[.theta] is constant. Thus a circular disk projected under gravity in a
vertical plane spins with constant angular velocity, whilst its centre
describes a parabola.
We may apply the equations (9) to the case of a solid of revolution
rolling with its axis horizontal on a plane of inclination [alpha]. If
the axis of x be taken parallel to the slope of the plane, with x
increasing downwards, we have
M[:x] = Mg sin [alpha] - F, 0 = Mg cos [alpha] - R, M[kappa]²[:theta] = Fa (10)
where [kappa] is the radius of gyration about the axis of symmetry, a
is the constant distance of G from the plane, and R, F are the normal
and tangential components of the reaction of the plane, as shown in
fig. 74. We have also the kinematical relation [.x] = a[.theta]. Hence
a² [kappa]²
[:x] = ------------- g sin [alpha], R = Mg cos [alpha], F = ------------- Mg sin [alpha]. (11)
[kappa]² + a² [kappa]² + a²
The acceleration of G is therefore less than in the case of
frictionless sliding in the ratio a²/([kappa]² + a²). For a
homogeneous sphere this ratio is 5/7, for a uniform circular cylinder
or disk 2/3, for a circular hoop or a thin cylindrical shell ½.
The equation of energy for a rigid body has already been stated (in
effect) as a corollary from fundamental assumptions. It may also be
deduced from the principles of linear and angular momentum as embodied
in the equations (9). We have
M([.x][:x] + [.y][:]y) + l[.theta][:theta] + X[.x] + Y[.y] + N[.theta], (12)
whence, integrating with respect to t,
½ M([.x]² + [.y]²) + ½I[.theta]² = [int](X dx + Y dy + Nd[theta]) + const. (13)
The left-hand side is the kinetic energy of the whole mass, supposed
concentrated at G and moving with this point, together with the kinetic
energy of the motion relative to G (§ 15); and the right-hand member
represents the integral work done by the extraneous forces in the
successive infinitesimal displacements into which the motion may be
resolved.
[Illustration: FIG. 75.]
The formula (13) may be easily verified in the case of the compound
pendulum, or of the solid rolling down an incline. As another example,
suppose we have a circular cylinder whose mass-centre is at an
excentric point, rolling on a horizontal plane. This includes the case
of a compound pendulum in which the knife-edge is replaced by a
cylindrical pin. If [alpha] be the radius of the cylinder, h the
distance of G from its axis (O), [kappa] the radius of gyration about
a longitudinal axis through G, and [theta] the inclination of OG to
the vertical, the kinetic energy is 1/2M[kappa]²[.theta]² +
½M·CG²·[.theta]², by § 3, since the body is turning about the line of
contact (C) as instantaneous axis, and the potential energy is--Mgh
cos [theta]. The equation of energy is therefore
½ M([kappa]² + [alpha]² + h² - 2 ah cos [theta]) [.theta]² - Mgh cos [theta] - const. (14)
Whenever, as in the preceding examples, a body or a system of bodies, is
subject to constraints which leave it virtually only one degree of
freedom, the equation of energy is sufficient for the complete
determination of the motion. If q be any variable co-ordinate defining
the position or (in the case of a system of bodies) the configuration,
the velocity of each particle at any instant will be proportional to
[.q], and the total kinetic energy may be expressed in the form ½A[.q]²,
where A is in general a function of q [cf. equation (14)]. This
coefficient A is called the coefficient of inertia, or the reduced
inertia of the system, referred to the co-ordinate q.
[Illustration: FIG. 76.]
Thus in the case of a railway truck travelling with velocity u the
kinetic energy is ½(M + m[kappa]²/[alpha]²)u², where M is the total
mass, [alpha] the radius and [kappa] the radius of gyration of each
wheel, and m is the sum of the masses of the wheels; the reduced
inertia is therefore M + m[kappa]²/[alpha]². Again, take the system
composed of the flywheel, connecting rod, and piston of a
steam-engine. We have here a limiting case of three-bar motion (§ 3),
and the instantaneous centre J of the connecting-rod PQ will have the
position shown in the figure. The velocities of P and Q will be in the
ratio of JP to JQ, or OR to OQ; the velocity of the piston is
therefore y[.theta], where y = OR. Hence if, for simplicity, we
neglect the inertia of the connecting-rod, the kinetic energy will be
½(I + My²)[.theta]², where I is the moment of inertia of the flywheel,
and M is the mass of the piston. The effect of the mass of the piston
is therefore to increase the apparent moment of inertia of the
flywheel by the variable amount My². If, on the other hand, we take OP
(= x) as our variable, the kinetic energy is 1/2(M + I/y²)[.x]². We
may also say, therefore, that the effect of the flywheel is to
increase the apparent mass of the piston by the amount I/y²; this
becomes infinite at the "dead-points" where the crank is in line with
the connecting-rod.
If the system be "conservative," we have
½ Aq² + V = const., (15)
where V is the potential energy. If we differentiate this with respect
to t, and divide out by [.q], we obtain
dA dV
A[:q] + ½ -- q² + -- = 0 (16)
dq dq
as the equation of motion of the system with the unknown reactions (if
any) eliminated. For equilibrium this must be satisfied by [.q] = O;
this requires that dV/dq = 0, i.e. the potential energy must be
"stationary." To examine the effect of a small disturbance from
equilibrium we put V = f(q), and write q = q0 + [eta], where q0 is a
root of f´(q0) = 0 and [eta] is small. Neglecting terms of the second
order in [eta] we have dV/dq = f´(q) = f´´(q0)·[eta], and the equation
(16) reduces to
A[:eta] + f´´(q0)[eta] = 0, (17)
where A may be supposed to be constant and to have the value
corresponding to q = q0. Hence if f´´(q0) > 0, i.e. if V is a minimum in
the configuration of equilibrium, the variation of [eta] is
simple-harmonic, and the period is 2[pi][root][A/f´´(q0)]. This depends
only on the constitution of the system, whereas the amplitude and epoch
will vary with the initial circumstances. If f´´(q0) < 0, the solution
of (17) will involve real exponentials, and [eta] will in general
increase until the neglect of the terms of the second order is no longer
justified. The configuration q = q0, is then unstable.
As an example of the method, we may take the problem to which equation
(14) relates. If we differentiate, and divide by [theta], and retain
only the terms of the first order in [theta], we obtain
{x² + (h - [alpha])²} [:theta] + gh[theta] = 0, (18)
as the equation of small oscillations about the position [theta] = 0.
The length of the equivalent simple pendulum is {[kappa]² + (h -
[alpha])²}/h.
The equations which express the change of motion (in two dimensions) due
to an instantaneous impulse are of the forms
M(u´- u) = [xi], M([nu]´ - [nu]) = [eta], I([omega]´ - [omega]) = [nu]. (19)
[Illustration: FIG. 77.]
Here u´, [nu]´ are the values of the component velocities of G just
before, and u, [nu] their values just after, the impulse, whilst
[omega]´, [omega] denote the corresponding angular velocities. Further,
[xi], [eta] are the time-integrals of the forces parallel to the
co-ordinate axes, and [nu] is the time-integral of their moment about G.
Suppose, for example, that a rigid lamina at rest, but free to move, is
struck by an instantaneous impulse F in a given line. Evidently G will
begin to move parallel to the line of F; let its initial velocity be u´,
and let [omega]´ be the initial angular velocity. Then Mu´ = F,
I[omega]´ = F·GP, where GP is the perpendicular from G to the line of F.
If PG be produced to any point C, the initial velocity of the point C of
the lamina will be
u´ - [omega]´·GC = (F/M)·(I - GC·CP/[kappa]²),
where [kappa]² is the radius of gyration about G. The initial centre of
rotation will therefore be at C, provided GC·GP = [kappa]². If this
condition be satisfied there would be no impulsive reaction at C even if
this point were fixed. The point P is therefore called the _centre of
percussion_ for the axis at C. It will be noted that the relation
between C and P is the same as that which connects the centres of
suspension and oscillation in the compound pendulum.
§ 18. _Equations of Motion in Three Dimensions._--It was proved in § 7
that a body moving about a fixed point O can be brought from its
position at time t to its position at time t + [delta]t by an
infinitesimal rotation [epsilon] about some axis through O; and the
limiting position of this axis, when [delta]t is infinitely small, was
called the "instantaneous axis." The limiting value of the ratio
[epsilon]/[delta]t is called the _angular velocity_ of the body; we
denote it by [omega]. If [xi], [eta], [zeta] are the components of
[epsilon] about rectangular co-ordinate axes through O, the limiting
values of [xi]/[delta]t, [eta]/[delta]t, [zeta]/[delta]t are called the
_component angular velocities_; we denote them by p, q, r. If l, m, n be
the direction-cosines of the instantaneous axis we have
p = l[omega], q = m[omega], r = n[omega], (1)
p² + q² + r² = [omega]². (2)
If we draw a vector OJ to represent the angular velocity, then J traces
out a certain curve in the body, called the _polhode_, and a certain
curve in space, called the _herpolhode_. The cones generated by the
instantaneous axis in the body and in space are called the polhode and
herpolhode cones, respectively; in the actual motion the former cone
rolls on the latter (§ 7).
[Illustration: FIG. 78.]
