Article

Probabilistic Fatigue Damage Estimation for Rotorcraft Life-Limited Components

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  • University of Maryland
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Abstract

Condition-based maintenance programs for modern helicopters rely on algorithmic techniques to estimate the useful life remaining for life-limited components. Regime-recognition-based condition-based maintenance programs involve a regime recognition step and a damage estimation step in which damage is calculated based on the identified regimes. Recently, new probabilistic regime recognition algorithms have been developed that produce probability distributions over the regimes, rather than deterministic regime classifications. However, to date, there has been no method to convert regime distributions to damage estimates. This paper proposes a technique to compute a probability distribution over the fatigue damage for each life-limited component directly from the regime probability distributions. The method treats the incurred damage at a given time as a random variable and accumulates the total damage incurred as a sum of random variables. The damage distribution at each time is computed from the regime distribution and the regime damage rates. A primary advantage of the approach is that it captures uncertainty in the regime recognition process by treating damage as a random variable rather than a deterministic value. Simulation results illustrate the benefit of the probabilistic approach over a deterministic method, particularly for flights where there is significant uncertainty in the flown regimes.

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Investigators have weakened considerably the moment requirements (A2) and (A3) while retaining strong restrictions on the dependence. However, in many situations of practical interest, assumptions (A) hold but neither strong dependence restrictions nor strong stationarity restrictions seem to apply. Thus it is important to have theorems which take advantage of assumptions (A) when they hold, in order to utilize conclusion (1.2) without recourse to severe additional assumptions. A basic theorem in this regard is offered in Section 4. It is unfortunate that the additional assumptions required, while relatively mild, are not particularly amenable to verification, with present theory. This difficulty is alleviated somewhat by the strong intuitive appeal of the conditions. The variety of ways in which the assumption of independence may be relaxed in itself poses a problem. It is difficult to compare the results of sundry investigations in central limit theory because of the ad hoc nature of the suppositions made in each instance. In Section 2 we explore the relationships among certain alternative dependence restrictions, some introduced in the present paper and some already in the literature. Conditions involving the moments of sums $\sum^{a+n}_{a+1} X_i$ are treated in detail in Section 3. The central limit theorems available for sums of dependent rv's embrace diverse areas of application. The results of Bernstein [2] and Loeve [14], [15] have limited applicability within the class of sequences satisfying assumptions (A). A result that is apropos is one of Hoeffding and Robbins [11] for $m$-dependent sequences (defined in Section 2). In addition to assumptions (A1) and (A3), their theorem requires that, defining $A_a^2 = E(X^2_{a+m}) + 2 \sum^m_1E(X_{a+m-j}X_{a+m},$ \begin{equation*}\tag{H}\lim_{n\rightarrow\infty} n^{-1}\sum^n_{i = 1} A^2_{a+i} = A^2 \text{exists uniformly in} a (n \rightarrow \infty).\end{equation*} Now it can be shown easily that conditions (A2) and (H) are equivalent in the case of an $m$-dependent sequence satisfying (A1) and (A3). Therefore, a formulation relevant to assumptions (A) is THEOREM 1.1 (Hoeffding-Robbins). If $\{ X_i\}$ is an $m$-dependent sequence satisfying assumptions (A), then it has the central limit property. In the case of a weakly stationary (with mean zero, say) $m$-dependent sequence, the assumptions of the theorem are satisfied except for (A3), which then is a mild additional restriction. For applications in which the existence of moments is not presupposed, e.g., strictly stationary sequences, Theorem 1.1 has been extended by Diananda [6], [7], [8] and Orey [16] in a series of results reducing the moment requirements while retaining the assumption of $m$-dependence. In the present paper the interest is in extensions relaxing the $m$-dependence assumption. A result of Ibragimov [12] in this regard implies THEOREM 1.2 (Ibragimov). If $\{ X_i\}$ is a strictly stationary sequence satisfying assumptions (A) and regularity condition (I), then it has the central limit property. (Condition (I) is defined in Section 2.) Other extensions under condition (I) but not involving stationarity assumptions are Corollary 4.1.3 and Theorem 7.2 below. See also Rosenblatt [17]. Other extensions for strictly stationary sequences, further reducing the dependence restrictions, appear in [12] and [13] and Sections 5 and 6 below. Section 2 is devoted to dependence restrictions. The restrictions (2.1), (2.2) and (2.3), later utilized in Theorem 4.1, are introduced and shown to be closely related to assumptions (A). Although conditional expectations are involved in (2.2) and (2.3), the