3. Physics based modular approach

#### 3.1. Aspects of physics based modelling

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

methods for aerodynamic and structural analysis are summarised in Figure 2.

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

oped based on traditional aircraft models.

52 Flight Physics - Models, Techniques and Technologies

<sup>η</sup>€ <sup>þ</sup> <sup>ω</sup><sup>2</sup>

<sup>η</sup><sup>i</sup> <sup>¼</sup> <sup>Q</sup>η<sup>i</sup> Mi

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 devel-

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

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].

(3)

The case for developing physics based simulation models and the motivation to move away from the classical formulations that rely on stability and control derivatives stems from the need for flight dynamic insight at the early conceptual design of highly integrated concepts. For such concepts, a database of stability and control derivatives such as Heffley and Jewell [17] does not exist. Moreover, these concepts integrate numerous technologies, such as active folding wingtips for flight and loads control [18] for which empirical methods also do not exist. The modelling and simulation of airframe aerodynamics alone can be complex, but a further layer of complexity is added when considering flexible aircraft for which, the inertial, aerodynamic and structural models need to be coupled. Multiple calculation points, known as structural nodes and aerodynamic panels, must be defined around the airframe and used to capture local flow physics. The structural model must be coupled with the aerodynamics model so that aerodynamic forces and moments acting on the structure modify the effective shape of the aircraft. To complete such an aeroelastic coupling, the updated shape is used to compute the aerodynamic loading for the next iteration.

This additional layer of complexity and iteration process requires a clear definition of methods used when investigating aircraft flight dynamics. These can be broadly divided into two categories:


For a given problem, multiple approaches can be adopted depending on the needs of the user or the key characteristics of the simulation framework. For example, the structural dynamics of the aircraft can be captured through the integration of a full Finite Element (FE) model with high fidelity, or with a simple beam, or 'stick' model. Within the latter method, multiple sublayers of complexity can be added depending on the mathematical formulation being used. A direct solving method, which is the most intuitive as it is based on discrete structural loads and nodes, will also be the most laborious and computationally heavy for a high number of structural elements. Alternatively, the modal approach restricted to frequency ranges of interest will be more efficient for linear deformations. In High Altitude Long Endurance (HALE) aircraft or HAR Wing concepts, structural nonlinearities can also become a physical phenomenon that must be captured by the model. Nonlinearities may be relevant only for specific modes and parts of the structure so that optimal solving methods can be identified as well.

Similarly, centre of gravity (CG) position and inertial terms will vary with structural flexibility and displacement. Therefore, acceptable or desired fidelity must be identified. For example, assuming a fixed CG and inertia can lead to significant simplifications in the EoM. However, this may be incorrect for HALE configurations where most of the mass lies in the flexible wing that undergoes large deformations.

Multiple methods to capture the aerodynamic loads acting on the aircraft have also been developed for different levels of fidelity; from simple lifting line theory, use of Engineering Science Data Unit (ESDU) to more complex UVLM and further to more expensive CFD based processes. The desired accuracy and performance can be optimised depending on the purpose of the framework. Dynamic stall models can also be added for a more accurate simulation of high angle of attack or flow detachment scenarios [19]. CFD simulations are at the higher fidelity end of the spectrum and can be used for construction of the aerodynamic databases [20].

#### 3.2. Modular simulation

need for flight dynamic insight at the early conceptual design of highly integrated concepts. For such concepts, a database of stability and control derivatives such as Heffley and Jewell [17] does not exist. Moreover, these concepts integrate numerous technologies, such as active folding wingtips for flight and loads control [18] for which empirical methods also do not exist. The modelling and simulation of airframe aerodynamics alone can be complex, but a further layer of complexity is added when considering flexible aircraft for which, the inertial, aerodynamic and structural models need to be coupled. Multiple calculation points, known as structural nodes and aerodynamic panels, must be defined around the airframe and used to capture local flow physics. The structural model must be coupled with the aerodynamics model so that aerodynamic forces and moments acting on the structure modify the effective shape of the aircraft. To complete such an aeroelastic coupling, the updated shape is used to compute the

