Minimization of Enclosed Acoustic Energy Density by means of Active Noise Control Methods: Comparison between Maximum Absorption and Minimum Energy Methods

Diplomarbeit Minimization of Enclosed Acoustic Energy Density by means of Active Noise Control Methods: Comparison between Maximum Absorption and Minimum Energy Methods vorgelegt von Jordi Sobrino Gil Kopernikusstraße. 30 10245 Berlin Matr.-Nr. 0320565 eingereicht beim Fachgebiet f¨ ur Technische Akustik der Technischen Universit¨ at Berlin am 30. Juni 2009 Betreuender Universit¨ atsprofessor: Prof. Dr.-Ing. M. M¨ oser Declaration of Academic Honesty I hereby declare to have written this Master Thesis on my own, having used only the listed resources and tools. Jordi Sobrino Gil Berlin, 30th of June 2009. Acknowledgements I would like to thank my family for their support and guidance who made possible my stay in Berlin, specially my parents Francisco Sobrino Marqu´es and Maria Isabel Gil P´erez, for their loving and unconditional support, my brother David Sobrino Gil, grandparents and uncles. Likewise, I am thankful to all the people that I have met in my life, specially those from school and college, who gave me friendship and support. Within them I am most grateful to the people from Gr`acia and Telecogresca. Thanks to the people I have met in Berlin and those who came to visit me, they have made me feel comfortable and like being between a family in these late months. Last but not least, I would really like to thank professor Dr. Ing. Michael M¨ oser for giving me the chance to write my Master Thesis in the Technische Akustik Department of the Technische Universit¨ at Berlin, and his supervising, and also to Marco Norambuena for his help and advise during this time. Abstract Active Noise Control (ANC) is a buzzword in acoustics today. Applied correctly, ANC methods have proved to be better insulators in low frequency noise, than usual absorbers. However, when a broad band noise needs to be controlled, such active methods have difficulties at high frequencies. The main objective of this master thesis is to minimize Acoustic Energy Density (AED) of a broad band signal by means of an active control method. A comparison between such method and an absorption one is done, so as to determine which has better broad band behavior. Furthermore, experimental results are compared with theoretical simulations. Contents Contents i List of Acronyms iii List of Tables iv List of Figures v List of Variables vii 1 Introduction 1 1.1 Motivation............................. 1 1.2 Objective and structure of the thesis . . . . . . . . . . . . . . 2 2 Theoretical Background 3 2.1 Thenatureofsound ....................... 3 2.1.1 Sound propagation . . . . . . . . . . . . . . . . . . . . 3 2.2 Energy propagation and energy density . . . . . . . . . . . . . 5 2.3 Kundt’stube ........................... 6 2.3.1 Sound propagation inside Kundt’s tube . . . . . . . . . 7 2.3.2 Planewaves........................ 7 2.3.3 Complex Amplitude Coefficient (CAC), β....... 8 2.4 Noisecontrol ........................... 9 2.4.1 Energy density calculation . . . . . . . . . . . . . . . . 10 2.4.2 Active method: Minimum Energy Method (MEM) . . . 11 2.4.3 Active method: Maximum Absorption Method (MAM) 12 2.5 Theoretical simulations . . . . . . . . . . . . . . . . . . . . . . 13 2.5.1 Experimentation set-up modelling . . . . . . . . . . . . 13 2.5.2 Closed-end simulation . . . . . . . . . . . . . . . . . . 15 2.5.3 Minimum Energy Method and Maximum Absorption Method’s simulations . . . . . . . . . . . . . . . . . . . 16 2.5.4 Methods Comparison . . . . . . . . . . . . . . . . . . . 18 i CONTENTS Jordi Sobrino Gil 3 Measurements and results 21 3.1 Experimentation Set up:Signal-generation and signal-measuring equipment............................. 21 3.2 Calibration ............................ 22 3.3 Measurement Methods . . . . . . . . . . . . . . . . . . . . . . 23 3.3.1 Auto-Spectral Method (ASM) . . . . . . . . . . . . . . 24 3.3.2 Auto-and Cross-Spectral Method (ACSM) . . . . . . . 24 3.3.3 Comparison and method election . . . . . . . . . . . . 25 3.4 Closed-end Measure . . . . . . . . . . . . . . . . . . . . . . . . 26 3.4.1 Experimentation set up: Closed-end measure . . . . . . 26 3.4.2 Measurement procedure . . . . . . . . . . . . . . . . . 26 3.4.3 Results........................... 27 3.5 Minimum Energy Method (MEM) Measure . . . . . . . . . . . 29 3.5.1 Experimentation set up: MEM . . . . . . . . . . . . . 29 3.5.2 Measurement procedure . . . . . . . . . . . . . . . . . 30 3.5.3 results........................... 30 3.6 Maximum Absorption Method (MAM) Measure . . . . . . . . 31 3.6.1 Experimentation set up: MAM . . . . . . . . . . . . . 32 3.6.2 Measurement procedure . . . . . . . . . . . . . . . . . 32 3.6.3 Results........................... 33 3.7 MethodComparison ....................... 33 4 Conclusion 41 A Enlarged Figures 42 A.1 Theoretical Simulations . . . . . . . . . . . . . . . . . . . . . . 42 A.2 Measurements........................... 44 B Matlab Scripts 51 B.1 Theoretical Simulation script . . . . . . . . . . . . . . . . . . . 51 B.2 CalibrationScript......................... 53 B.3 AcquisitionScript......................... 54 B.4 Auto-Spectral Method (ASM) and Auto-and Cross-Spectral Method (ACSM) comparison script . . . . . . . . . . . . . . . 56 B.5 AED calculation by means of ACSM . . . . . . . . . . . . . . 58 ii List of Acronyms AED Acoustic Energy Density ANC Active Noise Control ASM Auto-Spectral Method ACSM Auto-and Cross-Spectral Method BnK Br¨uel and Kjær CAC Complex Amplitude Coefficient MAM Maximum Absorption Method MEM Minimum Energy Method iii Chapter 2 Theoretical Background 2.1 The nature of sound The phenomenon of sound consists of pressure oscillations travelling across a medium, such fluids, gasses or solids. These oscillations propagate forming waves and they do so at what it is called speed of sound, which depends on the medium the propagation passing through. The speed of sound in gasses at 20oC is approximately 343 m·s−1, whereas in fresh water also at 20oC the speed of sound is approximately 1,482 m·s−1, and in steel is about 5.960 m·s−1. 