ABSTRACT
The free oscillations of a physical system governed by a second order con-
stant coefficient equation are described by
d2x
dt2 2zV0
dx
dt V20 x0:
If there is no friction or electrical resistance removing energy from a physical
system described by the above equation there is said to be no damping and
z/0, so the equation reduces to
d2x
dt2 V20 x0:
The characteristic equation for this differential equation is
l2V200, so l9iV0,
and the general solution (complementary function) is
xA sinV0tB cosV0t:
Let us rewrite this result as
x(A2B2)1=2
A
(A2 B2)1=2 sinV0t
B
(A2 B2)1=2 cosV0t
,
and then set
a(A2B2)1=2, cos o A
(A2 B2)1=2 and sin o
B
(A2 B2)1=2 ,
so it becomes
xa (sinV0t cos ocosV0t sin o):
Using the trigonometric identity
sin(PQ)sin P cos Qcos P sin Q
and setting P/V0t , Q/o , we see that
xa sin (Vto):
Fig. 154
amplitude of the oscillation (the maximum magnitude of x), V0 the angular
frequency, T/2p /V0 the period of the oscillation, F/1/T/V0/2p the frequency of the oscillation (the number of cycles in a unit time) and o the phase
of the oscillations. The relationship between a , T, V0 and o is shown in Fig.
