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Reconstruction of blockage in a duct from single spectrum
Applied Aeoustic,s 41 (1994) 229 236
Reconstruction of Blockage in a Duct from Single Spectrum Wu Qunli School of Mechanical and Production Engineering, Nanyang Technological University, Singapore 2263, Singapore (Received 28 October 1992; revised version received 8 February 1993; accepted l0 February 1993)
A BS TRA C T In some industrial practices it is important to detect blockages developing in the ducts and pipeline systems, especially in the nuclear power industry where safeO' is vital. In this paper, the properties of eigenfrequeney shifts o f a duct with blockages are studied. It is .found that the sum of the eigenfrequency shifts over a number of modes is close to zero for the duct with the blockage. A new method is proposed to reconstruct the blockage location and its size in the duct. The method requires only one spectrum o f the blocked duct under asymmetrical boundao' condition. The experimental results show that the new method can be as accurate as the method which requires four spectra data.
1 INTRODUCTION In some industrial practices it is important to detect blockages developing in ducts and pipeline systems, especially in the nuclear power industries where the safety is vital. Domis 1 applied the eigenfrequency shift idea to detect blockages in the sodium-cooled fast reactor subassemblies. He notices that the eigenfrequencies will be shifted from the original values when a blockage develops in the duct. He used eigenfrequency shifts as an early warning of the blockages. Based on an inverse problem approach, Wu and Fricke z concluded that blockages in a duct can be determined from two sets o f the eigenfrequency shifts under two b o u n d a r y conditions. This method is 229 Applied Acoustics 0003-682X/94/$07.00 © 1994 ElsevierScience Limited, England. Printed in Great Britain
Wu Qunli
230
further extended for the determination of the blockages in two-dimensional cavities? Using the eigenfrequency shifts to reconstruct the blockages in the duct requires at least two sets of spectral data, one contains the eigenfrequencies of the duct with the blockages and the other contains the baseline eigenfrequencies of the duct. From these two sets of the eigenfrequencies, the eigenfrequency shifts can be calculated. For some practical applications the baseline eigenfrequencies are difficult to obtain and the temperature changes may also cause the eigenfrequency shifts. In this paper, a method for reconstructing the blockages in the duct from a single spectrum is investigated. The method is able to compensate for the temperature effects on the eigenfrequencies.
2 THEORETICAL ANALYSIS Consider a oneidimensional model of a uniform duct with a blockage, as shown in Fig. 1. Let A o and Ab(x) be the cross-sectional areas of the duct and blockage, respectively. The blockage partitions the duct into three segments with lengths L1, L 2 and L 3, where L 2 is the blockage length. The following non-dimensional quantities are defined: )~h = L z / L ,
fl(x) = Ab(x)/A o,
Cb = (LI + L z / 2 ) / L
L is the total length of the duct. fl(x) is the blockage function. When L 1 < _ x < L 2, fl(x)=fl, and fl(x)=0 when x<_L~ and x > L 2 . By applying one-dimensional wave equation and the boundary conditions, the eigenvalue equation for blocked duct closed at x = 0 and open at x = L is derived 4 sin (kfLt){(1 -- fl) c o s ( k f L 2 ) sin (kfL3) + sin (kfL2) c o s (kfL3) } + (1 - fl) cos (k/L0{(1 - fl) sin (k/L2) sin (kfL3) - c o s ( k f L 2 ) c o s ( k f L 2 ) I --- 0
(1)
where k f = 2rcv/c is the wavenumber and v is the frequency and c is the speed I~ I~
L
closed I
~1 -I
i
Ao
L2
L~I_ I-
D,~,~oo~,~,~a
L~ - - - ~ -I
Ab(x )
X
Fig. !.
Duct with a uniform blockage.
I open
Reconstruction of blockage in a duct
231
o f the sound. In the limiting case of fl = 0, eqn (1) admits the expected elementary form for the empty straight duct, i.e. cos ( k r L ) = 0 Assuming the blockage is small, i.e. fl << 1, the eigenfrequency solutions of eqn (1) can be expressed as v. = v ° + b l f l +
e(fl2),
n = 1, 2, 3 ....
