Двигатели внутреннего сгорания. 2010. №2 PDF (Size:7770 МБ)

Рабочие процессы ДВСSmoothing the statistical series by a logarithmnormal distributionOften it happens to be desirable to compare toexperimental data such a distribution of a randomvalue x in an interval (a, b), so that some function g(x)would have a normal distribution, i. e. for the class ofdistributions by Captain [9, p. 613]:whereΦxΦ x( x)( )( x)⎡ g − μ ⎤x = Φu⎢ ⎥ , (3)⎢⎣σg⎥⎦– integral of probabilities,normal function of distribution, μ – expectation,standard deviation.The density of the distribution:ϕx( x)=σg⎧1 ⎪exp⎨−2π⎪⎩12⎡ g⎢⎢⎣( x)Φ u –σ g –2⎫− μ ⎤ ⎪ dg⎥ ⎬ . (4)σg⎥⎦⎪ dx⎭The random variable x, described by distribution(3, 4), can be considered as the limit of the sequence ofthe random valuesxr+ 1 xr+ zr+ 1( x )= h (r = 0, 1, 2, …),each of which is the result of an action of small independentimpulses z 1 , z 1 , …, z 1 , that satisfy conditions ofthe limit theorems (by Chebyshev, central limit theorem),moreoverr∑ − 1i=0xi+ 1 − xidxz1 + z2+ . . . + zr =≈ = g( x)h ∫.rx( x ) h( x)In particular, if h(x) = x, that is the effect of animpulse action is proportional to already achievedvalue of the random variable, and x 0 = 1, theng(x) = lnx, and we have logarithm normal distributionix0( x ≤ 0)⎧ 0,⎪( ) ⎨ ⎪⎧2ϕ x = 1 1 ⎡lnx − μ ⎤ ⎪⎫(5)⎪ exp⎨−⎢ ⎥ ⎬ ( x > 0).⎩xσ2π2 ⎪⎩ ⎣ σ ⎦ ⎪⎭According to the shape of the histogram4Nkponthe fig. 3, and theoretical reasons (5), we chooseμ = ln12,661 and σ = 0,5765 in such a way that thecurve f(x) on the fig. 3 would have similar shape to the4N kp.The hypothesis test of a logarithm normal distributionTo test the hypothesis about accordance betweenthe theoretical distribution (5) and assumed experimentaldata we use one of the most commonly used tests:the chi-square goodness-of-fit test.The Pearson criterion χ 2 is calculated by the formula[8, p. 152]χ2= nk *( pi− pi)∑i=1pi2, (6)where k – the number of intervals, p i – theoreticalprobability for random variable D k to get to the specifiedi th interval of the testing distribution given by thedensity (5).Or on the condition of (2) the measure of diversity(6) has the view ofU = χ2=k∑i=1( m − np )inpii2. (7)Distribution of χ 2 (6, 7) depends on a parameter r,which is called “the number of degrees of freedom” ofthe given distribution. It is calculated by the formular = k − s , (8)where s – the number of imposed restrictions.In this problem setting it is just onek*p ii=1∑= 1. (9)In the table 3 it is illustrated the results of calculations.The sum of p i :kp ii=1∑= 0,986086 ≠ 1becausethe normalizing condition for the case of the logarithmnormal distribution (5) means∞∫0( x)f dx = 1,which accomplishes.The number of degrees of freedom (8) with respectto the restriction (9) at the k = 16r = 16 −1= 15 .Using special tables composed for χ 2 distributionfor the values of χ 2 = 10,24 and r = 15 we get probabilitiesp = 0,8 at the χ 2 = 10,31 and p = 0,9 at theχ 2 = 8,55.Thus with the probability more than 0,8 the diversitiesbetween experimental and theoretical dataoccur due to random factors and the hypothesis of thelogarithm normal distribution of fuel droplets sizes in aspray with the parameters of μ = ln12,661 andσ = 0,5765 can be recognized as the one that does notcontradict to the experimental data.38ДВИГАТЕЛИ ВНУТРЕННЕГО СГОРАНИЯ 2'2010