The special case where both cones are right circular and [omega] 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 [alpha]be the semi-angle of the rolling cone, [beta] the
constant inclination of OC to OZ, and [.psi] 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
[omega] sin [alpha] = ±[.psi] sin [beta], (3)
where the lower sign belongs to the third case. The earth's
precessional motion is of this latter type, the angles being [alpha] =
.0087´´, [beta] = 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
2T = [Sigma] {m([omega]·PN)²} = [omega]²·[Sigma](m·PN²) = I[omega]², (4)
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
I = Al² + Bm² + Cn² - 2Fmn - 2Gnl - 2Hlm,
and therefore by (1),
2T = Ap² + Bq² + Cr² - 2Fqr - 2Grp - 2Hpq. (5)
Again, if x, y, z be the co-ordinates of P, the component velocities of
m are
qz - ry, rx - pz, py - qx, (6)
by § 7 (5); hence, if [lambda], [mu], [nu] be now used to denote the
component angular momenta about the co-ordinate axes, we have [lambda] =
[Sigma][m(py - qx)y - m(rx - pz)z], with two similar formulae, or
[dP]T \
[lambda] = Ap - Hq - Gr= -----, |
[dP]p |
|
[dP]T |
[mu] = -Hp + Bq - Fr = -----, > (7)
[dP]q |
|
[dP]T |
[nu] = -Gp - Fq + Cr = -----. |
[dP]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
2T = Ap² + Bq² + Cr², (8)
[lambda] = Ap, [mu] = Bq, [nu] = Cr. (9)
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 [lambda], [mu], [nu]. 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 [Gamma] be the resultant angular momentum, so
that
[lambda]² + [mu]² + [nu]² = [Gamma]², (10)
the length of the perpendicular OH on the tangent plane at J is
Ap p Bq q Cr r 2T [rho]
OH = ------- · -------[rho] + ------- · -------[rho] + ------- · -------[rho] = ------- · -------, (11)
[Gamma] [omega] [Gamma] [omega] [Gamma] [omega] [Gamma] [omega]
where [rho] = 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
u + qz - ry, v + rx - pz, w + py - qx (12)
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
2T = M0(u² + v² + w²) + Ap² + Bq² + Cr² - 2Fqr - 2Grp - 2Hpg, (13)
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 [xi], [eta], [zeta] of linear momentum are
[dP]T [dP]T [dP]T
[xi] = M0u = -----, [eta] = M0v = -----, [zeta] = M0w = -----, (14)
[dP]u [dP]v [dP]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 ([xi], [eta], [zeta], [lambda], [mu], [nu])
change the motion from (u, v, w, p, q, r) to (u´, v´, w´, p´, q´, r´) we
have
M0(u´- u) = [xi], M0(v´- v) = [eta], M0(w´- w) = [zeta], \
> (15)
A(p´ - p) = [lambda], B(q´- q) = [mu], C(r´- r) = [nu], /
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
T´- T = [xi] · ½(u + u´) + [eta] · ½(v + v´) + [zeta] · ½(w + w´),
+ [lambda] · ½(p + p´) + [mu] · ½(q + q´) + [nu] · ½(r + r´). (16)
The factors of [xi], [eta], [zeta], [lambda], [mu], [nu] 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
½M([kappa]² + Cq²)[omega]´², if C be the instantaneous centre; this is
seen to be equal to ½F·[omega]´·CP, where [omega]´·CP represents the
initial velocity of P.
The equations of continuous motion of a solid are obtained by
substituting the values of [xi], [eta], [zeta], [lambda], [mu], [nu]
from (14) and (7) in the general equations
d[xi] d[eta] d[zeta] \
----- = X, ------ = Y, ------- = Z, |
dt dt dt |
> (17)
d[lambda] d[mu] d[nu] |
--------- = L, ----- = M, ----- = N, |
dt dt dt /
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.
[Illustration: FIG. 79.]
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
M0[.u] = X, M0[.v] = Y, M0[.w] = Z,
C[.p] = L, C[.q] = M, C[.r] = N, (18)
where C is the constant moment of inertia about any axis through the
mass-centre. Take, for example, the case of a sphere rolling on a
plane; and let the axes Ox, Oy be drawn through the centre parallel to
the plane, so that the equation of the latter is z = -a. We will
suppose that the extraneous forces consist of a known force (X, Y, Z)
at the centre, and of the reactions (F1, F2, R) at the point of
contact. Hence
M0[.u] = X + F1, M0[.v] = Y + F2, 0 = Z + R, \
C[.p] = F2a, C[.q] = -F1a, C[.r] = 0. / (19)
The last equation shows that the angular velocity about the normal to
the plane is constant. Again, since the point of the sphere which is
in contact with the plane is instantaneously at rest, we have the
geometrical relations
u + qa = 0, v + pa = 0, w = 0, (20)
by (12). Eliminating p, q, we get
(M0 + Ca^-2)[.u] = X, (M0 + Ca^-2)[.v] = Y. (21)
The acceleration of the centre is therefore the same as if the plane
were smooth and the mass of the sphere were increased by C/[alpha]².
Thus the centre of a sphere rolling under gravity on a plane of
inclination a describes a parabola with an acceleration
g sin [alpha]/(1 + C/Ma²)
parallel to the lines of greatest slope.
Take next the case of a sphere rolling on a fixed spherical surface.
Let a be the radius of the rolling sphere, c that of the spherical
surface which is the locus of its centre, and let x, y, z be the
co-ordinates of this centre relative to axes through O, the centre of
the fixed sphere. If the only extraneous forces are the reactions (P,
Q, R) at the point of contact, we have
M0[:x] = P, M0[.y] = Q, M0[:z] = R, \
|
a a a > (22)
Cp = ---(yR - zQ), C[.q] = ---(zP - xR), C[.r] = ---(xQ - yP), |
c c c /
the standard case being that where the rolling sphere is outside the
fixed surface. The opposite case is obtained by reversing the sign of
a. We have also the geometrical relations
[.x] = (a/c)(qz - ry), [.y] = (a/c)(rx - pz), [.z] = (a/c)(py - gx), (23)
If we eliminate P, Q, R from (22), the resulting equations are
integrable with respect to t; thus
M0a M0a
p = - ---(y[.z] - z[.y]) + [alpha], q = - ---(z[.x] - x[.z]) + [beta],
Cc Cc
M0a
r = - ---(x[.y] - y[.x]) + [gamma], (24)
Cc
where [alpha], [beta], [gamma] are arbitrary constants. Substituting
in (23) we find
/ M0a²\ a / M0a²\ a
( 1 + ---- )[.x] = ---([beta]z - [gamma]y), ( 1 + ---- )[.y] = ---([gamma]x - [alpha]z),
\ C / c \ C / c
/ M0a²\ a
( 1 + ---- )[.z] = ---([alpha]y - [beta]x). (25)
\ C / c
Hence [alpha][.x] + [beta][.y] + [gamma][.z] = 0, or
[alpha]x + [beta]y + [gamma]z = const.; (26)
which shows that the centre of the rolling sphere describes a circle.
If the axis of z be taken normal to the plane of this circle we have
[alpha] = 0, [beta] = 0, and
/ M0a²\ a / M0a²\ a
( 1 + ---- )[.x] = -[gamma]--- y, ( 1 + ----- )[.y] = [gamma]--- x. (27)
\ C / c \ C / c
The solution of these equations is of the type
x = b cos ([sigma][tau] + [epsilon]), y = b sin ([sigma][iota] + [epsilon]), (28)
where b, [epsilon] are arbitrary, and
[gamma]a/c
[sigma]= ---------- (29)
1 + M0a²/C
The circle is described with the constant angular velocity [sigma].
When the gravity of the rolling sphere is to be taken into account the
preceding method is not in general convenient, unless the whole motion
of G is small. As an example of this latter type, suppose that a
sphere is placed on the highest point of a fixed sphere and set
spinning about the vertical diameter with the angular velocity n; it
will appear that under a certain condition the motion of G consequent
on a slight disturbance will be oscillatory. If Oz be drawn vertically
upwards, then in the beginning of the disturbed motion the quantities
x, y, p, q, P, Q will all be small. Hence, omitting terms of the
second order, we find
M0[:x] = P, M0[.y] = Q, R = M0g, \
> (30)
C[.p] = -(M0ga/c)y + aQ, C[.q] = (M0ga/c)x - aP, C[.r] = 0. /
The last equation shows that the component r of the angular velocity
retains (to the first order) the constant value n. The geometrical
relations reduce to
[.x] = aq - (na/c)y, [.y] = -ap + (na/c)x. (31)
Eliminating p, g, P, Q, we obtain the equations
(C + M0a²)[:x] + (Cna/c)y - (M0ga²/c)x = 0, }
(C + M0a²)[:y] - (Cna/c)x - (M0ga²/c)y = 0, } (32)
which are both contained in
_ _
| d² Cna d M0ga² |
|(C + M0a²)--- - i --- --- - ----- | (x + iy) = 0. (33)
|_ dt² c dt c _|
This has two solutions of the type x + iy = [alpha]e^{i([sigma]t +
[epsilon])}, where [alpha], [epsilon] are arbitrary, and [sigma] is a
root of the quadratic
(C + M0a²)[sigma]² - (Cna/c)[sigma] + M0ga²/c = 0. (34)
If
n² > (4Mgc/C) (1 + M0a²/C), (35)
both roots are real, and have the same sign as n. The motion of G then
consists of two superposed circular vibrations of the type
x = [alpha] cos ([sigma]t + [epsilon]), y = [alpha] sin ([sigma]t + [epsilon]), (36)
in each of which the direction of revolution is the same as that of
the initial spin of the sphere. It follows therefore that the original
position is stable provided the spin n exceed the limit defined by
(35). The case of a sphere spinning about a vertical axis at the
lowest point of a spherical bowl is obtained by reversing the signs of
[alpha] and c. It appears that this position is always stable.
It is to be remarked, however, that in the first form of the problem
the stability above investigated is practically of a limited or
temporary kind. The slightest frictional forces--such as the
resistance of the air--even if they act in lines through the centre of
the rolling sphere, and so do not directly affect its angular
momentum, will cause the centre gradually to descend in an
ever-widening spiral path.
§ 19. _Free Motion of a Solid._--Before proceeding to further problems
of motion under extraneous forces it is convenient to investigate the
free motion of a solid relative to its mass-centre O, in the most
general case. This is the same as the motion about a fixed point under
the action of extraneous forces which have zero moment about that point.
