Copyright © 1998 Society of Automotive Engineers, Inc. ABSTRACT. Practical issues to ... know more about NAH, and the technician or engineer acquiring and .... high degree of complexity in the frequency domain, while a simple point source ...
99NV-174
Practical Aspects of Making NAH Measurements S.M. Dumbacher and D.L. Brown University of Cincinnati
J.R. Blough Michigan Technological University/KRC
R.W. Bono The Modal Shop, Inc. Copyright © 1998 Society of Automotive Engineers, Inc.
ABSTRACT Practical issues to consider when making measurements for Nearfield Acoustical Holography (NAH) analysis are addressed. These include microphone spacing and placement from the test surface, number of microphones and array size, reference microphone number and placement, and filtering of the data. NAH has become an accepted analysis tool so that several commercial packages are available. Its application is limited to test surfaces that are fairly planar, lending itself well to tire testing, front of dash testing, engine face testing, etc. In order to achieve accurate NAH results, the measurement and analysis process must be clearly understood on a practical level. Understanding the advantages and limitations of NAH and the measurement parameters required of it will allow the user to determine if NAH is applicable to a particular test object and environment. INTRODUCTION The term acoustical holography refers to the calculation of acoustic quantities in 3D space based on pressure measurements from a 2D surface. Conventional acoustical holography equations were formulated at the same time as optical holography equations, with the development of the laser. Conventional acoustical holography measurements were made in the farfield of the sources. With planar pressure waves in the farfield, one node point of a wave has to be crossed in order to resolve the wave. Because of this, the maximum source resolution of conventional acoustical holography was a half wavelength.
The 2D pressure measurements of Nearfield Acoustical Holography (NAH) were made in the nearfield, rather than the farfield. By making nearfield measurements, localized information about the sources was obtained that allowed for sub-wavelength source resolution. With NAH, the acoustic quantities can be reconstructed on 2D surfaces parallel to the measurement surface, from the source surface out into the farfield. The reconstructed pressure is calculated as a function of the measured pressure and a propagation function that describes how sound propagates in space. The reconstructed velocity is a function of the gradient of the reconstructed pressure. The reconstructed intensity is a function of the reconstructed velocity and pressure. The sound power and reactivity index can also be computed. NAH can be used to locate sources at the surface, or to determine how the sound field propagates away from the surface and into the farfield. NAH is an alternative to intensity scans made with an intensity probe. Anyone who is familiar with making intensity scans knows that they are time consuming, tedious, possibly dangerous, and valid only for steady state operating conditions. The advantages of NAH include more consistent spatial data sets, the possibility of an automated measurement process, the evaluation of transient or steady-state noise sources, and the availability of 3D acoustic quantities. With an array of microphones, the test setup time is longer, but taking multiple operating conditions only takes as long as scanning the array. References for the NAH theory are included in [1-4]. References for comparisons of NAH and intensity probe scans are included in [5-6]. References for error studies of NAH are included in [3,7]. References for several applications of NAH are included in [8-15].
This paper aims at the entry level engineer wanting to know more about NAH, and the technician or engineer acquiring and processing the data. In an effort to understand the basics of NAH, the paper is practical rather than theoretical. Rules of thumb are given where applicable. HOW NAH WORKS Most of the details of the NAH analysis are “hidden” from the user of a commercial NAH software package, meaning that the user is not required to understand them in order to analyze the data. However, understanding the basics of the NAH analysis allows the user to understand why certain measurement parameters are related, and what happens when those parameters are changed. Understanding the basics of NAH requires an understanding of linear algebra and the Fourier transform.
transforms the 2D spatial pressure measurements, which are P(x,y) for a plane, into the wavenumber domain P(kx,ky), where kx and ky are components of the wavenumber k. Since the properties of the FFT hold for any domain transform, the time/frequency transform analogies are presented to aid in the understanding of the spatial/wavenumber transform. The requirements of the FFT and 2D FFT are listed below. The 2D FFT requires 2D surfaces. The source surface and the microphone array must be surfaces that can be described with 2 dimensions, meaning planes or cylinders. The FFT requires evenly spaced measurement points. For the 2D FFT, the measurement points must be evenly spaced in each dimension. In the time/frequency transform, the spacing between the time points ∆t must be a constant
∆t = const.
