Shock and Vibration

Volume 2016, Article ID 5090719, 17 pages

http://dx.doi.org/10.1155/2016/5090719

## Geometrically Nonlinear Aeroelastic Stability Analysis and Wind Tunnel Test Validation of a Very Flexible Wing

^{1}School of Aeronautics Science and Engineering, Beihang University, Beijing, China^{2}Department of Aerospace Engineering, University of Bristol, Bristol, UK

Received 21 December 2015; Revised 22 March 2016; Accepted 3 April 2016

Academic Editor: Samuel da Silva

Copyright © 2016 Changchuan Xie 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

VFAs (very flexible aircraft) have begun to attract significant attention because of their good flight performances and significant application potentials; however, they also bring some challenges to researchers due to their unusual lightweight designs and large elastic deformations. A framework for the geometrically nonlinear aeroelastic stability analysis of very flexible wings is constructed in this paper to illustrate the unique aeroelastic characteristics and convenient use of these designs in engineering analysis. The nonlinear aeroelastic analysis model includes the geometrically nonlinear structure finite elements and steady and unsteady nonplanar aerodynamic computations (i.e., the nonplanar vortex lattice method and nonplanar doublet-lattice method). Fully nonlinear methods are used to analyse static aeroelastic features, and linearized structural dynamic equations are established at the structural nonlinear equilibrium state to estimate the stability of the system through the quasimode of the stressed and deformed structure. The exact flutter boundary is searched via an iterative procedure. A wind tunnel test is conducted to validate this theoretical analysis framework, and reasonable agreement is obtained. Both the analysis and test results indicate that the geometric nonlinearity of very flexible wings presents significantly different aeroelastic characteristics under different load cases with large deformations.

#### 1. Introduction

Large-aspect-ratio wings produce a high lift-drag ratio and good flight performance for high-altitude long-endurance unmanned aerial vehicles (HALE UAVs). Advanced composite materials make wing structures lightweight but also introduce large elastic deformations during aerodynamic loads in flight, which induces significant geometric nonlinearity in aeroelasticity and flight dynamics. In particular, after the NASA Helios mishap, the shortcomings of traditional linear analysis were deemed unsuitable for VFAs with large deflections, and the following recommendation was made by the investigators of the accident [1]:

[There is a need to] develop more advanced, multidisciplinary (structures, aeroelastic, aerodynamics, atmospheric, materials, propulsion, controls, etc.) time domain analysis methods appropriate to highly flexible, morphing vehicles.

Structural geometric nonlinearity, a key feature of HALE UAVs, yield a nonlinear relationship between displacement and strain due to large elastic deformations. Because the structural stiffness and equilibrium equations all depend on instantaneous structural deflections and load condition, the aerodynamic calculations and dynamic equations should be established for a large deformed state [2]. Thus, geometrically nonlinear aeroelasticity could be defined as a subdiscipline of aeroelasticity that considers nonlinear large structural deformations and the aerodynamics on curved aerosurfaces simultaneously.

In traditional linear analysis, aircraft were not particularly flexible, and the geometric nonlinearity were not significant; thus, linear structural finite elements based on the small displacement assumption combined with the planar doublet-lattice method [3] were widely used in engineering analyses and even imbedded in commercial software [4]. As aircraft have become more flexible, researchers have found linear aeroelastic analysis methods to be inaccurate and many new analysis methods for VFAs have been brought up. Hodges developed a nonlinear geometric exact beam element [5], reducing the order of structural nonlinearity, to analyse the nonlinear aeroelastic response of VFAs. He reported that structural geometric nonlinearity had a significant effect on the structural dynamics and dynamic aeroelastic characteristics of a high aspect ratio wing. The geometrically exact calculation of the angle of attack and aerodynamically consistent application of the air loads was also important for accurate aeroelastic characterisation [6]. Combined with Peter’s 2D inflow theory [7], an aeroelastic analysis toolbox named NATASHA was developed to analyse the nonlinear behaviour of VFAs. The nonlinear beam model and reduced order model (ROM) combined with 2D or quasi-3D aerodynamic theory help researchers to predict VFA performance [8–10] but prevent wide application in industry because a real structure is overly complex (i.e., it cannot be simply represented with a beam model); thus, it is necessary to find a 3D aerodynamic code to manage VFA aerodynamic computations. To obtain accurate aerodynamic loads, some researchers used CFD/CSD [11] methods to analyse the geometric nonlinearity of VFAs. Smith et al. [12] loosely coupled the Euler solver with geometric exact beam structural analysis and investigated the effects of adding aerodynamic nonlinearity to the elastic behaviour of high aspect ratio wings. For VFA stability analysis, considering the high computing costs of time domain aerodynamic computations and the importance of highlighting basic aeroelastic principles for unconventional wings with high aspect ratios [13], dynamic flexible motion of the system can be assumed with small amplitudes around a nonlinear static equilibrium state [14]; thus the linearized method and frequency domain solution are still a valuable approach in preliminary designs and even in the detailed design stage. Panel aerodynamic methods have been well understood and widely used in engineering design; extending the panel aerodynamic code into 3-Dimentional application can make geometrically nonlinear aeroelastic analysis easy to accept in industrial applications.

