Statistics of extreme values of stochastic processes generated by rotating machines and associated probabilistic reliability model

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1 ème Congrès Français de Mécanique Bordeaux 6-3 August 3 Statistics of extreme values of stochastic processes generated by rotating machines and associated probabilistic reliability model B. COIN a a. Nexter Systems Allées des Marronniers 78 Versailles Cedex Résumé : es processus aléatoires de contrainte relevés sur les structures mécaniques des machines tournantes sont à structure non gaussienne. En effet ces derniers sont logiquement composés d'un processus périodique déterministe centré sur les harmoniques de la fréquence de rotation du rotor et sur lequel vient couramment se superposer un processus aléatoire à moyenne nulle de nature gaussienne. 'objectif de l'article est donc de présenter dans un premier temps les fondements mathématiques des modèles de valeurs extrêmes adaptés à leur nature spécifique. Puis dans un deuxième temps les modèles probabilistes proposés seront comparés entre eux afin d'en étudier leur degré de similitude et de conservatisme. Sur la base de la théorie de umbel retenue comme approche asymptotique des valeurs extrêmes un modèle statistique des valeurs extrêmes des processus aléatoires composites Sinus plus Bruit est alors proposé et discuté. Abstract: The random stress processes recorded on the mechanical structures of rotating machines have non-aussian structure. These processes are logically composed of a deterministic periodic process centred on the harmonics of the rotor rotation frequency on which is routinely superposed a aussian zero-mean random process. This article first presents the mathematical principles of extreme value models adapted to their specific nature. The proposed probabilistic models are then compared with each other in order to examine their degree of similarity and conservatism. On the basis of umbel s theory adopted as an asymptotic approach to extreme values a statistical model of extreme values of sine plus noise composite random processes is proposed and discussed. Key words: Stochastic process aussian and Non-aussian Rotating machine Statistics of extreme values of a sine plus noise process Introduction The random stress processes recorded on the mechanical structures of rotating machines are non-aussian in structure [] and consequently necessitate the use of probabilistic dimensioning approaches adapted to their specific nature. These technical considerations mean that structures at extreme stresses can no longer be dimensioned on the basis of conventional probabilistic theories developed historically in the context of stationary ergodic and aussian processes and widely incorporated into FE (Finite Element design tools []. On the basis of previous work on the dimensioning of mechanical structures subjected to the vibratory excitations produced by tracked vehicles [3] an analytical and numerical approach to the local imum values induced by this type of sine plus noise composite process is proposed. This local statistical approach is then used to examine the statistics of the overall ima of such processes which by nature have a great impact on the dimensioning of mechanical structures when the fundamental modes of such structures are locked on one of the rotational harmonics of the rotating machine under study. First the mathematical principles of the extreme value models adapted to the non-aussian probabilistic

2 ème Congrès Français de Mécanique Bordeaux 6-3 August 3 nature of this type of sine plus noise composite process are set out. The proposed probabilistic models are then compared with each other in order to examine their degree of similarity and conservatism. On the basis of umbel s theory adopted as an asymptotic approach to extreme values a statistical model of extreme values of sine plus noise composite random processes is proposed and discussed. ocal ima probability density The dimensioning of mechanical structures at extreme stresses subjected to a random excitation process has been discussed extensively by many authors in the case of aussian random processes whether narrow band or wide band [4] [5]. In contrast few authors have taken an interest in the case of composite random processes consisting of a periodic random process superposed on a wide-band aussian random process. iven the substantial complexity of the problem here it is proposed to deliberately limit the study to the case of narrow-band sine plus noise composite random processes consisting of a sinusoidal stress superposed on a narrow-band random stress and locked on a mean frequency f close to the sinusoidal frequency of the rotating process considered. For successful dimensioning of structures subjected to this type of narrow-band sine plus noise composite x t ξ the probability density function of the local ima of such a stress process random stress process ( should first be established. To accomplish this we show that it is then possible to use the analytical x t ξ. expression of the envelope z ( t ξ of this narrow-band process ( From these technical considerations it becomes clear that the analytical model of the envelope of the narrowband random process considered is accessible using the theory of analytical signals and thus of complex envelopes [6] through the Hilbert transform. This principle was applied by the author in 99 to establish the expression for the probability density of the local ima of narrow-band sine plus noise composite random processes [9].. Rice s analytical model The analytical model of this local ima probability density is defined as: where z z z. S p ( z =.exp.exp ( a. I for z σ. σ σ σ is the variance of the narrow-band random stress process ( ( b t ξ and S is the amplitude of the sinusoidal random process s ( t ξ of frequency f. The dimensionless coefficient σ represents the severity of the sine plus noise composite process x( t ξ = s ( t ξ + b( t ξ as it is the square of the signalto-noise ratio rms values. a S = (. σ This local ima probability density shown in Figure below is thus the product of two simple analytical functions: RAY z z p ( z =.exp ( and h a a = exp ( a. I ( a where a = z. S / σ σ. σ (3 and where I a represents the modified Bessel function of the sec ond king ( Fig. ( RAY The probability density p ( z represents the Rayleigh distribution for the narrow-band pure random case and h( a a represents the multiplicative function for the case of sine plus noise narrow-band processes. This probability density p ( z defined by expression ( is also referred to in the literature as the Rice function after its originator. This probability function was defined during his work on radio communication

