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On acoustic very near field measurements
Mechanical Systems and Signal Processing 40 (2013) 194–207
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On acoustic very near field measurements J. Prezelj a,n, P. Lipar a, A. Belšak b, M. Čudina a a b
University of Ljubljana, Faculty of Mechanical Engineering, Askerceva 6, SI-1000 Ljubljana, Slovenia University of Maribor, Faculty of Mechanical Engineering, Slovenia
a r t i c l e in f o
abstract
Article history: Received 3 February 2012 Received in revised form 30 October 2012 Accepted 11 May 2013 Available online 15 June 2013
Information about vibration modes is needed during the planning of noise control measures on different parts of machinery. A visualization of vibration modes is a starting point and different methods can be used to visualize vibration modes. Some methods which incorporate an inverse calculation of surface velocity from the sound pressure on some boundary have already been proposed, among others. A direct inverse method based on the discretized Rayleigh integral was used in our work, to demonstrate that microphones should be placed close to the vibrating structure to provide an acoustic transfer matrix with a low condition number. It is demonstrated, that there is practically no need for the calculation of the inverse matrix if microphones are placed in a very near field of the vibrating structure. A single microphone placed in a very near field together with a reference vibration sensor provides sufficient information for producing the images of vibration modes. Analytical results, numerical results, FEM simulations and measurement results were used to prove that properties of the sound pressure in a very near field permit a cost effective visualization of the vibration modes. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Acoustics Very near field Inverse methods Sound source visualization Near field acoustic holography
1. Introduction A reconstruction of the surface velocity on a complex vibrating structure is important because it pinpoints the modes which dominate the surface vibration. If the surface velocity is reconstructed, it can be used to predict the sound pressure field and the radiated sound power, which may be more valuable than the surface velocity reconstruction itself [1]. A prediction of the surface velocity of the vibrating structure based on the sound pressure measurements in the near field is well known. An analysis of vibrations with such an approach can be traced back to 1958 [2]. Such methods are of particular interest when the structure is rotating or moving, and if it is too light or too hot to be instrumented by accelerometers. The described methods are commonly used [3–5], although a detailed analysis and validation were not available. The development of multi-channel data acquisition systems made more sophisticated methods like the Nearfield Acoustic Holography (NAH), the Helmholtz differential equation (HELS), and Inverse numerical acoustics (INA) applicable. Advances in computers and digital imaging technology have allowed 3D digital image correlation (DIC) methods to measure the shape and deformation of a vibrating structure, [6]. At the same time, a development in the laser technology made a laser scanning Doppler vibrometer readily available. Laser scanning vibrometers are used for vibration mode identification because they provide accurate results of the measured surface velocity component [7,8]. All methods are quite costly due to the dedicated equipment and software; therefore more cost efficient method was taken into consideration for our applications and use in the laboratory. The presented method is similar to NAH, however it ensures well conditioned
n
Corresponding author. Tel.:+386 40 837647. E-mail address: [email protected] (J. Prezelj).
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J. Prezelj et al. / Mechanical Systems and Signal Processing 40 (2013) 194–207
Observation plane P
195
Vibrating Surface V m
R1,1 R2,1 R3,1
dx
n=1,2,...N
dS RN,1
dy n
m=1,2,...M
h Fig. 1. Geometry of the theoretical setup.
transfer matrix without a need for the regularization. Theoretical background, numerical simulations and some measurement results obtained with this method are presented in this paper. 1.1. Sound pressure field generated by a vibrating plate Let us consider an arbitrary vibrating surface V in an acoustic free field. A complex vibrating surface can be divided on many elementary surfaces dS. Each elementary surface dS on the arbitrary shaped surface can be regarded as a simple point source of an outgoing sound wave [9–11], if the wavelength of the generated sound is much longer than the dimensions of this elementary source dS. Sound pressure, radiated from an arbitrary shaped surface can be calculated on the parallel plane P by using the Rayleigh surface integral. The sound pressure in any point P(x0,y0,z0) in a free field can be expressed with Eq. (1), pðx0 ; y0 ; z0 Þ ¼
iωρ0 e−ikR ∬S vðx; yÞdS; 2π R
ð1Þ
A discretization of the vibrating surface, together with a discretization of the parallel observation plane is presented in Fig. 1. Sound pressure on the plane P is observed and values of sound pressure at a given time and at a given position pi(xi,yi,zi) are written in a vector p. If the values of vibration velocity on the vibrating surface V are well known and described in a vector v, then a pressure plane described with a vector p can be calculated by using a discretized Rayleigh surface interval given in Eq. (2): pi ðxi ; yi ; zi Þ ¼
M N eikRm;n iωρ0 dx dy ∑ ∑ vm;n 2π m ¼ 1 n ¼ 1 Rm;n
ð2Þ
Sound pressure, on the plane P, and vibration velocity, on the vibrating surface V, can be written in a vector form. The influence of geometry setup can be described with the acoustic transfer matrix, (ATM) R. A simple equation can be used for the correlation between surface velocity and sound pressure (Eq. (3)). p ¼ Rv
ð3Þ
1.2. Inverse method for source strength estimation If sound pressure on plane P and the geometry of the vibrating structure are well defined, then a vibration velocity on the vibrating surface V can be calculated backward from Eq. (4). If we consider R−1 as an inverse of the ATM R, then a surface velocity vector v can be calculated from the measurements of sound pressure p in the parallel plane. v ¼ R−1 p
ð4Þ
However, the ATM matrix R should fulfill some conditions in order to have a useable inverse. If two vectors p and v have different sizes, the ATM is not square, and therefore the inverse matrix R−1 cannot be directly determined. There are quite a few methods to solve this problem. However, to simplify the approach we used both vectors (p and v) of the same size and kept the ATM R in a square form. A spatial resolution of the surface velocity is in such a case determined by the microphone grid size on the pressure plane. But, a square form of the ATM matrix is only a necessary condition for a direct invertibility approach, and it is not a sufficient condition yet. For the evaluation of the matrix invertibility, its condition number was used. A condition number is the ratio between the largest singular values of R to the smallest. Large condition numbers indicate a nearly singular matrix, and if the condition number of the matrix increases to values over 10.0E+16 the solution of
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the inverse problem is not certain. The condition number used in our case was calculated according to Eq. (5). κðRÞ≥
maxi jRii j mini jRii j
ð5Þ
A condition number of the ATM R depends – on the shape of the observation plane and on the distribution of the microphone locations. That is, on the geometry of the microphone array relative to the geometry of the source, – on the distance between the vibrating surface and the observation plane, marked with h in Fig. 1, – on the ratio between the wave number k of the generated sound and dimensions of the elementary surface on the source, marked with dx and dy in Fig. 1, – on the ratio between the wave number k and distance between the elementary surface dS and observation point, marked with Ri,j in Fig. 1, – on measurement noise, – and on some other factors [12]. The ATM R appears to be best conditioned when the number of sources and sensors is small, when the geometrical arrangement of sensors closely matches the assumed source array geometry, when the distance between the sources is the same as the distance between the sensors, when the sensor array is placed close to the source array and when the sensor array is positioned symmetrically with respect to the source array. The orthogonal simple microphone mesh was shown to be the optimal solution already in Refs. [12,13]. A few simulations were performed to analyze the condition number of the ATM for our case, where a square plate was used as a model. The results are shown in Fig. 2, and they clearly indicate that condition number is much smaller if the microphones are placed close to the vibrating surface, for low and for higher kdx (frequencies). The influence of the microphone distance h is more pronounced than the influence of the frequency. Results, therefore, clearly indicate that sound pressure should be measured as closely as possible to the complex vibrating sound source. Numerical experiments were performed to investigate how a distance of the observation plane influences the inverse calculation. In these numerical experiments a distribution of the surface velocity was determined from a sound pressure field by inverse calculation. Inverse calculations were performed from three different sound pressure fields, which were parallel to the vibrating surface and at distances h¼150 mm, h¼164 mm, and h¼178 mm above the vibrating surface. The surface was set to vibrate with 1000 Hz. Elementary surface dS¼ dxdy had a surface of 400 mm2. A grid of 22 22 elements covered a 440 mm width square plate. The results of the inverse calculation for three different conditions are presented in Fig. 3. If the observation plane is 150 mm or closer from the vibrating surface, the inverse calculation provides good results for the given conditions. If the observation plane is 178 mm or more from the vibrating surface, the inverse calculation cannot provide any results for a given conditions. This analysis illustrates that for an appropriate determination of the surface velocity based on the measurements of generated sound it is necessary to place a microphone in the vicinity of the vibrating surface. The question remains, how close to the vibrating surface a microphone should be placed to gather the appropriate information and in the same time not to significantly disturb the sound field itself.
Fig. 2. Condition number of the ATM R as a function of the distance and frequency.
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Fig. 3. Transformation from convergence to divergence of inverse surface velocity calculation using sound pressure when increasing the distance h, between the vibrating surface and the parallel plane of sound pressure.
