(SER ~ Report 1848 4 9°: y A Ca ate ~ errwvencs eA Hing TON? [ry a: Say avamancres | a bhody yy] | Wooc iia j | OS ASitution HYDROMECHANICS FORCED VIBRATION OF BEAMS AND FLUTTER OF BEAM-FOIL SYSTEMS O se AERODYNAMICS a PPS OO OS ESS ae oO mB E. H. Kennard i STRUCTURAL = Ralph C. Leibowitz me) MECHANICS. | j fe) APPLIED MATHEMATICS ACOUSTICS AND VIBRATION LABORATORY RESEARCH AND DEVELOPMENT REPORT oO — Ce BRATION = / NG 25 October 1964 Report 1848 $a Wb: [$48 FORCED VIBRATION OF BEAMS AND FLUTTER OF BEAM-FOIL SYSTEMS by E. H. Kennard and Ralph C. Leibowitz October 1964 Report 1848 (ee) ABSTRACT TABLE OF CONTENTS IN NSOWDNUICAMUONM —5000600060000000600000606000000000000000000000000 HARMONIC END FORCING OF AN UNDAMPED UNIFORM BEAM WITH SHEAR WINMZING 660000000000000000060000000000000000000060000000006000 HARMONIC END FORCING OF AN UNDAMPED UNIFORM BEAM WITHOUT SHUN WINNING 6160000000000009000000600000000000000000000000000 HARMONIC FORCING AT AN INTERMEDIATE POINT HARMONIC END FORCING OF A UNIFORM BEAM HAVING EXTERNAL AND INTERNAL DAMPING FLUTTER OF A RIGID FOIL ATTACHED TO A UNIFORM BEAM 6.1 DERIVATION OF EQUATIONS OF MOTION 6.2 CRITICAL FLUTTER SPEED 6.3 THE SINGULAR CASE: 6.4 DISCUSSION OF DAMPING EFFECTS 6.5 THREE © oHoll 6.52 5553) 6.6 SIMILITUDE TWO SIMPLE e@eceeeeeeeeeeoece eee ee eee eee ee ee eee eee eee ee ee eee eececeeenocee e@oceeseeeeeeceee eee e eee eee $ 785784585 kjk 3 O)5 e0HNClh 1 & © oao6 SPECIAL CASES iH} Q iT] (o) wv DAMPING DUE TO LIFT ONLY (c, 5 Seva eee HOLE RIGIDLY MOUNTED ON LHR BEAM) 25s LOW-FREQUENCY APPROXIMATION e@eeeeeeee eee eee eee eee eee ee eee eee eeeee eee eo ee 8 CASES OF FLUTTER 7.1 FOIL ATTACHED ELASTICALLY WITH DAMPING TO AN IMMOBILE BASE 7.2 FOIL ELASTICALLY MOUNTED WITH DAMPING ON A FREE RIGID BODY SUMMARY REFERENCES eoeeeeees ee eeeeeeeeeeeee2 ee ee8 8882822282222 228 Cee See ee ii Page 60 Figure Figure Figure Fi gure Fi gure LIST OF FIGURES Page Noncesiandl Momentsmonnanbeamilmnrnaannseren ease eisai ae Forces and Moments on a Beam Slice .......cccccccccccee CL Forces on and Motions of a Beam-Foil System Immersed dat ey (Mower FAALGN ) Miarnem eae ata crastel ae te asian ele hie ctieleramurs ey > Forces on and Motions of a Foil Attached Elastically to an Immobile Base 73} Forces on and Motions of a Foil Elastically Mounted with Damping on a Free Rigid Body atatal NOTATION A Cross-sectional area of stressed material Influence or response coefficients (for certain cases these quantities will have a prime or a bar above them) B Lift constant of hydrofoil (equals Tbs p) b Semichord length of hydrofoil Inertia coefficients (for certain cases these quantities will have a prime or a bar above them) C Lift constant of hydrofoil (equals B (3b Soe cosh qk: a constant Pine Pio? Dor? a0 © Internal damping constant; cos a2 CHC Still-water damping constants (structural damping plus viscous damping at S = 0) ¢c External damping constant External damping constant dx Thickness of beam slice (differential length) E Young's modulus of elasticity; lift constant of hydrofoil (cquals B (3 + e)’) EI Flexural rigidity of beam e A constant; base of natural logarithms (2.718.....); distance from midchord line to axis of rotation positive toward approaching stream F Force on beam Fr Oscillatory lift force on hydrofoil G Modulus of elasticity in shear; external moment or couple on beam positive in same direction of positive @ or 2¥. ox iv Equals G for shear modulus Distance from axis of rotation to effective center of mass of hydrofoil, positive when center of mass is displaced from axis toward the approaching stream, as in Figure 3 Distance from axis of rotation to center of mass of rigid body, positive toward approaching stream, as in Figure 5 Area moment of inertia of beam section about axis perpendicular to the plane of bending; effective moment of inertia of foil about axis through its center of mass and parallel to axis of rotation (used for foil not attached to beam) Effective moment of inertia of foil as just defined when foil is attached to a beam; similar moment of a rigid body Equals 15 + hom for foil attached to a beam y—-1 Numerical factor dependent upon geometry of beam cross section; K <1 Shear rigidity Elastic constants of structure connecting foil to beam; k is also the reduced frequency ee A constant in the lift formula assumed equal to 5 b- e5 distance from effective center of mass of foil to lift line Length of beam Length of hydrofoil Total moment acting on cross section of beam material lying toward x = 0, taken positive in same direction as G or 6 Moment due to bending of beam Moment due to shear warping of beam Total moment on hydrofoil about axis of rotation Effective mass of foil and of a rigid body, respectively (including virtual mass due to surrounding fluid) Shear force on beam taken positive with F and v 2 af aaa V Equals =~ ie NIVGU SIL 6e : aVVi+e°-¢ E respectively Uniform speed of stream of fluid; sinh Ink Critical flutter speed Sin q,2 Time Displacement of beam, positive in same direction as F; similar displacement of foil at axis of rotation Equals v at x = 0, or at line of attachment of foil Rectangular coordinates with x axis always parallel to the beam axis and y along a perpendicular principal axis When vibrations are due to bending flexibility only, (KAG =~) y is the equivalent rotation of a cross section of the beam about an axis perpendicular to the xy-plane due to bending and rigid-body motion; positive in direction of positive a x For a vibrating beam with finite bending and shear flexibility, y is no longer the slope of the beam but represents an equivalent rotation of the cross section about an axis perpendicular to the xy-plane. For the meaning of equivalent rotation, see Appendix A and Equation [A27] of Reference 11 Bending strain; a number allowing for effect of varia- tion in E or shape of cross section Specific gravity of fluid D| 1 Ppa Damping constant; equals — ——— (See Section 7) AZ PoA Equals ae angle of rotation of hydrofoil about axis of rotation lying in plane of foil and parallel to its length, positive in same direction as G, and measured from zero when both foil and beam are in their neutral positions Equals 3 at x = 0; local rotation of beam; angle of rotation of a rigid body; positive like 6 + iq); or 2 a8 ratio of length of any beam to standard beam Effective mass per unit length of beam (including an appropriate allowance for virtual mass of surrounding water) ue Equals to =a mite qe Density of fluid bathing the foil and density of pure water, respectively Bending stress; equals 2KAG A number! allowing for the effect of a change of either beam density or cross-sectional area of beam Equals Go - iwe) Circular frequency of vibration a" mn ‘ + {nary : “a wah hes rTM he au? nanan as Ae = * shy ; lat i os AERP airing a =n & URRY: * ary + yp ‘ sue > ; :: POE Tr deal tee y i Sys ae i + | oaae « a oY Ae oes 4 te 5 = 7 sine r at ’ < " ¢ uy ee x | cua aE. Voune, Wa ees ¥s ’ ‘ ect ete Pek bee i H 7 = s ‘ | e ABSTRACT Equations are developed for two-dimensional harmonic trans- verse forcing at any point of an undamped uniform beam having bending and shearing flexibilities and for harmonic end forcing of a uniform beam having also external and internal damping. Variations of beam influence and inertia coefficients with fre- quency are discussed for an undamped beam without shear warping. Equations are also developed for the determination of the flutter speed of a rigid hydrofoil flexibly attached to such a beam. The flutter process is illustrated by a detailed discus- sion of two simple cases of flutter. 1. INTRODUCTION Sea trials of USS FOREST SHERMAN (DD 931) indicated that during a steady horizontal maneuver severe vibrations were transmitted by the rudders to the wall To anticipate the occurrence of severe vibrations and possible control-surface flutter within the range of ship operating speeds, the authors undertook a series of studies of the flutter phenomenon. A theory was advanced for treating the vibration character- istics of a control surface (e.g., rudder or diving plane) hull system subject to hydrodynamic forces on the control SwiERACESo 2” This theory has I References are listed on page 92. This theory has been applied to USS ALBACORE (AGSS 569) and to the motor gunboat PGM. The results for these two cases will be reported separately. * been shown to have some degree of veririeatione The method of determina- tion of the hydroelastic parameters for these control surfaces is given in References 5, 6, and 7. Methods for determining certain damping terms from observations, originally omitted from the flutter equations in Reference 2, can now be included and are given in Reference 8. Related studies of the static and dynamic loads on the rudder of a ship during a steady horizontal maneuver, which is also of interest in the treatment of flutter, were undertaken in References 9 and 10. The Bureau of Ships, recognizing the parallel need for the explora- tion of the possibility of the occurrence of flutter in hydrofoil craft, requested the David Taylor Model Basin to undertake a similar hydro- elastic study for these craft. To achieve this objective, in this report equations are derived for predicting the critical flutter speed of a rigid foil flexibly attached to a uniform mass-elastic (i.e., nonrigid) free-free beam immersed in a fluid moving with uniform velocity. The analysis includes the two-dimensional quasi-steady expression for hydrodynamic force and moment on the foil. The relative influences upon the vibrations, critical flutter speeds, and frequencies of the values of the various parameters of the beam-foil system are discussed in detail. Certain special cases are also considered. A survey of the work performed in this report is given in the Summary. 2. HARMONIC END FORCING OF UNDAMPED UNIFORM BEAM WITH SHEAR WARPING Consider a uniform beam that is maintained in vibration by a trans-— verse force F and a couple G, acting in a principal plane containing the principal axes of all beam cross sections; see p. 188 of Reference 11. It will be assumed that F and G are both harmonic functions of the time; nonharmonic forcing is much more complicated. In developing the theory, F and G will be assumed to act at one end of the beam, taken as x = 0; generalization for other positions will be given later. Furthermore, F and G will be assumed to vary at the same frequency and in phase, so that F = A sin ut; G = B sin wt [la,b ] in terms of constants A and B. The resulting formulas can then be applied to a case in which the frequencies of F and G differ by first putting B = 0, then A = 0, and adding the two motions thus obtained. If, on the other hand, F and G have the same frequency but differ in phase (F = A sin wt, G = B sin(wt + $)),) it is readily seen in the same way that the general formulas are all valid as they stand (hence, for instance, in Equations [12a,b] v, and 8, are intermediate in phase between F and G). Let v(x, t) denote the displacement of the beam, positive in the same direction as F, and let G be positive in the same direction as positive dv/dx. F and G are assumed to be applied in such a way that significant distortton of the end of the beam, near x = 0, is avoided except for the normal distortions of cross sections accompanying bending and shear. Let the other end of the beam, at x = 2, be entirely free. The general harmonic case of a transverse force and couple acting on one end of the beam can be resolved into two sets of forces and couples like the set just described, with the two sets acting in perpendicular prinetpal planes. Although nonharmonte forcing is much more complicated, solutions to such forcing can be found using Fourier Series or Fourier Integral methods. Basic equations for the harmonic forcing are developed, including the shear-warping effect; however, rotary inertia is ignored because of its small effect in practice (see p. 187 of Reference 11). Let M denote the moment acting across any cross section on material lying toward x = 0, taken positive in the same direction as Gor 06, where @ denotes 3v/dx for convenience, P the corresponding shear force taken positive with F and v, and y the mass of the beam per unit length. Then the appropriate differential equations are as follows (see Equations [2.12] to [2.15] in Reference 11, where V = -P because V was taken positive toward negative v, and the Iz term is to be omitted) ; the reader can also find the basis for these equations in many texts on vibration theory: 2 ap AP 9M HOUR bs 8 BO (Bade. at ov oY. M =—=y + 20P; == 2c,d Bip ISD a ame (Bea) alt Here Bo is written for the constant usually denoted by KAG, the shear rigidity, and this parameter as well as yu and EI are assumed untform along the beam. In harmonic vibration, Equation [2a] becomes: Ci ancl WOW Si eee An equation containing M alone is easily deduced, thus: Ly ELIA omen = 9x >? oo Way ye 5 ox ov GP BO ON av Fe aoM Soa ORE See ie ol = Tar oj v ox x ax- ax° and, substituting this last expression for dy/ax in Equation [2d] and then substituting for v yields: ee ee eee EI wwe y 2 ox OX or 4 2 2 aM 4 fe oM ww [3a] 2ouw — ~ - —— M=0 axe Ox° ae * For convenience, introduce q and &, defined thus: 2 4 ww wu 2 Qh as Fl >? Se A o = oEIq {4a,b] q aa XS 2 Cpa: : Then, substituting Mee sin wt and also 2oyw = 2q°& into Equation [3a], and dividing through by M, yields the following equation as the necessary and sufficient condition for ): 4 4 og2 ex - gt =0 Thus, by solving c = a6" & ot y ape + at and taking first the minus and then the plus sign before the radical and again extracting the square root, it is found that either A= + iq,> or A= 2 G56 where as = Vi +e ar (3 8 a, -aV Vive? - [5a,b] *Compare with equations in Appendix [D2] of Reference 11. All square roots are meant to be postttve. The values i = tel MelCladanto sin a4* and cos q,x as alternative real factors, whereas \ = =z Gl, leads to sinh 45x and cosh q5x: The following useful auxiliary formulas are easily obtained: e e 2 2 2 2 2 aj + a5 = 2q Tet > a dae Ore 2a°E; ae al [6a,b,c] oe ae ae + : 1 2 1 oe She tata 3 a wr a = 2q! (i = BE) : [6d,e] sy) *SI5) 12 Thus, in harmonic motion, M can be expressed in terms of the four independent solutions of Equation [3a] as follows: sin a,x + €, cos q,x + e3 sinh ox + €) cosh aX fival in which each coefficient cr is the product of sin wt and an arbitrary constant. Corresponding