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A piezoelectric array for sensing radiation modes
Applied Acoustics 64 (2003) 669–680 www.elsevier.com/locate/apacoust
A piezoelectric array for sensing radiation modes Qibo Maoa,*, Boling Xua, Zhe Jiangb, Jian Gub a
State Laboratory of Modern Acoustics, Institute of Acoustics, PO Box 1169, Nanjing University, Nanjing, 210093, People’s Republic of China b Institute of Vibration and Noise, Jiangsu University, Zhengjiang, 212013, People’s Republic of China Received 20 June 2002; received in revised form 5 December 2002; accepted 14 January 2003
Abstract With an example of simply supported beam, this paper presents new experimental wok that is to sense radiation modes by a piezoelectric array. An array of rectangular segments of PVDF film attached on the surface of the beam are taken as sensor. The output signals of the PVDF films are multiplied by appropriate weights so that the weighted combinations of the outputs directly lead to the radiation mode amplitudes of the beam. The weight of every PVDF film sensor is independent of the type, magnitude and position of the external excitation. Experimental results are also presented to show the feasibility of this new type radiation mode sensor. # 2003 Elsevier Science Ltd. All rights reserved. Keywords: Piezoelectric; PVDF film; Radiation mode
1. Introduction The reduction of structure-borne sound is a persistent problem in acoustics. In practical noise control, passive noise control often makes noise elimination equipment very bulky and inefficient. Since 19800 s, there has been a great amount of research, and plentiful significant gains were obtained in the active structural acoustic control (ASAC) techniques [1]. One of the primary concerns in ASAC approaches is to choose the appropriate sensor. Since active control is most efficient at low frequencies, the development of appropriate structural error sensors used in ASAC should be based on the low frequency mechanisms of the sound radiation from vibrating structures. In many cases, positioning enough microphones in the acoustic field to provide a global estimate of the total sound radiation is often * Corresponding author. E-mail address: [email protected] (Q. Mao). 0003-682X/03/$ - see front matter # 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0003-682X(03)00019-7
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impractical to implement. Another sensing strategy is to use structural error sensors, such as piezoelectric sensors and acceleration sensors. Since piezoelectric sensors have the inherent advantage of integrating over their surface area, which leads to potentially more robust implementations as compared to implementations that use acceleration sensors. Piezoelectric sensors have attracted more and more attention in recent years. Since structural error sensors measure structural vibration information, the designed sensors should take the relationship between the structural response and acoustic response into consideration. Because the sound power is the function of surface velocity distribution, the different distribution of surface velocity makes different contribution to sound power. For example, as for simple supported plate, radiation efficiency of odd–odd order structure modes is higher than that of the other order modes. Therefore merely minimizing the vibration levels of the structure at chosen coordinates does not guarantee minimization of the radiated sound power, and possibly results in an increase in the acoustic response [2]. As a result, the reasonable sensor to be designed should be able to mainly detect these well-radiating velocity distributions. It has been widely accepted that structural error sensors can be developed based on so-called radiation modes [3–6]. Radiation modes are the base vectors orthogonal one another in vector space. Each base vector represents a possible radiation pattern. Each of these radiation modes has independent radiation efficiency. At low frequencies, the first radiation mode accounts for the majority of the sound power. The main advantage of describing the structure-borne sound by radiation modes is to eliminate complex coupling terms in the structural modes, and often a relatively small number of modes are required compared with structural modes, especially at low frequencies. This makes the calculation and active control of the structure-borne radiation much simpler. To design ASAC system by radiation modes, the key is to acquire the correlated information concerning the first few radiation modes which radiate most efficiently. Furthermore, at low frequency the first radiation mode accounts for the majority of the sound power radiation and can be detected by measuring the net volume velocity over the surface of the structure. Recent publications in Refs. [7–10] have introduced the concept of the active control of sound radiated from structures by sensing and controlling the volume displacement. These researches have made use of one or several shaped PVDF film to measure the volume displacement. In order to obtain reduction at more high frequencies, i.e., ka > 1, with k the wavenumber and a the characteristic radius of the structure, more radiation modes are needed than only the first mode. Snyder et al. [11] have implemented the active control of sound radiation from a simply supported plate using PVDF film shaped in order to observe the radiation modes of the plate. Cazzolato [12] presented shapes of distributed sensors of higher order radiation modes in enclosed spaces. The theory for designing radiation mode sensors with piezoelectric patch has been well developed. However, the shaped PVDF sensor implementation is often difficult because the design requires one layer of piezoelectric film for each radiation mode. Moreover, for higher order radiation mode the shape of PVDF sensor becomes very complex which makes it difficult to fabricate.