The question was first discussed by Euler (1750); the geometrical
representation to be given is due to Poinsot (1851).
The kinetic energy T of the motion relative to O will be constant. Now T
= ½I[omega]², where [omega] is the angular velocity and I is the moment
of inertia about the instantaneous axis. If [rho] be the radius-vector
OJ of the momental ellipsoid
Ax² + By² + Cz² = M[epsilon]^4 (1)
drawn in the direction of the instantaneous axis, we have I =
M[epsilon]^4/[rho]² (§ 11); hence [omega] varies as [rho]. The locus of
J may therefore be taken as the "polhode" (§ 18). Again, the vector
which represents the angular momentum with respect to O will be constant
in every respect. We have seen (§ 18) that this vector coincides in
direction with the perpendicular OH to the tangent plane of the momental
ellipsoid at J; also that
2T [rho]
OH = ------- · -------, (2)
[Gamma] [omega]
where [Gamma] is the resultant angular momentum about O. Since [omega]
varies as [rho], it follows that OH is constant, and the tangent plane
at J is therefore fixed in space. The motion of the body relative to O
is therefore completely represented if we imagine the momental ellipsoid
at O to roll without sliding on a plane fixed in space, with an angular
velocity proportional at each instant to the radius-vector of the point
of contact. The fixed plane is parallel to the invariable plane at O,
and the line OH is called the _invariable line_. The trace of the point
of contact J on the fixed plane is the "herpolhode."
If p, q, r be the component angular velocities about the principal axes
at O, we have
(A²p² + B²q² + C²r²)/[Gamma]² = (Ap² + Bq² + Cr²)/2T, (3)
each side being in fact equal to unity. At a point on the polhode cone x
: y : z = p : q : r, and the equation of this cone is therefore
/ [Gamma]²\ / [Gamma]²\ / [Gamma]²\
A²( 1 - -------- )x² + B²( 1 - -------- )y² + C²( 1 - -------- )z² = 0. (4)
\ 2AT / \ 2BT / \ 2CT /
Since 2AT - [Gamma]² = B (A - B)q² + C(A - C)r², it appears that if A >
B > C the coefficient of x² in (4) is positive, that of z² is negative,
whilst that of y² is positive or negative according as 2BT <> [Gamma]².
Hence the polhode cone surrounds the axis of greatest or least moment
according as 2BT <> [Gamma]². In the critical case of 2BT = [Gamma]² it
breaks up into two planes through the axis of mean moment (Oy). The
herpolhode curve in the fixed plane is obviously confined between two
concentric circles which it alternately touches; it is not in general a
re-entrant curve. It has been shown by De Sparre that, owing to the
limitation imposed on the possible forms of the momental ellipsoid by
the relation B + C > A, the curve has no points of inflexion. The
invariable line OH describes another cone in the body, called the
_invariable cone_. At any point of this we have x : y : z = Ap. Bq : Cr,
and the equation is therefore
/ [Gamma]²\ / [Gamma]²\ / [Gamma]²\
( 1 - -------- )x² + ( 1 - -------- )y² + ( 1 - -------- )z² = 0. (5)
\ 2AT / \ 2BT / \ 2CT /
[Illustration: FIG. 80.]
The signs of the coefficients follow the same rule as in the case of
(4). The possible forms of the invariable cone are indicated in fig. 80
by means of the intersections with a concentric spherical surface. In
the critical case of 2BT = [Gamma]² the cone degenerates into two
planes. It appears that if the body be sightly disturbed from a state of
rotation about the principal axis of greatest or least moment, the
invariable cone will closely surround this axis, which will therefore
never deviate far from the invariable line. If, on the other hand, the
body be slightly disturbed from a state of rotation about the mean axis
a wide deviation will take place. Hence a rotation about the axis of
greatest or least moment is reckoned as stable, a rotation about the
mean axis as unstable. The question is greatly simplified when two of
the principal moments are equal, say A = B. The polhode and herpolhode
cones are then right circular, and the motion is "precessional"
according to the definition of § 18. If [alpha] be the inclination of
the instantaneous axis to the axis of symmetry, [beta] the inclination
of the latter axis to the invariable line, we have
[Gamma] cos [beta] = C [omega] cos [alpha], [Gamma] sin [beta] = A [omega] sin [alpha], (6)
whence
A
tan [beta] = --- tan [alpha]. (7)
C
[Illustration: FIG. 81.]
Hence [beta] <> [alpha], and the circumstances are therefore those of
the first or second case in fig. 78, according as A <> C. If [psi] be
the rate at which the plane HOJ revolves about OH, we have
sin [alpha] C cos [alpha]
[psi] = ----------- [omega] = ------------- [omega], (8)
sin [beta] A cos [beta]
by § 18 (3). Also if [.chi] be the rate at which J describes the
polhode, we have [.psi] sin ([beta]-[alpha]) = [.chi] sin [beta], whence
sin([alpha] - [beta])
[.chi] = --------------------- [omega]. (9)
sin[alpha]
If the instantaneous axis only deviate slightly from the axis of
symmetry the angles [alpha], [beta] are small, and [.chi] = (A -
C)A·[omega]; the instantaneous axis therefore completes its revolution
in the body in the period
2[pi] A - C
------ = ----- [omega]. (10)
[.chi] A
In the case of the earth it is inferred from the independent
phenomenon of luni-solar precession that (C - A)/A = .00313. Hence if
the earth's axis of rotation deviates slightly from the axis of
figure, it should describe a cone about the latter in 320 sidereal
days. This would cause a periodic variation in the latitude of any
place on the earth's surface, as determined by astronomical methods.
There appears to be evidence of a slight periodic variation of
latitude, but the period would seem to be about fourteen months. The
discrepancy is attributed to a defect of rigidity in the earth. The
phenomenon is known as the _Eulerian nutation_, since it is supposed
to come under the free rotations first discussed by Euler.
§ 20. _Motion of a Solid of Revolution._--In the case of a solid of
revolution, or (more generally) whenever there is kinetic symmetry about
an axis through the mass-centre, or through a fixed point O, a number
of interesting problems can be treated almost directly from first
principles. It frequently happens that the extraneous forces have zero
moment about the axis of symmetry, as e.g. in the case of the flywheel
of a gyroscope if we neglect the friction at the bearings. The angular
velocity (r) about this axis is then constant. For we have seen that r
is constant when there are no extraneous forces; and r is evidently not
affected by an instantaneous impulse which leaves the angular momentum
Cr, about the axis of symmetry, unaltered. And a continuous force may be
regarded as the limit of a succession of infinitesimal instantaneous
impulses.
[Illustration: FIG. 82.]
Suppose, for example, that a flywheel is rotating with angular
velocity n about its axis, which is (say) horizontal, and that this
axis is made to rotate with the angular velocity [psi] in the
horizontal plane. The components of angular momentum about the axis of
the flywheel and about the vertical will be Cn and A [psi]
respectively, where A is the moment of inertia about any axis through
the mass-centre (or through the fixed point O) perpendicular to that
of symmetry. If [->OK] be the vector representing the former component
at time t, the vector which represents it at time t + [delta]t will be
[->OK´], equal to [->OK] in magnitude and making with it an angle
[delta][psi]. Hence [->KK´] ( = Cn [delta][psi]) will represent the
change in this component due to the extraneous forces. Hence, so far
as this component is concerned, the extraneous forces must supply a
couple of moment Cn[.psi] in a vertical plane through the axis of the
flywheel. If this couple be absent, the axis will be tilted out of the
horizontal plane in such a sense that the direction of the spin n
approximates to that of the azimuthal rotation [.psi]. The remaining
constituent of the extraneous forces is a couple A[:psi] about the
vertical; this vanishes if [.psi] is constant. If the axis of the
flywheel make an angle [theta] with the vertical, it is seen in like
manner that the required couple in the vertical plane through the axis
is Cn sin [theta] [.psi]. This matter can be strikingly illustrated
with an ordinary gyroscope, e.g. by making the larger movable ring in
fig. 37 rotate about its vertical diameter.
[Illustration: FIG. 83.]
If the direction of the axis of kinetic symmetry be specified by means
of the angular co-ordinates [theta], [psi] of § 7, then considering the
component velocities of the point C in fig. 83, which are [.theta] and
sin [theta][.psi] along and perpendicular to the meridian ZC, we see
that the component angular velocities about the lines OA´, OB´ are -sin
[theta] [.psi] and [.theta] respectively. Hence if the principal moments
of inertia at O be A, A, C, and if n be the constant angular velocity
about the axis OC, the kinetic energy is given by
2T = A ([.theta]² + sin² [theta][.psi]²) + Cn². (1)
Again, the components of angular momentum about OC, OA´ are Cn, -A sin
[theta] [.psi], and therefore the angular momentum ([mu], say) about OZ
is
[mu] = A sin² [theta][.psi] + Cn cos [theta]. (2)
We can hence deduce the condition of steady precessional motion in a
top. A solid of revolution is supposed to be free to turn about a fixed
point O on its axis of symmetry, its mass-centre G being in this axis at
a distance h from O. In fig. 83 OZ is supposed to be vertical, and OC is
the axis of the solid drawn in the direction OG. If [theta] is constant
the points C, A´ will in time [delta]t come to positions C´´, A´´ such
that CC´´ = sin [theta] [delta][psi], A´A´´ = cos [theta] [delta][psi],
and the angular momentum about OB´ will become Cn sin [theta]
[delta][psi] - A sin [theta] [.psi] · cos [theta] [delta][psi]. Equating
this to Mgh sin [theta] [delta]t, and dividing out by sin [theta], we
obtain
A cos [theta] [.psi]² - Cn[.psi] + Mgh = 0, (3)
as the condition in question. For given values of n and [theta] we have
two possible values of [.psi] provided n exceed a certain limit. With a
very rapid spin, or (more precisely) with Cn large in comparison with
[root](4AMgh cos [theta]), one value of [.psi] is small and the other
large, viz. the two values are Mgh/Cn and Cn/A cos [theta]
approximately. The absence of g from the latter expression indicates
that the circumstances of the rapid precession are very nearly those of
a free Eulerian rotation (§ 19), gravity playing only a subordinate
part.
