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F2 (X2+Y2)

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Originally appearing in Volume V14, Page 135 of the 1911 Encyclopedia Britannica.
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F2 (X2+Y2) =y12+y22+y32-G2, F(X+Yi) =FY3~ (P 2 +iv X xa) aeYt. Suppose xa--F is a repeated factor of X3, then y3=G, and X3=(x,-F)2[p'r p(xa+F)2+24'r'G(xa+F)-G2], (23) and putting x3-F=y, (d) 2 = r2y2 [4 'F 2+4 q-FG -Gz +2 (2 EF+q r qG) y+~ y2] , (24) so that the stability of this axial movement is secured if A=p pF2+4 r 9FG-Gz (25) is negative, and then the axis makes r./ (-A)/x nutations per second. Otherwise, if A is positive f dy rt =J y s,/ (A+2By+Cy2) I sh-1 J A' (A+2By+Cy2) _ I ch—1 A+By sl AA ch-i y ' (B2--AC) J Ash-1 yJ (B2_AC), (26) and the axis falls away ultimately from its original direction. A number of cases are worked out in the American Journal of Mathematics (1907), in which the motion is made algebraical by the use of the pseudo-elliptic integral. To give a simple instance, changing to the stereographic projection by putting tan 10=x, (Nxe9i) a n = (x+ 1)'f X 1 +i (x - I) ' X2, (27) Xi X2= tax4+2ax3r3(a+b)x2+2bxtb, (28) N3-=-- -8(a+b), (29) will give a possible state of motion of the axis of the body; and the motion of the centre may then be inferred from (22). 50. The theory preceding is of practical application in the investigation of the stability of the axial motion of a submarine boat, of the elongated gas bag of an airship, or of a spinning rifled projectile. In the steady motion under no force of such a body in a medium, the centre of gravity describes a helix, while the axis describes a cone round the direction of motion of the centre of gravity, and the couple causing precession is due to the displacement of the medium. In the absence of a medium the inertia of the body to translation is the same in all directions, and is measured by theweight W, and under no force the C.G. proceeds in a straight line, and the axis of rotation through the C.G. preserves its original direction, if a principal axis of the body; otherwise the axis describes a cone, right circular if the body has uniaxial symmetry, and a Poinsot cone in the general case. But the presence of the medium makes the effective inertia depend on the direction of motion with respect to the external shape of the body, and on W' the weight of fluid medium displaced. Consider, for example, a submarine boat under water; the inertia is different for axial and broadside motion, and may be represented by ci=W+W'a, c2=W+W'S, (1) where a, are numerical factors depending on the external shape; and if the C.G is moving with velocity V at an angle 4, with the axis, so that the axial and broadside component of velocity is u =V cos 0, v =V sin 0, the total momentum F of the medium, represented by the vector OF at an angle 0 with the axis, will have components, expressed in sec. lb, F cos B = c,g = (W +W'a) 8 cos 0, F sin B = czg = (W +W'f) 8 sin 0. (2) Suppose the body is kept from turning as it advances; after t seconds the C.G. will have moved from 0 to 0', where OO'=Vl; and at 0' the. momentum is the same in magnitude as before, but its vector is displaced from OF to O'F'. For the body alone the resultant of the components of momentum W g cos q, andW % sin ¢ is W 8 sec. lb, acting along 00', and so is unaltered. But the change of the resultant momentum F of the medium as well as of the body from the vector OF to O'F' requires an impulse couple, tending to increase the angle FOO', of magnitude, in sec.' foot-pounds F.00'.sin FOO' =FVt sin (0-¢), (4) equivalent to an incessant couple N=FV sin (0-¢) = (F sin B cos 4,-F cos 0 sin ¢)V = (c2 -ci) (V2/g) sin ¢ cos ¢ = W'(1-a)uv/g. (5) This N is the couple in foot-pounds changing the momentum of the medium, the momentum of the body alone remaining