2.1.1 Sound propagation Sound generation is generally attributed to the vibration of solid objects which induce vibration into the particles of air[6]. Particulary in gasses, sound propagation is affected by temperature and it obviously has an attenuation with the distance. However, for understanding how it propagates, one can focus only on the fundamentals and not consider these factors. Sound needs a medium in order to propagate but this medium needs to be not only elastic, but also massive. This second property is required since sound propagation intrinsically consists of local variations of medium’s density[7]. The way plane waves propagate through a certain part of the space with these properties, can be modelled as a spring-mass system. If for instance, a tube is filled with air, the whole volume of it can be divided in sections, and every section can be modelled as a spring and a mass element, as shown in figure 2.1. When a sound field is generated at one end, there is just one possible direction of propagation[8]. 3 Theoretical Background Jordi Sobrino Gil Figure 2.1: Sections of air in a gas column[8]. Jumping to the mechanical simile, the sound field would be a force. An initial impulse force applied to the system from left to right, would compress the first spring, pushing the next mass towards right. At the same time, the spring and the mass behind it would stop the first of moving ahead. Furthermore, since the masses have inertia, the second mass does not react at the same time as the force was applied, but with a certain delay. Therefore, and because of the inertia, the second mass would start moving gradually. At this point two observations should be done: 1. The translation process pushing the masses gradually towards right would be repeated for the following masses, compressing the springs one by one, and altogether, translating the alteration with a finite velocity. This velocity is what it is called the speed of sound, cand has been named at the beginning of the chapter. 2. A second phenomenon occurs when a mass of the chain is pushed towards right. The preceding spring is being actually stretched, thus stopping the mass of going further, pulling towards the opposite direction. Then, the mass follows a harmonic oscillator’s movement around an equilibrium point. This oscillation movement gives the masses another velocity, which is called particle velocity vand it is the velocity with which gas particles vibrate when a sound field excites them. Figure 2.2 shows an element of a column of gas being accelerated by elastic forces.The mass of it, is m=Sρ0∆x(2.1) where Sis the surface of the transversal section, ρ0is the density of the air, and ∆xthe longitude of the longitudinal section. The stiffness of the volume of gas is defined as s=Ey·S ∆x(2.2) where Eyis the Young’s Modulus or the Elasticity Modulus and it is an inner 4 Theoretical Background Jordi Sobrino Gil Figure 2.2: Element of a column of gas accelerated[8]. property of the material which define its elasticity. Ey=ρ0c2(2.3) To sum up, when a sound field is applied to a volume of gas, it induces the movement of the following volumes of gas, translating the alteration at a constant and finite speed, c. However, gas particles do not move forward, they are excited and start oscillating at a certain frequency following an oscillation movement around the equilibrium point, muffled by the around particles of the inner medium. 2.2 Energy propagation and energy density As seen in the section before, sound propagation is basically pressure waves propagating along a medium. This implies that the air molecule are subjected to local vibrations around the equilibrium point, thus meaning that the gas particles momentarily store local energy before the next particle starts vibrating due to propagation.[8] One can model the particle vibration movement by means of a spring-mass system, whose springs store potential energy, and whose masses have kinetic energy due to their motion. A mass mmoving at a velocity v, has a kinetic energy of Ekin =1 2m|v|2(2.4) A spring with a stiffness scompressed by a force Fhas a potential energy of Epot =1 2 |F|2 s(2.5) Using the gas density ρ0, one can easily get the Kinetic energy of a gasvolume’s element Ekin =1 2ρ0|v|2∆V(2.6) 5 Theoretical Background Jordi Sobrino Gil Applying now the same idea at the potential energy equation, considering that F=pS, where pis pressure, and s=EyS/∆x=ρ0c2∆xand taking into account equations 2.2 and 2.3 Epot =1 2|p|2S2∆x/(ρ0c2S) = 1 2 |p|2∆V ρ0c2(2.7) Therefore, the total amount of energy that a gas-volume element, excited by a sound pressure wave has, is E∆V=1 2½|p2| ρ0c2+ρ0|v|2¾∆V(2.8) Since this expression may differ widely, depending on the amount of particles that are excited, hence the volume, it is more common to work with the energy density, which gives a energy value per unit of volume. E= 1 2½|p|2 ρ0c2+ρ0|v|2¾(2.9) The total amount of energy stored in a volume of gas Vis therefore EV=E·V(2.10) It is important to say, for a better understanding, that the energy of a gas has a wave behavior, just as the pressure or the particle velocity[8]. If there is a progressive wave being propagated p=f(t−x/c)and p(x, t) v(x, t)=ρ0c then the energy results to be E(x, t) = p2 ρ0c2=1 ρ0c2f2(t−x/c) (2.11) The energy is propagating with the sound field, and like this one, it has a wave behavior[8]. 2.3 Kundt’s tube The tube of kundt is an apparatus very used in experimentation, mainly because it is easy to control and generate all the sound factors in the inside. Such experiments are performed in a rigid tube with circular or squared section, where the sound waves are driven along the axial direction. 