series for P, v, and dv/dx or 96 are easily obtained by substitution for M in equations previously written, differentiating the series for v to obtain 6. , At x= 2, P=M=0; whereas at x = 0, P = -F and M= -G, since F and G represent actions on material lying toward positive x from the terminal cross section and P and M represent actions toward negative x. # Thus, the boundary conditions for the present problem are (since P = -0M/dx): at x = 0 But F 9 M = -G ax at x = 2 ve 0) ; M=0 x Substituting the series for M and then setting first x = 0 and then x = 2 gives the four equations: * Compare with Equations [D8] of Reference 11. a@1 + %e3 = Tie Peo at cls -G [8a,b] cos g_.k - 1&5 sin q,% +) qe (cosh) qey q€), sinh q 2 = 0 [8c] sical 1 Oe 2 2 e, sin q,% + €5 cos a, + e3 sinh qo2 + e) cosh qo2 = 0 [8a] These equations can now be solved for e. It appears to be more useful, however, to relate F and G to values of v and av/ax at x = 0, denoted, respectively, by Ma and 86° Since ww°v = EIqtv = 9°M/ax-, 4 eu 3 ine) We aM : EIq/e = o-M fe) 2) fo) 9 3 ax /x = 0 2S) [set ©) or, substituting the series for M; 4 2 2 EIq YW. 2 eis + qe), [9a] 3 EIq "8, = -q]e) + a3e 3 [9b ] Solving Equations [8a,b] and [9a,b] for e gives: 2 e 2 h + = a, (a; qn )e a5F - EIq’e 2 2 LR) 4 (ay ar a5 e, = 5S se EIq ie 2 2 2 4 + = + an(ay + a5)e, = ajF + Elq 6, Pies: eel vie hot? 4 (a5 + a, )e), = -a56 + EIq va These values of e may then be substituted into Equations [8c,d] and eae ‘ 2 F ‘ multiplied through for convenience by q, + ase Hereafter, it will also be convenient to shorten the notation by writing: s = sin a2 B ec = cos qa, ° S = sinh Ink g C = cosh Ink Then Equations [8c,da], taken in reverse order, may be written: \ S \ aj 95 (Come; oS 2 \tie @. = (3s + — Aa Ss 6@))]] = zs 12 De g FIq° Ge HES = Ea) = = jp. (Ca o sS - l-c ae, on Da Ze E EIq2 Cop = ii. DSRS or SN sare Da =1- cC - E—sS * Compare with Equations [A67a,b] of Reference 11 with Y = r= 0. =| + ° = F Pons ” Pipisp © 2 Sane > alls [13a,b] Z V 2 Daa 1+eE tanec + q,cS) EIq 2 [sS + E(1-cC)] EIq 2 [(a#2e)sS - e(1-cc)] EIq° 2 1+e° (& SG so i cS) EIq ah qd o. Sis CC Pv 285 & Deo Here both q and & increase as w increases. ¢ is a dimensionless quantity and may serve as a relative measure of the effect of shear warping on the vibrations of the beam. The increase of & with tnereasing w means that the effect of shear warping increases with increasing frequency, as ts well known in other connections. Using a common terminology, the coefficients a B51> and a UL? “no? 09 may be called influence coeffictents for forcing of the beam at one end. KE Similarly bp by5, D,..> and oe might be called inertta coeffictents. al Because of the factors SHIN Lt Gini cos q,£in the formulas, the values of both sets of coefficients, as w increases from 0, range in roughly cyclic fashion over all values between -~ and +™. These coeffictents may be of use tn treating the vibrations of a system composed of a untform beam attached at one end to another structure. * See pp. 9 and 180 of Reference 11. we WAS ue refer to the influence of forces in producing vibrational aL displacements; the b's to the effects of beam mass in making F and G ij necessary. 3. HARMONIC END FORCING OF AN UNDAMPED UNIFORM BEAM WITHOUT SHEAR WARPING Because of the two independent parameters q and &€ in the formulas, the discussion of special cases is complicated. Accordingly, the varia- tion of the coefficients with w will be discussed in greater detail only in the case of —& = 0; that is, when shear warping is neglected. When € = O, the formulas become, with two useful additions: Woe ai F + aG 5 eae AyoF + a6 [lh4a,b] sC-cS 1 sS 1 ial 1-cC EIq3 12 Ail laa BIg” sC+cS 1 d £ l+cC 1 ane ee Ng ok SS ENS oO pe = Elq > = 22 1-cC q i 22 2 ~ TES (er) ?ql Se Died, § © 2 Baad, ? Paats [15a,b] C+cS 3 ss 2 Da. 2 oe mee 8 ban = = - EIq 11 we 12 ~ “Bil 1+cC sC-cS 2 1-cC 2h byl = aeces eld sil Din Ba 5- Oe imines (EI) q The added formulas are obtained with the use of Formula [1lb]. It will be noted that when — = 0, a GL. Ereel 1) = Ip an 9 Ae a ABs As qg increases, sinh qg and cosh gg soon become large and nearly equal; for example, sinh 3 = 10.018, cosh 3 = 10.068. Hence, at least from q2 = 2n upward, the following simplified formulas may be sufficiently accurate; they are obtained by dividing numerator and denominator by C and then replacing S/C by unity, provided — = 0: 10 s-c 1 s 1 EN he rere SV La ies CL he 11% ¢-(1/C) pqq3 12 21 e-(1/C) ite Spector Pa, 2 eng hee (16) aera Sp - 7 (R/C) me >? Sivas ~ Be eC) ga2e0 If also le| >> 1/C, the still simpler approximations become available: ee aL ee Baa 2 (een of oi) 2 @€ Se. 8 stem al 11 2 EIq> nt2 eal te" a 1 2 ]h ars = QR + —-> _: Be ——————————— Bop = (tan a2 +1) FT7 3 8418507840 (enya! Several interesting cases are now discussed in the order of increasing q&%, assuming that & = 0. (a) Uniform rigid beam, q& << 1: Smallness of q may arise from smallness of etther 2 or w, the latter occurring in q. Then use may be made of the series (note that when & = 0, Gly SC q): cos qk n il} @ dl 4 sin gf = at - Z(qt)>...3¢ 1 = 5(a2) + dy (ak) +: Be sal y sinh q% = qk + Hat) >... 50 cosh qk = 1 + =(a2) + ay ae n ul Keeping only the lowest power of qk: s: ‘ 4 i i eG Se Bs Lo es = (ql) 8 sS = (qf) os a2 3 SCitcSu= eae: sc - cS = 3 (a4) : : : : 2 In this case, it may be enlightening to return also to WW by sub- 4 we stituting q = 2S Then, provided 4% << 1: EI 11 = = — 5 a = a = —_——_— = 5 11 EIg'tg raft M2 2 mgt? moar sige C a ay Nyala 220 Ig $23 ussur 2 2 2 nt 3 2 =a Qw b = — g . = Q nal Se TM OS ae a ae 5 : J eS ae In Poa? uL is the total mass of the beam; in Pes 3 uX is its moment of inertia as a rigid body rotating about one end; in Pipe zoe? equals mass times distance from either end of the beam to the center of mass. The validity of Equations [15a,b] for F and G in this approximation is easily verified from elementary mechanics. Thus, when qf << 1, the beam behaves, as it should, approximately like a uniform rigid rod of length 2 and mass uk, 2 Dn eee As qk +0, a5? B15? pe and 811% - a5 all become infinite, 2 whereas b__, b b and b Db - bd all go to zero. aly ela per in 22 12 é In the following discussion of other cases, the notation is often shortened by denoting the first fraction in each formula preceding the EI fraction by the symbol for the coefficient with a bar over it. For sC - cS =~ a CC. example , a = - however, is fixed and cannot be varied by changing F and G. For, since a= 0, 2 811859 - ayo = O and, hence: 812 800 PR ay aee =- 0.735 q 11 12 Thus, the ratio of the right-hand members of Equations [l4a,b] is -0.735 q for all finite values of F and G; it follows that ie = -0.735 q also. If, however, the ratio F/G = + 0.735 q, then F/G = - 85/8415 - Bno/B155 so that a,,F +a,_G=a Ft a, 3 EN ee ALT: 12 12 c 0 and Equations [lha,b] give Ue » =o 2 The beam is then vibrating as if it were built-in at x =0. If it actually is built-in, F/G or 0.735 q is the ratio of the reactions -F and -G on the supporting structure. Further illumination results from considering what happens when cC is merely very close to -1l. Then b bj, and db are finite but large so dal” 22 that, in general, large forces are required to produce arbitrarily assigned values of ve and Oo: The same conclusion can be derived from * The equation cC = -1 is assumed to be exactly true, the value of qk being within 0.001 of that stated. Similarly, the same is assumed in other cases. 13 Equations [l4a,b] by noting that the result of solving [l4a] for F and substituting for F in [14>], or solving for G and substituting for G, is: : mine : : Since 841850 - B10 is almost zero when cC is close to -l1, moderate-sized F and G must leave Oi close to 855/814 and also to a /85- To produce considerably different values of 86/Vo requires large-sized F and G. (c) First sliding-end frequency: When qk,= 0.753n = 2.365, calculation shows that sC = - cS or (multiplied by cC) tan q% = - tanh q% and a5 = SsS58 8 Qi> = Oo (or 8 Ano = 0 Also, b,, = 0. If F = 0, then Equation [14>] gives 6, = 0. The beam is then vibrating as if free to slide transversely (aida I? 2 ©)) Ere se S Os however, it is constrained against rotation. Also, in this case, Vi cite AG and -G or -V,/815 represents the reaction upon the constraining structure. Different vibratory motions occur if F is not zero. (a) As q2 passes through 7 or 3.142, a, passes through zero and 12 changes sign (because sin q% does), and i> does likewise. Thus, at Op Sain ae ES Eh 6 ii" and 8, =a,,G. Calculation gives: 22 SS bhi se 8... 2 0.017 's Aea= O82 . 22 11 (e) First pin-end frequency: At q%= 1.2500 = 3.927, tan qk = tanh q 2% and 14 The interesting special case at this value of q% is a pin-ended beam vibrating with G= 0, v. = 0, and F= 84/810" If G # 0, then Vo = - 0.946 G/(EIq°). (f) Second built-in frequency: As q& increases toward 1.57, c is negative but decreases numerically until cC = -l. Because of the huge size of S and C, c must be very small; cC = -l at q2 = 1.4950 = 4.695, with Ch= OR OMin SS) = (C=542 7 tol three! fileures).) and ar = 26.9 ; a5 Se OT sis Bas = 27.8 0) AU=)0 whereas Boa 9 Dio» and Boo are all infinite. As in Case (b), it follows from 811855 - a = 0 that, in general, oN. is fixed at the value 815/811> or here at - 1.017 q. If, however, either F = G = 0 or F/G = 845/814 = + 1.017 q, then Vans oS as for a built-in beam. (@) A@gGhe ib0Ss 47523 Cs mia.Ss6,. Sa, = 55.6 11 12 2] (to 3 figures), and A = 1. (h) First free-free frequency: At (very nearly) q% = 1.5057 = 4.730, eC = 1, as for free-free vibration. Now, finally, it is the turn of a and es Coats s A to become infinite. Also, bjjb55 - bio = 0, hence b1/%y0 = Dy o/o0> so that the right-hand members of Equations [15a,b], if not zero, are in the ratio bi 10> and, if G # 0: F b a elle (cot q& + coth q%) q = 0.982 q Bio a finite quantity. Thus, when a beam is forced so that F = 0.982qG, the beam amplitude remains finite! An attempt to force with F/G in a a) different ratio, or with either F or G alone, will result in infinite amplitudes of vibration. With F = G = 0, however, the beam may execute a free vibration with b 8 —2 = - T= _ 9.982 g Vv b © 12 (i) At qt = 1.55n = 4.870: Enis Shpall ae = TOOr 3 ai 5.89, ane AES 1422, Ae qa increases further, the a's continue to decrease numerically. (3) Second slitding-end frequency: Just beyond q&’ = 1.75n or 5.50, tan q2 = = tanh q& and a4 Ho 26022 ajo = oO. Ano = QO, and A =- 1.02. The general situation is as in Case (b). (k) At qt = 2n = 6.283: a eo 1.004, B15 = 05 Ap = 1.004, and - 1.008. >| u During each further 27 range of qk (2m < qe < lh, kn < qh < 6m, etc.), a cycle of zero and infinite values of as occurs similar to that in the range 0 < q% < 2m except that during the first quadrant of each em range, two additional points occur that have no analog in the first 2n range. Successive points of interest are now spaced almost exactly 1/4 apart, and the peaks in the curve that contains the infinities become extremely narrow. The two additional points in the fifth quadrant and two other points illustrating the concentrated occurrence of large a's are as follows: 16 (1) Second ptn-end frequency: tan qk’ = tanh qf, qk = 2.251 = 7.07 (within 0.01). Here s = c = 0.707, € = 588, a3 0, 85 = - OOK on = 2,00, and NS os 100 Caiiuin OO). (m) At qt = 2.4900 = 7.823: a), = BS. Bae 32.7, a5 = 33-7, and RS o 1.08. (n) Second free-free frequency: cC = 1; q2 = 2.500n = 7.601 (within 0.001). Here a and a., are all infinite. In Wises, wacn iL? “Ae 20 Case (g), finite amplitudes may occur only if neither F nor G vanisies and F/G = he = 1.001 q, or if F=G=0. (0) Third built-in frequency: cC = -1; hence qt = 2.500257 = 7.655. H = 12 a. 2A. as 6h = 64 A= ; ere C = 1290 and 811 = 855 Bo 2,5 645, and A = O (exactly) = }. 1 To find q% and B)0> let eC = =<1; but C=>S, hence c = - 1290 = - 0.000775. Also c = -8, the slight angle beyond 2.5m or 7.85398, there- fore 8 = 0.000775 = 0.000247n and qk = 7.85398 + 0.000775 = 7.855 or om I plcOOOOs do IL . 5 SoS g- = (=) = - =(- S 6S. 2.500257. a), Se Ss (re ) 5 (-1290) = 645 ; Sy a peed is Nan en ve a any As zi Alternatively , B15 Takers sS = 5 S = 5 sinh 7.855 because cC = -l, BS ae Ny Soa 7665S 22 085D,) wk 7655 _ s=1. Hence, a), = 5 (e -e ) 52 = 1290 and Bio = 645. (9) Ae oO & 208i = 7.6659 © S W522. a = - 32.1, B15 S Shoe, B55 == 30.2, and A= - 0.95. Above q2 = 3, at least, curves representing Bi4> 15> and B55 lie close to those defined by an = tan qk -l, Qe = - tan q2, and iT B55 = tan q2+1. The effect of the small omitted term 1/C is chiefly to shift the position of the infinite values very slightly along the q2 axis, by 4qg = + 1/C, since, when q& << 1, cos qg - (1/C) = cos [qe + (1/C)]. At the free-free frequencies defined by eC = 1, resonance may be said to occur, since the beam can vibrate at these frequencies even if F=G=0. In the one-dimensional cases, an attempt to force at a reso- nant frequency necessarily results in an infinite amplitude. In a two- dimensional (or two degrees of freedom, (v,6@)) case like the case under discussion, however, and presumably in any multidimensional case, finite amplitudes will occur, provided the applied reactions are in certain ratios to each other. Conversely, at the built-in frequencies defined by cC = -1, there is "antiresonance.'' In a one-dimensional case of this kind, the forced amplitude cannot be budged from zero at the forcing point; however, in similar multidimensional cases, the amplitudes of displacement or rotation at the forcing point merely stand necessary in certain fixed ratios to each other, these ratios being independent of the ratios of the applied forces or moments. The detailed discussion of individual cases is greatly simplified when shear warping is neglected because then, in the formulas, only the single parameter q varies with frequency. If shear warping is included, the parameter € also varies and the variation of the coefficients becomes more complicated. It may reasonably be surmised, however, that in this case, also, the a's and b's will vary widely with frequency. In roughly cyclic fashion, the infinite a's will occur at the natural 18 free-free frequencies and the infinite b's at the frequencies for built-in * vibration. The presence of damping should replace the infinittes by finite peaks in the curves. * Practically speaking, the effect of shear is small below the first natural frequency, but must be considered beyond this frequency. iS) 4. HARMONIC FORCING AT AN INTERMEDIATE POINT When the external harmonic force F and moment G act at an intermediate point instead of at the end of the uniform beam, the problem is more complicated but can be handled by a double application of the equations for forcing at one end. First the fundamental beam equations are generalized so as to include distributed forces and moments of respective magnitudes PS and Ge per unit length, acting on the beam in the same principal plane; see Figure 1. On a slice dx thick there is then a net force dP + Fy dx and a net moment dM + G,dx so that, after dividing through by dx, the equations 1 of motion become in place of Equations [2a,b]: 2 av eros ae O50 Tm [16a,b] Rotary inertia is omitted here. Equations [2c,d] and the elastic analysis leading to them require no change. Now let the external forces and moments act only on a very thin slice of the beam. See Figure 2, where the thickness of the slice is greatly exaggerated for clarity and at its faces, the common device of drawing the beam as separated is adopted. Let the values of the shear force P and moment M in the beam be P' and M' at the left side of the slice (toward negative x) and P", M" at the right side. Then the associated reactions on the slice are -P' and -M' at the left side but P" and M" at the right. 