Page 150 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 5.2 Initial Conditions 137

Thus, these equations show that for the given initial conditions, most of the response
is due to (/)j. This is to be expected because the ratio of the initial displacements
0.50
1.00

is somewhat close to that of the first normal mode and quite different from that of the
second normal mode.
Example 5.2-2 Beating
If the coupled pendulum of Example 5.1-3 is set into motion with initial conditions
differing from those of the normal modes, the oscillations will contain both normal
modes simultaneously. For example, if the initial conditions are 0,(0) = A, 82(0) = 0,
and 0,(0) = ¿2(0) = 0, the equations of motion will be
^1(0 "" cos 0)2t
^2(0 cos (o^t — \ A cos
Consider the case in which the coupling spring is very weak, and show that a beating
phenomenon takes place between the two pendulums.
Solution: The preceding equations can be rewritten as follows:
^ ( (O] — (Oy \ ( 0) ] + 0)-> \

(i) = A cos I ----2---- 1 / cos I -----2— ^ I^
/ 0), —COy \ / 0), -h \

(t) = -A sin ( ---- 2---- ] ^ ---- 2— ^ ^
Because (to, - CO2) is very small, 0,(/) and 62(1) will behave like cos (it>, + co2)t/2 and
sin (it), T (x)2)t/2 with slowly varying amplitudes, as shown in Fig. 5.2-1. Since the
system is conservative, energy is transferred from one pendulum to the other.
The beating sound, which is often audible, is that of the peak amplitudes, which
repeat in tt radians. Thus,
/ w, — it>2 \ 2 it
----^----h- = 77- or T;, = ----------

\ 2 J ^ " i t ) , — it>2
The beat frequency is then given by the equation
277
A simple demonstration model is shown in Fig. 5.2-2.

Figure 5.2-1. Exchange of energy
between pendulums.