THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Aeronautical and Astronautical Engineering Final Report NATURAL FREQUENCIES IN COUPLED BENDING AND TORSION OF TWISTED ROTATING AND NONROTATING BLADES G. Isakson Jo Go Eisley ORA Project 05753 under contract with: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION CONTRACT NO. NsG-27-59 WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR January 1964

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONSv SUMMARY vii lo INTRODUCTION 1 2. BLADE ANALYSIS 3 Symbols 3 Basic Matrices 6 Method of Solution 14 Steady-State Deformation 15 Nondimensional Form 18 Numerical Results 27 Discussion of Results 27 53 SIMPLE MODEL ANALYSIS 34 Symbols 34 Description of the Model 36 Derivation of the Equations of Motion 37 Specialization to the Case of Constant Shaft Speed 42 Solution of the Pseudo-Static Problem 46 Formulation and Solution of the Linearized Equations 48 Discussion of Results 50 4. CONCLUDING REMARKS 63 APPENDIX A. DIFFERENTIAL EQUATIONS OF MOTION 64 Bo RESULTANT LOADINGS 66 Co DEFORMATION OF A BLADE SEGMENT 70 Do CENTRIFUGAL FORCE COUPLING 74 REFERENCES 77

LIST OF ILLUSTRATIONS TABLE Page Io Beam Properties 28 FIGURE 2olo Blade axeso 7 2.2. Nomenclature and sign convention for cross-section coordinates, displacements, bending moments, and shearso 7 2.53 Blade segment rotation. 7 2.4. Effect of twist on natural frequencies of nonrotating blades. 29 2.5o Natural frequencies of rotating blade Noo 1. 30 2.6. Natural frequencies of rotating blade No 2o 31 2.7. Effects of centrifugal force coupling on blade No. 2. 32 5301 Model coordinateso 37 3520 Effect of mass offset on pseudo-static displacements of model. 51 353. Effect of mass offset on natural vibration characteristics of modelo 53 354. Effect of mass offset on pseudo-static displacements of model. 54 355~ Effect of mass offset on natural vibration characteristics of model. 55 3.~6 Effect of bending hinge orientation on natural vibration characteristics of model. 56 357~ Effect of rotational velocity on pseudo-static displacements of model. 57 3~8e Effect of rotational velocity on natural vibration characteristics of model. 58 v

LIST OF ILLUSTRATIONS (Concluded) FIGURE Page 3.9. Response of model as determined by nonlinear and linearized differential equations. 60 vi

SUMMARY A Holzer-Myklestad type of procedure, using a matrix formulation, is developed for the determination of the natural vibration characteristics of a pretwisted rotating blade in coupled bending and torsion. The nonrotating blade is considered as a special caseo Results of a limited parametric study are presentedo It is found that in the case of the rotating blade there can be an appreciable effect of centrifugal forces in coupling the bending and torsional vibrationso In order to investigate the effects of Coriolis forces and the nonlinear effects of large angular displacements, a study is made on the basis of a simple modelo Numerical results indicate that the Coriolis forces may introduce substantial phase differences between bending and torsional vibration. Limited numerical results on the nonlinear effects indicate that these effects decrease slightly the frequency of the characteristic motions as determined from a linearized analysis and introduce some coupling between the characteristic motionso vii

1o INTRODUCTION In a previous report' the natural vibration characteristics of rotating twisted blades were studied for the special case of coincident mass and elastic axes. This eliminates coupling between bending and torsional vibration, and the problem was studied as one in bending vibration onlyo Bending deformation about both principal axes of the cross section was consideredo The present work represents an extension of this previous work to the case of noncoincident mass and elastic axes, that is, the case of coupled bending and torsion. This case has already been treated analytically in rather complete fashion in Refo 2, the problem being formulated in terms of governing differential equations and also in terms of energy principleso However, very few results are presented in that reference, and they are for a few special cases of a rather restrictive nature. In the present work a different analytical approach has been used. It involves essentially an extension of the Holzer-Myklestad method for determining the bending vibrational characteristics of a beam to the case at hando The Holzer-Myklestad method had previously been extended by Targoff3 to the case of bending of twisted rotating blades and applied in Refo 1. It was found to be particularly well-suited to automatic digital computation, and, for that reason, has been extended in the present work to include torsion as well, and has been applied in a limited parametric studyo An effect of centrifugal forces in coupling bending and torsional vibration, considered initially in Refo 2, is taken into account in the present work. It arises when the mass and elastic axes of the blade are not coincident. It should be remarked that the inclusion. of torsional deformation complicates the effects of pretwist and rotation considerablyo There may be a sizable steady-state or "pseudo static" torsional deformation of the rotating blade in some cases. This is due to centrifugal twisting moment which, in the case of negative pretwist and positive pitch, tends to twist the blade negatively, and also to the twisting moment associated with tensile stress in the longitudinal fibers, the so-called "centrifugal untwisting moment." These two effects oppose each other in the normal case, and the extent to which one or the other predominates depends primarilyy upon the amount of pretwist and the pitch setting of the blade. An analysis of this deformation and presentation of some results are given in Refo 4. Additional effects relate to a departure of the torsional stiffness from the value provided by Saint Venant theoryo This departure is associated with 1

inclination of the longitudinal fibers of the blade with respect to the elastic axis, due to both pretwist and torsional deformation. The normal stresses in these fibers can'be seen to have components in the plane of a cross section and to exert a torsional moment about the elastic axis. They arise from two sources. Firstly, there are normal stresses associated directly with torsional deformation that are present even in a nonrotating blade. These stresses may introduce a substantial nonlinearity into the torsional stiffness.3-5 Secondly, there are normal stresses associated with centrifugal forces, contributing to the torsional stiffness in a manner which is essentially linear for practical deformations that is, there is a linear relationship between torque and elastic twist.534 Some theoretical results for the case of torsional vibration, with some or all of these effects included, are presented in Refs.. 4 and 6. Because of the possibility of substantial pseudo-static torsional deformation and nonlinearity in the torsional stiffness, an accurate determination of the natural frequencies of vibration of a twisted blade should be based on linearization with respect to the pseudo-static deformation. This has not been done explicitly in generating the results presented in the present report. The values of pretwist selected must be interpreted to include pseudo-static deformation. This facilitates comparison with the results of Refo 1, where pseudo-static torsional deformation would have an influence on bending vibrational characteristics, and where the values of pretwist must be similarly interpreted to include such deformation. Another interesting aspect of the rotating blade vibration problem is discussed in Ref. 7. It is shown that Coriolis forces, or so-called "secondary inertia" forces, associated with the combined vibrational and rotational motion introduce a phase difference between the bending and torsional vibration. In order to investigate this effect more fully and to investigate the nonlinear effects of substantial angular displacements on the dynamic characteristics of a rotating blade, an additional study, reported in Section 3, was conducted on the basis of a simple model. The nonlinear effects considered are those associated with inertia forces. Nonlinearity in the torsional stiffness, as discussed above, and the effects of centrifugal tension on the pseudo-static deformation and on torsional stiffness are not included, although they could, in any extension of the present work, be included without undue complicationo 2

2o BLADE ANALYSIS SYMBOLS A = GJe + Tk2 + EB1(P')2 A = A/EIlo B1, B2 section constants defined in Appendix A C = EB2/A = C E Young s modulus EI1, EI2 bending stiffness about major and minor principal centroidal axes, respectively EI1 = EI1/EI1o, El2 = EI2/EI20 e distance between mass and elastic axis, positive when mass axis lies ahead e = e/R eA distance between area centroid of tensile member and elastic axis, positive when centroid lies ahead eA = A/R eo distance at root between elastic axis and axis about which blade is rotating, positive when elastic axis lies ahead eo = e0/R GJe effective torsional rigidity GJe = GJe/EI1o I, I mass moment of inertia of cross section about I and T axes, respectively, defined so that corresponding moments for an element dx are Igdx and I dx I: = I /OR2 II = /poR 5

kA polar radius of gyration of cross-sectional area effective in carrying tensile stresses about elastic axis kA = kA/R k5, k9 mass radii of gyration about 5 and T axes, respectively ~ length of blade segment 7Q 2V/R M1, M2 bending moment about major and minor principal axes of cross section, respectively, when centrifugal tension is assumed to act along undeformed position of elastic axis m mass of blade segment Px P., PP resultant loadings per unit length in the x,!,T directions, respectively Q resultant torque about elastic axis at any cross section qx' q), qn resultant torsional loadings per unit length about the x,.,y axes, respectively R blade radius T centrifugal tension, dx = e2T1 N T = Piixi i=l N T1 = Z Pi ixi i=l u displacement in the x direction V1, V2 shearing forces in the direction of the minor and major principal axes of the cross section, respectively x,y,z coordinate system which rotates with blade (Fig. 2~2) x = x/R Y = (E2B') /EIe 4

Y = Y/EIlo angle between major principal axis of cross section and plane of rotation, either in the undeformed or pseudo-static state -P dp/dx P' = P'R Ad. increment in e between blade segments Y2 = Ello/EI2 b, ~ 8displacements of the elastic axis in the y and z directions, y z respectively 81, 82 displacements of the elastic axis in the direction of the minor and major principal axes of the cross section, respectively A, rj coordinates in direction of minor and major principal axes, respectively 9 total twist in blade between x = 0 and x = R, Q =-RP' X = u pOR4/EIlo, = n JpoR4/EIio p mass per unit length of blade torsional displacement, positive when leading edge is up natural frequency of blade vibration ni ~ rotational velocity [ ] rectangular matrix [ } column matrix Other symbols are defined in the texto Subscripts n order of natural mode 5

