Method and aircraft with a device for predicting flutter and buffeton set

Abstract

To predict flutter and buffeton set in a body (2) moving in a flow (3), structural properties of the body (2) are described in a structural model (13) comprising a mass matrix (M), a stiffness matrix (K), and a damping matrix (B), and from which at least one structural eigenmode of the body (2) is determined under the assumption that no aerodynamic forces act on the body (2). Aerodynamic properties of the body (2) in the flow (3) are described in an aerodynamic model (14) under the initial assumptions that the body (2) performs movements (15) according to the at least one structural eigenmode and that the flow (3) has no influence on the movements of the body (2). At least one dominant flow eigenmode of the flow (3) and aerodynamic forces acting on the body (2) are determined from the aerodynamic model (14) under the initial assumptions.The aerodynamic model (14) is then converted into an aeroelastic model (12) by coupling it with the structural model (13), whereby during a first sweep (22) while keeping the dynamic pressure (q. dyn ) the mass matrix (M) is first scaled with a large scaling factor (q m ), which is then successively reduced to 1, and the at least one dominant flow eigenmode is tracked with regard to its eigenvalue (16). Subsequently, the dynamic pressure (q dyn ) is gradually increased, while the scaling factor (q m ) remains equal to 1, wherein the at least one structural eigenmode and the at least one dominant flow eigenmode are tracked with regard to their eigenvalues ​​(16, 18, 20).

Description

[0001] The invention relates to a method for predicting flutter and buffetonset in a body moving in a flow. Furthermore, the invention relates to an aircraft with a device for implementing the method.

[0002] Flutter is a dynamic instability of a body moving in a flow. Flutter is based on a structural eigenmode of the body. Buffet is an aerodynamic bifurcation of a flow that flows around a body. A buffet can be described as the condition in which the linearized behavior of the flow becomes unstable on the side of the bifurcation, where a steady state is stable. Buffet is based on a flow eigenmode of the flow. In this description and the appended patent claims, the term buffet refers to the condition in which this flow eigenmode becomes unstable, regardless of whether the flow eigenmode is influenced by body movements or not. Flutter and buffet are limiting factors for the operational range of an aircraft in the transonic range up to high Mach numbers.In the transonic regime, a flow transition from a laminar to a turbulent state and shock-induced flow separations occur. The latter are initially accompanied by damped low-frequency flow eigenmodes, which can transform into the so-called transonic buffet.

[0003] The fluid dynamic feedback mechanisms underlying flutter and buffeting are largely unknown. To date, no satisfactory theory exists to predict the frequency and damping of these eigenmodes based on a steady background flow field. STATE OF THE ART

[0004] From Houtman, J. and Timme, S., "Global stability analysis of elastic aircraft in edge-of-the-envelope flow," Journal of Fluid Mechanics, Col. 967, A4, 2023, https: / / doi.org / 10.101 7 / jfm.2023.413, it is known that the occurrence of flutter and buffeting can be solved as a complete eigenvalue problem. This requires the calculation of the eigenvalues ​​of a very large matrix. Iterative procedures are necessary because not all eigenvalues ​​can be calculated at once. In addition, the user must specify an initial eigenvalue, which subsequently approximates the desired solution.

[0005] From J. Nitzsche, L. Ringer, C. Kaiser, and H. Hennings, "Fuild-mode flutter in plane transonic flows," International Forum on Aeroelasticity and Structural Dynamics (IFASD), 2019, it is known that, to describe a linearized flutter stability problem in the pre-buffet region, the feedback of generalized aerodynamic forces (GAF) on a structural model can be approximated using a reduced-order aerodynamic model in the time domain. The generalized aerodynamic forces are previously determined from an aerodynamic model. This approach requires time-domain calculations and an analysis of the convergence of the obtained solutions with respect to the excitation amplitude.

