus (pounds/inch2) h = thickness of plate (inches) v = Poisson's ratio (inches/inch) p = density of plate (pound-second /inch4) FIGURE 3. GEOMETRY OF ELLIPTICAL PLATE 3.1. DEFLECTIONS IN ELLIPTICAL PLATES 4 The equation governing the bending of plates (Reference 1) is V w = q/D. The load to which the plate is subjected is due entirely to the centrifugal forces in the plate (neglecting gravity), of which the normal component is distributed as shown in Figure 4. The load intensity on any element dA (Figure 4) is defined as the normal force per unit area. Hence, normal component of centrifugal force pw 2hx dA 2 The differential equation to be solved is therefore 4 pw2hx V w 2D 5

Institute of Science and Technology The University of Michigan dV = hdA /// / // // // // FIGURE 4. LOAD DISTRIBUTION ON ROTATING ELLIPTICAL PLATE Since the quantity D is the flexural rigidity and is given by the relationship Eh3 D= 2 12(1 - v2) the final equation is 4 12 p2(1 - v2)x V w= 2Eh 2Eh The boundary conditions are: dw at x = a, w= = A solution which satisfies the boundary conditions and also the d A solution which satisfies the boundary conditions and also the differential equation is 2pc (1 - 2) w= 2 4Eh 1 1 x 2 a 2 5 a 2+a 4 a a b a b b - - -irl r~ i " Since the major axis has the greatest dimension, the maximum deflection (W) will occur along the major axis. Differentiation of w with respect to x for cos 0 = 1 and sin 0 = 0 yields x=~a, a X = ~a, ~i:V5 6

Institute of Science and Technology The University of Michigan dw for d = 0. Since the deflection at x = ~a is zero, the maximum deflection must occur at dx a x = ~/. Substituting this into the deflection equation, _ 2 5 2F _ 8 pwa5(1 - v2) a = v2I 2 L5 26Eh2 for =. b Using the above relationships, the deflection may be calculated for any portion of the plate. The deflections along the major and minor axes for crown glass and fused-quartz elliptical plates are given in Figure 8. 3.2. BENDING STRESSES IN ELLIPTICAL PLATES The bending stresses created in the plate may be calculated from the following relationships: M = moment causing stresses in the x-direction x -D (I_+ vaa2W and a = normal stress in the x-direction x M x(h/2) h3/12 or 6D /2w 2w\;x _=-2 2 2) h ax ay / Combining the above with the solution for w obtained from the preceding calculation, the final result is _pco a 1 8 x\3 /x\y\2 a\4 54 +1()2 2 1 (x)+ (a) 4 4h- 4[ x)2 ( )2 a)2 3()+) (a) a 2 [ \a/ \a b a/ \a/ b 7

Institute of Science and Technology T he University of M i ch i a Differentiation yields the result that the maximum stress occurs along the major axis at a 2 The value of the radical is very close to unity for reasonable values of Poisson's ratio (v < < 1), so the maximum bending stress occurs near the point of maximum deflection i. e., atx= ). 5/ Hence, the maximum bending stress in the x-direction is 4 (23a (x)maximum ~13J 5 4h (1 + v) Similarly, from the relationships M = moment causing stresses in the y-direction =-D e-j + x = normal stress in the y-direction y M (h/2) 6M (h3/12) h2 or / 2 2\ -6D /aw a2w Cr 2 12 V 2 Y h y ax the normal stress in the y-direction is + h()3- 4 1 () 2 (a) (b) J [(a) b) + /a) This function reaches its maximum at y=0 8

Institute of Science and Technology The University of Michigan and + 2 [ + 3v()2] 1 + 5vy- 1 a which, for reasonable values of v, is approximately x = ~. Hence 3. (8 )8(p a (13 v) y) maximum 39 ) (1 + v) Since the ratio of the maximum stresses is (x)maximum_ 27 (a ). 20 y maximum the stresses are of the same order of magnitude. 3.3. SHEARING STRESSES IN ELLIPTICAL PLATES The shearing stresses created in the plate are calculated from the following: T = shearing stress of yz-plane in the y-direction xy E h / (a2w + )( 2 \a x ay For the elliptical plate, the result is a a 3(a aX T = r23'Y [)(b)2[1 xy 8h\2 2/a\2 / It can be determined by differentiation that the maximum value of this function occurs along the y-axis at y ~a 3i /b\2a" The maximum shearing stress is therefore (xy)m m 8 w 4h (1 - xy maximum 39 r6 4h (1 - v) 9

Institute of Science and Technology The University of Michigan 3.4. MAXIMUM TOTAL STRESS The greatest total stress in the plate is of interest in that, if the plate cannot withstand the stress, regardless of how small the deflections, it obviously cannot suit the purpose for which it is intended. The total bending stress at a point is given by the expression = ) + (a )2 The centrifugal stress in a plate whose edges are restrained from expansion is given by cr 2 p1a2 2(3 +v) ()2 ( +) p+ a +v)a r 16 \ The total stress will then be this maximum occurring at (a, 0). 3. 5. SAMPLE CALCULATIONS Assume that the plate is made of fused quartz and has the following constants. a = 4. 5 inches b a =2 h = 0. 5 inch E = 10. 12 x 106psi v= 0. 144 inch 50071 Co = radians/ second -4 2 4 2 4 p = 2. 07 x 10 pound-second /inch B p=o a 2 13 W=~ p (1-) = ~0. 233 x 10-3 inch L 26Eh i