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Recent, Preumont et al. [13,14] used a set of independent piezoelectric patches bonded to the regular mesh and evaluated the volume velocity by using an adaptive linear combiner. The theory proposed in references [13,14] is meant to overcome the difficulties by using shaped PVDF sensors. Berkhoff [15] presented Analytical and numerical work on piezoelectric array sensors that directly lead to the higher order radiation modes. The segmented design of modal sensors does not require spatially continuous shaping of the film and does not require a separate layer of sensor film for each mode. The use of an array of sensors seems to be a more practical way than shaped sensors. It appears that no paper has been published concerning the experimental work of sensing higher order radiation modes by a piezoelectric array. With a beam example, this paper presents an experimental demonstration of the higher order radiation mode sensors that builds upon results from the above papers. In this paper, segments of piezoelectric film made of polyvinylidene fluoride (PVDF) is chosen because it gives little loading on light structures, and is easy to cut into desired segments. This paper is organized as follows. The first gives a brief review of the relationship between structure response and output signals of the PVDF film segments. The second continues with the theoretical development on piezoelectric array sensors, which directly lead to the higher order radiation modes with simple supported boundary condition for a vibrating beam. Finally, the method is experimentally verified.
2. Array of piezoelectric segments Consider a beam with N rectangular piezoelectric film segments equally attached on the top surface, shown in Fig. 1. As referred to Lee and Moon’s work [16], The output charge qen of every PVDF sensor can be expressed as follows, ð h þ h f Lx @2 wðxÞ qen ¼ Fn ðxÞ e31 dx ð1Þ @x2 2 0 where hf is the PVDF sensor thickness, e31 is the PVDF sensor stress/charge coefficient, w(x) is the displacement of the beam, Fn(x) is the nth PVDF film shape function.
Fig. 1. Beam with an array PVDF segments.
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The PVDF film shape function of the nth segment gives Py xn Px 4 x 4 xn Fn ðxÞ ¼ 0 otherwise Substituting Eq. (2) into Eq. (1), ð h þ h f xn @2 wðxÞ qen ¼ e31 Py dx 2 2 xn Px @x
ð2Þ
ð3Þ
The output current of every PVDF, In(t) can be obtained by differentiation of the charge qen with respect to time, ð dqen h þ h f xn @2 vðxÞ ¼ e31 Py dx ð4Þ In ¼ 2 dt 2 xn Px @x The velocity distribution of the simply supported beam can be represented by a series of expansion. M X m Am sin x ð5Þ vðxÞ ¼ Lx m¼1 where Am represents the mth modal velocity coefficients. M is the index for the highest order structural mode. Substituting Eq. (5) into Eq. (4), 2 ð xn M M X h þ hf X m mx Py Am e31 sin Am Km;n ð6Þ In ¼ dx ¼ Lx Lx 2 xn Px m¼1 m¼1 2 ð xn h þ hf m m Py e31 sin x dx. where Km;n ¼ Lx Lx 2 xn Px Eq.(6) can be written in the matrix form I ¼ KA
ð7Þ
where A represents an M1 vector of modal velocity. K is an MN matrix. Modal velocity vector A is computed by inverse of K A ¼ K1 I
ð8Þ
Note that the K matrix must be square. From Eq. (7), it is clear that to structural mode number M=N segments PVDF film are required.