[0006] D. Quero, P. Vuillemin, and C. Poussot-Vassal, "A generalized eigenvalue solution to the flutter stability problem with true damping: The pL method," Journal of Fluids and Structures, 103266. doi: 10.1016 / j.jfluidstructs.2021.103266, 2021, describe a generalized eigenvalue solution to the flutter stability problem that takes damping into account. In this method, called the pL method, the nonlinear eigenvalue problem is transformed into a linearly generalized eigenvalue formulation by interpolating the nonlinear aerodynamic term using Loewner and shifted Loewner matrices, avoiding any approximation. Such a procedure is also known from D. Quero, P. Vuillemin and C. Poussot-Vassal, “Improved mode tracking for the pL flutter solution method based on aeroelastic derivatives,” in IFASD, Madrid, Spain, 2022. OBJECT OF THE INVENTION

[0007] The invention is based on the object of determining all aeroelastic eigenvalues ​​and eigenvectors relevant for the flutteron set and the buffeton set at once, thus eliminating the need for an estimation for the sought solution of the buffeton set. SOLUTION

[0008] The object of the invention is achieved by a method having the features of independent patent claim 1. Dependent patent claims 2 to 12 relate to preferred embodiments of the method according to the invention. Patent claims 13 and 14 relate to applications of the method according to the invention to an aircraft. And patent claim 15 is directed to an aircraft with a device for automatically implementing such a method. DESCRIPTION OF THE INVENTION

[0009] In a method according to the invention for predicting flutter and buffetonset in a body moving in a flow, structural properties of the body are described in a structural model comprising a preferably generalized mass matrix, a preferably generalized stiffness matrix, and a preferably generalized damping matrix. The body can be a body that deforms during its movements in the flow, in particular an elastically deforming body. Furthermore, the body can be a body that is supported during its movements in the flow, in particular elastically supported. However, the body can also move in the flow as a rigid body without any support at all. Depending on the structural properties of the body, entries on the main diagonals of the stiffness matrix and / or the damping matrix can also be zero.From the structural model, at least one structural eigenmode of the body is determined under the assumption that no aerodynamic forces act on the body. Several or even all structural eigenmodes of the body can also be determined under this assumption and considered in the following.

[0010] Aerodynamic properties of the body in the flow are described in an aerodynamic model. This is done under the initial assumption that the body moves according to at least one structural eigenmode and that the flow has no influence on the body's movements. At least one dominant flow eigenmode of the flow and aerodynamic forces acting on the body are determined from the aerodynamic model under the initial assumption. The at least one dominant flow eigenmode is the flow eigenmode of the flow that is responsible for the buffet on set. This is typically the most dominant flow eigenmode in the transonic region of the flow.By describing the aerodynamic properties of the body under the assumption that the body moves in the flow according to the at least one structural eigenmode, but that the flow does not affect these movements of the body, the aerodynamic model takes into account the effects of the at least one structural eigenmode on the flow, but no feedback of the aerodynamic forces on the movements of the body.

[0011] By coupling with the structural model, the aerodynamic model is converted into an aeroelastic model that also considers the effects of aerodynamic forces on the body's motion. During an initial sweep, the dynamic pressure, which is a measure of the relative velocity of the flow relative to the body, is kept constant. Either the preferably generalized mass matrix is ​​first multiplied by a large scaling factor, which is then successively reduced to 1, or the preferably generalized stiffness matrix and the preferably generalized damping matrix, if at least one of these two matrices has non-zero entries, particularly on its main diagonal, are first multiplied by a small scaling factor, which is then successively increased to 1.Using the scaling factor, the feedback of the aerodynamic forces on the body's movements is initially limited, so that the influence of the at least one structural eigenmode on the at least one dominant flow eigenmode is initially only small. By successively approaching the scaling factor to 1, the feedback between the aerodynamic forces and the body's movements is brought closer to its actual extent, and thus its effects on the at least one structural eigenmode and the at least one dominant flow eigenmode are adapted to the actual conditions. During the first sweep, at least the at least one dominant flow eigenmode is tracked, at least with regard to the development of its eigenvalue.The at least one dominant flow eigenmode can additionally be tracked with respect to any evolution of the flow eigenmode of its eigenvalue, and the at least one structural eigenmode can be tracked in the same way as the at least one dominant flow eigenmode.

[0012] During a second sweep, the dynamic pressure is then gradually increased while the scaling factor remains equal to 1 in order to observe the effects of a higher relative velocity of the flow relative to the body. The at least one structural eigenmode and the at least one dominant flow eigenmode are tracked, at least with regard to their eigenvalues. During this tracking, a flutteron set is detected when a real part of an eigenvalue of the at least one structural eigenmode becomes positive. A buffeton set is detected when a real part of an eigenvalue of the at least one dominant flow eigenmode becomes positive. During the second sweep, instead of the dynamic pressure, another physical quantity can be gradually changed, with which the dynamic pressure in the flow gradually increases.