0 value at x = O T value at x = R ( )', ( )" differentiation with respect to x BASIC MATRICES The governing differential equations of motion for a rotating blade with offset mass and elastic axes have been derived and are reported in Ref. 2. These equations are repeated in Appendix A. In the present report these equations have been adapted to a matrix formulation which permits rapid numerical analysis. This method is essentially an extension of the one presented in Refo lo The coordinate axes of the blade are shown in Fig. 201o The cross section coordinates and displacements are shown in Figo 2.20 The blade is divided into a number of spanwise segments, not necessarily equal in lengtho The mass of each segment is assumed concentrated at its center, and the bending stiffnesses, EI1 and EI2, the torsional stiffness, GJe, and the angle of incidence, P, are assumed constant between masses, appropriate average values being selectedo The built-in twist is accounted for by relative rotations of adjacent uniform bays (between masses) about a spanwise axis, the change in angle AP being equal to the total twist in a segment and occurring just outboard of the mass (Fig. 2.3). The quantities V1, M1, 56, 61, V2, M2, 62, 52, Q, and ~ (Fig. 2.2), which apply when the beam is at its maximum displacement in a free vibration, are defined at stations along the beam and may be represented at any station in the form of a column matrix: M1 V2 6a The elements of this matrix will vary along the beam in such a manner that the variation can be considered to occur in a series of stepso Moving from the 6

z o-,^-__ = M ELASTIC AXIS s|ea / L-TENSION AXIS I^-^ ^-c.g. AXIS Fig. 2.1. Blade axes., 81I,VIM2 e.8,.Vg.M. 27'22 z x,Q Ml (a) (b) Fig. 2.2. Nomenclature and sign convention for cross-section coordinates, displacements, bending moments, and shears. z B)n L Mn+1' Fig. 2.3. Blade se t Fig. 2.3. Blade segment rotation. 7

tip toward the root of the beam, the change in [A] occurring from a station immediately outboard of one mass to a station immediately outboard of the next mass can be broken down into three steps, the first involving movement across the mass, the second involving movement from one end to the other of a weightless uniform bay, and the third involving movement across the discontinuity in P. The relationship between the (A) matrices as they apply at the two extremes of this travel can be represented as follows: (A)n+l - [R][E][F][A]n (2.2) where [F], [E], and [R] are rectangular matrices representing linear relationships corresponding to the three steps discussed previously. The [F] matrix, relating the (A) matrices on either side of a concentrated mass, is written as follows: 1 0 0 F14 0 0 0 Fa1 0 F11o 0 1 F23 F24 0 0 0 0 0 0 O 0 1 0 0 0 0 0 0 0 O 0 0 1 0 0 0 0 0 0 0 0 0 F54 1 0 0 F58 0 F51o 0 0 0 0 0 1 F67 F8e 0 0 (2) 0 0 0 0 0 1 0 0 0 O 0 0 0 0 0 0 1 0 0 0 0 0 F94 0 0 0 F98 1 F910 0 0 0 0 0 0 0 0 0 1 where F14 = ~p(W2+Q2sin2 ) F18 = - 1p2 sin D cos D Fllo = ~pe[(c-+a22(sin2D-cos2 ) }- peSo~2cos D p ~soP F23 = 12 + ~In(2+a2) (2.4) 8

F24 = - p XSf F210 = -plexf2 F54 = F18 F58 = pX(2+ 2cos2 ) F510 = - 2pLeQ2sin P cos p - p~Q2eo sin P F67.= _ + (W2+g) F68 = F24 2 2 F94 = p~e(o +n2 sin P) F98 = - pleS2sin P cos X F91o = ~(I++Iq)w + (I-I%)(cosS2-sin2a)~ -1peeo2cos P (2.4) The derivation of the elements of this matrix is given in detail in Appendix B, except for the contribution of centrifugal force coupling, which is treated separately in Appendix D. It is seen that only the shear forces, bending moments, and torque are changed, since there are no discontinuities in slope or displacement. The changes in shear force are due partly to the inertia force associated with the vibrational motion of the mass and partly to the component of centrifugal force normal to the undeformed position of the elastic axis. Part of the change in torque is related to the change in shear, since the mass and elastic axes do not coincide, and part is due to the inertia force associated with the torsional vibrational motion. The change in bending moment, except that associated with centrifugal force coupling, is fictitious and arises from a special feature of the analysis. This feature involves the replacement of the component of the centrifugal force parallel to the undeformed position of the elastic axis by an equal force along the line of the undeformed axis and an appropriate couple to provide static equivalence. The changes in bending moment indicated in the [F] matrix are then due only to the applied couple, the moment due to the force applied along the undeformed axis being accounted for in the [E] matrix. When moments due to both sources are considered, the discontinuity in bending moment disappears. Note that, on the basis of this procedure, the bending moment at any station is not M, but rather M plus the moment of the tensile force T acting along the undeformed elastic axis. The elements in the [E] matrix are found by the solution of the differential equations of combined bending and torsion of the weightless uniform bay between masses. These equations and their solutions are given in Appen9

dix C. The resulting [E] matrix is: 1 0 0 0 0 0 0 0 0 0 E21 1 0 0 0 0 0 0 0 0 E31 E32 E33 E34 E35 E36 E37 E38 E39 E310 E41 E42 E43 E44 E45 E46 E47 E48 E49 E4 10 O 0 0 0 1 0 0 0 0 0 E = (2.5) 0 0 0 0 E65 1 0 0 0 0 E71 E72 E73 E74 E75 E76 E77 E78 E79 0 E81 E82 E83 E84 E85 E86 E87 E88 E89 0 O 0 0 0 0 0 O 0 1 0 EIoI E102 E103 E104 E105 E106 E107 E108 E109 E1010 where, if we define 2 2 p = ( (Pi-a) (p2-a) (a1-a3) (p-p ) p _ (pl-a3)(p2-a3) (a1-a3)(p1-P2) Ep (PI 1 P-a33)(p -aal)J (a-a3) (p - p2) the components of E are given below. The quantities Pi, ai, and fi are defined in Appendix C. E2- = Q E32 = - -P3 sinh Pl~ + IP2 sinh P2A (2.7) EIpL EIiP2 E33 = - P3 cosh PlQ + P2 cosh P2Q 10

E34 = alEIlE32 Es5 = a2P4 - sinh pli sinh P2 a3a4EI2 1 P2 E36 = a2P (cosh pl~-cosh P2~) a3a4EiD E37 = a3EI2E35 E38 = a3EI2E36 a2 2 ] E39 = 72 [-=(pl-a3)p2 cosh pl +(pia3)p' cosh P2~ + a2a3 f 2(pi-p2) L f2 E310 = a2EIIE32 + E36W(N) E41 1 (-a38P3a P)- sinh pl1 f2EI1 Pi (2.7) + [a3a8(P3+1)-aP2aP4] - sinh P2~-asa8I} E42 = I1 L -os P)+a3~),3c-+ P cosh P2 + cosh 1 (al"a~3) PE2 2 2 EI = (a-a,3) PPiP2 Pi 22 E43 = EI1E32 2 2 E44 = Pl P + aPcosh P2 2a2 2 2 2 2 2 E45 2= L2E2 P -P2) + p2 cosh Pi~ - pjcosh P2j a3a4plP2EIa E46 = E35 E47 = a3E12E45 E48 = E37 11

E49 = a2 2 [-(p-a3) P2 sinh pl~+(P2-a3) P1 sinh P21 + af2a3 f2(pl-P~) L Pi P2 2 E4o1 = a2EIlE42 + E46W(N) E65 = ~ E71 = -a4f2 [(a2a6P2-a3a8sP)Pl sinh Pl1 + (-a2a6P3+a3a8Pl)p2 sinh P22] E72 = -a4P1 (cosh pl1-cosh P22) a2EI1 E73 = a4P (-P sinh Pli+P2 sinh P2~) a2 E74 = alEI1E72 E75 1= (E77-1) a3EI2 (2.7) E76 = a1 (P2pl sinh Pll-P3P2 sinh P22) a3EI2 E77 = P2 cosh Pl2 - P3 cosh P22 E78 = a3EI2E76 2 2 E79 = 1b4P 2- [-(p1-ai) sinh p22 + (p2-al) sinh P2] f2(pl) 2 E710 = a2EIiE72 + E76W() E81 = a4f [(a2a6P2-a3a8Pl)cosh pli + (-a2a6P3+a3a8P1)cosh p22-a2a6] a1afEIifLE E82 = a4 1 sinh pl - p sinh P2 a2EI1 Pl P2 E83 = EIBE72 12

E84 = a 8EI1E82 E85 = (E87-~) a3EI2 1 E8 a=3EI2 (E88s-1) E87 sinh pl P Psinh P22 Pia P2 E88 = P2 cosh piQ - P3 cosh P22 E89 = a4 F -(p2-a)p, cosh pl~+(p=al)p% cosh P2] + ala4 f2(p2-pl) L E810 = a2EI1Es2 + E86W(N) EloI = a1f | 1 (Pa (a2a6P2-a3as8P) sinh Pli a2f2EII _ Pi - (P2a3) (a2a6P3-a3as8P) sinh P2~+a2a3a6e] (2.7) EIo0 =.p. [3 a3(p -p )-(p-al -a3)pc1osh P22 2 1 p - E103 = P1 (pl-a sinh -a) sinh P2 as Pi P2 El04 = alEIlElo2 E105 a = a4- 2EI2 al(Pl-p2)+(pl-al)p2 cosh p-(p-al )P cosh P22] a3a4Pip22EI2| E106 =aP4 ( p -a1) sinh Pli - (p2a sinh P2 a3a4EI2 P P2 E107 = a3EI2E105 Elos108 = asEIElo06 13

(Pl-P2) P (P -P2)P2 - = - 2 2 2 (p2 pa3)p22 1 (p-aj)(p-a3)p2 sinh p + 2-al)( sinh P2 1 a F 2 )2 2 EIoIO = aPl (Pa-a3 p2-a) + P1 -(p-a3p ch Ep1Zp (P~-as)-a3 p-al) + P1 -(P%,-a3)P2 cosh pll P1P2 (a1-a3) 2 2 |^ (N) + (p2-a3s) cosh p2:+ El06W!! Note that 51 and 52 are positive for increasing deflection in the positive x direction. The [R] matrix serves to rotate the coordinate axes through the angle A5 and is written as follows: cos Ap O O O -sin Ap O O O O O O cos Ap O O O -sin Ap O O O O O cos Ap O O O -sin Ap O O O O O O cos Ap O O 0 -sin Ap 0 0 sin AP O O O cos A O O O O O R = 0 sin Ap O O O cos A O O O O0 O O sin A 0 0 0O cos Ap O 0 - O O O sin AB O O O cos Ap 0 0 O O O O O O0 0 1 0 0 0 0 0 0 0 0 0 0 1 (2.8) METHOD OF SOLUTION By a successive multiplication of the appropriate matrices, a linear relationship can be established between the (A) matrices at the root and tip of the beam WArroot = [C][(Atip (2.9) Recognizing that the shears, bending moments, and torque are zero at the tip of the beam, the (A)tip matrix can be reduced to a five-element matrix, and the corresponding five columns can be eliminated from the first [F] matrix at the tip of the beam; successive multiplications will then yield a 10 x 5 matrix product. In order to satisfy the boundary conditions at the root of the beam, the determinant of a 5 x 5 matrix formed from appropriate elements of the [C] matrix must equal zero. For example, for a cantilever blade the third, fourth, 14

seventh, eighth, and tenth rows form the 5 x 5 determinant, and for a fully articulated blade with torsional restraint the second, fourth, sixth, eighth, and tenth rows form the determinant. Other boundary conditions, such as elastic restraint at the root, can be handled easily. The elements of this determinant will be polynomials in c2, and upon expansion a polynomial equation in ai will be obtained. In principle, the natural frequencies of the blade could be determined by solving for the roots of this equation; however, such a procedure is far toocumbersome to be feasible. A more practical procedure involves the introduction of trial values of D into the various [F] matrices and evaluating the elements of all matrices numerically~ The matrix multiplications can then be carried out numerically, and the appropriate determinant evaluatedo The value of this determinant, which may be termed the "residual," may then be plotted versus C or ou and the location of the zeros of the residual will determine the natural frequencies of the bladeo STEADY-STATE DEFORMATION As pointed out in the introduction, there may be a sizable steady-state or "pseudo-static" torsional deformation of the rotating blade in some cases. The loadings which produce this deformation are given in Appendix B along with those induced by the lateral and torsional vibratory motion. It is possible to determine this pseudo-static deformation and to then find the natural frequencies based on linearization with respect to the pseudo-static deformation. In the numerical results which follow this has not been done explicitlyo The values of pretwist selected should be interpreted to include the pseudo-static deformationo In order to determine the pseudo-static deformation let us define the following matrices: [A]i = column matrix of blade variables just outboard of mass i =(i column matrix of blade variables just inboard of mass i [Fl], = matrix [F] with CD = O, across mass i (d) = column matrix of steady state quantities across mass i ([g] = column matrix of steady state quantities across bay between masses i and i+l [D]i = [R]j[E]1 across bay between masses i and i+l. 15

