任意不規則騒音・振動システムの多変量状態確率に関する有限展開項型統一処理方法 : 方法論の設立とその実験的確認

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We are well aware of the fact that whole informations on the statistical property regarding the state variables of a stochastic environmental system (such as noise and vibratory nuisace system) can be in principle derived by finding firstly the multivariate joint probability density function of their variables. Moreover, the irregular signals (e. g. , street noise, machine or structure vibration) often appearing in actual engineering fields exhibit various kinds of probability distributions which are not necessarily limited to the usual Gaussian distribution due to the diversified causes of fluctuations. In such cases, a unified expression of the c. d. f. is required, by which the fluctuation mode in course of time of the individual irregular phenomenon under consideration does not influence too much on the whole functional form but is concretely reflected only in its internal parameters. Then, it is required to choose statistical expansion series expression, the expansion coefficients of which reflect the first and higher order statistical concepts which are necessary to express the phenomenon. 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 irregular fluctuation. Furthermore, in practical applications, since a statistical expansion expression is inevitably employed in the form of a finite number of expansion terms, the exact correction to the truncation error is always important. From this point of view, the unified explicit expression of the joint probability density function for the state variables of various kinds of noise and vibratory system is given in the general form of statistical orthogonal or non-orthogonal expansion series with remainder term through the generalization of our previous result on the joint probability density function in a form of multivariate statistical Hermite or Laguerre series expansion and the expansion expression of joint probability density function by P. I. Kuznestov et al. by using multidimensional Hermite polynomials and quasi-moment functions. Hereupon, the joint probability density function which can be arbitrarily chosen in advance for the convenience of noise and vibratory nuisance analysis is taken into the first term, and the effect of the fluctuation pattern of state variables on the joint distribution form is concretely reflected in the second and higher terms in the above expansion expression. In a concrete form, when irregular noise or vibration Z(t) of arbitrary distribution type can be considered as the sum of two different irregular processes X(t) and U(t) as a result of the natural internal structure of the fluctuation or the analytically artificial division of the fluctuation, unified statisticalt treatment for the c. d. f. of the resultant irregular 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 show Gaussian type distribution). First, let us introduce an arbitrary function ψ(Z)(ZΔ___=(Z(t_1), Z(t_2), …Z(t_k)) with the property of Eq. (1) and consider its expectation value, Eq. (2). Applying the multi-variate Taylor series expansion with a remainder term to Eq. (2), Eq. (3) can be obtained. Our main problem is how to derive the multi-variate probability density function P(Z) in the form of finite expansion terms based on statistical informations of X(t) and U(t). After somewhat complicated derivation, we obtain the expansion expression, Eq. (12). Next, multi-variate joint moment <Z^<m_2>_1 Z^<m_2>_2…Z^<m_k>_k> is derived in the general form (cf. Eq. (20)), which can be connected with power, bi- and poly-frequency spectra. Furthermore, we have experimentally confirmed the validity of our theory not only by means of digital simulation but also by actual
社団法人日本音響学会の論文
1978-07-01

任意不規則騒音・振動システムの多変量状態確率に関する有限展開項型統一処理方法 : 方法論の設立とその実験的確認

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