S22 = 1 cos a, (25)
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 o=a+x in (24) and neglect terms of the second order in x. We find, after some reductions,
z+(1+3 cos' a) n2x=o; (z6)
this shows that the variation of x is simpleharmonic, with the period
2rN (I +3 cos2 a) . S2
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 labt21P, where a, b are the semiaxes.
To apply the equations (it) 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 coordinates employed should be competent to specify the position of every particle. If X, µ, v be the components of momentum, we have
a=g• =AA,
= • = A sin2o1+C (¢+cos o,G) cos o, (27)
v=ate =C (q+cos elk).
The meaning of these quantities is easily recognized; thus a is the angular momentum about a horizontal axis normal to the plane of o, is the angular momentum about the vertical OZ, and v isthe angular momentum about the axis of symmetry. • If M be the total mass, the potential energy is V=Mgh cos 0, if OZ be drawn vertically upwards. Hence the equations (II) become
A6—A sing coso+G2+C(c6+coso'G)+G sin 0=Mgh sin 0,
d/dt. (A sin2 o;t', +C ($+cos o%1%) cos of =o, (28) d/dt. (C ( +cos B,G) } = o,
of which the last two express the constancy of the momenta µ, v. Hence
A8 —A sin 0 cos 00+v sin B¢ = Mgh sin o,
A sin C '+v cos B=µ. (29)
S
If we eliminate 4, • 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 0 be small the equations (29) may be written
6M —owe —vz—4 1 ghe,
02 w = Const.,,
( J
At. "=¢—zAAt.
av/aq,.=o. (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 coordinates. 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 coordinate 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 4142, ... qn 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 coordinates q1, q2, . . .q,, 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 = c11g12+c2g22.+ ... +2c12gig2+ •i (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„ are called coefficients of stability. We may further treat the coefficients of inertia a,,, ar, of § 22 (r) as constants. The Lagrangian equations of motion are then of the type
alr$1+a,els+ . . +a„rgn+cirgl+C2rg2+ • • • +cnrgR=Qr, (3) where Q, now stands for a component of extraneous force. In a free oscillation we have (21, Q2, . . . Q,,=o, and if we assume
qr = Are' a0, (4)
we obtain n equations of the type
(Ctr—a'air) Al+(C2r—e a2,.) A2+ . . +(cnr—a2a„r) An=o. (5)
(23)
where
(30) (31)
(22) § 14 (15) (16) that the motion of such a point will be ellipticharSince 8, e, are the polar coordinates (in a horizontal plane) of a point on the axis of symmetry, relative to an initial line which revolves
with constant angular velocity v/2A, we see by comparison with monic superposed on a uniform rotation v/2A, provided v2>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,) be one of equilibrium, the equations (14) of § 22 must be satisfied by
qi, q2, . . . 4,t= o, whence
Eliminating 'the n — 1 ratios A 1:A 2: : An we obtain the determinantal equation
(012) =0,
where
cll —a'au, C21 —a2a2l, , cnl —?2an1 C12 — Q2a12, c22 020122, .. , cn2  03(1,4
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).
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 0, 4, be the inclinations of the two strings to the vertical, we have, approximately,
2T=Ma292+m(a9+b )2, 2V =MgaO2+mg(a02+b¢2).
The equations (3) take the forms
0 +µb;+g8 =0, (lo) ab+b.+g4,=o.
where g =m/(M+m). Hence))
(a"a0+(a22_0)b~ _=o. (1I) The frequency equation is therefore
(a2 g/a) (a2 — gib) — µo4 = o. (12)
The roots of this quadratic in a2 are easily seen to be'real and positive. If M be large compared with m, u is small, and the roots are g/a and gib, 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 (02) of the corresponding period. In the second mode, M is nearly at rest [as appears from the second of equations (Ii)], whilst m swings almost like the bob of a simple pendulum of length b. Whatever the ratio M/m, the two values of o2 can never be exactly equal, but they are approximately equal if a, b are nearly equal and p 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, p is nearly equal to unity, and the roots of (12) are a2=g/(a+b) and a2=mg/M.(a+b)/ab, approximately.
