2. Fundamentos teóricos
2.4. Acústica del oboe
Section 3.2 introduced the microstructure idealisation where the material pore structure is modelled by a network of cylindrical pipes connecting spheres and the particle interaction contacts are modelled by mechanical line elements with circular cross-sections. Spheres, pipes and mechanical element cross-section radii are randomly chosen. The assignment of the radii is based on random number generations according to a chosen statistical distribution. The distributions used in the present thesis are the lognormal and the linear ones. The shape, the input parameters and the process of the random radii values generation will be presented.
3.3.1
Lognormal distribution
The lognormal distribution is a continuous distribution of a positive variable. Here, the variable is denoted by r which represents the radius of a pipe, a sphere or a mechanical element cross-section. The logarithm of the lognormal variable exhibits a normal distribution. The parameters that determine the two distributions are linked as
ζ = ln p rm 1 + COV2L ! (3.2) σ = q ln(1 + COV2L) (3.3)
where ζ and σ are the mean and the standard deviation of the normal distribution of ln r and rm and COVLare the mean and the coefficient of variation of the lognormal distribution of r. COVL is evaluated as
COVL = s rm
where s is the standard deviation of the lognormal distribution of r. The mathematical expression of the Probability Density Function (PDF) for r is
Ω(r) = 1 rσ√2πe −(ln r − ζ) 2 2σ2 (3.5)
The shape of the PDF is schematically presented in Figure 3.6. The vertical axis presents the frequency that a radius value (the random variable) is generated. Since the random variable values in a lognormal distribution span from 0 to infinity, it was decided to truncate each distribution in order to limit radii to a range that made physical sense. Examples of these imposed minimum and maximum radii, rminand rmaxrespectively, where the distribution was truncated, are presented in Figure 3.6. Note that the mean value of the distribution, rm, is greater than the value corresponding to the peak of the PDF curve, because of the asymmetry of the lognormal PDF.
Figure 3.6: Schematic lognormal Probability Density Function: frequency versus radius r, where rmis the mean and the distribution is truncated at rminand rmax.
The generation of a random variable would involve the use of the inverse of the PDF function where the frequency would be given as an input and the random variable as an output. However, with the exception of the peak, two different values of the random variable correspond to each value of frequency. The standard approach for generating random variable values according to
a given distribution is therefore to use of the inverse of the Cumulative Distribution Function (CDF). The lognormal CDF is schematically presented in Figure 3.7. The CDF begins from zero, increases, monotonically and asymptotically approaches a value of 1 as r tends to infinity and is expressed by the function
Φ(r) = 1 2 1 + erf (ln r − ζ) σ√2 (3.6)
where erf() denotes the error function defined as
erf(r) = √2 π r Z 0 e−t2dt (3.7)
Figure 3.7: Schematic lognormal Cumulative Distribution Function: probability versus radius r and the minimum, maximum and mean rmin, rmaxand r
m, respectively.
The process of the random variable generation for a lognormal distribution is described below.
1. The input parameters concerning the lognormal distribution of r (rm and COVL) and its truncation (rminand rmax) are chosen.
2. Random numbers with a uniform distribution over in the interval (0, 1) are generated using a random number generator algorithm (Knuth, 1982).
3. Each number generated is used as input for the inverse lognormal CDF (inverse of (3.6)) and a corresponding r value is generated.
4. A final check is performed for the generation of the lognormal variable according to the expression below. r = rmin if r ≤ rmin r if rmin< r < rmax rmax if r ≥ rmax (3.8)
3.3.2
Linear distribution
The linear distribution owes its name to the shape of its CDF (Figure 3.8). The probability of the generation of a radius P (r), varies from 0 to 1, as the variable r varies between minimum and maximum values rmaxand rmin, respectively. The CDF function is
P (r) = aLinr − bLin (3.9)
where aLin= 1/(rmax− rmin) and bLin= rmin/(rmax− rmin). The mean rm of the linear distri- bution is given by
rm =
rmin+ rmax
2 (3.10)
The corresponding PDF is presented in Figure 3.9. The PDF curve is a constant of value of 1/(rmax− rmin) and, as with the lognormal distribution, its inverse cannot be used to generate random variables that follow this distribution. Therefore, the inverse of the CDF is used for this purpose, that is
P−1= r = P + bLin
Figure 3.8: Schematic linear Cumulative Distribution Function: probability versus radius r and the minimum and maximum rminand rmax, respectively.
The process of generating random variables that follow the linear distribution is
1. Choose aLinand bLinas inputs based on chosen rminand rmax.
2. Generate uniformly distributed random numbers P in the (0,1) interval.
3. Use the random number P in (3.11) to generate the random variable r that follows the linear distribution.
3.3.3
Input parameters
The input parameters for the lognormal distribution are rm, COVL, rminand rmax. The symbols of the lognormal input parameters for the different entities i.e. sphere, pipe and contact area radii are presented in Table 3.1.
Figure 3.9: Schematic linear Probability Density Function: frequency versus radius r and the minimum and maximum rminand rmax, respectively.
Table 3.1: Lognormal distribution input parameters.
Entity rm COVL rmin rmax Sphere rsm COVs rmin
s rmaxs Pipe rpm COVp rpmin rmaxp Contact area rmech COVmech rmechmin rmaxmech
For the linear distribution, the input parameters are aLinand bLinas presented in (3.11). The corresponding symbols for each entity are presented in Table 3.2.