III. On Torsional Oscillations of Wires. By Dr. W. PEDDIE, Physical Laboratory, Edinburgh University*.

Sketch of Previous Work †.

THE present subject is part of the more general one-the

Td in rigid solid. That strain, or part

of a strain, which disappears wholly on the removal of the distorting stress, is called temporary strain: that which is observed after the complete removal of the distorting stress, is called permanent strain or set, though it may, and usually does, diminish as the time which has elapsed since the complete removal of distorting stress increases. The latter effect, together with the converse effect of the gradual increase of set under continued constant stress, is called after-action by the Germans.

In 1835 (Pogg. Ann.) Weber investigated the laws of permanent set of a stretched fibre. In 1837 (B. A. Report ; see also B. A. Reports, 1843, 1844) Hodgkinson, as the result of experiments on cast iron, came to the conclusion that "the maxim of loading bodies within the elastic limit has no foundation in nature"; that is, permanent set is produced by any stress however small. In 1842 (Ann. de Chim. et Phys.) and 1848 (Pogg. Ann. Ergbd. ii.) Wertheim observed that permanent set occurred in a previously undistorted body as the result of any stress however small; and in 1848 (Camb. & Dubl. Math. Journ.) this limitation to Hodgkinson's statement was independently pointed out, as a deduction from theory, by Prof. James Thomson. On the other hand, a body previously distorted in a given sense may be again distorted to a smaller, or the same, extent in the same sense without the production of new permanent set.

Between 1858 and 1862 G. Wiedemann made statical experiments on the torsion of rods, in the course of which he verified Wertheim's observation; and in 1880 (Phil. Mag. vol. ix.) he published the results of more extended experiments of the same kind. One of these results is that, after repeated twistings, alternately in opposite directions, by a given couple, the set of a rod becomes constant; and, if the rod be again twisted, by increasing couples, in the direction of the last twist, the strain (measured from the position of set) is

* Communicated by the Author, having been read before the Royal Society of Edinburgh, December 18, 1893.

This sketch is a mere outline. Fuller references will be found in the papers quoted.

practically proportional to the stress so long as the original value of the couple is not exceeded. Another is that the reversed couple produces a greater strain, measured from the last set, than does the equal direct couple; the difference in this case corresponds to the change of zero produced by the reversal of the couple-that is, to the set. Again, by repeated reversals of twist under a given couple, the total torsion and the set diminish to fixed minimum values. Also, in the case of torsion in one direction, the values of the total torsion and of the set increase at increasing rates as the couple increases, and the latter relatively at a greater rate than the former: these values for a given couple increase to maxima by repeated applications of the couple, and this increase is also relatively greater in the set than in the total torsion. Wiedemann remarks that the approach of the position of final set to that of final total torsion in this case is a phenomenon of the same kind as the narrowing of the limits of total torsion and of set by repeated reversals of a given couple; the only difference is that the negative couple is zero. He calls the process by which the wire is brought into the steady state as regards total torsion and set the process of accommodation.

In 1865 (Proc. Roy. Soc. Lond.) Lord Kelvin described results obtained from the observation of torsional oscillations of wires. He discovered the phenomenon of "elastic fatigue,' and found that the diminution of the range of oscillation, per equal number of oscillations, followed the law of compound interest when the range was very much smaller than the palpable limits of elasticity. Tomlinson's observations (Phil. Trans. 1886) support this conclusion.

Present Observations and Results.

So far as I am aware, no attempt has been made to find the law of decrease of the range of oscillation when it is so large that it is accompanied by marked set-set which may amount to a large fraction of the total range. Lord Kelvin's observations were purposely made upon small oscillations in order to avoid the disturbances which are introduced when the oscillations are large. Because of the known intimate dependence of the instantaneous state of strain of a body under given stresses upon all the previous strains to which it has been subjected, it might be supposed that it would be absolutely impossible to deduce with certainty any general law of decay of large oscillations. In other words, any systematic arrangement of conditions might seem to be unattainable because of the possible intrusion of arbitrary and uncontrollable, perhaps even untraceable, conditions. As a matter of fact, I have

found that, in all my observations hitherto made, such arbitrariness is notably absent; and I have been able to obtain an extremely accurate empirical formula for the representation of the results.