Then it follows that -i —{ =:i l _ 1] (2.10) and.(A-j+ [D] (1). (A).T.= L-j -- L 1. (2.11) Starting at the root where (Airoot = (A)n+l we have, 1 -F= (ho]],h j n- (2.12) where I -j [jH] (hi ] = Lt ED]' (g ['[Fo 1 1 j LLOr J i=1, Equation (2.11) may be written [(Aroot = [H][A)i + (h). (2.12a) Satisfying the boundary conditions at the root and the tip of the blade, Eq. (2.12) may be reduced to [H(K)] [A(l)) = (h(K), (2.13) 16

where 61 (A(')) = &2 2 (2.14) 62 and where K = 1 corresponds to a fixed root, and K = 2 corresponds to a fully articulated root (M1 = 2 = 0)o [H(K)] is a square matrix of order 5, and (h(K)3 is a five element column matrix. In the case of a fixed root, [H(1)] is obtained by deleting rows 1, 2, 5, 6, and 9 and columns 1, 2, 5, 6, and 9 from [H], and (h(l)) is obtained by deleting rows 1, 2, 5, 6, and 9 from (h)o Similarly, in the case of a fully articulated blade, [H(2)] is obtained by deleting rows 1, 3, 5, 7, 9 and columns 1, 2, 5, 6, and 9 from [H], and [h(2)jis obtained by deleting rows 1, 3. 5, 7, and 9 from (hjo Equation (2o12) may be solved for [A(l)}, and [A) then determined for all stations by applying Eqso (2~9) and (2o10), starting at the tip and progressing toward the rooto The matrices (d) and (gJ are each ten-element column matrices which are obtained from the steady state terms in Appendices B and Co From Appendix B, d! = - JQ2 sin P(e0+e cos P) d= d3 d7 = d8 = d=o = 0 d5 = pL2 cos P(eo+e cos P) (2o15) de =-p x e.Q2 ds = (I-I ) ~ 2 sin P cos P - pleeo0s sin P Appendix C shows that the (g) matrix can be derived from the [E] matrix if the terms involving MP and Q are extracted and M2 and Q are replaced by TeA and TkOP, respectivelyo Thus, giA - A + E E (2.16) 17

NONDIMENSIONAL FORM It is convenient and desirable to treat the problem in nondimensional form. The (A) matrix can be redefined in terms of nondimensional forces, moments, and deformations as follows: v1R2/EIl M1R/EI1o V2R2/EIlo (A) = 2R/E1 (2.17) 52/R QR/Eo10 The corresponding nondimensional form for the [F] matrix follows: 1 0 0 F14 0 0 0 F1i 0 F110 0 1 F23 F24 0 0 0 0 0 0 O 0 1 0 0 0 0 0 0 0 O 0 1 0 0 0 0 0 0 0 0 0 0 F54 1 0 0 F58 0 F510 [F] 0 0 0 0 0 1 F67 F8 0 0 (2.18) 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 F94 0 0 0 F98 1 F91o 0 0 0 0 0 0 0 0 0 1 where 18

-2 2 2 2 F14 = p(k +t sin ) Fl = -p~ sin: cos Flo= ple([ +( (sin P-cos P)) - poe0p. cos P F23 = P + 2i(X2+2) 12 F24 =- *xi~t F21o = - k p ex F54 = F18 (21o9) *_ _ —- o 2 2 Fs8 ='2(2+~ cos P) Fslo = - pep sin (2e cos P+e) -3 2 Fee = +:i(% +[ ) F68 = 24 - o ^'- 2 2 Fs_4 = e pe(X +k sin" ) F98 =" pZeesin p cos P F910 = (:+In)X2 + p cos 2P(I- I: ) - o cos P 19

And the corresponding [E] matrix 1 0 0 0 0 0 0 0 0 0 E21 1 0 0 0 0 0 0 0 0 E31 E32 E33 E34 E35 E36 E37 E38 E39 E310 E41 E42 E43 E44 E45 E46 E47 E48 E49 E410 O 0 0 0 1 0 0 0 0 0 [E] = 0 0 0 0 E65 1 0 0 0 0 E71 E72 E73 E74 E75 E76 E77 E78 E79 E710 E81 E82 E83 E84 E85 E86 E87 E88 E89 E810 Elo1 E102 E103 E104 E105 E106 E107 E108 E109 1 (2.20) where E21 = ~ E31 = - -I i2a6P4-a3a8P3)cosh p, + [ -a2a6P4+a3 8(P3+) ]cosh p2Q-a3a8 - EI3 P2 E32 = - sinh pij - -- sinh p27 EIlpi EI.32 E33 = - P3 cosh P 23. + P2 coshT'2Q E34 = a1EIIE32 (2.21) E35 = s- Zl h sinh ~ - sinh 2 a3a4EI2 P2 2a - E36 = 2i4 (cosh Pl~-cosh p2 ) a3a4EIJ2 3EI 2 E35 E37 = - a3BI23 72 y3EI2 E38 = E2 36 B39 (2 a3 E39 = -a 2 p (p.-a3)p2 cosh Pm ~-(-Pa-3 )pP cosh P2 - _a2a3 20

E31s = a2EIE32 + 2W(NE36 E41 f2EJ1 L - sinhI-l~ f4 = -_ (-a3a8P3+a2a6P4) 3 sinh Pi f2EI EL P1 + [s3a(P3+l)-asa6P4] sinh P2e3a Th 1 r Pl (P2&a3)+a3 (P2 ~al ) P3 - P2 - E42= _- 2 cosh P1 +P2 cosh P2] EI1 L ('al.a3)pYp2 pi p2 E43 = EIlE32 E44 = 3 ej- cosh P1 + - cosh P2Y -2-2 -2 -2 PlP2 P P2 E45 a2PY _ - -P2)+p2 cosh lQ'- -p cosh P2 l asa4pipE I2 E47 = - 2 E45 E48 = E37 (2.21) E4 = [ — pa_) P_ sinh P 2+(p2-a ) P1 sinh P2 + PI 2 (2 _~~~~ _(N) E41o = a2EI1E43 + P W E46 E65 = ~ =71 [ - --- ra2aeP2+as3aPi )pi sinh ITS+(7a12P3-3sa85 )p2 sinh P2I a2f2EIL E72- a4P (cosh iT -cosh p2QT) E73 _= s nh -P+- sinh +p sinh P ) Ev4 = a1EI1E72 21

E75 = -- (1-E77) a3EI2 - 2 E76 = _ (-P2pl sinh pile+P3p2 sinh p2A) a3EI2 E77 = P2 cosh P1Q - P3 cosh P2a E'8 = E78 2 E76 Y7 E79 — a E79 = _Ap12 pl-al)sinh -pl-(p2-al)sinh P2i - -2- 2E710 = a2EI1E72 + t W E76 E s ( 31 = _2a4;(a2a6eP23a3Pa)cosh p I+(-a2a6P3+a3a8Pl )cosh P2T-a2ai f2EIJ E82 = -E sinh PI + 1 sinh P2 -a2ET \PP p2 E83 = EI1E72 (2o21) E84 = alEIaE82 E85 = = - (t+Q) a3EI2 2 E86 - 7 (E88-1) a3EI2 - P2' - - p h E87 = sih P + P sinh P + Pm P2 E88 = P2 cosh p1i - P3 cosh P2~ E89 = (-)p cosh — a l)p| coshp+ c + aa4 f(P1"P2) fs _ - - a~(N)_ E810 = a2Ei1E82 + [ W E86 22

E101 = 1 -a3) (a27ae2- as P )sinh p, (faEIn P2p] Epo 3 = (a 3) (saa6 a-asasP) sisnh psi P2 P Pe _2 cos. _ * 0 - [ (-2 L E105 = -a2 L[ app +)(pJ-a1)p2 cosh p-ilpj al)p cosh P22] Eo03 = _ _ __ - sinh IJ. P2-al sinh Tr| a2 L Pi P2 El04 = a DEiEo2 ~~a3 -sa^~P~i~~P2Es~I2 L (2.21):1.~P - six4 22iJ zlos = f{ — sinha -. Pl +o~ 4 (P Di_)P osh P -a, -_- - T~.Sn 2 = a3a4EI2 P- P2 a' E0O7 = - a~sEI~Elo, E10o8 =aE2E+ ( P z P (ap PP() si P _ a^ la i) +( F|(<3))2 co^h Psh Pi":2 t(ai-a32 PiJL + (Pla;3P% cosh plj$+ 2W(E =~ 21~2 ai = s IT2 =aa = I -3 EIl 2aU EIl ~3

T a3 =e _y2 1 EI2 a 4 = - a6 = U 1 j - a7 = -y fEI2 7 a8 = A() fo = A - Y = as - a4a7 202 1 fL 2 1 - AT + -) + TY a2a6-as(a +a3 )+a la4a'iEI, IiEI e EI f2 = [14y2 T (ThA-M2[2) = a3(alas-a2a6) (2.22) EI EI2 f2 2 _f2 521 = - fl + (X) fO 2fo A fo0 0 _ P2 = _ (fl 2~ -22 - -_ __2 _2: p (Pl-a3)(P2 a-l) i=1 (ai-a3)(P-2 2 2 p = (Pl-a3)(P2-a3) (al-a3)(pDl-p2) n n W(N) = - cos n I ipieixi sin i sin sin n iPieixi cos Bi i=l i=l 24

The (d) matrix has the components d = -_ pp2 sin P(eo+e cos I) d2 = d3 = d4 = d7 = d8 = do = 0 d5 = P2 cos (eo +e cos ) d6 = - pxe 2 d9 = (lsin I) cs sin cos sin and the components of [g} are gi = ~(Ei2eTA+EiTkA ) ~ (2.24) For the case of zero rotational velocity, the [F] matrix is obtained directly by substitution of [i = 0. When this substitution is made in the [E] matrix, some of the elements are found to be of indeterminate form and a limiting process must be applied. This results in: E21 = ~ E31 2 (2.25) E33 = 1 E34 = 0 - _ i3 25

E42 = - E31 E43 = - E44 = = 1 E65 = ~ E75 -2I2 - E76 =- Ei) EI2 E77 =1 E78 = O E79 = - Ig (A) _!2A3 ( A7 (2.25) E85 - 6EI - E86 = - 75 E87 = - E88 = 1 Cy22 /A \ E89 - 8 2 EI2 \jA E105 = E89 El06 = E79 26