6 Theoretical Background Jordi Sobrino Gil 2.3.1 Sound propagation inside Kundt’s tube The common set up is to place a speaker on one side of the tube connected to a signal or noise generator, and close the other side with a sample of what is willing to study. The sound field inside can be divided in two, the progressive incoming wave produced by the loudspeaker, and the one reflected by the closed-end’s surface. p=p0©e−jkx +rejkxª(2.12) where ris the reflection coefficient,p−/p+defined as the relation between the reflected pressure and the incoming pressure. p+=p0e−jkx p−=p0rejkx r=Rejϕ Splitting the incoming wave in two parts, one completely reflected and the second one its difference p+=p0re−jkx +p0(1 −r)e−jkx The total field is composed by a stationary wave and a progressive wave p=p0r(e−jkx +ejkx) + p0(1 −r)e−jkx = 2p0rcos(kx) | {z } stationary +p0(1 −r)e−jkx | {z } progressive The stationary or standing waves remain in a constant position, meaning the the maximums and minimums of pressure are spread along the tube in the same position. The fact of having reflection in the inside and therefore a standing wave, leads to a dependency of RMS-measured values with position. In case of prefect reflection r= 1 ⇒˜p2=p2 0cos2(kx) (2.13) 2.3.2 Plane waves In order to ensure that only plane waves are propagated along the tube and no other mode is being excited, it must be ensured that one works under the frequency of cut-on. Above the cut-on frequency, vertical modes are also excited thus letting particle velocity and pressure propagate in more than just one direction. From Mser (2005), cut-on frequency in a circular tube can be obtained as fn=xn c 2πa (2.14) with n= 1, its corresponding coefficient xnshown in table 2.1, and aas the radius of the tube[8]. Whenever under this frequency, one can assure that 7 Theoretical Background Jordi Sobrino Gil n1 2 3 4 5 Xn1.841 3.054 3.832 4.201 5.331 Table 2.1: Coefficient xnused to obtain cut-on frequencies[8]. only planar waves are being guided inside of it. Figure 2.3 shows a drawing of how pressure waves would propagate being the intense-coloured vertical lines, the highest pressure fronts, and the whitened ones their opposite. An example of multimode propagation is shown in figure 2.4 which shows how, working within the corresponding frequencies between n= 1 and n= 2, next mode would also be excited and instead of planar waves, pressure waves would have a different horizontal distribution. Figure 2.3: Planar wave propagation inside Kundt’s tube. Figure 2.4: Multimode propagatoin inside Kundt’s tube. 2.3.3 Complex Amplitude Coefficient (CAC), β When a tonal sound wave propagates along a tube produced by a loudspeaker, air vibrates with a certain velocity. Such vibrations are induced by the mechanical movement of loudspeaker, which has a certain velocity, v1. If a second speaker is placed at the other end of the tube, generating another sound wave at the same frequency, this second speaker follows another movement with its corresponding velocity, v2. The relation between the velocities of movement of both loudspeakers is called Complex Amplitude Coefficient (CAC) and it is represented with the Greek letter β. 8 Theoretical Background Jordi Sobrino Gil β=v2 v1 v2=βv1(2.15) The coefficient is complex since the velocity of the two sources can differ not Figure 2.5: Complex Amplitude Coefficient (CAC) only in amplitude, but also in phase. 2.4 Noise control Noise control is a passive or active means of reducing sound emissions. On one hand, passive methods consist of applying materials to absorb, isolate or insulate. Damping materials can also be used in order to prevents transmission of vibration energy from a source to a receiver. On the other hand, Active Noise Control (ANC) is also known as noise cancellation, active noise reduction or antinoise, and differs from passive methods in that a powered system is involved. An example of ANC method would be a system generating an inverted phase sound to an original, so as to have phase cancellation. An example of it is shown in figure 2.6 where in this case and linking with the parameter seen in the preceding section, CAC would have the value of β=ejπ . Figure 2.6: Active Noise Control example: Tone Cancellation. 9 Theoretical Background Jordi Sobrino Gil 2.4.1 Energy density calculation Having a set up like described in section 2.3.1, we define V0as the initial velocity with which the system of gas is excited at x= 0. At the end of the tube, they will have a velocity of βV0as shown in the figure 2.5. Considering reflection at the end of the tube, the pressure equation can be expressed as p=ρcV0nAe−jkx +Bejkxo(2.16) where there is a travelling wave in the positive direction with a complex factor of amplitude and phase in it Ae−jkx produced by the speaker, and a travelling wave in the negative direction with another complex factor which represents reflection Bejkx. One can obtain the particle velocity applying the next formula to equation 2.16 vx=j ρω dp dx obtaining v=V0nAe−jkx −Bejkxo(2.17) As seen in section 2.3.3, velocity at the ends of the tube is first V0at x= 0 and βV0at x=l. Applying boundary conditions to equation 2.17 at x= 0 V0=V0(A−B) A−B= 1 (2.18) and at x=l β=Ae−jkl −Bejkl (2.19) To solve the system, equation 2.18 is multiplied by −e−jkl and by −ejkl, and added respectively to equation 2.19 in order to find the coefficients B and A. Then, B and A result B=−β−e−jkl ejkl −e−jkl (2.20) A=−β−ejkl ejkl −e−jkl (2.21) Replacing such values into equations 2.16 and 2.17 the following result for pressure and particle velocity is obtained p=ρcV0 ejkl −e−jkl n(ejkl −β)e−jkx + (e−jkl −β)ejkxo(2.22) 10 Theoretical Background Jordi Sobrino Gil v=V0 ejkl −e−jkl n(ejkl −β)e−jkx + (e−jkl −β)ejkxo(2.23) and working on the exponential functions bearing in mind that cos(t) = (et+e−t)/2 and sin(t)=(et−e−t)/2j, the pressure and velocity inside the tube are: p=jρcV0 sin kl nβcos