the same; the medium reacts on the body with the same couple N in the opposite direction, tending when c2-ci is positive to set the body broadside to the advance. An oblate flattened body, like a disk or plate, has cx-ci negative, so that the medium steers the body axially; this may be verified by a plate dropped in water, and a leaf or disk or rocket-stick or piece of paper falling in air. A card will show the influence of the couple N if projected with a spin in its plane, when it will be found to change its aspect in the air. An elongated body like a ship has c2-c1 positive, and the couple N tends to disturb the axial movement and makes it unstable, so that a steamer requires to be steered by constant attention at the helm. Consider a submarine boat or airship moving freely with the direction of the resultant momentum horizontal, and the axis at a slight inclination 0. With no reserve of buoyancy W =W', and the couple N, tending to increase 0, has the effect of diminishing the metacentric height by h ft. vertical, where Wh tan 0 =N = (c2-ci)Z % tan B, (6) _c2—c1 C, u2=(~— )I+a u2 h- W czg a1+19 g 51. An elongated shot is made to preserve its axial flight through the air by giving it the spin sufficient for stability, without which it would turn broadside to its advance; a top in the same way is made to stand upright on the point in the position of equilibrium, unstable statically but dynamically stable if the spin is sufficient; and the investigation proceeds in the same way for the two problems (see GYRoscoPE). The effective angular inertia of the body in the medium is now required; denote it by C1 about the axis of the figure, and by C; about a diameter of the mean section. A rotation about the axis of a figure of revolution does not set the medium in motion, so that C, is the moment of inertia of the body about the axis, denoted by Wk;. But if Wk2 is the moment of inertia of the body about a mean diameter, and w the angular velocity about it generated by an impluse couple M, and M' is the couple required to set the surrounding medium in motion, supposed of effective radius of gyration k', Wkpa=M-M', W'k'2w=M', (1) (Wk? +W'k'2)w =M, (2) C2 = Wkl +W'k'2 = (W+W's)kz, (3) in which we have put k'2=ek2, where e is a numerical factor depending on the shape. (13) (14) (15) (16) (17) (3) (7) jHYDRODYNAM1CS If the shot is spinning. about its axis with angular velocity p, and is precessing steadily at a rate µ about a line parallel to the resultant momentum F at an angle 0, the velocity of the vector of angular momentum, as in the case of a top, is CI pp sin 0- Csµ% sin B cos 0 ; (4) and equating this to the impressed couple (multiplied by g), that is, to gN = (el -c2)c2u2 tan B, (5) and dividing out sin 0, which equated to zero would imply perfect centring, we obtain C2µ2 cos e- Cif' µ + (c2-ci) 72u2 sec 9 = 0. (6) The least admissible value of p is that which makes the roots equal of this quadratic in µ, and then µ = Cl Ip sec e, the roots would be imaginary for a value of p smaller than given by C;p2-4(c2-ci)c, c C2u2=o. (8) 2 c l =4(C2-ci)i CI . If the shot is moving as if fired from a gun of calibre d inches, in which the rifling makes one turn in a pitch of n calibres or nd inches, so that the angle S of the rifling is given by tan S = ad/ad = id p/u, If a denotes the density of the metal, and if the shell has a cavity homothetic with the external ellipsoidal shape, a fraction f of the linear scale; then the volume of a round shot being *rd', and 'grd3x of a shot x calibres long W.