The former root makes 0=¢, nearly; in the corresponding normal mode m oscillates like the bob of a simple pendulum of length a+b. In the second mode aB+bq,=o, 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
Qr = a,B+a; 9'+a,"B"+ (13)
where
0=C cos(ot+E), 6'=C' cos(e'1+E), 0"=C" cos(a"t+E), . . . (14)
provided a2, on, a"2,. . . are the n roots of (6). The coefficients of 0, 8', 0", . . . in (13) satisfy the conjugate or orthogonal
relations
allalal'+a22a2a2'+ . . . +a12(a1a2'+a2a1')+ . . . =0, (15)
cll al+C22a2a8 +. . +c12(al a2' 1 aia1')+ . . . =0, (16) provided the symbols a,., a,' correspond to two distinct roots a2, on of (6). To prove these relations, we replace the symbols Al, A2, ....A,, in (5) by al, as, . . . an respectively, multiply the resulting equations by al', a'2, . . . a',,, 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 a2 and 01'2 are unequal. The actual determination of C, C', C", . . . and e, e', a", . . . in terms of the initial conditions is as follows. If we write
C cos e=H,—C sin e=K, (17) we must have
a,H+a,'H'+a,"H"+ . . =[Q,lo,
aa,H+01'01, H'+a"as„H'+ . . . [Q,lo, (18) 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 coordinates 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 =¢92+¢'9'Z+a"Bn2+ . . . , (9)
2V=ce2+c'9'2+c"B„2+. . ..f 1 The new coordinates 0, 0', 0" . . . are called the normal coordinates 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 coordinates 0, 0', 0", . . . have prescribed ratios to one another, we have, from (19),
2 = c92+c'8'2=c"B"2} . (20)
a 0101+a'8'2+a"B"2+ . .
This shows that the value of a2 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 0', 0", . . . are small compared with 0),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 (ql, q2,
T (gl,,g2, ... q,,)
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 et values of az are all real and positive.
(6) (7)
Cln—aea's, c2,,—0262,,, . . , cnn—a2ann
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 a2 are all real and positive. For any particular root, the equations (5) determine the ratios of the quantities Al, A2, . . . An, the absolute values being alone arbitrary; these quantities are in fact proportional to the minors of any one row in the determinate 0(012). By combining the solutions corresponding to a pair of equal and opposite values of a we obtain a solution in real form:
q,.= Car cos (011+E), (8) where a,, a2 . . . a, are a determinate series of quantites having to one another the abovementioned ratios, whilst the constants C, e are arbitrary. This solution, taken by itself, represents a
o motion in which each particle of the system (since its displacements parallel to Cartesian coordinate axes are linear functions of the q's) executes a simple vibration of period 21r/a. 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 phaseconstant (e) being alone arbitrary. A vibration of this character is called a normal mode of vibration of the system; the number et 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
m of the equation (6) are equal. In the case of a
multiple root the minors of A(a2) all vanish, and
i the basis for the determination of the quantities a,.
Flo. 85.
(9)
The case of three degrees of freedom is instructive on account of the geometrical analogies. With a view to these we may write
2T =ai2+by2+ci2}zfy2+2gz}2h±/;
2V = Axe +By2 +Cz2 +2Fyz +2Gzx+2 Hxy. 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 a12, we have
Ax1+H)'1+Gz1=o,2(axl+hYl+agzi),
HxI+By'+Fzi =a12(hxl+by' + fzi), (23) Gxl+Fyl+Czl =°12 (gxl+fyl + cz1),
provided xl : yl: zi 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
xlax +Ylay +z1aV =o; (24)
and if this be denoted by a22 we have a second system of equations similar to (23). The remaining value a32 is the value of (22) when x: y: z are chosen so as to satisfy (24) and
x2ax +Y2ay +z —=o. (25) az 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 a 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 simpleharmonic type cos (at+e); 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 Qr equal to ei°t multiplied by a complex coefficient; owing to the linearity of the equations the factor el°t will run through them all, and need not always be exhibited. For a system of one degree of freedom we have
a4 +cq = Q, (z6)
and therefore on the present supposition as to the nature of Q
q Q)
c —a'a
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 a 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 a2 is great q tends to the value Q/u2a, the same as if the potential energy were ignored. When there are n degrees of freedom we have from (3)
(Clr— ama2r)gl + (C2r— a2a2r)g2 + . . . +(ens— eanr)gn = Q,., (28)
and therefore
A (a2) . Qr = al,Q1 + a2rQ2 + ... + anrQn, (29) where air, a2r, . . . am 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 27r/a, and all the particles pass simultaneously through their equilibrium positions. The amplitude becomes very great when a2 approximates to a root of (6), i.e. when the imposed period nearly coincides with one of the free periods. Since ar,=a,r, the coefficient of Q, in the expression for qr is identical with that of Q,. in the expression for q,. 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),
aq+b4+cq =Q, (30)
XV'II. 32where 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 aT i~ aV
dt aqr+BlrQ1+U2rQ2+ +BnrQn+agr=Qr. (31)
If we put
br,=bsr=(Br.+Bar), ars= —OBr=(Br.—Bsr), this may be written
d aT aF RR aV
dt aQr+aQr+N1r4i+152qa42+ • . . +19nr4s+agr=Qr, (33)
provided
2F=611412+622422+ . . + 26124142+ • • • (34) The terms due to Fin (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 Ors 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 yr and sum with respect to r from I to n, we obtain, in virtue of the relations Ors= —Osr, Orr=o,
at (T + V) = 2F + Q141 + Q242 + ... + Qn4n. (35) This shows that mechanical energy is lost at the rate 2 F per unit time. The function F is therefore called by Lord Rayleigh the dissipation function.
