2. Review of past methods

An extended version of the Collar's triangle shown in Figure 1 highlights the physical phenomena that need to be integrated for accurate modelling and simulation of flexible aircraft. Traditionally the flight dynamics community has focused on the link between inertial dynamics and aerodynamics and it assumes structural dynamics to occur at far higher frequencies than those of rigid-body dynamics. The vice versa is true for the structural dynamics community who have mainly focused on specific loads cases for sizing airframe components. The development of aircraft such as the Boeing 747 [7], which was exceptionally large, and the Rockwell B-1 [8] with its flexible fuselage made it necessary for flight dynamics and structural dynamics to be integrated. The work done by Schmidt and Waszak [9] is an early example of such an integrated modelling approach carried out from a flight dynamicist's perspective. The approach retains the inertial components of the classical nonlinear six degree of freedom (6-DoF) equations [1, 2].

Figure 1. Extended Collar's triangle.

such as aeroelasticity and flight dynamics, must now integrate. This chapter aims to present the methods used for developing modelling and simulation tools that are needed to facilitate such an

The traditional approach to modelling and simulation of aircraft flight dynamics has framed the problem in the form of the equations of motion (EoM) that couple nonlinear inertial components with quasi-linear aerodynamic models [1, 2]. This has been found to be satisfactory when modelling the flight dynamics of rigid aircraft, but the assumptions of linearity in the method used to formulate the aerodynamic model remains the primary limitation of this approach. Typically, this limitation is the cause of significant uncertainty early in the aircraft design process where engineers can only resort to either empirical methods or panel based methods. For conventional tube and wing configurations, the civil aviation industry has developed and modified these methods based on extensive testing and operational data. On the other hand, the radical configurations seen in the military domain rely on significant effort put towards the identification of aerodynamic characteristics and validation of models during the expensive flight test phase. The latter may often span the entire service life of the

Accurate modelling and simulation of novel concepts aimed to address today's societal concerns is needed to enable the multidisciplinary approach necessary for design. However, it cannot resort to the knowledge gained either from significant operational data or extensive flight test data. As a result it can only rely on a physics based approach and moreover, this approach needs to be modular if it is to assist in the necessary multidisciplinary design process. Within this chapter, a brief review of past methods for modelling and simulation of flexible aircraft is presented before the physics based modular approach is discussed. This is followed by details of the methods needed to integrate aerodynamics, structural dynamics and flight dynamics within a single simulation framework. Finally, the reader is presented with two test cases that demonstrate the use of such a framework in aircraft design. The Cranfield

An extended version of the Collar's triangle shown in Figure 1 highlights the physical phenomena that need to be integrated for accurate modelling and simulation of flexible aircraft. Traditionally the flight dynamics community has focused on the link between inertial dynamics and aerodynamics and it assumes structural dynamics to occur at far higher frequencies than those of rigid-body dynamics. The vice versa is true for the structural dynamics community who have mainly focused on specific loads cases for sizing airframe components. The development of aircraft such as the Boeing 747 [7], which was exceptionally large, and the Rockwell B-1 [8] with its flexible fuselage made it necessary for flight dynamics and structural dynamics to be integrated. The work done by Schmidt and Waszak [9] is an early example of such an integrated modelling approach carried out from a flight dynamicist's perspective. The approach retains the inertial components of the classical nonlinear six degree of freedom (6-DoF) equations [1, 2].

LM) [5, 6] forms the basis of the discussion presented

integrated approach, especially focusing on large flexible aircraft.

50 Flight Physics - Models, Techniques and Technologies

aircraft [3, 4].

in this chapter.

Accelerated Aeroplane Loads Model (CA2

2. Review of past methods

However, the aeroelastic effects are introduced by the addition of states related to each aeroelastic mode. Assuming that the free vibration modes are available, these make a set of orthogonal functions. The modal representation of the airframe is often obtained through the use of beam element models of the structure and the use of structural analysis software such as NASTRAN. Thus the airframe deformation e(x,y,z,t) can be described in terms of the mode shape ϕi(x, y, z) and the general displacement coordinate ηi(t), as follow:

$$e(\mathbf{x}, y, z, t) = \sum\_{i=1}^{\bullet} \Phi\_i(\mathbf{x}, y, z)\eta\_i(t) \tag{1}$$