[0013] In the method according to the invention, all considered eigenmodes, their eigenvalues, and their evolutions in the transonic range are determined simultaneously. No estimations, in particular no estimation of the Buffeton set, are required.

[0014] The eigenvalues ​​of the at least one structural eigenmode and the at least one dominant flow eigenmode can be easily calculated as eigenvalues ​​of the aeroelastic model using a linear eigenvalue solver. Specifically, the applicant's efficient Linear Frequency Domain (LFD) solvers CFD Code Strategy TAU and CODA can be used for this purpose. However, the use of a linear eigenvalue solver requires a state-space realization of the aerodynamic model, from which the aerodynamic forces acting on the body can be determined as generalized aerodynamic forces (GAF) in the state space.

[0015] In the method according to the invention, a state-space realization must be implemented in which eigenvalues ​​representing the aerodynamic forces may have a non-zero imaginary part. For the dominant flow eigenmode in the transonic regime, this imaginary part corresponds to a buffet frequency. Furthermore, a resulting rational aerodynamic transfer function in the state-space realization must have a non-proper term that describes the effect of the time derivative of body movements on the aerodynamic forces.

[0016] After the state-space realization, the at least one dominant flow eigenmode can be determined by evaluating residuals and poles obtained from state-space matrices. When implementing the method according to the invention, it has proven particularly advantageous if the at least one dominant flow eigenmode is determined using a partial fractional expansion of an aerodynamic transfer function, with aerodynamic terms being represented on the basis of Loewner matrices and shifted Loewner matrices. This procedure corresponds to the basically known pL method. In addition, a residualization of the high frequencies can be introduced, as is basically known from D. Quero, P. Vuillemin and C. Poussot-Vassal, "A generalized state-space aeroservoelastic model based on tangential interpolation," Aerospace 6 (1), 9. doi: 10.3390 / aerospace6010009, 2019.

[0017] In the method according to the invention, it is advantageous, but not mandatory, to apply a continuation method that uses information about aeroelastic sensitivities with respect to the scaling factor during the first sweep to track at least the at least one dominant flow eigenmode. Such a continuation method utilizes the previous change in the respective eigenmode and / or its eigenvalue as a function of the scaling factor to determine an expected value of the eigenvalue for the current scaling factor. Thus, an eigenvalue of the aerodynamic model determined with the aid of an eigenvalue solver can be assigned to a specific eigenmode with high accuracy because it is closest to the associated expected value. The use of the information about aeroelastic sensitivities with respect to the scaling factor makes the tracking of at least the at least one dominant flow eigenmode more robust.

[0018] The dynamic pressure value, which is kept constant during the first sweep and used as the initial value for the dynamic pressure during the second sweep, must be adjusted to a reasonable starting point for the second sweep. This dynamic pressure value should therefore be below the transonic region of the flow. A small dynamic pressure value during the first sweep proves to be favorable, and in concrete terms, it can be a maximum of 1 × 10 -1 , preferably 1 × 10 -1 Pa and even more preferably 1 × 10 -3 Pa. However, it is not advantageous to use a very small value of dynamic pressure during the first sweep of less than 1 × 10 -4 or even less than 1 × 10 -5 Pa to use.

[0019] The large scaling factor at the beginning of the first sweep can be at least 1 × 10 4 , preferably at least 1 × 10 5and even more preferably at least 1 × 10 6 while the small scaling factor at the beginning of the first sweep is at most 1 × 10 -4 , preferably 1 × 10 -5 and even more preferably 1 × 10 -6 can amount to.

[0020] Furthermore, it is preferred if the scaling factor in the first sweep is brought closer to one in smaller steps, specifically in greatly smaller steps, the width of the steps decreasing by more than 99.9% and preferably by more than 99.99% or even more.

[0021] During the second sweep, it is not required, but preferred, to apply a continuation method to track the at least one structural eigenmode and the at least one dominant flow eigenmode, which takes into account information about aeroelastic sensitivities with respect to the dynamic pressure. Here, the continuation method exploits the previous change of the respective eigenmode and / or its eigenvalue as a function of the dynamic pressure to determine an expectation of the eigenvalue for the current dynamic pressure. The use of these expectation values ​​makes the continuation method more robust, especially when the aeroelastic model has multiple eigenmodes with closely spaced eigenvalues.

