. Theory of structures and strength of materials. izontal through O as the axis of x, the originO being chosen so that p,AO = H—mp, (i) p being the weight of a unit of length of the cable, and H thehorizontal pull at A. (^ -. .t M- CURVE OF CABLE. 707 in ox AO x?, the parameter, or modulus, of the catenary, andOG is the directrix. X, y be the co-ordinates of any point P, the length ofthe arc AP being s. Draw the tangent PT and the ordinate PN. The triangle PNT is evidently a triangle of forces for theportion ^/*, PiV representing the weight of AP{y\z.,ps), PI / ^ X \ y \ /-^Ay-= o^c Ap th
. Theory of structures and strength of materials. izontal through O as the axis of x, the originO being chosen so that p,AO = H—mp, (i) p being the weight of a unit of length of the cable, and H thehorizontal pull at A. (^ -. .t M- CURVE OF CABLE. 707 in ox AO x?, the parameter, or modulus, of the catenary, andOG is the directrix. X, y be the co-ordinates of any point P, the length ofthe arc AP being s. Draw the tangent PT and the ordinate PN. The triangle PNT is evidently a triangle of forces for theportion ^/*, PiV representing the weight of AP{y\z.,ps), PI / ^ X \ y \ /-^Ay-= o^c Ap the tangential pull T at P, and NT the horizontal pull H atA. ±=t^nPTN=^^ = ^=± dx TN H m (2) which gives the differential equation to the may be easily integrated as follows: 7x = \l^[^=\l+-;;^=Tn^TT^, (3) or ds Vs + vi dxin x .. log {s + Vs + m) = -+ C, c being a constant of integration. When .r = o, .f = o, and therefore log in = , {^ /Sy^ i^^-^^^^—-^^^^T^ •^ ^ •- d log _ / y --(.^--i _, V V 1^^ i^i-^i^-f r 2.^ ^ -y. 5-^*1^^ ~^ S - UX u . ^^ THEORY OF STRUCTURES. .^ V or or j + l/j* + m^ = met Again, s=—{e^-e«^) (4) and hence, -x*-t-^//,^-^!u-------= ^(^ _+£_iL- (5) V The constant, of integration is zero, since y ^ in whenX =0. The last equation is the equation to the catenary, while eq.(4) gives the length of the arc AP, By equations (4) and (5), T^ ~5r/-^ -^^^j - f=s-\-ire (6)k tk V r Draw iWI/perpendicular to PT, and let the angle PTN — ^,i and ^ ^^ iJ/i\^ = PNcose=y = ;;2, . . (8) . \|. ^ + ^^ ^ >^ . f) dy s , since tan a = ^ = —, ,^i_ ?V^ ax in ^-^ V->~ «i Thus, the triangle PMN possesses the property that theside PM is equal to the length of the arc AP, and the sideMN\s equal to the modulus vi (= AO). The area APNO /?* lii^ - £ ydx = —{e « — ^ «) = ;;w = 2 X triangle PMN. ^ 141^^e)^- /^^^6?^,>-,.u^G-- CUUVE OF radius of curvature, p, at P 709 1 ? + (!)?
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Keywords: ., bookcentury1800, bookdecade1890, bookpublishernewyo, bookyear1896