Institute of Science and Technology The University of Michigan 4 pw a (x)maximum- ~134- 4 (1 + ) = ~406 psi 8 co a (y) aximum- 3 (p4 (1 + v) = ~350 psi y maximum 39- 4h (Txy)maximum= 3 (p-s 22) ) = ~ 187 psi p maximum 13 (1 + 13 (5 (ma ximum 13,) (1+ v ) +m(5 + v) = 753.3 psi 4 ROTATING CIRCULAR PLATES (CLAMPED EDGES) A plate of circular shape is rotated about an axis passing through its center, the axis of rotation being inclined at 45~ to the surface of the plate, as shown in Figure 5. The plate edge is described2 2te 2 is described by the equation x + y = a FIGURE 5. GEOMETRY OF CIRCULAR PLATE The constants which describe the plate are: w = deflection of plate (inches) c = angular velocity of plate (radians/second) q = load intensity (pounds/inch ) h = thickness of plate (inches) D = flexural rigidity of plate (inch-pounds) E = Young's modulus (pounds/inch ) v = Poisson's ratio (inches/inch) p = density of plate (pound-second /inch 11

Institute of Science and Technology T he University of M i ch i a 4.1. DEFLECTIONS IN CIRCULAR PLATES 4 The equation governing the bending of plates is V w = q/D. Using the concept of normal load intensity described in Section 3. 1, the general solution (Reference 1) is p2 a 2 r 5 3 w + A - + B - 32Eh2 La) +() +D(a) en(a ) cos 9 aw At the center of the plate, w and ~ are both finite. Hence, C = D = 0. Also, along the edge ar of the plate, aw w= =0 Using these boundary conditions, the final result becomes w = pwa ( r 22 ) ( = pw2a5 r1 j [i()2 m oso 32Eh2 a a It should be noted that this is the same result as would be obtained by letting a = b, x = r cos 0, and y = r sin 0 in the equation of the elliptical plate (Section 3.1). Hence, the validity of the equations is verified. a The maximum value of this function occurs at 0 = 0 and r = ~. That is, 25 l6Eh 2 4.2. SAMPLE CALCULATION Assume that the plate is made of fused quartz and has the following constants: a = 4. 5 inches h = 0. 5 inch E =10.12 x 106 psi v = 0. 144 inch/inch 5007w c= radians/second 3 -4 2 4 p = 2. 07 x 10 pound-second /inch Then F 5 8 pw2a5(W= 25f- a 2-5)= 2 0. 363 x 10 inch 1 l6Eh" 12

Institute of Science and Technology The University of Michigan 5 ROTATING ELLIPTICAL PLATES (SIMPLY SUPPORTED EDGES) Section 3 discussed the mechanics of rotating elliptical plates with clamped edges. However, in the general case, the edge of the plate is neither clamped nor simply supported, but rather is supported in some intermediate manner. Hence, the simply supported case should also be considered. The required deflection curve must satisfy the relations 4 V w = q/D and w (boundary)= 0 m (boundary)= 0 After an unsuccessful attempt to find an exact solution to these equations, an approximation was used in order to determine the magnitude of deflection in a simply supported rotating elliptical plate. The condition which is difficult to satisfy in the simply supported case is that which specifies that the moment be zero at the edge. A method of obtaining this is to find the deflection (Figure 6) caused by the edge moment in the clamped plate and subtract the result from the solution for the clamped-edge case. From Sections 3. 1 and 3.2, the deflection and moment equations for the clamped-edge case are W = p 2a(1 - (x)2 (a) ab) (a1 4EhM = pw ha ___ 1 ____x_ + -~ha a/ ~I a 13

Institute of Science and Technology The University of Michigan Institute of Science and Technology The University of Michigan and M~(p~ha3~~ 1 2 xwa 4 [ (x) ()2()2a 2 + 3v2 ( -Ml(^^::::=:::::^M FIGURE 6. PLATE ACTED UPON BY A BENDING MOMENT ONLY Since it was desired to find the deflection curves for lines passing through the point of maximum deflection, i. e., the "equal and opposite" moments were applied at the points (, a- ) and (a, 0) as shown in Figure 7. It should be noted that the results a will be valid only for the lines y = 0 and x - FIGURE 7. LOCATION OF MOMENTS APPLIED TO AN ELLIPTICAL PLATE The existing moment at (a, 0) in the clamped-edge plate is, for a = v tb, 2 3 pw ha I 1 78 14

Institute of Science and Technology The University of Michigan At the point, the themoment is pow ha /8 + Iv _ 2 - 78 5 where these moments act as shown in Figure 7. Hence, the deflection of a plate acted upon only by the moments M = -M1 and M = -M2 may be computed, and the results superimposed upon the clamped-edge deflection curves to obtain the solution for the simply supported plate. From Section 3.2, the moments are related to the deflections by the expressions M x _M 1 = Dx2+ vaT2 X~ 1 2ax ay and M =-M = -D/2 +va2) My -2 a3y2 ax2/ These may be combined to give 2 M1 + M2 = D(1 + v) for which the general solution is -M1 - VM2 2 M2- vM 1 2 w= X-1 f x + + ax + by + _2D(1 - v\ _ 2D(1 - v2) 1 1 1 The constants are evaluated from the following boundary conditions: (1) At (0, 0), w = (2) At (a, 0), w = (3) At ( ), w=O From condition (1), cl = 0. From condition (2), M - M 2 a 1 = -a 2 L2D(1 - v )J 15

