- Email: [email protected]
Least square error method to estimate individual power of noise sources under simultaneous operating conditions
ARTICLE IN PRESS
International Journal of Industrial Ergonomics 35 (2005) 755–760 www.elsevier.com/locate/ergon
Least square error method to estimate individual power of noise sources under simultaneous operating conditions Shih-Yi Lua,, Ying-Jong Hongb a
Institute of Occupational Safety and Health, Council of Labor Affairs, No. 99 Lane 407, Hengke Rd., Shijr City, Taipei 221, Taiwan, ROC b Department of Safety, Health, and Environmental Engineering, National United University, Taiwan, ROC Received 2 February 2004; accepted 12 January 2005 Available online 21 April 2005
Abstract People have been aware of the connection between noise and hearing loss for centuries. Hearing loss is not the only adverse effect of occupational noise, but also a number of non-auditory effects may endanger worker’s safety. This paper proposes an approach enable engineers to point out quantitatively the noisiest source for modification, while multiple machines are operating simultaneously. The model with the point source and spherical radiation in a free field was adopted to formulate the problem. The proposed method requires input data that includes the coordinates (x, y) of the noise sources and the locations where the sound pressure level is measured, and the measured sound pressure levels. Then, the method of least squares was applying to estimate system parameters. Finally, a set of solutions that represents the sound power of noise sources can be found and these solutions have minimum error in least squares sense. With help of sound powers obtained, engineers would be able to evaluate noise distribution whenever re-layout workplace in future. Relevance to industry: Industry workers suffer psychological and physical stress as well as hearing loss due to industrial noise. Although noise source control can be profound process sometimes, it would be the most effective way to eliminate noise level on source. Therefore, identifying dominant source of noise shall be the first step to overcome the noise problem in industry. r 2005 Elsevier B.V. All rights reserved. Keywords: Noise assessment; Noise control; Sources contribution; Noise analysis
1. Introduction People have been aware of the connection between noise and hearing loss for centuries.
Hearing loss is not the only adverse effect of occupational noise, but also a number of nonauditory effects may endanger worker’s safety (Berger et al., 1997; East Kodak Company, 1983;
Corresponding author.
E-mail address: [email protected] (S.-Y. Lu). 0169-8141/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ergon.2005.01.011
ARTICLE IN PRESS 756
S.-Y. Lu, Y.-J. Hong / International Journal of Industrial Ergonomics 35 (2005) 755–760
Konz, 1990). Despite general recognition of the fact that noise causes health problems, workers felt that it was a necessary price to pay in an industrial job and managers felt that the noise was either too difficult or too costly to control. Generally, the handling of workplace noise problem consists of two phases: (1) noise evaluation, and (2) noise control. The first step toward solving noise problem is to conduct noise survey. This necessitates measuring the noise level and securing complete information on employee exposure time in the noisy environment and in any other environment to which the employee might be exposed during the working day. Once appropriate, accurate sound level measurements are made, measured value should be compared with the noise regulations or sound level criterion correct for the situation. After determining that reduction of the employee’s noise exposure should be investigated, the manager must then consider various measures for controlling exposure, such as engineering controls, administrative controls, and hearing protection devices. The preferred method of controlling excessive exposure to noise is engineering controls wherever feasible. It may not only eliminate the requirements of hearing protection, audiometric testing, and limitation of exposure time, but also may improve speech communication and reduce annoyance. When more than one noise source is involved in an area, it is essential to reduce the noisiest source if effective reduction is to be achieved. The amount of sound radiated by a source is determined by its sound power. The sound power is independent of distance or environment, and the main use of sound power is for the noise rating of machines. The sound power can be useful information to guide the design of new factory and the layout of existing one. Many potential noise problems can be eliminated in the design stage. If the sound power of noise sources becomes available, the sound pressure level at any locations in the field can be calculated, as well. By estimating the sound power of noise sources, the source order ranked list can be generated. From list the most effective candidates for noise control can be assessed. At this point, we are readily to perform some evaluation to determine which one the noisiest
machine really is, as a preliminary step in performing noise control. A truly simple, but most illuminating technique is to turn individual machine on and off and to measure and observe the resulting sound level at the position of interest. Such measurements and observations may reveal the one or two machines that are exceptionally noisy. However, we just discussed identification of the source of a problem noise in situation where it is possible to turn production equipment on and off. Often, the noise control engineer is faced with the task of making the necessary identification without the luxury of equipment being operated to his convenience. This paper presents a method capable of estimating the sound power of noise sources by using sound pressure meter, while there are multiple sound sources being operated simultaneously in workplace. A mathematical model was developed in this study, and the least squares method was adopting to solve the problem. The method requires input data that includes the coordinates (x, y) of the noise sources and the locations where the sound pressure level is measured, and the measured sound pressure levels. Then, a set of source powers can be found out, and this solution has minimum error in least squares sense.