3 ème Congrès Français de Mécanique Bordeaux 6-3 August 3 [7] and is a generalisation of the Rayleigh function used to describe the behaviour of a radio signal propagating along several paths before being received by a radar antenna.. Dimensionless expression of the Rice function In order to better define the behaviour of the Rice function it is useful to present the function in its dimensionless form by normalising the random amplitude of the envelope ( ξ with respect to the rms value σ of the noise alone. The dimensionless random variable is written ( ( / following dimensionless expression of the Rice function: η pη ( η = η.exp.exp a. I η.. a for η ( ( η ξ = ξ σ and leads to the (4 Dimensionless p.d.f of the ima of a narrow-band sine plus noise composite random stress process ero-order modified Bessel function of the first kind 7 6 Rice (a² = Normalised Rice p.d.f Rice (a² = 4 Rice (a² = Rice (a² = Rice (a² = 4 Rice (a² = 8 Function I (x Maxima values normalised with respect to the rms value of narrow-band noise abscissa x FI. Normalised Rice function FI. Bessel function I ( x At this stage it is important to note that the entire Rice distribution shifts to the right (towards high normalised ima values as the dimensionless coefficient a of the process increases..3 Asymptotic expression of the Rice function In order to successfully study the dimensioning of structures at the extreme stresses induced by these narrowband sine plus noise composite random processes the analytical expressions of the asymptomatic functions of the local ima must be defined. Using the asymptotic expression [8] of the zero-order modified Bessel function of the first kind the normalised asymptotic expression of the Rice function can then be obtained defined as: η η pη ( η /..exp / 4 η >> (. a. π (. a 3 (. a / It can be seen that this dimensionless asymptotic function is an exponential decay function and depends only on the dimensionless coefficient a defined in equation ( above. 3 ocal ima distribution function In order to successfully study the dimensioning of mechanical structures with respect to the extreme stresses of these narrow-band sine plus noise composite random processes it is now essential to be able to obtain the expression for the local ima distribution function in order to access the probability of exceeding a given threshold z = β. σ (see Figure 3 by the envelope z ( t ξ of the composite process considered x( t ξ. z P ( z = p u. du it can be Based on Figure 4 below and on the definition of the distribution function ( u= shown that this function is derived simply from the first-order Marcum function notated ( / (5 Q β and