[0022] In the second sweep, the dynamic pressure can be increased in pressure steps between 0.1 and 0.001 Pa. Preferably, the pressure steps are on the order of 0.01 Pa. The width of the pressure steps can remain the same during the second sweep or, for example, when strong aeroelastic sensitivities with respect to the dynamic pressure occur in the transonic range, can be reduced.

[0023] In particular, the method according to the invention can be applied to an aircraft as a body moving in a flow, wherein the flutter on set and the buffet on set are predicted in order to define a limit of the aircraft's flight envelope at higher Mach numbers. The method according to the invention can already be applied during the development of an aircraft, i.e., before its first flight tests. However, the method according to the invention can also be used, for example, by adapting the structural model to the current loading condition of the aircraft, to update the flight envelope depending on the aircraft's loading condition.In this way, for example, a large distance to the flutter and buffet set can be maintained whatever the aircraft's loading condition, but excessive distance and thus excessive restriction of the flight envelope can be avoided, as can be associated with a flight envelope that is independent of the aircraft's loading condition.

[0024] An aircraft according to the invention has a device for automatically implementing the method according to the invention. This device can be connected in such a way that it automatically updates a flight envelope stored in the aircraft's flight control system. Furthermore, the device can be connected to sensors for automatically detecting the aircraft's load status.

[0025] Advantageous further developments of the invention emerge from the patent claims, the description and the drawings.

[0026] The advantages of features and combinations of several features mentioned in the description are merely exemplary and can be used alternatively or cumulatively without the advantages necessarily having to be achieved by embodiments according to the invention.

[0027] With regard to the disclosure content – ​​not the scope of protection – of the original application documents and the patent, the following applies: Further features can be found in the drawings – in particular the illustrated geometries and the relative dimensions of several components to one another, as well as their relative arrangement and operative connection. The combination of features of different embodiments of the invention or features of different patent claims is also possible, deviating from the selected references of the patent claims, and is hereby suggested. This also applies to features that are illustrated in separate drawings or mentioned in their description. These features can also be combined with features of different patent claims.Likewise, features listed in the patent claims may be omitted for further embodiments of the invention, but this does not apply to the independent patent claims of the granted patent.

[0028] The number of features mentioned in the patent claims and the description is to be understood as meaning that exactly this number or a greater number than the stated number is present, without the need for the explicit use of the adverb "at least." The features mentioned in the patent claims may be supplemented by further features or may be the only features present in the subject matter of the respective patent claim.

[0029] The reference signs contained in the patent claims do not represent a limitation of the scope of the subject-matter protected by the patent claims. They serve only the purpose of making the patent claims easier to understand. BRIEF DESCRIPTION OF THE CHARACTERS

[0030] In the following, the invention is further explained and described with reference to preferred embodiments shown in the figures. Fig. 1 shows a wing as an example of a body moving in a flow, Fig. 2 shows an aerodynamic net 11 near the wing according to Fig. 1. Fig. 3 shows the coupling of a structural model and an aerodynamic model for an aeroelastic model of the body in the flow according to Fig. 1 and Fig. 2. Fig. Figure 4 illustrates the sequence of two sweeps of a scaling factor on the one hand and of the dynamic pressure on the other hand in the method according to the invention. Fig. Figure 5 shows the evolution of the complex eigenvalue of the most dominant flow eigenmode with decreasing scaling factor in the first sweep according to Fig. 4 in a first application example. Fig. Figure 6 shows the evolution of the complex eigenvalues ​​of two structural eigenmodes and the most dominant flow eigenmode with increasing value of the dynamic pressure during the second sweep according to Fig. 4 with the same first application example as in Fig. 5. Fig. Figure 7 shows the evolution of the complex eigenvalue of the most dominant flow eigenmode with decreasing scaling factor in the first sweep according to Fig. 4 in a second application example; and Fig. Figure 8 shows the evolution of the complex eigenvalues ​​of two structural eigenmodes and the most dominant flow eigenmode with increasing value of the dynamic pressure during the second sweep according to Fig. 4 for the same second application example as in Fig. 7. FIGURE DESCRIPTION

[0031] Fig. 1 is a sketch of an airfoil 1 with the idealized structural elements as an example of a moving body 2 in a flow 3. At a Mach number of the flow 3 of 0.73, a Reynolds number of 3,000,000 and an angle of attack of the airfoil 3 with respect to the flow 3 of 4°, a limited supersonic flow region 4 forms above the upper side of the airfoil 1, which terminates with a shock wave 5. Behind the shock wave 5, a detached shear layer 6 of the flow 3 forms. Double arrows 7 and 8 indicate that the shock wave 5 and the shear layer 6 move during buffeting of the flow 3. For the airfoil 1, two structural degrees of freedom of movement, lift 9 and pitch 10, are indicated by the associated stiffnesses.