2. Formulation of the problem The sound power level Lw (dB re 1 pW) radiated by a point source is related to the sound pressure level Lp (dB re 20 mPa) at a distance r(m) by the following equation (Jenson et al., 1978; Beranek, 1971; Wilson, 1989) Q Lp ¼ Lw þ 10 log , (1) 4pr2 where Q is the directivity factor of the source. Eq. (1) would be known as spherical radiation in a free field, a relationship fundamental to source diagnosis. This relationship is valid for airborne sound at most temperature and pressure conditions. A schematic diagram with m noise sources and n measurement locations is shown in Fig. 1. Each noise source and measurement location is donated by the letter si and mj, respectively, where
ARTICLE IN PRESS S.-Y. Lu, Y.-J. Hong / International Journal of Industrial Ergonomics 35 (2005) 755–760
i ¼ 1; 2; . . . ; m and j ¼ 1; 2; . . . ; n. The distance between the source si and the measurement location mj is rij. The component Lpij of each source si contributing to the measurement location mj can be calculated with help of Eq. (2): Qi Lpij ¼ Lwi þ 10 log , (2) 4prij where Lwi is sound power of the source si. For uncorrelated noise sources, the total meansquare sound pressure is simply the sum of component source mean-square pressures. Detailed discussion of the theory can be found in Wilson, 1989 and Beranek, 1971. Let the contribution to the sound at jth location be Pr msij, corresponding to the source si contributing to the measurement location mj, and the total sound pressure level Lpj at measurement point j is given by " !, # m X 2 2 Prmsij Lpj ¼ 10 log Pref i¼1
¼ 10 log
m X
!
100:1Lpij ;
j ¼ 1; 2; . . . ; n,
ð3Þ
i¼1
where Pref is reference pressure 20 106 Pa.
757
Substituting Eq. (2) into Eq. (3), we rewrite Eq. (3) as follows: m X Qi 0:1Lpj 0:1Lwi ¼ 10 ; j ¼ 1; 2; . . . ; n: 10 4prij i¼1 (4) Qi ¼ aij , 100:1Lwi ¼ xi , and Eq. Let 100:1Lpj ¼ bj , 4pr ij (4) can be formulated into a set of linear equations. Rewrite in vector notation: 3 0 1 2 a11 a12 a1m b1 7 B C 6 B b2 C 6 a21 a22 a2m 7 7 B C 6 C 7 6 B¼B .. 7, .. B .. C; A ¼ 6 .. B . C 6 . . 7 . 5 @ A 4 an1 an2 anm bn 0 1 x1 B C B x2 C B C C X ¼B B .. C. B . C @ A xm
Then B ¼ AX .
(5)
If we assume that the measured sound pressure levels at measurement locations are Lp1 ; Lp2 ; Lp3 ; . . . ; Lpn , and the natural approach to solve inverse problem is to search for the sound powers Lwi, i ¼ 1; 2; . . . ; m, by minimizing in a square-sum sense.
3. Solving linear systems with method of least squares
Fig. 1. Schematic of sound field with m noise sources and n measurement locations.