This additional layer of complexity and iteration process requires a clear definition of methods used when investigating aircraft flight dynamics. These can be broadly divided into two

a. Low fidelity models used in particular for flight simulation and preliminary design studies. These allow for a rapid flight dynamic analysis and may allow parameters to be

b. High fidelity computationally expensive models which are used to consolidate the results obtained via low fidelity simulations and help in the investigation of specific problems

For a given problem, multiple approaches can be adopted depending on the needs of the user or the key characteristics of the simulation framework. For example, the structural dynamics of the aircraft can be captured through the integration of a full Finite Element (FE) model with high fidelity, or with a simple beam, or 'stick' model. Within the latter method, multiple sublayers of complexity can be added depending on the mathematical formulation being used. A direct solving method, which is the most intuitive as it is based on discrete structural loads and nodes, will also be the most laborious and computationally heavy for a high number of structural elements. Alternatively, the modal approach restricted to frequency ranges of interest will be more efficient for linear deformations. In High Altitude Long Endurance (HALE) aircraft or HAR Wing concepts, structural nonlinearities can also become a physical phenomenon that must be captured by the model. Nonlinearities may be relevant only for specific modes and parts of the structure so that optimal solving methods can be identified as well.

Similarly, centre of gravity (CG) position and inertial terms will vary with structural flexibility and displacement. Therefore, acceptable or desired fidelity must be identified. For example, assuming a fixed CG and inertia can lead to significant simplifications in the EoM. However, this may be incorrect for HALE configurations where most of the mass lies in the flexible wing

Multiple methods to capture the aerodynamic loads acting on the aircraft have also been developed for different levels of fidelity; from simple lifting line theory, use of Engineering Science Data Unit (ESDU) to more complex UVLM and further to more expensive CFD based

modified for identifying and quantifying possible optimised solutions.

aerodynamic loading for the next iteration.

54 Flight Physics - Models, Techniques and Technologies

where low fidelity simulation is not accurate.

that undergoes large deformations.

categories:

The objectives and scope of the problem being considered will undoubtedly dictate which mathematical formulation is selected. For instance, the aerodynamic forces can be calculated using either a Modified Strip Theory (MST) or a UVLM method [21] depending on the fidelity requirements and the available computational power. The structural deflection of the wing can be assumed either linear through an Euler-Bernoulli model or nonlinear with a Timoshenko model [22]. Various atmospheric disturbance models [23] are also implemented so that flight simulations with or without gusts and turbulence are possible for specific gust loads and flight control research. Flight control laws and actuation models of a variety of control surfaces can be used if the user wishes to investigate and develop optimal control or loads alleviation laws. The gravity and navigation model allows for trajectory and autopilot if required. Specialised hardware can be used to accelerate the model and reach real time performances suitable for pilot in the loop simulations at 50 Hz, paving the way for handling quality analysis of flexible aircraft concepts. So far a number of different modelling approaches towards flight dynamics modelling of flexible aircraft have been introduced. This section focuses on the possible problems and issues that emerge when integrating the various elements of such a framework and discusses the need for modularisation.

The basic components required for building a simulation framework are as follows:


Figure 3 illustrates the links between each of the modules and their relative dependencies.

Figure 3. Links between each modules of the simulation framework.

Adopting a modular approach allows for a more versatile framework that can be used to study different configurations and scenarios. Moreover, it allows the adoption of multiple approaches to solve particular mathematical or physical problems. The overhead effort required to develop a modular framework, which primarily takes the form of software engineering, is justified by the end result. If carefully managed a versatile framework that allows solvers and models to be treated in a plug-and-play fashion is achievable. An example of a modular framework is given in Figure 4. The CA<sup>2</sup> LM framework offers the user multiple options in most of the different mathematical models. The modular approach was considered at the early stages of framework development, and has allowed continuous development aiming for a versatile academic research tool.

#### 4. Framework setup for CA<sup>2</sup> LM

#### 4.1. Wing aerodynamic modelling

There are numerous ways in which wing aerodynamics can be modelled for flexible wings, such as directly via CFD using RANS simulations or steady or unsteady VLM. However, given that there can be thousands of cases that need to be considered for flight loads, computationally cheap alternatives are needed. Within the CA<sup>2</sup> LM framework, the aerodynamics module Flight Dynamic Modelling and Simulation of Large Flexible Aircraft http://dx.doi.org/10.5772/intechopen.71050 57

Figure 4. CA2 LM framework overall modular architecture.