3. Estimation of radiation modes with piezoelectric sensors The beam is divided into J elements with equal area. The vector of normal velocities of each of these elements is denoted as v, the sound power can be expressed as W ¼ vH Rv, where superscript H denotes the complex conjugate transpose. The matrix R is a JJ matrix giving the real part of the acoustic transfer impedance
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between each pair of elemental. As the matrix R is real, symmetric and positive definite [5], the following eigenvalue decomposition is always possible R ¼ QK QT , where superscript T is denoted transpose. K is a diagonal matrix of eigenvalues lk , decreasing monotonously along the diagonal. Q=[Q1, Q2, . . ., Qk,. . ., QJ] is a group of orthogonal surface vibration patterns, Qk is a real vector representing the kth radiation mode shape. The amplitudes of the kth radiation mode can be expressed as [5,6] yk ¼ QTk v
ð9Þ
So the sound power may be modified to give W ¼ vH QKQT v ¼
J X 2 lk yk
ð10Þ
k¼1
Although these radiation modes are orthogonal in mathematical sense, they are generally different from the structural vibration modes. Eq. (10) shows that each radiation mode contributes to the sound power independently. While structural modes do not radiate sound independently since the sound power due to one structural mode depends on the amplitudes of other structural modes [5,9]. It has been
Fig. 2. The mode shapes of the first four radiation modes of the beam.
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demonstrated that the radiation mode shapes can be chosen to be independent of frequency [17,18]. The first four radiation mode shapes of beam (310 mm long25 mm wide) can be found in Fig. 2. It is possible to construct radiation mode sensors using linear combiners whose outputs emulate radiation mode amplitudes of the vibrating beam. Fig. 3 summarizes the principle of the analytical radiation mode sensors. Each radiation mode sensor consists of an array of weighs that multiply the outputs of the segments. The kth radiation mode sensors is an array of gains Wk(n), where k denotes the radiation mode number and n denotes the segment number to which the gain is connected. The output of each segment is denoted by In. The output of the k radiation mode sensor y^ k is the weighted sum of the segment outputs, y^ k ¼
N X Wk ðnÞIn ¼ Wk I
ð11Þ
n¼1
The task now is to determine the weight matrix Wk so that the filter outputs emulate the kth radiation mode’s amplitude. Substituting Eq. (8) into Eq. (5), then written in the matrix form v ¼ UK1 I
ð12Þ
where U is an JM matrix which depend on the structural mode shapes. Substitute Eq. (12) into Eq. (9), that is
Fig. 3. Principle of the analytical radiation mode sensors.
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Q. Mao et al. / Applied Acoustics 64 (2003) 669–680 Table 1 The physical properties of PVDF film Parameter
PVDF film
Length Px (mm) Width Py (mm) Desity (kg/m3) Poisson’s ratio Young’s modulus (N/m2) Piezo Strain Constant d31 (m/V) Maximum Operating Voltage (V/mm) Relative permittivity "/"0
30 12 1.78103 0.28 2–4109 231012 750 12
Table 2 The first six natural frequencies of beam (Hz) 1
2
3
4
5
6
81.4
325.6
732.6
1302.4
2023.0
2930.4
Fig. 4. Sensor weight Wk(n) for transforming nine segment outputs into the first four radiation mode sensors.
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yk ¼ QTk UK1 I
ð13Þ
Comparing Eq. (13) with Eq. (11), for simplicity, we set y^ k ¼ yk giving the result Wk ¼ QTk UK1
ð14Þ
Fig. 5. PVDF film and signal conditioner.
Fig. 6. Magnitude response from force located at xd=90 mm to sensor outputs. Solid lines: experimental; dashed lined: analytical.
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4. Experimental validation To verify the above conclusions about sensor development, an steel beam 310 mm long25 mm wide3.5 mm thick was equally covered with N=9 segments of PVDF film. Thin metal shim was used to connect each end of the beam to the flame. This shim gave almost no resistance to bending, but was very stiff in the beam’s transverse direction. Therefore, the shim was assumed to be a simply supported boundary condition. The physical properties of the PVDF film in the experiments are listed in Table 1. The first six natural frequencies of beam are list in Table 2. The weights of every PVDF film for sensing the first four radiation modes are shown in Fig. 4. These weights have been normalized so that the maximum absolute value is unity. A B&K4810 shaker, which is attached at the back of the beam, was used to provide the primary driving force. And a B&K8200 force transducer installed between the shaker’s stinger and the beam was used to measure the input force. HP35665A dynamic signal analyzer was used to perform the sweep-sine measurement. Sweptsine excitation was used to ensure clean transfer functions and prevent aliasing. Due
Fig. 7. Magnitude response from force located at xd=125 mm to sensor outputs. Solid lines: experimental; dashed line: analytical.
