Estimation of the divergence characteristics of a jet transport aircraft wing using numerical method

Aeroelasticity Wing divergence speed Aerodynamics Strip theory Torsional influence coefficient Numerical Method Matrix iteration The goal of the present paper work is to demonstrate the use of numerical matrix iteration technique to obtain the divergence speed of a jet transport aircraft wing by employing aerodynamic strip theory. Aerodynamics strip theory is employed to obtain the divergence speeds for a finite (Three Dimensional) wing and for an infinite (Two dimensional) wing by matrix iteration technique. The aircraft wing is divided in to a number of Multhopp’s stations. The elastic property of the wing of a typical jet transport is considered for this analysis. Assuming a straight elastic axis, the matrix of torsional influence coefficients associated with Multhopp’s stations has been computed. A MATLAB code is used to iterate the matrix to arrive at the required convergent approximate solution. The solution converges about after ten iterations of the matrix. It is observed that torsional divergence speed estimated on the basis of strip theory without finite span correction is about 18% lower than the divergence speed estimated on the basis of strip theory with finite span correction. Two-dimensional torsional divergence analysis based on strip theory yields conservative torsional divergence speed. A tentative increase of 20 % in torsional stiffness resulted in about 15.5 percent increase in torsional divergence speed of a three-dimensional wing. This shows that divergence speed of a wing is directly proportional to the square root of torsional stiffness. This corroborates the result obtained for a two-dimensional wing. The result of the findings will be mandatory to high performance modern airplane designers for aeroelastic analysis. 10 11

The goal of the present paper work is to demonstrate the use of numerical matrix iteration technique to obtain the divergence speed of a jet transport aircraft wing by employing aerodynamic strip theory. Aerodynamics strip theory is employed to obtain the divergence speeds for a finite (Three Dimensional) wing and for an infinite (Two dimensional) wing by matrix iteration technique. The aircraft wing is divided in to a number of Multhopp's stations. The elastic property of the wing of a typical jet transport is considered for this analysis. Assuming a straight elastic axis, the matrix of torsional influence coefficients associated with Multhopp's stations has been computed. A MATLAB code is used to iterate the matrix to arrive at the required convergent approximate solution. The solution converges about after ten iterations of the matrix. It is observed that torsional divergence speed estimated on the basis of strip theory without finite span correction is about 18% lower than the divergence speed estimated on the basis of strip theory with finite span correction. Two-dimensional torsional divergence analysis based on strip theory yields conservative torsional divergence speed. A tentative increase of 20 % in torsional stiffness resulted in about 15.5 percent increase in torsional divergence speed of a three-dimensional wing. This shows that divergence speed of a wing is directly proportional to the square root of torsional stiffness. This corroborates the result obtained for a two-dimensional wing. The result of the findings will be mandatory to high performance modern airplane designers for aeroelastic analysis.

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Aero-elasticity is a multidisciplinary science which studies the mutual interaction among aerodynamic, inertia, 21 and elastic forces and the influence of these interaction upon airplane design. No aircraft structure is completely rigid, 22 so when it is subjected to aerodynamic forces it will normally deflect by a small amount [1]. This effect can become 23 very important at high speeds because any change in the shape of the body can cause the applied aerodynamic forces    center of twist and aerodynamic center. Raising the divergence of a wing by increasing torsional stiffness is a costly 45 process at the expense of considerable weight. An approach more frequently employed by designers is to proportion 3 the wing structurally so as to move the center of twist forward and thus reduce the aerodynamic eccentricity. For example, a straight wing, which carries its torsional load by a D-box has a forward center of twist location and 48 subsequently a high divergence [4,14].

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The main aim of this research article is to demonstrate the use of numerical matrix iteration technique to 50 characterize the divergence speed of a jet transport aircraft wing by employing aerodynamic strip theory.