4 ème Congrès Français de Mécanique Bordeaux 6-3 August 3 defined as follows: z ( ξ ( ( η ( ξ β σ ( β P ( z = Pr ob t z = p u. du = p u. du Pr ob = z / = Q u= u= z Where : Q v.exp. I. v. dv and. a v + ( β = ( = v= β (6 First-order Marcum function for arguments Alpha et Béta Q (AlphaBéta Q ( β < Q ( β with < for β fixé =. ( a ββ ( =. a > β : Normalised limit threshold Q ( β Q ( β Marcum values 8 Alpha= Alpha=5 6 Alpha= Alpha=5 Alpha= 4 Alpha=5 Alpha=3 Alpha=35 Alpha=4 Alpha=45 Alpha= Argument Béta of the first-order Marcum function FI. 3 Probability of exceeding threshold FI. 4 Marcum function ( Q β At this stage it is important to note that for a given risk of exceeding the threshold the normalised threshold level β is higher the greater the value of the dimensionless coefficient a of the sine plus noise random process. 3. Series expressions of the first-order Marcum function The first-order Marcum function Q ( β is by definition between and and decreases continuously as shown in Figure 4 above. It is expressed in the form of an unbounded integral which can be put in the form of convergent infinite series [9] [] as below: γ ( ( ( [ ] Q ( β = exp.. γ k + β / with k! = Γ k + after 9 k k =.( k! ( a x is the complementary amma function derived from the amma function Γ ( a a ( = ( and from the incomplete amma function γ a x t.exp t. dt n + β β Q ( β = exp.. In (. β after [ ] n= (8 where I x is the nth order modified Bessel function of the first kind n ( 3. Asymptotic expression of the first-order Marcum function Based as previously on the asymptotic expression of the zero-order modified Bessel function of the first kind and also on the asymptotic function of the complementary error function [8] an asymptotic expression of Q β can be defined which is very similar to that described by Nuttall [] in work reported in April ( 97. Assuming that the normalised limit threshold β is high enough to satisfy the expression ( v ( v β v.exp. dv. exp. dv the following expression can be written: v β v= β = x t= (7 4

5 ème Congrès Français de Mécanique Bordeaux 6-3 August 3 Q ( β ( β β β β.. erfc..exp high β high β ( β / >> ( β >> / and ( β >> +. π.( β where erfc( x =. exp ( t. dt is the complementary error function π t= x (9 4 Design of structures such that the local stress imum occurs on average once over the load time T of the process By definition the local imum stress of the composite process ( once over the given load time T of the process is given by the expression: x t ξ likely to occur on average Normalised imum stress Normalised local imum stress according to the number of load cycles n at various values of a² a² = a² = a² = 4 a² = 6 a² = 8 a² = a² = a² = 4 a² = 6 a² = 8 a² = E+ 5E+3 E+4 E+4 E+4 3E+4 3E+4 Number of load cycles FI. 5 Normalised local ima β ( a n = / σ ( = σ. β a n where β is the root of ( β = ( β ( the function g n Q n with =. a and n = f. T fixed ( It is also shown that : β a = n =.ln( n ( Pure random And it is shown that : ( β a n = = ( a (b ( (a The value of β thus represents the normalised local imum stress which corresponds in fact to the threshold of the local ima of the composite stress process considered and the risk of exceeding this threshold is equal to the reciprocal of the mean number of load cycles n f. T β a n for different dimensionless values of a as a function of n [.3E 4 cycles] =. The function ( + is shown in Figure 5. Companies involved in the dimensioning of mechanical structures of rotating machines generally apply conventional processing techniques using the power spectral density (PSD concept and assuming normal distributions of the stochastic processes studied even though they are not appropriate given the statistical nature of the composite processes. To identify the design errors that can be made in the mechanical strength of structures when using a stochastic approach that is not appropriate for the statistical nature of the physical stress processes occurring in rotating machines in what follows a comparison is proposed between the stochastic approach developed previously ( and the approach using aussian processes ( ˆ applying constant rms values in these two probabilistic approaches. On completion of the calculations Figure 6 and expressions ( and (3 below are obtained for this comparison model (of design at extreme values between the sine plus noise (S+N composite approach and the aussian approach at iso-energy. 5

