(14)
provided σ2, σ′2, σ″2, ... are the n roots of (6). The coefficients of θ, θ′, θ″, ... in (13) satisfy theconjugateororthogonalrelations
a11α1α1′ + a22α2α2′ + ... + a12(α1α2′ + α2α1′) + ... = 0,
(15)
c11α1α1′ + c22α2α2′ + ... + c12(α1α2′ + α2α1′) + ... = 0,
(16)
provided the symbols αr, αr′ correspond to two distinct roots σ2, σ′2of (6). To prove these relations, we replace the symbols A1, A2, ... Anin (5) by α1, α2, ... αnrespectively, 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 σ2and σ′2are unequal. The actual determination of C, C′, C″, ... and ε, ε′, ε″, ... in terms of the initial conditions is as follows. If we write
C cos ε = H, −C sin ε = K,
(17)
we must have
(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 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 = aθ̇2+ a′θ̇′2+ a″θ̇″2+ ...,2V = cθ2+ c′θ′2+ c″θ″2+ ....
(19)
The new co-ordinates θ, θ′, θ″ ... are called thenormalco-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 θ, θ′, θ″, ... have prescribed ratios to one another, we have, from (19),
(20)
This shows that the value of σ2for 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 θ′, θ″, ... are small compared with θ), 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
where the denominator stands for the same homogeneous quadratic function of the q’s that T is for the q̇’s. It is easy to construct in this connexion a proof that the n values of σ2are 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 write2T = aẋ2+ bẏ2+ cż2+ 2fẏż + 2gżẋ + 2hẋẏ,2V = Ax2+ By2+ Cz2+ 2Fyz + 2Gzx + 2Hxy.(21)It is obvious that the ratioV (x, y, z)T (x, y, z)(22)must have a least value, which is moreover positive, since the numerator and denominator are both essentially positive. Denoting this value by σ12, we haveAx1+ Hy1+ Gz1= σ12(ax1+ hy1+ ∂gz1),Hx1+ By1+ Fz1= σ12(hx1+ by1+ fz1),Gx1+ Fy1+ Cz1= σ12(gx1+ fy1+ cz1),(23)provided x1: y1: z1be 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 conditionx1∂V+ y1∂V+ z1∂V= 0;∂x∂y∂z(24)and if this be denoted by σ22we have a second system of equations similar to (23). The remaining value σ22is the value of (22) when x : y : z arc chosen so as to satisfy (24) andx2∂V+ y2∂V+ z2∂V= 0;∂x∂y∂z(25)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 σ are proportional to the ratios of the lengths of corresponding diameters of the two quadrics.
The case of three degrees of freedom is instructive on account of the geometrical analogies. With a view to these we may write
(21)
It is obvious that the ratio
(22)
must have a least value, which is moreover positive, since the numerator and denominator are both essentially positive. Denoting this value by σ12, we have
(23)
provided x1: y1: z1be 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
(24)
and if this be denoted by σ22we have a second system of equations similar to (23). The remaining value σ22is the value of (22) when x : y : z arc chosen so as to satisfy (24) and
(25)
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 σ are proportional to the ratios of the lengths of corresponding diameters of the two quadrics.
We proceed to theforced vibrationsof the system. The typical case is where the extraneous forces are of the simple-harmonic type cos (σt + ε); 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 eiσtmultiplied by a complex coefficient; owing to the linearity of the equations the factor eiσtwill run through them all, and need not always be exhibited. For a system of one degree of freedom we have
aq̈ + cq = Q,
(26)
and therefore on the present supposition as to the nature of Q
(27)
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 σ 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 σ2is great q tends to the value −Q/σ2a, the same as if the potential energy were ignored. When there are n degrees of freedom we have from
(3)
(c1r− σ2a2r) q1+ (c22r− σ2a2r) q2+ ... + (cnr− σ2anr) qn= Qr,
(28)
and therefore
Δ(σ2) · qr= a1rQ1+ a2rQ2+ ... + anrQn,
(29)
where a1r, a2r, ... anrare 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 2π/σ, and all the particles pass simultaneously through their equilibrium positions. The amplitude becomes very great when σ2approximates to a root of (6),i.e.when the imposed period nearly coincides with one of the free periods. Since ars= asr, the coefficient of Qsin the expression for qris identical with that of Qrin the expression for qs. 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),
aq̈ + bq̇ + 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
(31)
If we put
brs= bsr=1⁄2(Brs+ Bsr), βrs= −βsr=1⁄2(Brs− Bsr),
(32)
this may be written
(33)
provided
2F = b11q̇12+ b22q̇22+ ... + 2b12q̇1q̇2+ ...