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to the high input impedance of the PVDF sensor, the sensor output signal was first conditioned and then measured with HP35665A dynamic signal analyzer. The circuit representation of the film and signal conditioner is shown in Fig. 5. The output signal is the product of the PVDF current and the feedback resistance Rf. All measurements were limited to the frequency range below 2000 Hz. The transfer functions between force and the segments were measured. All these experimental transfer functions were scaled to equivalent to the corresponding theoretical ones at 170 Hz. Then these transfer functions multiplied by the corresponding PVDF segment weights of the kth mode results in single kth radiation mode responses. Figs. 6–8 present the magnitudes of the transfer functions between force and multiplier outputs for three different locations of the excitation force respectively. Each of these outputs is intended to emulate a radiation mode amplitude, as shown in Eq. (13). As illustrated, when the excitation force position is changed, the surface velocity distribution is also changed. It is shown that these radiation mode sensors are sensitive to this change of the surface velocity distribution. It is important to note that weight coefficient Wk given by Eq. (14) is independent of the type and position of the excitation.
Fig. 8. Magnitude response from force located at xd=165 mm to sensor outputs. Solid lines: experimental; dashed line: analytical.
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Notice that the experimental outputs are close to the predicted outputs for the low frequency. However, the sensor accuracy is decreased in higher frequency range. It is important to note that the sensors are designed using the first nine structure modes since only nine segments PVDF film covered on beam. So the contribution of higher order structure modes is not considered in the weight of the sensors. If the number of PVDF segment is added, accuracy of the sensors can be increased because more order structural mode is taken into account. Another reason for inaccuracy of the sensors could be due to the errors in positioning the sensors on the beam. However, in general the experiment showed reasonable modal filtering effects.
5. Conclusions This paper presents a technique to measure the radiation modes from a onedimensional structure using an array of PVDF segments. The outputs of the PVDF segments can be transformed into the radiation mode amplitudes by multiplying them by appropriate weights. The weight of the sensors is independent of the type and position of the excitation. This sensor system uses PVDF segments of very simple shape with independent conditioning electronics for every segment. One advantage of the method is that when boundary condition of structure is changed, weights can be adjusted to compensate for the sensor imperfection in stead of changing the shape and location of PVDF sensors. The other advantage is that this method can be easily extended in the case of other boundary condition and 2D structures. The experimental results show the feasibility of this new type of radiation mode sensor.
References [1] Clark RL, Fuller CR. A model reference approach for implementing active structural acoustic control. Journal of the Acoustical Society of America 1992;92(3):1534–44. [2] Burdisso RA, Fuller CR. Theory of feedforward controlled system eigenproperites. Journal of Sound and Vibration 1992;153(4):437–51. [3] Borgiotti GV. The power radiated by a vibrating body in an acoustic fluid and its determination from boundary measurements. Journal of the Acoustical Society of America 1990;88(4):1884–93. [4] Photiais DM. The relationships of singular value decomposition to wave-vector by three-dimensional structures. Journal of the Acoustical Society of America 1990;88(4):1152–9. [5] Elliott SJ, Johnson ME. Radiation modes and the active control of sound power. Journal of the Acoustical Society of America 1993;94(4):2194–204. [6] Mao Q, Jiang Z. Research active structural acoustic control by radiation modes. Acta Acustica (China) 2001;26(3):277–81. [7] Mao Q, Jiang Z. Active control of sound radiation from simply supported beam using piezoelectric sensors. Acta Acustica (China) 2001;26(6):527–31. [8] Marcellin BZ, Naghshineh K, Kamman JW. Narrow band active control of sound radiated from a baffled beam using local volume displacement minimization. Applied Acoustics 2001;62:47–64. [9] Johnson ME, Elloitt SJ. Active control of sound radiation using volume velocity cancellation. Journal of the Acoustical Society of America 1995;98(4):2174–86. [10] Charette F, Berry A, Guigou C. Active control of sound radiation from a plate using a polyvinylidence
