'-'<f'

Journal ofScience & Technology 100 (2014) 020-025

Problem of Elastic Deformation for Aircraft Wings with the Variation of Velocity and Incidence Angle

Hoang ThiBich Ngoc^*, Dinh Van Phong', Nguyen JManh Hung', Nguyen Hong Son''^ '

'Hanoi Umversity ofScience and Technology, No. I, Dai Co Viel Str.. Hai Ba Trung, Ha Noi, Viet Nam

-Hanoi University of Industry I Received: Februaiy 28, 2014; accepted: April 22. 2014

Abstract | The structure of aircraft wing is a closed tube affected by aerodynamic forces. Aerodynamic forces

distnbuted on the upper side and lower side of wing are determined by a 3D numerical method which are external forces and considered as known input parameters. The problem of elastic deformation for the wing is solved with a 3D degenerate model, discretized and approximated by finite element method. The built program allows calculating and determining the location and value of the most dangerous stress on the wings which depends on the value and direction of the free flow velocity with a certain structure of wing, from that it allows determining the limit of elastic deformation for wings depending on the velocity anil the incidence angle of the free flow.

Keywords; 3D wing, Numerical methods, Aerodynamic forces, Elastic deformahon.

1. Introduction

The structure of aircraft wing has the shape of aerodynamic profile followed the velocity direction and it is a closed tube, so the shell thickness is very small compared lo the rest of the field size. This is the basis for the choice of degenerate 3D models in structural calculations. Aerodynamic load on the wing is very large, larger than the weight of the entire aircraft. Aerodynamic load distnbution is caused by the pressure difference between the upper side and the lower side of wing. With a defined structure, distributed load varies due to the velocity and the direction of motion (angle of incidence). If aerodynamic problem allows us to identify three- dimensional load distribution on the wing, the elastic deformation problem will facilitate the development of the model on the three-dimensional structure. In the limited framework of this article, only the method solving the elastic deformation of the wing is introduced. Details of the method determining the 3D aerodynamic forces are refened to [1]. Results of aerodynamic pressure distribution on the wing are extemal force given to solve elastic deformation.

Results of elastic deformation problems are worked on analysis of aspects related to limitation of elastic deformation of the wing,

2. Method of Calculation 2.1. Calculation of aerodynamic forces

The determination of the forces on the 3D aerodynamic profile wing considering the thickness

of wing and length of the finite wing span needed to be done by a numerical method to determine the pressure distribution on the upper side and the lower side of each specific field. Hereby, calculations are carried by the combined doublet and source singularity distiibution method (Fig, 1),

Fig. 1. For determining aerodynamic forces on 3D wing by using doublet - source singularities

Velocity potentials at a point P(x,y,z) induced from sources of constant strength O and doublets of constant strength |J ananged on the discrete elements of the upper side and the lower side are as followed;

(Ps{x,y,z)^-—\ dS

4.T J

{^(x-x,f+(y-y^y+z' zdS '\^{x-x,)-+{y-y^)'+z'T

(1)

The velocity components (u,v,w) at a pomt are determined from the derivatives of velocity f

• Corresponding author: Tel: (+S4) 912.313 350

Email: ngoc.hoangthibich@hust.edu.vn

'"'-'<f'|-f)

Journal ofScience & Technology 100 (2014) 020-025

[ du dv dvj 1

•,,,£•-)= — , — , — \;y„,

\^dx dy dz ) 5w 5u _ dw dv dx dz ' ^~ dy dz

du dv dy dx From the velocity calculated by (2), the lift and the pressure distribution on the wing can be easily identified. The algorithm and compuler programs of 3D aerodynamic calculation on 3D wing by the method of combined doublet and source singularity distnbution are verified wilh high accuracy is presented in [ I ] .

2.2. Calculation of structure by a degenerate 3D model using finite element method

2.2.1. Total potential energy

Balanced differential equations of the solid determine the relationship between stress and extemal force. Based on the relationship between stiess and strain (r = D£ {with o is the stress vector, E is the strain vector (£^,£-^,6'j arc normal stiains, /„,,/,„,/:,, are shear strains), D is a matnx of matenal characteristic} and the relationship between displacement u - [ u v wl and strain E :

( • ^ V ' ^ l

and apply the principle of minimum total potential energy to solve the problem of elastic deformation The total potential energy fl has form [2]:

n = | j C T ^ £ d F - J u ^ f , . d K - j u X d S - ^ u , ^ p , (3) where p, is point force al a node i with displacement u,; fv IS the vector of volume force; fs is the vector of area force; V and S is the volume and the cross- sectional area of the object,

2 2 2. Degenerate 3D models using finile element methods

Shell thickness is much smaller m size compared to the other dimensions of the wing, so it can be reviewed under degenerate 3D models. At each considered node, 6 degrees of freedom degenerate to 5 degrees of freedom. Wings are discretized into small panels with tetrahedral sides and approximated by the finite element method [3].