kx −cos k(x−l)o(2.24) v=V0 sin klnβsin kx −sin k(x−l)o(2.25) Eventually, replacing in equation 2.9, E=Ek+Epot =1 2ρ|v|2+1 2 1 ρc2|p|2 and one achieves the energy density expression in terms of CAC: E=ρV 2 0 2 sin2kl¯¯¯βsin kx −sin k(x−l)¯¯¯ 2+ρV 2 0 2 sin2kl¯¯¯βcos kx −cos k(x−l)¯¯¯ 2 Developing both squared expressions of potential and kinetic energies multiplying them by their conjugate, and taking into account that cos(a−b) = cos acos b+ sin asin band also that β=βr+jβi E=ρV 2 0 2 sin2klnβ2 r+β2 i−2βrcos kl + 1o(2.26) 2.4.2 Active method: Minimum Energy Method (MEM) Minimum energy methods are not always those which produce the anti-phase phenomena. One can try to minimize the average acoustic-energy density inside the tube, by means of optimizing the amplitude complex coefficient seen in section 2.3.3. Therefore, from equation 2.26, optimal βis achieved by ∂Etot(β) ∂βi = 0 ⇔(2βi,2βr−2 cos kl) = (0,0) βio= 0 βro= cos kl (2.27) In conclusion, one can minimize the average acoustic energy density forcing the Complex Amplitude Coefficient (CAC) to be optimal. A secondary loudspeaker is used in order to achieve such value of velocity at the end of the 11 Theoretical Background Jordi Sobrino Gil tube. Replacing βioand βrovalues in equation 2.26 ,the value of the average acoustic energy density inside the tube is Etotactive =ρV 2 0 2 sin2klncos2kl −2 cos2kl + 1o= =ρV 2 0 2 sin2kl³1−cos2kl´=1 2ρV 2 0(2.28) Etotactive =1 2ρV 2 0 2.4.3 Active method: Maximum Absorption Method (MAM) If an absorption method is used at the end of the tube, it is mandatory to re-define the equations of pressure and velocity that reign the interior of it. Such necessity comes from the fact that there is no reflected sound wave inside, and therefore it has no sense to consider a reflected wave reigned by Bejkl. The sound field consists of progressives sound waves incoming from the speaker with a constant effective value. If a loudspeaker is generating a tone inside a tube ended with maximum absorption, no sound wave is reflected. Therefore, the pressure field inside the tube can be described as a travelling wave in the positive direction multiplied by a complex factor of amplitude and de-phase. p=ρcV0Ae−jkx (2.29) Velocity is obtained by means of equation 2.4.1 v=V0Ae−jkx (2.30) Applying boundary conditions at x= 0 V0=V0A⇒A= 1 Then, using the general energy density equation (2.9) Etotpassive =1 2(ρV 2 0+1 ρc2ρ2c2V2 0)=ρV 2 0(2.31) Etotpassive =ρV 2 0 12 Theoretical Background Jordi Sobrino Gil Figure 2.11: Closed-end, MAM and MEM Simulations. In order to eliminate the influence of the speaker, and the unknown particle velocity, closed-end simulation’s values are subtracted from both active methods, thus leading to true and comparable values that can be taken into account. In figure 2.12 are shown such values, and once again, both methods differing 3dB as one could expect based on the theoretical development4. Due tot the subtracting of the closed-end graph which had maximum peaks at odd numbers of n at equation 2.32, from both maximum absorption and minimum energy methods, which had minimum peaks at even values of n, the result is a graph with minimum peaks at both even and odd values of n. The frequencies where extrema are located, are specified in table 2.4. Frequency (Hz) 98.9 197.6 296.4 395.2 494 n12345 Frequency (Hz) 592.7 691.5 790.3 889.1 987.9 n6 7 8 9 10 Table 2.4: Isolation Peaks. Even though the difference between both methods give dimensionless values at the y axis, they can be understood as the levels both methods would add to a certain signal inside the tube if any acoustic signal was introduced inside of it. 4See appendix A.2 in page 45 for consulting the Matlab script 19 Theoretical Background Jordi Sobrino Gil Figure 2.12: Method Comparison. 20 Chapter 3 Measurements and results In order to compare both active methods, three measurements need to be done. 1. Closed-end tube: So as to know the transfer function of the tube and the loudspeaker at each frequency. Its details are found in section 3.4. 2. Minimum Energy Method: A secondary loudspeaker is used at the other end of the tube in order to assure that minimum-energy conditions seen in section 2.4.2 are generated. It is explained in section 3.5. 3. Maximum Absorption Method: Maximum-Absorption conditions are assured at the end of the tube by means of an adaptive filter which eliminate the reflected wave thus leaving inside the tube only travelling waves in the positive direction. Its details are specified in section 3.6. 3.1 Experimentation Set up:Signal-generation and signal-measuring equipment Both the generation equipment which produce the excitation signal, and the measuring equipment which acquires pressure level’s data, are common for the three different measures. As seen in figure 3.1, on one hand the generation equipment consists of a function generator Voltcraft FG708S, a common power stage and loudspeaker. The loudspeaker is placed at the first end of the tube, where only pure tones will be propagated. On the other hand, the measuring equipment consists of two condenser microphones Br¨uel and Kjær (BnK) type 4181 connected to a preamplifier BnK type 2668, and two measuring amplifiers also BnK type 2610 at reception stage. A condenser microphone must be combined with a preamplifier 21 Measurements and results Jordi Sobrino Gil to provide impedance conversion, some filtering, and the capability to drive relatively long cables without significant signal degradation[10]. A cylindric Figure 3.1: Generation and measuring set up diagram. tube was built to perform the three measures, as well as both loudspeaker boxes. The dimensions of the tube are 1.736 m long and 0.075 m of diameter. It has 2 wholes in the middle of it, distanced 0.05 m, in order to place the microphones, as explained in section 3.3. The cut-on frequency according to equation 2.14 is, fn=xn c 2h= 0.59 343 0.075 = 2698.26[Hz] 3.2 Calibration The microphones and the entire measurement chain are calibrated at regular intervals. The calibration provides traceability and proven accuracy to the system. Therefore, before a measure is taken, a calibration procedure is done1. 