Vibrational velocity estimation by means of PNAH

Transcription

1 Vibrational velocity estimation by means of PNAH S. Potgieter DCT Bachelor Final Project Supervisor: dr.ir. I. Lopez Arteaga Technische Universiteit Eindhoven Department Mechanical Engineering Dynamics and Control Technology Group Eindhoven, June, 2009

2 Special thanks to Rick Scholte Rens Kodde Peter Hamels Pieter Nuij for their help during this project. 2

3 Contents 1 Introduction Investigation Theory of NAH Angular Spectrum Velocity Description of the measurement setup Performed experiments Data processing steps Processing the holography data Leakage Border padding L-curve regularization Processing the accelerometer data Results 29 6 Conclusion 33 Bibliography 36 A Derivation of normal component of velocity 39 B FFT mfile 41 C Cut out mfile 45 D Design drawings of the plate 47 E Measurement result plots 49 3

4 4 Contents

5 Chapter 1 Introduction Sound plays a big role in day to day life. Crisp sound from the speaker in your cell phone for easy communication, loud, vibrating beats in a dance club that resonate in your chest, the high pitched beeping noise coming from the television or the annoying zooming noise in the trains of the Nederlandse Spoorwegen: sometimes we cannot get enough of sound and sometimes you want to chuck the TV out the window just to stop it from beeping. One of the applications of Near-field Acoustical Holography (NAH) is the precise detection of sound sources in, for example, a cell phone, enabling product developers to take more effective measures to increase or reduce the sound coming from their device [1]. Unnecessary vibrations in the cell phone can thus be located and removed, resulting in less prematurely chucked phones. The term holography usually refers to a context of the use of light as the medium for presenting information and stems from the Greek words holos and gramma, meaning whole and message [14]. When referring to sound, which is a wave of pressure in air, the whole message refers to the use of both the amplitude of the wave and the phase of this wave with respect to a reference. The idea of using both phase and amplitude information in this way was developed in the 1940 s by Gabor [14] [16], but what unlocked the potential of acoustical holography was the use of the pressure information very close to the sound source, or in the near-field. This concept was introduced in the 1980 s by Williams and Maynard [8], who showed that the theoretical resolution of acoustical holography could be greatly increased by using evanescent waves; waves with larger wavenumbers than their propagating counterparts and which only occur in the near-field. Thus Near-field Acoustical Holography was introduced. Inherent to the math behind NAH, however, is a critical balance of two exponential terms, which makes NAH very sensitive to noise in the measured data. The figurative pressure was on, therefore, to find ways to manipulate the data so that its full potential could be exploited. This report is an effort to see how far along the road of accuracy recent developments have brought NAH. The foundation of Near-field Acoustical Holography is that once the pressure in a single plane between 0 z is known, the pressure field over any three-dimensional volume from z = 0 to infinity can be calculated without any more information. To obtain the pressure in a certain two-dimensional plane, say z = z h, microphones are used, placed in an array at height z h from the source. This z h is called the hologram plane. The position of the source plane is defined as 5

6 Chapter 1. Introduction z = 0 (see Figure 1.1). Figure 1.1: Schematic representation of the source plane, on z = 0, and the plane where the microphones are positioned: the hologram plane z = z h [9]. 1.1 Investigation The main question to be investigated in this project was: How accurately can the velocity of an oscillating piston be determined with planar NAH? More investigated questions were: 1. What happens to the accuracy if the amplitude of vibration is altered? 2. What happens to the accuracy if the distance between the source and the measurement plane is altered? 3. How do border effects (turbulence, acoustical short-circuitry, etc.) influence the measurement? Hereto a setup was made with which measurements could be done that would answer the questions, the concept of NAH needed to be understood and the research questions answered. It was chosen to investigate only planar Near-field Acoustic Holography, the stepping stone to other geometries [18, Chap. 3.1]. Therefore a rigid plate, vibrated using a shaker, was selected as the object to be measured. 6

7 Chapter 1. Introduction This report is organized as follows: in Chapter 2 the theory behind NAH is explained. Chapter 3 starts with a description of the measurement setup and an explanation of the choices behind it. This is followed by a description of the performed experiments. The steps that are needed to process the measurement data are presented in Chapter 4 and in Chapter 5 the results of these steps are given and the research questions answered. Finally, a conclusion in Chapter 6 ends the report. 7

8 8 Chapter 1. Introduction

9 Chapter 2 Theory of NAH Sound is an oscillation of pressure transmitted through a solid, liquid, or gas, composed of frequencies within the range of hearing and of a level sufficiently strong to be heard [4]. Of course, many pressure oscillations in liquids, solids or gases are not of an audible level and so do not produce sound, but they are still measurable and are therefore useful for acoustical holography. Two equations that govern the oscillations of acoustic pressure are 1. the wave equation 2. and the simplified Euler s equation of conservation of momentum. The linearized, homogeneous acoustic wave equation can be derived from the continuity equation, the equation of state (ideal gas law) and the equation of conservation of momentum [12, Chap ] and is given by (2.1). 2 p 1 c 2 2 p t 2 = 0 (2.1) Herein p [Pa] is an infinitesimal variation of pressure from its equilibrium value, dependent on three-dimensional space and time p p(x, y, z, t) and c [ m s ] is the speed of sound in the medium under consideration (air in this report). The right hand side of (2.1) indicates that there are no sources in the volume in which the equation is valid. The simplified Euler s equation of conservation of momentum is given by [18, page 15] ρ 0 v t = p (2.2) where ρ 0 is the density of the medium (in this case air) and v is the velocity vector of the particles of the medium with components u, v and ẇ. v = uî + vĵ + ẇˆk (2.3) Equations (2.1) and (2.2) are the only link needed to the physical world to make acoustical holography work. The following steps are all methods of calculation. 9

10 Chapter 2. Theory of NAH It simplifies the necessary calculations a lot if these are done in the frequency domain and the wavenumber domain, k. The value k is called the acoustic wave number, defined as k = ω c [ rad m ] and represents the number of waves per unit of distance [9, page 8], analogous to the frequency f which is the number of waves per unit of time. For both the frequency domain and the wavenumber domain, the Fourier transform is needed. Some important theorems regarding the Fourier transform which will be used later are given in (2.4), 2.5 and (2.6) [10]. F {p(t)} = F 1 { p(ω)} = 1 2π p(t) e iωt dt = p(ω) (2.4) p(ω) e iωt dω (2.5) F { (n) p t (n) } = (iω) (n) p(ω) (2.6) The wave number k is related to x (in one dimension) through the Fourier transform in the same way as the frequency f or angular speed ω = 2π f is to t, as can be seen in (2.7) [18, page 1] compared to (2.4). F {p(x)} = p(x) e ikxx dx = p(k x ) (2.7) According to Williams [18], applying the Fourier transforms (2.4) and (2.6) to the wave equation (2.1) results in (2.8), which is also called the Helmholtz equation, and applying (2.6) to Euler s equation (2.2) results in (2.9). 2 p(x, y, z, ω) + k 2 p(x, y, z, ω) = 0 (2.8) iωρ 0 v = p(x, y, z, ω) (2.9) A solution to the Helmholtz equation (2.8) is given by (2.10). p(x, y, z, ω) = A(x, y, z, ω)e i(k xx+k y y+k z z) (2.10) With k 2 = k 2 x + k 2 y + k 2 z (2.11) The exponential term in (2.10) is the wave form, and A(ω) is an arbitrary constant. If we consider only one wave, and therefore only one frequency, then k = ω c = 2π f c = constant. The variable of interest here is k z, since k x and k y can be measured in the hologram plane (see Figure 1.1 on page 6). There can be two situations: 1. k 2 x+k 2 y k 2 resulting in k z = k 2 k 2 x k 2 y being real and positive. Because the observation is made in z > 0 and we have assumed that there are only sources in z 0, there can be no observed plane waves traveling in the negative z-direction. Therefore k z has to be positive. 2. kx 2 + k 2 y > k 2 resulting in k z = k 2 kx 2 k 2 y = i kx 2 + k 2 y k 2 = ik z being imaginary. 10

