International Journal of Aerospace Engineering

Volume 2016 (2016), Article ID 4805817, 13 pages

http://dx.doi.org/10.1155/2016/4805817

## Flight Loads Prediction of High Aspect Ratio Wing Aircraft Using Multibody Dynamics

^{1}Department of Aerospace Engineering, Faculty of Engineering, University of Bristol, Bristol BS8 1TR, UK^{2}Aerospace Centre of Competence, Siemens PLM Software, 3001 Leuven, Belgium

Received 10 August 2016; Revised 4 November 2016; Accepted 29 November 2016

Academic Editor: Kenneth M. Sobel

Copyright © 2016 Michele Castellani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A framework based on multibody dynamics has been developed for the static and dynamic aeroelastic analyses of flexible high aspect ratio wing aircraft subject to structural geometric nonlinearities. Multibody dynamics allows kinematic nonlinearities and nonlinear relationships in the forces definition and is an efficient and promising methodology to model high aspect ratio wings, which are known to be prone to structural nonlinear effects because of the high deflections in flight. The multibody dynamics framework developed employs quasi-steady aerodynamics strip theory and discretizes the wing as a series of rigid bodies interconnected by beam elements, representative of the stiffness distribution, which can undergo arbitrarily large displacements and rotations. The method is applied to a flexible high aspect ratio wing commercial aircraft and both trim and gust response analyses are performed in order to calculate flight loads. These results are then compared to those obtained with the standard linear aeroelastic approach provided by the Finite Element Solver Nastran. Nonlinear effects come into play mainly because of the need of taking into account the large deflections of the wing for flight loads computation and of considering the aerodynamic forces as follower forces.

#### 1. Introduction

In recent years there has been a strong push in the aviation world towards the reduction of fuel consumption and the design of ecoefficient aircraft. Many research initiatives are currently addressed to investigate and develop design solutions that would lead to achieve these goals. The improvement of aerodynamic performance is at the forefront of these efforts and one of the most promising concepts being sought is the design of high aspect ratio wings. High aspect ratio wings can lead to significant fuel savings due to the reduction in induced drag. For future designs, a number of high aspect ratio wing configurations are currently being considered and both Airbus [1] and Boeing [2] have published their own concepts.

High aspect ratio wings nevertheless suffer from certain structural drawbacks. Due to the large span, the bending moment increases, resulting in higher structural weight. In order to achieve an effective performance benefit, a lightweight wing design is needed, which in turn leads to very flexible structures, where geometric nonlinearities due to large displacements cannot be neglected anymore. The greater flexibility and lower structural natural frequencies could also result in a strong coupling between structural dynamics and rigid body (flight mechanics) modes leading to undesirable effects on the handling qualities.

The move away from a linear behavior means that a nonconventional approach needs to be taken for the loads and aeroelastic analysis, in order to deal with geometric nonlinearities, and also the nonlinear aerodynamics and flight mechanics characteristics [3]. The ability to predict accurately limit loads, including these nonlinear effects, from the conceptual design phase onwards is paramount in achieving an optimized structural sizing and eventually reaching success with these configurations.

A great deal of work has considered the aeroelasticity of very flexible aircraft [4–11]. Most approaches have used nonlinear beam models coupled to aerodynamic models ranging from strip theory to unsteady vortex lattice method and CFD. However, less focus has instead been devoted to the use of multibody simulation for the modelling of high aspect ratio wings, the two most relevant pieces of work being those presented by Krüger [7] and Zhao and Ren [9]. Recently, Castellani et al. [12] developed two nonlinear methodologies, based, respectively, on nonlinear Finite Element Method (FEM) and multibody dynamics, for the static aeroelastic trim analyses including structural nonlinearities and applied these to a very flexible High-Altitude Long Endurance Unmanned Aerial Vehicle test case.

In this work, a framework based upon multibody dynamics is developed for the static and dynamic aeroelastic analyses of high aspect ratio wing aircraft including structural nonlinearities. The nonlinearities considered are the so-called geometric nonlinearities, arising because of the large deflections that a flexible high aspect ratio wing undergoes when loaded. Following this assumption, a further source of nonlinearity that must be introduced is the follower nature of the aerodynamic forces.