At a node k of the element c, displacement vector q'' is defmed as follows:

q'=[», „, a, e, #,]* (4)

where UpW^,!/, are length displacements and ^,,^2 are angular displacements Two -dimensional shape function (^,?7) and one-dimensional shape function

(^) are used to define the coordinate system (t, q, ^) that describes parameters in each point.

Fig. 2. Number of degrees of freedom in degenerate 3D model

Two-dimensional shape functions N"" are:

N,=i(l+5 )(l+ti); N,=i(l-i; )(1+TI)

One-dimensional shape function H*^ is;

H'=i[(i + 0 ( l - 0 - ( l - 0 ( l + 0]||(x,')'-(x.'')'|

where ^ indicates the position of the reference of surface and it values from -1 to 1; ^ = 0 denotes the mid-surface; index t and b denote the upper and the lower of the shell. Positions at k are represented by using shape functions-

+ | ; A f ' K , ; 7 ) « * ( f ) l ' „ ' ; i = l,2,3

where K,/ is the unit vector:

(5)

V,:

Displacements at k are;

",(^.'7.f) = X'V'(4^,'7)",' + (6)

where u^ is the displacement along the axis x^, w/

is the displacement at k; Q, and Oj ^^e rotation angles.

The relationship between displacements and defoi-mations are £ = Bq , with B is the matrix which transforms the degrees of fireedom via the shape

Journal of Science & Technology 100 (2014) 020-025 fiinction. Considenng the discrete element e, the

potential energy of deformation of element is V =-\a^£ dV (Vu is the volume of element).

2 /

Defining the stiffness matrix of element K = [ B ' ' D B dV . Then U = - q '"K q and the total potential energy for the elements is;

n = - q '^K Q . - q / T (7) where fe is the node force vector of element. Apply the principle of minimum total potential energy dn/dg — 0 and perform the tiansplantation of matrix for matnces Ke and fc in order to detenmne the displacement q and thereby determine the stress a.

Stress in each element is determined by the standard of Von Mises-

te

<^,JH<', J„)'t(t^, 0-„)' + 6(CT^+cr 1.) .

H

(8) 3. Applications

3.1. Stress distribution on the wing with different velocities

The velocity direction is determined by the incidence angle a. The value of the velocity is expressed through a dimensionless quantity being Mach number.

Fig. 3 presents results of dimensionless pressure on the wing with span b/c = 5, profile Naca 0009, incidence angle a = 4", free Mach number M^ = 0.5. Pressure coefficient is determined by dimensionless pressure difference:

Cpon lower surface

Cp on upper surface ROOT Fig. 3. Aerodynamic pressiue coefficient on wing b/c=5, N0009, a=4°, M«=0.5

where p^ is the pressure of firee flow, p is the density of fluid, p is the local pressure on the wing;

Pressure distribution on the lower -wing surfaces' is larger than on upper wing surface that creates the lifl of wing.

In the elastic deformation problem, the wing is discretized into elements -with dimensions being small enough e. The pressure p multiplied by surface area (shell) As of element creates force exerted on each element that is:

f = J9„.As,j (10)

Grids of the aerodynamic problem and the elastic problem are not always similar, so it is necessary to use an interpolation for determining the value of node force.

Structural calculations are made with a closed hollow wing with shell thickness t = 0.0 Im.

Duralumin material has elastic modulus E = 7 , 3 I e I 0 N / m and allowable stress [ a ] ^ 8e7 N / m ^ , Poisson ratio u = 0 . 3 3 . Results presented m Fig. 4 are stress distributions on the upper half shell and the lower half shell of wing under the infiuence of aerodynamic forces shown in Fig. 3. Along the wingspan, stiess gets maximum in the root section of wing (clamped position).

To better quantify, consider 2D graph of Von- Mises stress distiibution on the most dangerous section of the wing root, and results are shown in Fig.

5. As can be seen, for M „ = 0 . 5 , the maximal stiesses of both upper and lower sides have exceeded the allowable limit.

Also with the same structure, a reduction of the fi-ee fiow velocity is carried, M^ = 0 . 4 . Results of Von-Mises stiess distribution on the wing root

x1D^N/m2

Fig. 4. Von-Mises stress on the -wing, M»=0.5

Journal ofScience & Technology 100 (2014) 020-025

Fig. S. Von-Mises stress on root section, Mic=0.5

Fig. 7. Von-Mises sfress on root section, Ma.=0.3 Critical stress

Chord (m)

Fig. 9. Von-Mises stiess on root section, a^°

section are shown in Fig, 6. It is observed that the maximal stresses on both upper and lower sides of root section are less than allowable stress.

With smaller value of free flow velocity M„ = 0.3, results of stress distribution on the wing root section are shown in Fig. 7. The maximal stiess is much smaller than for the case of M^ = 0 . 4 . Positions of the maximum value of stiess on the upper and the lower sides along the chord direction are different, but it is slightly different.

