Building Acoustics

Oscillating systems 11

viscous damping small enough to make it oscillate. We assume that x(t) is the
displacement of the mass when we initially move it from its stable position and then
release it. We may write

0

( ) cos(2 ) for 0

and ( ) 0 otherwise.

xt Aeat ft t

xt

= − π >

=

How quickly the motion “dies out” is determined by the constant a and Figure 1.7 shows
a time section where a is 1 s-1 and 50 s-1, respectively. The amplitude A is set equal to 1.0
and the frequency f 0 is 25 Hz for both curves^1. The modulus of the Fourier transform will
be

()

1

2

222

22222 222

0

4

().

416

af

Xf A

aff af

π

ππ

⎡⎤

⎢⎥+

=⋅⎢⎥

⎢⎥⎡⎤+−+

⎣⎦⎣⎦

0.00 0.05 0.10 0.15 0.20 0.25

Time t (s)

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

x(

t)

Figure 1.7 Time function of a damped oscillation (see the expression for x(t) above). The constant a is equal to
1 (thin line) and 50 (thick line), respectively.

This expression is shown in Figure 1.8 for the two values of a. It should be noted
that the ordinate scale is logarithmic in contrast to that used in Figure 1.6. As expected
we do get a very narrow spectrum around f 0 when the system has low damping. Setting a
= 1 s-1 reduces the amplitude to 1/10 of the starting value after a time 2.3 seconds, i.e.
after some 60 periods. When a is 50 s-1 the amplitude is down to 1/10 just after one
period and the spectrum is then much broader.

(^1) The frequency f
0 will not be independent of the damping of a real system. However, with the chosen variation
in the damping coefficient a the variation in f 0 will be approximately 5%.