11 Chapter 2. Theory of NAH The second situation results in the solution to the Helmholtz equation (2.10) being [18], [9]: p(x, y, z, ω) = A(x, y, z, ω)e k zz e i(k xx+k y y) (2.12) which decays exponentially as the wave travels in the positive z-direction. This type of wave is called an evanescent wave, while the wave described by (2.10) and k 2 x + k 2 y k 2 has a constant pressure amplitude and is called a propagating or plane wave. The difference between propagating and evanescent waves can be made clear with more ease in 2D than in 3D, so let k y = 0 for the moment. The wavelength λ [m] of a propagating wave traveling in the x-z-plane is defined as λ = c f = 2πc ω = 2πc kc = 2π k (2.13) If we were to tilt this wave so that it travels along the x-axis (see Figure 2.1), its wave number would be exactly k = k x. This means that if a k x can still be measured such that k x > k (and therefore c > c x and λ > λ x ), there travels on or with the propagating wave some wave(s) with smaller wavelengths; disturbances, if you will. These are evanescent waves and die out very quickly. Figure 2.1: Plane wave traveling normal to x-axis [18] Physically, evanescent waves are pressure fluctuations caused by the interaction of individual air particles which do not have enough energy to create a wave with a wavelength of λ, and therefore a wavenumber of k. Say a plate at z s = 0 vibrates as a whole at f = 1000 Hz in air of 20 C. If this plate gives enough energy to the air particles it collides with on z > z s, these particles = m in the positive z-direction. This is a propagating wave with wavenumber k = 2π/λ = rad/m. Some air particles the plate collides with will not have enough energy to cause a propagating wave in the positive z- direction and will lose their energy in the form of a pressure flux in the x, y and z directions. The will cause a propagating wave with wavelength λ = c f =

12 Chapter 2. Theory of NAH wavelengths of these pressure fluctuations are a lot smaller than λ, and therefore, according to (2.13), k evanescent > k. Because of their exponentially decaying nature, evanescent waves can only be measured very close to the source, or in the so called near-field. Farther away from the source their amplitudes are so small that they cannot be distinguished from noise. Evanescent waves are, however, a very important source of information for acoustical holography, because of their short wavelengths λ. Measuring these increases the theoretical resolution of holography tremendously. The resolution of a field is defined by how rapidly the field varies in space. If in any direction (for example the x-direction) there are no wavenumbers larger than a certain k max = 2π/λ min, the minimum distance over which the field varies in space, or the resolution distance, is defined by Williams [8] as R π/k max [m]. Say then that you measure only the propagating waves for which k 2 x + k 2 y k 2 holds, the maximum value of k x or k y is k. As can be seen in equation (2.14), measuring only propagating waves limits the theoretical resolution of holography therefore to the wavelength of the radiation. R max = π = πλ min k max 2π = λ min 2 (2.14) Evanescent waves, however, can have arbitrarily large wavenumbers so, in theory, R max can be tremendously increased by measuring these. The actual resolution of a measured hologram however, depends very much on how the measurements are performed. The choice of the cut-off k-number, the dynamic range of the field, the distance between the microphones the pressure is measured with, the distance between the source and the microphones (height of z) etc. are all important factors that determine the actual resolution of the holography [16]. 2.1 Angular Spectrum The solution to the Helmholtz equation, equation (2.10), will now be written in a different form. The pressure distribution p(x, y, z) can be written as the sum of many propagating and evanescent waves. Instead of the arbitrary constant A(x, y, z, ω) the exponential term describing the wave form e i(k xx+k y y+k z z) in (2.10) is multiplied with a coefficient P(k x, k y ), called the angular spectrum, according to Williams [18]. This coefficient accounts for the different phases and amplitudes of the waves in the pressure distribution so is in fact the pressure distribution in wavenumberdomain. Furthermore, a continuum of possible wavenumbers is expected and in theory x and y can extend over an infinite space so the sum of waves can be written as an integral. Therefore, Williams gives the total pressure distribution as p(x, y, z) = 1 4π 2 dk x dk y P(k x, k y )e i(k xx+k y y+k z z) (2.15) 1 where the constant makes (2.15) analogous to a two-dimensional inverse Fourier transform (see (2.5)). Once P(k x, k y ) is known, (2.15) can be used to calculate the pressure field over 4π 2 12

13 Chapter 2. Theory of NAH the three-dimensional volume from z = 0 to infinity, without any more information. This statement is written as P(k x, k y, z) = P(k x, k y, z h )e ik z(z z h ) (2.16) This means that if the pressure distribution or angular spectrum in a plane z = z h is known, one can construct the pressure distribution on an arbitrary plane z by multiplication with a so called propagator, in this case e ik z(z z h ). This is also valid for evanescent waves for which the angular spectrum (2.16) reduces to P(k x, k y, z) = P(k x, k y, z h )e k z(z z h ) (2.17) 2.2 Velocity Analogous to the way equation (2.16) was found from the linearized, homogenous wave equation (2.1), Euler s equation of momentum (2.2) can be Fourier transformed to find for the normal component of velocity k z Ẇ(k x, k y, z) = ρ 0 ck P(k x, k y, z h )e ik z(z z h ) (2.18) = G(k x, k y, z z h )P(k x, k y, z h ) (2.19) where G(k x, k y, z z h ) = k z ρ 0 ck eik z(z z h ) is called the velocity propagator [18, Chap. 3.2]. For a derivation of (2.19), see Appendix A. It is this normal component of velocity and its inverse transform ẇ(x, y, z) = Fx 1 Fy 1 [G(k x, k y, z z h )P(k x, k y, z h )] (2.20) that is the object of interest in this report. To find it, one has to consider one of two situations: either z z h or z < z h. If the pressure at a point, plane or 3D object farther away from the source than where the measurement of the pressure distribution is taking place is of interest, i.e. if z z h, the solution is a forward problem, and can be solved using the Rayleigh integrals (see [18, Chap. 2.10]). However, if the velocity or pressure at a point, plane or 3D object nearer to the source is of interest, as is the case in this report, then z < z h, which makes G the inverse velocity propagator and the therefore whole problem becomes inverse. In this situation the Rayleigh integrals are of no help and Near-field Acoustical Holography (NAH) can offer a solution. The mathematics behind NAH are summarized in (2.20) [18, Chap. 3.2], so the theory behind NAH ends there. The complications, however, do not. Consider that the resolution of acoustical holography is improved considerably if evanescent waves (for which k z = ik z and therefore (2.17) holds) are taken into account, and consider the inverse velocity propagator for evanescent waves G(k x, k y, z z h ) = k z ρ 0 ck ek z(z h z) (2.21) 13