1. By means of a calibrator like a BnK type 4231 with an output of 94 dB SPL2at 1 KHz is used to adjust the measuring amplifiers at precisely 94 dB SPL with the calibration knob. 1Script in Annex B.2 in page 53 294 dBSPL is equivalent to 1 Pa 22 Measurements and results Jordi Sobrino Gil 2. Using the acquisition card and therefore the whole measuring chain, five samples of one second are taken in order to calculate the effective voltage value at 94 dB SPL that the computer is reading. The sampling frequency is 44.100 KHz. 3. The mean of every sample is taken away in order to eliminate a possible bias. Then the root mean value of each sample is calculated by means of the equation Vrms =v u u t1 N N X i=1 x2 i=rx2 1+x2 ”+· · · +x2 N N Eventually, a total Vrms is calculated as the average of the five samples. 4. On every measure taken afterwards, a calibration coefficient is applied to the data, obtained with the following conversion factor pcalib =Vrmsmeas Vrms94dB ·1 Vrms94dB [Pa] where the first dimensionless factor escalates the values of the data whereas the second factor gives the unit conversion. The calibration of microphones was done at 94 dB, and afterwards checked at 104 dB and 114 dB with positive results. 3.3 Measurement Methods There are several methods and configurations to measure acoustic pressure. The method chosen for the study of this Thesis’s concern is the twomicrophone method, for it is not relatively complicated. It is mainly used in the determination of acoustic absorption and other related acoustic properties. The two-microphone technique uses a tube with a sound source at one end and the sample placed at the other. A broadband signal is applied to the sound source and the transfer function between two microphones, specially calibrated to minimize amplitude and phase errors between the channels, is measured. Provided only plane waves propagate in the tube past the microphones, the analytical transfer function between the two stations allows the decomposition of the field into forward and backward travelling waves[11]. Nevertheless, it is energy density what it is willing to be studied. By means of frequency domain methods, an Acoustic Energy Density (AED) expressed in the frequency domain is obtained, by adding the weighted sum of the auto-spectral densities of both pressure and particle velocity. 23 Measurements and results Jordi Sobrino Gil 3.3.1 Auto-Spectral Method (ASM) This method calculates the time-average AED estimate as the weighted sum of the auto-spectral densities of pressure and particle velocity[12] Figure 3.2: Auto-Spectral AED Method. The instant energy density is needed in order to calculate the estimation: ED(t) = [p1(t) + p2(t)]2 8ρc2+Rt −∞[p1(τ)−p2(τ)dτ]2 8ρh2 where eventually ED=k(Gpp +Gvv)with k =1 2ρc2 3.3.2 Auto-and Cross-Spectral Method (ACSM) The second method also calculates the time-average AED estimate, but using the derived expression in terms of the auto-and cross-spectral densities weighted sum of the two pressure readings.[12] In these second case, the expression of the energy density corresponds the single-sided time-averaged AED spectral density estimate expression, in terms of single-sided sprectral densities. [12] ED1−sided(w)≈³1 8ρc2+1 8ρω2h2´(Gp1p1(w) + Gp2p2(w)) +³1 8ρc2−1 8ρω2h2´(2Re[Gp1p2]) where Gxy(w) = 2Sxy(w) = lim T→∞ 2 TE[XH(w, T)Y(w, T)] 24 Measurements and results Jordi Sobrino Gil Figure 3.3: Auto-And Cross-Spectral Method for AED calculation. 3.3.3 Comparison and method election Both methods were tested in the tube, in order to decide which one was optimum for the paper’s purposes. A white-noise signal generator was connected to the loudspeaker. At the other end a reflecting surface was placed, and frequency range was from 0 Hz to 1 KHz. Due to the randomness of the signal, several samples3were taken so as to be able to minimize the effects of bias and variance. Both methods were applied to each sample and eventually, an average was made on every auto-spectral and cross-spectral densities4. Figure 3.4 shows how both methods coincide from barely frequency 0 Hz. The cause of the poorly adjustment at the lowest frequencies is that, on one hand both ASM and ACSM have few samples to make the average. On the other hand, the distance between microphones5is too small to detect variations in pressure waves at low frequencies, with long wavelengths. One should think that the corresponding wavelength of a 10 Hz frequency is 34 meters, approximately. However, above 100 Hz, the difference between both is smaller than 0.02dB and decreases steadily. As a conclusion, and concerning to speed processing terms, ACSM is more efficient, not only because it processes data about 10 times faster, but also because spectrum analyzers generally calculate single-sided spectra.[12] 3Ten samples, each of one second 4Script in Annex B.4 in page 56 50.05 m 25 Measurements and results Jordi Sobrino Gil Figure 3.4: ASM and ACSM under white noise testing. 3.4 Closed-end Measure The main purpose of this measure is to know the frequency response of the whole system: The frequency response of the loudspeaker, and the resonant frequencies of the tube. Thus, the closed-end measure is useful to eliminate the influence of the loudspeaker in the forthcoming ones, that is to say with the MEM measure and MAM measure. 3.4.1 Experimentation set up: Closed-end measure For the closed-end measure barely no changes are needed in the montage, regarding the set-up shown in figure 3.1. The second end of the tube is blocked with a reflecting surface, seen in figure 3.5. 