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[11]
[12] [13] [14] [15] [16] [17] [18]
Q. Mao et al. / Applied Acoustics 64 (2003) 669–680 fluoride volume displacement sensor. Journal of the Acoustical Society of America 1998;103(3):1493– 503. Snyder SD, Tanaka N, Kikushima Y. The use of optimally shaped piezo-electric film sensors in the active control of free field structural radiation, part 1: feedforward control. Journal of Vibration and Acoustics 1995;117:311–22. Cazzolato BS, Hansen CH. Structural radiation mode sensing for active control of sound radiation into enclosed spaces. Journal of the Acoustical Society of America 1999;106(6):3732–5. Preumont A, Francois A, Dubru S. Piezoelectric array sensing for real-time, broad-band sound radiation measurement. Journal of Vibration and Acoustics 1999;121:446–52. Francois A, Man PD, Preumont A. Piezoelectric array sensing of volume displacement: a hardware demonstration. Journal of Sound and Vibration 2001;244(3):395–405. Berkhoff AP. Piezoelectric sensor configuration for active structural acoustic control. Journal of Sound and Vibration 2001;246(1):175–83. Lee CK, Moon FC. Modal sensors/actuator. Journal of Applied Mechanics 1990;57:434–41. Gibbs GP, Clark RL, Cox DE. Radiation modal expansion: application to active structural acoustic control. Journal of the Acoustical Society of America 2000;107(1):332–9. Berkhoff AP. Sensor scheme design for active structural acoustic control. Journal of the Acoustical Society of America 2000;108(3):1037–45.
A piezoelectric array for sensing radiation modes Qibo Maoa,*, Boling Xua, Zhe Jiangb, Jian Gub a
State Laboratory of Modern Acoustics, Institute of Acoustics, PO Box 1169, Nanjing University, Nanjing, 210093, People’s Republic of China b Institute of Vibration and Noise, Jiangsu University, Zhengjiang, 212013, People’s Republic of China Received 20 June 2002; received in revised form 5 December 2002; accepted 14 January 2003
Abstract With an example of simply supported beam, this paper presents new experimental wok that is to sense radiation modes by a piezoelectric array. An array of rectangular segments of PVDF film attached on the surface of the beam are taken as sensor. The output signals of the PVDF films are multiplied by appropriate weights so that the weighted combinations of the outputs directly lead to the radiation mode amplitudes of the beam. The weight of every PVDF film sensor is independent of the type, magnitude and position of the external excitation. Experimental results are also presented to show the feasibility of this new type radiation mode sensor. # 2003 Elsevier Science Ltd. All rights reserved. Keywords: Piezoelectric; PVDF film; Radiation mode
1. Introduction The reduction of structure-borne sound is a persistent problem in acoustics. In practical noise control, passive noise control often makes noise elimination equipment very bulky and inefficient. Since 19800 s, there has been a great amount of research, and plentiful significant gains were obtained in the active structural acoustic control (ASAC) techniques [1]. One of the primary concerns in ASAC approaches is to choose the appropriate sensor. Since active control is most efficient at low frequencies, the development of appropriate structural error sensors used in ASAC should be based on the low frequency mechanisms of the sound radiation from vibrating structures. In many cases, positioning enough microphones in the acoustic field to provide a global estimate of the total sound radiation is often * Corresponding author. E-mail address: [email protected] (Q. Mao). 0003-682X/03/$ - see front matter # 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0003-682X(03)00019-7
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impractical to implement. Another sensing strategy is to use structural error sensors, such as piezoelectric sensors and acceleration sensors. Since piezoelectric sensors have the inherent advantage of integrating over their surface area, which leads to potentially more robust implementations as compared to implementations that use acceleration sensors. Piezoelectric sensors have attracted more and more attention in recent years. Since structural error sensors measure structural vibration information, the designed sensors should take the relationship between the structural response and acoustic response into consideration. Because the sound power is the function of surface velocity distribution, the different distribution of surface velocity makes different contribution to sound power. For example, as for simple supported plate, radiation efficiency of odd–odd order structure modes is higher than that of the other order modes. Therefore merely minimizing