Many methods (Draper and Smith, 1981; Kahaner et al., 1989) have been proposed for solving system of linear equations, Eq. (5). The model developed above allows an estimation of m unknowns from n equations. If there are equations fewer than m independent unknowns, the model is under-determined, and we cannot determine the parameters uniquely. The model may contain equal number of unknowns and equations, while
ARTICLE IN PRESS S.-Y. Lu, Y.-J. Hong / International Journal of Industrial Ergonomics 35 (2005) 755–760
758
the number of the measurement locations is equal to the number of noise sources. Therefore, three basic techniques that are Cramer’s rule, inverse matrix method, and Gauss elimination method can be used to solve the problem. The Gauss elimination method is, by far, the most widely used, since it can be applied to all systems of linear equations. However, for certain (usually small) system of linear equations the other two techniques may be used to advantage. Nevertheless, it is unavoidable that there are always errors in the measurements, and the measured sound level at measurement locations may be the parameter that the error is mostly like to occur. To achieve better estimation of the power of the noise sources, we may pick measurement locations more than the number of noise sources. Therefore, the model becomes an over-determined system (i.e., redundancy), which allows us to use statistical methods to reduce uncertainty in the derived parameters. There is a procedure that can provide a best solution under these circumstances. The ‘‘best’’ can be specified in a least squares sense that is similar to linear regression. The least squares problem can be interpreted graphically as minimizing the vertical distance from data points to the model. Underlying this idea is the assumption that all the errors in the approximation correspond to errors in the observations bj. Choose estimates x1 ; x2 ; . . . ; xm , to minimize ¼
n X j¼1
bj
m X
!2 aij xi
:
i¼1
Differentiating with respect to x1 ; x2 ; . . . ; xm , leads to the set of m simultaneous linear equations in m unknowns. These equations are known as the normal equations. The normal equations may be written in vector notation as AT B ¼ AT AX . Usually assume ATA invertible, then X ¼ ðAT AÞ1 AT B. The solution of xi can be obtained, the sound powers Lwi shall be found, accordingly.
4. Estimation of noise source power The procedures for estimation of the source power can be described as follows: (1) Determining the coordinates of the locations of the noise source Since the computation requires an assumption of a point source, the noise source si, locations must be represented by a point on x–y plane. By selecting any one point on plane as reference origin (picking source or measurement location is recommended), each source can be expressed as a pair of x and y coordinates which are all measured from the reference point. (2) Determining directivity of the noise sources Radiation of noise energy from source is dependent on direction. For an ideal (nondirectional) point source in full space, the directivity factor is Q ¼ 1. If an ideal point source is located on an acoustically hard surface, then Q ¼ 2 for the half-space above surface. (3) Determining coordinates of the measurement locations and the sound pressure levels at these locations The coordinate (xj,yj) with reference to origin and measured sound pressure level Lpj at jth measurement location, j ¼ 1; 2; . . . ; n, need to be determined, then, the distances between the sources and the measurement locations can be calculated in accordingly. (4) Computing sound contribution and combined total sound at measurement locations With help of above three steps information as inputs, using Eq. (2) the sound contribution of each individual source to the measurement location can be obtained. Using Eq. (3), sum up those individual contributions, and then the computation can obtain the sound pressure level Lpj at each measurement location. (5) Computing the sound power of noise sources through statistical software It is similar to the linear regression problems for finding the coefficients of model equations, and these coefficients represent the power of noise sources in our model.
ARTICLE IN PRESS S.-Y. Lu, Y.-J. Hong / International Journal of Industrial Ergonomics 35 (2005) 755–760
5. Example To demonstrate the application for estimating the sound power of noise sources, please refer to Fig. 2, there are two machines on factory floor, and the sound pressure levels are measured at four measurement locations randomly distributed on the floor. The coordinates (x,y) of the two machines are (0,0) and (2,1), and the four measurement locations are (1,0), (1,1), (0,2), (2,3). In addition, the measured sound pressure levels are 93.6, 87.9, 85.0 and 92.1 dB at the abovementioned four measurement locations. The distances between machines (noise sources) and measurement locations may be given by r11 r12 r13 r14 r21
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð0 1Þ2 þ ð0 0Þ2 ¼ 1 m, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð0 0Þ2 þ ð0 2Þ2 ¼ 2 m, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi ¼ ð0 2Þ2 þ ð0 3Þ2 ¼ 13 m, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ¼ ð0 1Þ2 þ ð0 1Þ2 ¼ 2 m, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ¼ ð2 1Þ2 þ ð1 0Þ2 ¼ 2 m,
759
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ð2 0Þ2 þ ð1 2Þ2 ¼ 5 m, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð2 2Þ2 þ ð1 3Þ2 ¼ 2 m, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð2 1Þ2 þ ð1 1Þ2 ¼ 1 m.
