There are many ways of analysing sound data. The methods broadly fall into two categories:
• Single-value indices
• Frequency-dependent indices.
4.9.1 Single-value indices: pressure–time history
This is a two-dimensional plot of calibrated pressure (Pa) on the vertical axis against time (s) on the horizontal axis. Such plots are useful as a pre- liminary check on the quality of the data. The root mean square pressure is given by
p p
rms≈ max 2
(4.9) The sound pressure level is given by
L p
p p
rms ref
= ⎡ dB
⎣⎢ ⎤
20log10 ⎦⎥ (4.10)
where pref= 20 μPa (20 × 10−6Pa).
The time-varying sound pressure level offers a compact means of display- ing a fl uctuating sound fi eld on a two-dimensional plot. It is commonly used for both interior and exterior vehicle noise levels as well as for internal combustion (IC) engines and other machines with a wide operating speed range including cooling fans, alternators, pumps and injectors. Frequency weightings may be applied to the data (A, B and C weightings). It is usual to use the fast time weighting (rather than impulse or slow). In any case, it is important to state the use of any weightings in the vertical axis level – the usual form being, for instance, Lpf dBA at 1m from the source. This means fast time weighting and A frequency weighting applied to the raw data. The main application of pressure–time history is the total level of constant- speed tests including idle tests and pass-by noise.
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4.9.2 Frequency-dependent index methods:
frequency spectrum
The frequency spectrum is commonly used in vehicle tests for octave band analysis and order tracking. In order to understand the octave band analy- sis, narrowband fi lters are illustrated as below.
Noise bandwidth of narrowband fi lters
Before discussing frequency-dependent indices, the noise bandwidth Bn of a narrowband fi lter must be defi ned. The noise bandwidth Bn is defi ned as the bandwidth of the ideal fi lter that would pass the same signal power as the real fi lter when each is driven by stationary random noise:
Bn= − =f2 f1 ∞
∫
H f( )2df0
(4.11) In the ideal fi lter the modulus of the amplitude transfer function H(f) is zero outside the pass band and unity within the pass band:
H f x ( ) = xout
in
(4.12) where the over-score denotes a complex quantity.
A real fi lter will have an amplitude transfer function that is not unity right across the pass band and does not go immediately to zero outside the pass band.
Two common classes of narrowband fi lter are used for analysis of sound data:
• The constant bandwidth type where Bn is the same for all fi lter centre frequencies fc
• The constant percentage bandwidth type where Bn is a constant percent- age of fc throughout the frequency range.
Constant percentage bandwidth fi lters
The constant percentage bandwidth class will be considered fi rst. Commonly available fi lters of this type have widths of one octave, 1/3 octave, 1/12 octave and 1/24 octave as shown in Table 4.5.
The levels of narrowbands that fi t within the bandwidth of coarser fi lters (for instance, the three 1/3-octave bands that fi t within the bandwidth of the one-octave fi lter) may be combined to give the band levels of the coarser fi lter. The combination of band levels must be done by logarithmic addition:
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L L
i n
total= ⎡ i dB
⎣⎢
⎤
⎦⎥
∑
=10 10 10 10 1
log (4.13)
Following this logic, 1/24-octave bands may be combined to give a 1/12- octave spectrum, the 1/12 bands may be combined to give a 1/3-octave spectrum and so on, until the overall level is obtained (single index). The main application of constant percentage bandwidth fi lters is octave fre- quency spectrum analysis of constant-speed tests and order tracking analy- sis of run-up and run-down tests. Plate I (between pages 114 and 115) shows a constant-speed 1/12 octave noise spectrum of a vehicle.
Constant bandwidth frequency analysis methods
One of the most commonly used outputs from a constant bandwidth spec- trum analyser is the power spectral density (psd). In this case, the word
‘power’ is perhaps a misnomer as in general the psd has units of volts squared per hertz (V2/Hz) rather than watts per hertz (W/Hz). It may have true units of power if calibrated accordingly. For instance, in the far free- acoustic fi eld:
W P S
= rms2c
ρ0 (4.14)
where S is the surface area of the wave-front (m2), ρ0 the undisturbed air density (kg/m3) and c the speed of sound (m/s).