(2)
0
where v, and v, are the eigenfrequencies of the duct with and without the blockage, and v,o are also called the baseline eigenfrequencies, e(fl 2) represents the higher order terms. Inserting eqn (2) into eqn (1), the first-order eigenfrequency shifts due to a uniform blockage in the closed-open duct are obtained: 0 =
Av(n) = v, - v,
Cfl
47r[ cos ((2n - 1)Tt~b)sin ((n -- 1)rc2b), n = 1,2,3 ....
(3)
Equation (3) shows that the frequency shifts depend on the three geometric parameters, namely fl, ~b and 2b- For a given blockage, the eigenfrequency shifts are the function of the mode order n. It has been shown that the eigenfrequency shifts vary with mode order quasi-periodically. 3 For different mode order, n, the eigenfrequency shifts can be positive or negative. Taking summation of both sides of eqn (3), gives N
N
N
2 X°c 2 v. =
n=l
~. + ~
n=l
cos ((2n - 1)rC~b)sin ((n -- ½)rC2b)
(4)
n=l
The second term of the right-hand side of eqn (4) is the summation of the eigenfrequency shifts which is much smaller than the first term when the number o f mode, N, is large. Neglecting the small term of eqn (4), the following approximation is obtained: N
N
2S
o
v. =
n=l
~.
(5)
n=l
Equation (5) indicates that the sum of eigenfrequencies of blocked duct equals the sum of the eigenfrequencies o f unblocked under the same medium. The baseline eigenfrequencies, v°, of the uniform duct follow a certain rule. For the closed-open end conditions, this is Viio ~
(2n
1)v°,
n = 1 , 2 , 3 ....
(6)
232
Wu Qunli
where v° is the eigenfrequency of the first mode. Inserting eqn (6) into eqn (5), gives
'Z N
, 0K--( 2 n ~'n
1)~-
%,
n = 1,2,3,
(7)
"''
n-|
Equation (7) shows that the baseline eigenfrequencies of the unblocked duct can be estimated from the eigenfrequencies of the blocked duct. Thus, the eigenfrequency shifts become N 4
~
n-I
F r o m the result of the previous study, 2 the blockage function, [3(x), can be reconstructed from N
2 1-exp
-
~6c°sl-
/~(x) =
l~.
\
l~(x) > 0
. =,
-
0
(9)
fl(x) < 0
where it ° =(2~zv°/c) 2 is the baseline eigenvalues of the unblocked duct, 7. = / t . - / 1 ° is the eigenvalue shifts due to the blockage a n d / t . = (2~zv./c)2. Because the baseline eigenvalues can be estimated from eqn (7), it can be seen that the blockage location and its size can be reconstructed from a single spectrum which contains the eigenfrequencies of the blocked duct, %. The temperature changes will change the speed of the sound in the duct and cause the eigenfrequency shifts. For the reconstruction of the blockage, it is important to obtain the eigenfrequencies of blocked and unblocked duct under the same temperature, otherwise the additional eigenfrequency shifts will be added to the eigenfrequency shifts due to the blockage. Let v°(tl) be the baseline eigenfrequencies measured at temperature t I and v.(t2) be the eigenfrequencies of the blocked duct measured at temperature t 2. The eigenfrequency shifts due to blockage and temperature are ~'.(t2) - v~'(t,) = (v.(t2) - ,'~'i~2)) - v°(t2) A~
(10)
C2
where Ac = cl - c2, cl and c2 are the speed of the sound under temperature t 1 and t2. Take summation of the both sides ofeqn (10) and make usage ofeqns (5) and (6), then, N
v°(t2)=(2n
- 1)~-
v,(t2)
(11)
Reconstruction of blockage in a duct
233
It can be seen that the eigenfrequency shifts induced by the temperature variation can be filtered when eqn (7) is used for estimation of the baseline eigenfrequencies. The percentage error for using eqn (7) can be estimated. The percentage error E(N) caused by neglecting the second term in eqn (4) is 1
cfl
N
N 2 4rcL .=~1 cos ((2n -- 1)n~b) sin ((n -- 0"5)rcAb) fl e(N) = vO < --~ where v° =
c/(4L).