Institute of Science and Technology T he University of M i ch i a Finally, from condition (3) M 1I- M M2D2 - M.I D(1 - v 2D(1 - v For a = Vb, the solution for the deflection is therefore -/1- 2a 5Y +(8 + v) x (x) -- ))+ _(- ) - ( Once again, it should be noted that this equation applies only along the lines y = 0 and = a V5~~~~~~~~~~ The deflection curve for a simply supported elliptical plate (along y = 0) is given in Figure 8. 0. 0028 a = 4. 5 inches ^/ ~ ~ \ h = 0. 5 inch 0. 0024 \ 5007T radians W = 3 second Q _____ __ \ I 0. 0020 Light Borate Crown =S / n - I \ (Simply Supported) Z 0. 0016 Fused Quartz 0 Ev/ ip / ^ ~ I(Simply Supported) ~ 0. 012 Light Borate Crown | 0. 000 8 — ---- <(Clamped-Edge) C~ 0. 0008.000// ~~V 0 0.2 0.4 0.6 0.8 1.0 (x/ a) FIGURE 8. DEFLECTIONS OF SIMPLY SUPPORTED AND CLAMPED-EDGE ELLIPTICAL PLATES

Institute of Science and Technology The University of Michigan 6 COMPARISON of PLATE MATERIALS from VIEWPOINT of RIGIDITY The deflection equation for a rotating elliptical plate, as discussed in Section 3, contains essentially two constants: P(1 - 2) (1) 4 (8 Q L ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I clamped edge simply supported edge 2 5 w a (2) h2 The first of these depends only upon the material, whereas the second is a function of the plate geometry and rotational speed. 6.1. SURVEY OF MATERIALS A comparison of the material-dependent constants for different plate materials can be easily made by using numbers from a reliable engineering handbook. The magnitude of the constants will be directly proportional to the magnitudes of the deflections for a fixed- speed and-geometry situation. Table I is a summary of the physical constants for seven plate materials. TABLE I. PHYSICAL CONSTANTS FOR PLATE MATERIALS MATERIAL px 104 v Ex 10-6 (1 - v2) x 1010 E x Aluminum (98. 3 Rolled) Fused Quartz Molybdenum Steel (C. 38 Annealed) Magnesium Light Borate Crown Celluloid 2.55 0.47 10.10 2.07 9.02 I7.35 1.64 2. 11 0.14 0.00 0.24 0.25 0.27 10.12 42. 7 29 01 6.06 6.60 0. 197 0.201 0.211 0.239 0. 254 0.296 P [I +(5A )j x 10 0. 425 0.355 0.362 0. 440 0. 471 0. 554 3.46 0.41 0.53 5.431 11. 172 *See Reference 2. 17

Institute of Science and Technology The University of Michigan It can be seen from Table I that (with the exception of celluloid) the selection of a plate material cannot be made on the basis of rigidity. However, other criteria (such as reflectivity, ultimate strength, and grain size) may be such as to confer preference upon a particular material for a given application. 6.2. RELATIONSHIP OF PLATE DIMENSIONS Since the material constants do not offer much versatility for restricting plate deflections, the geometric constant 2 (for constant speed) will be considered. It is clear that, for a given h a5 material and speed, the deflection is proportional to 2. Hence, if a deflection w is calculated 5 h 5 5 for a particular - the corresponding deflection w for any other 5 may be obtained by inh 5 2 spection: (w) = (a i )( h). The relationship between a and h may be illustrated by a plot of versus for (w = constant, as shown in Figure 9. A decrease in deflection by approx2 imately a factor of 10 is obtained by decreasing a to- a or by increasing h to 3h, whereas doing both at once gives a decrease in deflection by a factor of almost 100. The indication here is that a change in size is far more effective than a change in material as far as deflections are concerned. 1.2 - \\ ~ ~ —(-^~)=10 1. C a X 0. 6 0. 4 0. 2 0 0.25 0.50 0.75 1.00 1.25 1.50 h 0 FIGURE 9. DIAMETER-THICKNESS RELATIONSHIPS FOR ROTATING PLATES 18

Institute of Science and Technology T he University of M i ch i a 7 ROTATING TRIANGULAR PRISMS The prism to be considered here has a cross section which is an equilateral triangle and a length which is greater than its greatest cross sectional dimension (Figure 10). The prism is acted upon by centrifugal forces resulting from rotation about an axis passing through its centroid. 7.1. DEFLECTIONS IN ROTATING TRIANGULAR PRISMS Using the same notations for stress as before, the equations of equilibrium for the prism are r 1 r6 r a +r - + +R=O ar r ae r and 1 a0' aTr8 2Tr= + + =0 r ae ar r where R is the body force (per unit volume) in the prism. The body force may be expressed in terms of a potential function V: R av ar 1 22 V = -pw r A system of stresses which satisfies equilibrium is i1 a2 + 12 r r r 2 2 r ae 2 a 2 ar FIGURE 10. GEOMETRY OF ROTATING PRISM 19

