Akimov Vladimir Nikolaevich – Doctor of Engineering Sciences, Associate Professor, Deputy General Director for Science – Chief Designer. Science research interests: rocket engineering, missile control systems.

Dolgoprudny.

Kostyukov Aleksandr Aleksandrovich – Head of Department. Science research interests: rocket engineering, missile flight dynamics.

Dolgoprudny.

Kravchuk Evgeny Nikolaevich – Lead Design Engineer. Science research interests: missile flight dynamics, missile stabilization systems.

Dolgoprudny.

Rozantsev Konstantin Olegovich – Lead Design Engineer. Science research interests: missile aerodynamics, numerical simulation.

Dolgoprudny.

This research paper is intended to refine the aerodynamic moment of the missile based on an analysis of flight tests and results of gas dynamics software computations. The paper compares mathematical simulation results with flight test data in order to demonstrate an improved convergence due to the proposed refinement.

В статье проведено уточнение аэродинамического момента ракеты на основе анализа летных испытаний и результатов расчетов в программе вычислительной газодинамики. Приведены результаты имитационного математического моделирования в сравнении с данными из летных испытаний, показывающие повышение сходимости за счет предложенного уточнения.

The conformity of theoretical aerodynamic characteristics to experimental data (gathered during flight tests), in particular, aerodynamic characteristics that describe the aerodynamic moment action, is very important in design and development of modern and advanced manoeuvrable unmanned aerial vehicles such as surface-to-air missiles (SAM).

Aerodynamic forces acting on the SAM as well as dynamic characteristics of the SAM as a controlled object are determined using theoretical dynamic characteristics. That is why the accuracy of determination of aerodynamic characteristics, in particular, the aerodynamic moment, has a direct effect on the optimal choice of the SAM design and load-carrying layout and, therefore, on attaining minimum dimension and weight characteristics of the SAM, as well as on the quality of transient processes that take place when the SAM performs its functional tasks in compliance with design specification requirements.

It is a challenging and time-consuming problem to determine the aerodynamic characteristics of the SAM because of wide ranges of various flight conditions (ranges of variation of Mach number, angles of attack, aerodynamic control surface deflection, aerodynamic roll angles).

Moreover, the highest priority task is to accurately determine theoretical aerodynamic characteristics in those flight conditions when the SAM is statically unstable. In this case, errors in determination of aerodynamic characteristics may cause the loss of stability or manoeuvrability deterioration.

This paper redefines the aerodynamic moment by the example of an axisymmetrical SAM. This missile is a single-stage missile of the conventional aerodynamic design, with the XX-type (cruciform) arrangement of its wings and aerodynamic control surfaces.

The aerodynamic moment is described by means of equations in three projections on the axes of the OXYZ bound coordinate system (BCS). These projections are designated Mх, Мy, Мz [

Fig. 1. Bound coordinate system OXYZ

The projection of moment Mх is referred to as the roll moment (moment relative to the longitudinal axis ОХ), Мy is the yaw moment (moment relative to the lateral axis ОY), Мz – pitch moment (moment relative to the lateral axis ОZ) [

Moments Mх, Мy, Мz depend on flight speed and altitude, spatial angle of attack, aerodynamic roll angle, and aerodynamic control surface deflection angles. Besides, moment values are affected by the angular velocity of the SAM body, as well as by derivatives of the spatial angle of attack, and derivatives of aerodynamic control surface deflection angles.

According to [2–5], in design and development of manoeuvrable axisymmetrical unmanned aerial vehicles (including SAM) and in the SAM engineering design, it is assumed that the aerodynamic moment acts in the SAM symmetry planes, i.e. the planes of moments Mх, Мy, Мz pass through the longitudinal axis (Fig. 1).

