Chapter XIX: Part II: Kinetics (4)

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."