Page:EB1911 - Volume 12.djvu/801

As H is moved along the tangent line HQ, a series of states of motion can be determined, and drawn with accuracy.

Fig. 13.

11. Equation (5) § 3 with slight modification will serve with the same notation for the steady rolling motion at a constant inclination α to the vertical of a body of revolution, such as a disk, hoop, wheel, cask, wine-glass, plate, dish, bowl, spinning top, gyrostat, or bicycle, on a horizontal plane, or a surface of revolution, as a coin in a conical lamp-shade.

The point O is now the intersection of the axis GC′ with the vertical through the centre B of the horizontal circle described by the centre of gravity, and through the centre M of the horizontal circle described by P, the point of contact (fig. 13). Collected into a particle at G, the body swings round the vertical OB as a conical pendulum, of height AB or GL equal to g/μ² = λ, and GA would be the direction of the thread, of tension gM(GA/GL) dynes. The reaction with the plane at P will be an equal parallel force; and its moment round G will provide the couple which causes the velocity of the vector of angular momentum appropriate to the steady motion; and this moment will be gM·Gm dyne-cm. or ergs, if the reaction at P cuts GB in m.

Draw GR perpendicular to GK to meet the horizontal AL in R, and draw RQC′K perpendicular to the axis Gz, and KC perpendicular to LG.

The velocity of the vector GK of angular momentum is μ times the horizontal component, and

horizontal component /Aμ sin α = KC/KC′,

(1)

so that

gM·Gm = Aμ² sin α(KC/KC′),

(2)

A	=	KC′		g	Gm = GQ·Gm.
M		KC		μ2 sin α

(3)

The instantaneous axis of rotation of the case of a gyrostat would be OP; drawing GI parallel to OP, and KK′ parallel to OG, making tan K′GC′ = (A/C) tan IGC’₁; then if GK represents the resultant angular momentum, K′K will represent the part of it due to the rotation of the fly-wheel. Thus in the figure for the body rolling as a solid, with the fly-wheel clamped, the points m and Q move to the other side of G. The gyrostat may be supposed swung round the vertical at the end of a thread PA′ fastened at A′ where Pm produced cuts the vertical AB, and again at the point where it crosses the axis GO. The discussion of the small oscillation superposed on the state of steady motion requisite for stability is given in the next paragraph.

12. In the theoretical discussion of the general motion General motion of
a gyrostat rolling on
a plane. of a gyrostat rolling on a horizontal plane the safe and shortest plan apparently is to write down the most general equations of motion, and afterwards to introduce any special condition.

Drawing through G the centre of gravity any three rectangular axes Gx, Gy, Gz, the notation employed is

u, v, w,	the components of linear velocity of G;
p, q, r,	the components of angular velocity about the axes;
h1, h2, h3,	the components of angular momentum;
θ1, θ2, θ3,	the components of angular velocity of the coordinate axes;
x, y, z,	the co-ordinates of the point of contact with the horizontal plane;
X, Y, Z,	the components of the reaction of the plane;
α, β, γ,	the direction cosines of the downward vertical.

The geometrical equations, expressing that the point of contact is at rest on the plane, are

u − ry + qz = 0,

(1)

v − pz + rx = 0,

(2)

w − qx + py = 0.

(3)

The dynamical equations are

du/dt − θ₃v + θ₂w = gα + X/M,

(4)

dv/dt − θ₁w + θ₂u = gβ + Y/M,

(5)

dw/dt − θ₂u + θ₁v = gγ + Z/M,

(6)

and

dh₁/dt − θ₃h₂ + θ₂h₃ = yZ − zY,

(7)

dh₂/dt − θ₁h₃ + θ₃h₁ = zX − xZ,

(8)

dh₃/dt − θ₂h₁ + θ₁h₂ = xY − yX.

(9)

In the special case of the gyrostat where the surface is of revolution round Gz, and the body is kinetically symmetrical about Gz, we take Gy horizontal and Gzx through the point of contact so that y = 0; and denoting the angle between Gz and the downward vertical by θ (fig. 13)

α = sin θ,   β = 0,   γ = cos θ.

