Civil Engineering Reference
In-Depth Information
Examples will be given in section
5.7
where we shall include matrices representing
porous materials based on models treated in section
5.5.
Acoustical
Electrical
Mechanical
Closed volume
Capacitor
Spring
Short tube
Inductor
Mass
Collection of
n
arrow tubes
Resistor
Viscous damper
Figure 5.7
Analogue components in acoustic, electrical and mechanical systems.
5.4.1
Simple analogues
Figure 5.7
shows one of the simple analogies one may use for the relationship between
acoustical, electrical and mechanical systems, the so-called impedance analogy. This
implies that that the sound pressure in the acoustical system is equal to a voltage in the
electrical system and to a force in the mechanical system. Correspondingly, an acoustical
particle velocity (or volume velocity) will be a current and a vibration velocity. The
relationship between acoustic impedance, specific acoustic impedance and mechanical
impedance will then be:
p
ZZ
s
ec
Z
=
=
=
,
(5.16)
a
2
vS
⋅
S
S
where
S
is an area (m
2
).
As mentioned above, we shall start with a very simple acoustical system, the
Helmholtz resonator, to illustrate the use of such analogies. In its simplest form, this may
be considered as a harmonic oscillator; mechanically speaking it is a simple mass-spring
system. The mass will be an oscillating air column in a tube (or in a slot) being driven by
the sound pressure, this mass is coupled to a spring represented by a closed volume of air
(see
Figure 5.8
a)). We then have to find the spring stiffness, the mass and the damping
coefficient, expressed by acoustical quantities, to calculate the resonance frequency and
the energy dissipation.
5.4.1.1
The stiffness of a closed volume
We may find the mechanical stiffness of a closed volume by using the equation
P
⋅
V
γ
=
constant, giving the relationship between pressure and volume under adiabatic
conditions. However, it may be more appropriate from an acoustical viewpoint to use the
