Popular Science Monthly
��111
��by fastening to it, near the bottom, a fairly large piece of cardboard, in such a way that it will act as a brake. The swings of the pendulum will die away much faster than before; that is, the damping will be increased on account of the increased frictional resistance of the fan. On such a more strongly damped pendulum (assuming that the oscillation is started by letting the pendulum begin from the point lo), the successive maximum swings to the left (at the ends of the first, second, third and later periods), may be as follows: 8.0 (end of 1st); 6.4 (end of 2nd); 5.1 (end of 3rd); 4.1; 3.3; 2.6; 2.1, etc. It is seen at once that now the swings decrease much more rapidly. This is even more vivid when Fig. 3, which shows the motions Oi the second pendulum, is inspected; the rapid fall of the broken line along the top, which indicates the damping, should be noted especially. The constant ratio or damping factor, whose value is an indication of the damp- ing, may be found as before by dividing the first maximum amplitude by the second, the second by the third, etc. This gives us: 10/8 = 8/6.4 = 6.4 = 5.1 = etc. = 1.25. Since this ratio is larger than before the brake was added to the pendulum, we have an arithmetical proof that the damping is larger.
So far we have considered only the "damping" of the oscillation system; what is the "logarithmic decrement?" Nothing more nor less than the natural logarithm of the constant ratio which has just been figured out. These logarithms, or special numbers, for several different ratios, are given in the following table:
��Ratio I
1.05 I. II 1. 16 1.22 1-25 1.28
1-35
��Logarithm 0.00 0.05 O.IO
0.15
0.20 0.22 0.25 0.30
��By looking up the ratio i.i, which was that of the first pendulum, in the table it is seen that the logarithmic decrement of that arrangement was a trifle under 0.1 per period; similarly, for the second pendulum (which had a damping factor
��of 1.25), the decrement is found to be 0.22 per complete period.
Although the examples just given are purely mechanical, damping in electric circuits is of the same character. Let us consider the circuit of Fig. 4, which has connected in series a condenser C,
���/fid/if\ ^^S- 3. Swings of second pendulum
an inductance L, a resistance R and a special current indicator /. This indica- tor is of the sort which will show the amount and direction of the current flowing through the circuit at any instant, as would a Braun-tube oscillo- graph. If C is charged to a certain potential and then is allowed to dis- charge through the oscillation circuit by the sudden closing of switch S, the result will be a free oscillating current through L, I and R. As was shown in the March article of this series, the frequency and time period of this free oscillation can be figured out from a simple rule, if one knows the inductance and capacity of the circuit. The thing important to this discussion is not the period of frequency, however, but the rate at which the free oscillation dies away. If the oscillograph / is arranged to make an actual photograph of the oscillation current-effects (which is en- tirely feasible, even on very high frequencies), the result will be a curve of the sort shown in Figs. 2 and 3; if the capacity and inductance, or either of them, are in- creased, the time period will be lengthened and the curves will spread out more along the horizontal line. If the voltage applied to the condenser before the switch S is closed is made larger, the current flowing will be in- creased and the highest and lowest
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J
��Fig. 4. Oscillograph Circuit
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