Based on the above discussion, in this paper, a practicable geometrically nonlinear analysis system is established and a wind tunnel test is conducted to validate the analysis results. Nonlinear finite element method (FEM) is utilized so that the method could be easily used in industry. Actually there are some refined modelling methods like intrinsic beam [5], strain-based beam and plate elements [15], nonlinear substructure, ROM, and so forth. These methods are well developed theoretically and can illustrate the flexible structural characteristics to different degrees. Considering the practical problems we have met, the structures are often so complex that these modelling methods are not convenient to apply. Additionally, these simplified methods cannot well reflect the original structure especially in detailed parts. Moreover, FEM is the most often used modelling method in practical analysis and it will be very efficient and convenient if the FEM model can be directly used to analyse the VFAs structural geometric nonlinearity. The purpose of this paper is to establish an analysis framework easy to implement and able to reveal some nonlinear aeroelasticity of flexible wings. Based on the concerns above, the nonlinear FEM is responsible for structural nonlinear analysis. As to the aerodynamic computation, the importance of the nonplanar aerosurface effect and exact aerodynamic modelling consistent with structural deflection comes from our experience and many reference papers. Structural nonlinearity and nonplanar aerodynamic compotation must be considered simultaneously. Considering the easy programming and good inheritance of conventionally used linear method, NVLM (nonplanar vortex lattice method) and NDLM (nonplanar doublet-lattice method) are adopted and they can well present the nonplanar effect of aerodynamic for flexible wings in stability analysis. All of these methods together can describe the geometric nonlinearity of both the aerodynamics and structure of VFAs and can be conveniently used in industrial design. Wind tunnel test under different load cases indicates that different deformations under varying flight states result in different flutter speeds that may alter the aircraft’s flight envelope. Reasonable agreement between the analysis results and test results has been obtained, and all the results demonstrate that the static aeroelastic response and flutter characteristics are quite different from the results obtained from linear analysis.

#### 2. Theory

##### 2.1. Structure Geometric Nonlinearity

Large structural excursions induced by very flexible wings when undergoing aerodynamic loads in the air prevent the use of linear methods based on the small displacement assumption and call for geometrically nonlinear structural analysis. Geometric nonlinearity are based on the kinematic description of the body, and the strain on the wing should be defined in terms of local displacement of the wing for dynamic motion. These result in the nonlinear geometric equations including the quadric term of the displacement differential and require the nonlinear force equilibrium equation established on the deformed state of the structure. Structural geometrically nonlinear problems are often solved by the maturely developed nonlinear incremental finite element method [16] and two formulae called the total Lagrange formulation (TLF) and the updated Lagrange formulation (ULF) [17], which are well known. The ULF is used in this study, and the primary equations are presented briefly below. The core method of structural analysis has already imbedded in commercial software, so it is convenient to be used in engineering analysis.

The relationship between the nonlinear Lagrange/Green strain and displacement iswhere is the partial derivative of displacement component to the coordinate at time .

Despite a large elastic deformation, the material is still within the elastic limitation for a small strain, so the conjugate Kirchhoff stress tensor at time* t* satisfieswhere is the direction cosine of a small area element at time and is the corresponding surface force in which the follower force effect is considered. The linear elastic constitutive relation is given as follows:where is the elastic tensor, which has a different form for isotropic or anisotropic material.

The finite element method (FEM) based on energy principles is an effective approach to solve structural problems. For geometric nonlinear problems, considering the follower force effect, the incremental FEM is used. The strain can be decomposed into a linear part and a nonlinear part of the current displacement:The stress can also be decomposed in increments, where represents the equilibrium stress at time and represents the incremental stress to be calculated at each time step:The integral equation is established by linearization in each incremental step:where is the incremental outer force including the aerodynamic force, engine thrust, and gravity, at the new time step. Considering a number of shape functions, the relationship between strain and deformation is presented asSubstituting these shape functions into (6) leads to the element governing equation [18]: where is the incremental outer force including the aerodynamic force, engine thrust, and gravity at the new time step. The stiffness matrix in (8) could be decomposed into a linear part and nonlinear part . The linear part is only related to the structure itself, whereas the nonlinear part is related to the deflected configuration and strain quality, which should be updated in each computation step.

The corresponding dynamic equation can be expressed aswhere is the structural displacement acceleration vector at new time step. The assumption of a small amplitude vibration around the static equilibrium state is suitable for many dynamic problems, including the flexible wing structural dynamic stability:where is the large deflecting equilibrium deformation from (8) and is a small vibration deformation. According to (9) and the static equilibrium condition, the linearized structural quasimode can be obtained by generalized diagonalization, and the vibration equation of the system under steady forces reduces towhere is the inertial matrix of the structure at the nonlinear static equilibrium configuration and is the corresponding stiffness matrix. Both of these parameters are nonlinear functions of and vary under different equilibrium states, which is a key feature of geometric nonlinear structures. The mode shapes and frequencies can be deduced from (11).

Introducing the harmonic oscillating assumption , the vibration equation can be written as where is the vibration circular frequency and is the vibration mode matrix. If (12) has all-nonzero solutions, that demands That is a generalized eigenvalue problem about and . Solving (13), the vibration circular frequency should be obtained. Substituting into (12), structural eigenvector (structural mode shape) can be obtained.

##### 2.2. Nonplanar Vortex Lattice Method

The modelling of flexible aircraft with significant structural deformations requires the incorporation of structural dynamics and aerodynamics in a unified framework. The aerodynamic loads in (9) should be computed on a deformed configuration. This section summarizes the primary characteristics of the NVLM and its application to a curved aerosurface. The NVLM is derived from three-dimensional potential flow theory and is suitable for most normal situations that VAFs may encounter. Because the simple programming effort is required, NVLM is easy to combine with structural dynamic computations to obtain the response results for aeroelastic structures. Additionally, the exact boundary condition is satisfied on the real wing surface in the NVLM, which can have camber and various platform shapes. Thus, the NVLM is convenient for use with very flexible wings, whose aerodynamic surfaces are subjected to large structural deformations [19]. The NVLM is implemented using vortex ring quadrilateral elements to discretize the curved lifting surface along with the wing’s deformation, as shown in Figure 1.