= srd3x(I (20) Wki2 = krd3xdo(1 -fs)a, (21) Wk22 = srd3xP2o (t -J5)e. (22) If p denotes the density of the air or medium W' ='ird3xp, W'_ I p _ W 1-j-3 a' k12 I I-f5 k22 x2+1 d2ol-f32 tan2S =a(fl-a)i0 fs), in which e/p may be replaced by 800 times the S.G. of the metal, taking water as 800 times denser than air on the average, in round numbers, and formula (10) may be written n tan S=r, or n6 =18o, when S is a small angle, and given in degrees. From this formula (26) the table following has been calculated by A. G. Hadcock, and the results are in agreement with practical experience. (7) (9) (Jo) (23) (24) (25) (26) Table of Rifling for Stability of an Elongated Projectile, x Calibres long, giving S Rifling, and n the Pitch of Rifling in Calibres. HYDROMECIIANICS 21 Cast-iron Common Shell Palliser Shell Solid Steel Bullet Solid Lead Bullet f = 1, S.G. 7.2. f = a, S.G. 8. f =o, S.G. 8. f =o, S.G. 10.9. x $-a S n S n S n S n 1 •o o•0000 0° o' Infinity o° o' Infinity 0° o' Infinity o° o' Infinity 2.0 0.4942 2 49 63.87 2 32 7P08 2 29 72.21 2 08 84'29 2'5 0.6056 3 46 47'91 3 23 53'32 3 19 54'17 2 51 63'24 3.0 0.6819 4 41 38'45 4 13 42'79 4 09 43.47 3 38 50.74 3'5 0.7370 5 35 32'13 5 02 35'75 4 58 36.33 4 15 42'40 4.0 0.7782 6 30 27.60 5 51 30.72 5 45 31.21 4 56 36'43 4.5 o•8ioo 7 24 24.20 6 40 26.93 6 32 27.36 5 37 3P94 5.0 x.8351 8 16 21.56 7 28 23.98 7 21 2436 6 18 28.44 6.0 0.8721 10 05 17.67 9 04 19.67 8 56 19'98 7 40 23'33 10.0 09395 16 57 1x•31 15 19 11.47 15 05 11.65 13 00 13.60 Infinity I•0000 90 00 0.00 90 00 0.00 90 00 0.00 90 00 0.00 the Angle of which is the ratio of the linear velocity of rotation ;dp to u, the velocity of advance, tan2S= =d2p2 = (c2aci ci C2d2 n 4u )cz _W' 1+W a (1 +ws) (r) 2 -w(l;-a) w' ki (II) 1+w-f3 (~) For a shot in air the ratio W'/W is so small that the square may be neglected, and formula (II) can be replaced for practical purpose in artillery by tan' S=yam?=W(R-a)\T)/ (d)4' (12) if then we can calculate a, or 9-a for the external shape of the shot, this equation will give the value of S and n required for stability of flight in the air. The ellipsoid is the only shape for which a and (3 have so far been determined analytically, as shown already in § 44, so we "must restrict our calculation to an egg-shaped bullet, bounded by a prolate ellipsoid of revolution, in which, with b=c, Ao= oo ab"-dX oo ab2dX (t3) J o (a2+X)d[4(a2+k)(b%+a)2]=f 0 2(a2+t;)3h2(b2+~-' Ao+2Bo= 1, (14) Ao Bo _1-Ao_ I a I-Ao' =I-B0 I+Ao 1+2a. (I5) The length of the shot being denoted by land the calibre by d, and the length in calibres by x l/d=2a/2b=x, (16) Ax i (17) =(x2_1)3jzch'x•-x-12Bo- 42I)212en-sX. f"e4I' (18) x'Ao h2Bo- sh-1 x%(- 1) I ,I (2x _I)Iog[x- .41(x2-1)]. -NI (19) 52. In the steady motion the centre of the shot describes a helix, with axial velocity ucos0+vsin0 _ (I +22tan' 0)ucos0 6--asu (1) and transverse velocity using-vcos6.= (1- Li) usineti -a)usine; (2) and the time of completing a turn of the spiral is 2r/µ. (3) When µ has the critical value in (7), u = p Cicose = (x2+1)cose, 53. The Motion of a Perforated Solid in Liquid.-In the preceding investigation, the liquid stops dead when the body is brought to rest; and when the body is in motion the surrounding liquid moves in a uniform manner with respect to axes fixed in the body, and the force experienced by the body from the pressure of the liquid on its surface is the opposite of that required to change the motion of the liquid; this has been expressed by the dynamical equations given above. But if the body is perforated, the liquid can circulate through a hole, in reentrant stream lines linked with the body, even while the body is at rest; and no reaction from the surface can influence this circulation, which may be supposed started in the ideal manner described in § 29, by the application of impulsive pressure across an ideal membrane closing the hole, by "means of ideal mechanism connected with the body. The body is held fixed, and the reaction of the mechanism and the resultant of the impulsive pressure on the surface are a measure of the impulse, linear i;, and angular a, p, v, required to start the circulation. which makes the circumference of the cylinder