E107 = 0 E-os = 0 (2.25) E109 = = A+Y NUMERICAL RESULTS A program of computations was performed for two representative cantilever bladeso The properties of these blades are given in Table 1. These blades were chosen to have the same bending properties as blades for which numerical results are reported in Refo lo For these two, the section constants I1, I2, B1, B2, GJe, kA correspond to those for a thin-walled rectangular section, and it was assumed that some nonstructural mass was distributed in such a way as to provide an offset between the mass and elastic axes, and to provide sufficient mass moment of inertia to made the uncoupled first torsional frequency and the second uncoupled flapwise frequency coincideo The result in both cases is a lightly coupled system as far as flapwise bending-torsion is concernedo In both cases the blades were divided into ten segments, the cantilever root condition was applied, and the four lowest frequencies were determined. A range of values of pretwist and rotational velocity were chosen, and the results are presented in Figso 2o4-2.o6 In addition, results for beam No. 2 with rotary inertia neglected are presented in Fig. 26., and with centrifugal force coupling neglected in Figo 2.7~ DISCUSSION OF RESULTS The influence of twist on the natural frequencies of nonrotating blades is shown in Figo 2o4o The fundamental frequency in each case is almost completely unaffectedo The higher frequencies are affected by the coupling between flapwise bending and. torsion, and by all three types of deformation when twist is introducedo The combined effects of rotation and twist on blade No. 1 are illustrated in Figo 2.5o In the untwisted, nonrotating case the fundamental mode is identified as predominantly flapwise bending; the second mode is uncoupled chordwise bending; and the third and fourth modes are coupled flapwise bending and torsion. The effect of rotation and twist is to couple the first two modes. A comparison of the results in Figo 2.5 with results in Refo 1 for a beam with the same bending properties but with torsion neglected shows that essentially no change has been introduced by the presence of torsiono The fourth frequency in Figo 205 differs slightly from the third frequency for the beam in Refo 1, 27

TABLE I BEAM PROPERTIES Beam No. 1 2 h -------- t h/L 2.285-10-1 6.225-10-2 h/R 1.000.10-2 1.000.10 2 e 0.25 L 0.15 L _ 0 0 kA 0.015522 0.04926 PT 0 0 21.0010 1.000.10 1.00010-2 Il[in.4] 2.355 h3t 8.1987 h3t I2[in.4] (2.355.10)h3t (8.1987*102)h3t Bl[in.6] 32.692 h5t 15.345 h5t B2[in.5] 0 0 GJe 1.153 1.405 Ir 2.190'10-5 2.402 105 I:. 5.832-10-3 7.114-10- 3 *Aluminum is assumed.. 28

30~ 8 150 0~ I I I - I 0 10 20 30 40 50 A 300 8 150 0~L 0 - I IL I 0 10 20 30 40 50 A Fig. 2.4. Effect of twist on natural frequencies of nonrotating blades. 29

'^T rQ - 0 a) ~ X~ ~N o... r0 0) 4) rc 0%^.~ 0^~ oo \ \ b0 3o ^ 1-1 C) /4 50

Icr~~~-~ ~O To O O O O > w o 0 N 0 1cr~~~~~~~~~~~~~~~~~~~~~~~~ zz~~~~~~~r d \'0~~~~~ 00\~ N0 CI A_ 0 C, 31 OO ~ 1 5]_ fn ^ ^"^ x -p ^^\ - 8 ^^S ^Kcsj ^ o'^X~~3

8 Cc o ~0,- \ o \ \ - NH 0) r<L H 8 rlPI< J ao ~~CDM~~~~l~~r cri CD a C) 0O) N \g 52

while the third frequency in Figo 205 is a new one introduced by the presence of torsion. It is seen that twist has little effect on the third and fourth frequencies shown in Figo 2~50 A small difference between the results in Ref. 1 and the present results is introduced by the inclusion of rotary inertia in the present analysis The results for blade No. 2 presented in Fig, 206 show that the fundamental frequency is essentially unchanged by the presence of torsion when compared with results for a similar blade reported in Refo lo For this blade the fundamental mode is predominantly flapwise bendingo The second and third modes for the untwisted, nonrotating blade are coupled flapwise bending and torsion, and the fourth mode is uncoupled chordwise bendingo When rotation and twist are added, the three higher modes exhibit considerable coupling, and it becomes difficult to reach any general conclusions~ When compared with the results in Refo 1 for a similar blade without torsion, it is seen that the effect of the presence of torsion is to introduce a new frequency and to modify the other two frequencies a moderate amounto rThat these two frequencies are not modified more by the presence of torsion is to be expected since the coupling for this blade (and also for blade Noo 1), as represented by the amount of offset between the mass and elastic axes, is relatively smallo The neglect of rotary inertia has a negligible effect on the natural frequencies except for those cases in which there is chordwise bendingo For example, in the case of the untwisted beam Noo 2 the only frequency which is appreciably affected is the uncoupled chordwise bending frequencyo The magnitude of this effect is shown in Figo 206o The effects of centrifugal force coupling on beam Noo 2 are shown i.n Fig~ 207o Curves with and without centrifugal force coupling are shown for ~ = 0~ and 30~ Only the ~ = 0~ case is shown for the second frequency to avoid confusion in plottingo The 0 = 30~ case for the second frequency is modified by a slightly smaller amount The ~ = 0~ case which. represents uncoupled chordwise bending (the fourth. frequency for the nonrotating beam) is essentially unaffected, as is the ~ = 30~ case for the third coupled frequencyo It can be seen from these results that centrifugal force coupling can have an appreciable effect on some- of the vibration characteristicso 55

3. SIMPLE MODEL ANALYSIS SYMBOLS ala2 functions defined immediately following Eq. (3.13) e offset of mass c.g. from supporting rod, positive forward e nondimensional form of e, e/r ijk unit vectors along the x,y,z axes respectively I moment of inertia of mass m about supporting rod I0 moment of inertia of mass m about its own c.g. IF moment of inertia of flywheel K kinetic energy of system kg stiffness of bending spring k~ stiffness of torsion spring mi mass MT shaft torque MT nondimensional form of MT, MT/mr2 2 p differential operator r length of supporting rod from shaft to mass m R radius vector from origin to element of mass dm t time U potential energy of system v velocity vector of mass element dm v magnitude of v 34

xyz axes fixed to supporting rod and mass assembly y y-coordinate of mass element dm XFYFZF stationary axes ~a orientation angle of bending hinge axis 3i ~ angle simulating built-in twist 7y'built-in coning angle phase lag of motion in /-coordinate relative to motion in -Q coordinate 0 elastic displacement about bending hinge Qs pseudo-static value of 0 Q departure of 0 from 0s go amplitude of 0, also initial value of 0 p nondimensional radius of gyration of mass m about supporting rod,,I /mr2 Po nondimensional radius of gyration of mass m about its own cogo. s Io/mr2 T nondimensional form of t,?t pt ~ elastic displacement about torsion hinge 8, ~ pseudo-static value of / departure of / from s I/0o initial value of / initial value of /' ji ~ amplitude of / angular displacement of shaft Ct angular velocity vector of x,y,z frame 3.5

Dx,Yly,Uz components of c along the x,y,z axes, respectively X natural frequency of characteristic oscillation El, )2 first and second natural frequencies of characteristic oscillation ~Q, rotational velocity of shaft DESCRIPTION OF THE MODEL In order to examine some effects of nonlinearity and Coriolis forces in the free vibrations of a rotating elastic blade in coupled bending and torsion and to consider the effects of certain parameters on the static deformation of the rotating blade, a simple model with a small number of degrees of freedom is set up and analyzed. The model consists of a rigid weightless rod on one end of which is mounted a mass and the other end of which is connected to a rotating shaft. The connection to the shaft is through a hinge with axis normal to the rod and set at an angle to the shaft. A spring, restraining motion about this hinge, simulates bending stiffness. In addition, the rod is free to rotate about its own axis against the action of a spring, which simulates torsional stiffness. The mass is assumed to be distributed along a line normal to the rod, simulating the major principal axis of a blade cross section, with its center of gravity displaced from the rod, simulating an offset of the mass axis of the blade from the elastic axis. The orientation of the model relative to a set of fixed axes and the generalized coordinates defining its configuration are shown in Fig. 3.1. The final orientation is reached by aligning the model initially with the fixed axes and then executing a sequence of rotations. The fixed axes X, YF, ZF form an orthogonal set oriented so that the xF-axis is coincident with the shaft centerline. Their origin is at the intersection of the rod and the shaft centerline and is coincident with the origin of the model axes x, y, z. The x-axis lies along the rod, the y-axis is parallel to the line along which the mass lies, and the z-axis completes the orthogonal set. The model is initially aligned so that the x, y, z axes are coincident with the xF, YF, ZF axes, respectively. The following rotations, positive in the right-handed sense, of the x, y, z frame are then executed in sequence: 1. A rotation about the zF-axis through the angle f to the position xl, yl, zl. 4 then defines the shaft rotation. 2. A rotation about the yl-axis through the angle -y to the position x2, Y2, Z2. -7 then defines a built-in coning angle. 36

Bending Hinge Axis ZF, Zl \ Y3,Y4 / Mass, parallel to / / Mass c.g. y-Axis Shaft,Axis / I Supporting Arm -— xx3 Hinge Axis v I X \ \XF Flywheel-l Fig. 3.1. Model coordinates. 35. A rotation about the x2-axis through the angle a to the position x3, Y3, Z3. The y3-axis then defines the position ofhe e hinge axis. 4. A rotation about the y3-axis through the angle -Q to the position x4, Y4, Z4. This represents a rotation about the hinge axis simulating bending displacement. 5. A rotation about the x4-axis through the angles P and i in sequence to the final position x, y, z. The angle P simulates built-in twist, and the angle ~ elastic twist. The angles a, P, and 7 are constants and constitute parameters in the problem. The angles 4, Q, and ~ are generalized coordinates representing shaft rotation, bending, and torsional displacement, respectively. DERIVATION OF THE EQUATIONS OF MOTION The equations governing the motion of the model are now derived using 37

Lagrange's equation. Toward this end it is necessary to obtain an expression for the kinetic energy of the system in terms of the generalized coordinates. Assuming a flywheel of moment of inertia IF to be mounted on the shaft, and defining m as the magnitude of the mass mounted on the rod, the kinetic energy of the system may be written, K = 1 IF2 + 1 v2dm (5.1) ~~K I + 1 2 2 where v is the magnitude of the velocity vector v of an element of the mass m. v may be developed from the relation, v = x R (3.2) where o is the angular velocity vector of the x, y, z frame and R is the radius vector of dm. Substituting = CWt + wyj + wk (35.53) R = ri + yj (.4) 4. 4 4 where i, j, k are unit vectors along the x, y, z axes, respectively, into Eq. (3.2) the following is obtained, v = - ywzi + rwozj + (yox-rey)k. (3.5) Thus, v = y oWZ + r Wz + (yox-ray)2 (3.6) and Eq. (3.1) may now be written, K = - I2 + (- mry+z) +. I(wx+ez) - mreo aWy (5.7) 38