1.3. Very near field If the piston is small compared with a wavelength of the sound it is emitting (ka51), it acts as a point source set a top of the reflecting plane [14]. Consequently, we can consider a vibrating part of the plate (that is one single antinode) as one individual piston. Let us consider a single antinode with diameter a, which vibrates with surface velocity v0, to be surrounded by a rigid plane. Based on this assumption, we can evaluate the frequency range in which the vibration identification can be performed by a sound pressure measurement in the vicinity of the vibrating surface. Near field and far field sound pressures on the antinode axis can be calculated by Eq. (6) [14,15] paxis ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ρ0 cv0 k 2 h þ a2 −h sin 2 πa2
ð6Þ
A distance from the antinode surface to the observation point is denoted with h, c is the speed of sound, ρ0 is the density of surrounding media and k is the wave number. Because we are interested in sound pressure on the antinode surface we can limit the height h in Eq. (6) toward zero to obtain an expression. paxis;VNF ¼ paxis ðh-0Þ ¼
2ρ0 cv0 ka sin 2 2 πa
ð7Þ
The pressure amplitude can be regarded as constant up to the frequency defined by a ratio a/λ¼0.26 (ka¼ 1.6), where sound pressure drops to 1 dB. This criterion is already accepted and used in measurement of loudspeakers [15], and it can be applied for the definition of the Very Near Field of other vibrating surfaces as well. For a low frequency conditions, where kao1, and for a far field conditions, where hba, Eq. (7) limits to the following form: paxis;FF ¼
ρ0 cv0 k π 2h
ð8Þ
However, if we observe Eq. (7) in very near field, h5a, and for low frequency condition kao1, it can be rewritten in the following form [15]: paxis;VNF ¼
ρ0 cv0 k π a
ð9Þ
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Fig. 4. Sound pressure along axis of a rigid circular piston radiating into a half-space freefield in frequency range ka 51 [15].
By dividing Eq. (9) with Eq. (8) we can obtain a relationship between sound pressure in a far field and sound pressure in a very near field on the axis of the piston. paxis;VNF ¼
2r ¼ paxis;FF a
ð10Þ
Eq. (10) shows, that sound pressure in a Very Near field is (kao1) directly proportional to the sound pressure in a far field at low frequency range. The relationship depends only on the ratio of the piston radius to the far field sample distance, and is independent from frequency. From a practical measurement standpoint of view, the very near field sound pressure and volume velocity are essentially independent of the environment into which the piston is radiating [15]. If the upper frequency of measurements is limited such that kao1, a division of Eq. (6) with Eq. (9) with a substitution of sin(X)≈X yields sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 paxis h h h þ a2 −h ð11Þ ¼ ¼ þ 1− a a a paxis;VNF Eq. (11) gives the low-frequency axial dependence of pressure on measuring distance normalized to the near field pressure occurring at h¼0. Plot of Eq. (11) is presented in Fig. 4 on a dB versus log(h/a) scale. Measurements correlate with this theoretical result very good, as discussed in Refs. [16,17]. To be within 1 dB of the very nearfield pressure, the pressure microphone must be no farther away from the center surface of the piston than 0.11a. For low frequency sound, far field conditions exist for distances beyond 2a. 2. Very near field above the vibrating plate Analytical results presented in Eqs. (9 and 11) and in Fig. 4 are based on an assumption, that individual antinode on the plate acts like a piston. This is a serious simplification, therefore an additional numerical calculation, FEM analysis and measurements were performed to confirm that such simplification can be applied and consequently yields to the applicability of measurements in a Very Near Field [3–5,15–19]. A sound pressure field, which is created above the vibrating surface, can be easily calculated for free field conditions, if the velocity pattern of the vibrating surface is well known. Velocity pattern of the vibrating structure was calculated numerically, following the procedure given in Ref. [20], and confirmed with measurements using a laser scanning vibrometer. Surface velocity pattern and acceleration pattern were subsequently included into a FEM analysis, and into the discrete Rayleigh integral given in Eq. (2), to estimate sound pressure in front of the plate with two different methods. FEM analysis was performed using a pressure acoustic model in a frequency domain. A 3D free tetrahedral mesh was used for the model. Edges of the 2 mm thick plate were described with hard surface boundary conditions. Both vibrating surfaces of the plate were described with surface normal acceleration boundary condition, according to the observed velocity pattern in Fig. 5. Front and back surface accelerations were out of phase, modeling a FFFF plate consisted from a number of dipole sources. A simple linear elastic fluid model was used for surrounding air. Air was bounded with a sphere with impedance match boundary conditions. The rate of sound pressure level decreases in front of the plate is presented in Fig. 5, for two points in front of the plate vibrating with a single mode; at the node in the center of the plate and in front of the first antinode above the center of the plate. Results, calculated with the discrete Rayleigh integral, and results obtained with FEM analysis are presented together. The diameter of the antinode, in front of which the sound pressure level decay was observed, is evaluated to have a diameter of 130 mm (Fig. 5B). 7 mm above the surface of this antinode a sound pressure level decreases for 1 dB. VNF in front of the single piston was defined and bounded with a decrease of the sound pressure level for 1 dB, which occurs at the ratio h/a ¼0.11. If the same criterion is applied for the vibrating plate at higher modes, a decrease for 1 dB can be observed at the lower value of ratio h/a ¼0.054. Results from the Rayleigh integral and results of FEM analysis confirm that the sound pressure level, in the frequency range of our interest, does not significantly vary with the distance, if it is observed very close to the vibrating surface.
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Fig. 5. Vibration of the plate square plate at 596 Hz measured with a laser scanning vibrometer (A), analytical solution for vibration mode (B), results of sound pressure level in the VNF in front of the plate obtained with FEM analysis and with numerical calculations using Eq. (2), (C).
A difference between the sound pressure level in front of the node and in front of the antinode is 20 dB up to 6 mm above the surface. This is a very important result, because it gives us 20 dB signal to noise ratio for visualization of mode vibrations. A difference between results obtained with FEM analysis and with numerical calculation using Eq. (2) is significant for heights above 10 mm. The difference originates from neglecting the radiation from the rear side of the plate and neglecting the diffraction and scattering of the sound when using the simple calculations using the Rayleigh integral. 3. Measurements In order to validate our method a thin, lightly damped square plate, and completely freely mounted was chosen for a test sample because its vibration modes and sound radiation can be relatively easy theoretically and numerically described and it is easy to handle in practice. Additionally, similar structures are routinely examined in industry via modal testing. Measurements of frequency response function are also performed under completely free boundary conditions. The scope of the measurements was limited by the following: 1. A thin square plate (400 400 2.5 mm3) was used as the test sample since elastic plates are found in many engineering applications. The example case is shown in Fig. 6; the free–free plate was excited by a point force at Xf ¼ M/2, Yf ¼ N/2. 2. No baffle is intentionally considered in our study to simulate industrial test conditions and to observe the acoustic short circuit effect. 3. Narrow band spectra (at resonant frequencies) up to 4000 Hz are considered. 4. Given the very near field location between microphone and surface of 3 mm, the Helmholtz number is less than 0.08 at 2000 Hz. The Rayleigh distance ranges from R0 ¼ 0.5 (a¼0.2 m, f ¼5000 Hz) to R0 ¼100,000 (a¼0.005 m, f ¼40 Hz). A square plate was hung from a supporting frame by means of elastic bands to achieve a free–free constraint condition. A shaker was mounted behind the plate and its output was fixed with a screw to the plate. Measurement setup is shown in Fig. 6. The plate was excited with a white noise in the frequency range between 40 Hz and 6 kHz. A 1/4" in. electret pressure microphone scanned 484 individual plate segments with dimension 18 18 mm2. A distance from the microphone to the vibrating plate surface was 3 mm. 20 s of sound pressure signal was recorded for each segment. The sampling frequency was 20 kHz and the resolution was 18 bit. During the measurement an FFT analysis was performed using Matlab and 484 spectra were recorded at the end of the measurement. At the same time a laser scanning vibrometer was used as a reference method for visualization of vibration modes.
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3 mm pressure microphone
FFFF suspended square steel plate
elastic rope
shaker
accelerometer
Fig. 6. Measurement setup.
The sound pressure spectrum was averaged over the whole vibrating plate with 484 individual segments. The averaged sound pressure spectrum, 3 mm from the plate surface, is presented in Fig. 9. Individual vibrating modes are very pronounced. Fig. 7 also indicates that good signal-to-noise ratio was achieved during the measurement and three most expressive modes are marked. Visualization results of vibration mode at 107 Hz are presented in Fig. 8, and of a vibration mode at 361 Hz are presented in Fig. 9. As expected, very good agreement between the analytical solution and the laser scanning measurements can be observed. Also, a very good agreement between the laser scanning measurements and measurements of sound pressure in a VNF can be observed. Visualization results from the sound pressure in a VNF exceed theoretical anticipations and they provide a picture with adequate resolution and correlation to the velocity field. This indicates an interaction in a VNF between the microphone membrane and the vibrating surface. The sound pressure, measured in a VNF and surface velocity are in very good agreement on the middle part of the plate. However, at the edge of the plate an acoustic short-circuiting occurs and suppresses the sound pressure level relative to the surface velocity level. A cross section plot of a surface velocity (measured with a laser scanning) and a cross section plot of sound pressure (measured in a VNF), at resonant mode at 107 Hz are presented in Figs. 10 and 11. Curves, which present measured sound pressure in a VNF, are scaled relative to the surface velocity measurements. A relative correlation between a surface velocity and a sound pressure in a VNF is very good for the area near the top of the plate, Fig. 10. A relative correlation between the surface velocity and sound pressure in a VNF is also very good for a cross section at the 1/3 of the plate length, Fig. 11. At the center of the plate, the correlation is very good and it exceeds anticipations. However, at the edges of the plate, where the un-baffled boundary condition in the form of an acoustic short circuit starts to dominate, the correlation is less precise. If the edge of the plate vibrates at its maximum, then the un-baffled boundary effect in the form of acoustic short circuit takes place. This is the only practical limitation of the mode visualization method with pressure scanning in a VNF. A visualization of the vibration mode in the vicinity of the free boundaries of a dipole source is therefore less accurate; however, useful results were obtained, even for un-baffled vibrating structure, which is the worst-case scenario.