14 Chapter 2. Theory of NAH This means that as k x and k y increase to infinity so will G(k x, k y, z z h ). In order for the product G(k x, k y, z z h )P(k x, k y, z h ) to remain finite, the angular spectrum P(k x, k y, z h ) has to decay faster with rising k x and k y than G, which is in fact the case for evanescent waves: P(k x, k y, z h ) = P(k x, k y, z)e k z(z h z) (2.22) This is a delicate balance, however. The measured data the angular spectrum is based upon is discreet and could be full of noise. Obtaining and processing the data in such a way that the balance is disrupted as little as possible is the challenge with NAH [14]. How accurately the velocity of an object can be calculated is a measure for how well the present methods do this and is the subject of this report. 14

15 Chapter 3 Description of the measurement setup To investigate the research questions given in the introduction, a measurement setup was built with which a plate was vibrated using a shaker (V201 Ling Dynamic Systems Vibrator). The velocity of this plate had to be determined using planar NAH and another method simultaneously, so that the velocity estimation by means of NAH could be compared to the velocity obtained with this other method. Because of the accuracy of laser interferometry, it would have been the first choice to measure the velocity of the plate with. However, because of the NAH setup, measuring simultaneously with interferometry and the microphone for NAH was impractical, and the use of accelerometers was decided upon. The output signal of the Kistler 8732A500 accelerometer and microphones were measured with a SigLab analyzer. The accelerometer was stuck to the back of the plate (between the plate and the shaker) with beeswax, and as close to the center as possible. The Kistler 8732A500 accelerometer was preferred above two other available accelerometers, the PCB 302A and the Kistler 8628B50, for its size and weight: it weighs only 1.1 gram, against 6.7 gram for the Kistler 8628B50 and 25 gram for the PCB 302A. This small accelerometer will therefore introduce negligible extra forces onto the plate. A plate size had to be chosen such that the first eigenfrequency of the plate was above the highest frequency the plate would be vibrated with. Using the Rayleigh-Ritz method of approximation, Olson [13] states that the first eigenfrequency of a circular plate with free edges and supported in the center is approximately f n = 0.172h r 2 E ρ(1 ν 2 ) with h the thickness of the plate, r the radius, ρ the density of the material of the plate, E the Young s modulus and ν the Poisson s ratio of the material. The range of measurement was determined somewhat by what the shaker is capable of, but the lower frequency-range was preferred, so the first eigenfrequency had to be above 500Hz. Estimated was that a plate of 50mm in diameter was about the smallest it could comfortably be to ensure that the plate would still give of enough radiation to be used with NAH. The necessary thickness of an aluminium plate with a radius r = m would then, according to (3.1), be h = (3.1) f n r 2 ( ) 1/2 E ρ(1 ν 2 (3.2) ) 15

16 Chapter 3. Description of the measurement setup = 500 ( ) 2 ( ( ) = 0.336mm The equation for a simply supported rectangular plate given in [2] and a circular plate fixed at the boundary given in [6, Chap ] give answers of the same order of magnitude, so this can be considered a valid value. However, driving the plate back and forth in the center would cause the edge to deflect; a phenomenon useful to validate the measurements with but unwanted otherwise. To keep this deflection at a minimum, therefore, a plate thickness of 2mm was chosen. A more complicated design of the plate, one that makes it possible for the accelerometer to be in the very center of the plate, was not merited by the simplicity of the envisioned experiments. Figure 3.1(a) shows a picture of the shaker, plate and accelerometer. The design drawings of the plate can be found in Appendix D. ) 1/2 (a) With small accelerometer. (b) With both accelerometers and suspension. Figure 3.1: Shaker and plate. The small Kistler accelerometer 8732A500 used in the measurements was calibrated with Kistler s most accurate accelerometer, the Vibration Standard 8076K. The transfer function between these accelerometers was measured using a random signal. The transfer function is defined as [17, Chap. 2] H(s) = Y(s) U(s) = L {y(t)} L {u(t)} (3.3) where y(t) and u(t) are respectively the output and input signal and L {y(t)} is the Laplace transform of y(t) (see [10, Chap. 3]). If the magnitude of this transfer function is equal to one, it would 16

17 Chapter 3. Description of the measurement setup mean that the magnitude of the input and output signals are equal to each other. Taking the input signal as the signal of Kistler s most accurate accelerometer and the output as the signal of small Kistler accelerometer, the transfer function gives a measure for how accurate the small Kistler is in comparison to the most accurate Kistler. The magnitude of the transfer function was close to one for most of the frequency range if the signal of the small Kistler was multiplied with a value of 925 [ m s 2 /Volt] (see Figure 3.2(a)). Furthermore, multiplied by this value the magnitude of the transfer function drops down to 0.98 just above 50Hz and then descends rapidly for lower frequencies. Therefore, the small Kistler accelerometer is not suited for measurements below 50Hz, and the sensitivity of the accelerometer for the frequency range of interest was 925 m s 2 /Volt. Using the transfer function, it has been assumed that the relation between the input and output is that of a linear system. To check this, the coherence function can be used [7]. Its value is one for the entire frequency range of interest (Figure 3.2(b)), so the relationship between input and output was indeed linear. The same calibration was performed for the ten times more sensitive sensor, the Kistler 8628B50, with similar results for the coherence and transfer function (Figures 3.2(c) and 3.2(d)), if its value was multiplied with a factor 102 m s 2 /Volt. To get a measure for how much inaccuracy was introduced by not placing the accelerometer in the center of the plate, the transfer function between the very sensitive Kistler 8628B50, placed on the top and in the center of the plate, and the small Kistler, placed where it would be in the actual measurements, namely on the back of the plate as close to the center as possible, was measured. A chirp signal of 50 to 500 Hz was used so the results which can be seen in Figure 3.3(a) are only valid for that frequency range. The results seen in Figure 3.3(a) are not perfect: a variation in the magnitude of the transfer function between 0.98 and 1.02 between 50 and 400Hz. This could be caused by slight sideways movements of the plate, by deflection of the edge of the plate or because of internal variations in the sensitivity of the two accelerometers. To see if possible sideways movements could be suppressed, an auxiliary suspension was mounted which provided additional stiffness in the plane perpendicular to the movement (Figure 3.1(b)). The transfer function of the measurement with auxiliary suspension is shown in Figure 3.3(b). The result shows a slightly better correlation for frequencies lower than 100Hz and higher than 300Hz, but new peaks are introduced around 200Hz. Furthermore, the amplitude of the resonance frequency at 500Hz, where the plate oscillates on the stiffness of the shaker coil, is reduced strongly. It would seem that the difference in the signals of the two accelerometers is influenced by sideways motion of the plate, which is suppressed by the auxiliary suspension for frequencies lower than 100Hz and higher than 300Hz but strengthened around 200Hz. However, the fact that the magnitude of the transfer function rises can be explained by the variations in internal variations in the sensitivity of the accelerometers. In Figure 3.2(a) can be seen that the sensitivity of the small Kistler rises somewhat from 50 to 550Hz. Furthermore, in Figure 3.2(c) can be seen that the sensitivity of the more accurate Kistler descends somewhat. Dividing the signal of the small accelerometer by that of the larger one therefore results in a more strongly rising magnitude like in Figures 3.3(a) and 3.3(b). The measurements were performed in a so-called anechoic chamber; a room that produces no echoes. The acoustical hologram was measured with a digital microphone (DigiSiMic TC100Z21A), attached to a robotic arm. The software of the measurement device designed by Rick Scholte uses this arm to move the microphone over an array of measurement points that can be defined by the 17