[Illustration: 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
[.psi], whilst its plane preserves a constant inclination [theta] to
the horizontal. The components of the reaction of the horizontal lane
will be Mc[.psi]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 Mc[.psi]². [alpha] sin [theta] - Mg[alpha] cos [theta], where
[alpha] is the radius of the disk. Equating this to the rate of
increase of the angular momentum about OB´, investigated as above, we
find
/ a \ a²
( C + Ma² + A --- cos [theta] ) [.psi]² = Mg --- cot [theta], (4)
\ c / c
where use has been made of the obvious relation n[alpha] = c[.psi]. If
c and [theta] be given this formula determines the value of [psi] for
which the motion will be steady.
In the case of the top, the equation of energy and the condition of
constant angular momentum ([mu]) about the vertical OZ are sufficient to
determine the motion of the axis. Thus, we have
½A ([.theta]² + sin² [theta][.psi]²) + ½Cn² + Mgh cos [theta] = const., (5)
A sin² [theta][.psi] + [nu] cos [theta] = [mu], (6)
where [nu] is written for Cn. From these [.psi] may be eliminated, and
on differentiating the resulting equation with respect to t we obtain
([mu] - [nu] cos [theta])([mu] cos [theta] - [nu])
A[:theta] - -------------------------------------------------- - Mgh sin [theta] = 0. (7)
A sin³ [theta]
If we put [:theta] = 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
[alpha] with the vertical, we write [theta] = [alpha] + [chi], and
neglect terms of the second order in [chi]. The result is of the form
[:chi] + [sigma]²[chi] = 0, (8)
where
[sigma]² = {([mu] - [nu] cos [alpha])² + 2([mu] - [nu] cos [alpha])([mu] cos [alpha] - [nu])
cos [alpha] + ([mu] cos [alpha] - [nu])²} / A² sin^4 [alpha]. (9)
When [nu] is large we have, for the "slow" precession [sigma] = [nu]/A,
and for the "rapid" precession [sigma] = A/[nu] cos [alpha] = [.psi],
approximately. Further, on examining the small variation in [.psi], 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 [alpha], the longer axis being in the plane of [theta].
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 [lambda], [mu], [nu]. 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 + [delta]t, 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[delta]t, q[delta]t, respectively.
Hence the altered angular momentum about Ox´ will be [lambda] +
[delta][lambda] + ([mu] + [delta][mu]) (-r[delta]t) + ([nu] +
[delta][nu]) q[delta]t. If L, M, N be the moments of the extraneous
forces about Ox, Oy, Oz this must be equal to [lambda] + L[delta]t.
Hence, and by symmetry, we obtain
d[lambda] \
--------- - r[nu] + q[nu] = L, |
dt |
|
d[mu] |
----- - p[nu] + r[lanbda] = M, > (1)
dt |
|
d[nu] |
----- - q[lambda] + p[nu] = 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 [lambda], [mu], [nu] = Ap, Bq, Cr, whence
dp \
A -- - (B - C) qr = L, |
dt |
|
dq |
B -- - (C - A) rp = M, > (2)
dt |
|
dr |
C -- - (A - B) pq = N. |
dt /
If we multiply these by p, q, r and add, we get
d
--- · ½(Ap² + Bq² + Cr²) = Lp + Mq + Nr, (3)
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 [omega] about a
fixed axis (l, m, n). Since p, q, r are then constant, the requisite
constraining couple is
L = (C - B) mn[omega]², M = (A - C) nl[omega]², N = (B - A) lm[omega]². (4)
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
dp \
A -- = (B - C) qr, |
dt |
|
dq |
B -- = (C - A) rp, > (5)
dt |
|
dr |
C -- = (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 + Cr²
and A²p² + B²q² + C²r² are both constant. The former is, in fact, equal
to 2T, and the latter to [Gamma]², where T is the kinetic energy and
[Gamma] 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
_
/ [phi] d[phi]
u = | ------------, [Delta][phi] = [root](1 - k² sin² [phi]),
_/ 0 [Delta][phi]
we have, in the notation of elliptic functions, [phi] = am u. If we
assume
p = p0 cos am ([sigma]t + [epsilon]), q = q0sin am ([sigma]t + [epsilon]),
r = r0[Delta] am ([sigma]t + [epsilon]), (7)
we find
[sigma]p0 [sigma]q0 k²[sigma]r0
[.p] = - --------- qr, [.q] = --------- rp, [.r] = - ----------- pq. (8)
q0r0 r0p0 p0q0
Hence (5) will be satisfied, provided
-[sigma]p0 B - C [sigma]q0 C - A -k²[sigma]r0 A - B
---------- = -----, --------- = -----, ------------ = -----. (9)
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, [sigma], [epsilon]. 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 [epsilon] =
0, and p0, r0 will become identical with the initial values of p, r.
The conditions (9) then lead to
A(A - C) (A - C)(B - C) A(A - B) p0²
q0² = -------- p0², [sigma]² = -------------- r0², k² = -------- · ---. (10)
B(B - C) AB C(B - C) r0²
For a real solution we must have k² < 1, which is equivalent to 2BT > [Gamma]². If the initial
conditions are such as to make 2BT < [Gamma]², we must interchange the
forms of p and r in (7). In the present case the instantaneous axis
returns to its initial position in the body whenever [phi] increases
by 2[pi], i.e. whenever t increases by 4K/[sigma], when K is the
"complete" elliptic integral of the first kind with respect to the
modulus k.
The elliptic functions degenerate into simpler forms when k² = 0 or k²
= 1. The former case arises when two of the principal moments are
equal; this has been sufficiently dealt with in § 19. If k² = 1, we
must have 2BT = [Gamma]². We have seen that the alternative 2BT <>
[Gamma]² determines whether the polhode cone surrounds the principal
axis of least or greatest moment. The case of 2BT = [Gamma]², exactly,
is therefore a critical case; it may be shown that the instantaneous
axis either coincides permanently with the axis of mean moment or
approaches it asymptotically.
When the origin of the moving axes is also in motion with a velocity
whose components are u, v, w, the dynamical equations are
d[xi] d[eta] d[zeta]
----- - r[eta] + q[zeta] = X, ------ - p[zeta] - r[chi] = Y, ------- - q[chi] + p[eta] = Z, (11)
dt dt dt
d[lambda] d[mu] \
--------- - r[mu] + q[nu] - w[eta] + v[zeta] = L, ----- - p[nu] + r[lambda]- u[zeta] + w[xi] = M, |
dt dt |
> (12)
d[nu] |
----- - q[lambda] + p[mu] - v[xi] + u[eta] = N. /
dt
To prove these, we may take fixed axes O´x´, O´y´, O´z´ coincident with
the moving axes at time t, and compare the linear and angular momenta
[xi] + [delta][xi], [eta] + [delta][eta], [zeta] + [delta][zeta],
[lambda] + [delta][lambda], [mu] + [delta][mu], [nu] + [delta][nu]
relative to the new position of the axes, Ox, Oy, Oz at time t +
[delta]t with the original momenta [xi], [eta], [zeta], [lambda], [mu],
[nu] relative to O´x´, O´y´, O´z´ at time t. As in the case of (2), the
equations are applicable to any dynamical system whatever. If the moving
origin coincide always with the mass-centre, we have [xi], [eta], [zeta]
= M0u, M0v, M0w, where M0 is the total mass, and the equations simplify.
When, in any problem, the values of u, v, w, p, q, r have been
determined as functions of t, it still remains to connect the moving
axes with some fixed frame of reference. It will be sufficient to take
the case of motion about a fixed point O; the angular co-ordinates
[theta], [phi], [psi] of Euler may then be used for the purpose.
Referring to fig. 36 we see that the angular velocities p, q, r of the
moving lines, OA, OB, OC about their instantaneous positions are
p = [.theta] sin [phi] - sin [theta] cos [phi][.psi], \
q = [.theta] cos [phi] + sin [theta] sin [phi][.psi], > (13)
r = [.phi] + cos [theta][.psi], /
by § 7 (3), (4). If OA, OB, OC be principal axes of inertia of a solid,
and if A, B, C denote the corresponding moments of inertia, the kinetic
energy is given by
2T = A([.theta] sin [phi] - sin [theta] cos [phi][.psi])² \
+ B([.theta] cos [phi] + sin [theta] sin [theta][psi])² > (14)
+ C([.phi] + cos [theta][.psi])². /
If A = B this reduces to
2T = A([.theta]² + sin² [theta][.psi]²) + C([.phi] + cos [theta][.psi])²; (15)
cf. § 20 (1).