4. Analysis of measurement results It is a traditional belief that measurements of acoustic quantities in a near field lead to a very low reproducibility and high measurement uncertainty. This is true because of a large number of acoustic minima and maxima which usually occur on a small area above the complex source. A position of the microphone has therefore more influence on the measured results as the radiated sound pressure itself. At the first glance, measurements in a very near field might suffer from the same effect; therefore an extensive analysis was performed exclusively to evaluate the reproducibility and uncertainty of method used in this paper. Parameters, which have significant influence on the measurement results, were identified from theoretical background, and studied by analyzing the measurement reproducibility;
effect of a distance between the vibrating surface and the observation point; (height of the microphone) acoustic effect of the semi-closed volume between the microphone and the vibrating plate effect of the microphone presence on the vibrations of the plate influence of the positioning accuracy of the microphone in the x- and y-directions along the vibrating surface on the vibration mode visualization influence of the microphone direction on the measured values of sound pressure
201
Sound pressure [mPa]
J. Prezelj et al. / Mechanical Systems and Signal Processing 40 (2013) 194–207
Frequency [Hz] Fig. 7. Averaged sound pressure spectrum, over the steel FFFF plate, excited by white noise.
Fig. 8. Results of vibration mode visualization for square plate at 107 Hz.
Fig. 9. Results of vibration mode visualization for square plate at 361 Hz.
Surface velocity [m/s] Sound pressure [Pa]
0.0025 Soundpressure [Pa]
Surfacevelocity [m/s]
0.003
0.002 0.0015 0.001 0.0005 0 0
0.2 0.4 0.6 0.8 Relative position of the plate
1
observed cross section
Fig. 10. Comparison between measured surface velocity and sound pressure in a very near field of a square plate at 107 Hz near the top of the plate.
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Surface velocity [m/s] Sound pressure [Pa]
0.002 Soundpressure[Pa]
Surfacevelocity[m/s]
0.0025
0.0015 0.001 0.0005 0
0
0.2 0.4 0.6 0.8 Relative position of the plate
1
observed cross section
Fig. 11. Comparison between measured surface velocity and sound pressure in a very near field of a square plate at 107 Hz near the center of the plate.
90 1 dB drop
Sound pressure level [dB]
80
Antinode at 596 Hz
70 Antinode at 361 Hz
60 50 40
Node at 596 Hz
30 20
Node at 361 Hz
10
VNF
NF
0 0.5 5 50 500 Distance from the surface [mm] Fig. 12. Sound pressure level as a function of the distance from the vibrating surface, measured in front of the node and antinode of the vibrating plate.
Additional FEM simulations were performed to analyze the
effect of the microphone presence on the sound pressure itself, and the dependence of sound pressure from the distance between the vibrating surface to the observation point; (height of the microphone)
4.1. A distance between the vibrating surface and the observation point Numerical calculations, analytical results and FEM analysis showed that sound pressure level generated by vibrating surface does not significantly depend on the distance from the surface in so called Very Near Field. The influence of the distance from the plate to the microphone on the measured sound pressure is negligible compared to the influence of the complexity of vibration modes. Sound pressure level in a VNF changes for a less than 1 dB if distance from the plate is within 3 mm; however, it changes for more than 20 dB if microphone is moved from the node to the antinode or vice versa. Additional measurements were performed to confirm such theoretical results, and to monitor the reproducibility of the measurement of sound pressure in a VNF. Measurements were performed with a setup shown in Fig. 6. A plate was hanged and the microphone fixture was moved to some arbitrary distance in front of the predetermined plate location. After the microphone was fixed, a distance between the vibrating surface and the microphone was measured. Spacers were used for measuring very small distances; 0.5 mm, 1 mm, 1.5 mm, 2 mm and 4 mm. Results for two modes at 361 Hz and 596 Hz are shown in Fig. 12. Results confirm that sound pressure level does not significantly depend on the distance from the vibrating plate in a VNF, and that its value significantly decreases outside the VNF region. The VNF region extends up to 10 mm above
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the vibrating surface if a 1 dB drop of sound pressure level is used for the limit between the VNF and Near Field [15] as shown in Fig. 12. Measured curves are very similar to results obtained with numerical calculations and FEM analysis in Fig. 5. Measured results are in fact even better than expected. The sound pressure level in front of the antinode is for more than 26 dB higher from the sound pressure level in front of the node, which gives us a comfortable range of 26 dB for mode visualization. Measured sound pressure level in the VNF of the vibrating plate is shown in Fig. 13 It is presented as a function of the height from the vibrating surface, for different frequency ranges between 125 Hz and 2000 Hz. Results are normalized to a decrease of sound pressure level from the estimated sound pressure level on the vibrating surface. Results show that the rate of the decrease of the sound pressure level in a VNF depends on frequency, and that the decrease rate is more pronounced in higher frequency range. Such results are expected and show that VNF is in higher frequency range limited. According to measuring results the VNF is only 3 mm above the vibrating surface at 2000 Hz. Fortunately, Eigen-modes of different structures are usually in lower frequency range.
4.2. Acoustics effect of the semi-closed volume between the microphone and the vibrating surface A volume of air, between the microphone membrane and the vibrating surface, acts like a coupling element if the microphone is placed very close to the vibrating surface. In order to analyze this effect a mechanical analogy to the acoustical problem is proposed and is shown in Fig. 14. A mass of the microphone membrane mmic is connected to the ground with a spring k1, which represents the stiffness of the membrane. Sound pressure between the membrane and the vibrating structure is established due to the resistance of the radiation mass toward the movement. A radiation mass is connected with two springs k2 and k3 toward a vibrating surface and toward a microphone membrane. These two springs present a stiffness of the air in the semi-closed volume;
Deviation of measured SPL regarding to estimated SPL on the vibrating surface [dB]
0
-0.5 125 Hz -1
500 Hz
-1.5
1000 Hz
2000 Hz
-2
2 4 6 Distance of the microphone from the vibrating surface [mm]
0
8
Fig. 13. Sound pressure level measured in front of the vibrating plate in a VNF, as a function of the height h from the vibrating surface.
k1 y, y, y
B1 mmic
k2
z, z, z
h x, x, x
mrad
z, z, z
y, y, y
x, x, x
B2
B3
k3
F(t)
D Fig. 14. Mechanical analogy to the acoustical problem.
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therefore k2 and k3 have the same values. The stiffness of the microphone membrane k1 is much higher than the stiffness of the air gap, k1 bk2 and k3. The effect of the damping is estimated in coefficients B1, B2, and B3. The coefficients of damping strongly depend on the height of the microphone h. The higher the microphone, the lower the value for damping, and vice versa. During our basic analysis, the damping was neglected due to very low particle velocities and practically no change in heat transfer conditions. The microphone membrane is very light and its typical resonant frequency is above 10 kHz, therefore, the microphone membrane can be regarded as a rigid object for the fluid behavior in a small volume in frequency ranges below 4000 Hz. For such a simplified case, a radiation mass of the vibrating surface under the microphone can be calculated together with the stiffness of the air for the semi-closed volume. Consequently, a resonant frequency of this semi-closed volume can be simply evaluated. The radiation mass can be calculated from the kinetic energy, which is introduced into the fluid by moving the surface under the microphone. Because the displacement of the surface is very small in comparison to the volume above, a linear approach can be used. Vibrating the surface below the microphone can be treated as a source of volume displacement with a cylindrical envelope. A radiation mass mrad can be calculated mrad ¼
πD3 ρ 16
ð12Þ
In Eq. (5) D represents the diameter of the microphone and ρ represents the density of the air. Stiffness of the air (k2 and k3) under the microphone can be calculated from the analogy to the acoustic resonator. In such an approach, the stiffness of the air trapped between the microphone membrane and vibrating surface is given by Eq. (6) !2 p0 πD2 k¼χ ð13Þ V0 4 where p0 presents a static pressure and V0 the volume of the air trapped between the microphone membrane and vibrating surface. From the stiffness and radiation mass we can estimate a resonant frequency ω0 (Eq. (7)) sffiffiffiffiffiffi p0 D ð14Þ ω0 ¼ πχ ρ V0 A resonant frequency depends on the ratio between the diameter of the microphone D, which is usually 1/2 in. or 1/4 in., and the volume of the air under the microphone, V0. The resonant frequency of the acoustic system increases if the microphone is getting closer and closer to the vibrating surface. The volume V0 is getting smaller and smaller, while the radiation mass is the same. A resonant frequency of the acoustic system is in any case much higher from the anticipated frequency range of our measurements. Additionally, by moving the microphone close to the vibrating surface we are gaining in the response. 4.3. Effect of the microphone presence on the sound pressure If a microphone is placed above the vibrating surface, free field conditions are no longer achieved because the microphone affects the sound field between the vibrating surface and a microphone membrane. By moving the microphone closer and closer to the vibrating surface, a sound field between the microphone membrane and the vibrating surface builds up. To analyze the influence of the microphone presence in the VNF on the sound pressure field itself, a few FEM simulations were performed. A square plate with dimensions 400 400 mm2 was simulated to vibrate as a rigid object with 1000 Hz, and the generated sound pressure level was observed at 1.5 mm above the plate. The sound pressure level across the plate is presented in Fig. 15A for perfect free field conditions. Seven microphones were later placed 3 mm above the vibrating surface. The sound pressure level was observed again at 1.5 mm above the vibrating surface. The sound pressure level across the plate with seven microphones in the VNF is presented in Fig. 15B. The influence of the microphone on the sound pressure field is only just noticeable. In order to observe the influence of the microphones, a difference between the two results is presented in Fig. 7. An influence of seven microphones on the measured sound pressure field itself can now be clearly observed. The level of influence depends on the frequency, and is typically stronger in higher frequency range. At 1 kHz the sound pressure field changes for about 70.4 dB. In the lower frequency range, the influence of the microphone on the sound pressure field is less pronounced. These results indicate that for a determination of the absolute value for the vibration velocity this effect should be taken into account. However, for the vibration mode visualization it practically plays no role. Acoustic propagation in fluids involves pressure, density, and temperature variations associated to potential movement (without any shear movement) in adiabatic conditions. However, in real fluids, diffusive processes occur (momentum diffusion due to the viscosity effect and thermal diffusion caused by heat conduction), especially near the boundaries of the fluid domain where acoustic boundary layers are developed. Such boundary layers are very thin (from about 0.2 mm at 20 Hz and to 20 um at 20 kHz for air. When this phenomenon is taken into account, the microphone proximity is limited to 0.2 mm [21].