[0032] Fig. 2 shows an aerodynamic mesh 11 near the wing 1, which for the example according to Fig. 1 in J. Nitzsche, L. Ringer, C. Kaiser, and H. Hennings, “Fuild-mode flutter in plane transonic flows,” International Forum on Aeroelasticity and Structural Dynamics (IFASD), 2019. The Spalart-Allmaras (SA) turbulence model was used, but any other available turbulence model can be used.

[0033] Fig. 3 outlines an aeroelastic model 12 of the body 2 in the flow 3, in which a structural model 13 describing the elastic properties of the body 2 and an aerodynamic model 14 describing the aerodynamic properties of the body 2 in the flow 3 are formed by aerodynamic forces F a on the body 2 and movements 15 of the body in the flow 3 are coupled.

[0034] The structural model 13 comprises a mass matrix M, a damping matrix B and a stiffness matrix K, whereby, depending on the structural properties of the body 2, entries on the main diagonals of the latter two matrices B and K can also be zero. The matrices M, B and K are quadratic and have a dimension corresponding to the number of degrees of freedom considered. The movements of the body 2 are described by a displacement vector x, which specifies displacements and rotations at various nodal positions of the body 2. The matrices M, B and K and the displacement vector x are related to the aerodynamic forces F by the following differential equation: a linked: Md2x(t)dt2+Bdx(t)dt+Kx(t)=Fa(t,x(t),dx(t)dt) Structural eigenmodes of body 2 are determined under the assumption that no aerodynamic forces act on body 2 (F a= 0). Often, the number of degrees of freedom can then be reduced by applying a Galerkin projection with a matrix containing a truncated set of these structural eigenmodes. In this case, the projected quantities are defined as generalized mass, damping, and stiffness matrices, and the aerodynamic forces F a referred to as generalized aerodynamic forces (GAF).

[0035] In the above-mentioned Fig. 1, the generalized mass matrix M hh and the generalized stiffness matrix K hh neglecting the generalized damping matrix B hh have the following entries: Mhh=[48.1056(kg)004.8106(kg*m2)] Khh=[2.5322*105(N / m)000.4502*105(N*m)]

[0036] The above differential equation shows couplings between the movements 15 described by the displacement vector x and the aerodynamic forces F a . The couplings depend on the movements 15 of the body 2, ie the displacement vector x and its time derivatives dx(t) / dt, while the displacement vector x in turn depends on the aerodynamic forces F acting on the structural model 13 a depends on how Fig. 3 is shown.

[0037] The aerodynamic forces F a(t,x(t),dx(t) / dt) depend on time t and the displacement vector x and its time derivatives dx(t) / dt. Various theories can be applied to determine the aerodynamic forces in the aerodynamic model 14. In this example, the unsteady Reynolds-averaged Navier-Stokes equations (URANS) are applied with the Spalart-Allmaras (SA) turbulence closure model. In particular, its linearized version around a stationary reference is considered. In this case, the aerodynamic forces additionally depend on the steady state and the non-dimensional Mach and Reynolds numbers. Various models with varying degrees of complexity can be considered. However, the determination of the Buffeton set requires flow eigenmodes whose eigenvalues ​​have an imaginary part. Thus, compressible potential flow methods are not suitable.