20 Figure 1 - Forces and Moments on a Beam : 2 tele a ‘ie De Figure 2 = Forces and Moments on a Beam Slice el Integration of Equation [l6a] through the slice gives: 2 9 9 Ee abe el] BE ace oll ae eee ae ox ; aL The first integral is negligible because of the thinness of the slice, whereas: Henee, the total extemal force is Ie a0 cy eM [17a] Similar treatment of Equation [16b] gives, since fPdx is negligible: Go we = iu" [17 ] Also, differentiation of Equation [2c], elimination of y by means of Equation [2a], and integration with respect to x give: | 2 ote) dx =| Sax + cof Ex x Here the M integral is negligible, whereas [ (96/ax)ax =I 6 Ol awineme 9' and 9" denote the slope of the beam or 3v/9x at the left and right sides of the slice, respectively. Hence, using Equation [17a]: US Seat aie!) SS eeie [18] The equations for a beam forced at one end can now be applied to the two sections into which the beam has been divided. Let the lengths of the sections be %' and 2", so that 2' + 2" = 2, the total length. Let all quantities referring to the left or %' section be distinguished by one prime and those referring to the 2%" section, which extends toward positive x, by two primes. * Compare with Equation ({18] of Reference 12. 22 The quantities Vie: 86> F, and G in the forcing equations now have the respective values for the £" section: vy" =v 3 oe = Q" I je ae jou g @M ee mM" Here F" and G" are the negatives of P" and M", which act on the slice; and vy is the transverse displacement of the beam at the slice. The 2' section, however, is forced on the end (AA') facing posttive x. A beam whose forced end faces negative x, as in the previously developed theory, can be brought into the £' position by a rotation through 180 degrees about an axis parallel to F. This rotation, however, reverses the spatial directions of e. and G. Hence, for the %' section: vw SY 8 cme =O ales iV DY g G' = - M! (Note that G becomes -G by the rotation, but -G' = M'.) Substitution into Equations [17a,b] then gives: ” pe Po So Coo Glo el [19a,b] These equations relate the actual external force and moment F and G to the quantities which were denoted by F and G in the equations for forcing at one end as applied to the two sections. Now, likewise, let the b's for the g' and g" sections be distinguished by one or two primes. Then Equations [13a,b] of the general end-forcing theory become, for the two sections: 0 = 0 +b! ig ja Sp? nay " PO Bre imac na YY ng ee G' =p! vtb! Q! ; q' = p" vy +b" ey Bal, 22 © al 22 © * Compare with Equation [19] of Reference 12 where G= 0. Note the use of two coordinate systems (Figures 7 and 8) in that reference. 23 In the formulas for b, £ is to be replaced by £' in calculating b> Dio» bv! dub esbutebyse cues Nemblensiaye 1! 5 1). 4 Ie beet. D1? an 50 ut by in calculating 1? Pye? Ps7? and op Thus, the quantities s, c, S, and C have different values for the two sections, but q and € are the same. Substituting oe = - 6' and 6. = 6", and then sub- stituting in Equations [19a,b] gives: ( F +p" ) v-pbd' 6' +H" oe" b! dat 11 12 12 CSS Se ap pe. OD es jh OP el al 22 22 The difference 6" - 6' represents a jwmp in slope due to shear warping (a jump being naturally assumed positive from negative toward positive x). Thus, the beam has no definite slope where F and G are applied (if shear warping is not neglected). If at this point it is desired to treat interaction with another structure, so that a unique value of 6 is needed, perhaps the mean of 9' and 9" may be used, or § = 5 (6' + 6") Then, replacing 9" - @' by -2oF as in Equation [18] yields: 1 Cees Ohne (e" - e' )= oto F 't 1 " ' e =et 5 - o') =e@-ofF The last equations for F and G then become: Lo ait. Spe i) ws (ol. Si? wes (ie? © DY Ne [20a] ! ol 12 ~ 12 ( i aa 1) 212 Seiat ape irs @as (pt ob! lye (b?_ oP )6 [20d ] 2 22 22 21 21 22 22 The final formulas derived by solving these equations, first for F and G, then, alternatively, for v and 6, are: ek Feb... Yeb 82 Geb. wtb. 6 [2la,b] ult 12 21 22 DD. Sl Gite My Sie wy eS ip! By alah a VIL b j2 12 12 Dy BS aol & HM gig. (ol 4S py 21 21 al an 22 a0) db S 0 eS + ob Tol. ow 22 50 20 12! 20 Do) DD Slop! 2 bY b ( 12 12) (The absence of primes on bia and Pio after 0 is correct.) ie, V = 8))F + aG ; 8 = a,4F + a,,G [22a,b ] Dona ast a AaB a ee eae i Doe i dI> Deen = Pa > Pa 7 OUP g = WigINOR) = Band a9! = tol inf ot) all Tt ee 22 = ) + " a 22 a P55, io} i) ' + ! (b Daa) +p! a ala DS) 1 (me, ae t] mn W ’ ks w (D1, Dip) 5 boy? 25 The coefficients Biai> bio> b51> and Dov are inertia coefficients and &)4> Bio» 85)> and B55 are response coefficients for the beam of length & under forcing at a distance %' from either end. If 60 = 0, so that € = 0 also, then boy = bio and oat = Coe? As a check, if &' +0, Equations [2la,b] reduce to the end-forcing equations for the &%'" section or F = Dea v+ OF 6" and G=b" 6" + Dee Cie as they must do. This reduction is not obvious, for 6 and Dip Bio bo : " tt " t and P55 do not become, respectively, equal to 6 , bia Dio 9 boy and O55" Some algebraic juggling is necessary to establish the reduction. 5. HARMONIC END FORCING OF A UNIFORM BEAM HAVING EXTERNAL AND INTERNAL DAMPING End forcing of an undamped uniform beam was considered previously in this report. The same problem will now be attacked with the beam subjected to both external and internal damping. Let the external damping be due to a uniform force per unit length of magnitude -cyav/dt, with wy denoting the mass per unit length of the beam, v its displacement parallel to a principal plane, and t the time. Here for convenience ta is written instead of the usual c. As in Reference 13, eG and 7, the internal damping is assumed to be due to a resistance to variation of the bending stress such that the instantaneous value of the stress is o = E(e + ndc/at), where E denotes Young's modulus, n a damping constant, ande the bending strain. (In ¥ Reference 13, En was denoted by y.) Thus, the moment M, due to bending is: 2 3 M, = EI iG +7 2) ox ata x * From elementary beam theory, if c is the distance from the neutral surface to the fiber whose strain is — andp is the radius of curvature of the elastic curve at the section for which the bending moment is M, then 2 3 m= 2a ig (e +n H) = eri ten 2% c c ax dtox because 1 5 1 : Se ee 8, 6 fe 5 ON ys eu e fo 2 e 2 9x dt ox I being the areal "moment of inertia" of the cross section and x denoting the distance along the beam. In an actual beam there may well be resistance to variation of the shear strains also. Inclusion of such an effect as an independent para- meter, however, greatly complicates the theory. Accordingly, the resistance of the shearing stresses to time variation is assumed to be such that the moment M, due to variation with time of shear warping along the beam retains its usual value as stated on p. 175 of Reference 11, namely in terms of the shearing force P, the area of cross section A, a dimensionless constant K depending on the shape of the cross section, and the modulus of rigidity G where G = E/[2(1 + v)]. The total moment (or “bending moment") M then equals Mi + Mo As usual, P = - 3M/ax, but the equation of motion now reads Be AZ) ee is a= te 7 OU Be Thus the basic equations for the damped uniform beam can be written as: poe eM [23] ox 2 3 2 = PN, oy LN Bl MS at Bil | KING — 2 a ox ot ox ox 2 2 EE A Sin AS os GS [25] 2 Pe) 2 ot ox It is assumed that an external force F and an external moment G act on the beam at one end, where x = 0, in the same principal plane with v. 28 F and v are taken positive in the same direction, and G is positive in the direction of dv/dx viewed as a rotation @. Then, the boundary conditions for a beam of length & are: oe sce Os Mase, Ms l pes [26a,b] [26c,a] " wo = " fo) | iH} 1 ac) " fo) Eke 5< The final results of the investigation will be relations between Yo and 86> the values of v and of 6 or dv/dx at x = 0, and F and G. (a) Harmonic motion. This problem is solved only for the case in which all time-dependent variables vary harmonically with time at circular frequency w. In developing the solution, complex quantities are employed because of their algebraic compactness. Complex quantities are distinguished from real ones by adding a bar over the symbol. All time= dependent complex variables are assumed to vary with time in proportion to aoa (i VS 7aL) 5 The complex analogs of Equations [24], [25], [26a,b,c,d] thus become: 2 = 2 = 3 g°v EI M egies = ‘i } y aa 2 [27] x gtax 4 ase d°y ou Ov. OCM =a 2 On ae ——= £0 [28] ato WB ace ye ein Se ee si at x = ag Cox a [29a, oe 2Sokhe Mes Og m= [29c,d] ox 29 Here F and G can be expressed in terms of real amplitudes A and B as: F = Ae i @ 5 BS [29e ,f] Furthermore, Equations [27] and [28] can also be written more Lwt explicitly because of the assumed time variation as ee » thus: Sa at ea = ; av 3 MUS tone (al ae ale) ey ee [27a | 9x 9x o_ \ — M (- po + iwep) v +25 = 0 [28a ] aX An equation for M alone can also be obtained by differentiating Equation [28a] relative to x and substituting from the resulting equation for se in Equation [27a]. Slightly rearranged, the result is: mee Ee eM eG We es (Sls) Ose" x Now it is convenient to introduce the notation: ae wot ge 2 mar OKAG AV) 1 - i(ce/w) _ se | (J sonal t= én i (C/w) + wn 1 + iwn BEE eh De ~ SD 1+wn alo a) i The symbols e and g represent real quantities whose values can be read from the last member of the equation, and always g > 0. The equation ob- tained for M, multiplied through by ae (e - ig), then reads: 4— 2— \ 2 —— + 2&q (e - ig) 22 -q ox ox (e = ig) M= 0 [30] 30 (Here, the symbols q and & represent the same real quantities as in the section on the undamped beam; and, if e = 1 and g = 0, Equation [30] becomes the same as Equation [3a] in that section.) Basie solutions for Equation [30] must now be sought. Assume that iwt + Ax M varies as e Then Equation [30] divided through by M gives for >| 4 2 ..\ 2 4 A + 2tq (e - ig) A -q (e - ig) =0 Solving as a quadratic in 1 ines + j 2 i= - eqe (e - ig) + (e2a" (e=ae\s + a (e = aT Since the square root indicated here has two possible values, each the negative of the other, plus or minus alternatives are indicated twice over, whereas < has really only two alternative values. It is conven- ient, therefore, to select a particular value of the square root, which will be denoted by a subscript + and will be defined presently. This square root is chosen so that when e = 1 and g = O it becomes the quantity ae Vi + ce occurring in the theory of the undamped beam. In this latter theory, it was found convenient to write NG = a when the positive sign in the plus or minus alternative was chosen but c =- at when the negative sign was chosen. Analoguously, the alternative values of ne obtained from the last equation are taken to be )°= - a, or as where: 2 2 onic a 2 @ { e(e - ig) + [e - ig + fe (e = ig) y. \ [31a] ee : ; 2 Penoeli/2 as = el eee = ae) ile = sta ee (estes ih [31b ] Sil An explicit expression for the square root occurring in Equations [3la,b] can be found from the general formula: 2 ib V 2 (a + iv)! = + (u+ =) : u= ° (2 + ae +d ) [32a,b] Here a and b are any two real numbers (except b =O if a < 0) and vo denotes, as usual, the positive square root of a positive real number. If M2 : ‘ : a < 0 and b = 0, (a + ib) Ee + iVja|. It will be convenient to denote 1/2 by (a + ip)! the value given by [32a] with use of the plus sign. 2 1/2 [Note that (a + int! (a + ib). =at+t+ib.] (at oy is a continuous function of a and b except for a discontinuous jump as b varies past zero with a < 0, the jump as b rises from negative to positive being readily 2 found to be from -iVlal toiVla] . [To verify [32a], solve [32b] for b and substitute in [32a] squared. ] Putting a = e and b = - g into Equations [32a,b] gives: 1/2 i IL V (es ig), awc aa ws ( eae é ) [33a,b] Since for a damped beam g > O, no difficulty can occur ife < 0 (which is quite unlikely). Also, from Equations [32a,b] with a =e + ee (Ga - ge), b=-g(1+ Bee), the square root indicated in Equations [3la,b] has the value [le -ig+ Ee (@< amen = = SG 2¢°e) [34a] + Ow where myat Ay eee) [3l] 32 Here, to lessen confusion, w is written in place of u but the lengthy values of a and b have not been inserted into Equation [34b]. For moderate damping, e > 0; in any case b < O and the formula holds ife>- (2S). Very likely, numerically greater values of negative e need not be admitted. If they are, the continuous values of the square root can be extended by choosing the negative sign in Formula [32a] and, hence, using the right-hand member of [34a] with signs reversed. 2 1/2 + + ay and a, may now be found as (a5) replace the symbols a, b, and-.u in Equations [32a,b] by a, 8, and r, and (a5) For clarity, respectively. Substitute from Equation [34a] for the square root in Equations [3la,b] and then, in the brace in each of these equations, collect the real terms as a and the imaginary terms as if. Then Equations [32a,b] give: apza(r +i), atetws @=-e [e+ (1+ 2¢e)) [350,b,c] Gl = if : S 5 . S65 — i Gy ol Wn ee) fe twos 8 ge [-6 + 7{1 + 2 e)) [354,e,f] where in either case 1 = r=Vi (aM + 6) and w is given by Equation [34b]. Here in both a, and a, a > 0 provided e > cope). For then in [34b], a=e+ Ee (e = e-)> ree > 0 so that we >a >Ee° and w > |Ee|. Otherwise it may be necessary again to extend the range of continuous values by reversing signs. 