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0.5
50 32 Hz
0.4
63 Hz 125 Hz 250 Hz 500 Hz
40
Sound pressure level difference [dB]
35 30
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20 15
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0.0
-0.2 125 Hz
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m
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0
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0
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-0.6
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45
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Fig. 15. A sound pressure level (A) over the vibrating square plate (B) in perfect free field conditions. A difference in sound pressure level (C) 1.5 mm above the vibrating square plate when nine microphones are placed 3 mm above the surface (D).
Additional measurements were performed in order to evaluate the influence of the microphone on the vibrations of the plate itself. An accelerometer was fixed on the same location as microphone, only on the opposite side. Microphone was slowly approaching to the vibrating surface while vibration levels were observed. Even when the microphone was placed only 0.5 mm above the vibrating plate, the vibration level did not change in observed frequency range from 40 Hz to 4000 Hz for more than 0.3 dB, which is evaluated uncertainty of measurement equipment. 4.4. Reproducibility A reproducibility of the mode visualization with measurements in a VNF was investigated for two significantly different objects; steel plate and bass bow. While a steel plate offers a perfect platform for testing the theory, the bass bow presents quite a challenge for the method, while there is not enough surface area on the round stick to effectively generate sound pressure. Both measured object were scanned for five times along the longest spatial dimension. Plate was scanned at 31 points across its width and the bass bow was scanned at 60 points along its length. Five averaged spectra across the plate and five averaged spectra across the bow were obtained. Each averaged spectra was obtained from 31 measured sound pressure spectra for the plate and from 60 measured spectra for the bass bow. Reproducibility is very good and Relative Standard Deviation (RSD) of sound pressure is less than 0.3 in the linear scale, across the frequency range up to 4 kHz and its average value is 0.075. Such a result confirms that the stability of the measurement equipment, including manual scanning, was sufficient for research and that measuring setup is working in a very stable manner. Reproducibility of vibration mode visualization on two different objects at single selected frequency is presented in Fig. 16. Results of scanning the sound pressure in a VNF above the vibrating bass bow are presented on the left side and results of scanning the plate are presented on the right. A scanning reproducibility is presented with a RSD as a spatial function in Fig. 16. RSD values are obtained from five repetitions. An average RSD value of bass bow measurements is 0.09, and only at certain points increases to values over 0.4. An average RSD value of the plate scanning is 0.14, and increases to over 0.3 at the edges of the plate. Numerical calculations, analytical results, FEM analysis and measurements showed that sound pressure level generated by vibrating surface does not significantly depend on the distance from the surface in so called Very Near Field. Consequently, the positioning error of the microphone distance from the plate (h) does not have the significant effect on measured sound pressure in comparison to the positioning error of the microphone location of the plate (x, y). 4.5. Uncertainty due to microphone directivity Scanning of the sound pressure over the vibrating objects in their VNF is in most cases performed manually. Surfaces are not always flat and in some cases microphone cannot be placed and directed perfectly perpendicular to the vibrating
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Fig. 16. Reproducibility of scanning the sound pressure in a Very Near field along the bass bow at 1500 Hz (left) and across the plate at 120 Hz (right).
2.0
Deviation [dB]
1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 5
500
5000
Frequency [Hz] Fig. 17. Deviation of results due to the direction error relative to the normal of the vibrating surface. Experimental results were performed on the vibrating plate. Center of the microphone was kept at the same distance from the plate 3.5 mm.
surface. Therefore an additional experiment was performed to analyze the influence of angle on measured sound pressure in a VNF. Results are presented in Fig. 17. Results show that the expected error originating from inaccurate directing the microphone toward the plate is less than 1 dB in frequency range below 2 kHz.
5. Conclusions The possibility of identifying vibration modes of a thin structure, with a non-contact measurement by using sound pressure, was investigated. Sound pressure above the vibrating surface depends on the velocity distribution of the vibrating surface. The inverse method for source strength estimation can be used to determine vibrating modes from the measurements of sound pressure on the parallel plane. It was demonstrated, that measurements of sound pressure should be performed close to the vibrating surface in order to provide suitable information for inverse calculations. Therefore, a very near field (VNF) was investigated. The VNF is limited by the thermal boundary layer of medium and with the medium viscosity on one side. On the other side, the VNF is limited by the geometry of the source, on the radiated sound wavelength and consequently by the sound pressure level decay vs. distance ratio. It was found out that an inverse calculation does not significantly improve the visualization result obtained directly from the sound pressure level if measurements of sound pressure are performed in a VNF. In order to find out how close to the vibrating surface a microphone should be placed, different aspects of the setup were checked. – A mechanical analogy was used to analyze the frequency response of the measuring setup. It was established that by placing a microphone closer to the surface, the upper frequency range of increases. The closer to the surface, the higher the applicable upper frequency range.
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– FEM analyses were performed to determine sound pressure amplitude very close to the vibrating surface. It was found out that the amplitude of sound pressure practically does not change up to 10 mm above the surface in a frequency range up to 4000 Hz. – An FEM analysis was performed to establish if the presence of the microphone affects the very near sound pressure field. It was found out that in a frequency range up to 2000 Hz the sound pressure between the vibrating plate increases by less than 0.3 dB. This is systematic error and does not affect the results of the vibration mode visualization. Encouraged by the results of the analysis, a visualization of square plate vibration modes was performed. After considering all limitations a measuring distance from the vibrating surface was set to 3 mm. An un-baffled and completely free steel plate was chosen as a test sample because it presents a complex real life situation. At the same time it reveals problems with acoustic short circuit at the edge of the un-baffled plate. Numerical calculations, analytical results, FEM analysis, and measurements showed that sound pressure level generated by vibrating surface does not significantly depend on the distance from the surface in so called Very Near Field. Consequently, the positioning error of the microphone distance from the plate (h) does not have the significant effect on measured sound pressure in comparison to the positioning error of the microphone location of the plate (x, y). The very good results of the vibration mode visualization by using a sound pressure scanning in a VNF were achieved for all modes below 4000 Hz. Correlation between the surface velocity and the sound pressure is poorer only at the edges of the plate, where an un-baffled boundary condition in the form of the acoustic short circuit starts to dominate. Sound pressure in the VNF of the plate at its free edges is suppressed relative to the surface velocity, and therefore the mode visualization is not exact. Such results indicate that sound pressure scanning in the VNF can be used for a vibration mode visualization of closed shells with some restrictions in the frequency range, background noise, and acoustic environment. The method is very cost effective, and only a bit time consuming. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
F. Martinus, D.W. Herrin, A.F. Seybert, Surface vibration reconstruction using inverse numerical acoustics, Proc. ITB Eng. Sci. 35B (1) (2003) 61–80. M.G. Prasad, J. Pattabiraman, B.V.A. Rao, On acoustic near field measurements, J. Sound Vib. 40 (3) (1975) 415–423. J. Žumer, et al., Characterization of the vibrations and structural noise of a suction units cover, Strojniški vestnik—J. Mech. Eng. 54 (12) (2008) 883–891. I. Salom, et al., Modelovanje oscilovanja zvona snimanjem zvuka u bliskom polju, 17. Telekomunikacioni forum TELFOR 2009 Srbija, Beograd, November 24–26, 2009. Zdenek Havranek, Visualization of sound fields on vibrating steel beam using linear microphone array, in: Proceedings of the 6th WSEAS International Conference on Signal Processing, Computational Geometry & Artificial Vision, Elounda, Greece, August 21–23, 2006, pp. 170–173. M.N. Helfricka, C. Niezreckia, P. Avitabilea, T. Schmidtb, 3D digital image correlation methods for full-field vibration measurement, Mech. Syst. Signal Process. 25 (3) (2011) 917–927. M. Martarelli, G.M. Reve, Laser Doppler vibrometry and near-field acoustic holography: different approaches for surface velocity distribution measurements, Mech. Syst. Signal Process. 20 (6) (2006) 1312–1321. Q. Leclere, B. Laulagnet, Particle velocity field measurement using an ultra light membrane, Appl. Acoust. 69 (2008) 302–310. H. Nelisse, O. Beslin, J. Nicolas, A generalized approach for the acoustic radiation from baffled or unbaffled plate with arbitrary boundary conditions, immersed in a light or heavy fluid, J. Sound Vib. 211 (2) (1998) 207–225. H.E. de Bree, V.B. Svetovoy, R. Raangs, R. Visser, The very near field—theory, simulations, and measurements of sound pressure and particle velocity in the very near field, ICSV-11, Petersburg, 2004. L.E. Kinsler, A.R. Frey, Fundamentals of Acoustics, Willey, New York, 1962. P.A. Nelson, S.H. Yoon, Estimation of acoustic source strength by inverse methods: Part I, conditioning of the inverse problem, J. Sound Vib. 233 (4) (2000) 639–664. Y. Kim, P.A. Nelson, Spatial resolution limits for the reconstruction of acoustic source strength by inverse methods, J. Sound Vib. 265 (3) (2003) 583–608. D.T. Blackstock, Fundamentals of Physical Acoustics, John Wiley & Sons, NewYork, 2000. D.B. Keele, Low frequency loudspeaker assessment by nearfield sound pressure measurements, J. Audio Eng. Soc. 22 (3) (1974). J. Prezelj, et al., Sound in the Very Near Field of the Vibrating Structures, in: Proceedings of Forum Acusticum 2011, 27 June–01 July, Aalborg, Denmark, European Acoustics Association, 2011, pp. 2303–2308. J. Prezelj, et al., Investigation on identifying structural modes by sound pressure signals, E I, Elektrotech. Inf. Tech. 126 (5) (2009) 194–199. J. Prezelj, M. Čudina, Very Near Field and Far Field Pressure Oscillations, Generated by the Housing of a Centrifugal Blower, in: Proceedings from Euronoise, Edinburg, Institute of Acoustics, 2009, pp. 1–8. L. Soria, V.C. Belgiovine, P. Aarnoutse, H.V. Auweraer, K. Janssens, Experimental Analysis of the Very Near Field Produced by Different Acoustic Sources, ISMA, 20–22 September, Katholieke Universiteit Leuven, Belgium, 2010. A.W. Leissa. Vibration of Plates, National Aeronautics and Space Administration SP-160, 1969. T. Lavergne, N. Joly, S. Durand, Acoustic thermal boundary layers and thermal boundary conditions on thin bodies: application to membranes and fibers, Forum Acusticum, Aalborg, Denmark, 2011.
