The boundary element method is potentially useful in the analysis
of both the exterior and interior acoustic fields of loudspeakers.
It is particularly attractive for the exterior problem as only
the outer surface of the loudspeaker (the cone and the cabinet)
require discretisation. Ideally, perhaps, the interior and exterior
acoustic fields and the vibration of the cone should all be coupled
together to give a complete model of the problem. Although this has
yet to be realised, it does demonstrate the attractiveness
of using boundary element-based methods for all the acoustic modelling.
For work on the application of the boundary element method
to the exterior acoustic problem of loudspeakers,
the reader is referred to Jones
[41] and Henwood [42].
In this section the interpolation method is
applied to the interior of the test loudspeaker of Jones [41].
The enclosure of a loudspeaker affects the sound radiated by the drive
units in two ways, namely by perturbing the motion of the diaphragm
and by causing the walls of the enclosure to vibrate. In addition,
if the enclosure is not air-tight but instead is ported,
it acts as a Helmholtz resonator [43].
The loudspeaker designer is therefore interested in predicting the
acoustic field generated within the enclosure with a view to
positioning the drive units and ports so as to optimise the
sound radiated by them, and to control the field itself through
the shape of the cabinet and the use of damping.
In this section a modal analysis of the air-tight interior of
a test axially symmetric loudspeaker is carried
out via the boundary element method,
as outlined in section 6. The results are compared with results from a
typical implementation of the finite element method [44].
As further
verification, the resonant frequencies given by the method are
compared with those obtained through physical experiment.
The effect of the fluid-loading of the air on the cone is
of some interest to loudspeaker designers.
The coupling of air external to the loudspeaker to the motion of
the cone is considered experimentally in Jones [41], wherein
the difference between the forced vibration of a cone in air and its
vibration in a vacuum is found to be negligible. However, it is
expected that the presence of the air inside the cabinet can have
a significant effect on the vibration of the cone
due to the relatively small, enclosed volume occupied by the air.
The greatest effect of air loading on the vibration of the cone occurs
when there are large changes in pressure over the surface of the
cone. The maximum change in pressure will be at acoustic resonant
frequencies and therefore the corresponding mode shapes are studied.
By applying the methods to an axisymmetric loudspeaker, the acoustic
properties may be examined
whilst reducing the dimension of the problem by one and thus
reducing the computational expense when compared to the full three-dimensional
analysis that is necessary for a general loudspeaker design.
In addition, considerable previous work has been done on the
structural vibration of axially symmetric loudspeaker drive units
(see Jones [41] and Jones and Henwood [45], for example).
In this work we show results for the cabinet shown in figure 2
which is basically a cylinder 120mm deep and 132mm in diameter. The loudspeaker
has a conical drive unit of radius 80mm
(similar to the one analysed in Jones et al [46] fitted.
Given these dimensions, the lowest cabinet resonance occurs around
1kHz, and the cabinet resonances can be observed in sound
pressure measurements outside the cabinet.
8.2 Particular implementations of computational methods
For the application of the finite element method the interior of the loudspeaker illustrated in figure 2
was divided into about one hundred triangles of approximately equal size.
The elements are three noded triangles swept through 2 p. The
basis functions are such that j is approximated by a linear
function on each triangle, defined by its values at the corner nodes.
The general method employed is outlined in section 3. Resonant frequencies
and mode shapes were computed through solving the linear eigenvalue problem
(13). The speed of sound c was taken to be 340ms-1,
the density of air r was taken to be 1.29kgm-3.
For the application of the boundary element method the boundary
of the generator of the loudspeaker, as shown in figure 2 is approximated
by 32 conical elements of approximately equal length along the generator.
The boundary elements thus have similar dimension to the finite elements.
In the boundary element method the boundary functions are approximated by
a constant on each element. Collocation was applied via subroutine H3ALC
in Kirkup [47] to obtain the matrices Ak in (23).
Solutions were sought in the k-ranges [0.0,5.0], [5.0,10.0], [10.0,15.0]
and so on. In each range quadratic interpolation of the matrix Ak was
applied so that the resulting linear eigenvalue problem has the form
(43). The approximations to the mode shapes are then obtained
at around 100 points in the interior through the summation (42)
for each point.
For the measurement of the resonant frequencies,
microphone readings of the sound pressure
were taken at various positions within the cabinet.
Measurements were made through all frequencies of interest, commencing
at 20Hz and going through to 20kHz. However, the natural decrease in current
due to the electrical circuit of the voice coil means that the signal
becomes very small above 5kHz (Jones et al, 1992b).
Figure 3 shows a typical spectrum of sound pressure at one point inside the loudspeaker.
The first cabinet resonance is seen at around 1.3kHz, and typically
around seven further peaks of varying strength are visible below 5kHz.
Above this frequency, the amplitude of the pressure decays due to
the fall off in amplifier current and the break up of the diaphragm
[48].
Firstly the five lowest resonant frequencies obtained through the application
of the finite element with the boundary element methods and the results
obtained by measurement are compared in table 3.
The peaks in the spectrum in figure 3 occur at the resonant frequencies and these
are read off to give the experimental results in the table.
Table 3: Computed and measured loudspeaker resonant frequencies
Mode
Finite Element
Boundary Element
Experimental
1
1458 Hz
1414 Hz
1318 Hz
2
1643 Hz
1590 Hz
1679 Hz
3
2326 Hz
2232 Hz
2133 Hz
4
2935 Hz
2815 Hz
2691 Hz
5
3008 Hz
2876 Hz
3306 Hz
It can be seen that the difference between the computed and the measured
results is generally within 10%,
which is acceptable given the
coarseness of the discretisation employed and the errors inherent
in the measurements.
The BEM and FEM results are generally within
4%
of each other, and since their efficiencies are similar, the BEM
is a viable alternative to the FEM.
The major contribution to the discrepancy between the measured and
calculated values is believed to be the simplicity of the model
chosen, which fails to include any internal structure to the
loudspeaker. In addition, the maximum pressure occurs at slightly
different frequencies for different microphone positions.
Figures 4 show contour plots of the mode shapes corresponding to the first, third
and fifth resonant frequencies obtained via the finite element method.
Figures 5 show the same results obtained via the boundary element method.
The values on the contours are arbitrary. The contour plot for FEM and
the BEM are very similar for each of the mode shapes.