Institute of Science and Technology The University of Michigan Institute of Science and Technology The University of Michigan and T rs -ar r a where 0 is a stress function. The general two-dimensional stress function in polar coordinates (Reference 2) is = A nn r + Br + Cr n r + Dr 0 + E a 3 + -rO sin 0 + (blr + a.r + br Qn r) cos 0 +-r1 cos& + - d1 r nr\+ C+ r cos 0 + (d1r + cir + dr n r) sin 0 00 +Z (n n n+2 + +a rn + b' r2 cos nO n=2 n n n n n=2 o0 + (cnrn +dnrn+2 + cr + dr ) sin no + 0 = [n n n n where 01 is the particular solution of the compatability equation 4 1 - 2v 2 1)V -V 2v)po)2 V l=-i This is the governing equation for the case of plane strain (no axial deformation). For plane stress (no axial stress) the compatability equation is V47 =-(1 - v)2V = 2(1- v)pw2 which, for v < < 1, is the same as for plane strain. The displacements in the prism are (for plane strain) u = [(f1 - )r - 0 e] dr +f(0) r E and E >[(1 - )) - vr do -J u d + g(r) 0B E 0 r 20

Institute of Science and Technology The University of Michigan From the symmetry of the prism, u (0) ()=(-) ( + ) and u ()= -u(-) = U (0 + ) The stress-displacement relationships may be derived as follows: a = Z a sin nO + Z b cos nO + c r n n n and a = Zd sin nO + 2 e cos n6 + f n n n where a, b, c, d, e, and f are not functions of 0. Hence u = 2 sin n |[(1 - i)a - vd dr + Z cos n f (1 - v)b - ve]dr + [(1 - )c vf ]dr 1 +v Ef(6)} and u + V ( cos no{va- ( -)d ]r +f[(1 - )a - vdn]dr}+ sin nn [(1 - v)e - vb]r n n n J n +[(1- )b n - ve dr} + r c+ f[(1-v)cn vf ]dr +Eg(r)\ 1 + v Since u (0) = -u (-0) and ur(0) = Ur(-0), c and f must be independent, or r and a = 6 6 0 r r n n n d = g(r) = O. Also, since the displacement at the center is zero, f(o) = 21

Institute of Science and Technology The University of Michigan Any terms which yield infinite stresses at the center must also be zero, so the final stress function is o0 = Br2 + r3n + b r32) cos 3n + n=1 n n Since the forces are only functions of r, let 1 - 2 4 1 t 12 OP) r Letting 1 - 2v (1 ) the associated stresses become 00 a = 2B - 3n(3n - l)a r 2 + (3n - 2)(3n - l)b r3 cos 3nO r L n j L n jj -(33 + 2 2 - 18 lp) r and oo, 2B + 2 1 f3n(3n - 1)a r3n-2] + (3n + 2)(3n - 1)b r3n] cos 3nr and the norm n=l strn ess is 1 + 3v 2- 2 r An approximate solution is obtained by using n = 1 and n = 2 in the above equations. Along the face of the prism, 2b r v3 sin 6 + cos 0 and the normal stress is a =a cos n r \31 6\3/s i ~ 8~ 22

Institute of Science and Technology The University of Michigan These conditions are applied at the four points indicated in Figure 11, yielding the following results: or =p) b 2 (0. 424 + 0. 286v) + (0. 151 + 0. 2l14)(-) 3r k 4 r - (0. 019 + 0. 054v) ( cos 30 + 0. 092 - 0. 005v)) - (0. 009 - 0. 0051) r cos 60 - (0. 375 + 0. 125v) and ao = pa2 b2 b(0. 424 + 0. 286v) - (0. 151 + 0. 214v)() - (0. 096 + 0. 268v) cos 30 -ro 0~0 oo^/~4 61 - (0.092 - 0.005v)( ) - (0.017 - 0. 009)(r) cos 60 - (0. 125 + 0. 375v) It may be shown by differentiation that the maximum stress will be the radial stress at 0.60 +0.86r 0= 0, (r) = 0. -+ 0-. 86v For small values of v this is r 0. 201 + 0. 352v b 0=0 - - f 12 /^ 3 6 FIGURE 11. POINTS USED IN SOLUTION OF PRISM EQUATIONS 23

Institute of Science and Technology The University of Michigan Institute of Science and Technology The University of Michigan The maximum radial stress is therefore (a 1 262 (1 +3v) ( r)maximum - w b (1 + 3) The radial displacement is given by the relationship U = p b(1 + ) (13. 56 + 9. 14v)(1 - 2)(b r 32E 2 4 + 2. 417 + 3.424v)() - (0.153 + 0. 429v) c os 30 + (0.588 - 0.029v)(1 - 4 3v) - (0 039 - 0.020v)(1 + v)( cos 60i 7T 7T 7T The above relationships are valid for 0 = 0, 1- 6- and 3 The maximum radial displacement is () p 2b2 (1 + v)(48 36v+38v2 (Ur)maximum = 32E at r = 2b and 0 = 0. 7. 2 VIBRATIONS IN ROTATING PRISMS Section 7. 1 discussed the stresses and deflections in a rotating triangular prism whose length is much greater than its greatest cross-sectional dimension. The sample calculations (Section 7. 3) indicate that the stresses and deflections caused by centrifugal forces in the member are very small in magnitude, even for applications involving large prisms rotating at high speeds. Another effect encountered in rotating members of this type is vibration caused by dynamic unbalance in the prism itself. Such an effect may be encountered in any rotating shaft whose length is greater than its cross-sectional dimensions. Reference 1 gives the expressions for the natural frequencies of clamped-end and simply supported beams in free vibration, involving a procedure developed by Lord Rayleigh. Using the constants L = length of prism I = moment of inertia of the prism cross section = y dA E = Young's modulus of elasticity m = mass of prism w = angular velocity of prism 24