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
On acoustic very near field measurements J. Prezelj a,n, P. Lipar a, A. Belšak b, M. Čudina a a b
University of Ljubljana, Faculty of Mechanical Engineering, Askerceva 6, SI-1000 Ljubljana, Slovenia University of Maribor, Faculty of Mechanical Engineering, Slovenia
a r t i c l e in f o
abstract
Article history: Received 3 February 2012 Received in revised form 30 October 2012 Accepted 11 May 2013 Available online 15 June 2013
Information about vibration modes is needed during the planning of noise control measures on different parts of machinery. A visualization of vibration modes is a starting point and different methods can be used to visualize vibration modes. Some methods which incorporate an inverse calculation of surface velocity from the sound pressure on some boundary have already been proposed, among others. A direct inverse method based on the discretized Rayleigh integral was used in our work, to demonstrate that microphones should be placed close to the vibrating structure to provide an acoustic transfer matrix with a low condition number. It is demonstrated, that there is practically no need for the calculation of the inverse matrix if microphones are placed in a very near field of the vibrating structure. A single microphone placed in a very near field together with a reference vibration sensor provides sufficient information for producing the images of vibration modes. Analytical results, numerical results, FEM simulations and measurement results were used to prove that properties of the sound pressure in a very near field permit a cost effective visualization of the vibration modes. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Acoustics Very near field Inverse methods Sound source visualization Near field acoustic holography
1. Introduction A reconstruction of the surface velocity on a complex vibrating structure is important because it pinpoints the modes which dominate the surface vibration. If the surface velocity is reconstructed, it can be used to predict the sound pressure field and the radiated sound power, which may be more valuable than the surface velocity reconstruction itself [1]. A prediction of the surface velocity of the vibrating structure based on the sound pressure measurements in the near field is well known. An analysis of vibrations with such an approach can be traced back to 1958 [2]. Such methods are of particular interest when the structure is rotating or moving, and if it is too light or too hot to be instrumented by accelerometers. The described methods are commonly used [3–5], although a detailed analysis and validation were not available. The development of multi-channel data acquisition systems made more sophisticated methods like the Nearfield Acoustic Holography (NAH), the Helmholtz differential equation (HELS), and Inverse numerical acoustics (INA) applicable. Advances in computers and digital imaging technology have allowed 3D digital image correlation (DIC) methods to measure the shape and deformation of a vibrating structure, [6]. At the same time, a development in the laser technology made a laser scanning Doppler vibrometer readily available. Laser scanning vibrometers are used for vibration mode identification because they provide accurate results of the measured surface velocity component [7,8]. All methods are quite costly due to the dedicated equipment and software; therefore more cost efficient method was taken into consideration for our applications and use in the laboratory. The presented method is similar to NAH, however it ensures well conditioned
n
Corresponding author. Tel.:+386 40 837647. E-mail address: [email protected] (J. Prezelj).
0888-3270/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ymssp.2013.05.008
J. Prezelj et al. / Mechanical Systems and Signal Processing 40 (2013) 194–207
Observation plane P
195
Vibrating Surface V m
R1,1 R2,1 R3,1
dx
n=1,2,...N
dS RN,1
dy n
m=1,2,...M
h Fig. 1. Geometry of the theoretical setup.
transfer matrix without a need for the regularization. Theoretical background, numerical simulations and some measurement results obtained with this method are presented in this paper. 1.1. Sound pressure field generated by a vibrating plate Let us consider an arbitrary vibrating surface V in an acoustic free field. A complex vibrating surface can be divided on many elementary surfaces dS. Each elementary surface dS on the arbitrary shaped surface can be regarded as a simple point source of an outgoing sound wave [9–11], if the wavelength of the generated sound is much longer than the dimensions of this elementary source dS. Sound pressure, radiated from an arbitrary shaped surface can be calculated on the parallel plane P by using the Rayleigh surface integral. The sound pressure in any point P(x0,y0,z0) in a free field can be expressed with Eq. (1), pðx0 ; y0 ; z0 Þ ¼
iωρ0 e−ikR ∬S vðx; yÞdS; 2π R
ð1Þ
A discretization of the vibrating surface, together with a discretization of the parallel observation plane is presented in Fig. 1. Sound pressure on the plane P is observed and values of sound pressure at a given time and at a given position pi(xi,yi,zi) are written in a vector p. If the values of vibration velocity on the vibrating surface V are well known and described in a vector v, then a pressure plane described with a vector p can be calculated by using a discretized Rayleigh surface interval given in Eq. (2): pi ðxi ; yi ; zi Þ ¼
M N eikRm;n iωρ0 dx dy ∑ ∑ vm;n 2π m ¼ 1 n ¼ 1 Rm;n
ð2Þ
Sound pressure, on the plane P, and vibration velocity, on the vibrating surface V, can be written in a vector form. The influence of geometry setup can be described with the acoustic transfer matrix, (ATM) R. A simple equation can be used for the correlation between surface velocity and sound pressure (Eq. (3)). p ¼ Rv
ð3Þ
1.2. Inverse method for source strength estimation If sound pressure on plane P and the geometry of the vibrating structure are well defined, then a vibration velocity on the vibrating surface V can be calculated backward from Eq. (4). If we consider R−1 as an inverse of the ATM R, then a surface velocity vector v can be calculated from the measurements of sound pressure p in the parallel plane. v ¼ R−1 p
ð4Þ
However, the ATM matrix R should fulfill some conditions in order to have a useable inverse. If two vectors p and v have different sizes, the ATM is not square, and therefore the inverse matrix R−1 cannot be directly determined. There are quite a few methods to solve this problem. However, to simplify the approach we used both vectors (p and v) of the same size and kept the ATM R in a square form. A spatial resolution of the surface velocity is in such a case determined by the microphone grid size on the pressure plane. But, a square form of the ATM matrix is only a necessary condition for a direct invertibility approach, and it is not a sufficient condition yet. For the evaluation of the matrix invertibility, its condition number was used. A condition number is the ratio between the largest singular values of R to the smallest. Large condition numbers indicate a nearly singular matrix, and if the condition number of the matrix increases to values over 10.0E+16 the solution of
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the inverse problem is not certain. The condition number used in our case was calculated according to Eq. (5). κðRÞ≥
maxi jRii j mini jRii j
ð5Þ
A condition number of the ATM R depends – on the shape of the observation plane and on the distribution of the microphone locations. That is, on the geometry of the microphone array relative to the geometry of the source, – on the distance between the vibrating surface and the observation plane, marked with h in Fig. 1, – on the ratio between the wave number k of the generated sound and dimensions of the elementary surface on the source, marked with dx and dy in Fig. 1, – on the ratio between the wave number k and distance between the elementary surface dS and observation point, marked with Ri,j in Fig. 1, – on measurement noise, – and on some other factors [12]. The ATM R appears to be best conditioned when the number of sources and sensors is small, when the geometrical arrangement of sensors closely matches the assumed source array geometry, when the distance between the sources is the same as the distance between the sensors, when the sensor array is placed close to the source array and when the sensor array is positioned symmetrically with respect to the source array. The orthogonal simple microphone mesh was shown to be the optimal solution already in Refs. [12,13]. A few simulations were performed to analyze the condition number of the ATM for our case, where a square plate was used as a model. The results are shown in Fig. 2, and they clearly indicate that condition number is much smaller if the microphones are placed close to the vibrating surface, for low and for higher kdx (frequencies). The influence of the microphone distance h is more pronounced than the influence of the frequency. Results, therefore, clearly indicate that sound pressure should be measured as closely as possible to the complex vibrating sound source. Numerical experiments were performed to investigate how a distance of the observation plane influences the inverse calculation. In these numerical experiments a distribution of the surface velocity was determined from a sound pressure field by inverse calculation. Inverse calculations were performed from three different sound pressure fields, which were parallel to the vibrating surface and at distances h¼150 mm, h¼164 mm, and h¼178 mm above the vibrating surface. The surface was set to vibrate with 1000 Hz. Elementary surface dS¼ dxdy had a surface of 400 mm2. A grid of 22 22 elements covered a 440 mm width square plate. The results of the inverse calculation for three different conditions are presented in Fig. 3. If the observation plane is 150 mm or closer from the vibrating surface, the inverse calculation provides good results for the given conditions. If the observation plane is 178 mm or more from the vibrating surface, the inverse calculation cannot provide any results for a given conditions. This analysis illustrates that for an appropriate determination of the surface velocity based on the measurements of generated sound it is necessary to place a microphone in the vicinity of the vibrating surface. The question remains, how close to the vibrating surface a microphone should be placed to gather the appropriate information and in the same time not to significantly disturb the sound field itself.
