This study investigates the utilize of the iteration of matrix numerically to get the divergence speed of an aircraft tapered-unswept wing via using method of strip theory (without finite span correction and with finite span correction). The 2-D fluid flow across the wing airfoil section is assumed by the aerodynamic strip theory and then integrated over wing span. In the present work, first the tapered unswept wing is divided into four Multhopp’s stations. Then, elastic properties of this wing in relation to torsional influence coefficients related with this four Multhopp’s stations have been evaluated. Second, equations for equilibrium are constructed as integral equations. Then, an appropriate aerodynamic theory in the form of strip theory was addressed, as well as the technique of solution for determining the divergence speed. Finally, using strip theory, the integral equation has been expressed in matrix form. Iteration of matrix has been done using MATLAB environment to reach for the solution to converge. Also, an increase of 15% in torsional stiffness of aircraft wing has been considered to illustrate its effects on torsional divergence speed of aircraft wing. The attained results are going to be helpful in understanding of wing instability for modern aircraft designer.

Aero-elasticity is the study of the interactions of inertia, elastic and aerodynamic forces in a flexible structure and the phenomenon that might occur as a result [

Divergence is a static aeroelastic phenomena that includes the interplay of aerodynamic and elastic forces. The most prevalent type of divergence is wing torsional divergence, which happens when the local angle of the wing grows to the point where structural collapse occurs [

Because the wing in consideration is tapered, the chord changes over the wing span as shown in

The wing root chord length, C_{r} = 5.588 m.

The wing tip chord length, C_{t} = 2.794 m.

The span at any span length, _{i} = L × cos ϕ.

The wing span,

All dimensions are in meter.

Aerodynamic center = 25% chord.

Elastic axis = 35% chord.

From relation:

The span at any span length, _{i} = L × cos ϕ

Then:

Point 4 at zero from wing root.

Point 3 at 4.8601 m from wing root.

Point 2 at 8.9803 m from wing root.

Point 1 at 11.7333 m from wing root.

It is assumed that the wing under investigation is perfectly elastic. Means that when the external loads are removed, the wing structure retains its original shape. Experiments on airplane structures demonstrated that, within certain limitations, force and deflection are linearly connected. Even if the material that makes up the wing structure is stressed at a low level, elastic buckling in the skin of an aircraft wing structure can cause a discontinuity in the force deflection diagram [

The influence coefficients conception is used throughout the study, which takes into account the wing structural deflections caused by various loads. I’ll use the

According to superposition concept, the deflection of point of application i^{th} generalized force caused by n generalized forces provided by:

where _{ij} referred to the coefficients of flexibility influence. The aforementioned equation can be written as follows in matrix notation:

It can be written as follows in short matrix notation:

Symmetry is an important feature of influence coefficients and associated matrices. This characteristic is expressed as:

Consider matrix [C]’s properties, which are relevant to

There are four categories of elements:

∁^{δδ}: Linear deformation at i as a result of unit force at j.

∁^{αδ}: Linear deflection at

∁^{αα}: Linear deformation at i caused by unit moment at j.

∁^{δα}: Linear deflection at

The reciprocal theorem of Betti has to be applied for the matrix [C] to be symmetric:

In order to apply the energy approach to an aeroelastic system, the following strain energy formulas in relation to influence coefficients can be constructed with reference to

Replacing

Consider the wing section depicted in

As shown in

If the beam is free to deform during the application of twisting moments, the strain energy is totally attributable to shear stress and provided as:

Twist angle of the beam caused by specified distributed applied torque T(y) is determined by using Castigliano’s theorem to

Assuming for the shear flow distribution caused by T = 1 is represented as (

When we differentiate

Consider the cantilever wing in ^{θθ} (

As previously stated:

Thus, for η >

If the distribution of shear flow

According to _{4} = 0, y_{3} = 4.8601, y_{2} = 8.9803, and y_{1} = 11.7333 meters from the root. Because four wing stations are assumed in this work, the obtained matrix of torsional coefficients is (4 × 4) square matrix, illustrated as follows:

These coefficients’ values have been calculated from

[

Divergence, a static instability phenomenon, is caused by the combination of elastic and aerodynamic forces. As a result, calculating the matrix of aerodynamic coefficients and the matrix of structural stiffness is required for the divergence speed analysis. The current study is concerned with the torsional divergence of tapered un-swept wing, which is the most prevalent problem in aero-elasticity.

Take the simple un-swept wing at incidence () with twist center behind the center of aerodynamic as an example. The pressure distribution with major loads situated towards the nose causes the wing to twist in the nose up direction. Due to the structure’s imperfect rigidity, it twists and changes in form in relation to the wing root section. As seen in

The following assumes are made for simplicity’s sake:

Un-swept wings are distinguished via an Elastic-axis that is vertical to the symmetry plane of aircraft.

The wing’s chordwise sections maintains rigid; camber bending is to be negligible.