(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 asfrictionalordissipativeforces. The terms affected with the coefficients βrson the other hand are such as occur in “cyclic” systems with latent motion (Dynamics, §Analytical); they are called thegyrostatic terms. If we multiply (33) by q̇rand sum with respect to r from 1 to n, we obtain, in virtue of the relations βrs= −βsr, βrr= 0,
(35)
This shows that mechanical energy is lost at the rate 2F per unit time. The function F is therefore called by Lord Rayleigh thedissipation function.
If we omit the gyrostatic terms, and write qr= Creλt, we find, for a free vibration,
(a1rλ2+ b1rλ + c1r) C1+ (a2rλ2+ b2rλ + c2r) C2+ ...+ (anrλ2+ bnrλ + cnr) Cn= 0.
(36)
This leads to a determinantal equation in λ 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,
qr= Cαre−t/τcos (σt + ε − εr),
(37)
where σ, τ, αr, εrare determined by the constitution of the system, whilst C, ε 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 σ 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 eiσtthe 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 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 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 eiσt; 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 σ is at our disposal, and the solution gives us the laws of wave-propagation (seeWave). If, on the other hand, the body is finite, certain terminal conditions have to be satisfied. These limit the admissible values of σ, which are in general determinedby 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 Pδy/δx, approximately, if y denote the lateral displacement. Hence, forming the equation of motion of a mass-element, ρδx, we haveρ δx · ÿ = δ (P · ∂y/∂x).(38)Neglecting the vertical acceleration we have P = gρx, whence∂2y= g∂(x∂y).∂t2∂x∂x(39)Assuming that y varies as eiσtwe have∂(x∂y)+ ky = 0.∂x∂x(40)provided k = σ2/g. The solution of (40) which is finite for x = 0 is readily obtained in the form of a series, thusy = C(1 −kx+k2x2− ...)= CJ0(z),121222(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 σ are determined byσ2= gz2/4l, J0(z) = 0.(42)The function J0(z) has been tabulated; its lower roots are given byz/π= .7655, 1.7571, 2.7546,...,approximately, where the numbers tend to the form s −1⁄4. 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), andMechanik des Himmels; L. Poinsot,Éléments de statique(Paris, 1804), andThéorie 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 titleTheory 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 theEncyclopädie der mathematischen Wissenschaften(Leipzig). There is a French translation of this work. (See alsoDynamics.)
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 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 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 eiσt; 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 σ is at our disposal, and the solution gives us the laws of wave-propagation (seeWave). If, on the other hand, the body is finite, certain terminal conditions have to be satisfied. These limit the admissible values of σ, which are in general determinedby 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 Pδy/δx, approximately, if y denote the lateral displacement. Hence, forming the equation of motion of a mass-element, ρδx, we have
ρ δx · ÿ = δ (P · ∂y/∂x).
(38)
Neglecting the vertical acceleration we have P = gρx, whence
(39)
Assuming that y varies as eiσtwe have
(40)
provided k = σ2/g. The solution of (40) which is finite for x = 0 is readily obtained in the form of a series, thus
(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 σ are determined by
σ2= gz2/4l, J0(z) = 0.
(42)
The function J0(z) has been tabulated; its lower roots are given by
z/π= .7655, 1.7571, 2.7546,...,
approximately, where the numbers tend to the form s −1⁄4. 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), andMechanik des Himmels; L. Poinsot,Éléments de statique(Paris, 1804), andThéorie 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 titleTheory 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 theEncyclopädie der mathematischen Wissenschaften(Leipzig). There is a French translation of this work. (See alsoDynamics.)
(H. Lb.)
II.—Applied Mechanics1
§ 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.”
PART I.—OUTLINE OF THE THEORY OF STRUCTURES