5  Test Problems and Results

The test problems each have S a square joining the points (0,0) (1,0), (1,1) and (0,1) with three rigid faces, the remaining face, joining the points (1,0) and (1,1), is a uniform sheet of steel (density (s) = 7800, Young's modulus (E) = 209 ×109, Poisson's ratio n = 0.3) of thickness h. The steel is hinged to its joints on the square. The acoustic mediums considered are those of air and water. The test problems, which is illustrated in figure 2, may be regarded as a simplified loudspeaker. The vibratory properties of the isolated structure are taken from reference [24].

The mode shapes on S are given by
uj (p) . np  =  {
sinj pp2    (p1 = 1, p Î S),
0               (p1 < 1, p Î S).
For the integral equation method, the square is divided into N uniform elements. Hence
[U]ij  =  {
sin 4 j p(2i-1)
N
    i=1,2,..., N
4
  ; j=1,2,..M,
0            i = N
4
 +1,..,N ; j=1,2,...M.
 .
The resonant frequency of the jth mode is
kj = p2 h
c
 é
ë 		E
12 s(1- n2)
 ù
û 		[1/2]

 
   j2.
The functions lj(k) are given by
lj(k) = i sh c (k2 - kj2)
k
  .

5.1  Results with Air as the Acoustic Medium

Since the density of air is much less than that of steel, the steel sheet must be thin for its vibratory properties to be significantly influenced by the acoustic loading. The density of air (r) and the speed of sound (c) in air were assigned the values 1.29 and 331. The steel sheet was assigned three different thicknesses of 0.002, 0.001 and 0.0005 in turn and the number of boundary elements and modes used were N=8 and M=2, N=16 and M=4, N=32 and M=8 in turn. In all tests the resonant frequencies had Re(k) in the range [0,1] and this was the range over which the approximation (25) was made. Hence nine runs of the method were carried out. The computed results for the first and second resonant frequencies are given in the following tables.


Structure in Air : Frequency of 1st Mode (k*)
Thickness of the steel sheet
N|M ¯ h = 0.002 h = 0.001 h = 0.0005
In vacuo k*  ® 0.093415 0.046707 0.023354
8 2 0.08956 - i0.00105 0.04143 - i0.00120 0.01605 - i0.00122
16 4 0.09102 - i0.00078 0.04270 - i0.00074 0.01814 - i0.00061
32 8 0.09030 - i0.00075 0.04281 - i0.00072 0.01827 - i0.00061


Structure in Air : Frequency of 2nd Mode (k*)
Thickness of the steel sheet
N|M ¯ h = 0.002 h = 0.001 h = 0.0005
In vacuo k*  ® 0.373660 0.186830 0.093415
8 2 0.36893 - i0.00036 0.18225 - i0.00034 0.08879 - i0.00010
16 4 0.37009 - i0.00019 0.18336 - i0.00016 0.09001 - i0.00017
32 8 0.37035 - i0.00008 0.18359 - i0.00005 0.09024 - i0.00007


5.2  Results with Water as the Acoustic Medium

The density of water is approximately an eigth of the density of steel. Hence, even if the steel sheet is thick, its modal properties are significantly influenced by the acoustic loading. The density of water (r) and the speed of sound (c) in water were assigned the values 1000 and 1524. The steel sheet was assigned three different thicknesses of 0.02, 0.01 and 0.005 in turn and the number of boundary elements and modes used were N=8 and M=2, N=16 and M=4, N=32 and M=8 in turn. In all tests the resonant frequencies had Re(k) in the range [0,1] and this was the range over which the approximation (25) was made. The computed results for the first and second resonant frequencies are given in the following tables.

Structure in Water : Frequency of 1st Mode (k*)
Thickness of the steel sheet
N|M ¯ h = 0.02 h = 0.01 h = 0.005
In vacuo k*  ® 0.202889 0.101445 0.050722
8 2 0.06664 - i0.01038 0.01433 - i0.00269 0.00209 - i0.000421
16 4 0.07545 - i0.00916 0.02027 - i0.00218 0.00366 - i0.00028
32 8 0.07674 - i0.00913 0.02071 - i0.00225 0.00376 - i0.00031


Structure in Water : Frequency of 2nd Mode (k*)
Thickness of the steel sheet
N|M ¯ h = 0.02 h = 0.01 h = 0.005
In vacuo k*  ® 0.811557 0.405779 0.202889
8 2 0.46637 - i0.01151 0.18213 - i0.00560 0.06521 - i0.00074
16 4 0.51168 - i0.00960 0.20354 - i0.00343 0.07666 - i0.00183
32 8 0.52278 - i0.00526 0.20907 - i0.00130 0.07907 - i0.00084