Eliminating the n − 1 ratios A1 : A2 : . . . : An we obtain the
determinantal equation
where
| Δ(σ2) = | c11 − σ2a11, | c21 − σ2a21, | ..., | Cn1 − σ2anl |
| c12 − σ2a12, | c22 − σ2a22, | ..., | Cn2 − σ2an2 | |
| . | . | . . . | . | |
| . | . | . . . | . | |
| . | . | . . . | . | |
| c1n − σ2a1n, | c2n − σ2a2n, | . . ., | Cnn − σ2ann |
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 σ2 are all real and positive. For
any particular root, the equations (5) determine the ratios of
the quantities A1, A2, . . . An, the absolute values being alone
arbitrary; these quantities are in fact proportional to the minors
of any one row in the determinate Δ(σ2). By combining the
solutions corresponding to a pair of equal and opposite values
of σ we obtain a solution in real form:
 |
| Fig. 85. |
where a1, a2 . . . ar are a determinate series of quantities having to one another the above-mentioned ratios, whilst the constants C, ε 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 2π/σ. 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 (ε) 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 Δ(σ2) all vanish, and the basis for the determination of the quantities ar 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 θ, φ be the inclinations of the two strings to the vertical, we have, approximately,
| 2T = Ma2θ. 2 + m (aθ. + bψ. )2 | |
| 2V = Mgaθ2 + mg (aθ2 + bψ2). |
The equations (3) take the forms
| aθ.. + μbφ.. + gθ = 0, | |
| aθ.. + bφ.. + gφ = 0. |
where μ = m/(M + m). Hence
| (σ2 − g/a) aθ + μσ2bφ = 0, |
| σ2aθ + (σ2 − g/b) bφ = 0. |
The frequency equation is therefore
The roots of this quadratic in σ2 are easily seen to be real and positive. If M be large compared with m, μ 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 σ2 can never be exactly equal, but they are approximately equal if a, b are nearly equal and μ 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, μ is nearly equal to unity, and the roots of (12) are σ2 = g/(a + b) and σ2 = mg/M·(a + b)/ab, approximately. The former root makes θ = φ, nearly; in the corresponding normal mode m oscillates like the bob of a simple pendulum of length a + b. In the second mode aθ + bφ = 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
where
provided σ2, σ′2, σ″2, . . . are the n roots of (6). The coefficients of θ, θ′, θ″, . . . in (13) satisfy the conjugate or orthogonal relations
provided the symbols αr, αr′ correspond to two distinct roots σ2, σ′2 of (6). To prove these relations, we replace the symbols A1, A2, . . . An in (5) by α1, α2, . . . αn 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 σ2 and σ′2 are unequal. The actual determination of C, C′, C″, . . . and ε, ε′, ε″, . . . in terms of the initial conditions is as follows. If we write
we must have
| αrH + αr′H′ + αr″H″ + . . . | = [qr]0, |
| σαrH + σ′αr′H′ + σ″αr″H″ + . . . | = [q̇r]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
2V = cθ2 + c′θ′2 + c″θ″2 + . . . .
The new co-ordinates θ, θ′, θ″ . . . 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 θ, θ′, θ″, . . . have prescribed ratios to one another, we have, from (19),
| σ2 = | cθ2 + c′θ′2 = c″θ″2 + . . . | , |
| aθ2 + a′θ′2 + a″θ″2 + . . . |
This shows that the value of σ2 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
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
| V (q1, q2, . . . qn) | , |
| T (q1, q2, . . . qn) |
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 σ2 are all real and positive.