33 Without finding ay or Qo, however, the following useful relations can be inferred directly from the expressions given for ay and as in Equations [3la,b]: 2 2,1/2 a4 . a5 220 loo igs & (es ig J. [36a] as - 45 = 2ea°(e - ig) [360 ] —p-) -- 2 ye q}45 = 4 (esaels hen Ce. ao (eo ie) [36c] 2 + That the square root of (e - ig) occurring in 4145 really is the one 1/2 denoted by (e - ig), can easily be verified by means of the argument from continuity, which holds also for the individual values of a, and 5: Also: = is 2 a + as Ger (eo gS ee (6 5 anh) [36a] 3 «(33 Fe Nia 41 42 a Sp a = 2g? [1 + 2° (e - ig)] (e -ig)/? [36] Qo 4) an ae + In the further development of the analysis some relations among the exponential-trigonometric-hyperbolic functions are useful. A systematic list of some of the relations for sin iz, cos iz, sinh iz, cosh iz, onan ee, Sin (S257), Cos (se sp a7)ly Gein (se 247) 5 Gosia (Ces +y), sin (x + iy), cos (x + iy), sinh (x + iy), and cosh (x + iy) can be found in an elementary calculus textbook. Here z = x + iy, x and y being any real numbers or 0. con : ae =P =e The four possible values of X will then be, since AX = - qa) or Q5> = = = i + iq.x A= + iq, or *4,- The corresponding values of M are erie ey 9 . a 5 gat n= 2 -ax eee ae elt dane eulie 45 ° 34 Further work will be simplified, however, as in the theory of the undamped beam, if combinations of these four functions are used such that two of them vanish when x = 0. The latter two combinations of the exponential factors are: ay igq.x S -iq,x pee ries eA ea caaaee 5 (e- =a e a) 7 sinh iq,x = sin q)x; (ee SG ROS /b SS Sain qx (ira The general solution of Equation [30] may then be written thus: Mieeo* (a me ned GC so ect Sa wate (68) e ) Sina, x 5 COS qd, x 3 sin q, x , cosh 4x 3 Here a, 900 4, are four adjustable constants, probably complex. Substitution for M from Equation [38] and from Equations [29e,f] for F and G in the boundary equations, Equations [29a,b,c,da], then gives the following four equations, after canceling out eit, dq+tdq=A; a+d=-B 39a ,b] Ta SB a ~Th | (je) d. sin q & + d_cos a2 + a, sinh Ane + dj, cosh Gok = [39¢] 1 AL 2 dja, cos a2 - dq. : ty + =z 29, sin a,t d 2 tt ee ane 345 cosh qt da, sinh ak ) [39a] The treatment may now proceed formally just as for the undamped beam. Equations for the complex displacement v and the complex slope 8 (= dv/dx) are as follows (derived from Equation [28a]): 2 pho. ca Oe | ie u (w - iwc) v = —=; uw = ane) @ 8 == Substitute here for M from Equation [38] and then let x > 0. Write vy and 35 ~ Shuby = = _= int = eine 8, for the values of v and @ at x = 0, and also Va Vine and Io = Goak in terms of complex amplitudes van and Bans Then, after canceling out et’ it is found that: 2 OO sO -2-= - iwc) v = - Glos d 1 tw ) ee The ooh 2 ee -3- -3- = signe) S566. < Cel uw W Don q, 1 4, 3 Baier tdi 2) aie, Une aeeeiaie (Note that 9. sin q, /9x SS Ge bin Gk = O when x = 0, etc.) These two equations and Equations [39a,b] are easily solved for the d's with the following results: a, (a +45) 2, = a5 A-u (um - tue) O (@2 + a2) 4, = - 2 B-u (u? - ive) ¥,, dp (a2 + &) d,- a A+u Ga - fac) ea (e+ 2) a =- a Bey (ue - iwc) ¥, The d's can now be eliminated entirely by multiplying Equations [39c,d] by (af + as) and then substituting in these equations the values of (ae + as)a, Pea given by the last four equations. Also, introduce the >) >) >} following notation: al " Jen (ae 5 Aes) sinh a2 Q| TT s = sin a2 cosh ap! Cc = cos a2 Then Equations [39c,d] divided by 9 become: 36 (o} -2 -2 qq, _ q. SEN AN ao Lint = = S + me ae + (qqc + asc) on [hoa] Q Q I, qa, Bee may tee Ge Se (q,8 a,8) oa (Cc c) oa -2- =P fi Sa pane ay apa =a (q3C w age) * a345(a,5 = 18) = [40d ] These equations are easily solved formally for wee and Cn in terms of A and B. The determinant: of the coefficients of vis and 8 is: a BAIR REL, 28 LAY MON aN G5 De= (Coser ss@sg -33(2- =) Be SR or, using Formulas (36b,c]: De 8 [ht = ee Sees (6 6 ae) [41] The results of this solution may be written in the form: ree % =| Q e BS ah =e D aa = QD q qd, + =e ss yo te (2 > 6G) + 20,85 ]B } [42a] — 1 = —-— ga 7 224-8, (1 - &) + 0,881 QD = g.sc + q,cS 42b Q, (@,86 + a,¢8) Bf [420 ] where, using Equations [36a - e]: 37 2 ee : 2 4 Palle A ah & ay & Ba Kos ales B= KS =) ats) Vi Q, = a4 - 45 = 2a (e - ig) == 5 ay Qip y Ades a° (e - ig) ane 2 1/2 2 2s = aa (oe = de) [1 +2: (e - ig)] 145 x Thus, the analytic solution of the complex problem is formally completed. (b) Real results. For practical use, real relations must now be inferred from the complex relations. The original equations requiring real solution were Equations [24], [25], and [26a,b,c,d]. As final results, relations between V, and 6, and the applied force F and moment G appear to be the most useful. For convenience, the complex Equations [42a,b] may be summarized thus: Yen, 2 Wedge Hela) BO (any @ Beja) B 8 oq = (agg ia5)) A+ (ay, + dass) B the primed coefficients at ath? etc., being all real. Then Te = Tee! = (a), + ia},) here + (a), + ial.) Bee @, = 0,,¢ = (ai, + ia") Ae” + (a) + ial) Bet Here Vo and 6, are the values at x = 0 of v and @ (or dv/ax). 38 For v and 6 and also for M, expressions have been found that satisfy Equations [27a] and [28a] and, hence, also Equations [27] and [28]. But in Equations [27] and [28], all coefficients are real. Consequently, the real parts of v, 6, and M by themselves must satisfy these equations and must also be harmonic solutions of Equations [24] ana [25]. The real parts of the end displacements we and es must, therefore, represent v, and @, as functions of the time in a possible forced fo} vibration. Keeping only the real parts in both members of the last two formulas for Vo and 6, gives, since eek =NCOSMWitie een SHua GiGi: Wig O an A cos wt - ayy A sin wt + a5 B cos wt - alo B sin ut [43a] 6, = as5 A cos wt - a7 A sin wt + es B cos wt - a5 B sin wt [43b ] Selecting real parts in Equations [29e,f] gives as the corresponding real external force and moment: F = A cos wt 3; G= Bcos wt Thus the displacement Me and slope oS produced by force actions proportional to cos wt alone contain terms proportional to both cos wt and sin yt when damping is present. Equations [43a,b] might be accepted as the final expressions for end displacement and slope produced by harmonic end forcing. If, however, interactions of the beam with attached structures are to be considered, it may be preferable to avoid explicit mention of sin wt and cos wt. 39 Note that: a a AL, Gls 12 Cle = a' F + —— — + a! 5 —= = Yo aa ae a1 56 maT {44a] "! 851 dF Q5> aG = q! + —— — + g! —= ——— 8. a,F Rage 2.4,56 + a ae [4b ] The eight coefficients aa a3! oe ete., may be regarded as an extended set of influence coefficients for the damped beam under harmonic end forcing, corresponding to the four coefficients a (2 a ,a,anda ee eal 22 for the undamped beam. As an alternative, inertia coefficients corres- ponding to b> Do? eo? and Lae for the undamped beam could be calculated. The following extension of the results is perhaps obvious. F and G have been assumed to vibrate in phase, but this restriction ts eastly removed. Let B= 0. Then the motion of the beam is excited only by F, and all formulas will obviously ‘hold if wt is replaced by (wt + @) in which a is an arbitrary phase angle. Similarly, if A = 0, only Gis active and wt may be replaced by (ut + 8), in which 8 is another arbitrary phase angle. Since the equations of motion are linear, these two motions and the exciting force actions ean be superposed. Hence, Equations [43a,b] will remain valid if the external force and moment are assumed to be F = A cos (wt + a) ; G=Bcos (ut + 8) 4O and provided that in Equations [43a,b] wt is changed to wt + a in all A terms but to wt + 8 in all B terms, a and 8 being independently arbitrary. Equations [44a,b] require no change. (c) Calculation of a' , a’ , etc. The coefficients a!_, etc., in 11 dal 11 Equations [43a,b] and [44a,b] can be calculated by evaluating the coefficients of A and B in Equations [42a,b] and separating real and imaginary parts. In preparation for this calculation, each complex number may be replaced by the sum of a real and an imaginary part. Write a S69 © 46% o@ Ss @ os 4—n q q, le 3 dq, qd, 14, in which a> etc., represent qr or q8/(2r) in Formulas [35a,d]; then and from Formulas [37m-p] applied to the definitions of s, c, S, C: = sin qa, cosh ay + i cos a, sinh ay 2 c = cos a; 2 cosh ay - isin a; 2 sinh qa; ae- * ’ " ° ' * " S = sinh an cos Qn & + i cosh Qs & sin An & as ' " : : ' ° " C cosh an 2 cos q,% + i sinh qn & sin qn, & 2] Similarly , Qo Qn? and Q), can be separated into parts by using Equations [33a] and [34a]. Denominators can be handled most conveniently by transferring i to the numerator. Write D = D' + iD", determining D' and D" from Equation [41]. Then: Tal Leama Dar DEy Des Seay = ' ‘pi To. 4pw = D JD) cp al)o) D iD p'2 a p'2 av) apa Ane eo ANS eS PAM yt w + iwe Sane ~ QI BD oy mW (tr So sie) won (ay Se) The final calculation of Boo a etc., from the coefficients of A and B in Equations [42a,b] then requires only a massive process of multi- plication and collection of real and imaginary parts. Here, it does not seem worthwhile to elaborate the final expressions. he 6. FLUTTER OF A RIGID FOIL ATTACHED TO A UNIFORM BEAM 6.1 DERIVATION OF EQUATIONS OF MOTION A rigid foil attached to a uniform beam of length & will be considered in this section. The foil is assumed to be equivalent to a straight flat strip of rectangular shape, with length much greater than width, attached to the beam so that its length is perpendicular to a principal plane of the beam, with this plane being midway between the ends of the beam. In the undeflected position of foil and beam, the plane of the foil is assumed to pass through the principal axis of the beam. Elastic deflections of foil and beam, where they are connected, will be allowed but the affected part of the beam is assumed to be much smaller than the whole beam. The relatively small part of the foil that lies within the beam may be absent, being replaced perhaps by a through-shaft similar to the mounting of a pair of diving planes. There is a certain axis of rotation lying in the plane of the foil and parallel to its length about which relative rotation of foil and beam evokes only elastic moments G acting on the beam and -G on the foil. Let @ denote a small angle of rotation of the foil about the axis, measured from zero when both foil and beam are in their neutral positions; in addition, the local part of the beam itself, due to vibration and aside from distortions of the attaching structure, may be rotated similarly through a small angle ee The positive directions are assumed the same for 8, one and G. The foil may also have a small translational displacement v perpendicular to its plane and the beam at the axis for Ge a small 43 displacement Vo? associated with a force F on the beam and -F on the axopLaLe Ikens Ws var and F be positive in the same direction. In terms of elastic constants kK and kK, of the connecting structure: 1? & Tk (Wr 8 Cok, (@ = 6.) Figure 3 represents a section in the principal plane of the beam and shows positive values except where the associated symbol is preceded by a minus sign (as in -F). A heavy line represents a section through the foil plane; the foil itself, of width 2b, being perhaps reduced to a connecting structure here. The axis A in the foil plane and the axis a through the principal axis of the beam coincide when foil and beam are unde flected. Let beam and foil be immersed in a stream of fluid approaching at uniform speed S, as shown in Figure 3. Expressions for the resulting lift forces on the foil are adapted from Theodorsen's formulas for a uniform foil of infinite length vibrating harmonically. An approximation closer than the common steady-motion approximation is used (see p. 841 of Reference 4 (Appendix H)), the additional complexity being only moderate. In the case of ship rudders or control foils, Theodorsen's parameter 1/k or S/bw, in which b is the half-chord* length and w the circular frequency, is less than 1 (or at least < 2); then Theodorsen's functions” eA! Tongyel 'G" may be replaced without great error by their values at S = 0 or 1/2 and 0, respectively (see pp. 840-841 of Reference 4). The effects of certain other terms may be included in the effective mass m and the effective moment of inertia I, (taken about the effective center of mass); * Not to be confused with external force and moment respectively on beam. 4h Fluid reactions on foil = G Elastic reactions on foil ons on De oF Elastic react! Deflected beam axis Neutral axis for Gio, eee Neutral position of foil and beam axis 2b Figure 3 - Forces on and Motions of a Beam-Foil System Immersed in a Moving Fluid 45 m and I, are further defined later. There remain then a total resultant lift force F. and a total moment L * Mg about the 6 axis whose magnitudes can be written } e ‘ F, = BS 6 - BSv + CSé [45a] M, = Wee) a HES & E88 [45b ] Here B, C, L, and E, are constants (defined below), L having the dimension of length. a and My are positive in the respective directions of v and 6; v = dav/dt and 8 = a0/dat. Since My denotes the total moment about the 6 axis, Be may be supposed (for concreteness of thought) to act through this axis, as drawn in Figure 3 (i.e., Theodorsen gives the total force F and total moment M about a certain axis. Hence, we cause no error if we arbitrarily assume F to act through the axis.) According to Theodorsen's calculations; 2 hoe. B =B(=b +e) BS qe P L : c=5 (2b +e); L= M|eR ff in terms of the density o of the surrounding fluid, the half-chord length b and length 2, of the foil, and the distance e that the axis of rotation fs or the @ axis lies ahead of the midchord line (here ahead means toward the approaching stream). The first two terms of M, may be regarded as 8 arising from a force equal to the first two terms of F_ acting at the L forward quarter-chord point. With these values inserted, Equations [45a,b] might be called the low-speed approximation to Theodorsen's expressions. For a derivation of the approximation, see Reference 4. In this reference it is shown that if e = b/2, Equations [45a,b] agree with the "Modified ** These values of F_ and M, agree with those of Ea and My given, respectively, on pp. 841 and 842 of Reference }. 