Predicting structure properties at the initial design stage is especially important for products that will undergo vibration and impact when used. The paper presents a technique for computing dynamic loads on an aircraft rudder arm when it strikes an arrester as it unfolds in flight. We used a finite element software package to analyse the natural vibrations of the structure.

Прогнозирование характеристик конструкций на начальном этапе проектирования особенно важно для изделий, подверженных в процессе эксплуатации виброударным воздействиям. В статье предложена методика расчета динамических нагрузок консоли руля летательного аппарата при ударе об упор в процессе его раскрытия во время полета. Анализ собственных колебаний конструкции проведен посредством конечно-элементного программного комплекса.

Today, the growing arms market requires improvement of the existing weapons and systems and development of new technologies. Currently, special attention is paid to the creation of advanced aircraft (A/C), their performance depending, in particular, on correct operation of aircraft controls. The main characteristics affecting the aircraft flight performance include natural frequencies and eigenmodes, as well as shock strength of cantilevers.

The main aircraft controls include cantilevered flight control surfaces (Fig. 1), which are circumferentially mounted in two pairs in one cross-section of the aircraft body, equidistant from one another. In addition to aerodynamic loads, two dangerous situations may occur during operation of control surfaces, resulting in aircraft malfunction. The first situation is caused by control surface frequency resonance and the second one consists in possible structural collapse of the control surface when it hits a limit stop while unfolding and locking.

Dynamic tests are a traditional and rather efficient method determining amplitude-frequency response and shock loads acting on aircraft controls. However, at the stage of design and preparation works, theoretical computational methods are required to determine the system’s free vibration characteristics, as well as dynamic loads acting on the control surface in case of shock impact, and also to estimate the strength of control surface structural elements and control unit compartment.

The purpose of the study is to develop a numerical method for determining dynamic loads acting on the control surface when it hits a limit stop while unfolding and locking.

The analysis of structural behaviour includes two stages. The first stage consisted in computer-aided simulation of free vibrations the unfolded control surface is exposed to under standard aircraft flight conditions in order to determine natural frequencies and eigenmodes. For computations, we have applied a finite element analysis (FEA) software package. The second stage involved mathematical modelling of cantilevers of a control surface using a one-dimensional beam with variable cross-section exposed to shock loads. The method is based on solving a beam bending vibration equation in partial derivatives with regard to conversion of the moving control surface’s kinetic energy into the elastic strain energy generated in the control surface structure exposed to shock loads. The method has been implemented in the MathCAD engineering software package but thanks to its flexibility this method can be integrated into any software application. Both stages are interrelated on a software level: results of computer-aided simulation of control surface vibrations were automatically integrated into the analytical software in the MathCAD environment by creating an input file. Further, dynamic loads determined at the second stage are used for estimating the strength of the control surface and its attachments to the control unit compartment, as well as for analysing amplitude-phase and frequency response of the whole aircraft.

The conducted simulation described in this paper allowed to develop an integrated approach to the study of the aircraft control surfaces behaviour during unfolding.

Let us introduce Cartesian coordinates in such a way that the Ox axis is the longitudinal axis of the control surface cantilever coaxial with the control surface’s steering axis; the Oy axis is the control surface’s axis of rotation when unfolding and locking (axis of folding). The Oz axis is normal to the Oxy plane (see Fig. 1). Control surface span length is L = 0.4 m.

The control surface plate being the base of its hinged section is made of fibreglass based on quartz fibreglass cloth and the DT-10 binder [

Free structural vibrations have been investigated using computer-aided analysis methods in the FEA software package. Simulated boundary conditions corresponded to the unfolded state of the prototype: the control surface’s steering axis was fixed along the longitudinal Ox axis of the control surface, its rotation around the Oy axis was set free using a hinge, and the control surface cantilever rested upon the limit stop. The FE node grid was built using tetrahedral and hexahedral elements. The calculated natural vibration frequencies in terms of the 1st and the 2nd modes are ν1 = 101.85 Hz and ν2 = 333.54 Hz, respectively.

The table gives frequencies of the first four nontrivial modes of the control surface free vibrations. Fig. 3 shows the first and the second vibration modes corresponding to frequencies ν1 and ν2 in the form of deflections of the control surface cantilever normalized to the unity in the Oxz plane. Since a substantial part of the control surface cantilever weight is concentrated along its base (see Fig. 2), the vibration mode curve is flat, and an evident bending of the cantilever is observed only in the area near the free end. That is why a deflection between the bending points of the cantilever (shown in green) is not visible in Fig. 3, b.

Frequencies of the first four free vibration modes of control surface

We should note that the first, second, and fourth vibration modes of the cantilever with frequencies ν1, ν2 and ν4, respectively, describe structural vibrations relative to the bending axis Oy in the Oxz plane, while the third mode with frequency ν3 defines free vibrations of the cantilever relative to the control surface’s steering axis Ox in the Oyz plane (see Fig. 1).