§ 22. _Equations of Motion in Generalized Co-ordinates._--Suppose we
have a dynamical system composed of a finite number of material
particles or rigid bodies, whether free or constrained in any way, which
are subject to mutual forces and also to the action of any given
extraneous forces. The configuration of such a system can be completely
specified by means of a certain number (n) of independent quantities,
called the generalized co-ordinates of the system. These co-ordinates
may be chosen in an endless variety of ways, but their number is
determinate, and expresses the number of _degrees of freedom_ of the
system. We denote these co-ordinates by q1, q2, ... q_n. It is implied
in the above description of the system that the Cartesian co-ordinates
x, y, z of any particle of the system are known functions of the q's,
varying in form (of course) from particle to particle. Hence the kinetic
energy T is given by
__
2T = \ {m([.x]² + [.y]² + [.z]²)}
/__
= a11[.q]1² + a22[.q]2² + ... + 2a12[.q]1[.q]2 + ..., (1)
where
_ _
__ | { / [dP]x \² / [dP]y \² / [dP]z \² } | \
a_rr = \ | m { ( ------- ) + ( ------- ) + ( ------- ) } |, |
/__ |_ { \[dP]q_r/ \[dP]q_r/ \[dP]q_r/ } _| |
_ _ > (2)
__ | / [dP]x [dP]x [dP]y [dP]y [dP]z [dP]z \ | |
a_rs = \ | m ( ------- ------- + ------- ------- + ------- ------- ) | = a_sr. |
/__ |_ \[dP]q_r [dP]q_s [dP]q_r [dP]q_s [dP]q_r [dP]q_s/ _| /
Thus T is expressed as a homogeneous quadratic function of the
quantities [.q]1, [.q]2, ... [.q]_n, which are called the _generalized
components of velocity_. The coefficients a_rr, a_rs are called the
coefficients of inertia; they are not in general constants, being
functions of the q's and so variable with the configuration. Again, If
(X, Y, Z) be the force on m, the work done in an infinitesimal change of
configuration is
[Sigma](X[delta]x + Y[delta]y + Z[delta]z) = Q1[delta]q1 + Q2[delta]q2 + ... + Q_n[delta]q_n, (3)
where
/ [dP]x [dP]y [dP]z \
Q_r = [Sigma]( X------- + Y------- + Z------- ). (4)
\ [dP]q_r [dP]q_r [dP]q_r /
The quantities Q_r are called the _generalized components of force_.
The equations of motion of m being
m[:x] = X, m[:y] = Y, m[:z] = Z, (5)
we have
_ _
__ | / [dP]x [dP]y [dP]z \ |
\ | m ( [:x]------- + [:y]------- + [:z]------- ) | = Q_r. (6)
/__ |_ \ [dP]q_r [dP]q_r [dP]q_r / _|
Now
[dP]x [dP]x [dP]x
[.x] = ------[.q]1 + ------[.q]2 + ... + -------[.q]_n, (7)
[dP]q1 [dP]q2 [dP]q_n
whence
[dP][.x] [dP]x
---------- = -------. (8)
[dP][.q]_r [dP]q_r
Also
d / [dP]x \ [dP]²x [dP]²x [dP]²x [dP]x
-- ( ------- ) = ------------[.q]1 + -------------[.q]2 + ... + --------------[.q]_r = --------. (9)
dt \[dP]q_r/ [dP]q1[dP]q_r [dP]q2[dP]q_r [dP]q_n[dP]q_r [dP]q_r
Hence
[dP]x d / [dP]x \ d / [dP]x \ d / [dP][.x] \ [dP][.x]
[:x]------- = ---( [.x]------- ) - [.x]---( ------- ) = ---( [.x]---------- ) - [.x]--------. (10)
[dP]q_r dt \ [dP]q_r/ dt \[dP]q_r/ dt \ [dP][.q]_r/ [dP]q_r
By these and the similar transformations relating to y and z the
equation (6) takes the form
d / [dP]T \ [dP]T
--- ( ---------- ) - ------ = Q_r. (11)
dt \[dP][.q]_r/ [dP]q_r
If we put r = 1, 2, ... n in succession, we get the n independent
equations of motion of the system. These equations are due to Lagrange,
with whom indeed the first conception, as well as the establishment, of
a general dynamical method applicable to all systems whatever appears to
have originated. The above proof was given by Sir W. R. Hamilton (1835).
Lagrange's own proof will be found under DYNAMICS, § _Analytical_. In a
conservative system free from extraneous force we have
[Sigma](X [delta]x + Y [delta]y + Z [delta]z) = -[delta]V, (12)
where V is the potential energy. Hence
[dP]V
Q_r = - -------, (13)
[dP]q_r
and
d / [dP]T \ [dP]T [dP]V
--- ( ---------- ) - ----- = - -------. (14)
dt \[dP][.q]_r/ Vq_r [dP]q_r
If we imagine any given state of motion ([.q]1, [.q]2 ... [.q]_n)
through the configuration (q1, q2, ... q_n) to be generated
instantaneously from rest by the action of suitable impulsive forces, we
find on integrating (11) with respect to t over the infinitely short
duration of the impulse
[dP]T
---------- = Q_r´, (15)
[dP][.q]_r
where Q_r´ is the time integral of Q_r and so represents a _generalized
component of impulse_. By an obvious analogy, the expressions
[dP]T/[dP][.q]_r may be called the _generalized components of momentum_;
they are usually denoted by p_r thus
p_r = [dP]T/[dP][.q]_r = a_(1r)[.q]1 + a_(2r)[.q]2 + ... + a_(nr)[.q]_n. (16)
Since T is a homogeneous quadratic function of the velocities [.q]1,
[.q]2, ... [.q]_n, we have
[dP]T [dP]T [dP]T
2T = ---------[.q]1 + ---------[.q]2 + ... + ----------[.q]_n = p1[.q]2 + p2[.q]2 + ... + p_n[.q]_n. (17)
[dP][.q]1 [dP][.q]2 [dP][.q]_n
Hence
dT
2-- = [.p]1[.q]1 + [.p]2[.q]2 + ... [.p]_n[.q]_n \
dt |
|
+ [.p]1[:q]1 + [.p]2[:q]2 + ... + [.p]_n[:q]_n |
|
/ [dP]T \ / [dP]T \ / [dP]T \ |
= ( --------- + Q1 ) [.q]1 + ( --------- + Q2 ) [.q]2 + ... + ( ---------- + Q_n )[.q]_n > (18)
\[dP][.q]1 / \[dP][.q]2 / \[dP][.q]_n / |
|
[dP]T [dP]T [dP]T |
+ ---------[:q]1 + ---------[:q]2 + ... ----------[:q]_n |
[dP][.q]1 [dP][.q]2 [dP][.q]_n |
|
dT |
= -- + Q1[.q]1 + Q2[.q]2 + ... + Q_n[.q]_n, /
dt
or
dT
-- = Q1[.q]1 + Q2[.q]2 + ... + Q_n[.q]_n. (19)
dt
This equation expresses that the kinetic energy is increasing at a rate
equal to that at which work is being done by the forces. In the case of
a conservative system free from extraneous force it becomes the equation
of energy
d
--- (T + V) = 0, or T + V = const., (20)
dt
in virtue of (13).
As a first application of Lagrange's formula (11) we may form the
equations of motion of a particle in spherical polar co-ordinates. Let
r be the distance of a point P from a fixed origin O, [theta] the
angle which OP makes with a fixed direction OZ, [psi] the azimuth of
the plane ZOP relative to some fixed plane through OZ. The
displacements of P due to small variations of these co-ordinates are
[dP]r along OP, r [delta][theta] perpendicular to OP in the plane ZOP,
and r sin [theta] [delta][psi] perpendicular to this plane. The
component velocities in these directions are therefore [.r],
r[.theta], r sin [theta][.psi], and if m be the mass of a moving
particle at P we have
2T = m([.r]² + r²[.theta]² + r² sin² [theta][.psi]²). (21)
Hence the formula (11) gives
m([:r] - r[.theta]² - r sin² [theta][.psi]²) = R, \
|
d |
---(mr²[.theta]) - mr² · sin [theta] cos [theta][.psi]² = [Theta], > (22)
dt |
|
d |
---(mr² sin² [theta][.psi]) = [Psi]. /
dt
The quantities R, [Theta], [Psi] are the coefficients in the
expression R [delta]r + [Theta] [delta][theta] + [Psi] [delta][psi]
for the work done in an infinitely small displacement; viz. R is the
radial component of force, [Theta] is the moment about a line through
O perpendicular to the plane ZOP, and [Psi] is the moment about OZ. In
the case of the spherical pendulum we have r = l, [Theta] = - mgl sin
[theta], [Psi] = 0, if OZ be drawn vertically downwards, and therefore
g \
[:theta] - sin [theta] cos [theta][.psi]² = - --- sin [theta], |
l > (23)
|
sin² [theta][.psi] = h, /
where h is a constant. The latter equation expresses that the angular
momentum ml² sin² [theta][.psi] about the vertical OZ is constant. By
elimination of [.psi] we obtain
g
[:theta] - h² cos² [theta] / sin^3[theta] = - --- sin [theta]. (24)
l
If the particle describes a horizontal circle of angular radius
[alpha] with constant angular velocity [Omega], we have [.omega] = 0,
h = [Omega]² sin [alpha], and therefore
g
[Omega]² = --- cos [alpha], (25)
l
as is otherwise evident from the elementary theory of uniform circular
motion. To investigate the small oscillations about this state of
steady motion we write [theta] = [alpha] + [chi] in (24) and neglect
terms of the second order in [chi]. We find, after some reductions,
[:chi] + (1 + 3 cos² [alpha]) [Omega]²[chi] = 0; (26)
this shows that the variation of [chi] is simple-harmonic, with the
period
2[pi]/[root](1 + 3 cos² [alpha])·[Omega]
As regards the most general motion of a spherical pendulum, it is
obvious that a particle moving under gravity on a smooth sphere cannot
pass through the highest or lowest point unless it describes a
vertical circle. In all other cases there must be an upper and a lower
limit to the altitude. Again, a vertical plane passing through O and a
point where the motion is horizontal is evidently a plane of symmetry
as regards the path. Hence the path will be confined between two
horizontal circles which it touches alternately, and the direction of
motion is never horizontal except at these circles. In the case of
disturbed steady motion, just considered, these circles are nearly
coincident. When both are near the lowest point the horizontal
projection of the path is approximately an ellipse, as shown in § 13;
a closer investigation shows that the ellipse is to be regarded as
revolving about its centre with the angular velocity 2/3 ab[Omega]/l²,
where a, b are the semi-axes.
