Books/MathForTheLayman/Vibrations

8: Vibrations

8A. Introduction

Vibrations are commonly seen in the motions of a clock pendulum, of a piano string, and of a weight (such as a plumb bob suspended on a rubber band or steel spring). If the bob is pulled straight down a certain distance from its rest position and released, it visibly oscillates above and below its rest position until it is brought to rest by friction in the air and in the suspension.

If b is a function that gives the position of the bob as a function of time, then its speed or velocity is given by b d.1 (the rate of change of position), and its acceleration is given by the second derivative b d.1 d.1 (the rate of change of its velocity).

Moreover, the acceleration is caused by the “restoring force” exerted by the spring, which is proportional to its extension as measured by the position b of the bob. In other words, the second derivative b d.1 is proportional to -b.

In the simplest case -b is equal to b d.1 d.1, and we seek a polynomial with this property. Consider the coefficients of the function fs introduced in Section 3A:

      c=:  0 1 0 _1r6 0 1r120 0 _1r5040
      der c
   1 0 _1r2 0 1r24 0 _1r720 0
      der der c
   0 _1 0 1r6 0 _1r120 0 0
      -der der c
   0 1 0 _1r6 0 1r120 0 0

The required pattern is clear: it is the same as for the exponential of Chapter 7, except that alternate coefficients are zero, and that the other coefficients alternate in sign. The function with this property is called the sine, commonly abbreviated to sin:

      sin=: 1 with o.
      sin t. i.12
   0 1 0 _1r6 0 1r120 0 _1r5040 0 1r362880 0
   _1r39916800
      sin t. 20 21 22 23
   0 1r51090942171709440000 0 _1r25852016738884976640000

8B. Harmonics

The function sin on (2:*]) applies the sin to the double of its argument, and therefore produces a vibration (or oscillation) of twice the frequency. It is called an harmonic of the oscillation sin. This may be seen in a combined plot of the two functions:

     x=: 1r10 * i.65
     plot x;(sin x) ,: (sin on (2:*]) x)

Exercises

   Replace the 2 in the function sin on (2:*]) by other integers to produce other harmonics, and plot the results.

   Plot the function cos and some of its harmonics.

   Plot the function cos against its harmonic cos on (2:*]).

8C. Decay

Because of "friction" of some sort, vibrations commonly decay, usually at an approximately exponential rate determined by a function of the form decay=: ^ on - on (] % [).

Exercises

   Plot the decaying oscillation sin * 6 with decay .

   Plot sine and cosine functions with rates of decay other than 6, and plot them together with their non-decaying counterparts.