r22 ¼ r23 r24
The contribution of each source to each measurement location can be calculated as follows:
Lp11 ¼ Lw1 þ 10 Lp12 ¼ Lw1 þ 10 Lp13 ¼ Lw1 þ 10 Lp14 ¼ Lw1 þ 10 Lp21 ¼ Lw2 þ 10 Lp22 ¼ Lw2 þ 10 Lp23 ¼ Lw2 þ 10 Lp24 ¼ Lw2 þ 10
2 log ¼ Lw1 8:0 dB, 4p 1 2 log ¼ Lw1 14:0 dB, 4p 4 2 log ¼ Lw1 19:1 dB, 4p 13 2 log ¼ Lw1 11:0 dB, 4p 2 2 log ¼ Lw2 11:0 dB, 4p 2 2 log ¼ Lw2 15:0 dB, 4p 5 2 log ¼ Lw2 14:0 dB, 4p 4 2 log ¼ Lw2 8:0 dB: 4p 1
Therefore, summing up all contributions of each source, the combined sound pressure levels at measurement location 1, 2, 3, and 4 can be denoted by Lp1, Lp2, Lp3, and Lp4, respectively. Where Lp1 ¼ 10 logð10ðLw1 8Þ=10 þ 10ðLw2 11Þ=10 Þ, Lp2 ¼ 10 logð10ðLw1 14Þ=10 þ 10ðLw2 15Þ=10 Þ, Lp3 ¼ 10 logð10ðLw1 19:1Þ=10 þ 10ðLw2 14Þ=10 Þ, Lp4 ¼ 10 logð10ðLw1 11Þ=10 þ 10ðLw2 8Þ=10 Þ.
Fig. 2. A factory floor with two noise sources and four measurement locations.
By using the least squares criterion in which only the errors in Lpi are considered statistical software can be adopted to solve this problem. Then, we may obtain the sound powers Lw1 and Lw2, standard errors and 95% confidence intervals, as
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follows: Lw1 ¼ 100:9 dB, Lw2 ¼ 96:0 dB, Standard error of Lw1 ¼ 78:2 dB Standard error of Lw2 ¼ 78:6 dB
reverberant sound field. The reverberant field alters the sound wave characteristics from those described above for free field. Although the procedure works very well in ideal cases (point source and free field), in our view this does not preclude the application of the method to practical cases.
95% confidence interval for Lw1 ¼ ½100:6 101:2 dB 95% confidence interval for Lw2 ¼ ½94:9 96:9 dB. References 6. Discussion and conclusion The use of the least squared error to identify dominant source from sound pressure measurements on sound field has been presented. The proposed approach provides an objective method to help engineers for finding the noisiest machine in factory, without using complicated instrument or subjective justification. However, it is necessary to further validate the idea for relatively low absorption room. Many industrial noise problems are complicated by the fact that the noise is confined in a room. Reflections from the walls, floor, ceiling, and equipment in the room create a
Beranek, L.L., 1971. Noise and Vibration Control. McGrawHill, New York. Berger, E.H., Ward, W.D., Morrill, J.C., Royster, L.H., 1997. Noise & Hearing Conservation Manual, fourth ed. American Industrial Hygiene Association, Fairfax, Virginia. Draper, N.R., Smith, H., 1981. Applied Regression Analysis. Wiley, New York. East Kodak Company, 1983. Ergonomic Design for People at Work. Van Nostrand Reinhold, New York. Jenson, P., Jokel, C.R., Miller, L.N., 1978. Industrial Noise Control Manual. U.S. Government Printing Office, Washington, DC. Kahaner, D., Moler, C., Nash, S., 1989. Numerical Method and Software. Prentice-Hall, Englewood Cliffs, NJ. Konz, S.A., 1990. Work Design: Industrial Ergonomics, third ed. Publishing Horizons, Scottsdale, Arizona. Wilson, C.E., 1989. Noise Control. Harper & Row, New York.