6 Theodorsen Analysis" described on p. 37 of Reference 14 by McGoldrick and Jewell. For greater generality, allowance is made for possible additional damping due to other causes by adding a force - eV acting on the foil along the same line as Fo and positive in the same direction, and also a moment - e586 about the 6 axis. Without too great a complexity, the more general expressions - en - C158 and — CoV - C8 could be Weeds but this was not thought worthwhile for the present purpose. Equations of motion may now be written for the foil. Let the effective center of the foil be at a distance h ahead of the @-axis, h being positive toward the approaching stream. The total effective mass of the foil (including virtual mass) will be denoted by m, and its total effective moment of inertia about an axis drawn through its center of mass and parallel to 6-axis by I): The displacement of its center of mass is v + h@ and the total upward force on the foil is - F - ev ap la: - F being ily? due to the attachment structure that exerts the force F on the beam. The total moment about a line drawn through the center of mass parallel to the 6 axis positive in the same direction as @ is, similarly: ao @e c,,8 + M, oii (o s c,v + F.) Hence m(v+h0) =-F- cv +F, 1,6 =-G-c59+M,-h (os Po cna FL) A simpler solution, however, is to add h times the first equation to the second equation, thus obtaining for the second equation: * See p. 78 of Reference 3, for example. C14 Coo and Cyo> Co, are direct and cross damping constants, respectively. AT hmv + (205 + hm) (Mewes ches cn6 + Mg: For brevity, write Then, inserting the expressions written previously for Fe and Mp: my + hmé + F + (ce) + BS) v - cSsé - BS-6 = 0 [46a] and hmv + 1.6 + G + LBSv + (c ° 2 i 5 a ES) 8 - LBS~6 = 0 [46d ] With these equations may be associated the F and G equations written previously, and also the equations furnished by beam-forcing theory for the response of the beam: F=k, (v-v,) 3; G=Ko (0 - 9) [47a,d] Vo = 82F + ayoG 3 8, = anF + annG [48a,b] Here a> Bio B51> and An are response coefficients to be calculated from the formulas for end forcing if the foil is attached at the end of the beam, otherwise from the formulas for forcing at an intermediate point. Thus are obtained six fundamental equations in the six variable functions of the time: v, 8, v_, 8 BR o> F, and G. At the critical flutter speed, all of these variables will vary harmonically at the same frequency but probably not in the same phase. It is only for such variation that the lift formulas are reliable. Equations involving only v and 6 as time variables are easily obtained by eliminating the other four variables. To shorten the notation write: 48 Then substitution for v, and 6, from Equations [48a,b] into [47a,b] gives: V, A, k5F + s5G = k,0 [49a,b] These equations may be solved for F and G in the form: DF = 5K) = a)5k,k58 [50a] DE ec An kK kev + s)k,8 [50b } D 8 8:8. = @ a. _k ik [50c] Le 12 2b dy 2 Direct substitution for F and G from Equations [50a,b] into [46a,b] then gives as equations of motion, provided D # 0: mv + hmé + (ce, + BS) v - csé ep Bana D~ (sokyV - ayok)k58) - BS = 0 [51a] hmv + 1,0 + LBSv + (ec, + ES) ) =i + + 6) - Oe= b Dp - (Ss -8,, kK KoV sk, ) LBS* 0 [51b ] Similar substitution for F and G in Equations [48a,b] gives, provided D # 0: coll ) s,) v + pola 6 ; 8, = Dla kv + (Qe D's, )8 [52a,b] Wa sured ie 211 The special case D = O requires other methods as will be explained later. Re) 6.2 CRITICAL FLUTTER SPEED For typical flutter to be possible, there must exist one or more values of the speed S at which a steady vibration of the foil and beam is possible in spite of existing damping actions. The lowest such speed is the critical flutter speed, which is the item of principal practical interest. To seek a critical flutter speed for the foil-beam system by the usual method, assume that in Equations [5la,b] v and @ are complex functions of the time proportional to er ut (a = Yi). Then v = iwv, v == wv, etc., and, after canceling one the remaining equations are linear in v and 6. Nonzero values of v or 6 are possible only if the determinant of the coefficients of v and 6 in these equations vanishes; ithatelsi it 2 =i) : 2 al OFet ails [-wm + D sok, + iw(e, + BS)} [-w I, + D sk, 5 is < iw(c,, + EB S)] = (eaSiny 6 oa =i Aa ee Be 5 fads) (Gio o Dom. ek eb Aas) = O eli 2 The real and imaginary parts of this equation must hold separately. The -2 imaginary equation may be divided by iw if w # 0, and the D term that occurs in it may be simplified by means of Equation [50c], thus: —P BO. ea D (sjs,kjk, = By 5857 ks) = 7 I 50 As a result the following two simultaneous equations in ne are obtained: h 2 -l w'at, - a {D [ms.k +15 _k + hm (a,, + a5) kjk, ] 2 Qe a + (c) + BS )(c,, + ES) + [(h - L) m + LC] s*} all iss o Oe, le 2 + s_L) BSS = 0 + D ok D 2 -w. [e,I. + cm + (I il, Ae hLm) BS + hmCS + mE S| =i Sal: + + + (c)s)k, c585k) D kK, (s La__k_) BS aL We al To RRS OM SRS oe MES SO = Cae Ore leaks 1 and S [53a] [53b] In calculations, however, it may be more convenient when D is small to multiply the equations through by D, thus: 2 -w {ms ky + T sok, + hm (ayo + A5,) ky ko + D (c, + BS) (c, -4 ahs + Dw mI | + kk, -k, (a,jk5 + spl) BS" = 0 c,s,k,+ Co8K, + k,, (s) + La, 5k, ) BS - a, ,k,CS 2 + s_k_E_S - Dw [ej I, +c oki E,, m+ (I, - hLm) BS 2 + hMCS + mE,S] - De, LBS“ = 0 + ES) +D [(h- L) m+ Lc ]Bs° \ [53c] [53a] It will be shown presently that Equations [53c,d] are valid even when D=0. Dal If a harmonic solution of the equations of motion, Equations [46a,b], exists when the stream speed is S, its circular frequency being w, then we and S must solve Equations [53c,d], or, if D # 0, Equations [53a,b]. Conversely, any solution of these equations in which a > O leads to a harmonic motion. The problem of finding such solutions is complicated by the fact that the parameters a4? a5? a 4° Oe and D all vary with w. In numerical computation, the only feasible procedure seems to be to assume successive values of i to calculate the paramaters for each value and then to attempt to find a value of S that satisfies both equations. Perhaps the work may be facilitated by rewriting the equations as quadratics in 3; perhaps for Equations [53a,b], with all coefficients reversed in sign for convenience, the following will result: 2 2 DS se) 1) S O Si teense =0) ha,b 1 2 cae Fal Balt hadi [5 ] where b. 2 8 [ie > Ca Ie je +s k ) + uw (hm - Lm + LC + £,)] i AN ® Qh b, 2 a Cee + c,B) Brn illeeli 2 ae 2 {pl b3 = - Dkk, + w )D [msjk, + I,s5k) +hm (a., + a_) k_k, J+ cc, $- ml ae 12 2 eanling 2 2 ) e = c_LB iL 1 o = = BOR (s, + La eS Goa kk 2 A SEAL WAL Pile ian? = } - + - hL B + hmC + mE E,Do's,k, + w [(I, m ) m dl Or where, to match Equations [53c,d] with signs reversed: 2 0) S + + - + 1 B [a,j k, Ls ,k, Dw (hm - Lm + LC EO] bo. = no (c_E_ + cB) } IL” IG } =i + + De Kk, mm) [ms k, Tsk, + hm (a,, + a,,) kk, + Desc, ] = DmI jw e = De LB It ease (s) + La, 5k, ) kB + 8.) kj Koc = sk) EL + w°D [ce = fia!) BS tee + mi, | 2 S56 oS + + mc CAS kK, c,8k w D (ec I, com) S must be a common root of both of the quadratic Equations [54a,b]. Two such equations can have a common root only for special values of the coefficients. If a common root exists, a simple formula is easily found by eliminating S-. thus, from Equation [54a,b]: 2 2 e, (b,S +b .S +b.) - by (e,S +e 5 3 S + e3) 2 (e,b, = B25) Sab. o Bde = @ [55] A general procedure for seeking a positive common root of Equations [54a,b] may now be given as follows: DS If, at the w chosen in calculating the parameters , 1°, - bj e5% OF determine S from Equation [55] using the second form of the left member; and, if S > 0, test this value by substituting it into Equation [54a] if d # 0 or into Equation [54b] if ey #0. If the chosen one of Equations [54a,b] is satisfied, simple reasoning from Equation [55], using the first form of the left member, shows that the other one of Equations [54a,b] is also satisfied, so that S is a common root. If this procedure cannot be used, one of the following procedures is available. Its?” Goldy 2: 19 y>o 120 = O, then e153 - bie = 0 also, to satisfy Equation [55]; otherwise there is no common root. If b, # 0 and ey #0 ictetolelows that e = (e bande. = (e_/b bd and, of course, e_ = (e_/b Deo a ; ( La! 5 3 ( a! a! 3° > 5 Cy ( al! 4! a Thus the two Equations [54a,b] are proportional and have the same roots, which may be found by solving either equation provided it contains S. If either >) =0Oore_ = 0 but not both, one of Equations [54a,b] reduces to O = O and the other must be solved for S. Finally, if b, = e, = 0, Equations [S4a,b] are, at most, linear in S;: and S determined from one of these equations may be tested in the other equation. In the very rare case that D5 = D5 = De = cat = @ = e, = 0, steady vibration of the system may occur at the assumed w with any value of S. If one of these procedures does not yield a common root for Equations [S4a,b], then there is no common root and steady vibration cannot occur at the chosen w. Furthermore, for our present purpose any negative S is to be rejected. Usually no acceptable S will be found; by repeated trials, using interpolation wherever useful, the rare values of Ae at which an 54 acceptable S occurs must be ascertained. For a complete solution, the entire range of positive values of w must be explored (see the case of small ql discussed later). 6.3 THE SINGULAR CASE: = 0, HENCE D = 0 Sy 7 Sipe ne 2 If a chosen value of w© makes D = O, the deduction given for Equations [53c,d] with corresponding values of by se e3 is not valid because at some stage it is necessary to divide by D. Equations [45a,b] through [50a,b] are all valid; but now Equations [50a,b] yield only the ratio v/®@ or 6/v, the two equations being obviously equivalent if S1> S5> Bio? and al are all nonzero. The same relations follow from Equations [51a,b] and [52a,b] if these equations are first multiplied through by D and D is then made zero. The relations that hold when D = 0 are, from Equations [50a,b]: S.7 SeaJk.6 9 ak ve 846 [56a,b] The difficulty may be overcome by choosing a different pair of variables from among v, 8, F, and G. When S$) # 0, a convenient choice is the pair v and G. From Equations [49a] and [56b]: s,F= kv - a,5k,G 5 s\6 = a,jkiv [57a,b ] Multiplication of Equations [46a,b] by Ss) and then elimination of @ and F by means of [57a,b] gives as equations of motion for the system in terms of v and G as variables: DD m(s4 + Ay ,kyh) v+ [sy (c, + BS) - @5)k,CS] Vv Se (Lo @ BS“) v-a 1 ok G=0 el a al to) a & (eis S ak (sjhm + a,)kjT, 1 1%) = fh jie LBS“v +s G=0 2nd at t and if v and G are proportional to oie : 2 2 {- ufm (Si) ee fe) IR S Cool i a) (6 (@. s 2S) 5 a ke CS v-a sk G=0 dL Jt 2a, ab 12) jal 2 2 2 hm + Te BS Oe Cee Basle) oy + iw [s, LBS + aK) (ce Equating to zero the determinant of the coefficients of v and G and dividing the imaginary part by iw yields: 2 2 s, [-w m (s, + Ay ,kjh) + Ky = 84k, BS ] 2 2 Sete ll (sola > a. tka it.) + a5,k, LBS" | iT] fo) PA Je ig ins > @ ik (@_ > iS) I) a © 2a at CAlmelt 2 L Now multiply these two equations by kK» substitute into them a a kk =s.s_ from D = 0, and then divide the equations thus obtained We Pi 2 We by s, on the assumption that sj #0. Equations [53c,d] with D set equal to zero are the result. 56 This proof fails, however, if S15 O. Then, as a third alternative, @ and F may be used as variables. For the present purpose, it is assumed 0 (since D = 0 and k._k, > 0). that s) = 0 and, hence, also that a_a = 1 12D He Then Equations [49b] and [56a] give the formulas: Substitution from these formulas into Equations [46a,b], multiplied by So gives, if 0= = 1/2155 LBS = 0. Equation [46a] gives F. Tf Qy5 = 85) = 0, then v = @ = O and, hence, by Equations [46a,b] F = G = 0; and by Equations [47a,b] in = Ge 3 Oo WSS eS Sg SOs = a,, = 0, there can be no harmonic vibration at the w for which these Ql parameters were calculated. Examination now shows that all of these conclusions as to w and S follow also from Equations [53c,d] when D = O and the parameters have the specified values. The general validity of Equations [53c,d] is thus established. 6.4 DISCUSSION OF DAMPING EFFECTS Components of force or moment on the foil that are proportional to velocities have a damping effect, positive or negative. Included here are the force — e1v and the components - BSv and csé in Fro assumed to act through the @ axis, whose velocity is v. Then there are also the moment - e568 and the terms - LBSv and - E,S6 in Mao acting at the velocity 8. These forces and moments do work on the beam-foil system at the total rate (see Section 8.3 of Reference 3 and Appendix H of Reference }): [- (c, + BS) v + CS6] v + (-c56 - LBSY - £88) 8 58 or of) 2° ee) = (cy + BS) v~ + (C - LB) Sve - (c, + E,S) i) This expression may be positive or negative. If, however, the Theodorsen relations are used, namely: B= mebL 3 ) a ( 1 2 S = + = — = = _— C a(3» e glk aie errs ED B a +e] then C - LB = B (b + 2e), and the rate of work is: 2 °2 re) E 02 IL | So es c 8 = BS Ise S (> <> AS) s78) +( oa + e] is) i S2hhn 26) E (2 ) al =- cv C58 - B|i|v- 5 bte] 6 With these values, therefore, if c, 20, ey 2 0, and BS > 0, there is no gain but, in general, a loss of energy from the beam-foil system during harmonic motion. Somehow, such a loss must be compensated for through work done by the BS“ terms in F, and Mp- L If s = 0, the Cy and C5 terms alone will cause a loss of energy unless Cy Sears 0; and when S = 0, the equations of motion are correct for any * type of motion (whether harmonic or not). Consequently any existing motion must die out unless maintained by additional forces. The same conclusion will probably hold at first as S increases from 0. Eventually , however, the rate of loss may decrease and become zero at a certain speed * The Theodorsen terms are derived only for harmontc motion. If, however, S = 0, the Theodorsen terms disappear and, in general, the equations of motion hold. General equations are needed to deal accurately with damped motion. 