the vibration levels of the structure at chosen coordinates does not guarantee minimization of the radiated sound power, and possibly results in an increase in the acoustic response [2]. As a result, the reasonable sensor to be designed should be able to mainly detect these well-radiating velocity distributions. It has been widely accepted that structural error sensors can be developed based on so-called radiation modes [3–6]. Radiation modes are the base vectors orthogonal one another in vector space. Each base vector represents a possible radiation pattern. Each of these radiation modes has independent radiation efficiency. At low frequencies, the first radiation mode accounts for the majority of the sound power. The main advantage of describing the structure-borne sound by radiation modes is to eliminate complex coupling terms in the structural modes, and often a relatively small number of modes are required compared with structural modes, especially at low frequencies. This makes the calculation and active control of the structure-borne radiation much simpler. To design ASAC system by radiation modes, the key is to acquire the correlated information concerning the first few radiation modes which radiate most efficiently. Furthermore, at low frequency the first radiation mode accounts for the majority of the sound power radiation and can be detected by measuring the net volume velocity over the surface of the structure. Recent publications in Refs. [7–10] have introduced the concept of the active control of sound radiated from structures by sensing and controlling the volume displacement. These researches have made use of one or several shaped PVDF film to measure the volume displacement. In order to obtain reduction at more high frequencies, i.e., ka > 1, with k the wavenumber and a the characteristic radius of the structure, more radiation modes are needed than only the first mode. Snyder et al. [11] have implemented the active control of sound radiation from a simply supported plate using PVDF film shaped in order to observe the radiation modes of the plate. Cazzolato [12] presented shapes of distributed sensors of higher order radiation modes in enclosed spaces. The theory for designing radiation mode sensors with piezoelectric patch has been well developed. However, the shaped PVDF sensor implementation is often difficult because the design requires one layer of piezoelectric film for each radiation mode. Moreover, for higher order radiation mode the shape of PVDF sensor becomes very complex which makes it difficult to fabricate.
Q. Mao et al. / Applied Acoustics 64 (2003) 669–680
671
Recent, Preumont et al. [13,14] used a set of independent piezoelectric patches bonded to the regular mesh and evaluated the volume velocity by using an adaptive linear combiner. The theory proposed in references [13,14] is meant to overcome the difficulties by using shaped PVDF sensors. Berkhoff [15] presented Analytical and numerical work on piezoelectric array sensors that directly lead to the higher order radiation modes. The segmented design of modal sensors does not require spatially continuous shaping of the film and does not require a separate layer of sensor film for each mode. The use of an array of sensors seems to be a more practical way than shaped sensors. It appears that no paper has been published concerning the experimental work of sensing higher order radiation modes by a piezoelectric array. With a beam example, this paper presents an experimental demonstration of the higher order radiation mode sensors that builds upon results from the above papers. In this paper, segments of piezoelectric film made of polyvinylidene fluoride (PVDF) is chosen because it gives little loading on light structures, and is easy to cut into desired segments. This paper is organized as follows. The first gives a brief review of the relationship between structure response and output signals of the PVDF film segments. The second continues with the theoretical development on piezoelectric array sensors, which directly lead to the higher order radiation modes with simple supported boundary condition for a vibrating beam. Finally, the method is experimentally verified.
2. Array of piezoelectric segments Consider a beam with N rectangular piezoelectric film segments equally attached on the top surface, shown in Fig. 1. As referred to Lee and Moon’s work [16], The output charge qen of every PVDF sensor can be expressed as follows, ð h þ h f Lx @2 wðxÞ qen ¼ Fn ðxÞ e31 dx ð1Þ @x2 2 0 where hf is the PVDF sensor thickness, e31 is the PVDF sensor stress/charge coefficient, w(x) is the displacement of the beam, Fn(x) is the nth PVDF film shape function.
Fig. 1. Beam with an array PVDF segments.
