騒音・振動の任意不規則レベル分布に対する有限展開項型統一処理方法

抄録

The random signals (e. g. , street noise, machine or structure vibration) often appearing in the actual engineering fields exhibit various kinds of prbability distributons apart from the usual Gaussian distribution due to the diversified causes of the fluctuations. When a general expression of the cumulative distribution function (agr, c. d. f. ) of such a random fluctuation is sought, we must give attention, particularly from fundamental and practical viewpoints, to the following considerations: i) A unified expression of the c. d. f. is required, which is not influenced too much on the whole but is concretely reflected in its internal parameters by the fluctuation mode with time of the individual random phenomenon under consideration. ii) In order to bring the above generality to the expression, it is better to choose a statistical expansion series expression whose expansion coefficients reflect the first and higher order statistical consepts which are necessary to explain the phenomenon. iii) From the standpoint of the convergence property of expansion expression to be used, the kind of c. d. f. which is chosen for the first term of the expansion expression is of vital importance, since this term describes the principal part of the random fluctuation. iv) In practical applications, since a statistical expansion expression will inevitably be employed in the form of a finite number of expansion terms, the exact correction to the truncation error is always important. From the above essential considerations, when a random noise or vibration Z(t) of arbitary distribution can be considered to be the sum of two different random processes X(t) and U(t) as a result of the natural internal structure of the fluctuation or the analytically artificial classification of the fluctuation, a unified statistical treatment for the c. d. f. of the resultant random fluctuation Z(t)(≜X(t)+U(t)) is introduced exactly in the form of finite expansion terms (here, X(t) and U(t) may be mutually correlated and need not always have a Gaussian type distribution). First, let us introduce an arbitary function φ(Z) with the property of Eq. (2) and consider its expectation value Eq. (3). Carrying out the Taylor series expansion with a remainder term for Eq. (4), Eqs. (5) and (6) can be obtained. Our main problem is how to derive the probability density function P(Z) in the form of finite expansion terms based on statistical information of X(t) and U(t). After a somewhat complicated derivation, we obtain the two expansion expressions, Eq. (16) when X(t) is statistically correlated with U(t) and Eq. (17) when X(t) is statistically independent of U(t). Needless to say, we can easily show that these results satisfy the above-mentioned properties (i)-(iv). Introducing the dimensionless variable Y≜(Z-μ)/σ, Eq. (16) can be rewritten in the universal form of Eq. (18). In the practical study of level fluctuations of random noise and vibration, the c. d. f. expressions (19), (20) and (21) are more important than the probability density expression (18). Also, our results contain the well-known expression of the c. d. f. for the non-stationary random process with a mean value fluctuation as a special case. Furthemore, we have experimentally confirmed the validity of our theory not only by means of digital simulation but also by experimentally obtained road traffic noise data in Hiroshima City. The experimental results clearly show the usefulness of the theory, especially the importance of the exact correction to the truncation error.