Building Acoustics

Excitation and response 41

ζ > 1 shows that the system is more than critically damped giving a solution

0 0021

xt() e−ξωt Aeωζ−−−tBeωζ.

⎡^21 t⎤

=+⎢ ⎥

⎣ ⎦

(2.17)

2.4.1.1 Free oscillations with hysteric damping

As stated in section 2.3.3.1, using viscous damping is not appropriate in modelling a
system with elastic components such as rubber, plastics etc. The damping is better
described as hysteretic, normally using the loss factor η as a characteristic quantity. In
our simple mass-spring system we shall remove the viscous damper and introduce
damping through a complex spring stiffness k

kk= (1 j+⋅η). (2.18)

The loss factor η will always be much less than one. For metals one will find η in the
range of 10-4 – 10-3, for rubber of the order 10-2. Equation (2.12) will then be replaced by
the following

2

2

d

(1 j ) 0.

d

x

mk x

t

⋅++⋅=η (2.19)

We now assume that the solution of this equation will have the same form as Equation
(2.15) but we shall express it using the complex form, x(t) = A⋅exp(jγ t). By insertion
into Equation (2.19) we easily solve for the exponent γ

(^0)
1
(1j) 1j 1j.
22
kk
mmη
ηη
γη ω
<<

⎛⎞⎛

=+⋅≈ +=+⎜⎟⎜

⎝⎠⎝

⎞

⎟

⎠

(2.20)

Hence, we obtain

1j 2 2 0

( ) e having a real solution: ( ) e cos( 0 ).

jo t

xtA xtA t

ω η η

ω

ω

⎛⎞⎜⎟+ −

≈=⎝⎠ ⋅ (2.21)

Compared with the solution (2.15) the damping ratio ζ is replaced by η/2 in the
exponential term. It should also be mentioned that other quantities are in use for
expressing the damping, such as the logarithmic decrement δ and the Q factor. Assuming
that the damping is small the relationship between all these quantities is as follows

1

2

Q

.

δ

ζη

π

=== (2.22)

The reverberation time T is used in building acoustics to express the damping of sound
in rooms. However, the concept is useful when dealing with vibration as well. By
definition the reverberation time T is the time elapsed before the energy in an oscillating
system is reduced to 10-6 of the initial value. As the energy is proportional to the square
of the vibration amplitude we may represent the definition by