59 Sos and it may safely be assumed that the motion will then be harmonic. As S is increased above Sao it is almost certain that the damping will be negative, so that any vibration tends to build up. At one or more higher speeds, harmonic motion may again become possible; however, this is of little practical interest. Thus S the lowest value of S at which steady c? harmonic motion can occur, constitutes the critical flutter speed that must not be exceeded in practical operation. 6.5 THREE SPECIAL CASES 6.5.1 Damping Due to Lift Only (c, = Css 0) life Cen Cons 0, so that damping arises only from terms in Fh and My > then S may be canceled out of Equations [53b] and [53d] entirely because every term is then linear in S. Also, only the se term occurs now in Equation [53c] or [53a] but not in Equations [53b] and [53d]. These features open the way to the following simpler mode of solution: having a assumed a value of w and calculated a__,a and B55? calculate a? lines hme eae from either [53b] or [53d]; if this value of we agrees with w as assumed, then harmonic motion is possible at thisw, provided S has a value that satisfies either Equations [53a] or [53c]. This value is easily found. 2 If the two values of w do not agree or if S turns out to be zero or negative, then the assumed w must be rejected. 6.5.2 Koil Rigidly Mounted on the Beam The equations may be adapted to the assumption of rigid mounting by letting k, > = and ky + >. Since s}/k, = (1/7) + al2 and s,/k, = (1/k,,) ial + a55 and D/(Gz ik) = (s1S5 = 81080} Ky ko) /(kyko) the following reduction lier 60 formulas hold as k.k, > @ alier2 ra sil tee ed > & 9 ky alent kK, 2e aus SD 2a -a = @ a 2 Pk. Se Kk, fe) Il ez WA Sa lee The limiting forms of most equations become evident after suitable mani- pulations to make these formulas applicable, Equations [53c,d] being divided through by kjk,- Equations [46a,b] and [48a,b] require no change. Equations [47a,b] divided by k, or Kk, give v. = v and 6, = 6. In Equations [49a,b], [50a,b], it [51a,b], [53a,b,c,d], and [54a,b], the final effect is simply to replace D by Do» S$} by €)], and So by apo, and to erase ky and k, everywhere, the 2 term kk, in [53c] being replaced by 1. The equations do not seem, however, to be simpler in this case or modified in a mathematically interesting way. 6.5.3 Low-Frequency Approximation In the theory of a uniform beam under forcing, as given in Section 2, formulas were found for a and a 22 ii? 210° 821? an terms of E, I, the length 2 of the beam, and two dimensionless parameters qk and & defined thus: \ - (2) aan 2o2e- = S [58a,b] : = = (a) EI eKAG, ae * Here wp is the effective mass of the beam per unit length, * wu includes the effect of virtual mass when the beam is immersed in a fluid. 61 E is Young's modulus, G. is the shear constant usually denoted by G, A is the area of the cross section, I is its area moment of inertia about its principal axis, and K is the usual shear-warping constant. If w is small enough, q&% and € will both be much less than unity. Shoaealal@ewisy 5 es Sa dl Erol § «—S al alse i (#2) *"° W << — ge vu 5 w << 2KAG, /7yET [59a,b] The second limit on w is obtained upon substituting for q& in the expression given for &€ and setting the result << l. When € is so small, all shear effects in the beam may be ignored and the approximations derived in Section 3 for qk << 1 may be used, namely: le 2 = a,, = a,, =- = [60a,b,c] ° 2 eae b) 2) >) &) ial mone 1 mene 2 2 3u2 Let w be small enough so that hk 12k 2 we << 1 : we << [61a,b] pL we Then, using the approximations for avn anda, la ea! >> 1 and lak, | >>aelt ence: After inserting the approximations into the formula: w= Se = Sipe ia 62 Upon substituting these approximations and also writing v for - aa and 6 for - n°@, Equations [50a,b] and approximate forms: [46a,b] divided by kk, take the a6 da Doo So F = uly + 1 432°6 ; C= 5 us vt 5 u07 [62a,b] 00 ee 2) 0 e m'v + h'm'@ - BS 6 + Ker + BS) v - CS@ = 0 [62c] him'y + 1,6 = WESee = ss (ef, + EFS) 8 = 0 [62a] where ~ 3 2 lus a Sim te MM. 9 Jali! 3 fot eS y= 2 i = I, + 3 [63a,b,c] Also, Equations [52a,b] become: ange ue-we Yoel & k 2k It ne 22 Sie L-w 8 = Ue este da 4 a Je Treated in the usual way (i.e., equating determinant to zero), Equations [62c,d] give as approximate conditions ae? {m'I'! i] ' 2 ie (alti) | mc - {h'm' + E, + L (@ o m?)]) BS” 2 © 2 ' w [(e, + BS) I, + c, LBS =0 for harmonic vibration: oy, Se 2 [6ha] + (ce, + E,S) m' + h'm'CS - Lh'm'BS] [64d ] These equations are simpler than Equations [53a,b,c,d], especially because w affects the coefficients here only through the factor w . 63 = The final search for a useable S, however, is about as complicated. A useful minimum for uw can be obtained from Equation [64a] provided 2 the coefficients of S and S are not negative, namely: Jl we 2 ¢)¢, [m'I'e = (h'm')2] Otherwise, the determination of a minimum w is more complicated. In Section 3 it was remarked that the approximate values of a1? B55? >? and a are the same as the values for a slender uniform rigid rod of 2 21 length 2 and mass uk forced at one end. For the rod, Ayo = 457° Thus the equations derived here for q& << 1 are applicable also to a foil attached at one end of such a rod. It is interesting also that the equations still hold if 2 = 0, which corresponds to removal of the beam altogether. Thus the foil may exhibit flutter all by itself. Usually, any low-frequency harmonic vibration of the foil that can occur with the beam attached can also occur without the beam, and at the same w and S, provided the mass of the foil is increased from m to m' and its moment of inertia from I to I'g, and provided h is changed to a value h' such that h'm' = hm + S wee. 6.6 SIMILITUDE An interesting question is the relative influence of the various parameters of the foil-beam system upon the cricitcal flutter speed. Relevant fotl parameters are n, I> h, and L; the lift constants B, C, bewandine)i-sandic=andaica meal sion and E, (or the foil constants p2& T 5 f? there are the attachment elasticities ky and ky. The influence of the (on i= beam is represented by the values of 84> 80> 41> and B50° Apparently , in general, the relative influence of these parameters can be ascertained only by making extensive numerical calculations. A major source of difficulty lies in the possible variation of the a over all real values. A simple form of similitude, however, is readily discovered. In Section 2 formulas are given for forced vibration of a wniform beam when F and G vibrate at the same circular frequency w but perhaps in different phases (for example, F = A sin wt, G = B sin (wt + ¢)). The mode of vibration of the beam was shown there to depend upon two dimensionless parameters qk and & whose definitions, as given in Equations [58a,b], may conveniently be rewritten as follows: EI all 2] EI 1 2 : -( 2KAG. +) ae) en -) = 2 (az) [65a,d] Here 2 denotes the beam length and q may have any positive value. The formulas obtained for the a5 can be written thus: al 3 2 2 = g pis, Z —— Q = = ORR SCS = ay 10 S — ee = 66 ae) a a b &e > Sia gre Ba = Gn ce 8 Poo an => (166c c,d] where a ales a. ©, and Os are complicated functions of q% and € but are 11 * otherwise independent of the beam parameters. To obtain simple similitude, variation of q& and € must be avoided. A group of foil-beam systems that can vibrate at the same q&% and € will be compared, and the comparison will include only vibrations of these systems * To infer Formulas [66a,b,c,d] from those given below Equations [12a,b] in Section 2, q, and qo being defined there by Equations [5a,b], note that q 2% and q,% are functions of q&% and € and, consequently, the same is true of the functions s,c, 5, and C. Elsewhere, replace q ang ® by (q,2)/2 or (a,2)/2, and aoe q@ by multiplying EI by (qet2/e 65 at a chosen q% and €. To facilitate thinking, choose one system out of the chosen group as a reference system and indicate the quantities that refer to it by adding a prime. For any other system of the group write & = \L', EI = eat(EI)". Thus within the group: Qt, oS EI PEAY with A and e« denoting arbitrary numbers that equal unity for the reference system. The factor 4 is introduced here because, in a simple change of scale with all dimensions increased in ratio A, I = ne The factor thus represents the effect on EI of any change in the shape of the beam and perhaps in E. Then, from Equations [66a,b], among the beams of the chosen group: g a 8 859 Cs [67a,b,c] To make & the same for all of the vibrations considered, the factor KAG, 2° /(EI) in Equation [65] must be the same for all beams. Hence, it must be that KAG, « ede. Also, p= pA in terms of the density ome of the beam material. Thus fe) A fe) tat es ae pk'G) KAG, ! ' ! irs ' ! 1 1 or fe) ' 1 Ros PENG T= ale) : Se ! Sy ' u oh KG, For the reference beam, tT = 1. Thus among the vibrating beams it must be that KAG, « e\- ee ts ae 5 He = [68a,b,c] 66 The variation of w is inferred from Equation [65b]. For any two beams made of the same material and with. cross sections of the same shape, so that KG, and t are the same, A« ane and I « ent, The validity of the fundamental Equations [46a,b], [47a,b], and [48a,b] for all beams must now be secured. Tentatively, the rule that all terms in a given equation must vary in the same ratio from one vibrating system to another will be followed. Equations [48a,b] read, for the reference system (denoted here by primes) and for any other chosen system, in view of Equations [67a,b,c]: ! _— q ' q i] 5 q —_ 1 1 i q Ya = Saal” Gig’ 3 Yo = Gah Goa sail ay az apo Voi F+oe; Qe, = SP + Se on Eh eh eA edo Here it can be assumed that the time factor in F is the same for all beams (perhaps "sin wt"). Furthermore, since the amplitude of any vibration may be arbitrarily varied, it does no harm to assume that the amplitudes are such that (a) ,/e) F is the same for all beams and hence equal to asf’: Then F <« edAF'. Also, according to the general rule being followed, the second term (afer) G must then equal ere so that G = ance. It follows that the phase of G is the same for all beams (perhaps G ¢« ene sin (wt + ¢), ¢ constant). Thus v= We for the vibrations considered, but substitution for F and G gives on = Ob/A. Summarizing, among the chosen vibrating beams: 2 Peeks @Gesr 3 Yi Gi 3 80 « 1/) [684a,e,f,¢] 67 It should be noted that Nis is in the same phase on all beams, and so is 8o> but these phases may differ from those of F and G. Consider next the beam-foil relations expressed by Equations [47a,b], which may be written: To make F/k, and C/k,, vary the same as Vv) and 85> respectively, assume Ky = Fk, © XG. “then Ricci 9 Kee ed 9) we vged ) 6 =e. 1 [een em) Turning now to the foil, assume that the values F', G', v', and 6! for the reference system satisfy Equations [46a,b] for a certain set of values of the foil parameters. Make these equations valid for all other systems of the group in such a way that all terms of each equation vary in the same ratio; that is, so that they are « F « eA in [46a] and « G« ed in [46b]. The necessary variation of the foil parameters and of S is found to be: D m TEX 3; hea 3 DUS 3 I «x TEA ec, «= en Volew ohne oh fae 83 co alfhe 2 3 4 B © Tene: Go Ten 8 ED = TEA 3 L«= > Ore, ator 35 G, etoyel EL have Theodorsen's values as stated following Equations [45a,b], then I) GK 2 ea ys Lp & terA/t 68 where ¢t is the specific gravity of the ambient fluid. Examples of the calculations leading to these expressions are: : ee 2 ss 2 2 ays : (a) Since v = - wv,mv « m v= (m/\°t) * 1; this is « eX if m« teh. I, is obtained similarly. (b) To make (c, + BS) v « ed, where v « wy « 1/(i\vYt), it is al necessary that (ce, + BS) « ene vt. This is most simply secured by making Cy « BS « aie YE o 2 2 2 Palani os : 2 (c) To make BS 6 « ek, let BS « cd. Dividing this by BS « ed” vr gives S « 1/vt, and dividing S « lv into BS = ed°vT gives B« eel”. (d) If Theodorsen's values for the lift are used, Sp) pee Gans SS) Sau “yi co i NM Jw whence b= C/B - VE /B =X ; e = (C/B) - (3/2) b= X Also, pL, = B/(mb) « teA. Write p = cP,» & being the specific gravity of the fluid bathing the foil and Py the density of pure water. Then, P. being the same for all foils, 2, « ted/t. fe As a spécial case, if the systems actually differ only in scale, all linear dimensions varying as A, then e = t = 1 and the similitude just D 2 described requires that m« > I, CIA on GN 5 Cy & a. Lepr, Be Nae Ca 3, E,, « 4 (orb «ea 2, « X), with S and ¢ remaining constant for fe all systems. Also, w = l1/d. 69 The general similitude described here might conceivably serve as the basis for a model test, although the limited variability of S is likely to be imeonvententen In practical cases the influence of shear warping tends to be small. It may be sufficiently accurate to omit this effect by letting K >= (hence — = 0). For this case write the relation u = (op A/o, A") u' in the form: Pb Thus te is replaced by n or t by n/e in all formulas, and Onk Ve. «\z ee 7 Since € «= RI/2", € can easily be varied without affecting A or n; for example, by changing all linear dimensions of the cross sections parallel to the principal plane of the beam in a ratio r and the perpendicular dimensions in the ratio 1/r, thus A is left unchanged. Variation of S in any desired ratio can be effected in the same manner. preKGe * With gq invariant, T= aD! KG - Since p,, K, and G, cannot vary much, ie) e then the variation of S is limited. 70 7. TWO SIMPLE CASES OF FLUTTER The mathematical theory of flutter is so complicated in realistic cases that it is not easy to form intuitive ideas of the flutter process. For this reason a certain interest may attach to simple cases that are easily understood. It apprears that typical flutter can occur only if at least two degrees of freedom are coupled together; and commonly, but not always, there is a difference of phase between the two motions. Two relatively simple examples are described in detail. Although the theory given here might conceivably be roughly applicable to some actual problem, the principal aim in devising these cases has been to make them simple. In both cases a rigid foil is assumed to be immersed in a uniform stream, and for the lift F_ on the foil the simple steady-motion approxi- L e ° * mation is used: we = — BSv + BS“6 In steady motion, 6@is conveniently assumed to represent merely a small inclination of the foil to the approaching stream and v, a slow perpendicular velocity of translation. This formula is known to provide a fair approximation also when 6 varies slowly, as is assumed here. Then @ must represent rotation about an axis parallel to the foil and perpen- dicular to the stream, while v may be regarded as denoting a displacement of a line on the foil located at the axis of rotation, v being * In Appendix H of Reference 4, it is shown that the effect of the 6 term is relatively small if pe is small. Hence, the 6 term is dropped here. 12 perpendicular to both the stream and the foil. The assumed axis of rotation may be displaced into any parallel position in the plane of the foil without affecting the motion provided an appropriate change is made in v. The foil is assumed to be attached to some structure but it is assumed to be so long that disturbance of the lift by the presence of the structure may reasonably be neglected. 