The studies performed are limited to structures undergoing large displacements, but small strains, so that the material constitutive law is still linear, and to attached subsonic flow, so that transonic and stall effects can be neglected.

The focus of this paper is on static and dynamic flight loads prediction, in accordance with the loads requirements set by airworthiness regulations (EASA CS-25 and FAR-25). Most of the research efforts dealing with structural nonlinearities in aeroelasticity have focused on the prediction of aeroelastic and flight dynamics instabilities; less focus has been instead devoted to the impact of geometric nonlinearities on flight loads and studies on this topic have been performed, for example, by Garcia [6] and De Breuker et al. [13]. There is therefore a need in the industry to develop tools and methodologies able to take into account these effects and assess their importance in the design of future high aspect ratio wing aircraft.

#### 2. Aeroelastic Modelling in Multibody Dynamics

Multibody dynamics simulation is a convenient tool capable of simulating multiphysics systems with arbitrary types of nonlinearities and both rigid and flexible components [14]. In the fixed-wing aeroelasticity field, it has been employed for the trim and simulation of manoeuvring flexible aircraft coupled with aerodynamic methods of various levels of fidelity [7, 15].

For the nonlinear aeroelasticity of very flexible aircraft, there have been applications of multibody simulation by Krüger [7] and Zhao and Ren [9], respectively, for the study of the flight mechanics stability of a HALE configuration and for the aeroelastic stability analysis and flight control in manoeuvres of a UAV-like flexible aircraft.

Multibody dynamics allows for arbitrary large displacements and rotations, generic force definition (follower and nonfollower) and inherent coupling between large rigid body motion, linked to flight mechanics, and elastic deformation, without the need of developing dedicated formulations. These are distinct advantages that make multibody dynamics attractive for the analysis of high aspect ratio wings including structurally nonlinear effects.

The multibody software employed for this work is LMS Virtual.Lab Motion v.13.1, a Commercial Off The Shelf (COTS) software developed by Siemens PLM [16].

In the following the equations of motion of a multibody system are briefly outlined (for more details see Shabana [14]). Each body is described by a set of Cartesian coordinates, identifying the location of its centre of gravity in the global reference frame. The vector of the generalized coordinates of the th body is thuswhere , , and are the Cartesian coordinates and , , , and the (redundant) Euler parameters used to describe the orientation of the body and to avoid the singularity occurring with other representations, for example, Euler angles.

The bodies in the system are connected together by joints and kinematic relationships, which are expressed as general nonlinear algebraic constraint equationsDifferentiating these equations twice with respect to time , one obtains the kinematic acceleration equationswhere . The dynamic equations of motion, for example, derived from Lagrange method, are, for the th body, written aswith mass matrix, vector of Lagrange multipliers, vector of generalized applied forces, and vector of velocity dependent terms. Adding the kinematic relationships to the equations of motion, a system of nonlinear Differential Algebraic Equations (DAE) describing the kinematics and dynamics of a multibody system is obtainedThese equations are nonlinear, as the matrices are a function of the vector of generalized coordinates itself, and are solved using a Backward Differentiation Formula integrator.

The bodies can be considered either as rigid or as flexible. The most common approach to model flexibility is a modal representation based on Component Mode Synthesis from FEM [14], which adds to the generalized coordinates the modal participation factors of each mode used to represent a body’s flexibility. This approach however limits the applicability to linear structures with small elastic displacements. Formulations based on nonlinear FE beams [17] and generic nonlinear FEM elements [18] have been also proposed to this purpose.

The work presented herein employs a simpler, yet efficient, approach to model a flexible wing with arbitrary large elastic displacements. It is based on the discretization of the wing by a series of rigid bodies, to which inertial properties are assigned, interconnected by beam force elements, representing the stiffness distribution. The CG of each body can have any arbitrary offset with respect to the elastic axis chordwise location. In the literature this modelling technique has been referred to as the Finite Segment approach [19] and has been successfully used for very flexible aircraft [7, 9]. Since the multibody formulation allows arbitrarily large rigid body motion, each wing section can undergo large displacements and rotations, and the ensuing internal forces are determined based on this displacement field. Each multibody beam element connects two consecutive rigid bodies and has a stiffness matrix derived from FE linear 6-degree-of-freedom (DOF) beam theory and the usual cross-sectional properties (, , and ) are assigned to it. The relative forces and moments exchanged between two connected bodies are calculated aswhere and are the relative displacements and velocities and are the linear stiffness and damping matrices. The stiffness matrix is a 6 × 6 symmetric matrix given byand the damping is taken as being proportional to the diagonal of the stiffness matrix by a damping factor ; that is, .

