MATHEMATICAL MODELS OF THE RADAR SIGNAL REFLECTED FROM A HELICOPTER MAIN ROTOR IN APPLICATION TO INVERSE SYNTHESIS OF ANTENNA APERTURE

Introduction. The problem of aircraft recognition can be solved on the basis of the formation of radar portraits that re-flect the characteristic structural features of aerial vehicles. Radar portraits based on images of the propellers of aerial vehicles have high informativeness. On the basis of such images, the number and relative position of the propeller blades, as well as the direction of their rotation, can be determined. Such images may be obtained on the basis of mathematical models constructed from reflected signals. Objective. The aim of the work is to develop mathematical models for the radar signal reflected from the main rotor of a helicopter applied to inverse synthetic aperture radar (ISAR). Methods and materials. ISAR processing is used to produce a radar image of a rotor in a radar with a monochromatic probing signal. The rotor blades in the models are approximated by different geometric shapes. The models used to describe the reflection from the rotors of helicopters differ significantly from those used to describe fixed-wing aircraft propellers. In the process of moving through the air, each helicopter rotor blade performs characteristic movements (flapping, lead-lag motion, torsion), as well as flexing in a vertical plane. Such movements and flexings of the blades influence the phase of the radar signal deflected from the main rotor. When developing an ISAR algorithm for imaging the main rotor, the phase change must accurately account for the reflected signal. Results. In the centimetre wavelength range, the mathematical model of the signal reflected from the main helicopter rotor as a system of blades is most accurately described if each blade is represented as a set of isotropic reflectors located on the leading and trailing edges of the main rotor blades. The distinct features of the actual signal are more closely approximated by a model that takes the flapping movements and curved shapes of the blades into account. Conclusion. The developed model, which takes the flapping movements and flexes of the helicopter main rotor blades into ac-count, can be used to improve the ISAR algorithms that provide radar imaging of aerial vehicles.

Introduction. Considered as an object of radar surveillance, the movements of a single-rotor helicopter's primary elements are complex, consisting not only the translational motion of the fuselage, but also of the translational-rotational motions of the main rotor (MR) and tail rotor, during which the angles of attack of their blades change. When developing Inverse Synthetic Aperture Radar (ISAR) algorithms for imaging the rotors of different helicopters, it is necessary to account for the laws describing changes in the phase shifts of signals reflected from the elements of each rotor. The most commonly-used mathematical model of reflected signals (RS) for MR, in which the blades are represented as cylinders [1], only partially meets this requirement. Therefore, in the present study we set out to develop mathematical models for signals reflected from helicopter main rotors differing in the form of their blades. In order to proceed with the modelling, it is assumed that an aircraft can be represented as a complex of reflectors [2] in the centimetre wavelength range of radio-waves with the overall signal reflected from it consisting in a superposition of signals reflected from each reflector.
Model of the temporal structure of a signal reflected from a helicopter's main rotor. A rectangular coordinate system   0XYZ is bounded with a radar sensor (RdS) (Fig. 1), with its origin coinciding with the centre of the RdS and axis 0X parallel to the helicopter velocity vector v.
The signal reflected from the rotor is represented as a combination of signals reflected from pointisotropic reflectors located on the surfaces of b.rot N blades. In general, the mathematical RS model on an antenna output is described by the expression [3]-[6]:    and cthe wavelength and the velocity of PS propagation, relatively.