(12)
For fl = 0"1 and N = 20, e(N) < 0-16%.
3 EXPERIMENTAL RESULTS The experimental studies were conducted in a 104-mm-diameter, 2000-mmlong duct. The duct cutoff frequency was about 2000 Hz. One end of the duct was fixed by a 13-mm-thick plastic glass cap with a 25-mm-diameter driver in the centre. The other end was closed with a 5-mm plastic cap for measuring the eigenfrequencies f , and f~,, and left open for measuring the o The experimental set-up was the same as shown eigenfrequencies v, and v,. in Ref. 2. The eigenfrequencies of the duct were measured by slowly increasing the frequency of the signal and recording the frequencies at which 30 20 I v
10 t--
t--
0
O"
E
-I0
O') Ld
-20
-30
0
J
i
i
5
10
15
Mode order
Fig. 2.
20
n
Comparison of the measured and estimated eigenfrequency shifts (2b = 0" 1, fl = 0'46, and ~b = 0-85). (O) Measured and (O) estimated from eqn (8).
Wu Qunli
234
the sound level peaked at the driver's end. The blockages were made from wood with uniform circular or rectangular cross-section. 0 From the measured eigenfrequencies, v,, the baseline eigenfrequencies, v,, were calculated from eqn (7). The eigenvalue shifts, 7,, were then obtained. Based on the eigenvalue shifts, 7,, the blockage function, /3(x), were reconstructed from eqn (9). The results were compared with the results obtained from four spectral data, I.e../,,/,, " . ,9 v, and v°. The details of the fourspectrum method were given in Ref. 2. 20
N
3Z
10
C
0
\
13-
03
;7,
-~0
-20
I
I 10
5 Mode
Fig. 3.
order
I 15
20
n
Comparison of the measured and estimated eigenfrequency shifts (2h=0"25, fl=0.23, and fh =0-355). (O) Measured and (O) estimated from eqn (8).
Figures 2 and 3 show the comparison of the measured eigenfrequency shifts, Av(n), with the eigenfrequency shifts estimated from eqn (8). The blockages used in the measurements are ":-b= 0"1,/3 = 0-46, and ~b = 0"85 for Fig. 2, and 2b = 0"25, [;~= 0-23, and ~b = 0"355 for Fig. 3. It can be seen that the agreements between the measured and estimated eigenfrequency shifts are good for N = 20. Figure 4 shows the comparison of the reconstructed blockage from single spectrum, v,, with the blockage reconstructed from four spectra,j~,,.~, v, and v°. The actual dimensions of the blockage are 2b = 0.1,/3 = 0.25, ~b =0"15. Figures 5 and 6 show the results for blockages of 2b = 0"04, /3 = 0"1 and ~ = 0"76 and blockage of )~h= 0"25,/3 = 0'23, and ~b = 0"355, respectively. It can be seen that the agreements between the single spectrum method and the four-spectrum method are excellent.
235
Reconstruction of blockage in a duct
0.5
0.4
0.,,'3
X
0.1 ~......-..-..~..~
0.0 -0.1 0.0
._~_
I
I
I
I
0.2
0.4
0.6
0.8
_
• .-~-~
1.0
x/l_
Fig. 4. Blockage reconstructed from single and four spectra (2h=0"l, /:t=0.25, and ~h = 0.15). (. . . . . ) Four spectra; ( - - ) single spectrum; and (...... ) actual blockage.
