 |
| Fig. 78. |
The special case where both cones are right circular and ω is constant is important in astronomy and also in mechanism (theory of bevel wheels). The “precession of the equinoxes” is due to the fact that the earth performs a motion of this kind about its centre, and the whole class of such motions has therefore been termed precessional. In fig. 78, which shows the various cases, OZ is the axis of the fixed and OC that of the rolling cone, and J is the point of contact of the polhode and herpolhode, which are of course both circles. If αbe the semi-angle of the rolling cone, β the constant inclination of OC to OZ, and ψ. the angular velocity with which the plane ZOC revolves about OZ, then, considering the velocity of a point in OC at unit distance from O, we have
where the lower sign belongs to the third case. The earth’s precessional motion is of this latter type, the angles being α = .0087″, β = 23° 28′.
If m be the mass of a particle at P, and PN the perpendicular to the instantaneous axis, the kinetic energy T is given by
where I is the moment of inertia about the instantaneous axis. With the same notation for moments and products of inertia as in § 11 (38), we have
and therefore by (1),
Again, if x, y, z be the co-ordinates of P, the component velocities of m are
by § 7 (5); hence, if λ, μ, ν be now used to denote the component angular momenta about the co-ordinate axes, we have λ = Σ {m (py − qx)y − m(rx − pz) z }, with two similar formulae, or
| λ = Ap −Hq − Gr = | ∂T | , |
| ∂p |
| μ = −Hp + Bq − Fr = | ∂T | , |
| ∂q |
| ν = −Gp − Fq + Cr = | ∂T | . |
| ∂r |
If the co-ordinate axes be taken to coincide with the principal
axes of inertia at O, at the instant under consideration, we have
the simpler formulae
It is to be carefully noticed that the axis of resultant angular momentum about O does not in general coincide with the instantaneous axis of rotation. The relation between these axes may be expressed by means of the momental ellipsoid at O. The equation of the latter, referred to its principal axes, being as in § 11 (41), the co-ordinates of the point J where it is met by the instantaneous axis are proportional to p, q, r, and the direction-cosines of the normal at J are therefore proportional to Ap, Bq, Cr, or λ, μ, ν. The axis of resultant angular momentum is therefore normal to the tangent plane at J, and does not coincide with OJ unless the latter be a principal axis. Again, if Γ be the resultant angular momentum, so that
the length of the perpendicular OH on the tangent plane at J is
| OH = | Ap | · | p | ρ + | Bq | · | q | ρ + | Cr | · | r | ρ = | 2T | · | ρ | , |
| Γ | ω | Γ | ω | Γ | ω | Γ | ω |
where ρ = OJ. This relation will be of use to us presently (§ 19).
The motion of a rigid body in the most general case may be specified by means of the component velocities u, v, w of any point O of it which is taken as base, and the component angular velocities p, q, r. The component velocities of any point whose co-ordinates relative to O are x, y, z are then
by § 7 (6). It is usually convenient to take as our base-point the mass-centre of the body. In this case the kinetic energy is given by
where M0 is the mass, and A, B, C, F, G, H are the moments and products of inertia with respect to the mass-centre; cf. § 15 (9).
The components ξ, η, ζ of linear momentum are
| ξ = M0u = | ∂T | , η = M0v = | ∂T | , ζ = M0w = | ∂T | |
| ∂u | ∂v | ∂w |
whilst those of the relative angular momentum are given by (7).
The preceding formulae are sufficient for the treatment of
instantaneous impulses. Thus if an impulse (ξ, η, ζ, λ, μ, ν)
change the motion from (u, v, w, p, q, r) to (u′, v′, w′, p′, q′, r ′)
we have
| M0 (u′ − u) = ξ, | M0 (v′ − v) = η, | M0(w′ − w) = ζ, | |
| A (p′ − p) = λ, | B (q′ − q) = μ, | C (r ′ − r) = ν, |
where, for simplicity, the co-ordinate axes are supposed to
coincide with the principal axes at the mass-centre. Hence
the change of kinetic energy is
The factors of ξ, η, ζ, λ, μ, ν on the right-hand side are proportional to the constituents of a possible infinitesimal displacement of the solid, and the whole expression is proportional (on the same scale) to the work done by the given system of impulsive forces in such a displacement. As in § 9 this must be equal to the total work done in such a displacement by the several forces, whatever they are, which make up the impulse. We are thus led to the following statement: the change of kinetic energy due to any system of impulsive forces is equal to the sum of the products of the several forces into the semi-sum of the initial and final velocities of their respective points of application, resolved in the directions of the forces. Thus in the problem of fig. 77 the kinetic energy generated is 12M (κ2 + Cq2)ω′2, if C be the instantaneous centre; this is seen to be equal to 12F·ω′·CP, where ω′·CP represents the initial velocity of P.
The equations of continuous motion of a solid are obtained by substituting the values of ξ, η, ζ, λ, μ, ν from (14) and (7) in the general equations
| dξ | = X, | dη | = Y, | dζ | = Z, | |
| dt | dt | dt | ||||
| dλ | = L, | dμ | = M, | dν | = N, | |
| dt | dt | dt |
 |
| Fig. 79. |
where (X, Y, Z, L, M, N) denotes the system of extraneous forces referred (like the momenta) to the mass-centre as base, the co-ordinate axes being of course fixed in direction. The resulting equations are not as a rule easy of application, owing to the fact that the moments and products of inertia A, B, C, F, G, H are not constants but vary in consequence of the changing orientation of the body with respect to the co-ordinate axes.
An exception occurs, however, in the case of a solid which is kinetically symmetrical (§ 11) about the mass-centre, e.g. a uniform sphere. The equations then take the forms
| M0u̇ = X, | M0v̇ = Y, | M0ẇ = Z, |
| Cṗ = L, | Cq̇ = M, | Cṙ = N, |
where C is the constant moment of inertia about any axis through