0 o.........U) c aD r- CHO 0 0 0 a >n ~ ~ -C 0 CC 0 r o w U-H. 0 0 ocO o CD - 0 C0 CCH ~ o 3 o o o. 0 r-" 0 1._ a.) -, 4-, _..-....... — +4 ~.H +P E "'G. +rC U) c coH o c CH CH) O O CLL r c I>O U I, O* r^ o.O c a -P_______ * (b' 4-4 r.H O 0 1 *H O U' 4-HD o? - bO 0 U) 0 D O hD0 C O 0* P U 0*H O U) O PA * (1 U QC 0 44 CH U) O ~, ~ J o o a O O; o o or r -P c0 Cc rd O M. D 0 I OU r q -H Oo r~. ~ c O H c 0 o i * 0- - I Hc 0 0 4 d c co 0 CH s i W + ) + 0 + o. U c) ac o [ 00U0 oH N. 0 +D 0 d uH W cj 0 r + c o c o 0 U) u U(^ 00 U)H ca 4 H H) U) 0 H U)'-~-O O - ~ Q a > Q t L.,. J *H -P 0H.,,.,, -P~ 0,. P 3 + + *H + O Cl QLi a cI cO a 0 3 O N0. 7 -+ 0D ^ * Q O- / ~ ^ rQ-l + ~ il U 0 +H U. + ~ co t c ~ (D o X ( D I _____ I + II OD CH > N — -- -H 0 0H O F (D 7 $ C: 0 O ePe^ ^o o*Q S 0 -O O -G +G c + N'- * O o % o c c o + *H0 ~~~~~~~ ~~~~So 01U~~ 01 0 U O~ >^ ftC O OO (3 O C-) U O O I O O 0 H c H c * 0 *H U 0 U) He + 11 CH > \-8 0 N. 3 o4-o. -r -. ri U (U O0 0 O o c U) o UD'-', 40 H Q o3 59

Neglecting gravity forces, the potential energy of the system may be written as follows, U = kG2 + 1 k2 (3 9) where kQ and kX are the spring constants of the springs restraining motion in 0 and B coordinates. Substitution of Eqs. (3~7), (3.8) and (3.9) into Lagrange's equation, d f( ) -aK + 0 (i=1,2,3) (35.10) dt \dqi i aqi where ql = -, q2 = ~, q3 =, yields the following differential equations, [mr2+I sin2(P+~)]Q + [mre cos(P+,)]* + -mr2 sin a cos y+mre a2 cos(p+() + I sin(P+)[(-sin(P+$)sin a cos y+al cos(fp+)} + E2I sin(P+/)(cos(P+O)sin a cos y+ al sin(P+())-2mrea2 sin(p+)(j) + [2I sin(P+z)cos(p+z() ] + [-rme sin(p+Z) ] 2 + mr2ala2+mre(al cos(p+O)sin U cos 7+(a2-a2)sin(p+O)) - Ia2 sin(p+ )[cos(p+$)sin ct cos y+al sin(p+O)}] 2 + k00 = 0 (3.11) 40

[mre cos(P+i)]~ + I + [Ia2-mre(cos(p+$)sin ca cos y+al sin(p+/)J] k + L2mrea2 sin(p+~)+2I sin(p+~)(cos(p+~)sin a cos y+al sin(p+$)}] Q + [-I sin(p+~)cos(p+) 162 + [mrea2(-sin(p+~)sin a cos y+al cos(p+)}) + I(-sin(p+~)sin a cos y+al cos(p+/)}(cos(p+0)sin a cos 7+ai sin(P+$)} 2 + k~ = O (3.12) E-mr2sin a cos y+mrea2 cos(P+~)+I sin(p+~)(-sin(p+~)sin a cos y+al cos (D+))]] + [Ia2-mre(cos(P+~)sinacos y+al sin(p+)})] + rIF+mr2(sin2 a cos2 y+a2)-2mrea2(cos(p+~)sin a cos y+al sin(p+B)J + Ia2+I(-sin(P+ )sin a cos 7+al cos(4+))21 + I-2mraala2+2mre[a2 sin(p+~)-al cos(p+~)sin a cos y-aia2 sin(p+~)} + 2Iaa2-2Ia2 cos(+sin(+)sin cos cos(+ )] + [-2mrea2(-sin( +$)sin a cos 7+al cos(P+~)) 2I(cos(p+~)si a cos 7+al sin(p+o))(-sin(+ )sin a cos y7+a cos(1P+))] + [2I -sin( )sin cos(+ sin(+)sin a cos +a os(P+)}]] 6 + [mre-al cos(P+O)-Ia2 sin(P+o)cos(p+o)]62 + mre(sin(p+-)sin a cos 7-al cos(p+$)1] 2 o (3.13) 41

where al = -sin Q sin 7 + cos 0 cos a cos 7 a2 - cos Q sin-7 + sin Q cos a cos 7 SPECIALIZATION TO THE CASE OF CONSTANT SHAFT SPEED The problem is now specialized to the case of constant rotational velocity of the shaft by setting r = 0, i = Q, and the equations are put into a nondimensional'form by defining the nondimensional parameters, - = ij 7m r e = I _ -b.- 1 / -t - d - I dig n ev and introducing the nondimensional time variable, T = t. It is seen that p is the nondimensional radius of gyration of the mass about the rod axis, e is the nondimensional offset of the mass center of gravity from the rod axis and wb and Lt are respectively the nondimensional natural frequencies in restrained bending and restrained torsion when the shaft is not rotating. Division of Eq. (3.11) by 2mr2 and Eq. (3.12) by 2 I now yields, 42

[1+p2 sin2(C+~)]@" + e cos(P+)" + -2p sin(p+$)(cos(p+$)sin Ca cos y+al sin(p+~)}-2ea2 sin(p+$)1,' + 2p2 sin(p+~)cos(p+Q)G'' - e sin(P+ )' 2 + aa2+e(aal cos(5+~)sin ca cos y+(a -a|)sin(p+)}) pa2 sin(p+)[(cos(p+O)sin e cos 7+al sin(p+0)] + -o2 = o (3514) + D- cos(p+ )@" + 2 2 a2 sin(P+$) + 2 sin(p+t)(cos(p+O)sin a cos y+al sin(p+0)}] Q' - sin(P+)cos(P+$)Q' + e2 a2(-sin(+)sin (+ cos y+al cos(P+$)) + (-sin(p+$)sin a cos 7+al cos(P+0)}(cos(P+$)sin a cos y+al sin(P+0)} + t = ~ ( 315) where primes denote differentiation with respect to r, and al and a2 are as defined in the preceding sectiono Recognizing that constant shaft speed represents the limiting case of infinite flywheel inertia, the term IF* in Eq. (3.13) can be seen to remain finite and equal to the shaft torque, which may then, from Eq. (3.13), be written in the following nondimensional form, 43

MT = -sin a cos y+ea2 cos((+() + p2 sin(p+B )(-sin(p+()sin a cos 7+al cos(p+z))] 0" + [c2a2-F[cos(+~)sin a cos 7+al sin(p+z))] " + [-2ala2+2e(a2 sin(p+8)-al cos(p+z)sin a cos 7-aja2 sin(p+$)) + 2p2ala2-2p2a2 os()i cos()-sin( )sin a cos y+a cos(+))] + -2ea2(-sin(p+~)sin a cos 7+al cos(p+Z)) - 2p2(cos(P+/)sin ac cos y+al sin(p+/)J(-sin(p+/)sin a cos y+al cos(p+1)} + 2p2 cos(p+) [-sin(p+()sin a cos 7+al cos((p+)]~'0' +- [eal cos(p+_)-_p2a2 sin(p+$)cos(p+O)]Q12 + e[sin(P+$)sin a cos y-al cos(pD+)]$'2 (3.16) where MT mr = 22 and MT is the dimensional torque. To facilitate solution, it is desirable to rearrange Eqs. (5.14) and (3.15) in the form, f10" + f2"3 = -f30'' - f4' - f,'2 - oQ - f (3.17) f7"1 + " = -faog - f9g12 -- f- 0 (3.18) where 44

fl = 1 + p2 sin2(p+$) f2 = e cos(p+~) fs = 2p sin(P+~)cos(p+$) f4 = -p2 sin(p+) (cos(p+z)sin a cos y+al sin(+ ) }-2ea2 sin(p+ ) f5 = -e sin(p+~) fe = ala2 + [(al cos(P+()sin ca cos y+(aj-a2)sin(p+)] p a2 sin(p+))(cos(p+O)sin a cos y+al sin(P+O)) f7:= - cos(p+$) 8 2= 2 a2 sin(p+~)+2 sin(p+0)(cos(p+$)sin a cos y+al sin(p+))) f = -sin(p+0)cos(p+0) I = f_- a2[-sin(p+))sin a cos y+al cos(p+0)) p+ (-sin(p+~)sin a cos y+a c )}os ( p+)( +os(+)sin a c os y+a sin(fP+)} Solving Eqsc (3.17) and (3518) for 0" and $" in terms of Q and 0 and their first derivatives yields the differential equations in the following form, 0, =_ 3 [=f73^ v f4 I -SfS2 -f6+f2P8 +f2fsg9 +f2Ot+f2flo] (3.19) -f2 — f + f74s'+f7f52+f7f6] (.20) u + f^f4o'+fSf-90 i2719+ (5.20) Equations (3519) and (3.20) are now in suitable form for solution on a digital or analog computer. 45

SOLUTION OF THE PSEUDO-STATIC PROBLEM It is of interest to determine the static configuration of the rotating model, that is, the static displacements under the action of centrifugal forces. This problem may be termed the pseudo-static problem. Its solution permits the setting up and solution of linearized differential equations for small motions about the pseudo-static configuration. The appropriate equations are obtained by eliminating all terms containing derivatives of 0 and O from Eqs. (3.17) and (3.18), yielding, E s + f6 = 0 (3.21) tis + fl=o =. (3.22) These equations are nonlinear, with f6 and flo being transcendental functions of the dependent variables. Since it is not feasible to obtain an analytical solution in closed form, the following iterative procedure was applied. Equations (3.21) and (3.22) are linearized with respect to departures AG and A$ from trial values 0n and On, respectively, of the variables, yielding, 2 2nDAG + fen + (f )AG + f 6f AO = 0 (3.23) ~l~0n + ~ A0 + fen + n + C Ad + fln + (f-) AG + (f) A = 0 (3.24) O2/ + "It + n+) n X ) n where subscript n denotes values at 0 = Gn, 9 = ~n. Equations (3.23) and (3.24) are now rearranged in the form, (2b+flln)A + fl2nAO = -fen - bn (3525) fl3nAQ + ('0+fl4n)A = -flon - O tn (.26) where 46