International Journal of Industrial Ergonomics 35 (2005) 755–760 www.elsevier.com/locate/ergon
Least square error method to estimate individual power of noise sources under simultaneous operating conditions Shih-Yi Lua,, Ying-Jong Hongb a
Institute of Occupational Safety and Health, Council of Labor Affairs, No. 99 Lane 407, Hengke Rd., Shijr City, Taipei 221, Taiwan, ROC b Department of Safety, Health, and Environmental Engineering, National United University, Taiwan, ROC Received 2 February 2004; accepted 12 January 2005 Available online 21 April 2005
Abstract People have been aware of the connection between noise and hearing loss for centuries. Hearing loss is not the only adverse effect of occupational noise, but also a number of non-auditory effects may endanger worker’s safety. This paper proposes an approach enable engineers to point out quantitatively the noisiest source for modification, while multiple machines are operating simultaneously. The model with the point source and spherical radiation in a free field was adopted to formulate the problem. The proposed method requires input data that includes the coordinates (x, y) of the noise sources and the locations where the sound pressure level is measured, and the measured sound pressure levels. Then, the method of least squares was applying to estimate system parameters. Finally, a set of solutions that represents the sound power of noise sources can be found and these solutions have minimum error in least squares sense. With help of sound powers obtained, engineers would be able to evaluate noise distribution whenever re-layout workplace in future. Relevance to industry: Industry workers suffer psychological and physical stress as well as hearing loss due to industrial noise. Although noise source control can be profound process sometimes, it would be the most effective way to eliminate noise level on source. Therefore, identifying dominant source of noise shall be the first step to overcome the noise problem in industry. r 2005 Elsevier B.V. All rights reserved. Keywords: Noise assessment; Noise control; Sources contribution; Noise analysis
1. Introduction People have been aware of the connection between noise and hearing loss for centuries.
Hearing loss is not the only adverse effect of occupational noise, but also a number of nonauditory effects may endanger worker’s safety (Berger et al., 1997; East Kodak Company, 1983;
Corresponding author.
E-mail address: [email protected] (S.-Y. Lu). 0169-8141/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ergon.2005.01.011
ARTICLE IN PRESS 756
S.-Y. Lu, Y.-J. Hong / International Journal of Industrial Ergonomics 35 (2005) 755–760
Konz, 1990). Despite general recognition of the fact that noise causes health problems, workers felt that it was a necessary price to pay in an industrial job and managers felt that the noise was either too difficult or too costly to control. Generally, the handling of workplace noise problem consists of two phases: (1) noise evaluation, and (2) noise control. The first step toward solving noise problem is to conduct noise survey. This necessitates measuring the noise level and securing complete information on employee exposure time in the noisy environment and in any other environment to which the employee might be exposed during the working day. Once appropriate, accurate sound level measurements are made, measured value should be compared with the noise regulations or sound level criterion correct for the situation. After determining that reduction of the employee’s noise exposure should be investigated, the manager must then consider various measures for controlling exposure, such as engineering controls, administrative controls, and hearing protection devices. The preferred method of controlling excessive exposure to noise is engineering controls wherever feasible. It may not only eliminate the requirements of hearing protection, audiometric testing, and limitation of exposure time, but also may improve speech communication and reduce annoyance. When more than one noise source is involved in an area, it is essential to reduce the noisiest source if effective reduction is to be achieved. The amount of sound radiated by a source is determined by its sound power. The sound power is independent of distance or environment, and the main use of sound power is for the noise rating of machines. The sound power can be useful information to guide the design of new factory and the layout of existing one. Many potential noise problems can be eliminated in the design stage. If the sound power of noise sources becomes available, the sound pressure level at any locations in the field can be calculated, as well. By estimating the sound power of noise sources, the source order ranked list can be generated. From list the most effective candidates for noise control can be assessed. At this point, we are readily to perform some evaluation to determine which one the noisiest
machine really is, as a preliminary step in performing noise control. A truly simple, but most illuminating technique is to turn individual machine on and off and to measure and observe the resulting sound level at the position of interest. Such measurements and observations may reveal the one or two machines that are exceptionally noisy. However, we just discussed identification of the source of a problem noise in situation where it is possible to turn production equipment on and off. Often, the noise control engineer is faced with the task of making the necessary identification without the luxury of equipment being operated to his convenience. This paper presents a method capable of estimating the sound power of noise sources by using sound pressure meter, while there are multiple sound sources being operated simultaneously in workplace. A mathematical model was developed in this study, and the least squares method was adopting to solve the problem. The method requires input data that includes the coordinates (x, y) of the noise sources and the locations where the sound pressure level is measured, and the measured sound pressure levels. Then, a set of source powers can be found out, and this solution has minimum error in least squares sense.