7.1 FOIL ATTACHED ELASTICALLY WITH DAMPING TO AN IMMOBILE BASE Let the foil be attached so that it can undergo displacements v and @ as described and, to simplify the formulas, locate the 6 axis so that the lift acts through it. When the foil is displaced, let an elastic force - k,v and a frictional force - c,v act on it along the same line as a a, F. and positive in the same direction. Also let an elastic moment - k,6 L and a frictional moment - cn 6 act on it about the 6 axis and positive in the same direction as 6. Let the foil be so positioned that the lift acts at a distance L from the center of mass of the foil,” L being positive when the lift is dis- placed from the center of mass toward the approaching stream. Denote by m the foil mass and by I its moment of sonatas about an axis drawn through its center of mass and parallel to the 6 axis. See Figure 4 where directions are shown for positive v, Vv, OQ, Wy Fh and L. * With § terms present, no single location for action of Fy, was con- sidered. In the present case such a location is specified. L here is not the same as in Equation [45b]. ** Note that here the moment of inertia of the foil is denoted by I instead of I,, to allow use of I, for a rigid body in Section 7.2. Te Figure 4 — Forces on and Motions of a Foil Attached Elastically to an Immobile Base ie Since the displacement of the center of mass of the foil is v - L6, then the equations of motion for the foil will be: Sane NE(s) eves Mee Nae une 0 C50 = 0 [69b ] 2 The v and 6 motions are thus coupled together inertially through the Lm terms. To search for harmonic motion, make the usual mathematical assumption . 2 te that ve 6 =< gas. Canceling en results in the equations: 2 : 2 2 [-wom + k, + iw Kors BS)] v + (w Lm - BS”) @=0 [70a] imum; & flea (i Siam) ok. > tae] © S 6 [700 ] 2 2 Equating to zero the determinant of these two equations gives: = 358) ) lene Geo Sa) & k, + iwe,] + iw (c > (eocm an his aL alt 2 Eien (ate = Ese) = © 74 Multiplying out, then equating to zero separately the real and imaginary parts and dividing the latter by iw, on the assumption that w # 0, yields: 2 ie ba - wo [mk + (I + ern) k 2 1 2 7 Gs (c) + BS) - LmBs ] + kk, = 0 [71a] =u" [(e, + BS)(I + ea) + cym] + (co) + 3S) Ik. > eile, = © [7lb ] The second equation can also be written (CHEB) picentcuk: Ae = pee a ee OMe ACD [71b! ] 2 + ik 4p JE te m (c, BS ) ( m) @, In both equations all symbols denote positive quantities except that it may be interesting to try to make Cy > Co» kK) > Ky or S zero. Hence, Equation [71b] cannot require a negative value of oan Equations [7la,b] fix ne and S but, in general, an algebraic solution is not practical. Substituting ne from [7lb'] into [7la] and clearing of fractions gives a quartic equation in S, or a cubic if either is or k, is 2 zero (or both), so that the term k_k, disappears and Equation [7la] can be v2 divided through by a 5 The most feasible method of numerical solution would probably be to assume a value of S, calculate aie by [71b'], and then calculate the value of the left member of [7la]. Successive approximation and interpolation would then be used in order to find the values of S for which Equation [7la] holds. The smallest positive value of S thus found will undoubtedly be a critical flutter speed, provided L # 0 so as to couple the v and @ motions. 12 If S = 0 but Equation [71b'] furnishes a positive value for ue then v and §@ will vibrate in a certain ratio, decreasing to zero with time if either Cc) > O or cy > 0. If, then S is made slightly positive, the term BSv will predominate over - BS“6 in Equation [69a] and will itself damp out the motion even if c = cy = 0. As S increases further, however, the damping effect may decrease because of the increase in the value of the - BS“ 6 term until, finally, at a certain speed S. harmonic vibration may again become possible. Then Sa is the critical flutter speed. As S increases above See the vibration will probably become unstable. In special cases further conclusions can be drawn, and it is these conclusions that may give a certain interest to the case of a foil mounted on a rigid base: (a) Assure cy = 0 but kp > 0, L #0, and either S > 0 or c, > 0 or alk both. Then the factor Cy + BS can be canceled out of Equation Alig) I)e size is found when substituting for awe that Kk, disappears from [7la]; finally Equations [7la,b'] can be written: k we = —+ g BS“ 2 EOP ERAT oem IE ee Jeri IC op lim [72a,b] Inspection of Equations [69a,b] shows why this case is so simple. If RE= nee, GS velwe specified in Equation [72b], the two 6 terms in Equations [69a] cancel each other. Hence, if k, > 0, v must either be zero or execute a damped oscillation ending in zero. Or, if k, = 0, v must als ultimately reduce to a constant value since the only other possible ut solution of [69a] is then the nonharmonic solution v «= e "’, where 76 mS (cy + BS) me In either case, ultimately v = 0 and the term - Lmv in Equation [69b] disappears; and the v and @ equations are independent of each other. According to Equation [69b], © vibrates steadily with w, as given by Equation [72a]. (1S) ta ke = 0 and either oS 0 or kK, = 0 or both, and if also at least one of cy and S is positive, then We = 0 according to Equation [7lb'], and no harmonic vibration with w # 0 is possible. When k,, = 0 and c, = 0, solutions of Equations [69a,b] are easily 2 found in the form of finite series in powers of t. The simplest of these may be mentioned; namely,@ = a and, if k, #0, v= BS“a PS. 2 OW as? 1 1 2 ik 2 @ bt GC wRES >O, ye ESmoum + 8B, where a and 8 are arbitrary a 1 ec, + BS 5 constants and t denotes the time. If ct # 0, FL = — BSv + BS 6 = BS-a = Re thee SOF (eo, © BS)2 6 BS"a, ao thet BSP G ESea = c Jt al Thus, in either case, a steady lift on the foil serves to balance another 1 v and Fo = cjV- force. (c) If L = 0, Equation [71a] does not contain BS* and it is more illuminating to return either to Equations [70a,b] or to the more specific equations of motion, Equations [69a,b], which now read: un : mv kv (ce, Ta Sk OS e.0 SO 2) 2 Now there are just two possibilities for steady harmonic vibration: * Here, this definition of u, is restricted to the case at hand and should not be confused with the definition of u generally used throughout this report. TI (a) Osis Wo eis Ms S03 WS 58 20 k Zoe (b) Cos OF k, > OF Wr = = Case (b) is not a typical flutter case since S may have any value, but it may serve to illustrate in a simple case a typical feature of flutter vibrations in that the term BS-6 forces a vibration of v in shifted phase, the energy abstracted by damping being furnished by the stream. This feature may be worth exploring in detail. By properly choosing the zero of time, 8 can be expressed thus: SC ain ws 2 Ue S k,/I where C denotes an arbitrary constant. The accompanying forced vibration of v can be written, in terms of amplitude factors a and 8 that remain to be found, as follows: v = aC sin wt + BC cos wt hence v = - wBC sin wt + waC cos wt as 2 v=- pane sin wt - w BC cos wt When these expressions are substituted into the mv equation, the sin wt and cos wt terms must balance separately. Hence, after canceling out (Y Ein (he he (C Cos (he? 2 2 A (5 - mw )a- (c, + BS) w8 = BS (for sin ut) (c) + BS) wa + (kK, Sim) BO (rope cos wt ) 78 Solving for a and 8 gives: where The phase angle of v relative to 6 is thus som (B/a) or - Seni ey, we, + BS) /(k) — mw Thts phase shtft enables BS76 to supply the energy aL lost from the fotl through the vy term. It may be worthwhile to verify this compensation. The rate of energy loss is (c_ + BS) We the rate of energy supply is ale 32 ene 2 WwW 20m, Were aac (a 2 } BS Beane +a cos wt - 208 sin wt cos wt). The time average, over a cycle, of sin’ wt or cos“wt is 1/2, whereas sin wt cos wt e) averages zero. Hence, averages (1/2) wees (ae + B~), but ae 4 Ga pl (sae The average rate of energy loss is, therefore: = (ic, = ES) ees wo (asa) On the other hand, ev = - wBCo sin” wt + ricto sin wt cos wt Thus the rate of energy gain, or BS“ ev, averages: dL = 2 = 5 wBC°BS* = = nee> De (ce, + BS) (BS-) This equals the average rate of energy loss. In coneluston, to have a typical flutter situation with a positive critical flutter speed, it is clearly necessary in the arrangement under 1g discussion that the center of mass of the foil does not lie on the axis of rotation; hence, that L # 0 and that, consequently, the v and 6 motions are mechanically coupled together. Furthermore, some elastic attachment WO) 1dXS INES aS MoCesSchay, Mines. Giese ie > @ Ofek, DS OZ alse Me 5 1 = 0, iy is 2 also necessary that the rotational motion be damped (c, > 0) to keep v from wandering off. 7.2 FOIL ELASTICALLY MOUNTED WITH DAMPING ON A FREE RIGID BODY Let the foil be attached to a free rigid body so that it will tur about an axis fixed in the body, drawn parallel to the foil axis and perpendicular to the stream. Let the center of mass of the foil lie on the axis, and denote the distances from the axis to the center of mass of the bode and to the center of lift on the foil by h, and L, respectively. These three points are assumed to lie on a line that is parallel to the stream in the undisplaced position of the system, and Ay and L are positive toward the approaching stream. Positive directions are shown in Figure 5. Let the body have a principal axis of inertia at its center of mass parallel to the axis of rotation, and denote the masses of foil and eae by m and m, and their moments of inertia about their center of mass by I and I)» respectively. Assume that motion of the axis of rotation in the direction of the stream is somehow prevented. * ho and m, refer now to the rigid body, not the foil; hence the change of notation from h and m used in previous sections for the foil. 80 Figure 5 - Forces on and Motions of a Foil Elastically Mounted with Damping on a Free Rigid Body In Section 7.2 only (see notations) m and m, denote the masses of the foil and body and I and I, their moments about their centers of mass, respectively. 81 The motions of foil and body will then be two dimensional. Let v denote displacement of the axis of rotation in a direction perpendicular to the foil and to the stream and let 6 and 86 denote the small rotations of foil and body, respectively, about the axis. 6 and 84 are zero in the undisplaced position of the system and positive from a direction opposite to that of the stream toward positive v; see Figure 2. Assume that a spring and also an internal damping mechanism are present, exerting turning moments - k (6 - 84) are Gee 6.) on the foil and k (6 - 84) +c (6 = 8) on the body. Assume also an external damping force - Cy acting on the foil along the same line as the lift Foe In general, there is a force F, positive toward positive v, acting on the body at the axis and a corresponding reaction -F acting on the foil. The displacement of the center of mass of the body is v + no 8: The primary equations of motion for foil and body will then be: WQS sk (0 = OS © (0 > Bo) vb (= Chas im) mv =- F-cvtFe a a (oho yer, £60 Sk (0) Se (60) ohF e) lomme) [e) Oo fo) (oe) Eliminating F and inserting F, = - BSY + BS-6 gives: (m +m) ¥ + homé, + (c, + BS) v - BS“6 = 0 [73a] fe Nea be ‘ 2 I6 +c (8 =6.) +k (0 0m) + L [(e, + BS) v - BS 6] =0 (73b ] 2) . we se is be: ¥ ee ms (me. + hem.) @,-¢ (6 8.) k (6 -@.) +homv = 0 (73c] A more simple equation may be obtained by subtracting L times Equation [73a] from the sum of Equations [73b] and [73c], namely: 82 This last equation can be integrated at once as follows: 2 at [hom eaipelival emits m,)] v+I60+ (0 ve JesaL h Lm, ) 6, =a + Bt where a and 8 are arbitrary constants and t denotes the time. A corresponding special solution of [73a,b,c] exists if c, = 0, namely: @=@ =y 3 v= 6 + ySt where y and 6 are constants which are easily expressible in terms of a and 8. This solution represents a steady lateral motion in which the foil does not disturb the eneame | By means of the integrated equation, it is possible to eliminate i). and thereby reduce the equations of motion from three to two. It seems simpler, however, to work with all three equations. In looking for harmonic motion, it is apparently less complicated to use as equations of motion Equations [73a] and [73c], with signs reversed, and Equation [74]. Assuming that in these equations v, 6, and 8, are Lot proportional toe , then canceling ge. and also dividing Equation [74] by -w-, gives the following result, provided a 53 O32 * The foil does not disturb the stream because v = 6 represents a steady displacement with 6 = 0, whereas @ = y and v = ySt represent a steady motion in which the foil slips chordwise through the water thus: Le Z as seen by observer as seen by water v = YS + 83 2 Z . } [-—w" (m + m,) + iw (cs + BS)] v - BS°6 - w ine. = 0 whomy + (k + iwc) 6 + fos (ts nom) - k - iwe] 6, = 0 [aon oi (> mi) yom So (Ge Shen Shim) 6. 6 O (ome) (e) (e) (0) (©) ©) © ie) The determinant of the coefficients of v, 6, and 8, in these equations has the value: (m + m,) + iw (ce. + BS)] (k + iwe) (I, + nom, = h Lm, ) D) ' I In -k- - + ( d om) iwe]{h m. L (m m,) “nee © wan mm (eS Amelia = mom) ome) (ome) 2° 0 O Beam (i Shen ohim) or fer (r & hon) ok © ame) (0) (©) fo) (ome) y' © oO (one) [-w"(m + m,) + iw (c, + BS)] Multiplying out and equating to zero separately the real part and the imaginary part divided by iw yields: wT [1, (m+ m) + hommy) - uf { E(T + Ig)(m + mg) + n2nmg] 2 2 @ (ie, > BS)(e ee kin thn) ae IS Gf ella. = 1G (Gas a) oO + wL (I, (m+ m,) + nomm) f= 0 [75a] -w° {(c, + BS) I (I, + hom) * c[(I + I5)(m + my) + nom] f F ) +k(c + BS)(I +I +h‘m -h Im a eo) [oume) Omm© on, provided ce > ONand cy teBsor sO; a 8h ete Sie oe Shen San) 6 use [nin 5 ih Ga om i) a e) (one) Oo oOo (oe) 2 Oo JS en GEE GGT OS UA. Go on en ee (7502 | (c + BS) I (I + mn )+e[(I1 +1 )(m+m) + hom ] a ° (ome) O ° © © * IC3¢ cc, = 0 and S = 0, Equation [75a] gives, as the natural frequency of vibration of the undamped system without the stream: By re I (m +m) Ww = Ww = ———.]) 