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Aviation History
1931
1931 - 0291.PDF
MARCH 27, 1931 THE AIRCRAFT ENGINEER SUPPLEMENT TO FLIGHT for : ub-bay 1. In the table, a positive or upward concen- tra> -d load " W " is marked with an arrow pointing in that direction, while an upward or + ve distributed loading " w " is denoted by placing a wavy line underneath the horizontal line joining CBA ; e.g., W2 is a + ^e concentrated load, and ws is a downward or negative distributed load. It is assumed that wJii^wJy.^, and M)v/(x22>we/^22- In the 5th line of the table the length " a " of a sub-bay has been converted into an angular measurement, or sub- sector, by multiplying by a factor 57-3[A. It is this angular measurement " a " in the polar diagram that corresponds to the linear value " a " on the actual spar. obviously negative : hjit is therefore directed towards the negative branch of h%. At h2 erect the perpendicular h2h3 = Ws/ni to take account of the concentrated load, and as W, this time is a downward or negative load, h2h3 is drawn to the left of its boundary line Ji^h. At h3 draw the fourth and last locus line K4, parallel to ll3, to cut B2B in Z4. Measure the angle 6 which this last locus line makes with BXB. (Note. —When there is no change of moment of inertia, all the locus lines make the same angle with B1B, aa in this case). It will have been noticed that at each boundary line a measurement along the line takes account of the change of " w," and a measurement perpendicular to the line takes Consider the bay AB and refer to Fig. 1. With OB hori- zontal, draw the angle AOB = (aj + aa + a3 + <x4), and then draw the boundary lines Of, Og and Oh between these angles. Produce all these lines beyond 0 to give an equal sector A1OB.1 The angle AOB1 wilKbe called " 0." The sector AOB now represents in angular measurement the length of the main bay AB, while the angles AO/, fOg, gOh and AOB represent the 8ub-bays 1,'2, 3 and 4, respectively, and, for any boundary line, the direction leading from the negative sector to the positive sector is a positive direction, e.g., f1/ is a positive direction, ff1 a negative one. Assume that MA and MB are both unknown. mA and mB being functions of MA and MB respectively, will therefore also be unknown. Let the length OwA represent the value of mA. At ?nA draw a perpendicular to OA cutting the first boundary line fxf in ft and the end boundary line or base line B'B in Zx. This perpendicular is marked lh and is called the first locus line. Measure off fjj2 equal to wj[i^ — wjy.^) to a convenient scale which will then be used throughout the construction (1 in. = 10,000 lb.-in. is a common one). When this is a negative quantity as it happens to be in this instance, fJi '3 measured towards the negative sector along the boun- line Z1/. At /2 draw a perpendicular fj3 to /*/ and equal i/Hi. The rule is that if Wx is positive (as it is here) s drawn to the right of /'/ looking along the positive ion of that boundary line. Both the discontinuities '« point having now been dealt with, we draw at /s a d locus line llt parallel to llt to cut the next boundary '9 in gi and the base line B*B in Za. Measure g^g2 equal •:,'Vi1 — w3l\xf), and as this quantity is obviously posi- since w3 is negative), gtfz is measured in the positive -on of the boundary line. At ga draw a perpendicular jual to W2/(i.x, which, being positive, is drawn to the s before. At g3 draw a third locus line ll3 parallel to 3Ut the next boundary line hlh in h\ and B*B in l3. ^i&i = (wj/jij2 — wjy.^) which, from Table I, is 272 10 v, tive dire- tighr account of the concentrated load W. At each boundary line also, the locus line has been shifted parallel to itself by an amount dependent on the length of these measurements. These shifts of the locus line are recorded on the base line B*B by the points lv l2, lz and lt. Shifts towards Bl are reckoned positive ; towards B, negative. Thus lxlt and lth are negative, while l3lt is positive. The important quantities are the lengths of these intercepts and the angle 0 which the last locus line makes with the baseline BXB. > The position of the last locus line is given by Oh an<i may be written in the form (OZt — l1l2 — ltla + lalt) or (mA sec (i — hh ~ hh + hh)- Note that the intercepts must be read off in their correct order. Thus, starting from llt the first intercept is l-J,i and, being directed towards B, is negative ; the second is lsl3 also negative. The last intercept JtZ4 is directed towards B1 and therefore positive. What may be called the first equation is now written :— ij" oB(wi Asec(i — Zjtj — 'J'J -f- hh ~i~ *"B) tan 6 = = giving SB in terms of mA and mB and, therefore, of MA and MB. The 2nd equation is SB = ; + RB, where RB is the ordinaryI -AB reaction at B due to the lateral loads, assuming pin joints at A and B and no end load. For example, if w were constant from A to B, and there were no concentrated loads, RB would wequal — Z AB- The last equation gives *B in terms of MA2 and MB- An exactly similar procedure is adopted for bay CB, which leads to an expression for tB in terms of Mo and MB. Equating the two values of »B (after changing sign) gives a relation between MA, MB and Mo.
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