3.4.2 Measurement procedure The measuring procedure was as follows 1. A constant level of power was selected for the three measures. Two requirements needed to be satisfied along the frequency sweep: the amount of power assured an acoustic pressure of 10 dB higher than the background-noise pressure level, but without saturating the measuring 26 Measurements and results Jordi Sobrino Gil Figure 3.5: Closed-end set up. system. The voltage applied at the main loudspeaker from that point on, V1was constant. 2. Starting from 100 Hz, a frequency sweep was done every 10 Hz up to 1000 Hz. 3. At every frequency step, five samples of one second were taken and stored6. 3.4.3 Results Data was analysed by means of the Auto-and Cross-Spectral Method (ACSM) and put altogether into the same graph, as shown in figure 3.6. Only the peak of every sample has true information of energy density values, due to the fact that only pure tones where generated. Ideally, one should see Dirac’s Delta functions every 10 Hz. Instead, these noisy tongues appear with no further consequences. Figure 3.7 shows the extraction of peak values without the influence of any other frequency component. 6Matlab script in Annex B.3, page 54 27 Measurements and results Jordi Sobrino Gil Figure 3.6: Closed-end samples at every 10 Hz processed by means of ACSM. Eventually, peak values corresponding to the resonant frequencies are presented in table 3.1. This peaks are caused by the standing wave explained in section 2.3.1. Figure 3.7: Peak extraction of the closed-end measure. 28 therefore more interesting that both methods have low levels. At these point it is interesting to focus on both maximum peaks of resonant frequencies, and minimum levels of AED, depicted by closed-end maximum and minimum extrema, respectively. •Focusing on the maximum extrema, one can see that the active method has a better behavior in minimizing the energy density in all those frequencies than the emulated passive one. MAM gives a lower average level of energy density in all the frequency span whereas MEM gives significant low levels, specially at alternated resonant extrema. Table 3.2 in page 39 shows the values at maximum resonant frequencies of both methods and the difference in dB’s between them. The difference between the values of both methods can be seen in figure 3.18. •At low AED levels, the minimum-energy method’s powering system is introducing more noise than the tube itself generates. This is due to the fact that MEM is a method conceived for minimizing the average AED through all frequencies, and at those specific frequencies at which the inner response of the system is low, the MEM is more inefficient since it introduces more noise than there already is. On the other hand, the levels of free field propagation that give the MAM are also higher than the destructive adding that take place at such points in the closed-end measure. Table 3.3 in page 40shows the values of both methods and the difference between them at minimum energy density level frequencies. A possible comparison between figure 3.14 and its corresponding figure in the theoretical simulation (Fig.2.11 on page 19) is hard to establish since the theoretical simulation is neither subjected to the conditions of propagation, nor loudspeaker’s frequency response. However, one can see that shapes of graphs and relative positions of them are not so different. In graph 3.14 values of energy density are real, but still depend on the frequency response of the loudspeaker. In order to eliminate such dependence, closed-end measured values are subtracted to the MEM and MAM measures as done in section 2.5.4. Result is shown in figure 3.15, and one can see what has been stated above10. The peaks where the resonant frequencies are 11, MEM has a better behavior. One should also notice that for low frequen10See Appendix A.4 in page 47 11see table 3.2 on page 39 35 Measurements and results Jordi Sobrino Gil cies MEM active method works better, while MAM has a better behavior approaching to 1 KHz. Figure 3.15: Subtraction of Closed-end measure values from MEM and MAM In order to validate the measurements, both graphs are compared to their theoretical simulations in figures 3.1612 and 3.1713, so as to see their similarities. In the minimum energy graph, one can see how the disadjustment between the two lines grows. These elevated values of difference are caused by the minimum pressure levels of stationary waves. In the absorption case these differences are not so strong, since the method is not affected to such waves. Eventually, the difference between both methods is presented in figure 3.1814. The theoretical difference of 3 dB’s is affected by the impreciseness of practice procedures, being the Minimum Energy Method (MEM) specially difficult. To sum up, the measures prove that the MEM is more effective at eliminating stationary waves, especially at low frequencies. It is clear that eliminating just one frequency is easier and much more effective with an antiphase wave generation. However, when covering a whole bandwidth it is not so effective and practical. Above the third or forth harmonic both methods become practically equivalent at high-level resonant frequencies, being absorption more effective in average. If a suitable material is found for the 12See Appendix A.5 in page 48 13See Appendix A.6 in page 49 14See Appendix A.7 in page 50 36 Measurements and results Jordi Sobrino Gil Figure 3.16: MEM measure validation Figure 3.17: MAM measure validation high range of frequencies, the active method could be replaced by a passive one, with the consequently efficiency improvement, since it would not involve any powering system. By any means, using absorption implies that no extra sound field is generated when there are destructive phase adding in the stationary waves along the tube, being more especially effective at high frequencies. In terms of Acoustic Energy Density (AED) reduction, both methods have practically the same average level, distancing just 0.7481 dB 37 Measurements and results Jordi Sobrino Gil Figure 3.18: MAM and MEM Difference the one from the other, which can be confusing due to the expected 3dB. This gap might be due to the difficulty of controlling the membrane velocity of the secondary speaker. It could be said that both methods have proved to be in some way equivalent, at the time of minimizing the Acoustic Energy Density (AED) in a whole bandwidth. Eventually, the last figure is supposed to be the homologue of figure 2.10, where the difference between both methods is presented. For values higher than 0, MEM method is more efficient. In the graph is clear how the efficiency of the active method decreases as the frequency increases. 