| ( | C | + x2 ) | dr | − xz sinθdΩdθ − Ωx (x sin θ − 2z cos θ − ρ sin2 θ) + rxρ cos θ = 0, |
| M | dθ |
| −xz | dr | + ( | A | + z2 ) sin θ | dΩ | − | h3 | + 2Ω ( | A | + z2 ) cos θ |
| dθ | M | dθ | M | M |
| ( | A | + x2 + z2 ) | dq | + q2p (x cos θ − z sin θ) − Ω | h3 | sin θ + Ω2 (AM + z2 ) |
| M | dt | M |
The steady motion and nutation superposed may be expressed by
where L, N, Q are small terms, involving a factor enti, to express the periodic nature of the nutation; and then if a, c denote the mean value of x, z, at the point of contact
Substituting these values in (C*) with dq/dt = −d2θ/dt2 = n2L, and ignoring products of the small terms, such as L2, LN, ...
| ( | A | + a2 + c2 ) Ln2 − (μ + N) ( | CR + K | + | CQ | ) (sin α + L cos α) |
| M | M | M |
| + (μ2 + 2μN) (A/M + c2 − 2Lρc sin α) (sin α cos α + L cos α) + (μ2 + 2μN) [ac − Lρ (a sin α − c sin α)] (sin2 α + L sin 2α) − (μ + N) (R + Q) (a + Lρcos α) [a sin α + c cos α + L (a cos α − c sin α)] − g (a cos α − c sin α) + gL (a sin α + c cos α − ρ) = 0, |
which is equivalent to
| −μ | CR + K | sin α + μ2 ( | A | + c2 ) sin αcos α |
| M | M |
the condition of steady motion; and
where
| D = ( | A | + a2 + c2 ) n2 − μ | CK + K | cos α − 2μ2ρc sin2 α cos α |
| M | M |
| + μ2 (A/M + c2) cos α − μ2ρ (a sin α − c cos α) sin2 α + μ2ac sin 2α − μRρ cos α (a sin α + c cos α) − μRa (a cos α − c sin α) + g (a sin α + c cos α − ρ), |
| E = −μ | C | sin α − μa (a sin α + c cos α), |
| M |
| F = − | CR + K | sin α + 2μ ( | A | + c2 ) sin α cos α |
| M | M |
With the same approximation (A*) and (B*) are equivalent to
| ( | C | + a2 ) | Q | − ac sin α | N | − μa (a sin α + 2c cos α − ρ sin2 α) + Raρ cos α = 0, |
| M | L | L |
| −ac | Q | + ( | A | + c2 ) sin α | N | − | CR + K | + 2μ ( | A | + c2 ) cos α |
| L | M | L | M | M |
The elimination of L, Q, N will lead to an equation for the determination of n2, and n2 must be positive for the motion to be stable.
If b is the radius of the horizontal circle described by G in steady motion round the centre B,
and drawing GL vertically upward of length λ = g/μ2, the height of the equivalent conical pendulum, the steady motion condition may be written
| = gM [bλ−1 (a sin α + c cos α) − a cos α + c sin α] |
LG produced cuts the plane in T.
Interpreted dynamically, the left-hand side of this equation represents the velocity of the vector of angular momentum about G, so that the right-hand side represents the moment of the applied force about G, in this case the reaction of the plane, which is parallel to GA, and equal to gM·GA/GL; and so the angle AGL must be less than the angle of friction, or slipping will take place.
Spinning upright, with α = 0, a = 0, we find F = 0, Q = 0, and
| − | CR + K | + 2μ ( | A | + c2 ) − Rcp = 0, |
| M | M |
| ( | A | + c2 ) n2 = μ | CR + K | − μ2 ( | A | + c2 ) + μRρc − g (c − ρ), |
| M | M | M |
| ( | A | + c2 ) | 2 | n2 = 14 ( | CK + R | + Rcρ ) | 2 | − g ( | A | + c2 ) (c − ρ). |
| M | M | M |
Thus for a top spinning upright on a rounded point, with K = 0, the stability requires that
where k, k′ are the radii of gyration about the axis Gz, and a perpendicular axis at a distance c from G; this reduces to the preceding case of § 3 (7) when ρ = 0.
Generally, with α = 0, but a ± 0, the condition (A) and (B) becomes
| ( | C | + a2 ) | Q | = 2μac − Raρ, | −ac | Q | = | CR + K | + Rcρ − 2μ ( | A | + a2 ), |
| M | L | L | M | M |
so that, eliminating Q/L,
| 2 [ | A | + c2 ) ( | C | + a2 ) − a2c2 ] μ = ( | C | + a2 ) ( | CR + K | ) + | C | Rcρ, |
| M | M | M | M | M |
the condition when a coin or platter is rolling nearly flat on the table.
Rolling along in a straight path, with α = 12π, c = 0, μ = 0, E = 0; and
| D = ( | A | + a2 ) n2 + g (a − ρ), |
| M |
| F = − | CR + K | − Ra2, |
| M |
| N | = − | D | = |
|
, | ||||
| L | F |
|
| ( | A | + a2 ) n2 = | (CR + K) | [ | C | + a2 ) R + | K | ] − g (a − ρ). |
| M | A | M | M |
Thus with K = 0, and rolling with velocity V = Ra, stability requires
| V2 | > | a − ρ | > 12 | A | a − ρ | , | |
| 2g | 2CA (CMa2 + 1) | C | CMa2 + 1 |
or the body must have acquired velocity greater than attained by rolling down a plane through a vertical height 12 (a − ρ) A/C.
On a sharp edge, with ρ = 0, a thin uniform disk or a thin ring requires
The gyrostat can hold itself upright on the plane without advance when R = 0, provided
For the stability of the monorail carriage of § 5 (6), ignoring the rotary inertia of the wheels by putting C = 0, and replacing K by G′ the theory above would require
| G′ | ( aV + | G′ | ) > gh. |
| A | A |
For further theory and experiments consult Routh, Advanced Rigid Dynamics, chap. v., and Thomson and Tait, Natural Philosophy, § 345; also Bourlet, Traité des bicycles (analysed in Appell, Mécanique rationnelle, ii. 297, and Carvallo, Journal de l’école polytechnique, 1900); Whipple, Quarterly Journal of Mathematics, vol. xxx., for mathematical theories of the bicycle, and other bodies.
14. Lord Kelvin has studied theoretically and experimentally the vibration of a chain of stretched gyrostats (Proc. London Math. Soc., 1875; J. Perry, Spinning Tops, Gyrostatic chain. for a diagram). Suppose each gyrostat to be equivalent dynamically to a fly-wheel of axial length 2a, and that each connecting link is a light cord or steel wire of length 2l, stretched to a tension T.
Denote by x, y the components of the slight displacement from the central straight line of the centre of a fly-wheel; and let p, q, 1 denote the direction cosines of the axis of a fly-wheel, and r, s, 1 the direction cosines of a link, distinguishing the different bodies by a suffix.
Then with the previous notation and to the order of approximation required,
to be employed in the dynamical equations
| dh1 | − θ3h2 + θ2h3 = L, ... |
| dt |
in which θ3h1 and θ3h2 can be omitted.
For the kth fly-wheel
and for the motion of translation
while the geometrical relations are
Putting
