Page:EB1911 - Volume 17.djvu/1012

The case of three degrees of freedom is instructive on account of the geometrical analogies. With a view to these we may write

2T = aẋ² + bẏ² + cż² + 2fẏż + 2gżẋ + 2hẋẏ,

2V = Ax² + By² + Cz² + 2Fyz + 2Gzx + 2Hxy.

(21)

It is obvious that the ratio

V (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 have

Ax1 + Hy1 + Gz1 = σ12 (ax1 + hy1 + ∂gz1),	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$
Hx1 + By1 + Fz1 = σ12 (hx1 + by1 + fz1),
Gx1 + Fy1 + Cz1 = σ12 (gx1 + fy1 + cz1),

(23)

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

x1	∂V	+ y1	∂V	+ z1	∂V	= 0;
	∂x		∂y		∂z

(24)

and if this be denoted by σ₂² we have a second system of equations similar to (23). The remaining value σ₂² is the value of (22) when x : y : z are chosen so as to satisfy (24) and

x2	∂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.

We proceed to the forced vibrations of 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 Q_r, equal to e^iσt multiplied by a complex coefficient; owing to the linearity of the equations the factor e^iσt will 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

q = Q/c − σ2a.

(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 σ² is 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)

(c_1r − σ²a_2r) q₁ + (c²_2r − σ²a_2r) q₂ + . . . + (c_nr − σ²a_nr) q_n = Qr,

(28)

and therefore

Δ(σ²) · q_r = a_1rQ₁ + a_2rQ₂ + . . . + a_nrQ_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 2π/σ, and all the particles pass simultaneously through their equilibrium positions. The amplitude becomes very great when σ² 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),

aq̈ + bq̣̇ + 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		∂T	+ B1rq̇1 + B2rq̇2 + . . . + Bnrq̇n +	∂V	= Qr.
dt		∂q̇r		∂qr

(31)

If we put

b_rs = b_sr = 1/2 (B_rs + B_sr),   β_rs = −β_sr = 1/2 (B_rs − B_sr),

(32)

this may be written

d		∂T	+	∂F	+ β1rq̇1 + β2rq̇2 + . . . + βnrq̇r +	∂V	= Qr,
dt		∂q̇r		∂q̇r		∂qr

(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 as frictional or dissipative forces. The terms affected with the coefficients β_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 q̇_r and sum with respect to r from 1 to n, we obtain, in virtue of the relations β_rs = −β_sr, β_rr = 0,

d/dt (T + V) = 2F + Q1q̇1 + Q2q̇2 + . . . + Qnq̇n.

(35)

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^λt, 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 = Cαr e−t/τ cos (σt + ε − εr),

(37)

where σ, τ, α_r, ε_r are 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 e^iσt 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^iσt; 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 (see Wave). 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 determined