To apply the equations (11) to the case of the top we start with the
expression (15) of § 21 for the kinetic energy, the simplified form
(1) of § 20 being for the present purpose inadmissible, since it is
essential that the generalized co-ordinates employed should be
competent to specify the position of every particle. If [lambda],
[mu], [nu] be the components of momentum, we have
[dP]T \
[lambda]= ------------ = A[.theta], |
[dP][.theta] |
|
[dP]T |
[mu] = ---------- = A sin² [theta][.psi] + C([.phi] + cos [theta][.psi]) cos [theta], > (27)
[dP][.psi] |
|
[dP]T |
[nu] = ---------- = C ([.theta] + cos [theta][.psi]). /
[dP][.phi]
The meaning of these quantities is easily recognized; thus [lambda] is
the angular momentum about a horizontal axis normal to the plane of
[theta], [mu] is the angular momentum about the vertical OZ, and [nu]
is the angular momentum about the axis of symmetry. If M be the total
mass, the potential energy is V = Mgh cos [theta], if OZ be drawn
vertically upwards. Hence the equations (11) become
A[:theta] - A sin [theta] cos [theta][.psi]² + C([.phi] + cos [theta][.psi]) [.psi] sin [theta] = Mgh sin [theta], \
d/dt · {A sin² [theta][.psi] + C([.phi] + cos [theta][.psi]) cos [theta]} = 0, > (28)
d/dt · {C([.phi] + cos [theta][.psi])} = 0, /
of which the last two express the constancy of the momenta [mu], [nu].
Hence
A[:theta] - A sin [theta] cos [theta][.psi]² + [nu] sin [theta][.psi] = Mgh sin [theta], \ (29)
A sin² [theta][.psi] + [nu] cos [theta] = [mu]. /
If we eliminate [.psi] we obtain the equation (7) of § 20. The theory
of disturbed precessional motion there outlined does not give a
convenient view of the oscillations of the axis about the vertical
position. If [theta] be small the equations (29) may be written
[nu]²- 4AMgh \
[:theta] - [theta][.omega]² = - ------------[theta], > (30)
4A² |
[theta]²[.omega] = const., /
where
[nu]
[omega] = [psi] - ---- t. (31)
2A
Since [theta], [omega] are the polar co-ordinates (in a horizontal
plane) of a point on the axis of symmetry, relative to an initial line
which revolves with constant angular velocity [nu]/2A, we see by
comparison with § 14 (15) (16) that the motion of such a point will be
elliptic-harmonic superposed on a uniform rotation [nu]/2A, provided
[nu]² > 4AMgh. This gives (in essentials) the theory of the
"gyroscopic pendulum."
§ 23. _Stability of Equilibrium. Theory of Vibrations._--If, in a
conservative system, the configuration (q1, q2, ... q_n) be one of
equilibrium, the equations (14) of § 22 must be satisfied by [.q]1,
[.q]2 ... [.q]_n = 0, whence
[dP]V / [dP]q_r = 0. (1)
A necessary and sufficient condition of equilibrium is therefore that
the value of the potential energy should be stationary for infinitesimal
variations of the co-ordinates. If, further, V be a minimum, the
equilibrium is necessarily stable, as was shown by P. G. L. Dirichlet
(1846). In the motion consequent on any slight disturbance the total
energy T + V is constant, and since T is essentially positive it follows
that V can never exceed its equilibrium value by more than a slight
amount, depending on the energy of the disturbance. This implies, on the
present hypothesis, that there is an upper limit to the deviation of
each co-ordinate from its equilibrium value; moreover, this limit
diminishes indefinitely with the energy of the original disturbance. No
such simple proof is available to show without qualification that the
above condition is _necessary_. If, however, we recognize the existence
of dissipative forces called into play by any motion whatever of the
system, the conclusion can be drawn as follows. However slight these
forces may be, the total energy T + V must continually diminish so long
as the velocities [.q]1, [.q]2, ... [.q]_n differ from zero. Hence if
the system be started from rest in a configuration for which V is less
than in the equilibrium configuration considered, this quantity must
still further decrease (since T cannot be negative), and it is evident
that either the system will finally come to rest in some other
equilibrium configuration, or V will in the long run diminish
indefinitely. This argument is due to Lord Kelvin and P. G. Tait (1879).
In discussing the small oscillations of a system about a configuration
of stable equilibrium it is convenient so to choose the generalized
cc-ordinates q1, q2, ... q_n that they shall vanish in the configuration
in question. The potential energy is then given with sufficient
approximation by an expression of the form
2V = c11q1² + c22q2² + ... + 2c12q1q2 + ..., (2)
a constant term being irrelevant, and the terms of the first order being
absent since the equilibrium value of V is stationary. The coefficients
c_rr, c_rs are called _coefficients of stability_. We may further treat
the coefficients of inertia a_rr, a_rs of § 22 (1) as constants. The
Lagrangian equations of motion are then of the type
a_(1r)[:q]1 + a_(2r)[:q]2 + ... + a_(nr)[:q]_n + c_(1r)q1 + c_(2r)q2 + ... + c_(nr)q_n = Q_r, (3)
where Q_r now stands for a component of extraneous force. In a _free
oscillation_ we have Q1, Q2, ... Q_n = 0, and if we assume
q_r = A_r e^(i[sigma]^t), (4)
we obtain n equations of the type
(c_(1r) - [sigma]²a_(1r)) A1 + (c_(2r) - [sigma]²a_(2r)) A2 + ... + (c_(nr) - [sigma]²a_nr) A_n = 0. (5)
Eliminating the n - 1 ratios A1 : A2 : ... : A_n we obtain the
determinantal equation
[Delta]([sigma]²) = 0, (6)
where
[Delta]([sigma]²) = | c11 - [sigma]²a11, c21 - [sigma]²a21, ..., C_(n1) - [sigma]²a_(nl) |
| c12 - [sigma]²a12, c22 - [sigma]²a22, ..., C_(n2) - [sigma]²a_(n2) |
| . . ... . |
| . . ... . | (7)
| . . ... . |
| c_(1n) - [sigma]²a{1n}, c_(2n) - [sigma]²a_(2n), ..., C_(nn) - [sigma]²a_(nn) |
The quadratic expression for T is essentially positive, and the same
holds with regard to V in virtue of the assumed stability. It may be
shown algebraically that under these conditions the n roots of the above
equation in [sigma]² are all real and positive. For any particular root,
the equations (5) determine the ratios of the quantities A1, A2, ...
A_n, the absolute values being alone arbitrary; these quantities are in
fact proportional to the minors of any one row in the determinate
[Delta]([sigma]²). By combining the solutions corresponding to a pair of
equal and opposite values of [sigma] we obtain a solution in real form:
q_r = C_(a_r) cos ([sigma]t + [epsilon]), (8)
where a1, a2 ... a_r are a determinate series of quantities having to
one another the above-mentioned ratios, whilst the constants C,
[epsilon] are arbitrary. This solution, taken by itself, represents a
motion in which each particle of the system (since its displacements
parallel to Cartesian co-ordinate axes are linear functions of the q's)
executes a simple vibration of period 2[pi]/[sigma]. The amplitudes of
oscillation of the various particles have definite ratios to one
another, and the phases are in agreement, the absolute amplitude
(depending on C) and the phase-constant ([epsilon]) being alone
arbitrary. A vibration of this character is called a _normal mode_ of
vibration of the system; the number n of such modes is equal to that of
the degrees of freedom possessed by the system. These statements require
some modification when two or more of the roots of the equation (6) are
equal. In the case of a multiple root the minors of [Delta]([sigma]²)
all vanish, and the basis for the determination of the quantities a_r
disappears. Two or more normal modes then become to some extent
indeterminate, and elliptic vibrations of the individual particles are
possible. An example is furnished by the spherical pendulum (§ 13).
[Illustration: FIG. 85.]
As an example of the method of determination of the normal modes we
may take the "double pendulum." A mass M hangs from a fixed point by a
string of length a, and a second mass m hangs from M by a string of
length b. For simplicity we will suppose that the motion is confined
to one vertical plane. If [theta], [phi] be the inclinations of the
two strings to the vertical, we have, approximately,
2T = Ma²[.theta]² + m(a[.theta] + b[.psi])² \ (9)
2V = Mga[theta]² + mg(a[theta]² + b[psi]²). /
The equations (3) take the forms
a[:theta] + [mu]b[:phi] + g[theta] = 0, \ (10)
a[:theta] + b[:phi] + g[phi] = 0. /
where [mu] = m/(M + m). Hence
([sigma]² - g/a)a[theta] + [mu][sigma]²b[phi] = 0, \ (11)
[sigma]²a[theta] + ([sigma]² - g/b)b[phi] = 0. /
The frequency equation is therefore
([sigma]² - g/a)([sigma]² - g/b) - [mu][sigma]^4 = 0. (12)
The roots of this quadratic in [sigma]² are easily seen to be real and
positive. If M be large compared with m, [mu] is small, and the roots
are g/a and g/b, approximately. In the normal mode corresponding to
the former root, M swings almost like the bob of a simple pendulum of
length a, being comparatively uninfluenced by the presence of m,
whilst m executes a "forced" vibration (§ 12) of the corresponding
period. In the second mode, M is nearly at rest [as appears from the
second of equations (11)], whilst m swings almost like the bob of a
simple pendulum of length b. Whatever the ratio M/m, the two values of
[sigma]² can never be exactly equal, but they are approximately equal
if a, b are nearly equal and [mu] is very small. A curious phenomenon
is then to be observed; the motion of each particle, being made up (in
general) of two superposed simple vibrations of nearly equal period,
is seen to fluctuate greatly in extent, and if the amplitudes be equal
we have periods of approximate rest, as in the case of "beats" in
acoustics. The vibration then appears to be transferred alternately
from m to M at regular intervals. If, on the other hand, M is small
compared with m, [mu] is nearly equal to unity, and the roots of (12)
are [sigma]² = g/(a + b) and [sigma]² = mg/M·(a + b)/ab,
approximately. The former root makes [theta] = [phi], nearly; in the
corresponding normal mode m oscillates like the bob of a simple
pendulum of length a + b. In the second mode a[theta] + b[phi] = 0,
nearly, so that m is approximately at rest. The oscillation of M then
resembles that of a particle at a distance a from one end of a string
of length a + b fixed at the ends and subject to a tension mg.
