nearly those of a free Eulerian rotation (§ 19), gravity playing only a subordinate part.
 |
| Fig. 84. |
Again, take the case of a circular disk rolling in steady motion on a horizontal plane. The centre O of the disk is supposed to describe a horizontal circle of radius c with the constant angular velocity ψ. , whilst its plane preserves a constant inclination θ to the horizontal. The components of the reaction of the horizontal lane will be Mcψ. 2 at right angles to the tangent line at the point of contact and Mg vertically upwards, and the moment of these about the horizontal diameter of the disk, which corresponds to OB′ in fig. 83, is Mcψ. 2. α sin θ − Mgα cos θ, where α is the radius of the disk. Equating this to the rate of increase of the angular momentum about OB′, investigated as above, we find
| ( C + Ma2 + A | a | cos θ ) ψ. 2 = Mg | a2 | cot θ, |
| c | c |
where use has been made of the obvious relation nα = cψ. . If c and θ be given this formula determines the value of ψ. for which the motion will be steady.
In the case of the top, the equation of energy and the condition of constant angular momentum (μ) about the vertical OZ are sufficient to determine the motion of the axis. Thus, we have
where ν is written for Cn. From these ψ. may be eliminated, and on differentiating the resulting equation with respect to t we obtain
| Aθ.. − | (μ − ν cos θ) (μ cos θ − ν) | − Mgh sin θ = 0. |
| A sin3 θ |
If we put θ.. = 0 we get the condition of steady precessional motion in a form equivalent to (3). To find the small oscillation about a state of steady precession in which the axis makes a constant angle α with the vertical, we write θ = α + χ, and neglect terms of the second order in χ. The result is of the form
where
(μ cos α − ν)2 } / A2 sin4 α.
When ν is large we have, for the “slow” precession σ = ν/A, and for the “rapid” precession σ = A/ν cos α = ψ. , approximately. Further, on examining the small variation in ψ. , it appears that in a slightly disturbed slow precession the motion of any point of the axis consists of a rapid circular vibration superposed on the steady precession, so that the resultant path has a trochoidal character. This is a type of motion commonly observed in a top spun in the ordinary way, although the successive undulations of the trochoid may be too small to be easily observed. In a slightly disturbed rapid precession the superposed vibration is elliptic-harmonic, with a period equal to that of the precession itself. The ratio of the axes of the ellipse is sec α, the longer axis being in the plane of θ. The result is that the axis of the top describes a circular cone about a fixed line making a small angle with the vertical. This is, in fact, the “invariable line” of the free Eulerian rotation with which (as already remarked) we are here virtually concerned. For the more general discussion of the motion of a top see Gyroscope.
§ 21. Moving Axes of Reference.—For the more general treatment of the kinetics of a rigid body it is usually convenient to adopt a system of moving axes. In order that the moments and products of inertia with respect to these axes may be constant, it is in general necessary to suppose them fixed in the solid.
We will assume for the present that the origin O is fixed. The moving axes Ox, Oy, Oz form a rigid frame of reference whose motion at time t may be specified by the three component angular velocities p, q, r. The components of angular momentum about Ox, Oy, Oz will be denoted as usual by λ, μ, ν. Now consider a system of fixed axes Ox′, Oy′, Oz′ chosen so as to coincide at the instant t with the moving system Ox, Oy, Oz. At the instant t + δt, Ox, Oy, Oz will no longer coincide with Ox′, Oy′, Oz′; in particular they will make with Ox′ angles whose cosines are, to the first order, 1, −r δt, qδt, respectively. Hence the altered angular momentum about Ox′ will be λ + δλ + (μ + δμ) (−r δt) + (ν + δν) qδt. If L, M, N be the moments of the extraneous forces about Ox, Oy, Oz this must be equal to λ + Lδt. Hence, and by symmetry, we obtain
| dλ | − r ν + qν = L, | |
| dt | ||
| dμ | − pν + r λ = M, | |
| dt | ||
| dν | − qλ + pν = N. | |
| dt |
These equations are applicable to any dynamical system whatever.
If we now apply them to the case of a rigid body moving
about a fixed point O, and make Ox, Oy, Oz coincide with the
principal axes of inertia at O, we have λ, μ, ν = Ap, Bq, Cr,
whence
| A | dp | − (B − C) qr = L, | |
| dt | |||
| B | dq | − (C − A) rp = M, | |
| C | dr | − (A − B) pq = N. | |
| dt |
If we multiply these by p, q, r and add, we get
| d | · 12 (Ap2 + Bq2 + Cr 2) = Lp + Mq + Nr, |
| dt |
which is (virtually) the equation of energy.
As a first application of the equations (2) take the case of a solid constrained to rotate with constant angular velocity ω about a fixed axis (l, m, n). Since p, q, r are then constant, the requisite constraining couple is
If we reverse the signs, we get the “centrifugal couple” exerted by the solid on its bearings. This couple vanishes when the axis of rotation is a principal axis at O, and in no other case (cf. § 17).
If in (2) we put, L, M, N = O we get the case of free rotation; thus
| A | dp | (B − C) qr, | |
| dt | |||
| B | dq | (C − A) rp, | |
| dt | |||
| C | dr | (A − B) pq. | |
| dt |
These equations are due to Euler, with whom the conception of
moving axes, and the application to the problem of free rotation,
originated. If we multiply them by p, q, r, respectively, or again
by Ap, Bq, Cr respectively, and add, we verify that the expressions
Ap2 + Bq2 + Cr 2 and A2p2 + B2q2 + C2r 2 are both constant.
The former is, in fact, equal to 2T, and the latter to Γ2, where
T is the kinetic energy and Γ the resultant angular momentum.
To complete the solution of (2) a third integral is required; this involves in general the use of elliptic functions. The problem has been the subject of numerous memoirs; we will here notice only the form of solution given by Rueb (1834), and at a later period by G. Kirchhoff (1875), If we write
| u = | dφ | , Δφ = √(1 − k2 sin2 φ), |
| Δφ |
we have, in the notation of elliptic functions, φ = am u. If we assume
we find
| ṗ = − | σp0 | qr, q̇ = | σq0 | rp, ṙ = | k2σr0 | pq. |
| q0r0 | r0p0 | p0q0 |
Hence (5) will be satisfied, provided
| −σp0 | = | B − C | , | σq0 | = | C − A | , | −k2σr0 | = | A − B | . |
| q0r0 | A | r0p0 | B | p0q0 | C |
These equations, together with the arbitrary initial values of p, q, r,
determine the six constants which we have denoted by p0, q0, r0, k2, σ, ε.
We will suppose that A > B > C. From the form of the polhode
curves referred to in § 19 it appears that the angular velocity q
about the axis of mean moment must vanish periodically. If we
adopt one of these epochs as the origin of t, we have ε = 0, and
p0, r0 will become identical with the initial values of p, r. The
conditions (9) then lead to
| q02 = | A (A − C) | p02, σ2 = | (A − C) (B − C) | r02, k2 = | A (A − B) | · | p02 | . |
| B (B − C) | AB | C (B − C) | r02 |