26 fll - ao a21 - a2 - e(a2 cos(P+/)sin a cos y+4ala2 sin (B+()} f12 = 2 2 = e[-al sin(p+~)sin OC cos y+(al-a2)cos(p+r)) - a2 [[cos2()p+)-sin2(3+,)}sin a cos y+2ai sin(p+0)cos(P+)]J f''13 -a sin( )sin cos y+(a-a) + 1flo f14 = a2(cos(p+~)sin o! cos y+al sin (P+()} p2 (cos(pB+)sin C cos y+al sin(p+~)) + (-sin(p+~)sin a cos y+al cos(p+1))2 Solution of Eqs. (3.25) and (3526) for AG and A~ yields, A~ = — *-nen ne (3-27) e3n -esneln+e4nf13n (328 e3n where 47

el = + fl e2 = C2 + f14 e3 = ele2 - f12f13 e4 = n +f6 b + e5 -= n + f o Equations (35.27) and (3.28) may be applied in conjunction with the iteration formulae, Qn+l = ~n + A@ (3.29)'n+l =:n + AB (3 3~) using as initial values, Q1 = 0, z1 = O. The process has been found to converge rapidly in the cases that have been considered in the present work. FORMULATION AND SOLUTION OF THE LINEARIZED EQUATIONS In order to assess the significance of nonlinear effects in the problem under consideration, it is desirable to obtain also solutions to linearized equations for small perturbations Q and 0 from the pseudo-static configuration. Application of small perturbation theory to Eqs. (3.17) and (3.18) yields, fls" + ( + f11)@Q + f2.S" + f4sO' + f12s = 0 (5.531) f7sQI + f8s' + fl3SQ + + (+f14s)7 = 0 (.532) in which subscript s denotes values corresponding to the pseudo-static configuration Q = Qs, 0 = Os. Putting Eqs. (3.31) and (3.32) into operator form, using symbol p to denote the differential operator, and expanding the determinant of coefficients, 48

the following characteristic equation is obtained, clp4+ + + C3 = 0 (.) where cl = fs - f2sf7S C2 = fls(Lt+fl4s) +L + flls f2SfS f7fl2s - f4SfS C3 = ( +flls))(+fl4s) f- f2s13Ss The terms in p and p3 are seen to vanish. The roots of this equation are Pi = c (-C2 + Jc2-4cc3) 2 1 P2 = (-c2 - cc22-4c c13) and the characteristic frequencies are given by il = (3.-4) M2 = p * (3.35) The characteristic mode shapes may be determined by assuming a solution of the form, 0 = Q0 cos AdT (3.36) = cos T + ~ sin ST = Sl cos (/T-.). (35.37) 49

introducing Eqs. (3.36) and (3.37) into Eq. (3.31), and equating the sum of the coefficients of the cos cmT and the sin cm- terms respectively to zero, the following result is obtained, To /Qo= (f2s 3 -f12s)(-fls(++flls) ( -.8) (f2 -U2f 12 ) 2 + f j i- — 2 (f2s-D -fl2s) + f4(s _ _2 2 __ where c = w1, W2~ The mode shapes may be expressed alternatively in the form, /P PO= 2 4+ 0 2 (3.40) = tan-1 ( ) (3.41) where, from Eq. (3.37) it is seen that z1/0o is the relative amplitude of displacements in the two coordinates and e is the phase lag of the oscillation in the /-coordinate relative to that in the O-coordinate. A solution involving only one characteristic mode of oscillation may be obtained by selecting as initial conditions, 0 = G 0 ++ G,' = (3 + (o, 8 = o (5.42) where 0o may be selected arbitrarily within the limitations imposed by the assumption of small perturbations, and.o and S$ are then determined from Eqs. (3.38) and (3.39). It should be noted that the existence of a phase difference between oscillation in the two coordinates is associated with the presence of the terms f4s f and f8sQ' in Eqs. (3.31) and (3.32) respectively. These terms originate in the terms in 4f and it in Eqs. (3.11) and (3.12), which are due to the presence of Coriolis forces. DISCUSSION OF RESULTS A series of computations on the simple model were performed using an auto50

matic digital computer. There computations were limited to the case of constant rotational velocity of the shaft and zero built-in coning angle (7 = 0). The pseudo-static configuration was determined by means of the iterative procedure developed earlier, and corresponding characteristics of the linearized system for small perturbations from this configuration were computed. In each case additional computations were performed in which the terms f4s' and fssQ' in Eqs. (3.31) and (3.32) were omitted. As discussed previously, these terms represent the influence of Coriolis forces, so that a comparison of results obtained with and without their inclusion provides a means of assessing the importance of the Coriolis forces. These results are presented in Figs. 3.2 to 3.8 inclusive. Figures 3.2 and 353 show the effect of varying the mass offset parameter e, with the parameter po, representing the nondimensional radius of gyration of the mass about its center of gravity, and the parameters, a, i, cb and wt being maintained constant. Since the parameter p must be varied accordingly, the maintenance of a constant value for dt implies that the variation of e does not involve merely a shifting of the mass relative to the supporting arm but involves also changes in m or k~ or both. The value of po selected for this case represents a rather extreme value, applicable to a short, wide blade. Figure 3.2 shows the substantial pseudo-static deformation occurring in 4 8 0.4 ) 5 w w 0, W Io~~~\ 0 0 0 0 0.1 0.2 0.3 -e Fig. 3.2. Effect of mass offset on pseudo-static displacements of model. O = 30o, P = 15~, ob = 1/3, wt = 1, po = 0.1732. 51

this case. It should be noted that the static twist decreases with increase in offset, when the center of gravity of the mass is behind the elastic axis. This occurs despite the fact that the relative values of the moment of inertia and torsional stiffness about the supporting arm remain the same because of the constancy of Et, which fact implies that centrifugal twisting moment, before deformation, remains the same. It must be concluded that the variation in twist is associated with a component of centrifugal force normal to the coning surface on which the supporting arm revolves. This effect is introduced through the term in mre+2 in Eq. (3.12) and terms deriving from it in later forms. It has been called "centrifugal force coupling" in Ref. 2,. and shown there to have a substantial effect on natural coupled frequencies of vibration. In the present case, since y 0, the coning of the supporting arm is associated solely with the displacement Gs. With positive Qs and a positive value for (a+P+~s), this effect opposes that of centrifugal twisting moment. It can be expected to be more pronounced in the case of blades with built-in coning angle. Figure 35.3 shows the effect of mass offset on the natural vibration characteristics of the system linearized with respect to the pseudo-static configuration. As can be expected, it is seen that the increased coupling between bending and torsion associated with increasing mass offset separates the natural frequencies and alters the natural mode shapes. It is seen also that the Coriolis forces introduce substantial phase differences between motion in the two coordinates, particularly in the case of the first or predominantly bending mode, where the phase angle is large throughout the range of e considered. In the case of the second mode, where torsional motion predominates, the phase angle is substantial only at small values of e. When e is zero the only coupling between bending and torsion is through the Coriolis forces, and the phase difference is then 90~, $ leading ~ by this amount in the case of the first mode and lagging by this amount in the case of the second mode. Furthermore, the Coriolis forces are seen to have a substantial effect on the mode shape of the first mode and a somewhat modest effect on the corresponding frequency. The corresponding effects on the second mode and frequency are seen to be small or negligible. It should be noted here that the apparent absence in some cases of curves associated with neglect of Coriolis forces is explained by the fact that such curves are indistinguishable from the corresponding solid-line curves, and the effect of these forces is thus very small. Figures 3.4 and 3.5 provide results corresponding to those of Figs. 3.2 and 35.53 for a different case, namely one involving a much smaller value of Po and consequently more realistic in relation to propeller or helicopter rotor blades. Similar trends are observed, except that Coriolis force effects are considerably reduced, but still substantial with respect to phase differences in the first mode. 52

First Mode I0 \ -- Coriolis Forces Neglected 0.93 - -0.3 60 — \ __ I ^^^^, 7 UJ 0.92 -0.2 a 40 ~~~~__ ~w - 80 0.91I 0.3 1.8 Second Mode 2.~0 —----- - I-e ------ Coriolis Forces Neglected // O<~.9 -0.3 0 -- 1.8 2 0 1. 5 0.1 20- O U 4 01 Q2 0.3 0 -e Fig. 5.3. Effect of mass offset on natural vibration characteristics of model. a 5 = 30O, = -150, (5 = 1/5, - =1, l o = 0.1752. 55

0.5 10 0.10 0.4 -8 0.08 U) wl CI) W 0.3 6 00.06 0 Pw 0 IQ 0.2 4'0.04 0.1 - 2 -0.02 0 - 0 0.01 0.02 0.03 0.04 0.05 -5 Fig. 3.4. Effect of mass offset on pseudo-static displacements of model. a = 30O, 3 = -15~, i = 1/3, at = 1, Po = 0.05. Figure 3.6 shows the effect of varying the bending hinge orientation angle while maintaining the orientation of the principal axis of the mass fixed. This involves varying a and ( so that a+: remains constant, and simulates a situation in which mean blade angle is kept constant while built-in twist is varied. All other parameters were maintained constant. Curves of Gs and zs are not shown, as variations in those parameters were small. For a variation of a from 15~ to 45~, Qs varied from 0.78o to 0.97~ and 4s varied from -6.50o to -6.60~. It is seen from Fig. 3.6 that first mode characteristics are affected very substantially by changes in a, the phase difference between the g and ~ motions especially varying over a very wide range. The effect on second mode characteristics is much smaller, although still considerable. Figures 3.7 and 3.8 show the effect of varying the rotational velocity of the shaft while other parameters remain constant. The information in Fig. 3.7 is principally of value in estimating the pseudo-static torsional deformation corresponding to a given rotational velocity. This deformation can be expected to depend primarily on the parameter at in the case of a blade without builtin coning angle, although from results discussed earlier it can be seen also to depend somewhat on the parameters e and Cb. From Fig. 3.7 it can be seen that 54

0.930 0.5 — 100 First Mode' ---- 0.928 C / -0.4 IMs 80a w 0.92 4 / -— Coriolis Forces Neglected 0.2 L 4 J w IJ~~ //r -J 0.924 - --- Coriolis Forces Neglected -0.2 L -40z - w I 0.922 / -~1'20 0.92o!00 2 O 0 0.01 0.02 0.03 0.04 0.05 -e 1.8 0.05 — 100 Second Mode 1.7 / - 0.04 -80 1.6-C V,-3/ /L0.03| —60a i3. 1/ n - 1.5- /z - -0.02uj - 40 oZ _> 1.4 ^^^/ -0.01 w-203 0 0.01 0.02 0.03 0.04 0.05 -5 Fig. 3.5. Effect of mass offset on natural vibration characteristics of model. a = 30~, P = -15~, ob = 1/3, ct = 1, po = 0.05. 55

1.2 0.5 200 First Mode 180(0 w. — Coriolis forces neglected LJw I. I 0 ~ \ /0 0.1o -60 10 1.j 1.1. 0 15 20 25 30 35 40 42 - DE00GREES 0.9 -0.2S - 80e 1. 6 \ -wI.161 0.8 3 0.I - J 40 "~.^-^_____ _-: 20 0.7' I I I I - I - 0 -J o a- DEGREES I.7 Second Mode 0o12 o e 1.- — 0.06- 2 w a -DEGREES characteristics ofe model. M+e 0=150,5b= 1/5, 5=1, p =0.2, 1. - 0.1. - o 56 > W 1.4i-w a. 1.3 I I I I I! 0 o. o4 - 15 20 25 30 35 40 45 a -DEGREES Fig. 3.6. Effect of bending hinge orientation on natural vibration characteristics of model. a + P = 150, ~b = 1/3, -jt p = 0.2, e = -0.1.