The equations of aerodynamic moment projections on the BSC axes are represented as follows [2–5]:

where mкоxα, mкоxδ – coefficients of roll moment (asymmetric airflow), which appears irrespective of the aerodynamic control surface deflection angle and is the total result of cross impact of the longitudinal control channels on the roll channel [

Roll moment coefficient mкоxα defines the asymmetric airflow moment caused by the asymmetric arrangement of trailing vortices from the SAM body that are located behind non-deflected lifting surfaces (δ = 0). Roll moment coefficient mкоxδ defines the asymmetric airflow moment caused by the asymmetric air flow around deflected windward and leeward aerodynamic control surfaces (δ ≠ 0) [

The equations of aerodynamic moment projections on the BCS axes are represented as follows, with the SAM non-rotating and control surface angles equal to zero:

(2)

We propose to redefine the aerodynamic moment projections (1) and (2) taking into account the aerodynamic moment acting out of the SAM symmetry planes.

According to theoretical mechanics [

(3) where уd, zd – coordinates of the point of aerodynamic force application from the ОХ axis; ум, zм – coordinates of the centre of mass of the ОХ axis; Fx – longitudinal component of aerodynamic force.

Fig. 2. Moment plane not passing through the symmetry axis

With regard to equations (3), aerodynamic moment projections (1) will have the following form:

(4)

Experience gained in engineering design and manufacturing of supersonic axisymmetrical SAMs proves that there are minor deflections of the centre of mass relative to the symmetry planes and coordinates ум and zм can be neglected. The equations of moments are represented as follows with the SAM not rotating and control surface angles equal to zero:

(5)

The comparative analysis of equations (2) and (5) shows that: – the asymmetric airflow moment caused by the asymmetric arrangement of trailing vortices from the SAM body relative to rear lifting surfaces is determined by the projections of normal aerodynamic force and the coordinates of the point of application of these forces, – the fact that SAM has moment Мх in various flight conditions shows that the centre of pressure of the aerodynamic force is out of symmetry planes, – in addition to the main moments from normal force projections in the OY and OZ planes (in longitudinal control channels), the longitudinal force moment acts as well.

According to the results of a comparative analysis of transient processes obtained during flight tests and by means of mathematical simulation, the aerodynamic moment of the SAM under consideration needs to be redefined.

Moreover, the SAM airframe flow analysis was conducted for various flight conditions using a computational gas dynamics software package. The problem is solved in 3D environment. The applicable mathematical model is based on solving Navier – Stokes averaged equations closed with the help of the SST k – ω turbulence model. According to calculation analysis results, the asymmetry of aerodynamic forces on the SAM airframe, including those on lifting surfaces, makes the aerodynamic force of the whole SAM act along the line that does not cross the longitudinal axis of the SAM, i.e. the aerodynamic moment projections act out of the SAM symmetry planes.

Transient processes were compared based on the results of simulation on the stabilisation system mathematical model (SSMM) [

It is worth mentioning that in order to improve the convergence of simulation and flight test results, and to select optimal settings of the stabilisation system, the relative coordinate of the SAM airframe’s centre of pressure was calculated with corrections depending of the engine operating time and ram air pressure. These corrections are mostly based on flight test results.

Figures 3–5 show SAM characteristics at two flight phases (а – powered phase, b – free-flight phase), where discrepancies in transient processes are registered (Fig. 3, 4). Accelerations and control surface deflection angles in longitudinal control channels , , , are determined in relation to their maximum values for the SAM and are normalised depending on the flight time relative to the engine operating time.

In these conditions, the SAM follows steplike asymmetric control commands (Fig. 3, lines 1, 2) and during this process it becomes statically unstable in the powered phase and statically stable in the free-flight phase (Fig. 5).

Simulation results show that according to longitudinal balance, different levels of control commands (relative accelerations in phase а) and in phase b) shall conform to different levels of control surface deflection in longitudinal control channels (in phase а – and in phase b –), but this conformity is not observed during launches. At the same time, significant discrepancies are observed in transient processes in the control channel with a lower level of command.

Therefore, the results of flight test analysis, results of simulation on the SSMM and computations by means of a computational gas dynamics software package show that the aerodynamic moment shall be redefined.