Adopting a modular approach allows for a more versatile framework that can be used to study different configurations and scenarios. Moreover, it allows the adoption of multiple approaches to solve particular mathematical or physical problems. The overhead effort required to develop a modular framework, which primarily takes the form of software engineering, is justified by the end result. If carefully managed a versatile framework that allows solvers and models to be treated in a plug-and-play fashion is achievable. An exam-

multiple options in most of the different mathematical models. The modular approach was considered at the early stages of framework development, and has allowed continuous

There are numerous ways in which wing aerodynamics can be modelled for flexible wings, such as directly via CFD using RANS simulations or steady or unsteady VLM. However, given that there can be thousands of cases that need to be considered for flight loads, computation-

LM framework offers the user

LM framework, the aerodynamics module

ple of a modular framework is given in Figure 4. The CA<sup>2</sup>

Figure 3. Links between each modules of the simulation framework.

56 Flight Physics - Models, Techniques and Technologies

development aiming for a versatile academic research tool.

ally cheap alternatives are needed. Within the CA<sup>2</sup>

LM

4. Framework setup for CA<sup>2</sup>

4.1. Wing aerodynamic modelling

contains the implementation of the MST based steady aerodynamics coupled with unsteady aerodynamic models [24].

To model the unsteady build-up of lift due to changes in angle of attack and airspeed, a statespace representation of the unsteady aerodynamics of the aerofoil has been implemented following the work done by Leishman and Nguyen [25]. This assumes an arbitrary motion of the aerofoil as combination of the indicial lift response and the superposition principle applying the well-known Duhamel's integral [26]. The following general two-pole approximation of the Wagner function has been adopted in CA<sup>2</sup> LM:

$$\Phi(\lambda) \approx 1 - \mathbf{A}\_1 e^{-b\_1 \lambda} - \mathbf{A}\_2 e^{-b\_2 \lambda} \tag{4}$$

where λ = 2Vt/c is the relative distance travelled by the aerofoil in terms of semi chords whilst A and b are the indicial response parameters that depend on the boundary conditions. Using the two-pole representation, Leishman and Nguyen developed the lift response to a change in angle of attack α(t) as follow:

$$
\begin{bmatrix}
\dot{\mathbf{x}}\_1 \\
\dot{\mathbf{x}}\_2
\end{bmatrix} = \frac{2V}{c} \begin{bmatrix}
0 & -b\_2
\end{bmatrix} \begin{bmatrix}
\mathbf{x}\_1 \\
\mathbf{x}\_2
\end{bmatrix} + \begin{bmatrix}
1 \\
1
\end{bmatrix} a(t) \tag{5}
$$

and the output equation of the normal force coefficient is given by:

$$\mathbf{C}\_{N}(t) = 2\pi \frac{\mathfrak{D}V}{c} [A\_1 b\_1 A\_2 b\_2] \begin{bmatrix} \mathfrak{x}\_1 \\ \mathfrak{x}\_2 \end{bmatrix} \tag{6}$$

Coefficients Ai and bi have been derived by Leishman in order to obtain the indicial response approximation for a two-dimensional subsonic flow [27]. However, since the Wagner indicial response cannot be applied to compressible flows, a correction introduced by Leishman and Beddoes [28], has been used including the Prandtl-Glauert coefficient <sup>β</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>M</sup><sup>2</sup> <sup>p</sup> . The full equation of unsteady aerodynamics is then described as:

$$
\phi(\tau) \approx 1 - 0.918e^{-0.366\phi^2} - 0.082e^{-0.102\phi^2} \tag{7}
$$

Increasing the number of poles of the Wagner function allows a closer approximation to be obtained, but at the cost of an increased number of states.

