ACOUSTIC CONTROL USING IMPEDANCE CHARACTERISTICS Eric Herrera Boeing Aerospace Corporation, Seattle Washington USA e-mail: eric.herrera@boeing.com Using the real and imaginary components of impedance, materials can be designed to provide optimal acoustic control. Theory will be discussed leading to appropriate experimental testing. Data will be shown to validate theory. The mathematical model developed will be shown to be validated using test data. The application of the theory/model in designing the optimal acoustic impedance material will be discussed. 1. Introduction Technology has enabled a myriad of possibilities for designing complex out of plane acoustic control devices although often these complex devices are designed via empirical data/experimentation, very costly/time consuming, and not quantitative thus not optimized. This represents a gap between manufacturing technology and quantitative approaches in R&D of complex acoustic devices. This paper proposes a method using the theory of acoustic impedance to enable a quantitative approach for optimizing complex acoustic devices and reviews limited empirical data verifying the approach suggested herein. The author would like to point out that the industry would be well benefited for further exploration/collaboration in relevant data collection/experimentation regarding this approach enabling design & optimization of complex acoustic devices. The acoustic impedance is known to consist of a real and imaginary characteristic. The real part can be properly considered as having a physical characteristic more spatially dependent, in this case attenuation. The imaginary part can be considered to be more characteristic of time (1/frequency), in this case phasing. Depending on the physical attributes of the interacting material(s) the real and imaginary characteristics can be made to be more or less cross correlated. As opposed to current models which relate the physical attributes using direct and dependent relationships (reference 7), the characteristic extremes for Linear materials can be characterised as primarily independent and thus generating a matrix of non-zero autospectra and relatively near zero cross correlation coefficients or primarily interdependent with non zero autospectra and non negligible cross correlation coefficients. The latter is what is desired to provide attenuation (real characteristic) over a controlled temporal (frequency) domain. Linear materials refer to materials which have a relatively constant flow resistance as a function of particle velocity. This paper will provide some detail of how the above concepts can be used to design acoustic control into linear materials using impedance characteristics. ICSV22, Florence (Italy) 12-16 July 2015 1
2. Acoustic Impedance real and imaginary relationship The mathematical formulation relating the real and imaginary characteristics of the Acoustic impedance can be derived from the transverse amplitude distribution function, F(y) for a propagating acoustic pressure disturbance (reference 1): (1) where k x = k + iχ and Y = k x /(k k x M x ), k is the free space wave-number, k x is the wave number in the x-direction and M x (a function of y for the sheared flow) is the Mach number of the mean flow in the x-direction. The pressure and the transverse particle velocity (in the y-direction see Fig. 1 below) are respectively: (2), v The particle velocity amplitude distribution function, G(y), can be related to F(y) by using the linear momentum equation for the fluid, reference [1]. This relationship is: (3) Figure 1: For air flow in the x direction we have M x = v x /c, v x being the mean flow velocity in x and c the speed of sound An acoustic consideration for the application here is to recognize that the main shear term in equation (1) M x (y) can be taken, in the limit, as constant and dropping to zero at the boundary surface. Therein is the opportunity to enable a closer inspection of the flow at the, relative to boundary layer, thin material liner layer. The references, [2], [3] and [4], show that with this limit assumption of the uniform plug flow model we can obtain correct results thus enabling the introduction of the no-flow acoustic impedance. Using the limit noted above a general solution of F(y) when M x (y) has a plug flow behavior is: (4), where the transverse wave number, k y, is defined by. The transverse wave number, k y, is determined by the boundary condition. The boundary condition being that the ratio of the acoustic pressure to the normal acoustic particle velocity (into the ICSV22, Florence, Italy, 12-16 July 2015 2
material) at the material, y = ±l y /2, must equal the specific normal acoustic impedance. In the application here the acoustic particle velocity in the linear material is effectively constrained to be normal to the face of the liner, as a purposely designed result of the construction of the liner material. The purposely designed result as noted herein is, in part, to simulate behavior of a perforate. Based on the above assumptions the impedance boundary condition at the liner material (y = ±l y /2) is; (5), where Z is the specific normal acoustic impedance ratio. Substitution for [F(y)]/ y, from equation (3) and simplification by using equation (4) with the plug flow model gives: (6), where the cot solution gives the even modes and the equivalent tan solution the odd modes. With the transverse wave-number now given by: (7). Solving equation (6) yields k y and substituting that result into (7) yields k x. Revisiting equation (2) with these substitutions it becomes clear that the pressure attenuation (real part of impedance) per unit length depends on the imaginary part of k x, from equation (7): (8). Parametric studies, reference [5], have indicated that nominal changes in the imaginary part of the impedance will result in a shift of the optimum attenuation (real part of impedance) frequency(s). Thus the material impedance needs to be chosen to maintain optimal attenuation over the desired frequency band(s). The conclusion here leads to the realization that more acoustic control is possible with a material when considering the real and imaginary parts of the acoustic impedance. 