38 Measurements and results Jordi Sobrino Gil Peak Frequency [Hz] 210 300 400 490 590 680 780 880 980 MAM level [dBJ ·m−3] -15.89 -13.44 -22.06 -22.62 -22.62 -20.78 -20.8 -22.02 -23.14 MEM level [dBJ ·m−3] -27.52 -13.73 -28.06 -26.26 -27.95 -21.03 -23.51 -22.47 -25.06 Difference [dB] 11.73 0.3 6 3.64 5.33 0.25 2.71 0.45 1.92 Table 3.2: Gap values between MEM and MAM of resonant frequencies 39 Measurements and results Jordi Sobrino Gil Peak Frequency [Hz] 170 260 320 450 525 640 730 830 940 MAM level [dBJ ·m−3] -25.5 -31.19 -36.28 -44.65 -48 -52.8 -55.46 -60.07 -63.56 MEM level [dBJ ·m−3] -34.74 -31.64 -36.58 -44.94 -45.4 -49.29 -63.94 -53.78 -60.93 Difference [dB] 9.24 0.45 0.3 0.29 -2.6 -3.51 8.48 -6.29 -2.63 Table 3.3: Gap values between MEM and MAM of destructive phase-adding frequencies 40 Chapter 4 Conclusion During this thesis two active noise control methods have been simulated. The first one emulated a passive method of by means of absorption by a control secondary source. The second one, also active and involving the control of the acoustic energy by means of a relation parameter between the velocities of both loudspeakers, signalling and control. Even though the theoretical explanation tried to start barely from scratch, the reader should be familiar with the basic concepts of physics and specially wave propagation and acoustics. Links to important literature are provided. Hence I think this thesis shows the strength and weaknesses of both active methods and proves them by experimentation. The goals defined in section 1.2 were reached. However, the results of experimentation did not match exactly with the theory stated in chapter 2, and that might be the biggest weakness of this thesis. Causes lay on inaccuracy of the instruments, specially the multimeter needed to control the amplitude of the secondary wave. Nevertheless, it has been demonstrated that both methods are equivalent, being the active one more effective at low frequencies and as effective as the passive one at high resonant frequencies, as seen in section 3.6.3 To conclude, the results of the thesis constitute what many bibliography state. It has been proved that active noise control methods are more efficient at low resonant frequencies whereas their efficiency decreases as the frequency increases. On the other hand, the absorption method has a better broad-band behavior. 41 Appendix A Enlarged Figures Important figures seen in the thesis are here enlarged. A.1 Theoretical Simulations 42 Enlarged Figures Jordi Sobrino Gil Figure A.1: Closed-end, MAM and MEM Simulations Enlarged. 43 A.2 Measurements 44 Appendix B Matlab Scripts B.1 Theoretical Simulation script %Declaration of variables clear all; clc; duration =1; Fs = 44100; T=1/Fs; L=duration*Fs; t =(0:L-1)’*T; nfft=Fs*duration; f=Fs/44.1*linspace(0,1,nfft); % t = linspace(0,1,nfft); w = 2*pi.*f’; % d=0.075; %Diameter of the tube [m] rho=1.21; %Air’s density [Kg*m^-3] % Delta_X=0.05; %Distance of Mic’s [mm] long = 1.736; %Longitude [m] c=343; %Sound velocity [m/s] K = (2*pi.*f’)/c; %Wavenumber Vo =1; x = long*linspace(0,1,nfft)’; ed = zeros(nfft,3); 51 %% -------- Closed-end Simulation -------- beta = zeros(length(f),1); %CAC Condition of Closed-end % From equations 2.22 and 2.23 for i=1:nfft p = (j*rho*c.*Vo.*(beta(i).*cos(K(i).*x) - cos(K(i).*(x-long))))./sin(K(i).*long); v = (Vo.*(beta(i).*sin(K(i).*x) - sin(K(i).*(x-long))))./sin(K(i).*long); ed(i,1) = abs(mean(p))^2./(2*rho*c^2) + (rho/2).*abs(mean(v))^2; end %% -------- Maximum Absorption Simulation -------- beta = exp(-j.*K.*long); % CAC condition of MAM % From equations 2.22 and 2.23 for i=1:nfft p = (j*rho*c.*Vo.*(beta(i).*cos(K(i).*x) - cos(K(i).*(x-long))))./sin(K(i).*long); v= (Vo.*(beta(i).*sin(K(i).*x) - sin(K(i).*(x-long))))./sin(K(i).*long); ed(i,2) = abs(mean(p))^2./(2*rho*c^2) + (rho/2).*abs(mean(v))^2; end %% -------- Minimum Energy Simulation -------- beta = cos(K*long); % CAC condition of MEM % From equations 2.22 and 2.23 for i=1:nfft 52 p = (j*rho*c.*Vo.*(beta(i).*cos(K(i).*x) - cos(K(i).*(x-long))))./sin(K(i).*long); v= (Vo.*(beta(i).*sin(K(i).*x) - sin(K(i).*(x-long))))./sin(K(i).*long); ed(i,3) = abs(mean(p))^2/(2*rho*c^2) + (rho/2).*abs(mean(v))^2; end save data_sim_teoriques; %% -------- Graphical Representation -------- ED = 10*log10(ed); Isolation_1 = ED(:,2) -ED(:,1); Isolation_2 = ED(:,3) - ED(:,1); figure; % plot(f,10*log10(abs(Isolation_1)),’r’, f,10*log10(abs(Isolation_2)),’b’) plot(f,Isolation_1,’r’,f,Isolation_2,’b’) grid on grid minor title(’Red=Max_Abs Blue=Min_En’) figure; plot(f,ED(:,1),’b’,f,ED(:,2),’r’,f,ED(:,3),’k’) grid on grid minor title(’Blue=Closed-end Red=Max_abs Blue=Min_e’) B.2 Calibration Script % Declaration of variables clear; clc; fs = 44100; duration =1; nfft = fs*duration; 53 data = zeros(44100,5); for j=1:5 ai = analoginput(’nidaq’,’Dev1’); ch = addchannel(ai,0:0); set(ai,’SampleRate’,fs) set(ai,’SamplesPerTrigger’,nfft) set(ai,’TriggerType’,’Manual’) set(ai,’Timeout’,5); start(ai) trigger(ai) wait(ai,duration+15); data(:,j) = getdata(ai); stop(ai); clear ai end Vrms = zeros(5,1); for j=1:5 data(:,j)=data(:,j) - mean(data(:,j)); data2= data(:,j).