restrictions are easily interpreted. It is found, under assumptions (A), that if (2.3) is sufficiently stringent, then (2.1) holds in a stringent form (Theorem 2.1). A link between regularity assumptions formulated in terms of joint probability distributions and those involving conditional expectations is established by Theorem 2.2 and corollaries. Implications of condition (I) are given in Theorem 2.3. Section 3 is devoted to the particular dependence restriction (2.1). Theorem 3.1 gives, under assumptions (A), a condition necessary and sufficient for (2.1) to hold in the most stringent form, (3.1). The remaining sections deal largely with central limit theorems. Section 4 obtains the basic result and its general implications. Sections 5, 6 and 7 exhibit particular results for weakly stationary sequences, sequences of martingale differences and bounded sequences. NOTATION AND CONVENTIONS. We shall denote by $\{ X_i\}^\infty_{-\infty}$ a sequence of rv's defined on a probability space. Let $\mathscr{M}_a ^b$ denote the $\sigma$-algebra generated by events of the form $\{(X_{i_1},\cdots, X_{i_k}) \varepsilon E\}$, where $a - 1 < i_1 < \cdots < i_k < b + 1$ and $E$ is a $k$-dimensional Borel set. We shall denote by $\mathscr{P}_a$ the $\sigma$-algebra $\mathscr{M}^a_{-\infty}$ of "past" events, i.e., generated by the rv's $\{ X_a, X_{a-1},\cdots\}$. Conditional expectation given a subfield $\mathscr{B}$ will be represented by $E(\cdot\mid\mathscr{B}),$ which is to be regarded as a function measurable ($\mathscr{B}$). All expectations will be assumed finite whenever expressed.
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1. The Report draws together insights gained from an examination of specialist views on the application of Health and Usage Monitoring Systems (HUMS) to militaty helicopters. These views were conveyed in discussions or correspondence with the author or in published works. Special attention is given to the views expressed by various manufacturers and military operators in the USA and the UK. 2. Multi-function systems now coming into service in some civil helicopters combine the functions of accident data recording and HUMS using common equipment. Similar systems are on order by some military operators and are being evaluated by others. 3. HUMS health monitoring technologies for the transmission and engine systems are fairly mature, and selected technologies are incorporated in current commercial HUMS. Rotor track and balance is also well handled in these HUMS but diagnosis of other rotor system faults has been identified as an area requiring much more research and development The verification of health diagnostics and the development of a suitable means of interfacing with militaty aircraft maintainers continue to be health monitoring areas requiring much more attention. 4. HUMS usage monitoring has received far less attention than health monitoring. This appears to have occurred because the main emphasis to this time, for civil helicopters, has been on airworthiness aspects rather than cost benefits. Usage monitoring in currently available HUMS is limited to exceedance monitoring. U sage monitoring appears to be regarded as being more important for militaty than for civil operators, probably because there is a perception that, in general, militaty operations are more severe and more difficult to quantify than civil operations. 5. Military operators see great airworthiness benefit from health and usage monitoring techniques which provide warnings of impending failures and ensure that fatigue life-limited components are replaced before the risk of failure becomes unacceptable, but consider the fitting of HUMS can only be justified if quantifiable cost benefits can be demonstrated. 6. A major concern of militaty operators is that HUMS will become a large generator of data requiring an unacceptably high level of ground support. The development and implementation of improved infonnation management strategies which address the specific requirements of the militaty environment are considered to be essential. The use of advanced infonnation management methods, such as artificial intelligence techniques, is being actively pursued by some leading HUMS developers. 7. Research currently being undertaken on the synthesis of loads on rotating components from loads measured in the static system, may overcome some of the major concerns relating to the practicality of measuring important structural loads in the operational environment The synthesis technique provides significant scope to place load sensors in benign locations and to minimise the number of sensors required. Developments in this area are likely to influence the technologies adopted for HUMS structural usage monitoring in the longer term. 8. A number of military working groups have been set up to investigate effectiveness or implementation issues for HUMS and accident data recorders. 9. Collaborative arrangements have been established under The Technical Cooperation Program, in the area of effectiveness of HUMS in the military environment. The application of HUMS for military helicopters is lagging that for civil helicopters, but military operators are seriously examining the effectiveness of such systems for their fleets. The material presented in this document is based mainly on the author's recent discussions with researchers, manufacturers and military operators. It outlines some of the important issues which operators face and some initiatives in the area. RAAF (DTA)