on which the helix is wrapped µ(u sino-vcoso)=2 (/3-a)(x2+I)sin' ocose nd - a) (x2+ I) sine ¢os 9, (4) and' the length of one turn of the helix 2r -(u cos 0+v sin 0) =nd(x2+1) ; thus for x=3, the length is to times the pitch of the rifling. (5) This impulse will remain of constant magnitude, and fixed relatively to the body, which thus experiences an additional reaction from the circulation which is the opposite of the force required to change the position in space of the circulation impulse; and these extra forces must be taken into account in the dynamical equations. An article may be consulted in the Phil. Meg., April 1893, by G. H. Bryan, in which the analytical equations of motion are deduced of a perforated solid in liquid, from considerations purely hydrodynamical. The effect of an external circulation of vortex motion on the motion of a cylinder has been investigated in § 29; a similar procedure will show the influence of circulation through a hole in a solid, taking as the simplest illustration a ring-shaped figure, with uniplanar motion, and denoting by E the resultant axial linear momentum"of the circulation. As the ring is moved from 0 to 0' in time t, with velocity Q, and angular velocity R, the components of liquid momentum change from aM'U +# and BM'V along Ox and Oy to aM'U'+i: and SM'V' along 0'x' and O'y', (1) the axis of the ring changing from Ox to 0'x'; and U=Qcos0, VQsin0, U'=Q cos (O —RI), V'=Q sin (0 —Rt), (2) so that the increase of the components of momentum, X,, Y,, and Ni, linear and angular, are X, _ (aM'U'+E) cos Rt—aM'U—E—13M'V' sin Rt = (a—$)M'Q sin_(o—Rt) sin Rt —Ever Rt (3) Yl = (aM'U'+E) sin Rt+RM'V' cos Rt—13M'V _ (a -13) M'Q cos (B —Rt) sin Rt+E Sin RT, (4) N, =[—(aM'U'+i;) sin (o—Rt)+RM'V' cos (0—Rt)IOO' [—(a—p)M'Qcos(0—Rt)sin (e—Rt)—Esin(o—ROM. (5) The components of force, X, Y, and N, acting on the liquid at 0, and reacting on the body, ate then X=lt. X,/t=(a—f3)M'QRsine= (a—13)M'VR, (6) Y=1t. Y,/t= (a—O)M'QRcose+ER=(a—$)M'UR+FR, (7) Z = It. Z,/t= —(a—MM'Q2sine cos°—EQsine =[—(a—13)M'U+E]V. (8) Now suppose the cylinder is free; the additional forces acting on the body are the components of kinetic reaction of the liquid _aM' ( —VR) —,BM' (dt +UR) , so that its equations of motion are M (dU—VR) (dttJ —VR) .... (a-13)M'VR, M (q+UR) = —$M' ( +UR) —(a—R)M'UR—ER,(11) Cddt = —eC dR+(a—B)M'UV+tV; and putting as before M+aM'=c,, M+BM'=c2, C+sC'=Cs, c,---c2VR=o, cz.d +(c,U+i;)R=o, dR cz dt — (c,U—c2U)V = o; showing the modification of the equations of plane motion, due to the component of the circulation. The integral of (14) and (15) may be written c1U+E=Fcoso, c2V= —FsinO, dx - =Ucoso—V sine =' cocls2o +F sictn2 o —#eoso, (In dt =U sine+V cos O = (F —~ sine cos o - -sine, C, d'O = (c— ) siuocoso—F sine=Fdt, (20) de F2cos2O F2sin2o FE Caa=Fy=A/ L-- c c +2 cos8+Hj;(2I) VVV i z i so that coso and y is an elliptic function of the time. When E is absent, dx/dl is always positive, and the centre of the body cannot describe loops; but with 1;, the influence may be great enough to make dx/dt change sign, and so loops occur, as shown in A. B. Basset's Hydrodynamics, i. 192, resembling the trochoidal curves, which can be looped, investigated in § 29 for the motion of a cylinder under gravity, when surrounded by a vortex. The branch of hydrodynamics which discusses wave motion in a liquid or gas is given now in the articles Somali and .W9 vz,; while the influence of viscosity is considered under HYDRAULICS. REFEREN'CEs.—For the history and references to the original memoirs see Report to the British Association, by G. G. Stokes (1846), and W. M. Hicks li882). See also the, Fortschritfe der Mathematik, and A. E. H. Love, " Hydrodynamik " in the Encyklop~gdie der mathematischen Wissenschaften (1901). (A. G. G.)
End of Article: F2 (X2+Y2)
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