Fig. 2. Condition number of the ATM R as a function of the distance and frequency.
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Fig. 3. Transformation from convergence to divergence of inverse surface velocity calculation using sound pressure when increasing the distance h, between the vibrating surface and the parallel plane of sound pressure.
1.3. Very near field If the piston is small compared with a wavelength of the sound it is emitting (ka51), it acts as a point source set a top of the reflecting plane [14]. Consequently, we can consider a vibrating part of the plate (that is one single antinode) as one individual piston. Let us consider a single antinode with diameter a, which vibrates with surface velocity v0, to be surrounded by a rigid plane. Based on this assumption, we can evaluate the frequency range in which the vibration identification can be performed by a sound pressure measurement in the vicinity of the vibrating surface. Near field and far field sound pressures on the antinode axis can be calculated by Eq. (6) [14,15] paxis ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ρ0 cv0 k 2 h þ a2 −h sin 2 πa2
ð6Þ
A distance from the antinode surface to the observation point is denoted with h, c is the speed of sound, ρ0 is the density of surrounding media and k is the wave number. Because we are interested in sound pressure on the antinode surface we can limit the height h in Eq. (6) toward zero to obtain an expression. paxis;VNF ¼ paxis ðh-0Þ ¼
2ρ0 cv0 ka sin 2 2 πa
ð7Þ
The pressure amplitude can be regarded as constant up to the frequency defined by a ratio a/λ¼0.26 (ka¼ 1.6), where sound pressure drops to 1 dB. This criterion is already accepted and used in measurement of loudspeakers [15], and it can be applied for the definition of the Very Near Field of other vibrating surfaces as well. For a low frequency conditions, where kao1, and for a far field conditions, where hba, Eq. (7) limits to the following form: paxis;FF ¼
ρ0 cv0 k π 2h
ð8Þ
However, if we observe Eq. (7) in very near field, h5a, and for low frequency condition kao1, it can be rewritten in the following form [15]: paxis;VNF ¼
ρ0 cv0 k π a
ð9Þ
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Fig. 4. Sound pressure along axis of a rigid circular piston radiating into a half-space freefield in frequency range ka 51 [15].
By dividing Eq. (9) with Eq. (8) we can obtain a relationship between sound pressure in a far field and sound pressure in a very near field on the axis of the piston. paxis;VNF ¼
2r ¼ paxis;FF a
ð10Þ
Eq. (10) shows, that sound pressure in a Very Near field is (kao1) directly proportional to the sound pressure in a far field at low frequency range. The relationship depends only on the ratio of the piston radius to the far field sample distance, and is independent from frequency. From a practical measurement standpoint of view, the very near field sound pressure and volume velocity are essentially independent of the environment into which the piston is radiating [15]. If the upper frequency of measurements is limited such that kao1, a division of Eq. (6) with Eq. (9) with a substitution of sin(X)≈X yields sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 paxis h h h þ a2 −h ð11Þ ¼ ¼ þ 1− a a a paxis;VNF Eq. (11) gives the low-frequency axial dependence of pressure on measuring distance normalized to the near field pressure occurring at h¼0. Plot of Eq. (11) is presented in Fig. 4 on a dB versus log(h/a) scale. Measurements correlate with this theoretical result very good, as discussed in Refs. [16,17]. To be within 1 dB of the very nearfield pressure, the pressure microphone must be no farther away from the center surface of the piston than 0.11a. For low frequency sound, far field conditions exist for distances beyond 2a. 2. Very near field above the vibrating plate Analytical results presented in Eqs. (9 and 11) and in Fig. 4 are based on an assumption, that individual antinode on the plate acts like a piston. This is a serious simplification, therefore an additional numerical calculation, FEM analysis and measurements were performed to confirm that such simplification can be applied and consequently yields to the applicability of measurements in a Very Near Field [3–5,15–19]. A sound pressure field, which is created above the vibrating surface, can be easily calculated for free field conditions, if the velocity pattern of the vibrating surface is well known. Velocity pattern of the vibrating structure was calculated numerically, following the procedure given in Ref. [20], and confirmed with measurements using a laser scanning vibrometer. Surface velocity pattern and acceleration pattern were subsequently included into a FEM analysis, and into the discrete Rayleigh integral given in Eq. (2), to estimate sound pressure in front of the plate with two different methods. FEM analysis was performed using a pressure acoustic model in a frequency domain. A 3D free tetrahedral mesh was used for the model. Edges of the 2 mm thick plate were described with hard surface boundary conditions. Both vibrating surfaces of the plate were described with surface normal acceleration boundary condition, according to the observed velocity pattern in Fig. 5. Front and back surface accelerations were out of phase, modeling a FFFF plate consisted from a number of dipole sources. A simple linear elastic fluid model was used for surrounding air. Air was bounded with a sphere with impedance match boundary conditions. The rate of sound pressure level decreases in front of the plate is presented in Fig. 5, for two points in front of the plate vibrating with a single mode; at the node in the center of the plate and in front of the first antinode above the center of the plate. Results, calculated with the discrete Rayleigh integral, and results obtained with FEM analysis are presented together. The diameter of the antinode, in front of which the sound pressure level decay was observed, is evaluated to have a diameter of 130 mm (Fig. 5B). 7 mm above the surface of this antinode a sound pressure level decreases for 1 dB. VNF in front of the single piston was defined and bounded with a decrease of the sound pressure level for 1 dB, which occurs at the ratio h/a ¼0.11. If the same criterion is applied for the vibrating plate at higher modes, a decrease for 1 dB can be observed at the lower value of ratio h/a ¼0.054. Results from the Rayleigh integral and results of FEM analysis confirm that the sound pressure level, in the frequency range of our interest, does not significantly vary with the distance, if it is observed very close to the vibrating surface.
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Fig. 5. Vibration of the plate square plate at 596 Hz measured with a laser scanning vibrometer (A), analytical solution for vibration mode (B), results of sound pressure level in the VNF in front of the plate obtained with FEM analysis and with numerical calculations using Eq. (2), (C).
A difference between the sound pressure level in front of the node and in front of the antinode is 20 dB up to 6 mm above the surface. This is a very important result, because it gives us 20 dB signal to noise ratio for visualization of mode vibrations. A difference between results obtained with FEM analysis and with numerical calculation using Eq. (2) is significant for heights above 10 mm. The difference originates from neglecting the radiation from the rear side of the plate and neglecting the diffraction and scattering of the sound when using the simple calculations using the Rayleigh integral. 3. Measurements In order to validate our method a thin, lightly damped square plate, and completely freely mounted was chosen for a test sample because its vibration modes and sound radiation can be relatively easy theoretically and numerically described and it is easy to handle in practice. Additionally, similar structures are routinely examined in industry via modal testing. Measurements of frequency response function are also performed under completely free boundary conditions. The scope of the measurements was limited by the following: 1. A thin square plate (400 400 2.5 mm3) was used as the test sample since elastic plates are found in many engineering applications. The example case is shown in Fig. 6; the free–free plate was excited by a point force at Xf ¼ M/2, Yf ¼ N/2. 2. No baffle is intentionally considered in our study to simulate industrial test conditions and to observe the acoustic short circuit effect. 3. Narrow band spectra (at resonant frequencies) up to 4000 Hz are considered. 4. Given the very near field location between microphone and surface of 3 mm, the Helmholtz number is less than 0.08 at 2000 Hz. The Rayleigh distance ranges from R0 ¼ 0.5 (a¼0.2 m, f ¼5000 Hz) to R0 ¼100,000 (a¼0.005 m, f ¼40 Hz). A square plate was hung from a supporting frame by means of elastic bands to achieve a free–free constraint condition. A shaker was mounted behind the plate and its output was fixed with a screw to the plate. Measurement setup is shown in Fig. 6. The plate was excited with a white noise in the frequency range between 40 Hz and 6 kHz. A 1/4" in. electret pressure microphone scanned 484 individual plate segments with dimension 18 18 mm2. A distance from the microphone to the vibrating plate surface was 3 mm. 20 s of sound pressure signal was recorded for each segment. The sampling frequency was 20 kHz and the resolution was 18 bit. During the measurement an FFT analysis was performed using Matlab and 484 spectra were recorded at the end of the measurement. At the same time a laser scanning vibrometer was used as a reference method for visualization of vibration modes.