Direction of Rotation
In order to ensure accurate radar imaging, it is necessary that the RS phase be represented correctly. Expressions (3)- (5) show that this phase is determined by the laws of change of the distances to the reflectors on the blades of the MR during rotation. The primary reflections from the MR are produced by the edges of its blades. Taking this into account, we will assume that the reflectors are placed not over the entire plane of the blade, but rather on its edges. Let us consider the laws of the change of the distances to the reflectors relative to the phase centre of RdS antenna for three options of the blade representation: Option 1as a combination of isotropic reflectors located on a straight line whose length corresponds to the blade length; Option 2as a combination of reflectors located on straight lines corresponding to the leading and the trailing edges of the blade; Option 3as a combination of reflectors located on the lines, which is bent due to flapping and nonuniform blade bending. Blade feathering is not considered.
The reflectors on the edges of the blades are isotropic in the range of hemispheres directed toward RdS for Options 2 and 3.
It is worth mentioning that a helicopter moves according to the orientation and magnitude of the thrust force vector of the main rotor with respect to the gravity force vector. One may assume that the thrust force vector is oriented perpendicular to the plane of the base of the cone described by the moving main rotor blades except when the orientation of this plane is changed by a pilot using a swashplate. The reflected signal model, which takes the orientation of the blade system into account; its slope and form during horizontal flight with constant altitude relating to RdS is of interest in the application of radar imaging techniques for the main rotor.
The distance to the reflector in the case of blade representation according to Option 1. Reflectors are located ( At the beginning of the analysis, the centre of rota- the blade chord. The distance to the reflectors of the blade with respect to the change of reflection characteristics during advancing and retreating of the blades relating to RdS is described by the expressions: retreating.
An expression for the condition corresponding to the change of the reflecting edges during advancing (retreating) of the blades for clockwise rotation direction (top view) (see Fig. 2  Trailing The distance to the reflector in the case of representation of the blade according to Option 3. The main rotor of the helicopter creates a lifting force and a horizontal thrust force. In general, each MR blade is installed on the central rotor head by a horizontal (flapping) hinge (FH), a vertical (drag hinge (DH) and a feathering hinge (FrH), with respect to which flapping movements (FM), dragging (lead-lag) and feathering (change of pitch angles) are carried out (Fig. 7) [7]- [10].
A blade flapping angle reaches the values of 12-15º that leads to the end of the blade being lifted to a sufficient height with respect to the rotation plane of the rotor head and influences the RS phase in centi-metre range. In addition, the free end of the blade bends in the vertical plane during rotation of the MR leading to a change of the backscattering diagram and the phase of refractions from the edges * .
Description of the flapping movement and features of blade construction. We will assume for simplification that the helicopter is oriented horizontally during the flight and that the rotation plane of the rotor shaft is parallel to the Earth's surface at the point at which the helicopter is located. With respect to this, a coordinate system 1 1 1 CX Y Z (Fig. 8), whose centre is the rotational centre of the rotor, is used for * By the phase structure of the signal reflected from the blade edge is meant a distribution of phases of the signals reflected from the edge fragments. e It is worth mentioning that the angle of the blade section setup is the pitch angle of the blade crosssection chord relative to the rotation plane, which is perpendicular to the propeller rotation axis.
The flapping angle and pitch angle of the blade in a stationary state of flight comprises the periodic functions of its angular position b .
 Therefore, this can be expanded in a Fourier series with respect to this parameter [7], [8], [11]: Top view Side view View from the end of the blade Fig. 8. Flapping motion of the blade   [7], [8], [11]:  (9) is complicated task. In the considered case, one could use the results obtained in the paper [11] assuming that a feathered blade is in the rectangular form, while a flapping controller is absent and the distribution of inductive velocities along the sweep away disk is uniform.
For these conditions:  Model of the signal reflected from the main rotor with respect to the flapping movements and bends. The rectangular coordinate system 0XYZ is used during the simulation (see Fig. 1). The edges of the main rotor are represented as a set of reflectors placed on the edge lines. Leading and trailing edges are described by a piecewise linear function within this model. For example, the leading edge of the main rotor of the Mi-2 helicopter is approximated (Fig. 10, a) by two sections whose lengths are ld1 R and ld2 R , respectively; the incidence angle of the second section relative to the first section is ld2  ; the trailing edge ( Fig. 10, b) is approximated by four sections, whose lengths are tr1 , , n R respectively (Fig. 10). Let us assume that the reflectors are placed on the edges at regular intervals . R  The quantity of the reflectors on the sections is determined by the rounding function: and coordinates of the ref.tr n th reflector by the expressions: It is worth mentioning that flight velocity v is negative during the helicopter's advancement to RdS and positive during retreat.
The distance to an arbitrary reflector with respect to the change of reflecting edges of the first blade during its advancing and retreating is described by the general expressions (7), (8) using new coordinates (10), (11)  All reflectors are isotropic within the approximation region.
The RS power for the leading edge is higher than the power for the trailing edge in the case of a helicopter moving away, and vice versa [15]- [18]. Thus, the RCS of the single reflector of the leading edge for Simulation results in the blade approximation according to Option 2. Fig. 12 shows: аreal part of RS, bthe energy spectrum of RS, c -RS for the advancing blade, and d -RS for the retreating blade.
Simulation results in the blade approximation according to Option 3. Analysis of the simulation results. RS for multiblade structure of MR comprises a set of pulses with frequency modulation. Each RS pulse consists of the chirp pulses adjacent to each other in the case of the piecewise linear approximation according to Option 3. Its quantity and modulation parameters are defined by the quantity of linear sections, their position on the edge, and propeller rotation frequency. RS pulse frequency is defined by multiplication of the blade quantity and the rotation frequency b.rot rot NF of the propeller (see Fig. 11, а; 12, а; 14, а).
The spectrum structure of the signal reflected from the MR is discrete (see Fig. 11, f). Spectrum components are divided by the intervals b.rot rot .
NF Peaks in the RS spectrum observed during simulation for approximation according to Option 3 (see Fig. 14 Correlation of results. A comparison of the simulation data and the experimental results shows that the RS model of the helicopter MR, which takes into account the flapping movements and the bended blade forms, is close to the real RS. Each pulse of the RS complex envelope (see Fig. 14, c and d; 15, c and d) consists of short pulses adjacent to each other. The quantity, duration and modulation parameters of these pulses are determined by the quantity, position, and orientation of linear sections on the corresponding edge. In particular, RS of the leading edge of the Mi-2 helicopter rotor consists of two pulses (see Fig. 14, c) during its advancement to RdS, while the RS of the trailing edge during its retreat from RdS consists of four partial pulses (see Fig. 14, d). The partial pulse duration is determined by the backscattering diagram sidelobe from the corresponding linear section of the blade edge (see Fig. 11, c and d; 12, c and d; Fig. 14, c and d; 15, c and d). Since the spectra of the signals reflected from advancing and retreating blades are placed symmetrically relative to the Doppler frequency signal reflected from the helicopter body, they have different levels (Fig. 12, b; 14, b; 15, e).
Conclusion. In the centimetre wavelength range, the mathematical model of the signal reflected from the main rotor considered as a system of the blades is described most precisely by representing of each blade as a set of isotropic reflectors located on leading and trailing blade edges. Taking the flapping movements and the bended forms of the blades into account allows the actual signal features to be maximally approximated, describing the signal phase structural change law more precisely, and, consequently, increasing the quality of the propeller radar imaging. The developed model could be used for improving the ISAR algorithms used to support the radar imaging.