By connecting the rate of twist to the applied torque as previously mentioned, the differential equation of torsional aero-elastic equilibrium of un-swept wing about its elastic axis is represented using

You might rewrite this as:

where (

Take a slender straight wing that is being affected by aerodynamic and inertial forces, as in

The utilized torque per unit span t(y) is provided in

where

When we combine

By using Castiglione’s theorem to the energy equation, the wing’s torsional deflection is calculated at any spanwise position y caused by torque t applied at span wise position η.

When

The angle of attack may be thought of as a superposition of an elastic twist and a rigid angle.

Additionally, local coefficient of lift may be expressed as follows:

The following integral equation is obtained by substituting

Similarly, we derive the following integral equation by replacing

This equation represents the required governing integral equation.

There are several approaches to modeling a wing’s spanwise lift distribution. This section will provide the strip theory for a discretized wing, which is the most basic. According to this approach, the wing is made up of a number of chordwise elements famed “strips,” and it is supposed that the lift coefficient on each of these pieces is proportional to the local angle of incidence α(y), and that the lift on one segment has no effect on other. In its most basic version, root and tip effects, as well as compressibility effects, are neglected. In reality, these presumptions predict a low air speed (M < 0.3) and a high aspect ratio (AR ≥ 6) for the wing. Take note, drag calculations cannot be performed using strip theory [

In

Θ is linear operator used to the lift distribution

According to the definition of strip theory, Θ is just
_{0} is the local 2-D slope of curve of lift coefficient. c is chord of the wing.

The dynamic pressure q_{d} with the least Eigen value, which is derived from equilibrium equation in its homogeneous-integral form, are used to calculate the torsional divergence speed of a 3-D wing [

To determine the divergence speed,

where

The governing

Wing chord at any span

If the previous numerical values are substituted, the following will result:

The following diagonal matrix is obtained by calculating the chord

The aerodynamic center (A.C.) of the wing is considered to be one-quarter of the chord taken from the leading edge (

The eccentricity, e = 0.35 C − 0.25 C = 0.1 C, which is the distance among the elastic axis and aerodynamic center as in

Therefore,

That results as diagonal matrix as follows:

Multhopp’s approximate quadrature is useful when dealing with functions derived from lifting line theory. utilizing the techniques and formula for a wing semi span

The following results are obtained by multiplying the respective matrix values based on

We derive the following simplified expression from

When we rearrange the previous expression, we obtain:

The matrix product [C] [E] now becomes:

By using MATLAB environment to iterate this matrix numerically, we obtained matrix [

For the strip theory modified for finite span to be useful and accurately forecast the divergence speed of a practical wing, namely, a finite wing, the following calculations must be conducted:

wing flat form aspect ratio:

As seen in the previous equation, the aspect ratio varies with wing span. As a result, the effective lift coefficient curve slope

Let’s now calculate the aspect ratio

Because the chord changes over the span, we must determine

Planform of a wing for the case study is as

We know that

Using

Because of symmetry

where

This means:

S

Therefore,

As a result, the strip theory corrected for finite span:

For the finite span correction, the divergence speed is:

Assume that the GJ is increased by 15% uniformly along the span of the wing. The matrix

The matrix product [C][E] now becomes:

By using MATLAB code, after nine iterations we obtained the result of the matrix [

Aerodynamic strip theory has been used throughout this research to determine the divergence speed of unswept-tapered wing (both with and without finite span correction). Divergence speed has been found to increase by approximately 18% when the finite span correction is used as
represented on

Method of analysis | Divergence speed, V_{D} (m/sec) |
---|---|

Strip theory (No finite span correction) | 472.8420 |

Strip theory (With finite span correction) | 557.9546 |

Strip theory (With increment of 15% of GJ and without finite span correction) | 507.0839 |

Strip theory (With increment of 15% of GJ and with finite span correction) | 598.3601 |

To investigate the influence of increasing up torsional stiffness on divergence speed, a 15% increase in torsional stiffness is tentatively examined, and the divergence speed is predicted via strip theory as well (without and with finite span correction). When the torsional stiffness is raised by 15%, the divergence speed increases by around 18% as illustrated on

To measure torsional deformation of wing under pure torsion, the torsional influence coefficient is utilized. Nevertheless, this matrix was created using the technique of matrix iteration to calculate the wing’s divergence speed. The unknowns in the governing integral equation are the elastic twist distribution and the lift coefficient, making the problem statically-indeterminate. Strip theory used to address this problem. The matrix technique is used to numerically solve the governing integral equation. The matrix has been iterated using MATLAB environment in order to converge on a solution. Strip theory has been used to calculate the wing’s divergence speed with and without finite span corrections. It has also been investigated how the wing’s torsional stiffness affects the speed of torsional divergence. The divergence speed predicted based on strip theory is found to be around 18 percent higher for (3-D wing) in comparison with (2-D wing). Based on strip theory, two-dimensional torsional divergence analysis offers conservative torsional divergence speed. A 15% increase in torsional stiffness caused around 18% improvement on divergence speed of a (3-D wing).

To sum up, this result demonstrates that the divergence speed of a wing is related to the square root of the torsional stiffness [

^{*}pi/8)^{*}pi/8)^{*}pi/8)];

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