Marc P. Mignolet

Recursive Simulation of Stationary Multivariate Random Processes—Part II

Stability and invertibility aspects of the AR to ARMA procedures developed in Part I in connection with simulation of multivariate random processes are addressed. A general criterion is proved for this purpose. Furthermore, several properties regarding the matching of the correlations at various time lags of the target and the simulated processes are shown. Finally, the reliability and efficiency of the discussed procedures are demonstrated by application to spectra encountered in earthquake engineering, offshore engineering, and wind engineering.

ARMA Monte Carlo simulation in probabilistic structural analysis

Autoregressive moving average (ARMA) systems for sysnthesizing realizations of stochastic processes are discussed in context with the technique of Monte Carlo simulation. Strictly autoregressive (AR) or strictly moving average (MA) systems are considered as special cases of the ARMA systems. Their applicability in wind, ocean, and earthquake engineering is briefly reviewed

Optimization of intentional mistuning patterns for the reduction of the forced response effects of unintentional mistuning: Formulation and assessment

The focus of the present investigation is on the use of intentional mistuning of bladed disks to reduce their sensitivity to unintentional random mistuning. The class of intentionally mistuned disks considered here is limited, for cost reasons, to arrangements of two types of blades (A and B, say). A two-step procedure is then described to optimize the arrangement of these blades around the disk to reduce the effects of unintentional mistuning. First, a pure optimization effort is undertaken to obtain the pattern(s) of the A and B blades that yields small/the smallest value of the largest amplitude of response to a given excitation in the absence of unintentional mistuning. Then, in the second step, a pattern screening technique based on a recently introduced measure of localization is used to determine which of the patterns does have a large/small sensitivity to random unintentional mistuning. In this manner, expensive Monte Carlo simulations can be eliminated. Examples of application involving both simple bladed disk models and a 17-blade industrial rotor clearly demonstrate the significant benefits of using this class of intentionally mistuned disks.

Modelling and prediction of machining errors using ARMAX and NARMAX structures

Forecasting compensatory control, which was first proposed by Wu [ASME J. Eng. Ind. 99 (1977) 708], has been successfully employed to improve the accuracy of workpieces in various machining operations. This low-cost approach is based on on-line stochastic modelling and error compensation. The degree of error improvement depends very much on the accuracy of the modelling technique, which can only be performed on-line in a real-time recursive manner. In this study, the effect of the control input (i.e. the cutting force) is considered in the development of the error models, and the formulation of recursive exogenous autoregressive moving average (ARMAX) models becomes necessary. The nonlinear ARMAX or NARMAX model is also used to represent this nonlinear process. ARMAX and NARMAX models of different autoregressive (AR), moving average (MA) and exogenous (X) orders are proposed and their identifications are based on the recursive extended least square (RELS) method and the neural network (NN) method, respectively. An analysis of the computational results has confirmed that the NARMAX model and the NN method are superior to the ARMAX model and the RELS method in forecasting future machining errors, as indicated by its higher combined coefficient of efficiency.

Maximum Amplification of Blade Response due to Mistuning: Localization and Mode Shape Aspects of the Worst Disks

This paper focuses on the determination and study of the maximum amplification of the steady state forced response of bladed disks due to mistuning. First, an optimization strategy is proposed in which partially mistuned bladed disks are considered as physical approximations of the worst case disk and the mistuned properties are sought to maximize the response of a specific blade. This approach is exemplified on both a reduced order model of a blisk and a single-degree-of-freedom per blade disk model an extensive parametric study of which is conducted with respect to blade-to-blade coupling, damping, and engine order. A mode shape-based formulation of the amplification factor is then developed to clarify the findings of the parametric study in the strong coupling/small damping limit. In this process, the upper bound of Whitehead is recovered for all engine orders and number of blades and the conditions under which this limit is exactly achieved or closely approached are clarified. This process also uncovers a simple yet reliable approximation of the resonant mode shapes and natural frequencies of the worst disk.

NONLINEAR REDUCED ORDER MODELING OF CURVED BEAMS: A COMPARISON OF METHODS

The accurate prediction of the response of aircraft panels subjected to strong acoustic excitation and high temperatures has been and remains an important requirement for the design of supersonic/hypersonic vehicles. One of the key challenges to achieving this prediction is the computationally efficient modeling of the complex physical processes taking place when the acoustic excitation is strong enough to induce a large, nonlinear geometric response of the panel. Even for a flat beam or plate, there exists a subtle energy exchange between the large transverse displacements resulting directly from the acoustic excitation and the much smaller in-plane motions nonlinearly induced by the change of geometry. The transverse displacements exhibit a stiffening behavior which is softened by the in-plane motions. Curved and buckled panels exhibit an even more complex behavior as they can “snap” through an unstable region leading to deformations that are drastically nonlinear, i.e. of the order of 10-100 thicknesses. The interplay of the curvature, acoustic excitation, and temperature leads to an occurrence of snap-through events varying from very rare, to frequent, to continuous. The present study demonstrates that reduced-order model can capture the complex snap-through behavior of shallow curved beams. Successful displacement comparisons are made for two different reduced-order modeling methods, for a single curved beam geometry considering combinations of thermal effects and loading.