[0038] To calculate the eigenvalues ​​of the aeroelastic model 12 using a linear eigenvalue solver, a state-space form of the GAF term is required. For this purpose, a state-space realization proposed for the pL method can be implemented. The resulting state space can be represented as described in D. Quero, P. Vuillemin, and C. Poussot-Vassal, "A generalized eigenvalue solution to the flutter stability problem with true damping: The pL method," Journal of Fluids and Structures, 103266. doi: 10.1016 / j.jfluidstructs.2021.103266, 2021. In addition, a residualization of high frequencies can be introduced, as is known from D. Quero, P. Vuillemin and C. Poussot-Vassal, “A generalized state-space aeroservoelastic model based on tangential interpolation”, Aerospace 6 (1), 9. doi: 10.3390 / aerospace6010009, 2019.Another state space realization can also be carried out, but it must fulfill two requirements:. - The eigenvalues ​​that specify the aerodynamic forces may have a non-zero imaginary part. This is of fundamental importance for the flow eigenmodes in the transonic regime, since the imaginary part represents the buffet frequency. In the Roger method (see K. Roger, "Airplane math modeling methods for active control design," Proceedings of the 44th AGARD Structures and Material Panel (AGARD-CP-228), 1977), and the Karpel method (see M. Karpel, "Design for active flutter suppression and gust alleviation using state-space aeroelastic modeling," Journal of Aircraft, Vol. 19, No. 3, 1982), only eigenvalues ​​with real parts are considered for the flow eigenmodes, so this requirement is not met. - The rational aerodynamic transfer function resulting from the state-space realization must contain a non-proper term that describes the effect of the time derivatives of the body's motion on the aerodynamic forces. Some known methods that consider eigenvalues ​​of the fluid mode with imaginary parts do not allow for the non-proper term, such as the Eigensystem Realization Algorithm (ERA). See J. Juang and R. Pappa, "An eigensystem realization algorithm for modal parameter identification and model reduction," Journal of Guidance, Vol. 8, No. 5, 1985.

[0039] After the state-space realization, the dominant flow eigenmodes can be extracted under the assumptions that the body 2 performs movements according to at least one structural eigenmode and that the flow has no influence on the body's movements. In the considered two-dimensional case according to Fig. 1, the most dominant flow eigenmode is typically sufficient for determining the Buffet onset. The determination can be performed using a criterion consisting of an evaluation of the residuals and poles obtained from the state-space matrices, as described by G. Sleijpen and J. Rommes, "Computing dominant poles of transfer functions," https: / / webspace.science.uu.nl / ~sleij101 / Temp / Sleijpen_Kyoto.pdf, last downloaded on December 21, 2023.

[0040] Now two sweeps 22 and 23 are performed as in Fig. 4 is illustrated.

[0041] In the first sweep 22, a scaling factor q m , with which the generalized mass matrix M hh multiplied by a value q m = 1 × 10 6 down to q m = 1. The width of the steps Δq m between two consecutive values ​​of the scaling factor q mcan initially be set higher, e.g., with Δq m = -1 × 10 3 , and can then be reduced step by step, e.g., down to Δq m = -0.05. A suitable tracking method can be applied to determine the most dominant flow modes with decreasing scaling factor q m The consideration of aeroelastic sensitivities with respect to the scaling factor q m may be useful for this purpose, but is not required. While the first sweep 22 is being performed, the dynamic pressure is set to the initial value desired for the second sweep 23, in this example to q dyn = 1 x 10 -3 Pa. For this example, Fig. 5 the evolution of the complex eigenvalue 16 of the most dominant flow eigenmode when the scaling factor q m is reduced, as indicated by an arrow 17.

[0042] In the second sweep 23, starting from the final value of the scaling factor q m in the first sweep 22, ie q m = 1, the dynamic pressure of q dyn = 1 × 10 -3 Pa to obtain the evolution of the stability behavior taking into account the structural eigenmodes and flow eigenmodes. A fixed pressure step size Δq dyn = 0.01 Pa. The pressure step size Δq dyn can also be varied, for example, manually or automatically using a predictor-corrector scheme. During the sweep 23, the structural eigenmodes and the flow eigenmodes are tracked. In this case, the aeroelastic sensitivities with respect to the dynamic pressure q dyn useful, see D. Quero, P. Vuillemin and C. Poussot-Vassal, “Improved mode tracking for the pL flutter solution method based on aeroelastic derivatives,” in IFASD, Madrid, Spain, 2022, but is not required.

[0043] Fig. Figure 6 shows the evolution of the eigenvalues ​​16, 18 and 20 of the most dominant flow eigenmode “Fluid mode 1” and the two structural eigenmodes “Pitch” (inclination 10) and “Heave” (lift 9) in the complex plane, with arrows 17, 19 and 21 pointing in the direction of the increasing dynamic pressure q dyn show. At the value q dyn = 1930.8 Pa, the real part of the eigenvalue 16 of the most dominant flow eigenmode crosses the imaginary axis, indicating a Buffeton set.