Institute of Science and Technology The University of Michigan the corresponding natural frequencies are f _I EI c 2 m 2 mL for the simply supported member, and 2 7T El f =2 I - N mL2 2 A mL for the clamped-end member. If the prism rotates at its natural frequency, any unbalance will cause resonance leading to an infinite amplitude of vibration. Due to the large magnitude of E, the natural frequencies are of the order of kilocycles. Since the amplitude of the vibration is always proportional to 2' then for f < < < f the amplitude will be very small. small. 1 - (f/fc)2 7.3. SAMPLE CALCULATIONS The triangular prism shown in Figure 12 is rotated about the centroid of its cross section, which is an equilateral triangle 2J3 inches on a side. The prism is 6 inches long, and is made of fused quartz with a density of 2. 07 x 10 (pound-second /inch ) and Young's modulus of 10.12 x 106 psi. (r)maxi = 1 1 b 2(1 + 3) = 9.44 psi r maximum 12pa 232 6 (u ). = PW b(1 + ) (48 - 36v + 38v2)= 12.6 x 106 inch r maximum 32E The mass of the prism is m = p [(2V3b)(3b)L = 64. 5 x 104 (pound-second2/inch) FIGURE 12. ROTATING TRIANGULAR PRISM 25

Institute of Science and Technology T he University of M i ch i a The moment of inertia of the cross section is I = y dA = 3.75 (inches) A The natural frequencies are therefore EIl f =- = 8.l17 kcs c1 2 L3 1 mL and f = _ i 18. 8 kcs c2 V3 mL REFERENCES 1. S. Timoshenko, Theory of Plates and Shells, 2nd ed., McGraw-Hill, New York, N. Y., 1959. 2. S. Timoshenko, Theory of Elasticity, McGraw-Hill, New York, N. Y., 1934 3. J. P. Den Hartog, Mechanical Vibrations, McGraw-Hill, New York, N. Y., 19340 26

Institute of Science and Technology The University of Michigan PROJECT MICHIGAN DISTRIBUTION LIST 7 1 February 1961-Effective Date Copy No. Addressee 1 Army Research Office, ORCD, DA Washington 25, D. C. ATTN: Research Support Division 2-3 Commanding General U. S. Army Combat Surveillance Agency 1124 N. Highland Street Arlington 1, Virginia 4-40 Commanding Officer U. S. Army Signal Research & Development Laboratory Fort Monmouth, New Jersey ATTN: SIGRA/SL-ADT 41-42 Commanding General U. S. Army Electronic Proving Ground Fort Huachuca, Arizona ATTN: Technical Library 43 Chief, Human Factors Research Division Office of the Chief of Research & Development Department of the Army, Washington 25, D. C. 44-45 Commander, Army Rocket & Guided Missile Agency Redstone Arsenal, Alabama ATTN: Technical Library, ORDXR-OTL 46-47 Commanding Officer U. S. Army Transportation Research Command Fort Eustis, Virginia ATTN: Research Reference Center 48 Commanding General Army Medical Research & Development Command Main Navy Building, Washington 25, D. C. ATTN: Neuropsychiatry & Psychophysiology Research Branch 49 Commanding Officer, Ordnance Weapons Command Rock Island, Illinois ATTN: ORDOW-GN 50-53 Director, U. S. Army Engineer Research & Development Laboratories Fort Belvoir, Virginia (50) ATTN: Chief, Topographic Engineer Department (51-52) ATTN: Chief, Electrical Engineering Department (53) ATTN: Technical Documents Center 54 Commandant, U. S. Army War College Carlisle Barracks, Pennsylvania ATTN: Library 55 Commandant, U. S. Army Command & General Staff College Fort Leavenworth, Kansas ATTN: Archives Copy No. Addressee 56 Commandant, U. S. Army Infantry School Fort Benning, Georgia ATTN: Combat Developments Office 57-58 Assistant Commandant U. S. Army Artillery & Missile School Fort Sill, Oklahoma 59 Assistant Commandant, U. S. Army Air Defense School Fort Bliss, Texas 60 Commandant, U. S. Army Engineer School Fort Belvoir, Virginia ATTN: ESSY-L 61 Commandant, U. S. Army Signal School Fort Monmouth, New Jersey ATTN: SIGFM/SC-DO 62 Commandant, U. S. Army Aviation School Fort Rucker, Alabama ATTN: CDO 63-64 President, U. S. Army Intelligence Board Fort Holabird, Baltimore 19, Maryland 65 Commanding Officer, U. S. Army Signal Electronic Research Unit, P. 0. Box 205 Mountain View, California 66-69 Office of Naval Research, Department of the Navy 17th & Constitution Avenue, N. W. Washington 25, D. C. (66-67) ATTN: Code 463 (68-69) ATTN: Code 461 70 The Hydrographer, U. S. NavyHydrographic Office Washington 25, D. C. ATTN: Code 4100 71 Chief, Bureau of Ships Department of the Navy, Washington 25, D. C. ATTN: Code 690 72-73 Director, U. S. Naval Research Laboratory Washington 25, D. C. ATTN: Code 2027 74 Commanding Officer, U. S. Navy Ordnance Laboratory Corona, California ATTN: Library 75 Commanding Officer & Director U. S. Navy Electronics Laboratory San Diego 52, California ATTN: Library 76-77 Department of the Air Force, Headquarters, USAF Washington 25, D. C. ATTN: AFOIN-1B1 78 Commander in Chief, Headquarters Strategic Air Command, OffuttAir Force Base, Nebraska ATTN: DINC 27