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Helicopter components whose fatigue lives are limited have prescribed Component Retirement Times (CRTs) that define the safe number of flying hours that they can remain in service. At the time a fleet of helicopters first enters service with a particular operator the CRTs are set by the manufacturer based on an assumed usage spectrum that has been defined with the agreement of the operator. The relationship between component loads and the various flight regimes that make up the usage spectrum is usually obtained via a flight load survey on the prototype helicopter. The Australian Defence Force discards components when they have reached their CRT. However, most components are, in effect, retired prematurely, due to the conservative nature of an assumed usage spectrum. At the other extreme, it is possible that some components could exceed their safe fatigue limit before reaching their CRT. The accurate determination of a loading history for dynamic components would have the potential to meet the two major objectives of reducing costs and improving safety. We review the literature for models that use fixed-component loads and flight parameters to model the loads in dynamics components. The reviewed papers naturally belong in one of three categories depending on model inputs: fixed-component loads, flight parameters, or a combination of fixed-component loads and flight parameters. A review of load variability reveals that even for the same aircraft under the same flight condition, the loading can vary dramatically due to pilot technique, altitude, and weight to name a few variables. For this reason, although the use of regime recognition models fleet-wide to obtain an estimate of the life fractions expended during flight on individual helicopters can produce results that are superior to the adoption of fixed CRTs, simply recognising the regime and its duration is insufficient to determine the true fraction of life expended for a component. A technique that develops a calibration matrix in the frequency domain, termed holometrics, appears promising. The holometrics technique was initially developed using only fixedcomponent loads, but was later extended to include flight parameters. Statistical methods such as significance tests and confidence intervals allow different modelling options to be prioritised. Neural network (NN) and multiple regression models were also extensively utilised by researchers. NNs were more computationally intensive than the regression models, but the NNs had better generalisation capabilities. Most studies focused on high load manoeuvres (since these produce the majority of the fatigue damage), and thus used filtering or data weighting to produce a high load bias. Adding the product of certain input parameters, such as swashplate servo position and accelerations also improved results significantly. Expert NN systems appear to be a promising area for further research. Fundamental loads modelling effects such as noise, rank deficiency, stability, and generalisation received no attention by most researchers. These essential questions need to be addressed if robust and accurate load estimation models are to be developed and implemented. A review of the literature for models that use fixed-component loads and flight parameters to determine loads in a dynamic component is presented. The reviewed papers naturally divide into one of three categories depending on the information they use to determine the load in the dynamic component. An initial section on load variability demonstrates that even for the same aircraft under the same flight condition, the loading can vary dramatically due to pilot technique, altitude, and weight to name a few variables. Neural networks, regression, and statistical indicators prove invaluable in developing load models. The review also demonstrated a lack of solutions to fundamental questions concerning loads modelling. DSTO
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This is an update of, and a supplement to, the author’s earlier survey paper [18] on basic properties of strong mixing conditions. That paper appeared in 1986 in a book containing survey papers on various types of dependence conditions and the limit theory under them. The survey here will include part (but not all) of the material in [18], and will also describe some relevant material that was not in that paper, especially some new discoveries and developments that have occurred since that paper was published. (Much of the new material described here involves “interlaced ” strong mixing conditions, in which the index sets are not restricted to “past ” and “future.”) At various places in this survey, open problems will be posed. There is a large literature on basic properties of strong mixing conditions. A survey such as this cannot do full justice to it. Here are a few references on important topics not covered in this survey. For the approximation of mixing sequences by martingale differences, see e.g. the book by Hall and Heyde [80]. For the direct approximation of mixing random variables by independent ones
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