Aeroelastic stability analysis using stochastic structural modifications
Introduction
Aerostructures are becoming increasingly lightweight and flexible thanks to the advent of composite materials with high specific strengths [1]. Although advantageous in terms of aerodynamic and fuel efficiency [2], this can lead to undesired aeroelastic phenomena that potentially compromise the safety of aircraft. One such phenomena is that of flutter, which is an unstable, self-excited vibration caused by the transfer of energy from the fluid to the structure [3,4]. Flutter is particularly dangerous since it can cause large structural deflections that expedite fatigue or even result in catastrophic structural failure [5]. It is crucial, therefore, that the conditions at which flutter occurs are accurately predicted.
Conventionally, the conditions that enable flutter to arise have been determined by means of computational modelling. In this approach, a finite-element model of the structure is coupled with an aerodynamic model and an eigenvalue analysis is performed [6]. Although well established and widely used, this method requires knowledge of both the model-form and its corresponding parameters. In practice, such knowledge is not known exactly and is merely estimated. Consequently, there are always discrepancies between the true and estimated flutter conditions.
In recent years, uncertainty quantification (UQ) techniques have been used in aeroelastic systems in order to approach the problem of flutter prediction from a probabilistic perspective [[7], [8], [9], [10], [11]]. Castravete and Ibrahim [12] first considered the effect of random stiffness parameters on the flutter speed by use of a Karhunen-Love expansion. Both a perturbation approach and Monte-Carlo simulations were used to compute the probability of aeroelastic instability. Likewise, Khodaparast et al. [13] considered the problem of uncertainty quantification using a first- and second-order perturbation method that relied upon a response surface, which took the form of a multivariate polynomial. Manan and Cooper [14], Scarth et al. [15] and Georgiou et al. [16] all used a polynomial chaos expansion, constructed by Latin Hypercube sampling, to create a surrogate model of the flutter speed variability, which arose from variable material and lay-up properties in composite materials. A Monte-Carlo simulation was then performed on the surrogate model, which was shown to significantly reduce computational time compared to direct Monte-Carlo. Allen and Maute [17] and Nikbay and Kuru [18] used a first-order reliability method to quantify variable aeroelastic responses and further used this to perform a reliability-based optimisation.
Although well-established, UQ in aeroelasticity has thus far concentrated primarily on introducing uncertainty into numerical models. This may be thought of as quantifying uncertainty during the design phase, i.e. pre-manufacture. To the best of the present authors’ knowledge, little research effort has been given to simulating the effect of uncertainty post-manufacture. Such uncertainty may arise from: i) manufacturing tolerances, and ii) damage and degradation of structural components. This is the subject of the present work.
In this paper, uncertainty is modelled as a direct structural modification [[19], [20], [21], [22], [23], [24], [25]] to a single set of receptances, which may be measured experimentally by means of conventional modal testing [26]. In this way, there is no need to model the structure, aerodynamics, nor their interaction since experimental data are used directly. The structural modification is represented as a stochastic matrix, which contains random mass, stiffness and damping parameters [27]. Assuming that the system (both the structure and coupled aerodynamics) is linear, the characteristic equation of the modified system is then obtained by using the Sherman-Morrison formula [28] in an iterative way. When the modification is single-rank and arises from a single mechanical element, it is shown that a graphical method can be used to determine the values of the random parameter that lead to flutter. The probability of flutter, in this case, is then evaluated simply by integrating the probability density function of the random parameter in the regions that lead to flutter. When the modification arises from more than one mechanical element, it is shown that the characteristic equation of the modified system can be projected onto the joint distribution of the random parameters and specific regions integrated in order to find the probability of flutter. To simplify the integration, it is shown that a first-order reliability method [29] can be used, which is computationally efficient. In contrast with conventional techniques, this method does not provide a direct estimate of the flutter speed probability distribution. Instead, it estimates the likelihood of flutter arising at a given airspeed. This allows for a Quantitative Risk Assessment (QRA) to be performed, for instance, in decision-making safety systems.
Section snippets
Structural modification theory in aeroelasticity
Consider a general n-degree-of-freedom, linear aeroelastic system governed bywhere are the structural mass, damping and stiffness matrices, respectively; is the vector of degrees-of-freedom; and is the external load arising from the aerodynamics. Taking the Laplace transform and dropping the initial value terms gives thatwhere is the complex variable.
According to Roger [30], the aerodynamic load may be approximated by a
Rank-one, single-element modification
When the structural modification is rank-one and arises from a single mechanical element, i.e. a single mass, spring or damper, Eq. (10) can be written in the formwhere is a vector with unit entries corresponding to the coordinates of the modification, and is the dynamic stiffness of the modification element [19]. The element's dynamic stiffness may always be separated so thatwhere is the random mass, stiffness or damping parameter; and is
Multiple-rank, multiple-element modification
Suppose that there are p random mechanical elements that act as a stochastic modification. The structural modification matrix may be written asIn general, the modification is rank-m (1 ≤ m ≤ n). However, it may be separated into the sum of contributory rank-one elements, as in Eq. (22). By definingthe modified receptance matrix is expressed asSince Eq. (24) is the inverse of a rank-one modification, the
Numerical examples
The methods developed in Sections 2 Structural modification theory in aeroelasticity, 3 Rank-one, single-element modification, 4 Multiple-rank, multiple-element modification are now applied on a two-degree-of-freedom, pitch-plunge, numerical aeroelastic model. The model is based on that of Theodorsen [34], which models unsteady aerodynamics using a potential flow approach. The equations of motion are given bywhere
Experimental setup
The methods developed were implemented on an experimental pitch-plunge aeroelastic model, which is shown in Fig. 7. The model consists of two main assemblies: i) the aerofoil, which is mounted to a test section inside the University of Liverpool's low-speed wind-tunnel; and ii) the external support structure. The aerofoil itself consists of five 3D-printed NACA0018 aerodynamic sectors with geometries and arrangements as shown in Fig. 8. Each sector is connected to two aluminium spars, which run
Conclusions
An experimental-based approach to uncertainty quantification in aeroelastic systems has been developed. Uncertainty is modelled as a single- or multiple-element dynamic stiffness modification to measured frequency response functions (FRFs). In the case of a single-element modification, parameter bounds that cause flutter are found graphically from the measured FRFs and the probability of flutter is calculated by integrating the probability density function of the random parameter between such
CRediT authorship contribution statement
L.J. Adamson: Conceptualization, Methodology, Formal analysis, Writing - original draft, Writing - review & editing. S. Fichera: Methodology, Formal analysis, Writing - review & editing, Supervision. J.E. Mottershead: Methodology, Formal analysis, Writing - review & editing, Supervision.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors gratefully acknowledge the financial support provided by the Engineering and Physical Sciences Research Council (EPSRC) grant EP/N017897/1. LJA also acknowledges the support provided by the Engineering and Physical Sciences Research Council (EPSRC) Doctoral Training Scholarship.
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