18 Chapter 3. Description of the measurement setup Magnitude Frequency [Hz] (a) Small Kistler: transfer function. Magnitude Frequency [Hz] (b) Small Kistler: coherence. Magnitude Frequency [Hz] (c) Larger Kistler: transfer function. Magnitude Frequency [Hz] (d) Larger Kistler: coherence. Figure 3.2: Calibration plots of the accelerometers with respect to Kistler s most accurate accelerometer. user (the hologram plane in Figure 1.1, page 6). The distance between two measurement points, the number of measurement points and therefore the size of the plane which holds the array can thus be set. Phase information of the pressure oscillation is not lost because the software compares the measured data to a reference, in this case the output signal received by the shaker. Because the Nyquist criterion for analogue to digital processing is equally valid for the spatial domain, according to Scholte [15] the minimal observable wavelength is λ min = 2 x (3.4) with x the distance between two measurement points. The distance between the measurement points is very important because (3.4) determines the maximum observable k-number and therefore the maximal resolution (see equation (2.14)): k max,ob = π x (3.5) 18

19 Chapter 3. Description of the measurement setup Magnitude Frequency [Hz] (a) Without suspension. Magnitude Frequency [Hz] (b) With suspension. Figure 3.3: Transfer functions: the signal of the small Kistler accelerometer divided by the signal of the more sensitive Kistler accelerometer. The hologram parameters for the performed measurements are listed in Table 3.1. A distance of 5mm between two consecutive measurement points was chosen, making the maximal observable wavenumber k max,ob = rad/m, and the minimal observable wavelength λ min = 10 mm. Furthermore, in order for the measurement array to cover the complete diameter of the plate and a little extra, the number of measurement points chosen was 16 in both the x and y direction and the center measurement point was chosen to be above the center of the plate. This made the plane that held the measurement array a square of 75mm by 75mm, with the circular plate (50mm in diameter) nicely in the middle. Table 3.1: Hologram measurement parameters. sampling distance 5 mm nr. of sampling points x 16 nr. of sampling points y 16 max. observable wavenumber rad/m min. observable wavelength 10 mm hologram size x 75 mm hologram size y 75 mm The digital microphone used in the measurements was calibrated with an indirect method. Because there was no suitable mount for the digital microphone on the Sound Level Calibrator (Type B&K 4230), an analog microphone (Sonion 8002) was calibrated with the Calibrator and placed next to the digital microphone in the anechoic chamber. A white noise signal from a speaker was measured with both microphones, and the transfer function between them (Figure 3.4(a)) was used to calibrate the digital microphone; similar to how the accelerometers were calibrated. In this way a value of 1.1 Volt/Pa was found for the digital microphone. The coherence of this measurement, shown in Figure 3.4(b), was not very good however; the amplitude of the 19

20 Chapter 3. Description of the measurement setup signal of the analog microphone seemed to vary much for different frequencies, no matter what the measured signal was, while that of the digital did not. Therefore, the value of 1.1 Volt/Pa comes with a large uncertainty Magnitude Magnitude Frequency [Hz] (a) Transfer function Frequency [Hz] (b) Coherence. Figure 3.4: Calibration plots of the digital microphone with respect to the analog microphone. 3.1 Performed experiments Two sets of measurements have been performed, the parameters of which are listed in Table 3.2. In the first, the pressure distribution has been measured at z = 1cm and z = 2cm from the plate with the plate excited by a chirp signal rising from 50Hz to 500Hz at an amplitude of 0.2 Volts peak. In the second experiment, the pressure distribution at z = 1cm from the plate was measured with the plate excited by a 90Hz sine wave and an amplitude of 0.2 Volts peak and 0.7 Volts peak. During these measurements the acceleration was also recorded. Both the acceleration data and the holography data had to be processed before it could be compared and the research questions answered. The processing of the holography data, especially, takes quite a few steps, most of which are automated in the software designed by Rick Scholte. experiment 1 experiment 2 hologram distance 1cm and 2cm 1cm signal chirp: Hz sine: 90Hz amplitude 0.2Volts peak 0.2 and 0.7Volts peak Table 3.2: Experiment parameters. 20

21 Chapter 4 Data processing steps 4.1 Processing the holography data The angular spectrum P(k x, k y, z h ) had to be obtained in such a way that the solutions to the inverse problems following from equations (2.17) and (2.19) do not blow up because of high wavenumbers in measured noise. The processing needed to do this consists of five steps which will be described briefly in the following. After these five steps the inverse problems can be solved and the pressure distribution and velocities at the source can be calculated. The accuracy of this calculation depends for a very large part on how much correct information eventually gets into the angular spectrum, therefore, the five post-processing steps are wholly aimed at getting all of the information into k-space and the frequency domain, without introducing errors, and then removing the influence of noise Leakage First of all, a temporal anti-leakage filter is applied to the data. Secondly, it is Fourier transformed to the frequency domain. The Fourier transform is an integration from minus infinity to infinity, but the data it is applied to is finite because it is not very practical to sample the data for an infinite amount of time. In the integration of the Fourier transform, the data that is sampled for a time T is simply assumed to be periodic, which can lead to sudden transitions from one period to the next (see Figure 4.1). The FFT (fast Fourier transform) tries to find sinusoids that can describe the sampled function as accurately as possible, therefore sinusoids with high frequencies are introduced to cover the sudden transition. The amplitudes of these higher frequencies are increased at the expense of the amplitude of the frequency that is actually present in the signal: leakage occurs. To reduce this phenomenon, a filter that forces the edge of the sampled data to zero is used. In this way, the periodicity assumed by the Fourier transform does not introduce new frequencies into the spectrum because there are no sudden transitions: the first period ends at zero and the second starts at zero. 21

22 Chapter 4. Data processing steps T time (s) Figure 4.1: A graphic representation of what the FFT integration does to a finitely sampled simple sine wave. T is the sampling time interval. A sudden transition from one period to the next occurs Border padding After the data has been Fourier transformed to the frequency domain, spatial border padding and/or zero padding and windowing is applied and the filtered, transformed and padded data is again Fourier transformed. This time the Fourier transform is spatial and in two dimensions to obtain the pressure distribution on the hologram plane P(k x, k y, z h ) as a function of k x and k y instead of x and y. Applying the filter that removes noise in the spatial domain is the fifth and final step. If this filter was applied before the Fourier transform, it would have resulted in a double convolution integral (see [3], [11] and [18, Chap 1.2]) which is hard to solve, but in the k-space, it results in a simple multiplication. More importantly, however, it would have diminished the amount of information that ended up in k-space while it is exactly this information that will later determine the result of the inverse propagation (think, for example, of the influence of k x, k y, k z and k z on the inverse velocity propagator (2.21)). Therefore, filtering for noise is done in the wavenumber domain. With NAH, border padding is applied in the two spatial dimensions, x and y, but the same principle can be applied in the time domain. Take for instance the simple sine function given in Figure 4.1 again. If the sample, taken in the sampling time T, is border padded, information at the border of the sampled data is used to extrapolate it outward [14, Chap 3.2]. This makes for a much larger sample. If a window, a function that forces the edges of the sample to zero so that no leakage occurs, is now applied, no actual data will be forced to zero, but only made-up extrapolated data. This way no higher frequencies are introduced by the Fourier transform and the data is not filtered: the complete sample has been transformed to the frequency domain, or k-space domain with NAH. 22