The results here given refer only to a single iron wire whose extremities were soldered into holes drilled axially in stout brass rods. The length of the wire was 89.1 centim., and its diameter was 0-1011 centim. The one rod was firmly clamped in a vertical position, with the wire suspended from it; and to the other rod was attached, symmetrically and horizontally, a heavy lead ring of considerable moment of inertia. In performing the observations, one experimenter increased the torsional oscillations of the system up to a predetermined maximum, taking care to avoid as far as possible any swing of the system like that of an ordinary pendulum. Whenever the required maximum oscillation was attained, the system was left to itself, except in so far as any marked swing of the latter kind was damped out in such a way as not to interfere, by friction or otherwise, with the torsional oscillations. Another observer commenced at once to take readings of the maximum elongation by means of a telescope placed a few yards off. The scale was fastened round the outer circumference of the lead ring, and a fixed pointer was placed close in front of it. At first readings were taken at the end of each complete oscillation; subsequently, as the time-rate of decay of the oscillations became less, readings were taken at the end of two, three, five, or more, complete oscillations. A curve was then plotted with the scale-readings as ordinates and the number of swings as abscissæ. The oscillations were found to be almost isochronous, so that the axis of abscissæ was practically a time-axis. In almost all cases the curve showed traces of ordinary pendulum oscillations, but a smooth curve could easily be drawn on the average through the observed points so as to avoid all such irregularities. It would serve no useful purpose to give here the full details of each experiment. Their general nature will be seen from the curves shown in fig. 1, and the special data given in Table II. will be found sufficient for each. In fig. 1 the curves give the data obtained from observation, and the points show positions calculated from the respective equations in Table I.

It was found that equations of the form

where a, b, and n are constants, applied with great accuracy in each case Table I. gives details on this point.

The curves C, G, H, I, J, K are omitted as they were made with an entirely different object and the readings did not cover the same range.

The actual value of the constant added to a depends upon the interval which elapses after starting the experiment until the first reading is taken. Thus the fact that in B this constant is greater by about unity than the similar constants in D, E, F, while the first reading in B is much less than the first readings in D, E, F, points to the conclusion that the first reading in that curve was taken one oscillation later than the first readings in the latter. We should therefore expect that the value will be small when the initial range is large, as in M. The values of the other two constants in the equation for M are much increased relatively to their values in the preceding equationsa fact which illustrates the dependence of the action at any stage upon the previous treatment of the wire. The decay of the oscillations is at first more rapid, afterwards more slow, than in the preceding experiments.

In P the phenomenon of elastic fatigue is very apparent. The conditions were practically the same in this experiment as in, for example, E, F, and O, with the exception that in P the wire was kept oscillating to the maximum extent for about half an hour before the observations began. The rate of decay of the oscillations is immensely increased at all the observed values of the range.

Putting aside the special experiments M and P, we find that, after the wire had once reached a steady state (in B), the steady state was maintained day after day; so that it was easy to repeat an experiment under practically the same conditions. Even the exceptional treatment in M did not prevent the return of the wire to its old condition before the experiment on the following day was performed. In only one case, L1, is there any exception, and this may have been due to a difference of temperature.

In the earlier experiments the initial range is said to be over 100. No exact record was kept, but the excess was considerable; the actual angle was probably about 125. In the experiment O the angle was maintained steadily at 125 for some time before the wire was let go and the observations were begun. The constants were not altered by this treatment; and this seems to indicate that the "after-action" under steady stress has little or no effect in these experiments, which are made under otherwise similar conditions.

The experiment C was performed on the same date as B ; so that D was performed after the wire had been at rest for