The power in each spectral band i is given by:
W S
cP
T P t t S
i= i = ⎡ i( ) c
⎣⎢ ⎤
⎦⎥ ×⎡
⎣⎢
⎤
( ) ⎦⎥
∞
∫
ρ0 ρ
2 2
0 0
1
rms d (4.15)
W Wi
i
= n
∑
= 1(4.16)
Table 4.5 Characteristics of constant percentage bandwidth fi lters (n = band number)
Filter Centre frequency fc (Hz)
Lower frequency f1 (Hz)
Upper frequency f2 (Hz)
Integer values of n for the audio range
Bandwidth around centre frequency (%) Octave 10n/10 10(n−1.5)/10 10(n+1.5)/10 12–43 69
1/3 octave 10n/10 10(n−0.5)/10 10(n+0.5)/10 12–43 23 1/12 octave 10(n+0.5)/40 10n/40 10(n+1)/40 48–172 6 1/24 octave 10(n+0.5)/80 10n/80 10(n+1)/80 96–344 3
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psdi
B i
i
B W
= → B
lim→ 0
2 2
0 (4.17)
where Bi is the bandwidth of band i.
The psd of the signal from a stationary random process is a smooth con- tinuous function of frequency. For a cyclic process the psd is not smooth as it consists of a series of harmonically related peaks.
Spectral analysis may be performed using either:
• contiguous fi lters (analogue or digital) or
• Fourier analysis.
The main application of constant bandwidth frequency analysis is frequency spectrum analysis of constant-speed tests and order tracking analysis of run-up and run-down tests.
Order tracking
When analysing the sound from rotating machinery such as internal com- bustion engines, it is common to use the order-tracking technique. The now obsolete but instructive analogue method was as follows:
1. Obtain an electrical signal that is proportional in some way to the speed of rotation of the machine, for instance a tachometer signal. Calculate the rotational frequency of the machine, f (Hz).
2. Set a constant percentage bandwidth fi lter (6% is common for 1/12 octave) to have fc equal to the rotational frequency of the machine, and organize by electrical means for it to follow or track changes in the rotational frequency f0.
3. Set other constant percentage bandwidth fi lters (6% is common for 1/12 octave) to follow, or track, changes in the rotational frequency, each one with fc set to a different order, or harmonic, of the rotational fre- quency of the machine f0:
fc= nf0 for n > 0, not necessarily an integer value
4. Plot the output from each fi lter against the rotational speed of the machine.
Order track analysis separates noise characteristics of the order of interest from overall noise. Figure 4.7 shows the engine combustion order spectrum of a vehicle.
Waterfall contour plot
Some spectrum analysers allow the use of a tachometer signal to produce a three-dimensional contour plot of frequency spectra against time or
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machine speed. These are known as waterfall contour plots as shown in Plate II (between pages 114 and 115).
Each horizontal ‘slice’ is an individual spectrum, gathered over a user- defi ned averaging period. Beware, with rapid changes in machine speed, that the averaging time for each spectrum will have to be short and there- fore individual band levels will only be estimates of the true band levels.
Vertical lines of peaks signify resonances – high amplitudes at particular frequencies that are independent of machine speed. Diverging lines of peaks signify different orders. The fi rst order is usually caused by imbal- ances or misalignments. For a four-stroke internal-combustion engine, half of the number of cylinders is the number of the combustion order, since for every two rotations all of the cylinders fi re once, which gives the number of cylinders for combustion pulses. For every rotation half of the cylinders have combustion pulses; therefore half of the number of cylinders is the combustion order of the engine.
The waterfall contour diagram is often used for evaluation of powertrain noise and vibration in the intake and exhaust development, engine and transmission mounting development, and prop shaft and differential mount- ing development.