[0044] For a second application example, which differs from the previous one by an angle of attack of the wing 1 of 0°, Fig. 7 the result of sweep 23 and Fig. 8 the result of the sweep 22, where an instability is caused by an eigenvalue 20 of the structural eigenmode “Heave” (lift 9)”, which changes the imaginary axis with increasing dynamic pressure at the value q dyn= 7102.8 Pa. This indicates a flutteron set. Thus, the same eigenvalue solver is capable of predicting both the flutter and buffeton sets. LIST OF REFERENCE SYMBOLS 1 wing 2 bodies 3 Current 4 Supersonic flow area 5 compression shock 6 Shear layer 7 Double arrow 8 double arrow 9 Lifting 10 Inclination 11 Aerodynamic mesh 12 Aeroelastic model 13 Structural model 14 Aerodynamic model 15 movements 16 Eigenvalue of the most dominant flow mode 17 Arrow 18 Eigenvalue of the structural eigenmode “Pitch” 19 Arrow 20 Eigenvalue of the structural eigenmode “Heave” 21 Arrow 22 sweeps 23 sweeps M mass matrix M hhGeneralized mass matrix K stiffness matrix K hh Generalized stiffness matrix B Damping matrix B hh Generalized damping matrix x displacement vector F a Aerodynamic force F ahh Generalized aerodynamic force q m Scaling factor Δq m Step size q dyn Dynamic pressure Δq dyn Printing step QUOTES CONTAINED IN THE DESCRIPTION

[0000] This list of documents submitted by the applicant was generated automatically and is included solely for the convenience of the reader. This list is not part of the German patent or utility model application. The DPMA assumes no liability for any errors or omissions. Cited non-patent literature

[0000] Houtman, J. and Timme, S., Global stability analysis of elastic aircraft in edge-of-theenvelope flow, Journal of Fluid Mechanics, Col. 967, A4, 2023, https: / / doi.org / 10.101 7 / jfm.2023.413

[0004] J. Nitzsche, L. Ringer, C. Kaiser und H. Hennings, „Fuild-mode flutter in plane transonic flows,“ International Forum on Aeroelasticity and Structural Dynamics (IFASD), 2019 [0005, 0032] D. Quero, P. Vuillemin und C. Poussot-Vassal, „A generalized eigenvalue solution to the flutter stability problem with true damping: The p-L method,“ Journal of Fluids and Structures, 103266. doi: 10.1016 / j.jfluidstructs.2021.103266, 2021

[0006] D. Quero, P. Vuillemin und C. Poussot-Vassal, „Improved mode tracking for the p-L flutter solution method based on aeroelastic derivatives,“ in IFASD, Madrid, Spanien, 2022 [0006, 0042] D. Quero, P. Vuillemin und C. Poussot-Vassal, „A generalized state-space aeroservoelastic model based on tangential interpolation,“ Aerospace 6 (1), 9. doi: 10.3390 / aerospace6010009, 2019

[0016] D. Quero, P. Vuillemin und C. Poussot-Vassal, „A generalized eigenvalue solution to the flutter stability problem with true damping: The p-L method“, Journal of Fluids and Structures, 103266. doi: 10.1016 / j.jfluidstructs.2021.103266, 2021

[0038] D. Quero, P. Vuillemin und C. Poussot-Vassal, „A generalized state-space aeroservoelastic model based on tangential interpolation“, Aerospace 6 (1), 9. doi: 10.3390 / aerospace6010009, 2019

[0038] K. Roger, „Airplane math modeling methods for active control design“, Proceedings of the 44th AGARD Structures and Material Panel (AGARD-CP-228), 1977

[0038] M. Karpel, “Design for active flutter suppression and gust alleviation using state-space aeroelastic modeling,” Journal of Aircraft, Vol. 19, No. 3, 1982

[0038] J. Juang and R. Pappa, “An eigensystem realization algorithm for modal parameter identification and model reduction,” Journal of Guidance, Vol. 8, No. 5, 1985

[0038] G. Sleijpen and J. Rommes, “Computing dominant poles of transfer functions,” https: / / webspace.science.uu.nl / ~sleij101 / Temp / Sleijpen_Kyoto.pdf, last downloaded on December 21, 2023

[0039]