1.6 16 1.4 \ -14 1.2- \2 LU 1.0 l I o a 0.8 8 0.6 6 0.4 - 0.2 2 0 0 0 0.2 0.4 0.6 0.8 1.0 fb Fig. 3.7. Effect of rotational velocity on pseudo-static displacements of model. ct/b = 3, = 455~, = -15, p = 0.2, e = -0.1. the torsional displacement will exceed 20% of the initial blade angle (a+P) if at is less than about 2, that is, if the rotational velocity is greater than about one-half the value of restrained torsional frequency corresponding to zero rotational velocity. Figure 5.8 indicates an increasing prominence of torsion relative to bending in both modes as rotational velocity becomes large. It indicates further a marked sensitivity of the phase difference between coordinates in the first mode to variation in rotational velocity, at least in a limited range of rotational velocity. The phase angle is seen to approach zero at large values of rotational velocity. A somewhat different situation is seen to exist in the case of the second mode, where the phase angle increases with increase in rotational velocity. Digital computer solutions to the nonlinear differential equations were also obtained, using a Runge-Kutta procedure. Initl itial conditions were established on the basis of natural vibration characteristics determined from the linearized equations, that is, by applying Eqs. (3.42), using values from Eqs. (3.38) and (3.39) and the pseudo-static displacements. With such initial con57

1.8 - 180 1.4 \ First Mode 1.6 160 --— Coriolis Forces / iq Neglected /1.4 -140 1.2 t \ 1.2 - 120 1.0- \ -1.0 -00 -I,.0 - -0.8 -80 1r -- \' -6-00.6 > 60 0.8 - — 0.4 J - 40 1 0.2 - 20 0.6.,:::- 0M- 0 0 0.2 0.4 0.6 0.8 1.0 ib 5 10.16 - 16 Second Mode i n 0.14'2 - 14 4 -2 -0.12 - 12 - \Q/, 7 0.10 I 0' 3 -\ ^ -0.08 < - 8 ~3 Z'c~J > 2 0.06 6 2 t- /,>^ — 0.04 - 4 2- / ^^^^ ^^"^=0.02 2 i 0 0 0.2 0.4 0.6 0.8 1.0 wb Fig. 3.8. Effect of rotational velocity on natural vibration characteristics of model. ~/w = 3, a = 45~0 p = -150, = 0.2, e = -0.1. 58

ditions, the linearized system responds in only one of the natural modes, and comparison with the corresponding response of the nonlinear system provides a means of assessing the extent to which nonlinear effects distort the motion. Results were obtained for only one case and are shown in Fig. 3.9. Figure 3.9(a) illustrates the response of the first mode when the initial bending displacement from the pseudo-static configuration is 10~o Displacement in the bending coordinate predominates in that mode and its time history is seen to be distorted only slightly by nonlinear effects. There is a slight increase in period and a very slight but irregular variation in amplitudeo The torsional response is seen to be strongly influenced by nonlinear effects in a manner which suggests that there is substantial coupling with the second mode. The slight variation in amplitude of the bending motion is likely associated with this couplingo The response in the second mode for an initial bending displacement of 2~ from the pseudo-static configuration is shown in Figo 3.9(b). In this case, displacement in the torsional coordinate predominates and has an amplitude of about 22.5~. It is seen that there is an appreciable increase in period caused by nonlinear effects, but otherwise only a slight distortion of the motion in both coordinateso Again, it is likely that this distortion is due to coupling with the first mode. The solutions were not carried far enough to ascertain whether there is a decay or divergence of the oscillations. The fact that such may exist is not inconceivable, in view of the fact that the system is not necessarily conservative. It has been seen to be conservative when linearized with respect to small perturbations from the pseudo-static configuration. However, with imposition of the condition of constant shaft rotational velocity it is a driven system, and it is possible that nonlinear effects may result in a transfer of energy to or from it through the shaft. The results obtained indicate that, at least for the case considered, any such divergence or decay will be small and probably represent a negligible effect in comparison with aerodynamic effects in the case of an actual bladeo It is possible that a different choice of parameters or the introduction of built-in coning may produce a different result. This requires further investigation. On the basis of the present results it appears that the effect of Coriolis forces is likely to have a greater practical significance than the effect of nonlinearity, particularly since it does not depend upon the existance of large motions. This relates mainly to the problem of blade flutter, since the flutter phenomenon is highly sensitive to phase differences between motion in bending and torsiono The phase differences associated with the presence of Coriolis forces may conceivably alter the balance in the flutter problem sufficiently to change the conditions for flutter significantlyo 59

? I f!" ^aI t I iI (2 ~, -- 0 -~ o 0)r^ ^\\0 o.o- U) CY /3 0 0 rd II ~~of').E r IC ~~) NO ^\ ^Z- - Z ~ I-^ "^ ^ "^^^_ 5 "o, /^^^ 0e-U d, 0. 60// ~~~~. Cl) / -p~ 000 - T C) 6o F\QE

cQ cr X.K Wi.VI_ o''~s 0 LL z 22.. -'-. CD W o. o LU) roc'- - 0 - 0 0 0 0 0 0 00 C: L C) t0 -IJ O, o o~: - I./ I 6i $3~S0 8 3S0 61/ O 5

Other effects which have not been considered in the present study, but which can be expected to be of considerable importance in some cases, are those of nonlinearity of the torsional spring and of centrifugal tension on torsional stiffness and on pseudo-static deformation, as discussed in the Introduction. Their introduction into the present analysis should not result in undue complication and would represent an appropriate and desirable extension of the present work. 62

40 CONCLUDING REMARKS A practical numerical method, suitable for implementation on an automatic digital computer, has been developed for determining the natural vibration characteristics of twisted rotating and nonrotating blades in coupled bending and torsion. A limited numerical study indicates that the method is an efficient one for including the effects of bending-torsion coupling and pretwist. The nature of the coupling is complicated and a much more extensive parametric study would be needed in order to draw general conclusionso It can be said, however, that centrifugal force coupling can have an appreciable effect when there is a substantial offset of the mass axis from the elastic axis. In order to investigate some effects of nonlinearity and Coriolis forces in the rotating blade vibration problemi, a study has been made of a simple model with a small number of degrees of freedom. Computations performed on this model indicate the following. (1) There is an effect of centrifugal force, apart from the familiar centrifugal twisting moment, on the torsional deformation when the mass axis of the blade is offset from the elastic axis. It may, in some cases, modify the static deformation of the rotating blade substantially, and tends to introduce additional coupling between bending and torsion when the blade is vibrating, as discussed also in the case of the continuous blade. (2) The presence of Coriolis forces causes a phase difference between the bending and torsional oscillations which is equal to 900 when the mass and elastic axes are coincidento This phase difference decreases when the mass and elastic axes are not coincident, but remains substantial in the case of a natural mode of the model consisting primarily of bendingo (3) Nonlinear effects for large motions tend to change the natural frequencies of the system slightly and introduce some coupling between the natural vibration modes associated with solution of the linearized equations. A limited amount of results did not provide any evidence of decay or divergence of the free vibrations of the model. 65

APPENDIX A DIFFERENTIAL EQUATIONS OF MOTION The differential equations for free motion of a rotating twisted blade with offset mass and elastic axes are, from Ref. 2, with some changes in notation, - [GJe+TkA+EB1(i, )2], - EBa (8y cos +5z sin ) ) + TeA(Sy sin P-5z cos P) + 2pxe(-by sin P+5z cos P) + 92pe(sin )5y +f 2p[(k-k)cos 2P+eeo cos B]j + p(k+k )Z - pe(y sin P-z cos ) =+ (TkAt) - ro[(ki-k2)sin P cos P+eeo sin $] [EI1 cos2 P+EI2 sin2 P)6 + (EI2-EIl)sin P(cos P)y TeAo cos P - EB2'' sin " - (Tz) - (2 pxe cos P) + p(5z+eC cos ) = (TeA sin D)" + (Q2pxe sin ) (Al) + p(6z+e$ cos ) = (TeA sin D) + ( 2pxe sin ) [(EI2-EIi)sin P(cos P)6z + (EI1 sin2 P+EI2 cos2 p)3y + TeA0 sin D - EB2'P' cos P)] - (Ty)' + (92pxeo sin P) + ~2pe$ sin P + p(5y-ee sin P) - pSy = +(TeA cos P) + (U2pxe cos P)' + 12p(eO+e cos p) An explanation of the origin of the various terms in the equations is given in Ref. 2. The integrals which define the section constants B1 and B2 64

are given below = ( 2+2 k2-)( 2+2)dA B2 _ / (^+1 (A2)-k dA other symbols are defined in the ist of sybol. The coordinate system All other symbols are defined in the list of symbols. The coordinate system is as shown in Figo 2.1. It should be noted that Eqs. (Al) are for small displacements from the undeformed configuration of the blade when it is not rotating. The analysis of the present report linearizes the problem with respect to small displacements from the steady-state deformed configuration of the rotating blade. 65

APPENDIX B RESULTANT LOADINGS The resultant loadings per unit length in the x, ~, and J directions have been obtained in Ref. 2 for a rotating twisted blade with offset mass and elastic axes. The loads include the inertial, centrifugal and Coriolis force terms. In the notation of the present report they are Px = -p(ui-tu) - 22pSl sin P + 2fp62 cos 8 - pe' pe + pe2 peb1' - ~2pe62 - 2Q2pe~ sin p + ef px pn = -p52 + ftp(-81 sin P cos P+52 cos2 P+eo cos P) - 2S2p cos P + fpe cos2 P - rpe~ sin P cos B - Speo$ sin $ - Q2pe$ sin P cos P + 2Qpe cos 2(-lft+G2) Dp = -p65 - Sf p(-51 sin2 P+62 sin Pcos P+esin P) + 2npip sin P - pe - ft2pe sin P cos P + f2peo sin2 - 2npe sin 2(-61t'+52) (Bl) - Spe0o cos - 22pe$ cos2, qx = -Qpe [(-56 sin P+52 cos P+eo) sin P + eo$ cos ] + pe(-51+2Mu sin P) - fQ [(I-I~) sin ~ cos ~ + (It-IQ) cos 2p] - (TI+IT)0 - 2(l:-IT)(-8. sin p-6.p'cos + cos P-52P' sin p)sin P cos P ~Ir2.2 pI2 CS2 p\(' " + 2( sin2 I cos )( cos P-1 sin sin - P+2P' cos )' = ^I4(61+62') - I1(ii+ 21') + 2QI1 cos P qf = -2pe(x+u) + peU - 2pen(-1. sin P+g2 cos P) _- +I 1.'i + n2I 2 + I88' - Ig82 + 2nIZ sin P 66