23 Chapter 4. Data processing steps L-curve regularization Step five, the filtering of the pressure distribution in k-space, was the only step of which the influence on the final result was investigated in this project. There are many types of filters of which a Tikhonov, an exponential and a modified exponential filter are examples, but the only one used here was the modified exponential filter. In Figure 4.2(a) an example of such a k-space filter can be seen. The cut-off wavenumber k co is one characteristic of the filter that can be varied and is defined as the point in k-space where the filter has dropped in magnitude by half [14, Chap ]. Another characteristic that can be varied in a modified exponential filter is the slope with which the filter drops from the all-pass region (magnitude 1) to the no-pass region (magnitude 0), but this had little effect on the measurements of this project. (a) Example of a k-space filter. (b) Example of an L-curve. Figure 4.2: Examples of an L-curve and k-space filter [9, Chap. 2.3]. The ideal k co was determined using a so called L-curve criterion. This is a log-log plot of two calculations done with P(k x, k y, z h ) and a filter with a certain k co (see Figure 4.2(b)). On the x-axis, the norm of the pressure distribution at the hologram plane minus the filtered pressure distribution at the hologram plane is represented. This is called the norm of the residual. To find this value, the filter with a certain k co is applied to the pressure distribution at the hologram plane and the result is subsequently subtracted from the measured pressure distribution at the hologram plane. The residual should be as small as possible, because the pressure distribution at the hologram plane is simply the correct pressure distribution and corrupting it is undesirable. On the y-axis, the norm of the pressure distribution at an arbitrary plane closer to the source is represented. Choosing a very large k co means that most of the wavenumbers will go through the filter: under filtering. Therefore the residual is very small; the filtered pressure distribution at the hologram plane is almost the same as the actual pressure distribution. However, the back propagation is blown up because of the exponential term in the inverse propagator and high k-numbers in noise that have not been filtered out. Therefore the norm of the pressure distribution at an arbitrary plane closer to the source than the hologram plane (i.e. z < z h ) is very large for small values of the residual. Choosing the k co too low, i.e. letting very little wavenumbers through (over filtering), results in a very large residual: the filtered pressure distribution is a distortion of the actual one. However, it also means the solution to the inverse problem does not blow up because of noise. 23

24 Chapter 4. Data processing steps The optimal k co is somewhere in the middle. The maximal slope of the L-curve should be a good point and can be determined mathematically. Experiments have shown though, that it is often not the optimal point [9, Chap 2.3] [14, Chap ], so the cut-off wavenumber was tweaked a little further in this project. Figure 4.3 shows pictures of the absolute value of the speed in z-direction at the source (the vibrating plate), calculated with a modified exponential filter with k co = rad/m (4.3(a)) and k co = rad/m (4.3(c)). Here the difference is most clearly visible. The high k-numbers that are still present after filtering with a high k co blow up the solution, while using a lower k co they are cut out and the solution is not blown up. 30 x x x Number of points in y direction Number of points in x direction Number of points in y direction Number of points in x direction (a) From above, k co = rad/m, at the source. (b) From the side, k co = rad/m, at the source. 30 x x Number of points in y direction Number of points in x direction x Number of points in y direction Number of points in x direction (c) Blown up. From above, k co = rad/m, at the source. (d) Blown up. From the side, k co = rad/m, at the source. Figure 4.3: Magnitude of the normal component of the speed ẇ(x, y, z) of the plate, vibrating at 100Hz and an amplitude of 0.2 Volts peak, calculated with high and low k co. Once the holography data has been filtered, padded and transformed enough to arrive at a pressure distribution that is as noise-free and as full of useful information as possible, the normal velocity at the source, ẇ(x, y, z s ), can be calculated using the inverse propagator G(k x, k y, z s z h ) from (2.20) and (2.19). Because it is still in the frequency domain, ẇ is not truly the normal velocity dependent on time. At one frequency, ẇ gives the amplitude and phase information of 24

25 Chapter 4. Data processing steps the variation of the velocity. The amplitude of ẇ is therefore the amplitude of the variation of the velocity at the source, but because this variation is a sinusoid, the amplitude of its variation is the maximal velocity at the source. This maximal velocity occurs as the piston travels through its center position. 4.2 Processing the accelerometer data The accelerometer data was used to determine the reference velocity. By assuming this reference to be true, the accuracy of the velocity estimation by means of NAH could be determined by comparing it to this reference. To arrive at the reference velocity, the acceleration data had to be processed: it had to be integrated and transformed to the frequency domain. Remember that ẇ(x, y, z) is a function of frequency, not of time, due to the Fourier transform to time (see appendix A). Therefore the velocity of the acceleration measurement also had to be in the frequency domain. Integration is a matter of dividing by iω or i2π f in the frequency domain [10], whereas integrating a discrete signal in the time domain is a slightly more complicated matter. An mfile was therefore written (given in Appendix B) to transform the data to the frequency domain before the integration, using a fast Fourier transform algorithm of MATLAB (fft). Inherent to this method are a few complications, so to validate if it was correctly applied, the FFT of the acceleration data calculated with the mfile was compared to the FFT of the acceleration data calculated by SigLab. In Figure 4.4 the results of a chirp signal rising from 100 to 1000Hz at 0.4 Volt peak can be seen. The frequency of SigLab s chirp signal did not increase continuously but in steps of 12.5Hz. This does not affect the experiment, however X: 87.5 Y: X: 87.5 Y: FFT 15 Yacc(f) Frequency (Hz) Frequency (Hz) (a) Calculated with SigLab. (b) Calculated with the mfile. Figure 4.4: The single-sided amplitude spectrum of the acceleration data. Comparing the amplitude, then, we see that the results obtained with SigLab are consequently a factor 2 lower than the results obtained with the mfile. This can be explained by the fact that SigLab gives the root mean square of the signal in the FFT, and the mfile does not. 25

26 Chapter 4. Data processing steps The root mean square of a signal y(t) is defined as [5] Y 1 t0 +T y T 2 (t) dt (4.1) t 0 For a sinusoidal function y(t) = A sin(ωt) Cathey [5] therefore states that its root mean square is exactly a factor 2 smaller than its amplitude: Y = A 2 (4.2) Therefore the mfile calculates the fast Fourier transform of the discrete acceleration signal correctly. It would have been easier to take the FFT as calculated by SigLab, but this was not recorded during all of the measurements, so some of the frequency spectra had to be calculated with the mfile. To validate that the integration by division with i2π f in the frequency domain was correct, the integration was also performed using a curve fitting tool of MATLAB (cftool). With this tool, an interpolant, shape-preserving curve was fitted through the acceleration data of the accelerometer so that the residuals were zero at all time and then this curve was integrated in the Analysis section of the tool. In this way, the signal was integrated in the time domain. The integrated curve was fast Fourier transformed and scaled in the same way as before. The results of the integration in frequency and in time domain can be compared in Figure Yspeed(f) X: 100 Y: Yspeed(f) X: 100 Y: Frequency (Hz) (a) Integrated in the frequency domain Frequency (Hz) (b) Integrated in the time domain. Figure 4.5: The single-sided amplitude spectrum of the integrated acceleration data. The amplitude of two arbitrary peaks at the same frequency differ only about 0.25%. Therefore we can state that the preferred integration method, dividing by i2π f in the frequency domain, is also correct. That there even is a difference in the amplitudes can be accounted for by the fact that the length of the vector of the time integrated signal is larger than that of the frequency integrated signal. Therefore time integrated signal has more points that describe the signal which 26

