340 AIRCRAFT ENGINEERING December 1943, By J . Lockwood Taylor, D.Sc. RDINARILY , for th e purpos e of strength value for dθ/dr as before. Further numerical calculations, as well as in estimating substitution yields, for th e same improved wood bending and torsional rigidity with a view blade previously taken as an example, a ratio t o deriving blade deflections and investigating of induced tension or compression to shear of flutter characteristics, the blade is regarded as 2·6 . In the practical case, the direction of flat (i.e. as if designed for zero pitch) ; any twist, considering centrifugal loads only, or th e effects of th e twiste d form of th e blade in caus same loads plus an aerodynamic C.P. in front ing departures from the classical bending and of mid-chord, is such as to cause compression so that for Q =0·1 M as above, the extr a strain torsion theories being regarded as secondary. a t the leading and trailing edges (the section energy is about 2·5 per cent of the bending I t has never, however, been proved that they face being flat), so that the maximum stress S.E., leading to this amount of reduction in th e are actually negligible, and an approximate quoted above is a relief of th e ordinar y bending effective bending rigidity. The blade in ques analysis indicates that they may in some cases stress. The addition to the ordinary bending tion is a fairly narrow one, and the correction become appreciable. stress a t mid-chord is abou t one-quarter of this, increases rapidly with blade width, as may be or 0·6 5 of th e maximu m torsion shear stress. seen from the formulae. Thus twice th e width Torsion Stresses Induced by Bending of chord means four times the torqu e (for given Increase in Torsional Rigidity These arise from the fact that the bending bending moment) and 16 times the added stresses in the blade under thrust must act in strain energy, which is then 40 per cent of th e By integration over the section, the direct a direction parallel to the blade surface in a bending S.E., a very appreciable correction strain energy can be found ; since this is an thi n blade, the surface of which is free from t o the rigidity. addition to the total energy at a given twist normal stress and tangential shear. This dα/dr, it implies an addition to the torsional means tha t the stress forces in th e leading por rigidity constant, Torque ÷ da/dr. Detailed Longitudinal Stresses Induced by Torsion tion of the blade are inclined a t a small angle calculations have been made for three types of Somewhat similar to th e effect of th e initial t o the blad e axis and t o th e forces in th e trail- section, a circular segment, a semi-ellipse, and, geometrical twist of th e blade form in causing ing portion, viewing the blade edgewise. The as a check on th e effect of asymmetry, an aero torsion shear stresses to be generated by the components of these forces in the plane of th e foil-type section. For th e segment, the added bending of the blade, a twist due to external blade section, normal to the axis, combine in torsional rigidity is 0·05Eb5t(dθ/dr)2; for the applied couples in practice, superposed on th e general to give a twisting moment, resulting semi-ellipse the value of the numerical constant initial twist of the blade as made, gives rise to in torsion shear stresses a t th e section, additive becomes 0·088 and for the aerofoil 0·074. longitudinal induced stresses. These however ar c t o those normally arising from the resultant The flat-blade torsional rigidity for the latter not linear over the section, but are parabolic- aerodynamic and centrifugal torque. ma y be taken to be 0·35 Nbt3, so that the ally distributed in the chordwise direction; proportionate increase due to induced direct The bend stresses are assumed to b e given b y there is therefore no actua l induced bend, apart stresses is 0·2 1 E/N.b4/t2.(dθ/dr)2. Again put th e usual formula, My/I, and it is readily seen from a small effect of asymmetry of section, ting E/N = 10, dθ/dr=0·7/R, this reduces to tha t the stress component causing twist is given, bu t the longitudinal stresses are quite appre 1·03 b4/t2R2; for the relatively narrow blade a t any point, by My/I.x.dθ/dr, where x is th e ciable, even for a blade of moderate width, and previously taken as a n example, substitution of distance, measured parallel to th e chord, from the corresponding strain energy is also im numerical values gives as the final result 23 th e axis of twist, and dθ/dr the blade twist, or portan t ; the effect is t o increase the torsional per cent extra torsional rigidity from the effect rat e of change of pitch angle with radius, as rigidity above the flat-blade value, in contrast considered. This increase in the value of the designed. The resulting twisting moment Q t o the bending rigidity, which has been seen rigidity usually taken for purposes of calcula follows b y integratio n in th e form M/I.dθ/dr∫yx2 t o be decreased for a twisted blade. tion would be substantially greater for a wider dA, the integra l being take n over th e area of th e I t is convenient to consider the strains in a blade, varying in proportion to the fourth section. The induced torsion stresses follow, blade subject to a torsion twist, da/dr, super power of th e blade-width. t o a first approximation, by applying the usual posed on an initial twist dθ/dr; the slope expression, 2Qt/J ; where t is maximu m thick of the leading or trailing edge of th e blade, a t a ness of blade section, and J its geometrical distance, say b from the blade axis, is easily rigidity constant (=41, nearly). A.R.C . Reports and seen to be b dθ/dr initially in relation to the I n terms of the width, 2b, and thickness, t, axis, in a n edge view (see Fig . 