The sum of the mode shapes is theoretically infinite but in practice, a finite number of mode shapes are selected in order to investigate the coupling of aeroelastic modes with rigid-body dynamics. The coupling between the rigid-body motion and elastic motion takes place through the forces and moments. The generic force and moment term can be described as function of the inputs (as in the general rigid equations of motion) and the generalised displacement η and its first derivative η\_, as follow:

$$F = f(\mu, \alpha, \delta, \dots, \eta, \dot{\eta}) \tag{2}$$

A new equation is then introduced to account for the elastic dynamics as:

$$
\ddot{\eta} + \omega^2 \eta\_i = \frac{Q\_{\eta i}}{M\_i} \tag{3}
$$

where Qη<sup>i</sup> and Mi are the generalised force and mass terms, respectively. This formulation allows the application of stability analysis and flight control methods that have been developed based on traditional aircraft models.

Since the work done by Waszak and Schmidt, modelling frameworks of varying complexity have been developed both in industry and academia. Industrial frameworks are highly complex and aimed at supporting certification activities. These often couple Computational Fluid Dynamics (CFD) with Computational Structural Modelling (CSM) and result in processes that provide the desired insight, but at a very high computational cost [10–12]. Much research has been carried out to reduce the computational cost and the effort needed to integrate CFD solvers with CSM packages. However, more often the approach has depended on the specific technical challenge faced by the designer. For example, a few CFD-CSM simulations may be carried out to provide a means of validation for Reduced Order Models (ROMs). The various methods for aerodynamic and structural analysis are summarised in Figure 2.

Academic research has shown the capability to link aeroelasticity with flight control and develop novel approaches to aeroservoelastic analysis of highly flexible configurations [13–15].

Figure 2. A non-exhaustive list of modelling methods ranked by complexity and fidelity.

Structural flexibility effects have been modelled through the implementation of a nonlinear structural dynamics formulation and aerodynamic contributions have been captured by means of an Unsteady Vortex Lattice Method (UVLM) code. Solving the geometrically-nonlinear beam equations in three different ways, Palacios et al. concluded that the intrinsic beam element model is more efficient regarding the computational time than the classical displacements and rotations based model. It has been shown that for certain geometries the intrinsic model required two times less operations per iteration due to simpler algorithms.

With regards to aerodynamic modelling Palacios et al. [14] showed that an indicial response based on the usual Pade approximation to Wagner's step response performs better at low reduced frequencies than the model based on a Glauert's expansion of the inflow velocity field. Three models—strip theory, strip theory with wingtip effects correction and UVLM have been compared for different reduced frequencies and wingtip deflections. It has been shown that at low reduced frequency wingtip effects is of high importance both for low and high aspect ratio wings. However, for the case of increased reduced frequencies there has been no agreement of results for low aspect ratio wing. On the other hand, for high aspect ratio wing the agreement between the UVLM and the strip theory without wingtip correction has been shown. Such an agreement has been expected as increasing wing aspect ratio tends to reduce the 3D effect over the wing. The dynamic stall effects have not been modelled in the examples, nevertheless they may be of a great importance for a highly flexible wing. It is important to notice at this point that, if such a dynamic stall model is required by the user, empirical methods are much easier to implement within 2D strip theory than within the UVLM. Palacios and Cesnik [13] included aerofoil deformations in both the structural and the aerodynamic models: A Ritz (finite-section) expansion includes cross-sectional structural deformations, while a Glauert's expansion accounts for deformations of the aerofoil camber line. Integration of both expansions into a single methodology provides a simple alternative to more complex two-dimensional and three-dimensional models for preliminary active aeroelastic analysis of High Aspect Ratio Wings (HARW).

Although the approach adopted by Palacios is computationally cheaper than coupled CFD-CSM, real time simulation is still not possible. The need for real time simulation of flexible aircraft arises from the concern that low frequency aeroelastic modes can potentially couple with rigid-body modes such as the aircraft's short period pitch oscillation and result in poor handling qualities due to unwanted aircraft-pilot coupling [16]. Furthermore, novel concepts for future aircraft, such as those based on blended-wing-body configurations, need detailed stability and control analysis early in the design stage. A real time pilot-in-the-loop simulation environment is therefore needed to identify and solve stability and control problems. The development of such a simulation model requires a trade-off between model fidelity and computational cost.