27 Chapter 4. Data processing steps the FFT can use to reconstruct the signal in the frequency domain. The reconstruction can therefore usually be done more accurately. The loss of amplitude at a certain frequency of the Fourier transformed signal in favor of the amplitude at another frequency due to the number of sampling points is termed leakage, and is explained further in Chapter

28 28 Chapter 4. Data processing steps

29 Chapter 5 Results Once the data is processed, the results can be examined. Plots of the holography data for some measurements can be seen in Figure 5.1. Plots of the rest of the hologram results can be found in Appendix E. Examining the holography data, we found that the magnitude of the velocity in z-direction of the layer of air right above the vibrating plate appeared to be cone-shaped (Figure 5.1(e)). This can be explained partially by the velocity in x and y direction (see Figures 5.1(c) and 5.1(d)). The farther from the center of the plate, the higher the magnitude of the velocities in x and y direction. Figure 5.1(f) shows a quiver3 plot, a plot of the length and direction of the velocity vectors in the form of arrows, which shows this relation. If the length of these vectors was the same for all points above the plate, it would mean that velocity in x- or y-direction would diminish velocity in the z-direction, making the vibrating plate a monopole [12, Chap 2.3.1]. However, the length of these vectors is not the same for all points above the plate. Apparently, not only does the direction of the velocity of the particles right above the plate depend on their position above the plate, but so does the magnitude of their velocity. This could be explained by the following. It stands to reason that as the plate moves upward, the air above it is displaced in the direction where it meets the least resistance. Below the plate, an underpressure is created, sucking in air from above the edges of the plate and farther below the plate. The velocity at the outside of the plate should be in the positive z-direction when that above the plate is in the negative direction and vice versa. This is supported by a plot of the real components of the velocity vibration (Figure E.1(e)). Furthermore, a plot of the pressure distribution at the source also shows a conic shape. The cone-shape of the magnitude of the velocity could therefore be explained by acoustical short-circuitry: particles at the edge of the plate move only a little to the side and can immediately fill the pressure void underneath the plate. Particles at the center of the plate cannot escape so easily and are forced upward, building up the pressure at the center of the plate. The velocity of the particles at the center of the plate can therefore be used for the comparison with the accelerometer data. 29

30 Chapter 5. Results (a) Magnitude of velocities in z-direction, seen from above. (b) Magnitude of velocities of a 90Hz 0.7Volts peak sine signal in z-direction, seen from above. k co = rad/m (c) Magnitude of velocities in x-direction, seen from above. (d) Magnitude of velocities in y-direction, seen from above Number of points in y direction Number of points in x direction (e) Magnitude of velocities in z-direction, seen from the side. (f) quiver3 plot of a sixth of the number of points in the data. Figure 5.1: All figures except 5.1(b) are plots of the values at the source of a chirp signal of 0.2Volts peak at 90Hz, calculated with k co = rad/m. Figures 5.1(a), 5.1(b), 5.1(e) and 5.1(f) show data that is cut out with an mfile (Appendix C) so that only points above the plate are visible. 30

31 Chapter 5. Results The results of accelerometer and holography data of all the measurements can be compared using Table 5.1, where the velocity given by the accelerometer is taken as the true velocity. The first two measurements have the smallest error, and these errors are of the expected size of about 10%. Both give an overestimation of the true velocity, while if only the distance of the microphones to the plate is changed, the errors change to an underestimation (measurements three and four). The reader is pointed to the fact that smaller cut-off wavenumbers k co had to be used for measurements three and four in order for the result not to be disturbed by noise. That this results in an underestimation of the velocity will be shown farther on in this chapter. The last two measurements (nos. five and six) were done to investigate what happens to the accuracy of the calculation of the velocity with NAH if the amplitude of the vibration is altered. A very large error in both measurements can be seen, which was unexpected. If Figure 5.1(a) is compared to Figure 5.1(b), it can also be seen that measurements five and six (for which a plot like Figure 5.1(b) is similar) do not show a nice, round, circular form of the velocity in the normal direction as measurements one to four do (similar to Figure 5.1(a)): the circle of measurements five and six has become a lopsided oval. This can only be explained if the plate was not perpendicular to the axis of the microphone during the measurement. Taking a lower k co does make the picture rounder, but drastically increases the error of the velocity calculation, therefore the strange results are not due to the choice of k co. The results of measurements five and six are therefore inconclusive. Theoretically, the accuracy with a larger amplitude should have been better. A larger amplitude at the same frequency means a larger amount of energy would have been transferred to the surrounding air, which would in turn have resulted in more easily obtainable pressure information. Noise would therefore have mattered less, relatively speaking, and a higher k co could have been used. Nr. Mic. dist. Amp. k Signal f [Hz] co Velocity, measured with Error [cm] [Volt] [rad/m] NAH [m/s] Acc. [m/s] 1 1 Chirp % 2 1 Chirp % 3 2 Chirp % 4 2 Chirp % 5 1 Sine % 6 1 Sine % Table 5.1: Measurement results. The second column shows the distance of the microphone from the center position of the plate. The numbers have been rounded for clarity, but the percentages of the error column are based on the unrounded values. The influence of the choice of the cut-off wavenumber k co can be seen when comparing the velocities calculated by NAH using several different k co (see Table 5.2) with each other. Two peaks in the normal velocity at the source can already be seen with a k co of 227 rad/m, which is a blowup of noise. The error seems to decrease for higher cut-off numbers, but this is all due to noise as the picture makes less and less sense the higher the cut-off number becomes. More random peaks are introduced and therefore the highest velocity in z-direction, which is given in Table 5.2, are values at these peaks. The plot of the result of the last cut-off number, that of result six, resembles Figure 4.3(c): destroyed by noise. Comparing result one to two, a drop in the value of the velocity calculated with NAH can be seen. This is also the reason for the lower accuracy 31

32 Chapter 5. Results of measurements three and four in Table 5.1. A lower k co is needed there, which results in less information in the pressure distribution and a larger error in the velocity. The fact that the error in measurement four is so small compared to that of measurement three could by explained by the fact that relative to the cut-off numbers of measurements one and two, k co of measurement four dropped less than that of measurement three. Nr. k co [rad/m] Velocity, measured with NAH [m/s] Acc. [m/s] Error % % % % % % Table 5.2: Measurement results for different values of k co. The distance of the microphones from the plate was 1cm, the amplitude and frequency of the sine wave vibration was resp. 0.7Volts peak and 90Hz. The numbers have been rounded for clarity, but the percentages of the error column are based on the unrounded values. 32

