This equation gives two real values of k, one greater than the larger and the other less than the smaller of the two quantities k, and ką. In the slower of the two possible forms of vibration there is a node only at the fixed end of the shaft, and the two masses twist always in the same direction; in the faster vibration the two masses are always twisting in opposite directions, and there is a node on the shaft between them. The distance λ of this node from the outer mass I, is given by

where k is the corresponding root of equation (12).

In cases in which the reciprocating parts of the engines are taken into consideration in calculating the frequency of vibration of the system, it is, of course, the moment of inertia of these parts about the centre of the crank-shaft that is required. In single-line engines, and in two-line engines with the cranks at 180°, this moment of inertia varies throughout a revolution, but in two-line engines with the cranks at 90° and three-line engines with the cranks at 120°, if the reciprocating masses are the same in each line, and if the obliquity of the connecting-rod is neglected, this moment of inertia is constant throughout a revolution. If M denote the mass of the reciprocating parts in each line, R denote the crank radius, and I denote the moment of inertia of the reciprocating parts about the centre of the crank-shaft, then in a two-line engine with cranks at 90°,

Most of the formulas in the Paper have been obtained by a method which seems novel and worthy of a brief explanation. Certain types of vibration, e.g., the fixed-free and the fixed-fixed vibrations of a single mass, are taken as fundamental, and from these the most complicated cases are built up.

Taking for example the case, illustrated in Fig. 7, of the longitudinal vibrations of two masses M, and M, on the shaft (E, σ, l), it is clear that a vibration is possible in which there is a node somewhere between the two masses. Let this be at a distance x from M, and therefore 1 - -x from M2. Then M, vibrates as if on a fixed-free shaft (E, o, x), M, as if on a fixed-free shaft (E, σ, 1 − x), and the frequency is the same in the two cases. Thus by equation (5), p. 378, the mass of the shaft being supposed negligible,

which is equivalent to equation (11), p. 381.

Taking, as a second example, the case illustrated in Fig. 20, of the torsional vibrations of two masses I, and I, on the composite shaft C1, C2, it is clear that a vibration is possible with a node somewhere between the two masses, say

at a distance z from I. Then I, vibrates as if on a fixed-free shaft C, (';),

Ca la

(l1⁄2 − x)'

and the frequency

and I, as if on a fixed-fixed composite shaft C1, is the same in the two cases. Thus by equations (10), p. 384, and (2), Appendix II., p. 397, assuming the inertia of the shaft itself negligible,

which on reduction gives equation (12), Appendix II.

As a third example, the free-free vibrations of a shaft with four masses,

tion C2, at a distance x from I, and therefore at a distance l, -x from I. From the frequency-equation of fixed-free vibrations in the portion to the left of the node, neglecting the inertia of the shaft itself, the following equation is obtained:

k2 k2

- x C2
=(-) + {-c G11 + } }) }

Adding these two equations, x is eliminated, and after some simple reductions the following frequency-equation is obtained :

2

(k2-k123) (k2-1232) (k2 —k3,2)—k,2 kg (k2 - k2,3)—k22 k ̧2 (k2 — k122) = 0; (1)

Regarding (1) as an equation in 2 of the third degree, it is easily shown that there are three real positive roots. Supposing k1 to be larger than k, the three values for k are:

Thus there are three real types of vibration, and the frequencies are:-the

smallest less than

l34 If k12

=

largest greater than

Σπ

34, i.e., if the free-free vibrations of the shaft C1

with inertias I, and I, at its ends have the same period as those of the shaft C, with inertias I, and I, at its ends, then k122 or k342 is likewise a root of equation (1). Equation (1) may also be obtained, with equal ease, by supposing a node to come

1

between, say, I, and I. By supposing I, infinite-i.e. zero-in equation (1)

I

the frequency-equation for three inertias I, I, I,, on a shaft fixed at the righthand end, is obtained.

If in the case illustrated in Fig. 10, it were required to find the length L of a shaft of given uniform diameter D which would give the same result as that shown in the figure it would be obtained from the equation

The total length (L + 1) can then be used in equations (12) and (13), p. 385.

[THE INST. C.E. VOL. CLXII.]

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(Paper No. 3551.)

"Note on the Underpinning of the Piers in the Reservoirs of the Galatz Waterworks, Roumania."

By WILLIAM MORRIS LANGFORD, Assoc. M. Inst. C.E.

FOR several years considerable leakage had been taking place from two covered masonry reservoirs at the Galatz Waterworks. The nature of the earth under their foundations was such that it afforded a sound base while in a dry condition, but when saturated with water it allowed the walls to settle slightly, and this increased the leakage. After carrying out certain repairs of a more or less superficial character, the directors of the Waterworks Company decided to line both compartments throughout with "Callender's pure bitumen sheeting."

Each reservoir consisted of a rectangular chamber, 95 feet in length by 74 feet in width, and 13 feet in depth below the overflow sill. The floor was a bed of concrete 2 feet 2 inches in thickness, the walls were built of rubble, and the roof was constructed of brickwork arches. The total width of the roof of each compartment was divided into seven spans, each running the whole length of the compartment. The abutment-walls were carried by six rows of eight piers, of rectangular section, and also built of brickwork, Figs. 1.

The sheet-bitumen was attached to the walls by means of numerous tie-bolts (Fig. 2), and was further supported by cementrendering laid on expanded-metal sheeting secured to the same tie-bolts, Fig. 3. This part of the work presented no special difficulty, but it was considered desirable to prevent leakage taking place downwards through the bodies of the piers, and the satisfactory sheathing of all the ninety-six piers, with their battered footings and superincumbent arches, would have been both difficult and expensive work. It was therefore decided to cut away the feet of all the piers, and to lay the sheet-bitumen flat on the concrete floor, rebuilding the pier-footings upon it.

In the meantime each pier, with its load of about 30 tons, had

to be supported. In order to do this, two special girders were prepared, each consisting of a steel channel, 10 inches by 4 inches by it inch in cross-section and 7 feet in length, having a piece of 3-inch steel angle-bar riveted to it in such a position as to form a

Loadbadbad

圓圓圓圓圓圓

圓圓圓圓圓圓

tee with the lower flange of the channel, as shown in Figs. 4. A groove was cut on each side of the pier to be dealt with, 2 feet 3 inches above the floor, and deep enough to take the projecting flange of the 3-inch angle-bar. The channels were held tightly

74.

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