0.3
0.2
X
0.1
0.0
-0.1 0.0
--
i 0.2
i 0.4
i 0.6
i 0.8
1.0
x/L
Fig. 5. Blockage reconstructed form single and four spectra (2b=0"04, fl=0.1, and ~h = 0.76). (. . . . . ) Four spectra; ( - - ) single spectrum; and (...... ) actual blockage.
236
Wu Qunli 0.4
0.3
0.2 x
0.1
0.0
i
-0.1
0.0
0.2
I
I
I
I
0.4
0.6
0.8
1.0
×/
Fig. 6. Blockage reconstructed form single and four spectra ()-b=0.25, fl=0.23, and ~b = 0"355). ( . . . . ) Four spectra: (-i single spectrum: and (...... ) actual blockage.
4 CONCLUSIONS T h e eigenfrequency shift properties o f a duct with a blockage are studied in this paper. It is found that the sums o f the eigenfrequencies o f the blocked and unblocked duct are equal when the n u m b e r o f m o d e s in the s u m m a t i o n is large. The eigenfrequencies for unblocked duct can be estimated from the eigenfrequencies o f blocked duct. T h e r e f o r e , the blockage l o c a t i o n can be r e c o n s t r u c t e d f r o m the single s p e c t r u m o f the b l o c k e d duct. T h e e x p e r i m e n t a l results show that the new m e t h o d can be as a c c u r a t e as the m e t h o d using f o u r spectra.
REFERENCES 1. Domis, D. A., Frequency dependence of acoustic resonances on blockage position in a fast reactor subassembly wrapper. J. Sound Vibration, 72 (1979) 443- 50. 2. Wu, Q. & Fricke, F., Determination of blockage locations and cross-sectional area in a duct by eigenfrequency shifts. J. Acoustical Soc. America, 87 (1990) 67-75. 3. Wu, Q. & Fricke, F., Determination of the size of an object and its location in a rectangular cavity by eigenfrequency shifts: first order approximation. J. Sound Vibration, 144 (1991) 131-47. 4. E1-Raheb, E. & Wagner, P., Acoustic propagation in rigid ducts with blockage. J. Acoustical Soc. America, 72 (1982) 1046-55.
Reconstruction of Blockage in a Duct from Single Spectrum Wu Qunli School of Mechanical and Production Engineering, Nanyang Technological University, Singapore 2263, Singapore (Received 28 October 1992; revised version received 8 February 1993; accepted l0 February 1993)
A BS TRA C T In some industrial practices it is important to detect blockages developing in the ducts and pipeline systems, especially in the nuclear power industry where safeO' is vital. In this paper, the properties of eigenfrequeney shifts o f a duct with blockages are studied. It is .found that the sum of the eigenfrequency shifts over a number of modes is close to zero for the duct with the blockage. A new method is proposed to reconstruct the blockage location and its size in the duct. The method requires only one spectrum o f the blocked duct under asymmetrical boundao' condition. The experimental results show that the new method can be as accurate as the method which requires four spectra data.
1 INTRODUCTION In some industrial practices it is important to detect blockages developing in ducts and pipeline systems, especially in the nuclear power industries where the safety is vital. Domis 1 applied the eigenfrequency shift idea to detect blockages in the sodium-cooled fast reactor subassemblies. He notices that the eigenfrequencies will be shifted from the original values when a blockage develops in the duct. He used eigenfrequency shifts as an early warning of the blockages. Based on an inverse problem approach, Wu and Fricke z concluded that blockages in a duct can be determined from two sets o f the eigenfrequency shifts under two b o u n d a r y conditions. This method is 229 Applied Acoustics 0003-682X/94/$07.00 © 1994 ElsevierScience Limited, England. Printed in Great Britain
Wu Qunli
230
further extended for the determination of the blockages in two-dimensional cavities? Using the eigenfrequency shifts to reconstruct the blockages in the duct requires at least two sets of spectral data, one contains the eigenfrequencies of the duct with the blockages and the other contains the baseline eigenfrequencies of the duct. From these two sets of the eigenfrequencies, the eigenfrequency shifts can be calculated. For some practical applications the baseline eigenfrequencies are difficult to obtain and the temperature changes may also cause the eigenfrequency shifts. In this paper, a method for reconstructing the blockages in the duct from a single spectrum is investigated. The method is able to compensate for the temperature effects on the eigenfrequencies.
