D. P. Onwood Purdue University at
Fort
Fort
Wayne
Wayne, Indiana 46805
A Lecture Demonstration of Normal Modes of Vibration
T h e concept of normal modes of vihration is sufficiently novel to many undergraduate students to warrant a simple demonstration. The apparatus used should be capable of being excited in any one of its normal modes, as well as a comhination, so as to demonstrate that an apparently complicated motion may he represented as the result of combining simple motions in various directions. Such an apparatus is a system of coupled meter rules, swinging on parallel knife edges mounted the same short distance from one cnd, and connected near to their tops by means of a weak spring (see figures). This system is more likely to be available than one involving an air track which is of course eminently suitable. The two normal modes of vibrations have frequency v,, for the oscillation in which both rules move in phase; and vz, for the motion in which the rules move in opposite phase. Since t,he restoring force is greater for the latter mode, a > VI. Any general motion of the system is obtainable by comhination of these modes. If the angles through which the two pendulums have swung a t any time t are 01, 02, then the first normal mode may be described by
and t,hesecond normal mode by 8, = Y
826
1 Journal
of Chemical Education
+ 20
(3)
+ ?y)
(4)
where X, Y, are amplitudes, and z, y, arbitrary phases. Any general vibration may now be represented by sum terms 8, =
X sin 2 r ( v d
ez
X sin 2s(ult
=
+ z ) + Y sin 2 = ( v d + y)
(5)
+ x) - Y sin Z a ( d + y )
(6)
A particularly easy case to reproduce experimentally is that in which one pendulum is held in its rest position while the other is held in a displaced position so that O1 = 0, O2 = A: the two are then released simultaneously. Then a t time t = 0, Ol = 0, 02 = A, d h / dt = 0, d&/dt = 0; which establishes that z = y = O;X = Y = A/Z,and
Since (vl - us) is very much less than v2), eqns. (9) and (10) represent vi(v, brations of frequency close to '/?(PI vz), and with varying amplitudes A cos ( n v2)d, A sin ( Y , - vZ)rt. Suitably chosen dimensions for the system result in the amplitude of the motion of each rule varying from zero to a maximum value within afcw seconds, enabling the cycle to be observed many times before damping effects become significant. Thanks are due to Professor R. E. Wise for practical assistance
+
~h~ opparotur ~ i t or,0wr h indicating motions for modes PI (left1 and uz (right).
sin 2&t
0, = - Y sin 2 r ( v 2 t
+