Institute of Science and Technology The University of Michigan Distribution List 7, 1 February 1961-Effective Date Copy No. Addressee 79 Aerospace Technical Intelligence Center U. S. Air Force Wright-Patterson AFB, Ohio ATTN: AFCIN-4Bla, Library 80-89 ASTIA (TIPCR) Arlington Hall Station, Arlington 12, Virginia 90-98 Commander, Wright Air Development Division Wright-Patterson AFB, Ohio (90-93) ATTN: WWDE (94) ATTN: WWAD-DIST (95) ATTN: WWRDLP-2 (96-98) ATTN: WWRNOO (Staff Physicist) 99-100 Commander, Rome Air Development Center (99) (100) 101-103 Griffiss AFB, New York ATTN: RCOIL-2 ATTN: RCWIP-3 Commander, AF Command & Control Development Division Laurence G. Hanscom Field Bedford, Massachusetts ATTN: CCRHA-Stop 36 104 APGC(PGTRI) Eglin Air Force Base, Florida 105-108 Central Intelligence Agency 2430 E Street, N. W. Washington 25, D. C. ATTN: OCR Mail Room 109-114 National Aeronautics & Space Administration 1520 H Street, N. W. Washington 25, D. C. 115 Combat Surveillance Project Cornell Aeronautical Laboratory, Inc. Box 168, Arlington 10, Virginia ATTN: Technical Library 116 The RAND Corporation 1700 Main Street Santa Monica, California ATTN: Library 117-118 Cornell Aeronautical Laboratory, Inc. 4455 Genesee Street Buffalo 21, New York ATTN: Librarian VIA: Bureau of Naval Weapons Representative 4455 Genesee Street Buffalo 21, New York 119-120 Director, Human Resources Research Office The George Washington University P. 0. Box 3596, Washington 7, D. C. ATTN: Library 121 Chief, U. S. Army Air Defense Human Research Unit Fort Bliss, Texas ATTN: Library Copy No. Addressee 122 Chief, U. S. Army Armor Human Research Unit Fort Knox, Kentucky ATTN: Security Officer 123 Chief, U. S. Army Infantry HumanResearch Unit P. O. Box 2086 Fort Benning, Georgia 124 Chief, USA Leadership Human Research Unit P. O. Box 2086, Presidio of Monterey, California 125 Chief Scientist, Department of the Army Office of the Chief Signal Officer Research & Development Division, SIGRD-2 Washington 25, D. C. 126 Columbia University Electronics Research Laboratories 632 W. 125th Street New York 27, New York ATTN: Technical Library VIA: Commander, Rome Air Development Center Griffiss AFB, New York ATTN: RCKCS 127 Coordinated Science Laboratory University of Illinois, Urbana Illinois ATTN: Librarian VIA: ONR Resident Representative 605 S. Goodwin Avenue Urbana, Illinois 128 Polytechnic Institute of Brooklyn 55 Johnson Street Brooklyn 1, New York ATTN: Microwave Research Institute Library VIA: Air Force Office of Scientific Research Washington 25, D. C. 129 Visibility Laboratory Scripps Institution of Oceanography University of California, San Diego 52, California VIA: ONR Resident Representative, University of California Scripps Institution of Oceanography, Bldg. 349 La Jolla, California 130 U. S. Army Aviation, Human Research Unit U. S. Continental Army Command P. O. Box 428, Fort Rucker, Alabama 131 Commanding General Quartermaster Research & Engineering Command U. S. Army, Natick, Massachusetts 132 Cooley Electronics Laboratory University of Michigan Research Institute Ann Arbor, Michigan ATTN: Director 133 U. S. Continental Army Command Liaison Officer, Project MICHIGAN The University of Michigan P. O. Box 618, Ann Arbor, Michigan 134 Commanding Officer, U. S. Army Liaison Group, Project MICHIGAN The University of Michigan P. O. Box 618, Ann Arbor, Michigan 28