Claims

[1] Method for predicting flutter and buffetonset for a body (2) moving in a flow (3), - wherein structural properties of the body (2) are described in a structural model (13) comprising a mass matrix (M), a generalized stiffness matrix (K) and a damping matrix (B), - wherein at least one structural eigenmode of the body (2) is determined from the structural model (13) under the assumption that no aerodynamic forces (F a ) act on the body (2), - wherein aerodynamic properties of the body (2) in the flow (3) are described in an aerodynamic model (14) under the initial assumptions that the body (2) performs movements (15) according to the at least one structural eigenmode and that the flow (3) has no influence on the movements (15) of the body (2), - wherein at least one dominant flow mode of the flow (3) and aerodynamic forces on the body (2) are determined from the aerodynamic model (14) under the initial assumptions, - wherein the aerodynamic model is converted into an aeroelastic model (12) by coupling the structural model (13), wherein during a first sweep (22) while keeping the dynamic pressure (qdyn) constant - either the mass matrix (M) is initially scaled with a large scaling factor (q m ), which is then successively reduced to 1, - or the stiffness matrix (K) and the damping matrix (B) are first multiplied by a small scaling factor, which is then successively increased to 1, wherein at least the at least one dominant flow eigenmode is tracked at least with regard to its eigenvalue (16), and - wherein the dynamic pressure (q dyn ) is successively increased, while the scaling factor (q m ) remains equal to 1, wherein the at least one structural eigenmode and the at least one dominant flow eigenmode are tracked at least with regard to their eigenvalues ​​(16, 18, 20). [2] Method according to claim 1, wherein during the second sweep (23) a flutteron set is detected by a real part of an eigenvalue of the at least one structural eigenmode becoming positive, and a buffeton set is detected by a real part of an eigenvalue of the at least one dominant flow eigenmode becoming positive. [3] Method according to claim 2, wherein the eigenvalues ​​of the at least one structural eigenmode and the at least one dominant flow eigenmode are calculated as eigenvalues ​​of the aeroelastic model using a linear eigenvalue solver. [4] Method according to one of the preceding claims, wherein the aerodynamic forces on the body (2) are determined from the aerodynamic model as generalized aerodynamic forces (GAF) in the state space. [5] Method according to claim 4, wherein the aerodynamic model is realized in state space, - where eigenvalues ​​representing the aerodynamic forces may have a non-zero imaginary part, which corresponds to a buffet frequency for the flow eigenmodes in the transonic region, and - where a resulting rational aerodynamic transfer function has a non-proper term which describes an effect of a time derivative of the movements of the body (2) on the aerodynamic forces. [6] The method of claim 5, wherein the at least one dominant flow eigenmode is determined by evaluating residuals and poles obtained from state space matrices. [7] Method according to claim 5 or 6, wherein the at least one dominant flow eigenmode is determined by means of a partial fractional expansion of an aerodynamic transfer function, wherein aerodynamic terms are represented on the basis of Loewner matrices and shifted Loewner matrices. [8] Method according to one of the preceding claims, wherein during the first sweep (22) for tracking at least the at least one dominant flow eigenmode, a first continuation method is applied which uses information on aeroelastic sensitivities with respect to the scaling factor (qm). [9] Method according to one of the preceding claims, - where a value of the dynamic pressure (q dyn) during the first sweep (22), which is used as the initial value of the second sweep (23), a maximum of 1 × 10 -1 Pa or 1 × 10 -2 Pa or 1 × 10 -3 Pa and / or - the large scaling factor at the beginning of the first sweep (22) being at least 1 × 10 4 or 1 × 10 5 or 1 × 10 6 or the small scaling factor at the beginning of the first sweep (22) is at most 1 × 10 -4 or 1 × 10 -5 or 1 × 10 -6 amounts. [10] Method according to one of the preceding claims, wherein in the first sweep (22) the scaling factor is brought closer to 1 in smaller steps, wherein a width of the steps optionally decreases by more than 99.9% or by more than 99.99%. [11] Method according to one of the preceding claims, wherein during the second sweep (23) for tracking the at least one structural eigenmode and the at least one dominant flow eigenmode, a second continuation method is applied which takes into account information about aeroelastic sensitivities with respect to the dynamic pressure (qdyn). [12] Method according to one of the preceding claims, wherein in the second sweep (23) the dynamic pressure (q dyn ) in pressure steps (Δq dyn ) between 0.1 Pa and 0.001 Pa. [13] A method according to any one of the preceding claims, wherein the body (2) moving in the flow (3) is an aircraft and wherein the flutter on set and the buffet on set are predicted to define a boundary of a flight envelope of the aircraft towards higher Mach numbers. [14] Method according to claim 13, wherein the structural model (13) is adapted to the current loading condition of the aircraft. [15] Aircraft with a device for automatically carrying out the method according to claim 13 or 14.

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