(10)

The components of angular momentum are

h₁ = Ap,   h₂ = Aq,   h₃ = Cr + K,

(11)

where A, C denote the moment of inertia about Gx, Gz, and K is the angular momentum of a fly-wheel fixed in the interior with its axis parallel to Gz; K is taken as constant during the motion.

The axis Gz being fixed in the body,

θ₁ = p,   θ₂ = q = −dθ/dt,   θ₃ = p cot θ.

(12)

With y = 0, (1), (2), (3) reduce to

u = −qz,   v = pz − rx,   w = qx;

(13)

and, denoting the radius of curvature of the meridian curve of the rolling surface by ρ,

dx	= ρ cos θ	dθ	= −qρ cos θ,	dz	= −ρ sin θ	dθ	= qρ sin θ;
dt		dt		dt		dt

(14)

so that

du	= −	dq	z − q2ρ sin θ,
dt		dt

(15)

dv	=	dp	z −	dr	x + pqρ sin θ + qrρ sin θ,
dt		dt		dt

(16)

dw	=	dq	x − q2ρ cos θ.
dt		dt

(17)

The dynamical equations (4) . . . (9) can now be reduced to

X	= −	dq	z − p2z cotθ + q2 (x − ρ sin θ) + prx cot θ − g sin θ,
M		dt

(18)

Y	=	dp	z −	dr	x − pq (x + z cot θ − ρ sin θ) + qrp cos θ,
M		dt		dt

(19)

Z	=	dq	x + q2 (z − ρ cos θ) + p2z − prx − g cos θ,
M		dt

(20)

−zY = A	dp	− Apq cot θ + qh3,
	dt

(21)

−zX − xZ = A	dq	+ Ap2 cot θ − ph3,
	dt

xY =	dh3	= C	dr	= −Cq	d	.
	dt		dt		dθ

(23)

Eliminating Y between (19) and (23),

(	C	+ x2 )	dr	− xz	dp	+ pqx (x + z cot θ − ρ sin θ) − qrxρ cos θ = 0,
	M		dt		dt

(24)

(	C	+ x2 )	dr	− xz	dp	− px (x + z cot θ − ρ sin θ) + rxρ cos θ = 0.
	M		dθ		dθ

(A)

Eliminating Y between (19) and (21)

(	A	+ z2 )	dp	− xz	dr	−	A	pq cot θ + q	h3
	M		dt		dt		M		M

− pqz (x + z cot θ − ρ sin θ) + qrzρ cos θ = 0,

(25)

−xz	dr	+ (	A	+ z2 )	dp	+	A	p cot θ −	h3
	dθ		M		dθ		M		M

+ pz (x + z cot θ − ρ sin θ) + rzρ cos θ = 0.

(B)

In the special case of a gyrostat rolling on the sharp edge of a circle passing through G, z = 0, ρ = 0, (A) and (B) reduce to

p = (	C	+ 1 )	dr	= (	1	+	1	)	dh3	,
	Mx2		dθ		Mx2		C		dθ

(26)

dp	+ p cot θ =	h3	,	d·p sin θ	=	h3 sin θ	;
dθ		A		dθ		A

(27)

d2h3	+	dh3	cot θ =	CMx2	h3,
dθ2		dθ		A (Mx2 + C)

(28)

a differential equation of a hypergeometric series, of the form of Legendre’s zonal harmonic of fractional order n, given by

n (n + 1) = CMx² / A (Mx² + C).

(29)

For a sharp point, x = 0, ρ = 0, and the previous equations are obtained of a spinning top.

The elimination of X and Z between (18) (20) (22), expressed symbolically as

(22) − z(18) + x(20) = 0,

(30)

gives

(	A	+ x2 + z2 )	dq	− p	h3	+ (	A	+ z2 ) p2 cot θ + p2xz
	M		dt		M		M

+ q²ρ (x cos θ − z sin θ) − prx (x + z cot θ) − g (x cos θ − z sin θ) = 0,

(C)

and this combined with (A) and (B) will lead to an equation the integral of which is the equation of energy.

13. The equations (A) (B) (C) are intractable in this general form; but the restricted case may be considered when the axis moves in steady motion at a constant inclination α to the vertical; and the stability is secured if a small nutation of the axis can be superposed.

It is convenient to put p = Ω sin θ, so that Ω is the angular velocity of the plane Gzx about the vertical; (A) (B) (C) become