33 Chapter 6 Conclusion The goal of this project was to find a measure for how accurately the velocity of a vibrating plate could be determined with planar Near-field Acoustical Holography (PNAH). Hereto the theory behind NAH has been investigated, a measurement setup made with which the research questions could be answered and experiments have been performed. In the experiments, the acceleration of an oscillating plate was measured with an accelerometer, while the pressure oscillation produced by the plate was measured with a digital microphone for the purpose of NAH. It has been shown, both theoretically and experimentally, what a challenge it is to come to accurate results with Near-field Acoustical Holography. The difficulty is caused by the exponential nature of the inverse calculation when evanescent waves are taken into account. Taking these evanescent waves into account, though, increases the accuracy of the calculation considerably. Although only three measurements are not a statistical foundation for the proposition that the error in the velocity calculated with NAH is about 10%, three of the measurements given in this report do support this proposition. It has not been possible to examine the effect of a change in amplitude in the oscillation of a flat, circular plate on the normal velocity, although increasing the amplitude should, theoretically, have increased the relative accuracy of the measurement. Increasing the distance between the microphone and the plate deteriorates the accuracy of the velocity measurement by means of NAH. This is caused by the higher cut-off wavenumber of the filter needed to keep measurement noise at bay. Acoustical short-circuitry at the border of the plate makes for a cone-shaped plot of the amplitude of the variation of the normal velocity and the pressure at the source. The velocity of the plate can be determined by taking the value at the center of the plate. The measurement setup described in this report could be recommended for further investigation of the accuracy of velocity measurements by means of planar NAH. Consideration should be taken, however, in mounting the plate in order for it to stay perpendicular to the axis of the microphone during the measurement. Furthermore, to truly be able to say what the accuracy of the velocity measurements is, a statistically more meaningful number of measurements should be taken. 33

34 34 Chapter 6. Conclusion

35 Bibliography [1] 20 May [2] 13 May [3] 5 June [4] The American Heritage Dictionary of the English Language. Boston: Houghton Mifflin Company, [5] Jimmie J. Cathey and S. A. Nasar. Schaum s outline of theory and problems of basic electrical engineering. McGraw-Hill Professional, [6] A.D. Dimarogonas and S. Haddad. Vibration for Engineers. Prentice-Hall, Inc., [7] H. Herlufsen. Dual Channel FFT Analysis (Part I). Brüel & Kjær Technical Review, [8] E.G. Williams J.D. Maynard and Y. Lee. Nearfield acoustic holography: I. Theory of generalized holography and the development of NAH. J. Acoust. Soc. Am. 78, [9] F. Kamalizadeh. Source identification of incoherent sound effects. Master s thesis, Eindhoven University of Technology, [10] J.J. Kok and M.J.G. van de Molengraft. Systeem Analyse, Department of Mechanical Engineering, TU/e. [11] J.J. Kok and M.J.G. van de Molengraft. Signaalanalyse, 4A250, Course year 2006/2007. Department of Mechanical Engineering, TU/e. [12] Michael P. Norton and Denis G. Karczub. Fundamentals of noise and vibration analysis for engineers. Cambridge University Press, [13] Harry F. Olson. Music, physics and engineering. Courier Dover Publications, [14] Rick Scholte. Fourier Based High-resolution Near-Field Sound Imaging. PhD thesis, Eindhoven University of Technology, [15] Rick Scholte and N.B. Roozen. High Resolution near-field acoustic holography [16] Xander Schuurbiers. Aliasing in PNAH, Bachelor Final Project, TU/e, [17] M. Steinbuch. Dictaat College Regeltechniek (4A550), Department of Mechanical Engineering, TU/e. 35

36 Bibliography [18] Earl G. Williams. Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography. ACADEMIC PRESS,

37 Appendices 37

38

39 Appendix A Derivation of normal component of velocity The Fourier transform to time of Euler s equation of conservation of momentum is [18, Chap. 2.4]: iωρ 0 v = p(x, y, z, ω) (A.1) Analogous to F x F y {p(x, y, z)} = P(k x, k y, z), (A.2) the following can be done with equation (A.1). If F x F y {ẇ(x, y, z)} = Ẇ(k x, k y, z) (A.3) and F x F y { p} = F x F y { î p x + ĵ p p + ˆk y z } = îik x P(k x, k y, z) + ĵik y P(k x, k y, z) + ˆkik z P(k x, k y, z) = i(k x î + k y ĵ + k zˆk)p(kx, k y, z) then F x F y {iωρ 0 v = p} = iωρ 0 ( U(k x, k y, z)î + V(k x, k y, z)ĵ + Ẇ(k x, k y, z)ˆk) = i(k x î + k y ĵ + k zˆk)p(kx, k y, z) U(k x, k y, z)î + V(k x, k y, z)ĵ + Ẇ(k x, k y, z)ˆk = 1 ρ 0 ck (k xî + k y ĵ + k zˆk)p(kx, k y, z) And combining this with (2.16) the normal component of velocity can be written as k z Ẇ(k x, k y, z) = ρ 0 ck P(k x, k y, z h )e ik z(z z h ) (A.4) = G(k x, k y, z z h )P(k x, k y, z h ) (A.5) where G(k x, k y, z z h ) = k z ρ 0 ck eik z(z z h ) is called the velocity propagator [18, Chap. 3.2]. 39

40 Appendix A. Derivation of normal component of velocity 40

41 Appendix B FFT mfile %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % fft of acceleration data and its integral %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % load the data file of SigLab. for example, if the data file is called % filename.vna, type 'load filename.vna mat' in matlab. There should be % a structure with the name 'SLm' in the workspace. Typing 'SLm' and hitting % enter shows the contents of the structure. In our measurements the % acceleration data was store under 'SLm.scmeas(3)': i.e. single channel % measurement at channel 3. For more information about how SigLab stores % its data, see the SigLab documentation. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % in SLm, rbw means resolution bandwith = df. % df*(length(slm.tdxvec))=sampling frequency % time between any two measurement points = dt % 1/dt = sampling frequency % In SLm structure: % channel 1: input signal % channel 2: microphones % channel 3: small kistler accelerometer % the multiplication factor to calculate the acceleration in m/s^2 from the % signal of the accelerometers in volt. % factor = [ ]; % m/^2/volt. small kistler: 925, black kistler: 102 % Bandwith was 1 khz, this is the frequency you can be sure of is measured % accurately. So f = Fs/(Fs/1000)=Fs/2.56. Nyquist factor plus extra is % therefore Nyq=2.56 ; % % time and acceleration xtime=slm.tdxvec; % time vector yacc=slm.scmeas(3).eu_val*slm.scmeas(3).tdmeas ; %eu_val is the % % calibrated sensitivity of the sensor (filled in in vna) 41