In order to redefine the aerodynamic moment (to refine the SSMM) in accordance with equations (4), (6), it is necessary to determine the coordinates of the aerodynamic force application хd yd, zd.

The easiest way to determine coordinates хd yd, zd and other aerodynamic characteristics in various flight conditions is to analyse threedimensional flow motion around the SAM airframe using a computational gas dynamics software.

Based on the analysis of 3D flow motion around the SAM airframe in various flight conditions with the SAM body non-rotating, the projections of the vector of aerodynamic forces (Fx, Fy, Fz) and the vector of aerodynamic moment (Mх, Мy, Мz) on the BCS axes are determined.

Based on equations (3) and using the vector addition rules, coordinates xd, уd, zd are determined:

(6)

where F – aerodynamic forces vector modulus; М – aerodynamic moment vector modulus; d – aerodynamic moment vector arm relative to SAM nose.

Figure 6 illustrates the coordinates determined by the above method as values normalised to the SAM length for some flight conditions.

In general, calculated values xd, as well as d are close to values xd applicable to the SSMM, which are obtained based on wind tunnel tests and refined according to flight test results.

The most interesting results are values уd, zd which define the extent to which the longitudinal component of aerodynamic force Fx affects moments Мy, Мz.

An analysis of calculated values уd, zd shows that coordinates depend on Mach number, spatial angle of attack, aerodynamic roll angle φп [

Based on multiple calculations by means of a computational flow dynamics software package for the SAM under consideration, table values of coordinates of the centre of pressure xd, уd, zd are presented. With these table values, the aerodynamic moment in the SSMM is redefined in accordance with expressions (4) and (5), while corrections depending on the engine operating time and ram air pressure are eliminated. Besides, the airframe flow analysis results allowed to form a mathematical model of aerodynamic forces and moments based on vane-by-vane representation of forces (moments) [

The results of simulation on the updated SSMM for the flight phases considered above (Fig. 3–5) are shown in Figures 7–10.

Figure 10 shows changes in relative coordinates by non-dimensional time.

The analysis of Figures 3–5 and 7–10 proves that redefining the aerodynamic moment projections in accordance with expressions (4), (5) (by eliminating the assumption of the symmetry of aerodynamic moment action) significantly improves the convergence of simulation and flight test results under conditions with antisymmetric flow around the airframe.

Simulation on the SSMM allowed to refine the stabilisation system settings, the effectiveness and correctness of which are confirmed by successful flight tests. According to statistic simulation, the selected settings of the stabilisation system allow to improve the guidance accuracy and the probability to kill an air attack target, especially in conditions of missile evasive manoeuvring.

The study of various flight phases by means of mathematical simulation on the SSMM and comparison with the results of flight test of the SAM under consideration shows the following: – redefining the aerodynamic moment in accordance with expressions (4) and (5) allows to improve the convergence of SAM aerodynamic and dynamic characteristics based on flight test and simulation results; – in flight conditions with symmetric flow around the airframe (pattern “+” – φп = 0, 90º, 180º, 270º and pattern “×” – φп = 45º, 135º, 225º, 315º), at which absolute values of coordinates уd and zd are close to each other or one of them is close to zero, the action of longitudinal force moment can be compensated by introducing corrections to the moment of normal force or to the moment caused by deflection of aerodynamic control surfaces. In engineering design, this is mostly used for eliminating discrepancies in aerodynamic characteristics. – in flight conditions with asymmetric flow around the airframe (at angles φп in ranges of 10º÷30º, 100º÷120º, 190º÷210º and 280º÷300º), at which absolute values of coordinates уd and zd may considerably differ from each other, it is practically impossible to compensate the action of the longitudinal force moment by introducing corrections to the moment of normal force or to the moment caused by deflection of aerodynamic control surfaces.

Therefore, redefining the surface-to-air missile’s aerodynamic moment assumed to be asymmetric allows to improve the agreement between design and experimental aerodynamic characteristics of the SAM, especially in flight conditions with asymmetric flow around the airframe.

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