Fig. 4 represents a graph (red curve) showing the function of cantilever deflections normalized to the unity, relative to the control surface span. Deflections are taken in the midplane of the control surface by the 1st vibration mode.

To identify the obtained results, we have implemented an additional numerical algorithm for calculating the first three natural vibration frequencies and modes of a one-dimensional cantilever beam, the properties of which are close to the real design characteristics of an aircraft control surface. The algorithm is based on a widely known step-by-step approximation method [

By implementing the proposed scheme regarding the structure under study, we obtained 3 vibration modes and the corresponding free vibration frequencies of the control surface cantilever. Frequency values are given in the table, and the 1st vibration mode is depicted in Fig. 4 (blue curve). We should note that the selected onedimensional algorithm allows to determine the cantilever beam’s vibration modes and frequencies only in one plane of vibration, in this case relative to the Oy axis, i.e. the axis of control surface unfolding in the Oxz plane of cantilever bending deflections (see Fig. 1). Herewith, vibration frequency ν3 determined using the FEA software package is lost in the Oyz plane.

The difference between numerical values of the first vibration frequency calculated using the FE method and the step-by-step approximation method is 3.75 %. This figure proves the adequacy of calculations using the FE method.

When the aircraft control surface unfolds in flight, its cantilevered section turns through an angle of ~90° relative to the aircraft longitudinal axis, and its support pad hits the limit stop, after which the cantilever is locked by retainers. The most dangerous factors in this situation are dynamic forces caused by shock impact on the support pad. Such forces may damage not only the control surface, but also the locking mechanism, attachment fittings between the control surface and the aircraft body, as well as the control unit compartment.

Simulation of the shock impact considered the fact that the control surface mostly unfolds in the Ozx plane (Fig. 1). In this case, ignoring the aerodynamic moment generated by the airflow during aircraft flight, the maximum deflection of the control surface cantilever at impact, when it is locked (fixed), is determined based on the equation for calculating the kinetic energy of the cantilever motion when the control surface is fully unfolded, and the maximum potential energy of bending strain of the cantilever’s hinged section. Kinetic energy of control surface’s rotation motion is as follows

where Jин – mass moment of inertia of the control surface’s hinged section cantilever relative to axis of rotation Oy (see Fig. 1); – maximum angular velocity of the cantilever at impact, depending on the control surface unfolding mechanism (spring-type, lever-type, link mechanism, etc.).

Potential energy of the control surface’s hinged section bending with bending stiffness EJ and deflections z = z(x) can be represented as follows [

Taking into account only the first free vibration mode of the control surface cantilever with frequency ν = V1 (see subsection “Control surface free vibration simulation”) prevailing over other modes at impact, the function of displacement of reference cross-sections of the cantilever midplane relative to the Oz axis when the control surface hits a limit stop, in accordance with the Fourier method, will be represented as follows [

z (t, x ) = zmax f (x)sin (2πνt), (2)

where zmax – maximum deflection at reference point, corresponding to the outermost free edge of the control surface span (at x = L);

f (x) – normalized natural vibration mode of the cantilever midplane by the first mode of vibration (see Fig. 4), which is called the first normal function and describes the shape of cantilever deflection;

sin(2πνt) – time component of free vibrations.

The elastic strain energy reaches its maximum at maximum time-dependent displacement z(t, x), i.e. under condition sin (2πνt) = 1, for example at t = 1 / (4 ν). Substituting (2) in (1), we can calculate the maximum value of potential energy of bending:

Comparing the above formula with the maximum potential energy of elastic force, we can note that multiplier is the cantilever stiffness reduced to the free end. Then Πmax = Спрz2max / 2 .

Similarly to vibrations of a mechanical system, let us assume that the first frequency of the cantilever’s natural vibration is related to the control surface characteristics as follows:

where Mпр – generalized weight reduced to the free end of the cantilever and calculated using formula [3, 5]

Here, m(x) – control surface mass per unit length distributed along the longitudinal axis. With reduced stiffness Cпp taken from (3) and substituted in the expression for (3) Πmax , we get

Using the equation of energy Т = Πmax, we can calculate the maximum deflection of the cantilever free edge upon hitting the limit stops while unfolding:

In order to calculate the distribution of the bending moment along the cantilever span, we will rely on the Timoshenko beam theory [

We will apply the equation of bending vibration of a one-dimensional beam with lateral cross-section F = F( x), density ρ = ρ( x) and bending stiffness EJ = EJ (x), cantilevered on axis of unfolding x = 0 in the common form [3, 4]:

formulated relative to cantilever deflections z (x, t), in unfolding plane Oxz. Substituting expression (6) in equation (7), we get

where ρF – cantilever mass per unit length ρF = m(x).