2. Formulation of the problem The sound power level Lw (dB re 1 pW) radiated by a point source is related to the sound pressure level Lp (dB re 20 mPa) at a distance r(m) by the following equation (Jenson et al., 1978; Beranek, 1971; Wilson, 1989) Q Lp ¼ Lw þ 10 log , (1) 4pr2 where Q is the directivity factor of the source. Eq. (1) would be known as spherical radiation in a free field, a relationship fundamental to source diagnosis. This relationship is valid for airborne sound at most temperature and pressure conditions. A schematic diagram with m noise sources and n measurement locations is shown in Fig. 1. Each noise source and measurement location is donated by the letter si and mj, respectively, where
ARTICLE IN PRESS S.-Y. Lu, Y.-J. Hong / International Journal of Industrial Ergonomics 35 (2005) 755–760
i ¼ 1; 2; . . . ; m and j ¼ 1; 2; . . . ; n. The distance between the source si and the measurement location mj is rij. The component Lpij of each source si contributing to the measurement location mj can be calculated with help of Eq. (2): Qi Lpij ¼ Lwi þ 10 log , (2) 4prij where Lwi is sound power of the source si. For uncorrelated noise sources, the total meansquare sound pressure is simply the sum of component source mean-square pressures. Detailed discussion of the theory can be found in Wilson, 1989 and Beranek, 1971. Let the contribution to the sound at jth location be Pr msij, corresponding to the source si contributing to the measurement location mj, and the total sound pressure level Lpj at measurement point j is given by " !, # m X 2 2 Prmsij Lpj ¼ 10 log Pref i¼1
¼ 10 log
m X
!
100:1Lpij ;
j ¼ 1; 2; . . . ; n,
ð3Þ
i¼1
where Pref is reference pressure 20 106 Pa.
757
Substituting Eq. (2) into Eq. (3), we rewrite Eq. (3) as follows: m X Qi 0:1Lpj 0:1Lwi ¼ 10 ; j ¼ 1; 2; . . . ; n: 10 4prij i¼1 (4) Qi ¼ aij , 100:1Lwi ¼ xi , and Eq. Let 100:1Lpj ¼ bj , 4pr ij (4) can be formulated into a set of linear equations. Rewrite in vector notation: 3 0 1 2 a11 a12 a1m b1 7 B C 6 B b2 C 6 a21 a22 a2m 7 7 B C 6 C 7 6 B¼B .. 7, .. B .. C; A ¼ 6 .. B . C 6 . . 7 . 5 @ A 4 an1 an2 anm bn 0 1 x1 B C B x2 C B C C X ¼B B .. C. B . C @ A xm
Then B ¼ AX .
(5)
If we assume that the measured sound pressure levels at measurement locations are Lp1 ; Lp2 ; Lp3 ; . . . ; Lpn , and the natural approach to solve inverse problem is to search for the sound powers Lwi, i ¼ 1; 2; . . . ; m, by minimizing in a square-sum sense.