The aerodynamic model is based on quasi-steady strip theory. Though more simplistic than higher-fidelity methods, this approach is suitable and still accurate for high aspect ratio wings. Besides, the assumption of quasi-steady aerodynamics is deemed acceptable because the first natural frequencies of a flexible high aspect ratio wing aircraft are generally low (refer to Table 2 for the natural frequencies of the test case considered in this work) and, considering the speeds forming the typical flight envelope of a commercial transport aircraft, the resulting reduced frequencies are also low.

To further support this choice, strip theory can be straightforwardly integrated with the wing Finite Segment representation because no interpolation process is required between the aerodynamic and structural meshes, and the aerodynamic forces and moment are in fact applied at the aerodynamic centre of each rigid body, which represents a strip.

The aerodynamic forces on each strip are given bywhere represents drag or lift and the aerodynamic pitching moment byUsing , , and to indicate the relative airflow velocities in body axes for each strip, the local angle of attack is calculated asand includes all the contributions due to the aircraft states (aircraft angle of attack, sideslip, and angular rates) and to the elastic deformation of each section.

The quasi-steady aerodynamics stems from two contributions: the first being the inclusion in the sectional of the kinematic boundary conditions due to the heave and pitch motion of the wing section and the second being the terms proportional to the angle of attack time derivative. As pointed out by Dowell [20], there is ambiguity in the definition of quasi-steady approximation; in this work, it is assumed that the quasi-steady approximation is an expansion in reduced frequency of the unsteady aerodynamics for sinusoidal motion truncated to the first power of frequency, which in time domain corresponds to the first-time derivative, represented by the term proportional to .

In order to compare the results of the multibody nonlinear approach to the linear FEM, which employs linear DLM aerodynamics, limiting the sources of discrepancies between the methodologies, equivalent strip theory coefficients are derived from the DLM aerodynamic matrix.

In the light of the quasi-steady approximation, an expansion, truncated to the first derivative, of the DLM unsteady aerodynamic matrix about zero reduced frequency is performed, such thatwhere is the complex reduced frequency and indicates the complex matrix, tabulated versus a set of reduced frequencies, relating aerodynamic forces on aerodynamic panels to a change in the local downwash, that is, local angle of attack. The expansion of about delivers real matrices which, in the time domain, relates the aerodynamic force on each panel, acting along the panel normal as per the DLM assumption, to the local angle of attack, the matrix , and to the time derivative of the local angle of attack, the matrix . This latter term is computed using finite differences aswhere is a value of reduced frequency sufficiently close to zero. In deriving (12), the following properties of have been used:(i)The matrix is assumed to be analytic and, as a result, satisfies the Cauchy-Riemann equations so that .(ii)The real part of is an even function of so that .(iii)The imaginary part of is an odd function of so that .

Equivalent sectional lift coefficients and are derived from this expansion by summing the matrix terms corresponding to a strip of chordwise panels along the wing span.

By computing the coefficients from a 3D aerodynamic method such as DLM it is possible to correct strip theory for the sweep angle and tip loss effects. In addition to the lift coefficients, a constant drag coefficient is assigned to each strip, representing the airfoil viscous drag.

#### 3. High Aspect Ratio Wing Aircraft Model

The multibody framework presented in this paper has been applied to a high aspect ratio wing aircraft representative of a future concept of a narrow-body commercial transport aircraft, similar to the Boeing Sugar Volt configuration [2] and depicted in Figure 1. It features a high wing with moderate sweep angle, two wing-mounted engines, and a conventional aluminium construction. The main geometric and inertial properties of the test case aircraft are given in Table 1.