1). More gener Memorand a of a typical blade section, Q can be calculated ally, at any point in th e section of abscissa x as 0·03 b3t2 M/I.dθ/dr or 0·31b2/t.Mdθ/dr on R. & M. No. 1,901. 26 October, 1942. parallel to the chord, the corresponding slope puttin g I=0·096 bt3. A value 0·7/R , where R Electrode Potentials of Metals, By A. H. of a line drawn longitudinally on th e blade is is the semi-diameter of the airscrew, may be Turnbull and H. C. Davis. (3s. 6d.) x dθ/dr. An applied twist dα/dr, supposed to take n for dθ/dr, leading to Q=0·22M b2/tR. be small in relation to dθ/dr, alters the length Potential differences are measured between certain Substituting numerical values of b/t, b/R for a of such a line, and in the strained position metals and a saturated calomel electrode used as a particular design of improved wood blade standard. The actual potential difference between (shown dotted) the longitudinal strain amounts Q is found to have th e valu e 0 • \M, the rati o of any pair of dissimilar metals is then obtained by t o x2 dθ/dr.da/dr. The induced stresses pro Q to M remaining roughly constant over the difference. portional to th e strain however, violate the con outer part of the blade. Since the torsion Approximately fifty metals, alloys and plated dition of zero resultant normal traction across modulus J/2t is about twice J/y, the torsion specimens have been examined both in the "a s th e section, and correcting for this the strain received " condition and after their surfaces were shear stress is in th e region of 5 per cent of th e becomes (x2—c) dθ/dr.dα/dr; the constant c abraded with emery paper. maximum bend stress. This may no t appear a has the value 0·2b2 for a segmental type of In th e first series of measurement sea-water was very high shear, bu t i t is t o be superposed on th e blade section, and 0·25b2 for a semi-ellipse. used as the electrolyte, while in the final series normally reckoned shear stress from the ai r an d There is a further small correction, linear in y 3 per cent sodium chloride solution was used. The centrifugal loads, and is i n fact of th e same sign potentials were measured at intervals over an im t o be applied in order to fulfil the condition as these, tending to reduce the natural twist mersion period of 24 hours. All the measurements of zero bending moment about the major axis of the blade and increase the ti p pitch angle, were made with the electrolyte at a temperature of of the section, and for sections not symmetrical and of a similar order of magnitude; it is also 22 deg. C. about the minor axis (i.e. aerofoils in general)' t o be remembere d tha t the shear strength of an y a correction linear in x has t o be made also, to wood material is considerably less than the Data Sheet No. 10 give zero bending moment about the minor tensile or compressive strength. axis. I t is greatly regretted that owing to a tem porary editorial lapse the title of Data Sheet Taking the value (x2—0·262) dθ/dr.dα/dr for Modification in Bending Rigidity No. 10, which appeared on p . 318 of th e Novem th e induced strain (disregarding th e small linear ber issue of AIRCRAFT ENGINEERING, read From th e induced torque due t o bending just corrections for the moment), an approximate calculated, the additional strain energy in the " Equivalent Air Speed and Macchi Number numerical value for th e maximu m longitudinal blade corresponding with the torsion shear Chart. " The reference should have been, of stress due t o torsion may b e deduced. I t is seen stresses may be derived, and hence the change t o occur a t the blade edge, x=b, and to have course, to Mach Number, as appeared on the in rigidity. The torsion strain energy per unit th e value 0·8b2E dθ/dr.dα/dr as compared with Chart itself. length of blade is Q2/2NJ compared with Nt dα/dr for th e torsion shear stress, giving a This Data Sheet should, in fact, have been M2/2EI, in the usual notation, N being the ratio of longitudinal strain (tension or com numbered 11. Data Sheet No. 10 (Compara shear modulus and E Young's modulus. A pression) to shear of 0·8b2/t.E/N.dθ/dr, or tive Hardness Values) appeared on p. 240 of value E/N of 10 and J/I of 4 may be taken, 0·5 6 P/tR.E/N on using the same numerical th e August 1942 issue.
Aircraft Engineering and Aerospace Technology – Emerald Publishing
Published: Dec 1, 1943