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The PVDF film shape function of the nth segment gives Py xn Px 4 x 4 xn Fn ðxÞ ¼ 0 otherwise Substituting Eq. (2) into Eq. (1), ð h þ h f xn @2 wðxÞ qen ¼ e31 Py dx 2 2 xn Px @x
ð2Þ
ð3Þ
The output current of every PVDF, In(t) can be obtained by differentiation of the charge qen with respect to time, ð dqen h þ h f xn @2 vðxÞ ¼ e31 Py dx ð4Þ In ¼ 2 dt 2 xn Px @x The velocity distribution of the simply supported beam can be represented by a series of expansion. M X m Am sin x ð5Þ vðxÞ ¼ Lx m¼1 where Am represents the mth modal velocity coefficients. M is the index for the highest order structural mode. Substituting Eq. (5) into Eq. (4), 2 ð xn M M X h þ hf X m mx Py Am e31 sin Am Km;n ð6Þ In ¼ dx ¼ Lx Lx 2 xn Px m¼1 m¼1 2 ð xn h þ hf m m Py e31 sin x dx. where Km;n ¼ Lx Lx 2 xn Px Eq.(6) can be written in the matrix form I ¼ KA
ð7Þ
where A represents an M1 vector of modal velocity. K is an MN matrix. Modal velocity vector A is computed by inverse of K A ¼ K1 I
ð8Þ
Note that the K matrix must be square. From Eq. (7), it is clear that to structural mode number M=N segments PVDF film are required.
3. Estimation of radiation modes with piezoelectric sensors The beam is divided into J elements with equal area. The vector of normal velocities of each of these elements is denoted as v, the sound power can be expressed as W ¼ vH Rv, where superscript H denotes the complex conjugate transpose. The matrix R is a JJ matrix giving the real part of the acoustic transfer impedance
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673
between each pair of elemental. As the matrix R is real, symmetric and positive definite [5], the following eigenvalue decomposition is always possible R ¼ QK QT , where superscript T is denoted transpose. K is a diagonal matrix of eigenvalues lk , decreasing monotonously along the diagonal. Q=[Q1, Q2, . . ., Qk,. . ., QJ] is a group of orthogonal surface vibration patterns, Qk is a real vector representing the kth radiation mode shape. The amplitudes of the kth radiation mode can be expressed as [5,6] yk ¼ QTk v
ð9Þ
So the sound power may be modified to give W ¼ vH QKQT v ¼
J X 2 lk yk
ð10Þ
k¼1
Although these radiation modes are orthogonal in mathematical sense, they are generally different from the structural vibration modes. Eq. (10) shows that each radiation mode contributes to the sound power independently. While structural modes do not radiate sound independently since the sound power due to one structural mode depends on the amplitudes of other structural modes [5,9]. It has been
Fig. 2. The mode shapes of the first four radiation modes of the beam.
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demonstrated that the radiation mode shapes can be chosen to be independent of frequency [17,18]. The first four radiation mode shapes of beam (310 mm long25 mm wide) can be found in Fig. 2. It is possible to construct radiation mode sensors using linear combiners whose outputs emulate radiation mode amplitudes of the vibrating beam. Fig. 3 summarizes the principle of the analytical radiation mode sensors. Each radiation mode sensor consists of an array of weighs that multiply the outputs of the segments. The kth radiation mode sensors is an array of gains Wk(n), where k denotes the radiation mode number and n denotes the segment number to which the gain is connected. The output of each segment is denoted by In. The output of the k radiation mode sensor y^ k is the weighted sum of the segment outputs, y^ k ¼
N X Wk ðnÞIn ¼ Wk I
ð11Þ
n¼1
The task now is to determine the weight matrix Wk so that the filter outputs emulate the kth radiation mode’s amplitude. Substituting Eq. (8) into Eq. (5), then written in the matrix form v ¼ UK1 I
ð12Þ
where U is an JM matrix which depend on the structural mode shapes. Substitute Eq. (12) into Eq. (9), that is
Fig. 3. Principle of the analytical radiation mode sensors.
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Q. Mao et al. / Applied Acoustics 64 (2003) 669–680 Table 1 The physical properties of PVDF film Parameter
PVDF film
Length Px (mm) Width Py (mm) Desity (kg/m3) Poisson’s ratio Young’s modulus (N/m2) Piezo Strain Constant d31 (m/V) Maximum Operating Voltage (V/mm) Relative permittivity "/"0
30 12 1.78103 0.28 2–4109 231012 750 12
Table 2 The first six natural frequencies of beam (Hz) 1
2
3
4
5
6
81.4
325.6
732.6
1302.4
2023.0
2930.4
Fig. 4. Sensor weight Wk(n) for transforming nine segment outputs into the first four radiation mode sensors.