3. Solving linear systems with method of least squares
Fig. 1. Schematic of sound field with m noise sources and n measurement locations.
Many methods (Draper and Smith, 1981; Kahaner et al., 1989) have been proposed for solving system of linear equations, Eq. (5). The model developed above allows an estimation of m unknowns from n equations. If there are equations fewer than m independent unknowns, the model is under-determined, and we cannot determine the parameters uniquely. The model may contain equal number of unknowns and equations, while
ARTICLE IN PRESS S.-Y. Lu, Y.-J. Hong / International Journal of Industrial Ergonomics 35 (2005) 755–760
758
the number of the measurement locations is equal to the number of noise sources. Therefore, three basic techniques that are Cramer’s rule, inverse matrix method, and Gauss elimination method can be used to solve the problem. The Gauss elimination method is, by far, the most widely used, since it can be applied to all systems of linear equations. However, for certain (usually small) system of linear equations the other two techniques may be used to advantage. Nevertheless, it is unavoidable that there are always errors in the measurements, and the measured sound level at measurement locations may be the parameter that the error is mostly like to occur. To achieve better estimation of the power of the noise sources, we may pick measurement locations more than the number of noise sources. Therefore, the model becomes an over-determined system (i.e., redundancy), which allows us to use statistical methods to reduce uncertainty in the derived parameters. There is a procedure that can provide a best solution under these circumstances. The ‘‘best’’ can be specified in a least squares sense that is similar to linear regression. The least squares problem can be interpreted graphically as minimizing the vertical distance from data points to the model. Underlying this idea is the assumption that all the errors in the approximation correspond to errors in the observations bj. Choose estimates x1 ; x2 ; . . . ; xm , to minimize ¼
n X j¼1
bj
m X
!2 aij xi
:
i¼1
Differentiating with respect to x1 ; x2 ; . . . ; xm , leads to the set of m simultaneous linear equations in m unknowns. These equations are known as the normal equations. The normal equations may be written in vector notation as AT B ¼ AT AX . Usually assume ATA invertible, then X ¼ ðAT AÞ1 AT B. The solution of xi can be obtained, the sound powers Lwi shall be found, accordingly.
4. Estimation of noise source power The procedures for estimation of the source power can be described as follows: (1) Determining the coordinates of the locations of the noise source Since the computation requires an assumption of a point source, the noise source si, locations must be represented by a point on x–y plane. By selecting any one point on plane as reference origin (picking source or measurement location is recommended), each source can be expressed as a pair of x and y coordinates which are all measured from the reference point. (2) Determining directivity of the noise sources Radiation of noise energy from source is dependent on direction. For an ideal (nondirectional) point source in full space, the directivity factor is Q ¼ 1. If an ideal point source is located on an acoustically hard surface, then Q ¼ 2 for the half-space above surface. (3) Determining coordinates of the measurement locations and the sound pressure levels at these locations The coordinate (xj,yj) with reference to origin and measured sound pressure level Lpj at jth measurement location, j ¼ 1; 2; . . . ; n, need to be determined, then, the distances between the sources and the measurement locations can be calculated in accordingly. (4) Computing sound contribution and combined total sound at measurement locations With help of above three steps information as inputs, using Eq. (2) the sound contribution of each individual source to the measurement location can be obtained. Using Eq. (3), sum up those individual contributions, and then the computation can obtain the sound pressure level Lpj at each measurement location. (5) Computing the sound power of noise sources through statistical software It is similar to the linear regression problems for finding the coefficients of model equations, and these coefficients represent the power of noise sources in our model.