AD Div. 6/3 Institute of Science and Technology, U. of Michigan, Ann Arbor MECHANICS OF ROTATING PLATES AND PRISMS by W. L. Brown and C. T. Yang. Rept. of Proj. MICHIGAN. Feb 61. 26 p. incl. illus., table, 3 refs. (Rept. no. 2900-251-T) (Contract DA-36-039 SC-78801) Unclassified report This is the report of an analysis of rotating plates and prisms from a theoretical viewpoint for the purpose of setting limits on the size and rotational speed of such bodies for scanner applications. The analysis was carried completely through for a fused-quartz elliptical plate rotating about an axis, in the plane which contains a normal to the plate and the major axis, at a 45~ angle to the major axis, and for a fused-quartz prism of triangular cross section rotating about an axis through its centroid, normal to the cross section. The analysis indicated that the stresses and displacements are quite large for a 0. 5-inch-thick plate with a 9-inch major axis and a 9/f2-inch minor axis, but negligible for a 6-inch prism whose cross section is (over) AD Div. 6/3 Institute of Science and Technology, U. of Michigan, Ann Arbor MECHANICS OF ROTATING PLATES AND PRISMS by W. L. Brown and C. T. Yang. Rept. of Proj. MICHIGAN. Feb 61. 26 p. incl. illus., table, 3 refs. (Rept. no. 2900-251-T) (Contract DA-36-039 SC-78801) Unclassified report This is the report of an analysis of rotating plates and prisms from a theoretical viewpoint for the purpose of setting limits on the size and rotational speed of such bodies for scanner applications. The analysis was carried completely through for a fused-quartz elliptical plate rotating about an axis, in the plane which contains a normal to the plate and the major axis, at a 45~ angle to the major axis, and for a fused-quartz prism of triangular cross section rotating about an axis through its centroid, normal to the cross section. The analysis indicated that the stresses and displacements are quite large for a O. 5-inch-thick plate with a 9-inch major axis and a 9/v2-inch minor axis, but negligible for a 6-inch prism whose cross section is (over) UNCLASSIFIED I. Title: Project MICHIGAN II. Brown, W. L., Yang, C. T. III. U. S. Army Signal Corps IV. Contract DA-36-039 SC-78801 Armed Services Technical Information Agency UNCLASSIFIED UNCLASSIFIED I. Title: Project MICHIGAN II. Brown, W. L., Yang, C. T. III. U. S. Army Signal Corps IV. Contract DA-36-039 SC-78801 Armed Services Technical Information Agency UNCLASSIFIED AD Div. 6/3 Institute of Science and Technology, U. of Michigan, Ann Arbor MECHANICS OF ROTATING PLATES AND PRISMS by W. L. Brown and C. T. Yang. Rept. of Proj. MICHIGAN. Feb 61. 26 p. incl. illus., table, 3 refs. (Rept. no. 2900-251-T) (Contract DA-36-039 SC-78801) Unclassified report This is the report of an analysis of rotating plates and prisms from a theoretical viewpoint for the purpose of setting limits on the size and rotational speed of such bodies for scanner applications. The analysis was carried completely through for a fused-quartz elliptical plate rotating about an axis, in the plane which contains a normal to the plate and the major axis, at a 45~ angle to the major axis, and for a fused-quartz prism of triangular cross section rotating about an axis through its centroid, normal to the cross section. The analysis indicated that the stresses and displacements are quite large for a 0. 5-inch-thick plate with a 9-inch major axis and a 9/12-inch minor axis, but negligible for a 6-inch prism whose cross section is (over) AD Div. 6/3 Institute of Science and Technology, U. of Michigan, Ann Arbor MECHANICS OF ROTATING PLATES AND PRISMS by W. L. Brown and C. T. Yang. Rept. of Proj. MICHIGAN. Feb 61. 26 p. incl. illus., table, 3 refs. (Rept. no. 2900-251-T) (Contract DA-36-039 SC-78801) Unclassified report This is the report of an analysis of rotating plates and prisms from a theoretical viewpoint for the purpose of setting limits on the size and rotational speed of such bodies for scanntr applications. The analysis was carried completely through for a fused-quartz elliptical plate rotating about an axis, in the plane which contains a normal to the plate and the major axis, at a 45~ angle to the major axis, and for a fused-quartz prism of triangular cross section rotating about an axis through its centroid, normal to the cross section. The analysis indicated that the stresses and displacements are quite large for a 0. 5-inch-thick plate with a 9-inch major axis and a 9/f2-inch minor axis, but negligible for a 6-inch prism whose cross section is (over) UNCLASSIFIED I. Title: Project MICHIGAN II. Brown, W. L., Yang, C. T. III. U. S. Army Signal Corps IV. Contract DA-36-039 SC-78801 Armed Services Technical Information Agency UNCLASSIFIED UNCLASSIFIED I. Title: Project MICHIGAN II. Brown, W. L., Yang, C. T. III. U. S. Army Signal Corps IV. Contract DA-36-039 SC-78801 Armed Services Technical Information Agency UNCLASSIFIED

AD an equilateral triangle of 3-inch altitude. The foregoing applies for rotational frequencies of approximately 100 cps, which is a typical operating speed for an infrared scanner. AD an equilateral triangle of 3-inch altitude. The foregoing applies for rotational frequencies of approximately 100 cps, which is a typical operating speed for an infrared scanner. UNCLASSIFIED DESCRIPTORS Infrared scanner Prisms Mechanics Rotation AD an equilateral triangle of 3-inch altitude. The foregoing applies for rotational frequencies of approximately 100 cps, which is a typical operating speed for an infrared scanner. UNCLASSIFIED UNCLASSIFIED DESCRIPTORS Infrared scanner Prisms Mechanics Rotation UNCLASSIFIED UNCLASSIFIED DESCRIPTORS Infrared scanner Prisms Mechanics Rotation UNCLASSIFIED DESCRIPTORS Infrared scanner Prisms Mechanics Rotation AD an equilateral triangle of 3-inch altitude. The foregoing applies for rotational frequencies of approximately 100 cps, which is a typical operating speed for an infrared scanner. UNCLASSIFIED UNCLASSIFIED