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Q. Mao et al. / Applied Acoustics 64 (2003) 669–680
yk ¼ QTk UK1 I
ð13Þ
Comparing Eq. (13) with Eq. (11), for simplicity, we set y^ k ¼ yk giving the result Wk ¼ QTk UK1
ð14Þ
Fig. 5. PVDF film and signal conditioner.
Fig. 6. Magnitude response from force located at xd=90 mm to sensor outputs. Solid lines: experimental; dashed lined: analytical.
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4. Experimental validation To verify the above conclusions about sensor development, an steel beam 310 mm long25 mm wide3.5 mm thick was equally covered with N=9 segments of PVDF film. Thin metal shim was used to connect each end of the beam to the flame. This shim gave almost no resistance to bending, but was very stiff in the beam’s transverse direction. Therefore, the shim was assumed to be a simply supported boundary condition. The physical properties of the PVDF film in the experiments are listed in Table 1. The first six natural frequencies of beam are list in Table 2. The weights of every PVDF film for sensing the first four radiation modes are shown in Fig. 4. These weights have been normalized so that the maximum absolute value is unity. A B&K4810 shaker, which is attached at the back of the beam, was used to provide the primary driving force. And a B&K8200 force transducer installed between the shaker’s stinger and the beam was used to measure the input force. HP35665A dynamic signal analyzer was used to perform the sweep-sine measurement. Sweptsine excitation was used to ensure clean transfer functions and prevent aliasing. Due
Fig. 7. Magnitude response from force located at xd=125 mm to sensor outputs. Solid lines: experimental; dashed line: analytical.
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to the high input impedance of the PVDF sensor, the sensor output signal was first conditioned and then measured with HP35665A dynamic signal analyzer. The circuit representation of the film and signal conditioner is shown in Fig. 5. The output signal is the product of the PVDF current and the feedback resistance Rf. All measurements were limited to the frequency range below 2000 Hz. The transfer functions between force and the segments were measured. All these experimental transfer functions were scaled to equivalent to the corresponding theoretical ones at 170 Hz. Then these transfer functions multiplied by the corresponding PVDF segment weights of the kth mode results in single kth radiation mode responses. Figs. 6–8 present the magnitudes of the transfer functions between force and multiplier outputs for three different locations of the excitation force respectively. Each of these outputs is intended to emulate a radiation mode amplitude, as shown in Eq. (13). As illustrated, when the excitation force position is changed, the surface velocity distribution is also changed. It is shown that these radiation mode sensors are sensitive to this change of the surface velocity distribution. It is important to note that weight coefficient Wk given by Eq. (14) is independent of the type and position of the excitation.
Fig. 8. Magnitude response from force located at xd=165 mm to sensor outputs. Solid lines: experimental; dashed line: analytical.
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Notice that the experimental outputs are close to the predicted outputs for the low frequency. However, the sensor accuracy is decreased in higher frequency range. It is important to note that the sensors are designed using the first nine structure modes since only nine segments PVDF film covered on beam. So the contribution of higher order structure modes is not considered in the weight of the sensors. If the number of PVDF segment is added, accuracy of the sensors can be increased because more order structural mode is taken into account. Another reason for inaccuracy of the sensors could be due to the errors in positioning the sensors on the beam. However, in general the experiment showed reasonable modal filtering effects.
5. Conclusions This paper presents a technique to measure the radiation modes from a onedimensional structure using an array of PVDF segments. The outputs of the PVDF segments can be transformed into the radiation mode amplitudes by multiplying them by appropriate weights. The weight of the sensors is independent of the type and position of the excitation. This sensor system uses PVDF segments of very simple shape with independent conditioning electronics for every segment. One advantage of the method is that when boundary condition of structure is changed, weights can be adjusted to compensate for the sensor imperfection in stead of changing the shape and location of PVDF sensors. The other advantage is that this method can be easily extended in the case of other boundary condition and 2D structures. The experimental results show the feasibility of this new type of radiation mode sensor.
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