Equation (51) can be transformed into Bessel's equation by the substitutions

To make dp/dx vanish at x=1 we must have A'=0, and to make p vanish at x=0 we must have J_(§)=0 at =l (W/B). Hence the critical length is given by the equation

The lowest root of this equation for 12 W/B is (7.91...), and we infer that the rod will be bent by its own weight if the length exceeds (2.83...)√√(B/W).

Greenhill (loc. cit. p. 405) has worked out a number of cases in which the rod is of varying section, and has applied his results to the explanation of the forms and growth of trees.

CHAPTER XX.

VIBRATIONS OF RODS. PROBLEMS OF DYNAMICAL RESISTANCE.

277. THE vibrations of thin rods or bars, straight and prismatic when unstressed, fall naturally into three classes: longitudinal, torsional, lateral. The "longitudinal" vibrations are characterized by the periodic extension and contraction of elements of the central-line, and, for this reason, they will sometimes be described as "extensional." The "lateral" vibrations are characterized by the periodic bending and straightening of portions of the central-line, as points of this line move to and fro at right angles to its unstrained direction; for this reason they will sometimes be described as "flexural." In Chapter XII. we investigated certain modes of vibration of a circular cylinder. Of these modes one class are of strictly torsional type, and other classes are effectively of extensional and flexural types when the length of the cylinder is large compared with the radius of its cross-section. We have now to explain how the theory of such vibrations for a thin rod of any form of cross-section can be deduced from the theory of Chapter XVIII.

In order to apply this theory it is necessary to assume that the ordinary approximations described in Articles 255 and 258 hold when the rod is vibrating. This assumption may be partially justified by the observation that the equations of motion are the same as equations of equilibrium under certain body forces-the reversed kinetic reactions. It then amounts to assuming that the mode of distribution of these forces is not such as to invalidate seriously the approximate equations (21), (22), (23) of Article 258. The assumption may be put in another form in the statement that, when the rod vibrates, the internal strain in the portion between two neighbouring cross-sections is the same as it would be if that portion were in equilibrium under tractions on its ends, which produce in it the instantaneous extension, twist and curvature. No complete justification of this assumption has been given, but it is supported by the results, already cited, which are obtained in the case of a circular cylinder. It seems to be legitimate to state that the assumption gives a better approximation in the case of the graver modes of vibration, which are the most important, than in

the case of the modes of greater frequency, and that, for the former, the approximation is quite sufficient.

The various modes of vibration have been investigated so fully by Lord Rayleigh* that it will be unnecessary here to do more than obtain the equations of vibration. After forming these equations we shall apply them to the discussion of some problems of dynamical resistance.

278. Extensional vibrations.

Let w be the displacement, parallel to the central-line, of the centroid of that cross-section which, in the equilibrium state, is at a distance s from some chosen point of the line. Then the extension is dw/ds, and the tension is Ew (w/ds), where E is Young's modulus, and the area of a crosssection. The kinetic reaction, estimated per unit of length of the rod, is pw (w/t2), where p is the density of the material. The equation of motion, formed in the same way as the equations of equilibrium in Article 254, is

The condition to be satisfied at a free end is dw/ds=0; at a fixed end w vanishes.

If we form the equation of motion by the energy-method (Article 115) we may take account of the inertia of the lateral motion+ by which the cross-sections are extended or contracted in their own planes. If x and y are the coordinates of any point in a crosssection, referred to axes drawn through its centroid, the lateral displacements are

-- σx (dwds), - oy (dwds),

where σ is Poisson's ratio. Hence the kinetic energy per unit of length is

where K is the radius of gyration of a cross-section about the central-line. The potential energy per unit of length is

2

and, therefore, the variational equation of motion is

dt ρω

Ow\2
δε

Ow

4 Εω

8 ƒ de [[ + pw {(~)* + * K * ( * * )}} - Ew()"]ds,
J

where the integration with respect to s is taken along the rod. In forming the variations we use the identities

+ The lateral strain is already taken into account when the tension is expressed as the product of E and w (w/os). If the longitudinal strain alone were considered the constant that enters into the expression for the tension would not be E but λ+2μ.

and, on integrating by parts, and equating to zero the coefficient of dw under the sign of double integration, we obtain the equation

By retaining the term poK2w/ds2dt2 we should obtain the correction of the velocity of wave-propagation which was found by Pochhammer and Chree (Article 201), or the correction of the frequency of free vibration which was calculated by Lord Rayleigh*.