42 Appendix B. FFT mfile %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % fft %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % see for references: % % help fft and then the doc file % % and % % notes/1700/1702.html % 3O4RH1.html?solution=1 % 3O4RH1 % notes/1700/1703.html % conversations and th % e fft/ % % and % % % elosery/matlab/matlab.pdf % % labexample.htm % tutor.pdf % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % fft of acceleration: channel3 L=length(yacc); Fs=SLm.rbw*L; NFFT = 2^nextpow2(L); % Next power of 2 from length of yacc Yacc = fft(yacc,nfft)/l ; % divide by L to normalize for the % length of the signal. f = Fs/Nyq*linspace(0,1,(NFFT/Nyq+1)); % f is the frequency vector, very important. without this you can't % tell what peak in the fft belongs with what frequency % Plot single sided amplitude spectrum. figure plot(f,2*abs(yacc(1:(nfft/nyq+1)))) % multiply by 2 because you throw % the negative frequency range away title('single Sided Amplitude Spectrum of Yacc(t)') xlabel('frequency (Hz)') ylabel(' Yacc(f) ') grid %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % speed % Fourier{int_{ inf}^{t} a(t) dt}=1/(i*2pi*f)* A(f) % so we have to divide by the angular frequency times i or i*2pi*f f2 = Fs*linspace(0,1,NFFT); Yspeed =Yacc./(i*2*pi*f2'); % dividing by 0 equals infinite, so first % element should be discarded or altered % otherwise it messes up the ifft. 42

43 Appendix B. FFT mfile Yspeed(1)=0; figure plot(f,2*abs(yspeed(1:(nfft/nyq+1)))) xlabel('frequency (Hz)') ylabel(' Yspeed(f) ') grid 43

44 44 Appendix B. FFT mfile

45 Appendix C Cut out mfile %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % cut out data on size of disc %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % load the *.mat file with the results of the nah measurement. there should % be a structure 'nah' in the workspace. what is of interest to us is % nah.freq. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % say we know the distance between two points in uz_planes_p2uz, and say % it's dx. % the midpoint of the signal appears to be very nicely approximateted by % the point where a maximum in the pressure occurs % circle: x = r*cos(alfa), y = r*sin(alfa) % > alfa = acos(x/r) > y = r*sin(acos(x/r)) x = (length(nah.freq.uz_planes_p2uz) 1)/15 ; dx = 5/x ; % [mm] distance between two points in uz_planes_p2uz % length(nah.freq.uz_planes_p2uz) = 121= x*15+1, where x 1 is the number of % interpolation points, determined by the value you set for zero and border % padding. x = (length(nah.freq.uz_planes_p2uz) 1)/15, % dx = 5/x, because distance between two measurement points is 5 mm r = 50/2 ; % [mm] radius of plate % helpmatrix and p should be as large as the matrix you take the % results from. In this case nah.freq.uz_planes_p2uz helpmatrix = zeros(size(nah.freq.uz_planes_p2uz)); p= ones(size(nah.freq.uz_planes_p2uz)); [max_value,center_x] = max(max(abs(nah.freq.uz_planes_p2uz(:,:,2)))); [max_value,center_y] = max(max(abs(nah.freq.uz_planes_p2uz(:,:,2))')); % I define the row in which the maximal value of speed occurs as the y % coordinate of the center of the plate, and the column in which the % maximal value occurs the x coordinate of the center. for j=0:round(r/dx) coordinatey = round(r/dx*sin(acos(j/(r/dx)))) ; 45

46 Appendix C. Cut out mfile end % by dividing j by r*dx we achieve that acos walks from 0 to 1. if j 1 % now we have the y coordinate as our x coordinate moves from 1 to the % edge of the plate (r/dx). This discribes one quarter of a circle. Of % course, we want the circle to start on position (center_x,center_y), % so we have to add center_x to j. helpmatrix((center_y coordinatey):(center_y+coordinatey),(j+center_x),:) = p((center_y coordinatey):(center_y+coordinatey),(j+center_x),:) ; % and we want the same coordinatey to apply for the left half of the % circle helpmatrix((center_y coordinatey):(center_y+coordinatey),(center_x j),:) = p((center_y coordinatey):(center_y+coordinatey),(center_x j),:) ; end % and the following includes the very center of the circle if j == 0 helpmatrix((center_y coordinatey):(center_y+coordinatey),center_x,:) = p((center_y coordinatey):(center_y+coordinatey),center_x,:) ; end cutoutz=helpmatrix.*nah.freq.uz_planes_p2uz ; cutoutx=helpmatrix.*nah.freq.ux_planes_p2ux ; cutouty=helpmatrix.*nah.freq.uy_planes_p2uy ; % plot results figure, surface(1.1*abs(cutoutz(:,:,1))), %1.1 volt/pascal is the calibrated value of the digital microphone shading interp axis([ ]) xlabel('number of points in x direction'), ylabel('number of points in y direction') colorbar 46

47 Appendix D Design drawings of the plate Figure D.1: The plate used in the measurements. 47

48 Appendix D. Design drawings of the plate Figure D.2: Plate from the side. Figure D.3: Plate from the top. 48

49 Appendix E Measurement result plots 49

50 Appendix E. Measurement result plots (a) Magnitude of velocity in z-direction, seen from above. (b) Magnitude of velocities in z-direction, seen from the side. (c) Magnitude of velocity in x-direction, seen from above. (d) Magnitude of velocity in y-direction, seen from above. (e) Real values of velocity in z-direction, seen from the side. Figure E.1: Plots of the values at the source of a chirp signal of 0.2Volts peak at 90Hz, calculated with k co = rad/m. Hologram plane distance 1cm. 50

51 Appendix E. Measurement result plots (a) Magnitude of velocity in z-direction, seen from above. (b) Magnitude of velocities in z-direction, seen from the side. (c) Magnitude of velocity in x-direction, seen from above. (d) Magnitude of velocity in y-direction, seen from above. Figure E.2: Plots of the values at the source of a chirp signal of 0.2Volts peak at 290Hz, calculated with k co = rad/m. Hologram plane distance 1cm. 51

52 Appendix E. Measurement result plots (a) Magnitude of velocity in z-direction, seen from above. (b) Magnitude of velocities in z-direction, seen from the side. (c) Magnitude of velocity in x-direction, seen from above. (d) Magnitude of velocity in y-direction, seen from above. Figure E.3: Plots of the values at the source of a chirp signal of 0.2Volts peak at 90Hz, calculated with k co = rad/m. Hologram plane distance 2cm. 52

53 Appendix E. Measurement result plots (a) Magnitude of velocity in z-direction, seen from above. (b) Magnitude of velocities in z-direction, seen from the side. (c) Magnitude of velocity in x-direction, seen from above. (d) Magnitude of velocity in y-direction, seen from above. Figure E.4: Plots of the values at the source of a chirp signal of 0.2Volts peak at 290Hz, calculated with k co = rad/m. Hologram plane distance 2cm. 53

54 Appendix E. Measurement result plots (a) Magnitude of velocity in z-direction, seen from above. (b) Magnitude of velocities in z-direction, seen from the side. (c) Magnitude of velocity in x-direction, seen from above. (d) Magnitude of velocity in y-direction, seen from above. Figure E.5: Plots of the values at the source of a sine signal of 0.2Volts peak at 90Hz, calculated with k co = rad/m. Hologram plane distance 1cm. 54

55 Appendix E. Measurement result plots (a) Magnitude of velocity in z-direction, seen from above. (b) Magnitude of velocities in z-direction, seen from the side. (c) Magnitude of velocity in x-direction, seen from above. (d) Magnitude of velocity in y-direction, seen from above. Figure E.6: Plots of the values at the source of a sine signal of 0.7Volts peak at 90Hz, calculated with k co = rad/m. Hologram plane distance 1cm. 55