THE WAVE EQUATION
(1a)
NAH starts by assuming the pressure satisfies the linear homogeneous acoustic wave equation, which describes how sound propagates in fluids.
In the spatial/wavenumber domain, the spacing of the microphones must be constant in a given direction. This is equivalent to
The pressure at any point in space exterior to the measurement surface can be obtained by solving the acoustic wave equation in free-field space (a sourcefree, reflection-free space). This equation is known as the 1st Rayleigh integral equation, and is a 2D “convolution” integral. It states that the pressure at a reconstruction surface is equal to the pressure at the measurement surface “convolved” with a function describing how sound propagates from the measurement surface to the reconstruction surface.
dx = const. dy = const.
Those familiar with the Fourier transform will recognize that the Fast Fourier Transform (FFT) is a computationally efficient method for solving a convolutional integral, since convolution in one domain is equal to a simple multiplication in the transformed domain. NAH uses the 2D FFT, which is a computationally efficient approach to solving the 2D convolution integral. THE 2D FFT Since NAH solves the 2D convolution integral using a 2D Fourier transform, understanding the basics of the FFT is helpful in understanding the basics of NAH. The Fourier transform assumes that a signal can be represented as a sum of sine and cosine waves. Many people are familiar with the Fourier transform from the time domain to the frequency domain. The Fourier transform is actually a valid transformation from any evenly spaced domain to another, and was originally developed by Fourier for heat conduction applications. The 2D FFT performs an FFT along each of the two dimensions of the data. The 2D FFT used by NAH
(1b)
where dx is the spacing of the microphones in the x direction and dy is the spacing of the microphones in the y direction. Note that dx does not have to equal dy. Shannon’s sampling theorem for the FFT states that the resolution or spacing of the measurement points is equal to one over the sampling frequency fsamp. The maximum frequency of interest fmax must be half the sampling frequency. This prevents a possible error in the FFT due to aliasing, which occurs when information at frequencies higher than fsamp wraps into the frequencies lower than fsamp. In the time/frequency transform, the relationship is
∆t =
1 f samp
=
1 2 f max
(2a)
For the spatial/wavenumber transform, avoiding spatial aliasing means that the spacing of the microphones must be less than or equal to a half wavelength of the highest frequency of interest. The relationship is
max(dx, dy ) ≤
c λ = 2 f max 2
(2b)
where c is the speed of sound and λ is the wavelength corresponding to the maximum frequency of interest.
The FFT assumes that the measured signal is either periodic or completely observable within the acquisition block. If this is violated, leakage occurs where energy from one frequency spills into adjacent frequencies. In the time/frequency transform, a time domain window, such as a hanning window or exponential window, is applied to the data to reduce leakage effects for signals that are not periodic or completely observed. For the spatial/wavenumber 2D FFT, avoiding leakage means that the pressure signals in the microphone array must be completely observed spatially within the microphone array. Therefore, the pressure should be zero at the edges of the grid to avoid spatial leakage. A 2D window can be applied to the spatial measurements to force the grid edges to zero. Keep in mind, however, that applying a window adds distortion to the system. In order to compute the Fourier coefficients accurately, a given number of measurement points must be used. For example, in the time/frequency transform, if there are only 4 time points, then there are only 4 spectral lines. All the represented frequencies must fall into these 4 spectral lines. In the spatial/wavenumber transform, the grid must have more than a few microphones in each direction, typically on the order of at least 12-16. The reason why this number is less than is typically used in a time/frequency transform is that a time signal can have a high degree of complexity in the frequency domain, while a simple point source or mode does not have a very complex spatial pattern in the wavenumber domain. EVANESCENT WAVES Measurements for NAH are made in the nearfield of the source surface, or less than a wavelength away, to capture the localized information about the sources. The localized information contains what are called evanescent waves, allowing NAH to have subwavelength source resolution. How NAH obtains subwavelength source resolution is explained as follows in the wavenumber domain. The wavenumber k, given by
k=
2πf c
2
2
e jk z z , 2
2
k z = k 2 − (k x + k y )
(5)
where z is the distance from the source surface. When kz is real, which occurs when k2 ≥(kx2+ky2), the propagating function is a pure phase term. This represents waves varying in phase as a function of space. When kz is purely imaginary, which occurs when k2
