The free oscillations of a physical system governed by a second order con-

stant coefficient equation are described by


dt2 2zV0


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


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



(A2 B2)1=2 sinV0t


(A2 B2)1=2 cosV0t


and then set

a(A2B2)1=2, cos o A

(A2 B2)1=2 and sin o


(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.