Nonlinear Reduced-Order Models for Thermoelastodynamic Response of Isotropic and Functionally Graded Panels

The focus of this investigation is on the development and validation of thermoelastic reduced-order models for the geometrically nonlinear response and temperature of heated structures. The reduced-order modeling approach is based on a modal-type expansion of both displacements and temperatures in the undeformed, unheated configuration. A set of coupled nonlinear differential equations governing the time-varying generalized coordinates of the response and temperature expansion are derived from finite thermoelasticity using a Galerkin approach. Furthermore, the selection of the basis functions to be used in these reduced-order models is discussed, and the numerical evaluation of the model coefficients is addressed. This approach is first validated on an isotropic beam subjected to both thermal effects and external loads. The thermal effects are large enough to induce a significant buckling of the panel, while the time-varying loads lead to snap-throughs ranging in frequency from infrequent to continuous. Validation to a functionally graded material panel in similar conditions is then performed. In both cases, the reduced-order modeling predicted temperatures and responses are found to very closely match their full finite element counterparts.

Identification of Mistuning Characteristics of Bladed Disks From Free Response Data— Part II

The focus of the present two-part investigation is on the estimation of the dynamic properties, i.e., masses, stiffnesses, natural frequencies, mode shapes and their statistical distributions, of turbomachine blades to be used in the accurate prediction of the forced response of mistuned bladed disks. As input to this process, it is assumed that the lowest natural frequencies of the blades alone have been experimentally measured, for example in a broach block test. Since the number of measurements is always less than the number of unknowns, this problem is indeterminate in nature. In this second part of the investigation, the maximum likelihood method (ML) will first be revisited and a thorough assessment of its reliability in a wide variety of conditions, including the presence of measurement noise, different distributions of blade structural properties, etc., will be conducted. Then, a technique that provides a bridge between the two identification methods introduced in Part I, i.e., the random modal stiffnesses (RMS) and maximum likelihood (ML) approaches, will be presented. This technique, termed the improved random modal stiffnesses (IRMS) method is based on the maximum likelihood concepts but yields a mistuning model similar to that of the random modal stiffnesses technique. Finally, the accuracy of the RMS, ML, and IRMS methods in predicting the forced response statistics of mistuned bladed disks will be investigated in the presence of close blade alone natural frequencies.

Local/Global Effects of Mistuning on the Forced Response of Bladed Disks

The focus of the percent investigation is on the assessment and modeling of the local (spanning only a few blades) and global (encompassing the entire disk) effects of mistiming on the forced response of bladed disks. To this end, the concept of localization is first revisited and a new measure of this effect is introduced in terms of the number of blades the mistuning of which actually affects the forced response of a central blade. Using this new metric, it is demonstrated that high responding blades typically exhibit a high level of localization and that the reverse is not necessarily true. Thus, localization is not only disk dependent but also varies from blade-to-blade on the same disk. This observation is then used to validate a partial mistiming approach to the determination of the maximum amplitude of response over the entire population of disks. The results of this study indicate that the largest amplification due to the mistuning occurs at very strong blade-to-blade coupling levels, at the contrary of a general perception, but is associated with large mistuning levels. Finally, the above phenomenological observations are used to devise a modeling technique of both local and global components of mistuning. An example of application is presented that demonstrates the high accuracy of this approach through the entire blade-to-blade coupling domain.

NONLINEAR REDUCED ORDER MODELING OF FLAT CANTILEVERED STRUCTURES

The focus of this investigation is on the development and validation of structural dynamic reduced order models for the geometrically nonlinear response of flat cantilevered structures, e.g. beams and plates. The specificities of cantilevered structures, as compared to those supported all around, are first highlighted. On this basis, extensions of existing reduced order modeling strategies are presented which provide a complete representation of the structural response, i.e., including both transverse and inplane displacement fields. Next, this methodology is successfully applied, first to a simple beam model and then to a flat wing both of which subjected to transverse loads. Finally, the response of a beam under combined transverse and inplane loads as well as its postbuckling behavior are also shown to be also accurately predicted by the reduced order model.