In the CA<sup>2</sup> LM framework the two-pole representation is used to find lift and pitching moment response with respect to a change in angle of attack α and pitch rate q for each section. The generic total normal force coefficient is given by [29, 30]:

$$\mathbf{C}\_{N}(t) = \mathbf{C}\_{N}^{\circ}(t) + \mathbf{C}\_{Na}^{nc}(t) + \mathbf{C}\_{Nq}^{nc}(t) \tag{8}$$

where the superscripts c and nc represent the circulatory and non-circulatory terms respectively. Once aerodynamic characteristics are obtained at each aerodynamic node, the results are extended along the wingspan applying the method defined by DeYoung and Harper [31]. This approach considers the lift line and its trailing vortex as continuous. The circulation strength, however, can be discretized in as many control points as desired. In the CA<sup>2</sup> LM framework the control points are assumed to be at the aerodynamic nodes. DeYoung and Harper stated that a number of seven control points is enough to correctly represent the span loading without any sharp discontinuities. As the lifting line is discretized in m nodes, the method allows the calculation of the aerodynamic coefficients as follows [29]:

$$\text{rcC}\_{lv} = \sum\_{n=1}^{m} A\_{vn} \text{G}\_{n} \alpha\_{n} \qquad n = 1, 2, \dots, m \tag{9}$$

where Avn is the influence matrix which defines the effect of the circulation in the node v to the downwash at node n. The load coefficient G is dimensionless circulation and describes the strength of the circulation at any node n. When the aerodynamic forces and moments at each node are obtained, the loads are transposed from nodal-axis to body-axis and summed to give the overall lift, drag and moment acting on the aircraft structure.

Following the same methodology used for the calculation of the drag, the pitching moment is comprised of circulatory and non-circulatory term, described as follow:

$$\mathbf{C}\_{M} = \mathbf{C}\_{M\_{a}}^{\epsilon} + \mathbf{C}\_{M\_{q}}^{\epsilon} + \mathbf{C}\_{M\_{a}}^{m} + \mathbf{C}\_{M\_{q}}^{m} \tag{10}$$

The drag is instead modelled as the sum of the zero-lift drag coefficient, CD<sup>0</sup> , and the pressure drag coefficient, CDP . The unsteady drag force has been defined by Leishman as:

$$\mathcal{C}\_{D} = \mathcal{C}\_{D\_0} + \mathcal{C}\_N \sin \alpha\_\varepsilon(t) - \eta\_c \mathcal{C}\_\varepsilon \cos \alpha\_\varepsilon(t) \tag{11}$$

where the effective angle of attack α<sup>e</sup> is function of both the states and it is described as:

$$\alpha\_{\varepsilon}(t) = \beta^2 \frac{2V}{\mathfrak{c}} \left( A\_1 b\_1 \mathfrak{x}\_1 + A\_2 b\_2 \mathfrak{x}\_2 \right) \tag{12}$$

and the chord force term is:

CNðÞ¼ t 2π

ϕ τð Þ ≈ 1 � 0:918e

CNðÞ¼ <sup>t</sup> Cc

method allows the calculation of the aerodynamic coefficients as follows [29]:

n¼1

cClv <sup>¼</sup> <sup>X</sup><sup>m</sup>

the overall lift, drag and moment acting on the aircraft structure.

comprised of circulatory and non-circulatory term, described as follow:

CM <sup>¼</sup> Cc

The drag is instead modelled as the sum of the zero-lift drag coefficient, CD<sup>0</sup>

equation of unsteady aerodynamics is then described as:

58 Flight Physics - Models, Techniques and Technologies

obtained, but at the cost of an increased number of states.

generic total normal force coefficient is given by [29, 30]:

In the CA<sup>2</sup>

drag coefficient, CDP

2V

Beddoes [28], has been used including the Prandtl-Glauert coefficient <sup>β</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Coefficients Ai and bi have been derived by Leishman in order to obtain the indicial response approximation for a two-dimensional subsonic flow [27]. However, since the Wagner indicial response cannot be applied to compressible flows, a correction introduced by Leishman and

�0:366β<sup>2</sup>

Increasing the number of poles of the Wagner function allows a closer approximation to be

response with respect to a change in angle of attack α and pitch rate q for each section. The