In the following, we eliminate terms in Px which are dependent on displacement variables and their derivatives, since these lead to nonlinearities in subsequent analysis, and terms in all force and moment expressions which involve u and its derivatives and first derivatives of the remaining displacement variables. We also eliminate terms involving I, since these terms arise when 51 and 52 are derivatives referred to the axes T and. rotating about the x axis. In the lumped parameter treatment, 51 and b5 can be considered to be derivatives with respect to locally fixed axeso Px = 2 px P = -PA2 -,2p sin P (cos PB1 + 2p (cos2 A52 n2p sin P(2e cos P+eo)s + cp cos P(e cos P+eo) P = -P6l + n2P (sin2 i - 22p sin (cos 5) 62 - pea + p 21e(sin2 p-cos2 p) - eo Cos - a2p sin P(e cos P+eo) (B2) qx = -pel 1 + Q2pe (sin2 05 - 2pe sin P (cos f62 -~ s peeo cos P+(I~-Ii) cos 2p - (I+I )Z - 2 sin P~ peeo+(I-IO) cos P q = Ini2 In I i f 2 i I ( P q = 2I I 1 - 2 - s pex Now, if we consider the matrix equation Ai(N = [F]NfA)N (B5) where (A) is 67

vi M1 Va (A) = M2 (B4) M2 and (A)N refers to the value of these quantities at station N just outboard of the mass, and [A)N refers to the values just inboard of the mass, it follows that (N) (N) +) + (N)(N) = VN) + p)(N) ((+2 si2 )6(N) - p(N) (N)2 c i N) + p(N)(N) [e(N)([a+(sin2 p-cos2 P)2}+eo02 cos p] (N) p(N) e(N)2 sin P(e(N)+e(N) cos ) (B5) A (N) (N ( + (N) (N) (N) (N)(N) Ml = Mq ~ - p( I t(N)' (N) 61 = 61 A (N) (N) 61 = 61 68

A (N) (N) + N(N)( V2 V2 +p = (N) - p(N)(N)f2 sin p (cos P)(N) +p(^)^(c)(^+ 2 (~) + (N) (N) ( 2+ C S2 (N) p(N) (N)~? sin 3(e(N)2 cos P+e))) i,(N) (N)s2,..,? (N)+ (N) + p (N) (N) cos (e ( ( cos ) A (N) (N) (N) (N) (N) (N) (N) Me Me + q I + Px (N) (N) (N)x(N)2 (N) (N) (N) (N) ( 2 ) (N)(N)) o^p +! (B)2f2 = M2 -p I x. — 2 p I x.e 2~ (u-+22)b2 Av(N) = t(N) (B5) ^(N) (N) (N) (N) Q = Q (N) po )o= sin2 ) (N)( p (N)) cossin 2 Q (N) (N) (N)(w+N) s in2 r- (N) [(II+i)((N);(N)w2 + (I-I.)(N)(cos2 5-sin2 )p(N)/2 p (N)e(N) (N)eo (N)2 cos ] (N) $(N) = (N) 69

APPENDIX C DEFORMATION OF A BLADE SEGMENT Consider a segment of a weightless beam for which the values of the moment, torque, shear, and tension at a station N are given as Ml(N), M2(N), Q(N), V1, V2 and T(N) Then moments and torques at other points along the segment are Ml = M1(N) + V1(N)s + T(N)b1 U(N)$ M2 = M2(N) + V2(N)s + + T(N) 2 + W(N) (C1) Q = Q(N) + ((N) where U(N) and W(N) are the contributions of the centrifugal force coupling as explained in Appendix D, and s is the longitudinal coordinate measured from station N toward the root. These same quantities in terms of 51 and 52 for a twisted blade are Mi = EIl 1+2p's2+P"S2-(P') F M = EI2 12-2p'51-5P6-(')22 - TeA - EB21'' (C2) Q = LGJe+Tk2+EBl('I)2] I - Tk2' + EB2P' [2-2p'51 - P"-(')2 For a straight segment Eqo (C2) reduces to M1 = EIl1t M2 = EI252 - TeA - EB2B'' Q = - [GJe+Tk+EBl(, )2] $' - TkA', + EB2(' )52 70

Equations (C1) and (C3) may be considered and put in the form 65 - al15 - a2 b = bls + b2 82 - a352 - a4' - a50 = bss + b4 (C4) a651 a75s + a8sa = -bs where (N) VI(N) al = ( b = EI) u(N) Mj(N) EIl EI1 T(N) V (N) a3 b3 = - EI2 Elm a4 EBB___ M2(N) a4' ^E b4 El2 a5 = W (N) Q(N) EI2 a6 = U a7 = EB2Pa a8 = A(N) = GJe + Tk + EBi(P')2 These equations may be solved as they stand, however, considerable simplification can be achieved at a modest sacrifice in accuracy by replacing the term a5$ by a5$(N). This modifies the second of Eqs. (C4) to read 56 a365 - a4A' = b3s + (C5) 71

where b6 = W(N) Q(N) 2(N) EI2 EI2 The characteristic equation for the new set becomes foP4 + fl2 + f2 = 0 (C6) where fo = (a8-a4a7) fi = a2a6 - as(al+a3) + ala4a7 f2 = ala3a8 - a2a3a6 The roots of this equation are then p =!, = - _2 + ( - 2 2 (C7) i'3 2 2-fo fo 1- N When the solution is carried out, with the elements of (A)(N) as initial values, and s is taken equal to 2, the following form for the solution results 10:(N+l) 10 1(N+) =. E E3jAj j=l 10 5(+1 = E4jAj j=l 10,(N+l) E (c8) 62 = EjAj (C8) j=l 72

10 (N+l) 10 2 = E8jAj j=l (N+) (c8) EloojAj j =1 where the Eij are presented in Eq. (2.6) in the body of the report and Aj are the elements of (A}(N) given in Eq. (21o). The remaining Eij elements are found from the following equations which apply across each bay (N+l) (N) V1 = V1 (N+l) (N) (N) M=' = M1 + V1 ~ (N+l) () (c9) V2 ) = V2 (C9) (N+l) (N) (N) M2e = M2 + V2 ~ Q(N+l) (N) It should be noted that the bending moment and torque quantities in Eqs. (C9) by definition do not include the contributions of the centrifugal force displaced to the elastic axis and of centrifugal couplingo These contributions are introduced separately through satisfaction of Eqs. (C1) in the solution for the deformation variables. 73

APPENDIX D CENTRIFUGAL FORCE COUPLING As shown in Ref. 2, there is a type of coupling betweep bending and torsion associated with the presence of centrifugal forces. Explicit consideration must be given to the derivation of the terms associated with this coupling. If xl is used to denote the station where centrifugal force is acting and x the station where bending moment is measured, the components of bending moment associated with offset of mass center from the elastic axis of the rotating blade may be written as follows: MI = - cos(P+O) | 2plxlel sin(Pl+lj)dxj R + sin(P+~) | 2plxlel cos(pl+lj)dxj (D1) R Ms = - sin(p+0) / plxlei cos (l+0j)dx R cos(p+O) R Q2plxlel cos(P+L)dx1 where subscript 1 refers to values at x1. Assuming 0 to be a small angle and eliminating higher order terms, Eqs. (Dl) become, R R Ml = - cos X e plxlel sin Pldxl + 0 sin P Q2plXlel sin Pldxl R cos P S 2pixleljl cos Pldxl R R + sin x Q p cos 1dpxlel cos dx + cos e cos dxl - sin P / n2pixleil$ sin Pldxl (D2) 74

R R M2 = - sin X s 2plxlel sin Pldxl - cos P/ 2plxlel sin Pldxl -sin XP 2plxlel1C cos Pldxl R R - cos p Q 2plxlel cos Pldxl + ~ sin P 02plxlel cos Pldxl x R + cos p ~ 2plxlell sin Pldxl (D2) In each case, the first and fourth terms represent steady-state moments. An examination of Appendix B shows that the term -2pex in q~ will give rise to these moments. They are taken into account through the element d6 in the [(d matrixo The third and sixth terms in each component represent the effect of torsional displacement of the blade mass on bending about the torsionally undisplaced positions of the T and ~ axes in the M; and M2 components respectively. This effect can be taken into account by an appropriate modification of the [F] matrix, incorporating a change in bending moment across each mass given by AM1 = - s22papxe~ (D5) EM' n2plxeW (D3) yielding the element F21o = - 2pxe (D4) The second and fifth terms represent the effect of centrifugal forces acting on the torsionally undisplaced masses between x and R on bending about the torsionally displaced r and 5 axes in the Ma and M2 components respectively. They must be taken into account in the development of the [E] matrix. In terms of the lumped mass model, the contribution to the bending moments in the bay between the nth and (n+l)th masses is 75

M1 = (in Pn ipixie sin i+cos Pn 2pi ixiei cos i =1 i =1 (N) (D5) o n n' M2 = pi-cos n PiiXiei sin Pi+sin Pn 22pilixiei cosi) (N) = W. There is correspondingly an effect of bending on torque. It is associated with the fact that with a bending slope 51 at station x the centrifugal force f22p1xldx on an element dxl at station xl outboard of station x has a component -22plxldxlj1 normal to the tr-axis in the plane of the cross section at station x. The moment arm of this force about the elastic axis is el cos(B-P1), so that the contribution to the torque from this source may be written, R Q = - 61 J eplxlel cos(P-Pl)dxl x R R - (sin P Plxlel sin lidxl+cos p Plxliei cos Pdx (D6) Applying this result to the lumped mass model, the contribution to the torque in the bay between the nth and (n+l)th masses is Q = n n = - t(i n 2pikixiei sin p +cos n Z) a2piAixiei cos P 31 i=l i=l (D7) and is taken into account in the development of the [E] matrix in Appendix C. 76

REFERENCES 1. Isakson, G., and Eisley, Jo G.: Natural Frequencies in Bending of Twisted Rotating Blades. NASA TN D-371, March, 1960. 2. Houbolt, Jo C., and Brooks, G. Wo: Differential Equations of Motion for Combined Flapwise Bending, Chordwise Bending, and Torsion of Twisted Nonuniform Rotor Bladeso NACA Repto 1346. (Supersedes NACA TN 3905.) 3. Targoff, W.: The Bending Vibrations of a Twisted Rotating Beamo Proco Third Midwestern Conf. on Solid Mecho (Ann Arbor, Micho, 1956), 1957, pp. 177-194; (Also WADC Tech. Repto 56-27 ) 4. Brady, W. G., and Targoff, W. P.: Uncoupled Torsional Vibrations of a Thin, Twisted, Rotating Beam. WADC Techo Repto 56-501, June, 1957. (ASTIA Document Nr. AD 1307860) 5. Niedenfuhr, Fo Wo: On the Possibility of Aeroelastic Reversal of Propeller Blades. Jour. Aero. Sciences, Vol. 22, No 6, June, 1955, PP458-4400 6. Bogdanoff, Jo Lo, and Horner, J. T.: Torsional Vibration of Rotating Twisted Bars. Jour. Aero. Sciences, Vol. 23, No. 4, April, 1956, pp. 303-305o 7. Bogdanoff, Jo L.: Influence of Secondary Inertia Terms on Natural Frequencies of Rotating Beams. Jouro Appo Mecho, Vol. 22, No. 4, December, 1955, ppo 587-591. 8. Doolin, Brian Fo: The Application of Matrix Methods to Coordinate Transformations Occurring in Systems Studies Involving Large Motions of Aircraft. NACA TN 3968, May, 1957o 77

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