3.2. Stress distribution on the wing with different incidence angles of velocity

In order to consider the influence of velocity direction, cases of different incidence angles (7 = 2°,4°,8° are calculated, and other parameters as

Chord (m)

Fig. 6. Von-Mises stress on root section, Mffi=0.4 6 " "

t" ry 1 3 If

S 2 f S 1,

y ' * 4 - . ^ ^ a —^ Lower

• Root section \ j

~—-—

Chord (m)

Fig. 8, Von-Mises stress on root section, a=2''

Fig. 10. Von-Mises stiess on the wing, a ^ S "

free flow velocity and wing structure are similar. For these cases, the cross section of wing is a kind of asyinmetnc profile Naca 2409, and free Mach number M ^ - 0 . 4 . Results in Fig. 8 are the distiibution of Von-Mises stresses on the wmg root section with « = 2 ° .

The maximal value of stiess on the upper side and the lower side is smaller than allowable stress.

Stress graph in Fig. 8 is different from ones in Fig, 5, 6, 7 that the difference of the maximal values of stress on the upper and the lower sides are larger, and the location of the maximum stress point foflowed the motion (along the wing chord) is much more different. This is caused by asymmetiic wing profile.

When the incidence angle of the free flow a—4°, the maximal stress on the wing strongly

Journal ofScience & Technology 100 (2014) 020-025

025

^ 0 0 5

'

Trailing edge

Leading edge

1 2 3

• • Upper

— Lower

4 5 Span(m)

Fig. I I . Displacement of wmg, a==8°

increased, as shown in graph results in Fig. 9. The maximal stress on the upper side reaches the allowable stiess.

If the incidence angle increases with the value a = 8 ° , the maximal stress on the wings gets nearly 150% allowable stress. Results shown m Fig.lO are Von-Mises stress distnbution on the upper and the lower surfaces of wing. The maximal stress on the upper surface is 13.561 NI m^ and on the lower surface is 11.5e7 N / m'.

Displacements of wing m the case of the incidence angle a = 8" are presented in Fig. 11.

Displacement graphs show that the wing bends strongly and twists negligibly (displacements of the leading edge and tiailing edge are not much different). For all six cases considered above, twisted deformation of wing are not significant, so this report does not realize the recalculation of aerodynamic forces when there is no great change for local incidence angles,

3.3. Comparison and verification

Computational program for calculating elastic deformation problem of the wing with a degenerate 3D model is verified by comparisons of present numerical results with published results.

In this report we present a case for the comparison that is a cylindrical shell affected by a force F^lOO lb at the lengthwise center Parameters of cylinder- radius R=5 in, shell thickness t=0.094 in, length L=I0.35 in, material of cylinder shell having elastic modulus E=I0.5e6 psi, Poisson's ratio u=0.3I25. With the cyhndrical model described above, Brogan [4] and Kwon [3] calculated displacement values and determined that the position of maximal displacement and the point of affected force are overlapped. Using the built code with degenerate 3D model to calculate the case of cyhndrical shell, we receive results similar to results of Brogan and Kwon with differences letter than 6%

( T a b l e l ) .

Fig. 12. Cylindrical shell model Table 1. Displacement at the point of affected force

4. Conclusions

In terms of static strength, elastic deformation of the wing under the infiuence of aerodynamic load needs to be addressed. Firstly, the twist elastic deformation which increases local incidence angles' and changes aerodynamic forces that should be recalculated. This problem will be presented in other reports. Secondly, making survey of stress distribution at dangerous positions should be carried, and this is exactly the report done. It is possible to draw some comments as follows.

- With a certain structure, the stress field is influenced very strongly not only by the velocity value but also the velocity direction. The incidence angle for the wing is normally less than 5 degrees, larger incidence angles are used only in the special conditions The selection of material related to flight speed factor is very important and necessary.

The sensitivity for the behavior of stmcture is caused by the specific characteristics of the aerodynanuc load. Aerodynamic load is a distribution force which changes the mode and the strength depended on the shape of wing, the velocity and direction of free flow. Thus, in order to solve the problem of wing structure, a good understanding of wing aerodynamics is much needed. It is known that, two important issues of the wing being aerodynamics and structure are two very different fields of applied mechanics.

The built program for calculating the elastic deformation problem with a degenerate 3D model allows extending applications for other studies with structural solutions to improve the stif&iess of wing when the aerodynamic li

Journal ofScience & Technology 100 (2014) 0204)25 References

[I] Nguyen M, H., Hoang B. N., Nguyen H. S., Calculating aerodynamic characteristics of swept-back wings. Proceedings of The 14"' Asia Congress of Fluid Mechanics, Hanoi, (2013) 132-137.

[2] Megson T. H. G., Aircraft stmctures for engineering students, Butterworth-Heinemann Eds., Oxford, 2003

[3] Kwon Y, W, Bang H. C , The finite element method using Matiab, CRC Press, USA, 2000 [4] Brogan, Y. W., Modems Contiol Theory,

Publisher Prentice Hall, USA, 1990 [5] T.A. Weisshaar, Static and Classical Dynamic

Aeroelasticity, Volume 3, Structural Mechanics, Section 3.2.2, Encyclopedia of Aerospace Engineering, John Wiley, 2010.