^2; Vrms(j)=sqrt(sum(data2)/fs); end Vrms_med=sum(Vrms)/5; B.3 Acquisition Script %% It gets 5 samples of 1 second %Declaration of variables clear all; clc; duration =1; Fs = 44100; T=1/Fs; L=duration*Fs; t=(0:L-1)*T; nfft=Fs*duration; f=Fs/2*linspace(0,1,nfft/2); d=0.075; %Diameter of the tube [m] 54 rho=1.21; %Air’s density [Kg*m^-3] Delta_X=0.05; %Distancia Mic’s [mm] c=343; %Sound velocity [m/s] N= 5; %Number of samples long = 1.736; %Longitude data = zeros(N,nfft,2); data_cal = zeros(1,Fs); %Data acquisition for j=1:N ai = analoginput(’nidaq’,’Dev1’); ch = addchannel(ai,0:1); set(ai,’SampleRate’,Fs); set(ai,’SamplesPerTrigger’,nfft); set(ai,’TriggerType’,’Manual’); set(ai,’Timeout’,50); start(ai); trigger(ai); wait(ai,duration+90); data(j,:,:) = getdata(ai); stop(ai); clear ai; pause(1) %Calibration ch 0, Mic1 Output Gain 10 data(j,:,1)=data(j,:,1) - mean(data(j,:,1)); %Factor_calibracion = sqrt(10); % If Input Section gain is -10dB %Factor_calibracion = 10; % If Input Section gain is -20dB %data(j,:,1) = Factor_calibracion.*data(j,:,1); %data(j,:,1) = data(j,:,1)./Factor_calibracion; data_cal = data(j,:,1).^2; Vrms_Mic1 = sqrt(sum(data_cal)/nfft); Escala_Mic1 = Vrms_Mic1/(0.1434^2); %Calib.Mic1 a 94 dBSPL data(j,:,1) = Escala_Mic1.*data(j,:,1); %Calibration ch1, Mic2 Output gain 10 data(j,:,2)=data(j,:,2) - mean(data(j,:,2)); %Factor_calibracion = sqrt(10); % If Input Section gain is -10dB %Factor_calibracion = 10; % If Input Section gain is -20dB %data(j,:,2) = Factor_calibracion.*data(j,:,2); 55 %data(j,:,2) = data(j,:,2)./Factor_calibracion; data_cal = data(j,:,2).^2; Vrms_Mic2 = sqrt(sum(data_cal)/nfft); Escala_Mic2 = Vrms_Mic2/(0.1584^2); %Calib Mic2 a 94 dBSPL data(j,:,2) = Escala_Mic2.*data(j,:,2); end clear data_cal save data_1000 B.4 Auto-Spectral Method (ASM) and Autoand Cross-Spectral Method (ACSM) comparison script %Declaration of variables clc; duration =1; Fs = 44100; T=1/Fs; L=duration*Fs; t=(0:L-1)*T; nfft=Fs*duration; f=Fs/2*linspace(0,1,nfft/2); d=0.075; %Diameter of the tube [m] rho=1.21; %Air’s density [Kg*m^-3] Delta_X=0.05; %Distancia Mic’s [mm] c=343; %Sound velocity [m/s] N= 5; %Number of samples load data %Particle velocity calculation v = zeros(nfft,N); for j=1:N int_a=data(j,:,2)-data(j,:,1); for t=1:L v(t,j) = sum(int_a(1:t)); end v(:,j) = T.*v(:,j)./(Delta_X*rho); end 56 clear int_a % Fourier Transform of Mic’s Pressure and Particle velocity P1=2.*fft(data(:,:,1)’,nfft)/nfft; P2=2.*fft(data(:,:,2)’,nfft)/nfft; Vel = (2*rho*c).*fft(v)/nfft; Pot = (P1 + P2)/2; % AED by means of ASM Gp1p1 = zeros(nfft,N); Gp2p2 = zeros(nfft,N); for j=1:N Gp1p1(:,j) = 2.*Pot(:,j).*conj(Pot(:,j)); Gp2p2(:,j) = 2.*Vel(:,j).*conj(Vel(:,j)); end Gp1p1_tot = sum(Gp1p1,2)./N; Gp2p2_tot = sum(Gp2p2,2)./N; ED_time = (Gp1p1_tot + Gp2p2_tot)./(2*rho*c^2); clear Gp1p1_tot; clear Gp2p2_tot; %AED by means of ASCM temp = zeros(nfft,1); Gp2p1 = zeros(nfft,N); for i=1:nfft temp(i,1) = 1/((2*pi*i)^2); end temp = temp./(2*rho*Delta_X^2); for j=1:N Gp1p1(:,j) = 2.*P1(:,j).*conj(P1(:,j)); Gp2p1(:,j) = 4.*real(P2(:,j).*conj(P1(:,j))); Gp2p2(:,j) = 2.*P2(:,j).*conj(P2(:,j)); end Gp1p1_tot_cross = sum(Gp1p1,2)/N; Gp2p1_tot_cross = sum(Gp2p1,2)/N; Gp2p2_tot_cross = sum(Gp2p2,2)/N; ED_freq = (Gp1p1_tot_cross+Gp2p1_tot_cross + Gp2p2_tot_cross)./(8*rho*c^2) + (Gp1p1_tot_cross+Gp2p2_tot_cross - 57 Gp2p1_tot_cross).*temp; clear Gp1p1; clear Gp2p1; clear Gp2p2; clear temp; clear Gp1p1_tot_cross; clear Gp2p1_tot_cross; clear Gp2p2_tot_cross; B.5 AED calculation by means of ACSM clear all clc; duration =1; Fs = 44100; T=1/Fs; L=duration*Fs; t=(0:L-1)*T; nfft=Fs*duration; f=Fs/2*linspace(0,1,nfft/2); d=0.075; %Diameter of the tube [m] rho=1.21; %Air’s density [Kg*m^-3] Delta_X=0.05; %Distancia Mic’s [mm] c=343; %Sound velocity [m/s] N= 5; %Number of samples freq = 100:5:1000; K=2*pi/c.*freq; Beta = cos(K*long); Coef = 0.033.*Beta; Resultat = [freq;Coef]; figure hold on grid on grid minor title(’AED by means od ASCM’) xlabel(’Frequency [Hz]’) ylabel(’Density of Energy [dBJ/m^3]’) tic for i=1:length(freq) 58 s = [’load data_’ int2str(freq(i)) ’.mat’]; eval(s) % Fourier Transform of Mic’s pressure. P1=2.*fft(data(:,:,1)’,nfft)/nfft; P2=2.*fft(data(:,:,2)’,nfft)/nfft; % AED by means of ACSM Gp1p1 = zeros(nfft,N); Gp2p2 = zeros(nfft,N); Gp2p1 = zeros(nfft,N); for j=1:N Gp1p1(:,j) = 2.*P1(:,j).*conj(P1(:,j)); Gp2p1(:,j) = 4.*real(P2(:,j).*conj(P1(:,j))); Gp2p2(:,j) = 2.*P2(:,j).*conj(P2(:,j)); end Gp1p1_tot_cross = sum(Gp1p1,2)/N; Gp2p1_tot_cross = sum(Gp2p1,2)/N; Gp2p2_tot_cross = sum(Gp2p2,2)/N; temp=1./((2*pi*(1:nfft)’).^2)./(2*rho*Delta_X^2); ED_freq = (Gp1p1_tot_cross+Gp2p1_tot_cross + Gp2p2_tot_cross)./(8*rho*c^2) + (Gp1p1_tot_cross+Gp2p2_tot_cross - Gp2p1_tot_cross).*temp; clear Gp1p1; clear Gp2p1; clear Gp2p2; clear temp; clear Gp1p1_tot_cross; clear Gp2p1_tot_cross; clear Gp2p2_tot_cross; end toc 59 Bibliography [1] American Society of Civil Engineers, American Society of Mechanical Engineers,Paper University of Michigan, 1948. [2] David A. Bies and Colin H. Hansen, Engineering Noise Control:Theory and Practice, 2003. [3] Colin H. Hansen, Understanding Active Noise Cancellation Routledge, 2001. [4] Sutton, T.J., Elliot, S.J., MeDonald, A.M. and Saunders, T.J. Active Control Of Road Noise Inside Vehicles Noise Control Engineering Journal, Vol. 42, No.4, July 1994. [5] Colin D. Kestell and Colin H. Hansen An overview of active noise control Department of Mechanical Engineering, University of Adelaide, Adelaide, Australia, 1998 [6] Frank Fahy. Foundations of Engineering Acoustics Institute of Sound and Vibration Research, University of Southampton, UK, 2001. [7] David T. Blackstock, Fundamentals of physical acoustics, Wiley-IEEE, 2000. [8] Michael M¨ oser Technische Akustik Fachgebieten der Technischen Akustik, Technische Universit¨ at, Berlin, 2005 [9] Bernd Page and Wolfgang Kreutzer. The Java Simulation Handbook. Simulating Discrete Event Systems with UML and Java. Shaker Verlag GmbH, Germany, 2005. [10] Br¨uel and Kjær Microphones and Conditioning Product Catalogue n.16, Denmark, 2009. 60