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Phenomena of wing divergence is a primary interest to the airplane structural designer. Divergence speed 52 of sweptback wings is inherently high. However, divergence speed of swept forward wings is so low that for this 53 reason alone, swept forward is practically ruled out as a design feature [15]. A plethora of research paper has been

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The findings showed that positive fiber angles produce divergence-free wings, but the flutter speeds were small 96 relative to negative fiber angle wings, which resulted a challenge to achieve composite tailoring that at the same time 97 realizes high-flutter and high-divergence boundaries. Poisson's integral equation was also involved in the static 98 stability of a thin plate [25]. In the research report of this paper, the point at which the static instability of a plate wing 99 that could be described by partial differential equations (PDE) are considered to be the divergence speeds which

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It is assumed that the wing of the transport aircraft under consideration is perfectly elastic. That is when the 142 external forces are removed, the wing structure resumes its original form. Experiments on aircraft structure have 143 shown that within certain limits, the force and the deflection are linearly related. In the skin of aircraft wing structure 144 elastic buckling may produce a discontinuity in the force deflection diagram even though the material that make up 145 the wing structure are stressed at relatively low level. Therefore, the elastic behavior of the wing structure is defined 146 in the range below the point of elastic buckling. individual forces and moments. This is stated by the principle of superposition, which is the base for the analysis of 152 linear system. In Fig 1, the symbol Q is assigned to the arbitrary force or moment called "generalized forces".

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Similarly, the symbol q is assigned to the linear or angular displacement of the point of application of each generalized 7 independent displacement of the system. Hence, they must not violate the geometric constraints imposed up on the 156 system. Such a constraint implies that wing deflection at the point of attachment to the fuselage must be zero.

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By the principle of superposition, the displacement of the point of application of the i th generalized force due to n 160 generalized forces is given by where the constants are called flexibility influence coefficients.

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In matrix notation, the above equation can be expressed as: (2) 165 In short matrix notation, it can be expressed as: Influence coefficients and their matrices have the important property of symmetry. This property is expressed by:

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If, during the application of twisting moments, the beam is free to warp, the strain energy is due entirely to shear stress 195 and given by: Application of Castigliano's theorem, to the equation (7) gives the angle of twist of the beam due to the given     The resulting matrix of torsional influence coefficient will be:

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2. Chord wise segments of the wing remains rigid; that is, camber bending is negligible.

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The differential equation of torsional aeroelastic equilibrium of a straight wing about its elastic axis is obtained by 288 relating the rate of twist to applied torque as discussed earlier equation (11) is given as follows: This can be rewritten as: where t(y)=applied torque per unit span 293 ( )=elastic twist distribution.

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Consider a slender straight wing subjected to aerodynamic and inertia forces as shown in Fig. 7.

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Combining equation (17) and (18), we have the following differential equation of equilibrium as follows:
The wing torsional deflection at any span wise location y due to torque t applied at span wise location is derived Introducing equation (18) in to equation (20), we obtain: We can regard the angle of attack as a superposition of a rigid angle and an elastic twist.

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The torsional divergence speed of a three-dimensional wing is determined from the lowest Eigen value of dynamic The homogenous form of equation (26)

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Where =the effective lift curve slope corrected for aspect ratio.

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The matrix form of equation (31)

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[4] where it is shown that for a particular four station configuration that we have adopted in this work has the form of

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The eccentricity e, the distance between the elastic axis and aerodynamic center, and is defined by e=0.35C-

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The divergence speed of a finite wingspan, without finite span correction has been determined. In order to be practical

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As we can see in the above equation the aspect ratio changes with span of the wing. So, when the finite wing is 471 considered, the effective lift coefficient curve slope will be changes.

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Now let's compute the aspect ratio: = ̅ . Since chord varies along the span, we need to calculate ̅ . But ̅ =

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For the present case the wing plan form looks the following.

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It is seen that divergence speed is about 18.28% higher when the finite span correction is applied. This means that the 529 method of analysis using strip theory without finite span correction is conservative since it yields divergence speed 530 which is less than that of a three-dimensional wing.

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To study the effect of torsional stiffness of the wig up on the divergence speed, a 20% increase in torsional stiffness 532 is tentatively considered and the divergence speed is estimated using strip theory with finite span correction. It is found 533 that divergence speed is increased by about 18.30% when the torsional stiffness is increased by 20%. This may be 534 attributed to the fact that divergence speed of a wing is directly proportional to the torsional stiffness of a wing.

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The divergence speed of the wing has been estimated using strip theory with and without finite span corrections. The 544 effect of torsional stiffness of the wing upon torsional divergence speed has also been studied.

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It is observed that torsional divergence speed is estimated on the basis of strip theory without finite span correction