The motion of the system consequent on arbitrary initial conditions may
be obtained by superposition of the n normal modes with suitable
amplitudes and phases. We have then
q_r = [alpha]_r[theta] + [alpha]_r´[theta]´ + [alpha]_r´´[theta]´´ + ..., (13)
where
[theta] = C cos ([sigma]t + [epsilon]), [theta]´
= C´ cos ([sigma]´t + [epsilon]), [theta]´´
= C´´ cos([sigma]´´t + [epsilon]), ... (14)
provided [sigma]², [sigma]´², [sigma]´´², ... are the n roots of (6).
The coefficients of [theta], [theta]´, [theta]´´, ... in (13) satisfy
the _conjugate_ or _orthogonal_ relations
a11[alpha]1[alpha]1´ + a22[alpha]2[alpha]2´ + ... + a12([alpha]1[alpha]2´ + [alpha]2[alpha]1´) + ... = 0, (15)
c11[alpha]1[alpha]1´ + c22[alpha]2[alpha]2´ + ... + c12([alpha]1[alpha]2´ + [alpha]2[alpha]1´) + ... = 0, (16)
provided the symbols [alpha]_r, [alpha]_r´ correspond to two distinct
roots [sigma]², [sigma]´² of (6). To prove these relations, we replace
the symbols A1, A2, ... A_n in (5) by [alpha]1, [alpha]2, ... [alpha]_n
respectively, multiply the resulting equations by a´1, a´2, ... a´_n, in
order, and add. The result, owing to its symmetry, must still hold if we
interchange accented and unaccented Greek letters, and by comparison we
deduce (15) and (16), provided [sigma]² and [sigma]´² are unequal. The
actual determination of C, C´, C´´, ... and [epsilon], [epsilon]´,
[epsilon]´´, ... in terms of the initial conditions is as follows. If we
write
C cos [epsilon] = H, -C sin [epsilon] = K, (17)
we must have
[alpha]_rH + [alpha]_r´H´ + [alpha]_r´´H´´ + ... = [q_r]0, \ (18)
[sigma][alpha]_rH + [sigma]´[alpha]_r´H´ + [sigma]´´[alpha]_r´´H´´ + ... = [[.q]_r]0, /
where the zero suffix indicates initial values. These equations can be
at once solved for H, H´, H´´, ... and K, K´, K´´, ... by means of the
orthogonal relations (15).
By a suitable choice of the generalized co-ordinates it is possible to
reduce T and V simultaneously to sums of squares. The transformation is
in fact effected by the assumption (13), in virtue of the relations (15)
(16), and we may write
2T = a[.theta]² + a´[.theta]´² + a´´[.theta]´´² + ..., \ (19)
2V = c[theta]² + c´[theta]´² + c´´[theta]´´² + .... /
The new co-ordinates [theta], [theta]´, [theta]´´ ... are called the
_normal_ co-ordinates of the system; in a normal mode of vibration one
of these varies alone. The physical characteristics of a normal mode are
that an impulse of a particular normal type generates an initial
velocity of that type only, and that a constant extraneous force of a
particular normal type maintains a displacement of that type only. The
normal modes are further distinguished by an important "stationary"
property, as regards the frequency. If we imagine the system reduced by
frictionless constraints to one degree of freedom, so that the
co-ordinates [theta], [theta]´, [theta]´´, ... have prescribed ratios to
one another, we have, from (19),
c[theta]² + c´[theta]´² = c´´[theta]´´² + ...
[sigma]² = ---------------------------------------------, (20)
a[theta]² + a´[theta]´² + a´´[theta]´´² + ...
This shows that the value of [sigma]² for the constrained mode is
intermediate to the greatest and least of the values c/a, c´/a´,
c´´/a´´, ... proper to the several normal modes. Also that if the
constrained mode differs little from a normal mode of free vibration
(e.g. if [theta]´, [theta]´´, ... are small compared with [theta]), the
change in the frequency is of the second order. This property can often
be utilized to estimate the frequency of the gravest normal mode of a
system, by means of an assumed approximate type, when the exact
determination would be difficult. It also appears that an estimate thus
obtained is necessarily too high.
From another point of view it is easily recognized that the equations
(5) are exactly those to which we are led in the ordinary process of
finding the stationary values of the function
V (q1, q2, ... q_n)
------------------------,
T (q1, q2, ... q_n)
where the denominator stands for the same homogeneous quadratic function
of the q's that T is for the [.q]'s. It is easy to construct in this
connexion a proof that the n values of [sigma]² are all real and
positive.
The case of three degrees of freedom is instructive on account of the
geometrical analogies. With a view to these we may write
2T= a[.x]² + b[.y]² + c[.z]² + 2f[.y][.z] + 2g[.z][.x] + 2h[.x][.y], \ (21)
2V = Ax² + By² + Cz² + 2Fyz + 2Gzx + 2Hxy. /
It is obvious that the ratio
V (x, y, z)
----------- (22)
T (x, y, z)
must have a least value, which is moreover positive, since the
numerator and denominator are both essentially positive. Denoting this
value by [sigma]1², we have
Ax1 + Hy1 + Gz1 = [sigma]1²(ax1 + hy1 + [dP]gz1), \
Hx1 + By1 + Fz1 = [sigma]1²(hx1 + by1 + fz1), > (23)
Gx1 + Fy1 + Cz1 = [sigma]1²(gx1 + fy1 + cz1), /
provided x1 : y1 : z1 be the corresponding values of the ratios x:y:z.
Again, the expression (22) will also have a least value when the
ratios x : y : z are subject to the condition
[dP]V [dP]V [dP]V
x1 ----- + y1 ----- + z1 ----- = 0; (24)
[dP]x [dP]y [dP]z
and if this be denoted by [sigma]2² we have a second system of
equations similar to (23). The remaining value [sigma]2² is the value
of (22) when x : y : z arc chosen so as to satisfy (24) and
[dP]V [dP]V [dP]V
x2 ----- + y2 ----- + z2 ----- = 0 (25)
[dP]x [dP]y [dP]z
The problem is identical with that of finding the common conjugate
diameters of the ellipsoids T(x, y, z) = const., V(x, y, z) = const.
If in (21) we imagine that x, y, z denote infinitesimal rotations of a
solid free to turn about a fixed point in a given field of force, it
appears that the three normal modes consist each of a rotation about
one of the three diameters aforesaid, and that the values of [sigma]
are proportional to the ratios of the lengths of corresponding
diameters of the two quadrics.
We proceed to the _forced vibrations_ of the system. The typical case is
where the extraneous forces are of the simple-harmonic type cos
([sigma]t + [epsilon]); the most general law of variation with time can
be derived from this by superposition, in virtue of Fourier's theorem.
Analytically, it is convenient to put Q_r, equal to e^(i[sigma]^t)
multiplied by a complex coefficient; owing to the linearity of the
equations the factor e^(i[sigma]^t) will run through them all, and need
not always be exhibited. For a system of one degree of freedom we have
a[:q] + cq = Q, (26)
and therefore on the present supposition as to the nature of Q
Q
q = -------------. (27)
c - [sigma]²a
This solution has been discussed to some extent in § 12, in connexion
with the forced oscillations of a pendulum. We may note further that
when [sigma] is small the displacement q has the "equilibrium value"
Q/c, the same as would be produced by a steady force equal to the
instantaneous value of the actual force, the inertia of the system being
inoperative. On the other hand, when [sigma]² is great q tends to the
value -Q/[sigma]²a, the same as if the potential energy were ignored.
When there are n degrees of freedom we have from (3)
(c_(1r) - [sigma]² a_(2r)) q1 + (c²_r - [sigma]² a_(2r)) q2 + ... + (c_(nr) - [sigma]² a_(nr)) q_n = Qr, (28)
and therefore
[Delta]([sigma]²)·q_r = a_(1r)Q1 + a_(2r)Q2 + ... + a_(nr)Q_n, (29)
where a_(1r), a_(2r), ... a_(nr) are the minors of the rth row of the
determinant (7). Every particle of the system executes in general a
simple vibration of the imposed period 2[pi]/[sigma], and all the
particles pass simultaneously through their equilibrium positions. The
amplitude becomes very great when [sigma]² approximates to a root of
(6), i.e. when the imposed period nearly coincides with one of the free
periods. Since a_(rs) = a_(sr), the coefficient of Q_s in the expression
for q_r is identical with that of Q_r in the expression for q_s. Various
important "reciprocal theorems" formulated by H. Helmholtz and Lord
Rayleigh are founded on this relation. Free vibrations must of course be
superposed on the forced vibrations given by (29) in order to obtain the
complete solution of the dynamical equations.
