integral of the first kind; and by inversion x_{3} is in elliptic function
of the time t. Now
y_{1} + y_{2}i | = | FG − x_{3}y_{3} + i √ X_{3} | , |
x_{1} + x_{2}i | x_{1}^{2} + x_{2}^{2} |
d | (x_{1} + x_{2}i) = −i [ (q′ − q) x_{3} + r′y_{3} ] + irx_{3} (y_{1} + y_{2}i), |
dt |
d | log (x_{1} + x_{2}i) = −(q′ − q) x_{3} − r′y_{3} + rx_{3} | FG − x_{3}y_{3} + i √ X_{3} | , |
dti | F^{2} − x_{3}^{2} |
d | log √ | x_{1} + x_{2}i | = −(q′ − q) x_{3} − (r′ − r) y_{3} − Fr | Fy_{3} − Gx_{3} | , |
dti | x_{1} − x_{2}i | F^{2} − x_{3}^{2} |
requiring the elliptic integral of the third kind; thence the expression of x_{1} + x_{2}i and y_{1} + y_{2}i.
Introducing Euler’s angles θ, φ, ψ,
x_{1} = F sin θ sin φ, x_{2} = F sin θ cos φ, x_{1} + x_{2}i = iF sin θε−ψi, x_{3} = F cos θ; |
sin θ | dψ | = P sin φ + Q cos φ, |
dt |
F sin^{2} θ | dψ | = | dT | x_{1} + | dT | x_{2} |
dt | dy_{1} | dy_{2} |
= (qx_{1} + ry_{1}) x_{1} + (qx_{2} + ry_{2}) x_{2} = q (x_{1}^{2} + x_{2}^{2}) + r (x_{1}y_{1} + x_{2}y_{2}) = qF^{2} sin^{2} θ + r (FG − x_{3}y_{3}), |
ψ − qFt = ∫ | FG − x_{3}y_{3} | Frdx_{3} | , | |
F^{2} − x_{3}^{2} | √ X_{3} |
elliptic integrals of the third kind.
Employing G. Kirchhoff’s expressions for X, Y, Z, the coordinates of the centre of the body,
F(X + Yi) = | Fy_{3} − Gx_{3} + i √ X_{3} | εψi. |
√ (F^{2} − x_{3}^{2}) |
Suppose x_{3} −F is a repeated factor of X_{3}, then y_{3} = G, and
X_{3} = (x_{3} − F)^{2} [ | p′ − p | (x_{3} + F)^{2} + 2 | q′ − q | G (x_{3} + F) − G^{2} ], |
r | r |
and putting x_{3} − F = y,
( | dy | ) | ^{2} | = r^{2}y^{2} [ 4 | p′ − p | F^{2} + 4 | q′ − q | FG − G^{2} + 2 ( 2 | p′ − p | F + | q′ − q | G ) y + | p′ − p | y^{2} ], |
dt | r | r | r | r | r |
so that the stability of this axial movement is secured if
A = 4 | p′ − p | F^{2} + 4 | q′ − q | FG − G^{2} |
r | r |
is negative, and then the axis makes r√(−A)/π nutations per second. Otherwise, if A is positive
rt = ∫ | dy |
y √ (A + 2By + Cy^{2}) |
= | 1 | sh^{−1} | √ A √ (A + 2By + Cy^{2}) | = | 1 | ch^{−1} | A + By | , | ||||
√ A | ch^{−1} | y√ (B^{2} ~ AC) | √A | sh^{−1} | y √ (B^{2} ~ AC) |
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 12θ = x,
X_{1} | = ± ax^{4} + 2ax^{3} ± 3 (a + b) x^{2} + 2bx ± b, |
X_{2} |
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 the weight 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
where α, β are numerical factors depending on the external shape; and if the C.G. is moving with velocity V at an angle φ with the axis, so that the axial and broadside component of velocity is u = V cos φ, v = V sin φ, the total momentum F of the medium, represented by the vector OF at an angle θ with the axis, will have components, expressed in sec. ℔,
F cos θ = c_{1} | u | = (W + W′α) | V | cos φ, F sin θ = c_{2} | v | = (W + W′β) | V | . |
g | g | g | g |
Suppose the body is kept from turning as it advances; after t
seconds the C.G. will have moved from O to O′, where OO′ = Vt;
and at O′ 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 | V | cos φ and W | V | sin φ is W | V | sec. ℔, |
g | g | g |
acting along OO′, 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
equivalent to an incessant couple
N = FV sin (θ − φ) = (F sin θ cos φ − F cos θ sin φ) V = (c_{2} − c_{1}) (V^{2} / g) sin φ cos φ = W′ (β − α) uv / g. |
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 c_{2} − c_{1} is positive to set the body broadside to the advance.
An oblate flattened body, like a disk or plate, has c_{2} − c_{1} 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 c_{2} − c_{1} 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 θ. With no reserve of buoyancy W = W′, and the couple N, tending to increase θ, has the effect of diminishing the metacentric height by h ft. vertical, where
Wh tan θ = N = (c_{2} − c_{1}) | c_{1} | u^{2} | tan θ, | |
c_{2} | g |
h = | c_{2} − c_{1} | c_{1} | u^{2} | = (β − α) | 1 + α | u^{2} | . | |||
W | c_{2} | g | 1 + β | 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 C_{1} about the axis of the figure, and by C_{2} 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_{1} is the moment of inertia of the body about the axis, denoted by Wk_{1}^{2}. But if Wk_{2}^{2} is the moment of inertia of the body about a mean diameter, and ω the angular velocity about it generated by an impulse couple M, and M′ is the couple required to set the surrounding medium in motion, supposed of effective radius of gyration k ′,
in which we have put k ′^{2} = εk^{2}, where ε is a numerical factor depending on the shape.