2 THEORETICAL ANALYSIS Consider a oneidimensional model of a uniform duct with a blockage, as shown in Fig. 1. Let A o and Ab(x) be the cross-sectional areas of the duct and blockage, respectively. The blockage partitions the duct into three segments with lengths L1, L 2 and L 3, where L 2 is the blockage length. The following non-dimensional quantities are defined: )~h = L z / L ,
fl(x) = Ab(x)/A o,
Cb = (LI + L z / 2 ) / L
L is the total length of the duct. fl(x) is the blockage function. When L 1 < _ x < L 2, fl(x)=fl, and fl(x)=0 when x<_L~ and x > L 2 . By applying one-dimensional wave equation and the boundary conditions, the eigenvalue equation for blocked duct closed at x = 0 and open at x = L is derived 4 sin (kfLt){(1 -- fl) c o s ( k f L 2 ) sin (kfL3) + sin (kfL2) c o s (kfL3) } + (1 - fl) cos (k/L0{(1 - fl) sin (k/L2) sin (kfL3) - c o s ( k f L 2 ) c o s ( k f L 2 ) I --- 0
(1)
where k f = 2rcv/c is the wavenumber and v is the frequency and c is the speed I~ I~
L
closed I
~1 -I
i
Ao
L2
L~I_ I-
D,~,~oo~,~,~a
L~ - - - ~ -I
Ab(x )
X
Fig. !.
Duct with a uniform blockage.
I open
Reconstruction of blockage in a duct
231
o f the sound. In the limiting case of fl = 0, eqn (1) admits the expected elementary form for the empty straight duct, i.e. cos ( k r L ) = 0 Assuming the blockage is small, i.e. fl << 1, the eigenfrequency solutions of eqn (1) can be expressed as v. = v ° + b l f l +
e(fl2),
n = 1, 2, 3 ....
(2)
0
where v, and v, are the eigenfrequencies of the duct with and without the blockage, and v,o are also called the baseline eigenfrequencies, e(fl 2) represents the higher order terms. Inserting eqn (2) into eqn (1), the first-order eigenfrequency shifts due to a uniform blockage in the closed-open duct are obtained: 0 =
Av(n) = v, - v,
Cfl
47r[ cos ((2n - 1)Tt~b)sin ((n -- 1)rc2b), n = 1,2,3 ....
(3)
Equation (3) shows that the frequency shifts depend on the three geometric parameters, namely fl, ~b and 2b- For a given blockage, the eigenfrequency shifts are the function of the mode order n. It has been shown that the eigenfrequency shifts vary with mode order quasi-periodically. 3 For different mode order, n, the eigenfrequency shifts can be positive or negative. Taking summation of both sides of eqn (3), gives N
N
N
2 X°c 2 v. =
n=l
~. + ~
n=l
cos ((2n - 1)rC~b)sin ((n -- ½)rC2b)
(4)
n=l
The second term of the right-hand side of eqn (4) is the summation of the eigenfrequency shifts which is much smaller than the first term when the number o f mode, N, is large. Neglecting the small term of eqn (4), the following approximation is obtained: N
N
2S
o
v. =
n=l
~.
(5)
n=l
Equation (5) indicates that the sum of eigenfrequencies of blocked duct equals the sum of the eigenfrequencies o f unblocked under the same medium. The baseline eigenfrequencies, v°, of the uniform duct follow a certain rule. For the closed-open end conditions, this is Viio ~
(2n
1)v°,
n = 1 , 2 , 3 ....
