Elements of analysis as applied to the mechanics of engineering and machinery . 22/ therefore constant. If, further, another normal ^ C be drawn at a second point Q.,infinitely near the point P, there results, in the point of intersectionof the two lines, the centre 0 of a circle to be described through thetwo points of contact P and (5, or the circle of gyration] and theportions OF and G Q of the normals are the radii of this circle, orthe radii of gyration. Of all the circles which may be drawnthrough P and Q, this is the one that most nearly coincides withthe circular element P Q^ and w
Elements of analysis as applied to the mechanics of engineering and machinery . 22/ therefore constant. If, further, another normal ^ C be drawn at a second point Q.,infinitely near the point P, there results, in the point of intersectionof the two lines, the centre 0 of a circle to be described through thetwo points of contact P and (5, or the circle of gyration] and theportions OF and G Q of the normals are the radii of this circle, orthe radii of gyration. Of all the circles which may be drawnthrough P and Q, this is the one that most nearly coincides withthe circular element P Q^ and we may therefore assume that its arcP Q coincides with that element. If we designate the radius of gyration CP = C Qhy r, the cir-cular arc AF by s, therefore its element P (^ by 8 s, and the tan-gential angle or arc of P T31 by a, therefore its element S U31— Art. 33.] ELEMENTS OF ANALYSIS. 51 S TM, Le.— UST= — PCQ, by 8«, we have simply, since there is F Q = GP . arc of the angle P C Q: 8s = — r8aj and couse- 8squently, the radius of gyration: r = — ^r-. Fig. ^^ T A M if 0 K Generally, « can only be determined by means of the equation of co-ordinates, since we put tang, a = -^, ox But there is, further, 8 tang, a = COS. a? dxand COS. a ?=--z^\ 8s ? 8 x^hence, we have 8« = cos. a? . 8 tang, a == —— . 8 tang. «, and G o 8s 8s^ ^ dx^ ds ds^ For a convex curve, we have r = -\- ^z-- = -{- -— 0(j^ cx^ c tang, a for a point of inflection, r ; and oo. For the co-ordinates AO = u and 0 G = v of the centre G ofgyration there is u = AM-\- HG = X ^ GP sin. GPH^i. e. u = x -{- r sin. «, andv= 0G = 31P — HP = y—GP cos. GPH, v = y— The continuous succession ofthe centres of gyration gives acurve which is called the evoluteof A P, and whose course is de-termined by the co-ordinates uand V. If the ellipse ADA^D^ , be brought into connectionwith a circle AB A^B^^ its co-or-dinates GM= X and MQ = ymay be expressed by the angleP
Size: 2420px × 1033px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1800, bookdecade1860, bookpublisherphiladelphiakingba