With regard to (2), acceleration ∂2 z (x,t) / ∂t2 in the cantilever cross-section with coordinate x will have a simpler form:

After integrating both parts of equation (8) over x twice, substituting (9) in it and taking an iterated integral, we find the expression for the bending moment at a particular point x ∈ [0; L] on the longitudinal axis of the cantilever:

The bending moment Mизг reaches its maximum time-dependent value at sin (2πνt) = -1, then

To simplify (10), let us single out conventional lateral layers of the cantilever of an arbitrarily small thickness, where functions m(ξ) and f (ξ) are assumed to be constant values. In this case, the inner integral in expression (10) is definitely taken in each segment of this type. As a result of transformation with regard to expression (5), the bending moment function can be represented in the following final form:

where Мпр – reduced weight calculated using formula (4).

In particular, if functions m(ξ) and f (ξ) are set in a discrete form and their approximation leads to a considerable error, it is more convenient to use expression (11) as the sum

where xi – longitudinal coordinate of the current cross-section of the cantilever;N – number of cantilever segments (layers), assuming that each segment has uniform distribution of weight and uniform deflection;– the j-th segment centre coordinate;Mj – the j-th segment weight;fj – deflection in the middle of the j-th segment.

The calculation result gives us a set of N values of the bending moment in the control surface cantilever cross-sections.

It should be noted that the obtained formulae for calculating moments Mизг maintain the relevance of the sum of products of the cantilever’s lateral forces in each segment at a certain distance from cross section x to the axis of rotation of the control surface’s hinged section.

Let us derive the formula for calculating a diagram showing the distribution of maximum lateral load per unit length along the control surface cantilever span. Based on ratios (2), (5), and (9), we can determine the maximum time-dependent linear acceleration a(x) at point x on the cantilever’s longitudinal axis:

In a qualitative sense, the graph of function a(x) will be apparently similar to the graph showing the control surface cantilever deflections at vibrations by the first mode f (x). Thus, the lateral load per unit length in cross-section x will be determined as the product of mass per unit length m (x) and accelerations a(x), i.e.

Let us demonstrate the represented method exemplified by calculation of the bending moment and load per unit length discussed in subsection “Control surface free vibration simulation” of the control surface structure upon hitting the limit stop while unfolding and locking.

The proposed formulae are implemented into the MathCAD engineering software package.At the angular velocity of = 23 rad/s, at which the cantilever moves towards the limit stop, calculated by specialists in aerodynamics and based on the specifics of aircraft aerodynamic flows in flight for the given shape of the control surface, the maximum deflection of the cantilever free end is Zmax = 20 mm according to formula (5). The maximum value of the bending moment of Мизг(x) = 1200 N·m is observed on the axis of rotation of the control surface’s hinged section. The bending moment on the free end is equal to zero. Fig. 5 shows the diagram of the bending moment along the control surface span.

When calculating the lateral load diagram, we should note that functions m (x) and f (x) are highly non-linear, while extremum m(x) is closer to the axis of rotation (see Fig. 2) than extremum f (x) on the free end of the cantilever (see Fig. 4). In accordance with formula (12), their product has a global maximum equal to qmax = 1515.9,9 kN/s2 at point X = 0,32 m. The diagram is shown in Fig. 6.

The obtained diagrams Мизг (x) and q (x) can be used for estimating the strength of the control surface cantilever, attachment fittings between the control surface and the aircraft body, as well as the aircraft control unit compartment in the area where control surfaces are attached.

For studying dynamic loads acting on the control surface structure upon hitting the limit stop while unfolding, we have proposed an analytical method for calculating the bending moment and lateral load generated at such impact. The advantage of the approach is the possibility to determine the specified dynamic loads in any cantilever cross-section required. This is possible because the method is applied with account for distribution of bending stiffness and mass per unit length along the control surface span. Frequencies and modes of free vibrations of the control surface required for calculating impact loads have been determined using the FEA software package and verified using a well-known method of step-by-step approximation.

We should note that unlike [3, 5], where structural weight and stiffness characteristics have stepwise variations along the cantilever length, the proposed analytical method allows to consider monotonic variation of the bending stiffness. The novelty of the described approach is that it enables a combination of the finite element method (FEM) and analytical calculations for determining dynamic loads and for analysing control surface characteristics without setting of bending stiffness values when conducting an analytical calculation, but determining them using the FEM. The proposed method is applied with account for the cantilever bending stiffness values in a more comprehensive sense: these values have been calculated for the structure under study taking into account contact interactions between the control surface and the limit stop along with rotation of the cantilever around the axis of folding.

The authors declare that there are no conflicts of interest present.