(6)
232
Wu Qunli
where v° is the eigenfrequency of the first mode. Inserting eqn (6) into eqn (5), gives
'Z N
, 0K--( 2 n ~'n
1)~-
%,
n = 1,2,3,
(7)
"''
n-|
Equation (7) shows that the baseline eigenfrequencies of the unblocked duct can be estimated from the eigenfrequencies of the blocked duct. Thus, the eigenfrequency shifts become N 4
~
n-I
F r o m the result of the previous study, 2 the blockage function, [3(x), can be reconstructed from N
2 1-exp
-
~6c°sl-
/~(x) =
l~.
\
l~(x) > 0
. =,
-
0
(9)
fl(x) < 0
where it ° =(2~zv°/c) 2 is the baseline eigenvalues of the unblocked duct, 7. = / t . - / 1 ° is the eigenvalue shifts due to the blockage a n d / t . = (2~zv./c)2. Because the baseline eigenvalues can be estimated from eqn (7), it can be seen that the blockage location and its size can be reconstructed from a single spectrum which contains the eigenfrequencies of the blocked duct, %. The temperature changes will change the speed of the sound in the duct and cause the eigenfrequency shifts. For the reconstruction of the blockage, it is important to obtain the eigenfrequencies of blocked and unblocked duct under the same temperature, otherwise the additional eigenfrequency shifts will be added to the eigenfrequency shifts due to the blockage. Let v°(tl) be the baseline eigenfrequencies measured at temperature t I and v.(t2) be the eigenfrequencies of the blocked duct measured at temperature t 2. The eigenfrequency shifts due to blockage and temperature are ~'.(t2) - v~'(t,) = (v.(t2) - ,'~'i~2)) - v°(t2) A~
(10)
C2
where Ac = cl - c2, cl and c2 are the speed of the sound under temperature t 1 and t2. Take summation of the both sides ofeqn (10) and make usage ofeqns (5) and (6), then, N
v°(t2)=(2n
- 1)~-
v,(t2)
(11)
Reconstruction of blockage in a duct
233
It can be seen that the eigenfrequency shifts induced by the temperature variation can be filtered when eqn (7) is used for estimation of the baseline eigenfrequencies. The percentage error for using eqn (7) can be estimated. The percentage error E(N) caused by neglecting the second term in eqn (4) is 1
cfl
N
N 2 4rcL .=~1 cos ((2n -- 1)n~b) sin ((n -- 0"5)rcAb) fl e(N) = vO < --~ where v° =
c/(4L).
(12)
For fl = 0"1 and N = 20, e(N) < 0-16%.
3 EXPERIMENTAL RESULTS The experimental studies were conducted in a 104-mm-diameter, 2000-mmlong duct. The duct cutoff frequency was about 2000 Hz. One end of the duct was fixed by a 13-mm-thick plastic glass cap with a 25-mm-diameter driver in the centre. The other end was closed with a 5-mm plastic cap for measuring the eigenfrequencies f , and f~,, and left open for measuring the o The experimental set-up was the same as shown eigenfrequencies v, and v,. in Ref. 2. The eigenfrequencies of the duct were measured by slowly increasing the frequency of the signal and recording the frequencies at which 30 20 I v
10 t--
t--
0
O"
E
-I0
O') Ld
-20
-30
0
J
i
i
5
10
15
Mode order
Fig. 2.
20
n
Comparison of the measured and estimated eigenfrequency shifts (2b = 0" 1, fl = 0'46, and ~b = 0-85). (O) Measured and (O) estimated from eqn (8).
Wu Qunli
234
the sound level peaked at the driver's end. The blockages were made from wood with uniform circular or rectangular cross-section. 0 From the measured eigenfrequencies, v,, the baseline eigenfrequencies, v,, were calculated from eqn (7). The eigenvalue shifts, 7,, were then obtained. Based on the eigenvalue shifts, 7,, the blockage function, /3(x), were reconstructed from eqn (9). The results were compared with the results obtained from four spectral data, I.e../,,/,, " . ,9 v, and v°. The details of the fourspectrum method were given in Ref. 2. 20
N
3Z
10
C
0
\
13-
03
;7,
-~0
-20
I
I 10
5 Mode
Fig. 3.
order
I 15
20
n
Comparison of the measured and estimated eigenfrequency shifts (2h=0"25, fl=0.23, and fh =0-355). (O) Measured and (O) estimated from eqn (8).