6 ème Congrès Français de Mécanique Bordeaux 6-3 August 3 Ratio of local imum stresses (S+N/N Comparison between the sine plus noise composite approach and the aussian approach at iso-energy a² = a² = a² = 4 a² = 6 a² = 8 a² = a² = a² = 4 a² = 6 a² = 8 a² = E+ 5E+3 E+4 E+4 E+4 3E+4 3E+4 Number of load cycles β ( a n ( + ( = ˆ a..ln n ( ( a and n > It is also shown that : ˆ ( Pure random (3 a = for any value of n FI. 6 Composite model normalised with respect to the aussian noise model at iso-energy At this stage it is important to note that a random approach such as PSD is always more conservative than the sine plus noise composite approach in terms of local imum stress however severe the stress process applied. This justifies the use of probabilistic structure dimensioning models appropriate for the statistics of the excitation processes with a view to minimum necessary dimensioning enabling the design to be optimised. 5 Design of structures in terms of the overall stress imum with defined risk δ of exceeding the threshold (controlled-risk dimensioning By definition the overall imum stress ( δ with defined risk δ of the composite process x( t ξ corresponds in fact to the highest imum stress of the composite process of duration T. This overall imum stress is then associated with a risk δ that the designer defines according to the level of reliability to be obtained for the product. In general the designers define a risk of exceeding the threshold δ of % in the case of standard structures and.% in the case of secured structures. As the probability density function of the composite random process local ima is described asymptotically by an exponential decrease it can be shown that the distribution function of the highest ima Yn ( ξ = MAX z ( tk ξ over the duration T of the composite process tends to a umbel function []: ( ( ( β ( β k= to n ( σ β ( ( F y exp exp. y y where y =. a n and = n. p y n high n Y n n n n n / n. with n =.exp where β = and =. a and n = f. T σ.. π. σ At this stage it can be seen that it is now possible to determine the design value ( (4 δ to which the designer has to design the structure for a risk δ that the designer defines according to the level of reliability required by the customer: ( = ( δ y n ln ln δ / n ( refer to [3] with (5 The comparative approach between the secured design approach with regard to the local imum ( δ with a defined risk δ is then described by the basic expression (6 below (in terms of umbel s asymptotic approximation and by Figures 7 and 8 associated with a risk δ of % and.% respectively. 6 and the overall imum approach (

7 ème Congrès Français de Mécanique Bordeaux 6-3 August 3 ( δ ln ln ( δ = > =. = / (6 n f T and a σ. β. n Note : This ratio depends on a n and δ ( fixed at % or % Comparison between the local and overall ima approaches (with defined risk of % for the case of a sine plus noise composite process Comparison between the local and overall ima approaches (with defined risk of.% for the case of a sine plus noise composite process 8 Ratio of imum stresses (Overall/ocal n=e+ n=e+ n=e+3 n=e+4 n=e+5 n=e+6 n=e+7 n=e+8 n=e+9 Ratio of imum stresses (Overall/ocal n=e+ n=e+ n=e+3 n=e+4 n=e+5 n=e+6 n=e+7 n=e+8 n=e Dimensionless coefficient a² Dimensionless coefficient a² FI. 7 Model with risk δ of % FI. 8 Model with risk δ of.% 6 Conclusion This work shows that the overall ima approach with defined risk δ of exceeding the threshold leads logically to securing of the design of structures subjected to random stress processes compared with a dimensioning approach using the local imum stresses used routinely by designers in dynamic dimensioning. In the case of sine plus noise composite processes it is shown that the higher the value of the dimensionless coefficient a² the lower this severity level for a given risk and number of cycles. astly it is shown clearly that for a given sine plus noise composite stress process (given value of dimensionless coefficient a² generated by a rotating machine the higher the number of cycles the lower the severity of the overall approach as shown in Figures 7 and 8 above. References [] J.S. Bendat Principles and Applications of Random Noise Theory John Wiley & Sons 958 [] MSC/FATIUE V8 User Manual MSC Corporation os Angeles CA 998 [3] B. Colin Tracklayers : a complex vibration environment Institute of Environmental Sciences 99 [4] D.E. Cartwright M.S onguet-higgins The distribution of the ima of a random function Proceedings of the Royal Society [5] A.. Davenport Note on the distribution of the largest value of a random function with application to gust loading J. Inst. Civ Eng [6] F. de Coulon Théorie et traitement des signaux Dunod Chapitre [7] S.O. Rice Mathematical Analysis of Random Noise Bell System Technical Journal vol [8] A. Angot Compléments de mathématiques à l usage des ingénieurs de l électrotechnique et des télécommunications Masson 98 [9] B. Colin Spectres des réponses extrêmes d un environnement sinus plus bruit Mécaniques Matériaux Electricité n 446 Octobre-Novembre 99 [] S.O. Rice Statistical properties of Random Noise Currents Bell System Technical Journal vol [] A.H. Nuttall Some Integrals Involving the Q-Function NUSC Technical Report [] E.J. umbel Statistics of Extremes Columbia University Press 958 [3] B. Colin Conception sécurisée des structures soumises aux valeurs extrêmes de processus stochastiques stationnaires Bell Mécanique & Industries DOI :.5 9 7