<sup>N</sup>ð Þþ <sup>t</sup> Cnc

where the superscripts c and nc represent the circulatory and non-circulatory terms respectively. Once aerodynamic characteristics are obtained at each aerodynamic node, the results are extended along the wingspan applying the method defined by DeYoung and Harper [31]. This approach considers the lift line and its trailing vortex as continuous. The circulation strength, however, can be discretized in as many control points as desired. In the CA<sup>2</sup>

framework the control points are assumed to be at the aerodynamic nodes. DeYoung and Harper stated that a number of seven control points is enough to correctly represent the span loading without any sharp discontinuities. As the lifting line is discretized in m nodes, the

where Avn is the influence matrix which defines the effect of the circulation in the node v to the downwash at node n. The load coefficient G is dimensionless circulation and describes the strength of the circulation at any node n. When the aerodynamic forces and moments at each node are obtained, the loads are transposed from nodal-axis to body-axis and summed to give

Following the same methodology used for the calculation of the drag, the pitching moment is

Mq <sup>þ</sup> Cnc

. The unsteady drag force has been defined by Leishman as:

<sup>M</sup><sup>α</sup> <sup>þ</sup> Cnc

<sup>M</sup><sup>α</sup> <sup>þ</sup> Cc

� 0:082e

LM framework the two-pole representation is used to find lift and pitching moment

<sup>N</sup>αð Þþ <sup>t</sup> <sup>C</sup>nc

�0:102β<sup>2</sup>

AvnGnαn, n ¼ 1, 2, …, m (9)

<sup>c</sup> ½ � <sup>A</sup>1b1A2b<sup>2</sup>

x1 x2 � �

(6)

(7)

LM

<sup>1</sup> � <sup>M</sup><sup>2</sup> <sup>p</sup> . The full

Nqð Þt (8)

Mq (10)

, and the pressure

$$\mathbb{C}\_{\epsilon}(t) = \frac{2}{\beta} \frac{\pi}{\alpha\_{\epsilon}^{2}} \alpha\_{\epsilon}^{2}(t) \tag{13}$$

As a real flow is unable to be fully attached in any real flow, the coefficient η<sup>c</sup> is used to account for the properties of the real flow.

#### 4.2. Structural modelling

Now all aerodynamic forces have to be applied to the structures of the aircraft. This is done in the structural dynamics modelling block.

Aerodynamic forces and moments, along with forces and moments due to gravity, are converted to modal forces F through modal transformation matrix Θ<sup>T</sup> m:

$$F\_i = \Theta\_m^T F\_{aero} \tag{14}$$

The next step is to solve the following structural equation of motion:

$$\frac{F\_i}{m\_i} = \ddot{\mathbf{x}}\_i + 2\zeta\omega\_n\dot{\mathbf{x}}\_i + \omega\_n^2\mathbf{x}\_i \tag{15}$$

where Fi represents the modal forces, mi the modal masses, ωn, <sup>i</sup> the modal natural frequencies, ζ the modal damping ratios, i is the modes number, xi, x\_i, x€<sup>i</sup> are the modal displacements, velocities and accelerations. To obtain the structural dynamics in modal form, the Normal Modes analysis solver SOL 103 from the NASTRAN finite element analysis program is used. Its output (modal masses, natural frequencies and modal transformation matrix) are used in the CA<sup>2</sup> LM framework to calculate structural deflections. The displacements, velocities and accelerations of each structural node can then be obtained using the transformation matrix.

As these deflections, velocities and accelerations are applied to aerodynamic frame, the interpolation between structural and aerodynamic nodes is executed.

The first 12 structural modes are considered in the CA<sup>2</sup> LM framework because the tool is designed to investigate interactions between aeroelasticity effects and flight dynamics phenomena that are typically at low frequencies. An illustration of an aircraft first four modes is given in Figure 5.

It is important to note that only small wingtip deflections (less than 10% of a wing semi-span) are modelled within CA<sup>2</sup> LM framework as linearly varying beam properties are assumed. However, recent developments in highly flexible aircraft [32] have introduced wingtip

Figure 5. First four modes of the AX-1 aircraft implemented in CA2 LM.

deflections of more than 25% of a wing semi-span. To investigate the effects of such high structural deformations on flight dynamics, a structural dynamics model capable of capturing the nonlinear phenomena due to large deformations is needed.