Figures 2 and 3 show the comparison of the measured eigenfrequency shifts, Av(n), with the eigenfrequency shifts estimated from eqn (8). The blockages used in the measurements are ":-b= 0"1,/3 = 0-46, and ~b = 0"85 for Fig. 2, and 2b = 0"25, [;~= 0-23, and ~b = 0"355 for Fig. 3. It can be seen that the agreements between the measured and estimated eigenfrequency shifts are good for N = 20. Figure 4 shows the comparison of the reconstructed blockage from single spectrum, v,, with the blockage reconstructed from four spectra,j~,,.~, v, and v°. The actual dimensions of the blockage are 2b = 0.1,/3 = 0.25, ~b =0"15. Figures 5 and 6 show the results for blockages of 2b = 0"04, /3 = 0"1 and ~ = 0"76 and blockage of )~h= 0"25,/3 = 0'23, and ~b = 0"355, respectively. It can be seen that the agreements between the single spectrum method and the four-spectrum method are excellent.
235
Reconstruction of blockage in a duct
0.5
0.4
0.,,'3
X
0.1 ~......-..-..~..~
0.0 -0.1 0.0
._~_
I
I
I
I
0.2
0.4
0.6
0.8
_
• .-~-~
1.0
x/l_
Fig. 4. Blockage reconstructed from single and four spectra (2h=0"l, /:t=0.25, and ~h = 0.15). (. . . . . ) Four spectra; ( - - ) single spectrum; and (...... ) actual blockage.
0.3
0.2
X
0.1
0.0
-0.1 0.0
--
i 0.2
i 0.4
i 0.6
i 0.8
1.0
x/L
Fig. 5. Blockage reconstructed form single and four spectra (2b=0"04, fl=0.1, and ~h = 0.76). (. . . . . ) Four spectra; ( - - ) single spectrum; and (...... ) actual blockage.
236
Wu Qunli 0.4
0.3
0.2 x
0.1
0.0
i
-0.1
0.0
0.2
I
I
I
I
0.4
0.6
0.8
1.0
×/
Fig. 6. Blockage reconstructed form single and four spectra ()-b=0.25, fl=0.23, and ~b = 0"355). ( . . . . ) Four spectra: (-i single spectrum: and (...... ) actual blockage.
4 CONCLUSIONS T h e eigenfrequency shift properties o f a duct with a blockage are studied in this paper. It is found that the sums o f the eigenfrequencies o f the blocked and unblocked duct are equal when the n u m b e r o f m o d e s in the s u m m a t i o n is large. The eigenfrequencies for unblocked duct can be estimated from the eigenfrequencies o f blocked duct. T h e r e f o r e , the blockage l o c a t i o n can be r e c o n s t r u c t e d f r o m the single s p e c t r u m o f the b l o c k e d duct. T h e e x p e r i m e n t a l results show that the new m e t h o d can be as a c c u r a t e as the m e t h o d using f o u r spectra.
REFERENCES 1. Domis, D. A., Frequency dependence of acoustic resonances on blockage position in a fast reactor subassembly wrapper. J. Sound Vibration, 72 (1979) 443- 50. 2. Wu, Q. & Fricke, F., Determination of blockage locations and cross-sectional area in a duct by eigenfrequency shifts. J. Acoustical Soc. America, 87 (1990) 67-75. 3. Wu, Q. & Fricke, F., Determination of the size of an object and its location in a rectangular cavity by eigenfrequency shifts: first order approximation. J. Sound Vibration, 144 (1991) 131-47. 4. E1-Raheb, E. & Wagner, P., Acoustic propagation in rigid ducts with blockage. J. Acoustical Soc. America, 72 (1982) 1046-55.