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3 mm pressure microphone
FFFF suspended square steel plate
elastic rope
shaker
accelerometer
Fig. 6. Measurement setup.
The sound pressure spectrum was averaged over the whole vibrating plate with 484 individual segments. The averaged sound pressure spectrum, 3 mm from the plate surface, is presented in Fig. 9. Individual vibrating modes are very pronounced. Fig. 7 also indicates that good signal-to-noise ratio was achieved during the measurement and three most expressive modes are marked. Visualization results of vibration mode at 107 Hz are presented in Fig. 8, and of a vibration mode at 361 Hz are presented in Fig. 9. As expected, very good agreement between the analytical solution and the laser scanning measurements can be observed. Also, a very good agreement between the laser scanning measurements and measurements of sound pressure in a VNF can be observed. Visualization results from the sound pressure in a VNF exceed theoretical anticipations and they provide a picture with adequate resolution and correlation to the velocity field. This indicates an interaction in a VNF between the microphone membrane and the vibrating surface. The sound pressure, measured in a VNF and surface velocity are in very good agreement on the middle part of the plate. However, at the edge of the plate an acoustic short-circuiting occurs and suppresses the sound pressure level relative to the surface velocity level. A cross section plot of a surface velocity (measured with a laser scanning) and a cross section plot of sound pressure (measured in a VNF), at resonant mode at 107 Hz are presented in Figs. 10 and 11. Curves, which present measured sound pressure in a VNF, are scaled relative to the surface velocity measurements. A relative correlation between a surface velocity and a sound pressure in a VNF is very good for the area near the top of the plate, Fig. 10. A relative correlation between the surface velocity and sound pressure in a VNF is also very good for a cross section at the 1/3 of the plate length, Fig. 11. At the center of the plate, the correlation is very good and it exceeds anticipations. However, at the edges of the plate, where the un-baffled boundary condition in the form of an acoustic short circuit starts to dominate, the correlation is less precise. If the edge of the plate vibrates at its maximum, then the un-baffled boundary effect in the form of acoustic short circuit takes place. This is the only practical limitation of the mode visualization method with pressure scanning in a VNF. A visualization of the vibration mode in the vicinity of the free boundaries of a dipole source is therefore less accurate; however, useful results were obtained, even for un-baffled vibrating structure, which is the worst-case scenario.
4. Analysis of measurement results It is a traditional belief that measurements of acoustic quantities in a near field lead to a very low reproducibility and high measurement uncertainty. This is true because of a large number of acoustic minima and maxima which usually occur on a small area above the complex source. A position of the microphone has therefore more influence on the measured results as the radiated sound pressure itself. At the first glance, measurements in a very near field might suffer from the same effect; therefore an extensive analysis was performed exclusively to evaluate the reproducibility and uncertainty of method used in this paper. Parameters, which have significant influence on the measurement results, were identified from theoretical background, and studied by analyzing the measurement reproducibility;
effect of a distance between the vibrating surface and the observation point; (height of the microphone) acoustic effect of the semi-closed volume between the microphone and the vibrating plate effect of the microphone presence on the vibrations of the plate influence of the positioning accuracy of the microphone in the x- and y-directions along the vibrating surface on the vibration mode visualization influence of the microphone direction on the measured values of sound pressure
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Frequency [Hz] Fig. 7. Averaged sound pressure spectrum, over the steel FFFF plate, excited by white noise.
Fig. 8. Results of vibration mode visualization for square plate at 107 Hz.
Fig. 9. Results of vibration mode visualization for square plate at 361 Hz.
Surface velocity [m/s] Sound pressure [Pa]
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Surfacevelocity [m/s]
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observed cross section
Fig. 10. Comparison between measured surface velocity and sound pressure in a very near field of a square plate at 107 Hz near the top of the plate.
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Surface velocity [m/s] Sound pressure [Pa]
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Surfacevelocity[m/s]
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0
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observed cross section
Fig. 11. Comparison between measured surface velocity and sound pressure in a very near field of a square plate at 107 Hz near the center of the plate.
90 1 dB drop
Sound pressure level [dB]
80
Antinode at 596 Hz
70 Antinode at 361 Hz
60 50 40
Node at 596 Hz
30 20
Node at 361 Hz
10
VNF
NF
0 0.5 5 50 500 Distance from the surface [mm] Fig. 12. Sound pressure level as a function of the distance from the vibrating surface, measured in front of the node and antinode of the vibrating plate.
Additional FEM simulations were performed to analyze the
effect of the microphone presence on the sound pressure itself, and the dependence of sound pressure from the distance between the vibrating surface to the observation point; (height of the microphone)
4.1. A distance between the vibrating surface and the observation point Numerical calculations, analytical results and FEM analysis showed that sound pressure level generated by vibrating surface does not significantly depend on the distance from the surface in so called Very Near Field. The influence of the distance from the plate to the microphone on the measured sound pressure is negligible compared to the influence of the complexity of vibration modes. Sound pressure level in a VNF changes for a less than 1 dB if distance from the plate is within 3 mm; however, it changes for more than 20 dB if microphone is moved from the node to the antinode or vice versa. Additional measurements were performed to confirm such theoretical results, and to monitor the reproducibility of the measurement of sound pressure in a VNF. Measurements were performed with a setup shown in Fig. 6. A plate was hanged and the microphone fixture was moved to some arbitrary distance in front of the predetermined plate location. After the microphone was fixed, a distance between the vibrating surface and the microphone was measured. Spacers were used for measuring very small distances; 0.5 mm, 1 mm, 1.5 mm, 2 mm and 4 mm. Results for two modes at 361 Hz and 596 Hz are shown in Fig. 12. Results confirm that sound pressure level does not significantly depend on the distance from the vibrating plate in a VNF, and that its value significantly decreases outside the VNF region. The VNF region extends up to 10 mm above
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the vibrating surface if a 1 dB drop of sound pressure level is used for the limit between the VNF and Near Field [15] as shown in Fig. 12. Measured curves are very similar to results obtained with numerical calculations and FEM analysis in Fig. 5. Measured results are in fact even better than expected. The sound pressure level in front of the antinode is for more than 26 dB higher from the sound pressure level in front of the node, which gives us a comfortable range of 26 dB for mode visualization. Measured sound pressure level in the VNF of the vibrating plate is shown in Fig. 13 It is presented as a function of the height from the vibrating surface, for different frequency ranges between 125 Hz and 2000 Hz. Results are normalized to a decrease of sound pressure level from the estimated sound pressure level on the vibrating surface. Results show that the rate of the decrease of the sound pressure level in a VNF depends on frequency, and that the decrease rate is more pronounced in higher frequency range. Such results are expected and show that VNF is in higher frequency range limited. According to measuring results the VNF is only 3 mm above the vibrating surface at 2000 Hz. Fortunately, Eigen-modes of different structures are usually in lower frequency range.
4.2. Acoustics effect of the semi-closed volume between the microphone and the vibrating surface A volume of air, between the microphone membrane and the vibrating surface, acts like a coupling element if the microphone is placed very close to the vibrating surface. In order to analyze this effect a mechanical analogy to the acoustical problem is proposed and is shown in Fig. 14. A mass of the microphone membrane mmic is connected to the ground with a spring k1, which represents the stiffness of the membrane. Sound pressure between the membrane and the vibrating structure is established due to the resistance of the radiation mass toward the movement. A radiation mass is connected with two springs k2 and k3 toward a vibrating surface and toward a microphone membrane. These two springs present a stiffness of the air in the semi-closed volume;
Deviation of measured SPL regarding to estimated SPL on the vibrating surface [dB]
0
-0.5 125 Hz -1
500 Hz
-1.5
1000 Hz
2000 Hz
-2
2 4 6 Distance of the microphone from the vibrating surface [mm]
0
8
Fig. 13. Sound pressure level measured in front of the vibrating plate in a VNF, as a function of the height h from the vibrating surface.
k1 y, y, y
B1 mmic
k2
z, z, z
h x, x, x
mrad
z, z, z
y, y, y
x, x, x
B2
B3
k3
F(t)
D Fig. 14. Mechanical analogy to the acoustical problem.