AD Div. 6/3 Institute of Science and Technology, U. of Michigan, Ann Arbor DYNAMIC BALANCING OF SCANNER DRUM AND GYROSCOPIC EFFECT DURING MANEUVERS OF AIRCRAFT by W. L. Brown and C. T. Yang. Rept. of Proj. MICHIGAN. Feb 61. 12 p. incl. illus. (Rept. no. 2900-250-T) (Contract DA-36-039 SC-78801) Unclassified report This report presents the equations for determining the dynamic balancing and gyroscopic effect of a rotating scanner drum assembly of the type used in the Project MICHIGAN wide-angle scanner. The parameters used in the numerical example are those of the wide-angle scanner, and the theoretical calculations of the dynamic balancing agree excellently with the actual mechanical dynamic balancing of the drum assembly. The calculations also indicate that gyroscopic forces in the drum are negligible even during maximum rate of turn and maximum rate of climb of the aircraft carrying the scanner. (over) AD Div. 6/3 Institute of Science and Technology, U. of Michigan, Ann Arbor DYNAMIC BALANCING OF SCANNER DRUM AND GYROSCOPIC EFFECT DURING MANEUVERS OF AIRCRAFT by W. L. Brown and C. T. Yang. Rept. of Proj. MICHIGAN. Feb 61. 12 p. incl. illus. (Rept. no. 2900-250-T) (Contract DA-36-039 SC-78801) Unclassified report This report presents the equations for determining the dynamic balancing and gyroscopic effect of a rotating scanner drum assembly of the type used in the Project MICHIGAN wide-angle scanner. The parameters used in the numerical example are those of the wide-angle scanner, and the theoretical calculations of the dynamic balancing agree excellently with the actual mechanical dynamic balancing of the drum assembly. The calculations also indicate that gyroscopic forces in the drum are negligible even during maximum rate of turn and maximum rate of climb of the aircraft carrying the scanner.... UNCLASSIFIED I. Title: Project MICHIGAN II. Brown, W. L., Yang, C. T. III. U. S. Army Signal Corps IV. Contract DA-36-039 SC-78801 Armed Services Technical Information Agency UNCLASSIFIED UNCLASSIFIED I. Title: Project MICHIGAN II. Brown, W. L., Yang, C. T. III. U. S. Army Signal Corps IV. Contract DA-36-039 SC-78801 Armed Services Technical Information Agency UNCLASSIFIED AD Div. 6/3 Institute of Science and Technology, U. of Michigan, Ann Arbor DYNAMIC BALANCING OF SCANNER DRUM AND GYROSCOPIC EFFECT DURING MANEUVERS OF AIRCRAFT by W. L. Brown and C. T. Yang. Rept. of Proj. MICHIGAN. Feb 61. 12 p. incl. illus. (Rept. no. 2900-250-T) (Contract DA-36-039 SC-78801) Unclassified report This report presents the equations for determining the dynamic balancing and gyroscopic effect of a rotating scanner drum assembly of the type used in the Project MICHIGAN wide-angle scanner. The parameters used in the numerical example are those of the wide-angle scanner, and the theoretical calculations of the dynamic balancing agree excellently with the actual mechanical dynamic balancing of the drum assembly. The calculations also indicate that gyroscopic forces in the drum are negligible even during maximum rate of turn and maximum rate of climb of the aircraft carrying the scanner. (over) AD Div. 6/3 Institute of Science and Technology, U. of Michigan, Ann Arbor DYNAMIC BALANCING OF SCANNER DRUM AND GYROSCOPIC EFFECT DURING MANEUVERS OF AIRCRAFT by W. L. Brown and C. T. Yang. Rept. of Proj. MICHIGAN. Feb 61. 12 p. incl. illus. (Rept. no. 2900-250-T) (Contract DA-36-039 SC-78801) Unclassified report This report presents the equations for determining the dynamic balancing and gyroscopic effect of a rotating scanner drum assembly of the type used in the Project MICHIGAN wide-angle scanner. The parameters used in the numerical example are those of the wide-angle scanner, and the theoretical calculations of the dynamic balancing agree excellently with the actual mechanical dynamic balancing of the drum assembly. The calculations also indicate that gyroscopic forces in the drum are negligible even during maximum rate of turn and maximum rate of climb of the aircraft carrying the scanner. (over (over) UNCLASSIFIED I. Title: Project MICHIGAN II. Brown, W. L., Yang, C. T. III. U. S. Army Signal Corps IV. Contract DA-36-039 SC-78801 Armed Services Technical Information Agency UNCLASSIFIED UNCLASSIFIED I. Title: Project MICHIGAN II. Brown, W. L., Yang, C. T. III. U. S. Army Signal Corps IV. Contract DA-36-039 SC-78801 Armed Services Technical Information Agency UNCLASSIFIED I over) I

AD AD UNCLASSIFIED DESCRIPTORS Infrared detection Infrared scanners Mathematical analysis Equations UNCLASSIFIED AD UNCLASSIFIED DESCRIPTORS Infrared detection Infrared scanners Mathematical analysis Equations UNCLASSIFIED UNCLASSIFIED DESCRIPTORS Infrared detection Infrared scanners Mathematical analysis Equations CO CO o c z O r 01 I o > O - o~ CO CA)___ 0) UNCLASSIFIED DESCRIPTORS Infrared detection Infrared scanners Mathematical analysis Equations AD UNCLASSIFIED UNCLASSIFIED