#### 4.3. Equations of motion

For large flexible aircraft, the centre of gravity (CG) position may vary significantly as a function of structural deformation. This is typically ignored in the classical EoM formulation for rigid aircraft [1, 2]. This issue together with continuously deforming aerodynamic and structural stations requires the careful definition of the axes systems for each module of the simulation framework. The selection of an appropriate axes system has been extensively discussed for many years [8, 33, 34]. Effectively there are two approaches that may be adopted: (1) use an arbitrary point on the aircraft also called the body axes centre (BAC) or, (2) adopt the mean axes system which has a floating point as the reference centre [35]. The latter has seen widespread application in research [9, 36] because its formulation minimises the coupling between rigid-body dynamics and aeroelastic modes. On the other hand, the axes system centre is often collocated with the CG which moves in phase with the flexible airframe, making the application of traditional flight dynamics analysis techniques more difficult. The idea of the mean axes system's inertial decoupling and complexity of its formulation has been questioned [34].

The CA<sup>2</sup> LM framework uses a fixed BAC as a reference centre for its flight dynamic axis system. This allows the framework to be used in both flexible and rigid modes and more importantly, it allows the integration of classical flight dynamics post-processing tools.

Flight Dynamic Modelling and Simulation of Large Flexible Aircraft http://dx.doi.org/10.5772/intechopen.71050 61

Figure 6. Motion of a body and its particle within the frame.

deflections of more than 25% of a wing semi-span. To investigate the effects of such high structural deformations on flight dynamics, a structural dynamics model capable of capturing

LM.

For large flexible aircraft, the centre of gravity (CG) position may vary significantly as a function of structural deformation. This is typically ignored in the classical EoM formulation for rigid aircraft [1, 2]. This issue together with continuously deforming aerodynamic and structural stations requires the careful definition of the axes systems for each module of the simulation framework. The selection of an appropriate axes system has been extensively discussed for many years [8, 33, 34]. Effectively there are two approaches that may be adopted: (1) use an arbitrary point on the aircraft also called the body axes centre (BAC) or, (2) adopt the mean axes system which has a floating point as the reference centre [35]. The latter has seen widespread application in research [9, 36] because its formulation minimises the coupling between rigid-body dynamics and aeroelastic modes. On the other hand, the axes system centre is often collocated with the CG which moves in phase with the flexible airframe, making the application of traditional flight dynamics analysis techniques more difficult. The idea of the mean axes system's inertial

LM framework uses a fixed BAC as a reference centre for its flight dynamic axis

system. This allows the framework to be used in both flexible and rigid modes and more

importantly, it allows the integration of classical flight dynamics post-processing tools.

the nonlinear phenomena due to large deformations is needed.

Figure 5. First four modes of the AX-1 aircraft implemented in CA2

60 Flight Physics - Models, Techniques and Technologies

decoupling and complexity of its formulation has been questioned [34].

4.3. Equations of motion

The CA<sup>2</sup>

The derivation of the EoM begins by considering a fixed node which is located away from the BAC, as shown in Figure 6. The velocities of this point can be expressed as:

$$
\begin{bmatrix} u \\ v \\ w \end{bmatrix} = \begin{bmatrix} \mathcal{U} + \dot{\mathcal{x}} \\ \mathcal{V} + \dot{\mathcal{y}} \\ \mathcal{W} + \dot{z} \end{bmatrix} - \begin{bmatrix} x \\ y \\ z \end{bmatrix} \times \begin{bmatrix} p \\ q \\ r \end{bmatrix} \tag{16}
$$

And therefore, the following accelerations can be obtained:

$$
\begin{bmatrix} a\_x \\ a\_y \\ a\_z \end{bmatrix} = \begin{bmatrix} \dot{u} \\ \dot{v} \\ \dot{w} \end{bmatrix} - \begin{bmatrix} u \\ v \\ w \end{bmatrix} \times \begin{bmatrix} p \\ q \\ r \end{bmatrix} \tag{17}
$$

The velocities U, V and W express the motion of the BAC, while x, y and z express the position of the node. The angular rates p, q and r represent the angular velocities of the overall aircraft. Merging both equations gives following accelerations expressions:

$$a\_x = \dot{\mathcal{U}} - rV + qW - x(q^2 + r^2) + y(pq - \dot{r}) + z(pr + \dot{q}) + \ddot{x} - 2r\dot{y} + 2q\dot{z} \tag{18}$$

$$a\_{\mathcal{Y}} = \dot{V} - p\mathcal{W} + r\mathcal{U} + \mathbf{x}(p\eta + \dot{r}) - y(p^2 + r^2) + z(qr - \dot{p}) + \ddot{y} - 2p\dot{z} + 2r\dot{\chi} \tag{19}$$

$$a\_z = \dot{W} - q\mathcal{U} + pV + \mathbf{x}(pr - \dot{q}) + y(qr + \dot{p}) - z(p^2 + q^2) + \ddot{z} - 2q\dot{\mathbf{x}} + 2p\dot{\mathbf{y}} \tag{20}$$