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therefore k2 and k3 have the same values. The stiffness of the microphone membrane k1 is much higher than the stiffness of the air gap, k1 bk2 and k3. The effect of the damping is estimated in coefficients B1, B2, and B3. The coefficients of damping strongly depend on the height of the microphone h. The higher the microphone, the lower the value for damping, and vice versa. During our basic analysis, the damping was neglected due to very low particle velocities and practically no change in heat transfer conditions. The microphone membrane is very light and its typical resonant frequency is above 10 kHz, therefore, the microphone membrane can be regarded as a rigid object for the fluid behavior in a small volume in frequency ranges below 4000 Hz. For such a simplified case, a radiation mass of the vibrating surface under the microphone can be calculated together with the stiffness of the air for the semi-closed volume. Consequently, a resonant frequency of this semi-closed volume can be simply evaluated. The radiation mass can be calculated from the kinetic energy, which is introduced into the fluid by moving the surface under the microphone. Because the displacement of the surface is very small in comparison to the volume above, a linear approach can be used. Vibrating the surface below the microphone can be treated as a source of volume displacement with a cylindrical envelope. A radiation mass mrad can be calculated mrad ¼
πD3 ρ 16
ð12Þ
In Eq. (5) D represents the diameter of the microphone and ρ represents the density of the air. Stiffness of the air (k2 and k3) under the microphone can be calculated from the analogy to the acoustic resonator. In such an approach, the stiffness of the air trapped between the microphone membrane and vibrating surface is given by Eq. (6) !2 p0 πD2 k¼χ ð13Þ V0 4 where p0 presents a static pressure and V0 the volume of the air trapped between the microphone membrane and vibrating surface. From the stiffness and radiation mass we can estimate a resonant frequency ω0 (Eq. (7)) sffiffiffiffiffiffi p0 D ð14Þ ω0 ¼ πχ ρ V0 A resonant frequency depends on the ratio between the diameter of the microphone D, which is usually 1/2 in. or 1/4 in., and the volume of the air under the microphone, V0. The resonant frequency of the acoustic system increases if the microphone is getting closer and closer to the vibrating surface. The volume V0 is getting smaller and smaller, while the radiation mass is the same. A resonant frequency of the acoustic system is in any case much higher from the anticipated frequency range of our measurements. Additionally, by moving the microphone close to the vibrating surface we are gaining in the response. 4.3. Effect of the microphone presence on the sound pressure If a microphone is placed above the vibrating surface, free field conditions are no longer achieved because the microphone affects the sound field between the vibrating surface and a microphone membrane. By moving the microphone closer and closer to the vibrating surface, a sound field between the microphone membrane and the vibrating surface builds up. To analyze the influence of the microphone presence in the VNF on the sound pressure field itself, a few FEM simulations were performed. A square plate with dimensions 400 400 mm2 was simulated to vibrate as a rigid object with 1000 Hz, and the generated sound pressure level was observed at 1.5 mm above the plate. The sound pressure level across the plate is presented in Fig. 15A for perfect free field conditions. Seven microphones were later placed 3 mm above the vibrating surface. The sound pressure level was observed again at 1.5 mm above the vibrating surface. The sound pressure level across the plate with seven microphones in the VNF is presented in Fig. 15B. The influence of the microphone on the sound pressure field is only just noticeable. In order to observe the influence of the microphones, a difference between the two results is presented in Fig. 7. An influence of seven microphones on the measured sound pressure field itself can now be clearly observed. The level of influence depends on the frequency, and is typically stronger in higher frequency range. At 1 kHz the sound pressure field changes for about 70.4 dB. In the lower frequency range, the influence of the microphone on the sound pressure field is less pronounced. These results indicate that for a determination of the absolute value for the vibration velocity this effect should be taken into account. However, for the vibration mode visualization it practically plays no role. Acoustic propagation in fluids involves pressure, density, and temperature variations associated to potential movement (without any shear movement) in adiabatic conditions. However, in real fluids, diffusive processes occur (momentum diffusion due to the viscosity effect and thermal diffusion caused by heat conduction), especially near the boundaries of the fluid domain where acoustic boundary layers are developed. Such boundary layers are very thin (from about 0.2 mm at 20 Hz and to 20 um at 20 kHz for air. When this phenomenon is taken into account, the microphone proximity is limited to 0.2 mm [21].
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0.5
50 32 Hz
0.4
63 Hz 125 Hz 250 Hz 500 Hz
40
Sound pressure level difference [dB]
35 30
1000 Hz
25
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20 15
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-0.2 125 Hz
250 Hz
500 Hz
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80
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m
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m
m
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m
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45
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Fig. 15. A sound pressure level (A) over the vibrating square plate (B) in perfect free field conditions. A difference in sound pressure level (C) 1.5 mm above the vibrating square plate when nine microphones are placed 3 mm above the surface (D).
Additional measurements were performed in order to evaluate the influence of the microphone on the vibrations of the plate itself. An accelerometer was fixed on the same location as microphone, only on the opposite side. Microphone was slowly approaching to the vibrating surface while vibration levels were observed. Even when the microphone was placed only 0.5 mm above the vibrating plate, the vibration level did not change in observed frequency range from 40 Hz to 4000 Hz for more than 0.3 dB, which is evaluated uncertainty of measurement equipment. 4.4. Reproducibility A reproducibility of the mode visualization with measurements in a VNF was investigated for two significantly different objects; steel plate and bass bow. While a steel plate offers a perfect platform for testing the theory, the bass bow presents quite a challenge for the method, while there is not enough surface area on the round stick to effectively generate sound pressure. Both measured object were scanned for five times along the longest spatial dimension. Plate was scanned at 31 points across its width and the bass bow was scanned at 60 points along its length. Five averaged spectra across the plate and five averaged spectra across the bow were obtained. Each averaged spectra was obtained from 31 measured sound pressure spectra for the plate and from 60 measured spectra for the bass bow. Reproducibility is very good and Relative Standard Deviation (RSD) of sound pressure is less than 0.3 in the linear scale, across the frequency range up to 4 kHz and its average value is 0.075. Such a result confirms that the stability of the measurement equipment, including manual scanning, was sufficient for research and that measuring setup is working in a very stable manner. Reproducibility of vibration mode visualization on two different objects at single selected frequency is presented in Fig. 16. Results of scanning the sound pressure in a VNF above the vibrating bass bow are presented on the left side and results of scanning the plate are presented on the right. A scanning reproducibility is presented with a RSD as a spatial function in Fig. 16. RSD values are obtained from five repetitions. An average RSD value of bass bow measurements is 0.09, and only at certain points increases to values over 0.4. An average RSD value of the plate scanning is 0.14, and increases to over 0.3 at the edges of the plate. Numerical calculations, analytical results, FEM analysis and measurements showed that sound pressure level generated by vibrating surface does not significantly depend on the distance from the surface in so called Very Near Field. Consequently, the positioning error of the microphone distance from the plate (h) does not have the significant effect on measured sound pressure in comparison to the positioning error of the microphone location of the plate (x, y). 4.5. Uncertainty due to microphone directivity Scanning of the sound pressure over the vibrating objects in their VNF is in most cases performed manually. Surfaces are not always flat and in some cases microphone cannot be placed and directed perfectly perpendicular to the vibrating
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Fig. 16. Reproducibility of scanning the sound pressure in a Very Near field along the bass bow at 1500 Hz (left) and across the plate at 120 Hz (right).
2.0
Deviation [dB]
1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 5
500
5000
Frequency [Hz] Fig. 17. Deviation of results due to the direction error relative to the normal of the vibrating surface. Experimental results were performed on the vibrating plate. Center of the microphone was kept at the same distance from the plate 3.5 mm.
surface. Therefore an additional experiment was performed to analyze the influence of angle on measured sound pressure in a VNF. Results are presented in Fig. 17. Results show that the expected error originating from inaccurate directing the microphone toward the plate is less than 1 dB in frequency range below 2 kHz.
5. Conclusions The possibility of identifying vibration modes of a thin structure, with a non-contact measurement by using sound pressure, was investigated. Sound pressure above the vibrating surface depends on the velocity distribution of the vibrating surface. The inverse method for source strength estimation can be used to determine vibrating modes from the measurements of sound pressure on the parallel plane. It was demonstrated, that measurements of sound pressure should be performed close to the vibrating surface in order to provide suitable information for inverse calculations. Therefore, a very near field (VNF) was investigated. The VNF is limited by the thermal boundary layer of medium and with the medium viscosity on one side. On the other side, the VNF is limited by the geometry of the source, on the radiated sound wavelength and consequently by the sound pressure level decay vs. distance ratio. It was found out that an inverse calculation does not significantly improve the visualization result obtained directly from the sound pressure level if measurements of sound pressure are performed in a VNF. In order to find out how close to the vibrating surface a microphone should be placed, different aspects of the setup were checked. – A mechanical analogy was used to analyze the frequency response of the measuring setup. It was established that by placing a microphone closer to the surface, the upper frequency range of increases. The closer to the surface, the higher the applicable upper frequency range.
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– FEM analyses were performed to determine sound pressure amplitude very close to the vibrating surface. It was found out that the amplitude of sound pressure practically does not change up to 10 mm above the surface in a frequency range up to 4000 Hz. – An FEM analysis was performed to establish if the presence of the microphone affects the very near sound pressure field. It was found out that in a frequency range up to 2000 Hz the sound pressure between the vibrating plate increases by less than 0.3 dB. This is systematic error and does not affect the results of the vibration mode visualization. Encouraged by the results of the analysis, a visualization of square plate vibration modes was performed. After considering all limitations a measuring distance from the vibrating surface was set to 3 mm. An un-baffled and completely free steel plate was chosen as a test sample because it presents a complex real life situation. At the same time it reveals problems with acoustic short circuit at the edge of the un-baffled plate. Numerical calculations, analytical results, FEM analysis, and measurements showed that sound pressure level generated by vibrating surface does not significantly depend on the distance from the surface in so called Very Near Field. Consequently, the positioning error of the microphone distance from the plate (h) does not have the significant effect on measured sound pressure in comparison to the positioning error of the microphone location of the plate (x, y). The very good results of the vibration mode visualization by using a sound pressure scanning in a VNF were achieved for all modes below 4000 Hz. Correlation between the surface velocity and the sound pressure is poorer only at the edges of the plate, where an un-baffled boundary condition in the form of the acoustic short circuit starts to dominate. Sound pressure in the VNF of the plate at its free edges is suppressed relative to the surface velocity, and therefore the mode visualization is not exact. Such results indicate that sound pressure scanning in the VNF can be used for a vibration mode visualization of closed shells with some restrictions in the frequency range, background noise, and acoustic environment. The method is very cost effective, and only a bit time consuming. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
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