3. Applying real and imaginary impedance characteristics to Materials The relationship mathematically, quantitatively, described above has been investigated qualitatively. Specifically, materials have been made taking into account the impedance characteristics noted above regarding attenuation and effective range of frequencies. The material sample shown below, Figure 2, depicts such a material. The Figure 2 is a top down perspective for flow considerations. How open the weave is of the material is one consideration not only the interstice (~perforate) size/shape but also the percent open area (POA). The interstice detail and POA will provide a phenomenon with respect to the flow analogous to a mass effect and thus likened to the resistance which is related to the real part of the impedance, equations [6,8], ICSV22, Florence, Italy, 12-16 July 2015 3
Figure 2: A material showing interstice at 50 and 100X magnification The same material as above is shown in Figure 3. The Figure 3 is a cross section perspective allowing a view at how the weave is constructed, note interest in mechanicals and flow. Of particular interest is how the flow (in terms of, mechanicals, force and velocity) is related to the reactance, imaginary part, of the impedance and as shown in equations [6, 8] related to the frequency(s) over which the attenuation is optimum. Figure 3: A material showing interstice The reactance can be shown to be the ratio between force and velocity which is directly proportional to frequency, reference [6, pp315]. The more the material reacts locally the more the material absorbs the energy-frequency information. The material construction thus considers how to best distribute that frequency by weaving, intertwining of the materials. The intertwining of these materials can be designed to best permit attenuation over a broad frequency range. The Figure 3 shows a segment of how intertwined this particular example material is. 3.1 Empirical results Various materials were made as suggested above and tested. A standard acoustic flowbench was used to measure the Rayl versus particle velocity. ICSV22, Florence, Italy, 12-16 July 2015 4
Figure 4: Rayl VS various intertwined materials The Figure 4 shows the flow resistance (in term of Rayl value), which is essentially resistance as a function of velocity. The flowbench for these tests uses DC flow so there is no frequency content. Thus the resistance measured in Figure 4 is only due to inertial losses. The interstice details and POA allow various resistance options which can be used for tuning the particular material option. Recall although there is a cross correlation relationship with resistance and reactance thus the most resistive material is not necessarily the best. The interdependence of the resistance and reactance as discussed above indicates it is a combination of resistance and reactance characteristics we seek in identifying optimal material for our purposes. Notional optimum impedance curves were created to illustrate the concept of identifying optimal material via impedance. Certainly true material impedance curves can be established using current technology. These notional optimum impedance curves can be considered the acceptable impedance limits which, when projected to a far field noise environment, provide optimal acoustic attenuation over a specified (controlled) broadband frequency range. Once again the Figure 4 essentially represents resistance but using the logic of the above discussion and reference [5] we find interest in the slopes of the various curves as a means to bring light to the reactance. Obviously the closer the slope is to unity (1) the more consistent the resistance is over velocity which is directly proportional to frequency. The Figure 5 shows impedance resistance and reactance for the same materials as Figure 4. Due to measurement difficulties a harmonic (red spike) with material S caused great variation in scale. The other material systems show similar resonance but not as severe. ICSV22, Florence, Italy, 12-16 July 2015 5
Figure 5: Impedance Resistance and Reactance The Figure 6 has the material S spike removed and is thus easier to discern the notional optimum impedance limits. The Figure 6 more clearly shows the material VC within the notional optimum impedance limits. Note all materials had the ~4kHz spike due to an air volume resonance in measurement. ICSV22, Florence, Italy, 12-16 July 2015 6
Figure 6: Rayl VS various intertwined materials At the time of writing this paper a measurement device is being implemented with intent to eliminate the 4 khz resonance thus providing data more correctly representing the in situ environment. Improved measurement without the 4 khz resonance is predicted to show material VC more closely within the notional optimum impedance limits. 3.2 Conclusion Based on the results of these impedance designed linear acoustic materials with particular interstices and intertwined materials it can be concluded that considering the resistance and reactance of acoustic impedance will result in optimal acoustic control. An improved measurement device is needed and underway to capture more precisely the in situ environment of interest. A forth coming paper is intended to update on those findings. ICSV22, Florence, Italy, 12-16 July 2015 7
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