Now applying Newton's second law with a nodal mass of δm the EoM can be obtained as follows:

$$
\begin{bmatrix} X \\ Y \\ Z \end{bmatrix} = \sum\_{i=1}^{N} \delta m\_i \begin{pmatrix} a\_x \\ a\_y \\ a\_z \end{pmatrix} = \overbrace{\sum\_{i=1}^{N} \delta m\_i \dot{v}\_0 + \sum\_{i=1}^{N} \delta m\_i \omega \times v\_0}^{\text{Rigid body dynamics force}}
$$

$$\begin{aligned} \underbrace{\begin{aligned} \text{Axes réferenc\\_point\\_offset} \\ \text{\\_Cartr\text{"signal force}} \\ + \sum\_{i=1}^{N} \delta m\_i \omega \times (\omega \times r\_i) + \sum\_{i=1}^{N} \delta m\_i \dot{\omega} \times r\_i}\_{\text{Inertial force}} \end{aligned} \tag{21}$$

$$\begin{aligned} \begin{bmatrix} L \\ M \\ N \end{bmatrix} &= \overbrace{\dot{I}\dot{\boldsymbol{\omega}} + \boldsymbol{\omega} \times (I\boldsymbol{\omega})}^{\text{Rigid body dynamics}} + \overbrace{\sum\_{i=1}^{N} \delta m\_{i} \boldsymbol{r}\_{i} \times \dot{\boldsymbol{v}}\_{0} + \sum\_{i=1}^{N} \delta m\_{i} \boldsymbol{r}\_{i} \times (\boldsymbol{\omega} \times \boldsymbol{v}\_{0})}^{\text{Array reference point offset}} \\ &+ \underbrace{\dot{I}\boldsymbol{\omega} + \boldsymbol{\omega} \times \sum\_{i=1}^{N} \delta m\_{i} (\boldsymbol{r}\_{i} \times \boldsymbol{v}\_{\text{rel},i}) + \sum\_{i=1}^{N} \delta m\_{i} \boldsymbol{r}\_{i} \times a\_{\text{rel},i}}\_{\text{Hermity effects}} \end{aligned} \tag{22}$$

The forces and moments on the left hand side of the above equations are the sum of the forces and moments obtained from the structural dynamics, aerodynamics and gravitational modules.

Flexibility effects

#### 4.4. Aeroelastic coupling and equations of motion integration

The previous sections have shown that each module within the simulation framework requires the definition of its own axis system and a separate means of modelling the aircraft, whether it is through a set of structural nodes or aerodynamic panels. This presents two issues that must be addressed before scenarios can be simulated: (1) node and panel distributions and densities need to be optimised based on the scope of the research and, (2) the structural nodes must be linked to aerodynamic nodes.

As seen in the previous section, the structural loads calculations rely on a set of structural nodes. Displacements, velocities and accelerations of each node are calculated in all 6 degrees of freedom.<sup>1</sup>

<sup>1</sup> It is possible to constrain specific degrees of freedom to reduce model complexity after a comparison study with the 6 DoF model. For stiff wings, structural rotation around the vertical axis can be neglected for example.