If we omit the gyrostatic terms, and write q,. =Crept, we find, for a free vibration,
(air12 + b1rX + Cl,) C1 + (a2,.X2 + b2rx + C2r) C2 +
+ (anrA' + bad. + C,,,.) Cn = o. (36) This leads to a determinantal equation in X whose 211 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,
qr=Care— t/r cos (at+e—er), (37)
where a, r, ar, er are determined by the constitution of the system, whilst C, e 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 a 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°t the variation of each coordinate is simpleharmonic, 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 coordinates qr; 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 onedimensional 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 etat; 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 a is at our disposal, and the solution gives us the laws of wavepropagation (see WAVE). If, on the other hand, the body is finite, certain terminal conditions have to be satisfied. These limit the admissible values of a, which are in general determined
II
(2I)
(27)
(32)
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 Pay/ax, approximately, if y denote the lateral displacement. Hence, forming the equation of motion of a masselement, pox, we have
pSx.p= S(P. ay/ax). (38)
Neglecting the vertical acceleration we have P=gpx, whence 82y a ay
ale =g ax ax) Assuming that y varies as e'ad we have
ax (x a) +ky=0. (40)
Provided k=oz/g. The solution of (40) which is finite for x=o is readily obtained in the form of a series, thus kx k2x2
y=C (1 f 1 2...) =CJo(z), (41)
in the notation of Bessel's functions, if z2=4kx. Since y must vanish at the upper end (x .=l), the admissible values of a are determined by
a2 =gz2/4l, Jo(z) =o. (42)
The function Jo (z) has been tabulated; its lower roots are given by
z/7'= •7655, 1.7571, 2'7546, ...,
approximately, where the numbers tend to the form s—i. 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=o); thus in the second mode there is a node at a distance .1901 from the lower end.
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, Traite de mecanique (1888—1898); P. Appell, Mecanique rationelle (Paris, vols. i. and ii., 2nd ed., 1902 and 1904; vol. iii., 1st ed., 1896); G. Kirchhoff, Vorlesungen fiber Mechanik (Leipzig, 1896) ; H. Helmholtz, Vorlesungen caber 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., Ziirich, 1895); A. Foppl, Technische Mechanik, vol. ii. (Leipzig, 1900) ; L. Henneberg, Statik des starren Systems (Darmstadt, 1886); M. Levy, La statique graphique (2nd ed., Paris, 886—1888); II. MiillerBreslau, 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
r44,resent time, with full bibliographical references, are given in the f.)urth volume (edited by Professor F. Klein) of the Encyclopddie der nw.thematischen Wissenschaften (Leipzig). There is a French transla
tion of this work. (See also DYNAMICS.) (H. LE.)
II.—APPLIED MECHANICS
§ I. The practical application of mechanics may be divided into two classes, according as the assemblages of material
1 In view of the great authority of the author, the late Professor Macquorn Rankine, it has been thought desirable to retain the greater part of this article as it appeared in the 9th edition of the Encyclo
paedia Britannica. Considerable additions, however, have been introduced in order to indicate subsequent developments of the subject; the new sections are numbered continuously with the old,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."
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