ARTICLE IN PRESS S.-Y. Lu, Y.-J. Hong / International Journal of Industrial Ergonomics 35 (2005) 755–760
5. Example To demonstrate the application for estimating the sound power of noise sources, please refer to Fig. 2, there are two machines on factory floor, and the sound pressure levels are measured at four measurement locations randomly distributed on the floor. The coordinates (x,y) of the two machines are (0,0) and (2,1), and the four measurement locations are (1,0), (1,1), (0,2), (2,3). In addition, the measured sound pressure levels are 93.6, 87.9, 85.0 and 92.1 dB at the abovementioned four measurement locations. The distances between machines (noise sources) and measurement locations may be given by r11 r12 r13 r14 r21
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð0 1Þ2 þ ð0 0Þ2 ¼ 1 m, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð0 0Þ2 þ ð0 2Þ2 ¼ 2 m, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi ¼ ð0 2Þ2 þ ð0 3Þ2 ¼ 13 m, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ¼ ð0 1Þ2 þ ð0 1Þ2 ¼ 2 m, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ¼ ð2 1Þ2 þ ð1 0Þ2 ¼ 2 m,
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qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ð2 0Þ2 þ ð1 2Þ2 ¼ 5 m, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð2 2Þ2 þ ð1 3Þ2 ¼ 2 m, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð2 1Þ2 þ ð1 1Þ2 ¼ 1 m.
r22 ¼ r23 r24
The contribution of each source to each measurement location can be calculated as follows:
Lp11 ¼ Lw1 þ 10 Lp12 ¼ Lw1 þ 10 Lp13 ¼ Lw1 þ 10 Lp14 ¼ Lw1 þ 10 Lp21 ¼ Lw2 þ 10 Lp22 ¼ Lw2 þ 10 Lp23 ¼ Lw2 þ 10 Lp24 ¼ Lw2 þ 10
2 log ¼ Lw1 8:0 dB, 4p 1 2 log ¼ Lw1 14:0 dB, 4p 4 2 log ¼ Lw1 19:1 dB, 4p 13 2 log ¼ Lw1 11:0 dB, 4p 2 2 log ¼ Lw2 11:0 dB, 4p 2 2 log ¼ Lw2 15:0 dB, 4p 5 2 log ¼ Lw2 14:0 dB, 4p 4 2 log ¼ Lw2 8:0 dB: 4p 1
Therefore, summing up all contributions of each source, the combined sound pressure levels at measurement location 1, 2, 3, and 4 can be denoted by Lp1, Lp2, Lp3, and Lp4, respectively. Where Lp1 ¼ 10 logð10ðLw1 8Þ=10 þ 10ðLw2 11Þ=10 Þ, Lp2 ¼ 10 logð10ðLw1 14Þ=10 þ 10ðLw2 15Þ=10 Þ, Lp3 ¼ 10 logð10ðLw1 19:1Þ=10 þ 10ðLw2 14Þ=10 Þ, Lp4 ¼ 10 logð10ðLw1 11Þ=10 þ 10ðLw2 8Þ=10 Þ.
Fig. 2. A factory floor with two noise sources and four measurement locations.
By using the least squares criterion in which only the errors in Lpi are considered statistical software can be adopted to solve this problem. Then, we may obtain the sound powers Lw1 and Lw2, standard errors and 95% confidence intervals, as
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S.-Y. Lu, Y.-J. Hong / International Journal of Industrial Ergonomics 35 (2005) 755–760
follows: Lw1 ¼ 100:9 dB, Lw2 ¼ 96:0 dB, Standard error of Lw1 ¼ 78:2 dB Standard error of Lw2 ¼ 78:6 dB
reverberant sound field. The reverberant field alters the sound wave characteristics from those described above for free field. Although the procedure works very well in ideal cases (point source and free field), in our view this does not preclude the application of the method to practical cases.
95% confidence interval for Lw1 ¼ ½100:6 101:2 dB 95% confidence interval for Lw2 ¼ ½94:9 96:9 dB. References 6. Discussion and conclusion The use of the least squared error to identify dominant source from sound pressure measurements on sound field has been presented. The proposed approach provides an objective method to help engineers for finding the noisiest machine in factory, without using complicated instrument or subjective justification. However, it is necessary to further validate the idea for relatively low absorption room. Many industrial noise problems are complicated by the fact that the noise is confined in a room. Reflections from the walls, floor, ceiling, and equipment in the room create a
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