Now applying Newton's second law with a nodal mass of δm the EoM can be obtained as

CCCA <sup>¼</sup> <sup>X</sup> N

> þ X N

> > i¼1

δmiω � vrel,i


i¼1

<sup>δ</sup>miv\_ <sup>0</sup> <sup>þ</sup><sup>X</sup>

δmiω\_ � ri zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{ Euler force

<sup>δ</sup>miri � <sup>v</sup>\_ <sup>0</sup> <sup>þ</sup><sup>X</sup>

δmið Þþ ri � vrel,i


N

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ Axes reference point offset

i¼1

X N

i¼1

δmiri � ð Þ ω � v<sup>0</sup>

δmiri � arel,i

N

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ Rigid body dynamics force

i¼1

δmiω � v<sup>0</sup>

(21)

(22)

ax

1

0

BBB@

δmiω � ð Þ ω � ri zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ Centrifugal force

> þ2 X N

> > i¼1


> þ X N

<sup>I</sup><sup>ω</sup> <sup>þ</sup> <sup>ω</sup> �<sup>X</sup>

i¼1

N

i¼1

The forces and moments on the left hand side of the above equations are the sum of the forces and moments obtained from the structural dynamics, aerodynamics and gravitational modules.

The previous sections have shown that each module within the simulation framework requires the definition of its own axis system and a separate means of modelling the aircraft, whether it is through a set of structural nodes or aerodynamic panels. This presents two issues that must be addressed before scenarios can be simulated: (1) node and panel distributions and densities need to be optimised based on the scope of the research and, (2) the structural nodes must be

As seen in the previous section, the structural loads calculations rely on a set of structural nodes. Displacements, velocities and accelerations of each node are calculated in all 6 degrees

It is possible to constrain specific degrees of freedom to reduce model complexity after a comparison study with the 6

DoF model. For stiff wings, structural rotation around the vertical axis can be neglected for example.

ay

az

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ Axes reference point offset

follows:

X

i¼1 δmi

δmiarel,i


<sup>5</sup> <sup>¼</sup> <sup>I</sup>ω\_ <sup>þ</sup> <sup>ω</sup> � ð Þ <sup>I</sup><sup>ω</sup> zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{ Rigid body dynamics

þ \_

4.4. Aeroelastic coupling and equations of motion integration

62 Flight Physics - Models, Techniques and Technologies

Y

Z

þ<sup>X</sup> N

þ<sup>X</sup> N

L M N

3 7 7

linked to aerodynamic nodes.

of freedom.<sup>1</sup>

1

i¼1

i¼1

Figure 7. Illustration of the different mass, structural node and aerodynamic station positions for the AX-1 aircraft.

Appropriate balance between accuracy and computational cost must be obtained using a convergence study to identify the optimal number of structural nodes and aerodynamic panels or strips. This number can vary with aircraft configuration and the type of flight dynamics being considered. However, the number of structural nodes may be different from the optimal number of aerodynamic stations. A modular simulation environment such as CA<sup>2</sup> LM allows the definition of different numbers of aerodynamic strips and structural nodes. The aerodynamic forces and moments calculated at the aerodynamic stations must then be transferred to the structural set of nodes using various interpolation methods. Similarly the structural displacements, velocities and accelerations calculated from the structural model must be transferred to the aerodynamic stations in order to calculate the local forces and moments with structural flexibility. An example of this coupling can be found in Figure 7 where both the structural node and aerodynamic station layout is illustrated for an example aircraft.

The EoM rely on the aircraft total forces and moments, acting around the centre of gravity of the vehicle. Therefore, the updated CG position due to structural deformation must be used to calculate the new global set of moments acting on the aircraft. Aerodynamic loading calculated at each aerodynamic station is merged and calculated at the temporary CG position. Only then can the coupling between the aerodynamic and structural block be made with the EoM.

The output of the EoM such as aircraft position, attitude and velocity can then be used by conventional atmospheric models to compute the dynamic pressure and other aerodynamic

Figure 8. Aircraft flexible structure overlaid with aerodynamic profiles and control surfaces for pilot input visualisation.

parameters used by the aerodynamic model, closing the main calculation loop. Similarly, the adequate gravity contribution can be computed with position (or altitude) and applied to the structural model.

Appropriate inputs, usually on aircraft control surface and thrust, should be linked to the model in the correct format. Control surface dynamics can be implemented for higher fidelity.

As each module is included in the simulation framework, correct integration testing must be conducted to verify that each modules are behaving as expected. Therefore, as the complexity of the framework increases, thorough testing also requires more effort. It can also be really helpful to have visual aids and illustrations of the simulation. For example, an illustration of aerodynamic station and structural node positions updated with structural flexibility at each time step can be found in Figure 8 and is very useful to visualise the modelled aircraft.