2 fo) I Te (m + m. ) + homm It should be noted that the mathematical complexity of Equations [75a,b] is no greater than that of Equations [7la,b] in the case of the foil attached elastically to an immobile base, although here three variables (v, 6, 6.) and three corresponding equations are involved as against two in the previous case. The reason for the simplicity here lies in the occurrence of the factor ie in all three terms of Equation [74]. It is not clear in what general class of systems this sort of reduction in complexity will occur. The smallest positive value of S for which both [75a] and [75b] are true will undoubtedly be a critical speed for the inception of flutter. At a lower speed any motion will be damped out due to the BSv term and, perhaps, other causes if S > 0; or, if S = 0, provided at least one of the constants c, and c is positive, a Thencase. ic) V0) ific —Onbuthce + sBSa> One Equataon [75b'] gives: 6 tel hen Sch m H( fo) (oe) 22 [76] 2 Is + aa € ! all * ce, = O means c = O or ene O or both. 85 The necessary value of ge may be found most easily by rearranging Equation [75a], giving the value of the determinant as follows: 2 2 2 2 -w (m+ m,){ (I, + hom, - Ihgm,) - Ilu” (1g + hom) - k)} a 2 + uTneMecear Think +4BSg laaenneme=) nun)! (e) ©) (e}) 6) (e) ORO) (0) ©) 2 2 + [hom - L(m+m,)] {wk hom, - BS oo [w°(I, + hom.) - k]} = 0 (Here the leading factors are the coefficients of v in the v, 6, 86 equations, the coefficients of v having been chosen because they contain the only imaginary term when c = 0.) Now the value of i given by Equation [76] causes the first brace (i.e., the coefficient of en + m,)) to vanish. The second brace can be made to vanish by assigning the proper value to BS“. Then it is easily seen that the third brace also vanishes. For, the ratio of the first term in the third brace to the first term in the second brace, and the similar ratio of the second terms, are, respectively; 2 (ir jam) a ik kg 2 ° oe) i 1 on = Eyam Oo (ome) (6) {(C) But the vanishing of the first brace makes these fractions equal. Hence, the third brace is proportional to the second, so that if the second is made zero, the third brace also vanishes, and, consequently, the whole determinant is zero. The value of BS° obtained in this way, when c = 0 but c, + is} > 0, 189 86 [77] m - Llhm IG. Ai Yay fo) fe) ©) © Incidentally, it may be remarked that if the values of Re and of BS- given by [76] and [77] are substituted into the v, 6, and 6. equations with ec = 0, it is found that If Equations [76] and [77] yield positive values for ae and sam then S is the critical flutter speed for the system. Even if Gris 0, any motion at a smaller speed will be damped out by the - BSv term ain) ithe deity A critical flutter speed will not exist for all values of ho and L. In Equations [76] and [77], only h, and L can be negative. Hence to make aif? and Se, as given by these equations, positive, it is necessary that: ee + nom, It ap i + hom - Lh om, > @ 2 aa -L>0O fo) (The sign of a fraction is not changed if the fraction is inverted or if numerator and denominator are multiplied or divided by the same number. ) If h_ > 0, the second inequality requires that: (e) ae Ih Sie oP fe) hm (ome) and then the first inequality is satisfied also. Here, L may be positive or negative. If h, < 0, substitute h, = - |no|- Then QT IE ae aC I {joo —2 ne = bp fe) m |h | ©’ © fe) ol im ae (o)" ©) with the left-hand inequality coming from the first of the basic inequalities >) and the right-hand inequality from the second. (Note that no°/|h| = [h, Thus . hy and L may both be positive, as drawn in Figure 4; if he < 0, L must be negative also. 88 8. SUMMARY A theory has been advanced for determining the vibrations, including flutter, of a hydrofoil craft. In general, the craft is treated as a rigid foil flexibly attached to a uniform mass-elastic free-free beam immersed in a fluid moving with wniform velocity. The hydrodynamic force and moment on the foil are represented by two-dimensional quasi-steady expressions. The sequence of steps undertaken in devising this theory is summarized as follows: 1. In Introduction the background and objective of this report is discussed. 2. In Section 2 influence and inertia coefficients are derived for a uniform beam that is maintained in harmonic vibration with two degrees of freedom by a transverse force F and a moment or couple G acting at one end of the beam and in the same principal plane of the beam. Shear warping of the beam is included but rotary inertia and damping are not. These coefficients are useful in treating the vibrations of a system composed of a uniform beam attached at one end to another structure (e.g., a foil). 3. In Section 3, for convenience of analysis, the formulation for these coefficients is simplified by neglecting shear warping. Various interesting cases are discussed, in the order of increasing frequency, for the vibrations of a forced, undamped, uniform beam without shear warping and rotary inertia. 4. In Section 4, the analysis is extended to provide formulas for these coefficients in the case where the external force F and the moment 89 G act at an tntermediate point instead of at an end of the uniform beam; shear warping is included but rotary inertia and damping are ignored. 5. In Section 5, the analysis is further extended to include the effects of both external (Rayleigh) and internal damping, the latter being associated with bending strains; shear warping is included but rotary inertia is ignored. It is shown that the equations derived are valid even if the external force F and the moment G are not vibrating in phase. Formulations are developed from which the final calculation of the co- efficients can be made. The procedure for making these calculations is clearly indicated. 6. In Section 6, flutter of a rigid foil flexibly attached to a uniform beam is considered, the entire system being immersed in a stream of fluid approaching at uniform speed. Expressions for the resulting lift forces and moment on the foil are adapted from Theodorsen's equations for a uniform foil of infinite length vibrating harmonically. An approximation closer than the common steady-motion approximation is used. For greater generality, allowance is made for still-water damping (structural plus fluid) as well as for hydrodynamic damping. Equations of motion for the foil are related (or coupled) to the equations previously derived for the response of the beam subject to harmonic loads; i.e., the coefficients previously derived for the beam and Hooke's law for the flexible connecting structure serve to couple the motions of the foil to the harmonic loads imposed upon it by the motions of the beam. A method is described for finding the critical flutter speed and frequency for this system, at which a steady vibration of the foil and beam is possible in 90 spite of existing damping actions. The effects of various damping terms are discussed and three special cases for which the determination of the critical flutter speed and frequency are of interest are treated. These cases occur for (1) damping which arises only from terms in the lift force and associated moment, damping due to friction in the surrounding fluid being excluded; (2) a foil rigidly mounted on a beam; and (3) a low- frequency approximation, in which shear effects in the beam may be ignored. For the latter case, it is shown that any low-frequency harmonic vibration of the foil that can occur with the beam attached can also occur without the beam and at the same frequency and stream speed provided certain simple changes specified in the text are made in the foil. In subsection 6.6, a simple form of similitude is described. The parameters of the foil-beam system are assumed to be changed in certain ratios with resulting changes in the frequency of vibration and in the critical flutter speed. The necessary restrictions on the ratios are specified in detail. The relations derived here may be of interest in designing a model to represent a much larger system. 7. In Section 7, two simple cases of flutter are treated in order to reduce the mathematical complexity to a point where intuitive ideas of the flutter process are relatively easy to arrive at. The cases discussed are for (1) a foil attached elastically to an immobile base; and (2) a foil elastically mounted with damping on a free rigid body. For these cases various concepts, relatively simple equations, and criteria for flutter are presented; and methods are described for determining the critical flutter speed and frequency from these equations. 91 REFERENCES 1. Antkowiak, E. T., "Hull Vibration in the DD-931 Class Destroyers," Boston Naval Shipyard Evaluation Report R-10 (20 Aug 1956). 2. Kilcullen, A., "Hydroelasticity Research Program of the Fundamental Research Branch (Code 771), Technical Note SML-760-64 (Aug 1963). 3. Kennard, E. H. and Leibowitz, R. C., "Theory of Rudder-Diving Plane- Ship Vibrations and Flutter, Including Methods of Solution," David Taylor Model Basin Report 1507 (Feb 1962). 4. Leibowitz, R. C. and Belz, D. J., "Comparison of Theory and Experiment for Marine Control-Surface Flutter,'’ David Taylor Model Basin Report 1567 (Aug 1962). Also paper presented at Fourth Symposium on Naval Hydrodynamics on "Ship Propulsion and Hydroelasticity" at Washington, D. C., 27-31 Aug 1962, Vol. 3 of ACR-73, Office of Naval Research. 5. Leibowitz, R. C. and Belz, D. J., "A Procedure for Computing the Hydro- elastic Parameters for a Rudder in a Free Stream," David Taylor Model Basin Report 1508 (Apr 1962). 6. Leibowitz, R. C. and Kilcullen, A., "A Preliminary Guide for the Experimental Determination of Structural Damping and Virtual Mass of Control Surfaces," Technical Note SML-760-47, (Feb 1963). 92 7. Leibowitz, R. C. and Kilcullen, A., "Experimental Determination of Structural and Still Water Damping and Virtual Mass of Control Surfaces ," David Taylor Model Basin Report 1836 (in preparation). 8. Leibowitz, R. C. and Kennard, E. H., "Theoretical and Experimental Determination of Damping Constants of One-to-Three Dimensional Vibrating Systems," David Taylor Model Basin Report 1770 (June 1964). 9. Leibowitz, R. C. and Strandhagen, A. G., "Theory of Static and Dynamic Loads on a Rudder in a Steady Turn, David Taylor Model Basin Report 1647 (Feb 1963). \ 10. Leibowitz, R. C. and Kennard, E. H., "Theory of Forces and Moments on Rudders during a Steady Horizontal Turn," David Taylor Model Basin Report (in preparation). ll. Leibowitz, R. C. and Kennard, E. H., "Theory of Freely Vibrating Nonuniform Beams, Including Methods of Solution and Application to Ships," David Taylor Model Basin Report 1317 (May 1961). 12. Leibowitz, R. C., "Natural Modes and Frequencies of Vertical Vibration " of a Beam with an Attached Sprung Mass," David Taylor Model Basin Report 1215 (Sep 1958). 13. Kennard, E. H., "Forced Vibrations of Beams and the Effect of Sprung Masses ," David Taylor Model Basin Report 955 (Jul 1955). 14. McGoldrick, R. T. and Jewell, D. A., "A Control-Surface Flutter Study in the Field of Naval Architecture," David Taylor Model Basin Report 1222 (Sep 1959). 93 : Litt : wie g. \) ‘ Se Oa aA Uk i : \) z sl ate) Uy ASP anaes Pi ea ile a nave 1% en eee | . . a S i J 9 perry 4 te ake it on 7 a fg TPP gual ate } he i Yas S is Leeks ye ¥ . Y vy ri hoe j Heh Wee. am Trae: AL a % A : } y if j ' . bit \ é . bv 4 ey . 7 re sre marge i : oie Hey Pre S Mac Oud . al tay TE ote erie Loe eee A i: 4 Sint a 5 i : Copies 12 we Ye BPP BP ee ee Pee PP a INITIAL DISTRIBUTION CHBUSHIPS 3 Tech Info Br (Code 210L) 1 Lab Mgt (Code 320) Applied Res (Code 340) Ship Sil Br (Code 345) Prelim Des Br (Code 420) Hull Des Br (Code 440) Sci & Res Sec (Code 442) Structures Sec (Code 443) CHONR 1 Math Sci Div (Code 430) 1 Fluid Dyn Br (Code 438) CHBUWEPS CO & DIR, USNMEL CO & DIR, USNMDL CDR, USNOL CDR, USNOTS, China Lake CDR, USNOTS, Pasadena DIR, USNRL CO, USNROTC & NAVADMINU MIT 0 in C, PGSCOL, Webb NAVSHIPYD LBEACH (Code 240) NAVSHIPYD PEARL (Code 240) NAVSHIPYD PUG (Code 240) NAVSHIPYD SFRAN (Code 240) NAVSHIPYD NORVA (Code 240) NAVSHIPYD PHILA (Code 240) NAVSHIPYD BSN (Code 240) NAVSHIPYD NYK 1 Des Supt (Code 240) 1 MATLAB (Code 912b) CMDT, USCG 1 Secy, Ship Struc Conm DIR, Natl BuStand ADM, MARAD DIR, NASA 1 Ship Struc Comm Pee ew 95 Copies 20 1 Sw ee DDC WHOL St. Anthony Falls Hydrau Lab MIT, Dept of NAME Webb Inst New York Unit 1 Dept of Meteorology 1 Fluid Mech Lab Inst of Hydrau Res, Univ of Iowa Sch of Engin & Arch, Catholic Univ Inst of Engin Res, Univ of California 1 Attn: Prof Schade Univ of Michigan | Exper Naval Tank 1 Dept of Engin Mech 1 Dr. Finn Michelson, Dept. of Nav Arch. Hudson Lab, Columbia Univ Univ of Notre Dame Attn: Prof. A. Strandhagen, Head, Dept of kng Med APL, JH Univ Fluid Dyn Res Grp, MIT 1 Mr. John Dugundsi 1 Mr. Holt Ashley Dept of Applied Mech, SWRI 1 Dr. H. Norman Abramson 1 Mr. Wen-Hwa Chu DIR, Davidson Lab, SIT 1 Mr. Charles J. Henry 1 Dr. Paul J. Kaplan Computer Eng Associates NNSB & DD Co. Attn: Mr. Montgomery Gen Dyn, EB Div Copies 2 SNAME 1 Hull Struc Comm bo Grumman Aircraft Eng Corp 2 Technical Research Group 2 Aerial Wax, Syosset, New York il J. G. Eng. Res. Associates, 3831 Menlo Drive, Baltimore 15, Md. Engineering Index, New York Dr. E. H. Kennard, 4057 Lenango Road, Claremont, Calif. il Dr. Theodore Theodorsen, Republic Aircraft Corp, Farmingdale, L. I., N.Y. Il Mr. I. E. Garrick, Langley Res Ctr, NASA, Langley Field, Virginia IL Migeeedicn Die CisSipl, Aeroelastic & Structures Res Lab, MIT ill Mr. Maurice Sevik, College of Engin & Arch, Penn State Univ, ORL all Mr. Alexander H. Flax, Cornel] Aero Lab, Inc. al Mr. R. T. McGoldrick Box 293, Sheffield, Mass. il MacNeal Schwendlek Corp, 2556 Mission Street San Marino, Calif Il Prof M. Landahl, Dept of Aero & Astro, MIT, Cambridge 39, Mass 2; Hydronautics Inc. il! Press.) 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