279. Let

Torsional vibrations.

denote the relative angular displacement of two cross-sections, so that as is the twist of the rod. The centroids of the sections are not displaced, but the component displacements of a point in a cross-section parallel to axes of x and y, chosen as before, are y and x. The torsional couple is C (ds), where C is the torsional rigidity. The moment of the kinetic reactions about the central-line, estimated per unit of length of the rod, is pwK2 (2/dt2). The equation of motion, formed in the same way as the third of the equations of equilibrium (11) of Article 254, is

The condition to be satisfied at a free end is ǝyǝs = 0; at a fixed end vanishes.

When we apply the energy-method, we may take account of the inertia of the motion by which the cross-sections are deformed into curved surfaces. Let be the torsion function for the section (Article 216). Then the longitudinal displacement is and the kinetic energy of the rod per unit of length is

The potential energy is C (dos), and the equation of vibration, formed as before, is

By inserting in this equation the values of C and fø2dw that belong to the section, we could obtain an equation of motion of the same form as (2) and could work out a correction for the velocity of wave-propagation and the frequency of any mode of vibration. In the case of a circular cylinder there is no correction and the velocity of propagation is that found in Article 200.

280. Flexural vibrations.

Let the rod vibrate in a principal plane, which we take to be that of (x, z) as defined in Article 252. Let u denote the displacement of the centroid of any section at right angles to the unstrained central-line. We may take the angle between this line and the tangent of the strained centralline to be duds, and the curvature to be d'u/ds2. The flexural couple G' is Boulos, where B= Ewk', k' being the radius of gyration of the cross-section

* Theory of Sound, § 157.

about an axis through its centroid at right angles to the plane of bending. The magnitude of the kinetic reaction, estimated per unit of length, is, for a first approximation, pw (d2u/dt2), and its direction is that of the displacement u. The longitudinal displacement of any point is - x (du/ds); and therefore the moment of the kinetic reactions, estimated per unit of length, about an axis perpendicular to the plane of bending is pwk' (u/sot). The equations of vibration formed in the same way as the second equation of each of the sets of equations of equilibrium (10) and (11) of Article 254 are

and, on eliminating N, we have the equation of vibration

If" rotatory inertia " is neglected we have the approximate equation

.(4)

.(5)

and the shearing force N at any section is -Ewk'u/ds. At a free end ô2u/s2 and Ə3u/ds3 vanish, at a clamped end u and duds vanish, at a "supported" end u and du/s2 vanish.

By retaining the term representing the effect of rotatory inertia we could obtain a correction of the velocity of wave-propagation, or of the frequency of vibration, of the same kind as those previously mentioned*. Another correction, which may be of the same degree of importance as this when the section of the rod does not possess kinetic symmetry, may be obtained by the energy-method, by taking account of the inertia of the motion by which the cross-sections are distorted in their own planes +. The components of displacement parallel to axes of x and y in the plane of the cross-section, the axis of x being in the plane of bending, are

and the kinetic energy per unit of length is expressed correctly to terms of the fourth order in the linear dimensions of the cross-section by the formula

where k is the radius of gyration of the cross-section about an axis through its centroid drawn in the plane of bending. The term in σ (k'2 - k2) depends on the inertia of the motion by which the cross-sections are distorted in their planes, and the term in '2 depends on the rotatory inertia. The potential energy is expressed by the formula

The cross-sections are distorted into curved surfaces and inclined obliquely to the strained central-line, but the inertia of these motions would give a much smaller correction.