In practice the vibrations of a system are more or less affected by
dissipative forces. In order to obtain at all events a qualitative
representation of these it is usual to introduce into the equations
frictional terms proportional to the velocities. Thus in the case of one
degree of freedom we have, in place of (26),
a[:q] + b[.q] + cq = Q, (30)
where a, b, c are positive. The solution of this has been sufficiently
discussed in § 12. In the case of multiple freedom, the equations of
small motion when modified by the introduction of terms proportional to
the velocities are of the type
d [dP]T [dP]V
--- ---------- + B_(1r)[.q]1 + B_(2r)[.q]2 + ... + B_(nr)[.q]_n + ------- = Q_r (31)
dt [dP][.q]_r [dP]q_r
If we put
b_(rs) = b_(sr) = ½[B_(rs) + B_(sr)], [beta]_(rs) = -[beta]_(sr) = ½[B_(rs) - B_(sr)], (32)
this may be written
d [dP]T [dP]F [dP]V
--- --------- + ---------- + [beta]_(1r)[.q]1 + [beta]_(2r)[.q]2 + ... + [beta]_(nr)[.q]_r + ------- (33)
dt [dP][.q]_r [dP][.q]_r [dP]q_r
provided
2F = b11[.q]1² + b22[.q]2² + ... + 2b12[.q]1[.q]2 + ... (34)
The terms due to F in (33) are such as would arise from frictional
resistances proportional to the absolute velocities of the particles, or
to mutual forces of resistance proportional to the relative velocities;
they are therefore classed as _frictional_ or _dissipative_ forces. The
terms affected with the coefficients [beta]_(rs) on the other hand are
such as occur in "cyclic" systems with latent motion (DYNAMICS, §
_Analytical_); they are called the _gyrostatic terms_. If we multiply
(33) by [.q]_r and sum with respect to r from 1 to n, we obtain, in
virtue of the relations [beta]_(rs) = -[beta]_(sr), [beta]_(rr) = 0,
d
---(T + V) = 2F + Q1[.q]1 + Q2[.q]2 + ... + Q_n[.q]_n. (35)
dt
This shows that mechanical energy is lost at the rate 2F per unit time.
The function F is therefore called by Lord Rayleigh the _dissipation
function_.
If we omit the gyrostatic terms, and write q_r = C_re^([lambda]t), we
find, for a free vibration,
[a_(1r)[lambda]² + b_(1r)[lambda] + c_(1r)] C1 + [a_(2r)[lambda]² + b_(2r)[lambda] + c_(2r)] C2 + ...
+ [a_(nr)[lambda]² + b_(nr)[lambda] + c_(nr)] C_n = 0. (36)
This leads to a determinantal equation in [lambda] whose 2n roots are
either real and negative, or complex with negative real parts, on the
present hypothesis that the functions T, V, F are all essentially
positive. If we combine the solutions corresponding to a pair of
conjugate complex roots, we obtain, in real form,
q_r = C[alpha]_re^(-t/[tau]) cos ([sigma]t + [epsilon] - [epsilon]_r), (37)
where [sigma], [tau], [alpha]_r, [epsilon]_r are determined by the
constitution of the system, whilst C, [epsilon] are arbitrary, and
independent of r. The n formulae of this type represent a normal mode of
free vibration: the individual particles revolve as a rule in elliptic
orbits which gradually contract according to the law indicated by the
exponential factor. If the friction be relatively small, all the normal
modes are of this character, and unless two or more values of [sigma]
are nearly equal the elliptic orbits are very elongated. The effect of
friction on the period is moreover of the second order.
In a forced vibration of e^(i[sigma]t) the variation of each co-ordinate
is simple-harmonic, with the prescribed period, but there is a
retardation of phase as compared with the force. If the friction be
small the amplitude becomes relatively very great if the imposed period
approximate to a free period. The validity of the "reciprocal theorems"
of Helmholtz and Lord Rayleigh, already referred to, is not affected by
frictional forces of the kind here considered.
The most important applications of the theory of vibrations are to the
case of continuous systems such as strings, bars, membranes, plates,
columns of air, where the number of degrees of freedom is infinite.
The series of equations of the type (3) is then replaced by a single
linear partial differential equation, or by a set of two or three such
equations, according to the number of dependent variables. These
variables represent the whole assemblage of generalized co-ordinates
q_r; they are continuous functions of the independent variables x, y,
z whose range of variation corresponds to that of the index r, and of
t. For example, in a one-dimensional system such as a string or a bar,
we have one dependent variable, and two independent variables x and t.
To determine the free oscillations we assume a time factor
e^(i[sigma]t); the equations then become linear differential equations
between the dependent variables of the problem and the independent
variables x, or x, y, or x, y, z as the case may be. If the range of
the independent variable or variables is unlimited, the value of
[sigma] is at our disposal, and the solution gives us the laws of
wave-propagation (see WAVE). If, on the other hand, the body is
finite, certain terminal conditions have to be satisfied. These limit
the admissible values of [sigma], which are in general determined by
a transcendental equation corresponding to the determinantal equation
(6).
Numerous examples of this procedure, and of the corresponding
treatment of forced oscillations, present themselves in theoretical
acoustics. It must suffice here to consider the small oscillations of
a chain hanging vertically from a fixed extremity. If x be measured
upwards from the lower end, the horizontal component of the tension P
at any point will be P[delta]y/[delta]x, approximately, if y denote
the lateral displacement. Hence, forming the equation of motion of a
mass-element, [rho][delta]x, we have
[rho][delta]x·[:y] = [delta]P·([dP]y/[dP]x). (38)
Neglecting the vertical acceleration we have P = g[rho]x, whence
[dP]²y [dP] / [dP]y \
------ = g ----- ( x ----- ). (39)
[dP]t² [dP]x \ [dP]x /
Assuming that y varies as e^(i[sigma]t) we have
[dP] / [dP]y \
----- ( x ----- ) + ky = 0 (40)
[dP]x \ [dP]x /
provided k = [sigma]²/g. The solution of (40) which is finite for x =
0 is readily obtained in the form of a series, thus
/ kx k²x² \
y = C ( 1 - -- + ---- - ... ) = CJ0(z), (41)
\ 1² 1²2² /
in the notation of Bessel's functions, if z² = 4kx. Since y must
vanish at the upper end (x = l), the admissible values of [sigma] are
determined by
[sigma]² = gz²/4l, J0(z) = 0. (42)
The function J0(z) has been tabulated; its lower roots are given by
z/[pi]= .7655, 1.7571, 2.7546,...,
approximately, where the numbers tend to the form s - ¼. The frequency
of the gravest mode is to that of a uniform bar in the ratio .9815
That this ratio should be less than unity agrees with the theory of
"constrained types" already given. In the higher normal modes there
are nodes or points of rest (y = 0); thus in the second mode there is
a node at a distance .190l from the lower end.
AUTHORITIES.--For indications as to the earlier history of the subject
see W. W. R. Ball, _Short Account of the History of Mathematics_; M.
Cantor, _Geschichte der Mathematik_ (Leipzig, 1880 ... ); J. Cox,
_Mechanics_ (Cambridge, 1904); E. Mach, _Die Mechanik in ihrer
Entwickelung_ (4th ed., Leipzig, 1901; Eng. trans.). Of the classical
treatises which have had a notable influence on the development of the
subject, and which may still be consulted with advantage, we may note
particularly, Sir I. Newton, _Philosophiae naturalis Principia
Mathematica_ (1st ed., London, 1687); J. L. Lagrange, _Mécanique
analytique_ (2nd ed., Paris, 1811-1815); P. S. Laplace, _Mécanique
céleste_ (Paris, 1799-1825); A. F. Möbius, _Lehrbuch der Statik_
(Leipzig, 1837), and _Mechanik des Himmels_; L. Poinsot, _Éléments de
statique_ (Paris, 1804), and _Théorie nouvelle de la rotation des
corps_ (Paris, 1834).
Of the more recent general treatises we may mention Sir W. Thomson
(Lord Kelvin) and P. G. Tait, _Natural Philosophy_ (2nd ed.,
Cambridge, 1879-1883); E. J. Routh, _Analytical Statics_ (2nd ed.,
Cambridge, 1896), _Dynamics of a Particle_ (Cambridge, 1898), _Rigid
Dynamics_ (6th ed., Cambridge 1905); G. Minchin, _Statics_ (4th ed.,
Oxford, 1888); A. E. H. Love, _Theoretical Mechanics_ (2nd ed.,
Cambridge, 1909); A. G. Webster, _Dynamics of Particles_, &c. (1904);
E. T. Whittaker, _Analytical Dynamics_ (Cambridge, 1904); L. Arnal,
_Traitê de mécanique_ (1888-1898); P. Appell, _Mécanique rationelle_
(Paris, vols. i. and ii., 2nd ed., 1902 and 1904; vol. iii., 1st ed.,
1896); G. Kirchhoff, _Vorlesungen über Mechanik_ (Leipzig, 1896); H.
Helmholtz, _Vorlesungen über theoretische Physik_, vol. i. (Leipzig,
1898); J. Somoff, _Theoretische Mechanik_ (Leipzig, 1878-1879).
The literature of graphical statics and its technical applications is
very extensive. We may mention K. Culmann, _Graphische Statik_ (2nd
ed., Zürich, 1895); A. Föppl, _Technische Mechanik_, vol. ii.
(Leipzig, 1900); L. Henneberg, _Statik des starren Systems_
(Darmstadt, 1886); M. Lévy, _La statique graphique_ (2nd ed., Paris,
1886-1888); H. Müller-Breslau, _Graphische Statik_ (3rd ed., Berlin,
1901). Sir R. S. Ball's highly original investigations in kinematics
and dynamics were published in collected form under the title _Theory
of Screws_ (Cambridge, 1900).
Detailed accounts of the developments of the various branches of the
subject from the beginning of the 19th century to the present time,
with full bibliographical references, are given in the fourth volume
(edited by Professor F. Klein) of the _Encyclopädie der mathematischen
Wissenschaften_ (Leipzig). There is a French translation of this work.
(See also DYNAMICS.) (H. Lb.)
II.--APPLIED MECHANICS[1]
§ 1. The practical application of mechanics may be divided into two
classes, according as the assemblages of material objects to which they
relate are intended to remain fixed or to move relatively to each
other--the former class being comprehended under the term "Theory of
Structures" and the latter under the term "Theory of Machines."