The paper describes the impact of aerodynamic coefficients on the ballistic target (BT) velocity and proposes a method of approximation of the dependence of ballistic target drag coefficient Cxa on the Mach number and angle of attack. The paper proves that the proposed approach allows to substantially reduce errors in drag coefficient simulation, but requires a more complicated calculation process.

В статье рассмотрено влияние аэродинамических коэффициентов на скорость баллистической цели (БЦ) и предложена методика аппроксимации зависимости коэффициента лобового сопротивления Cxa баллистических целей от числа Маха и угла атаки. Показано, что предложенный подход позволяет на порядок уменьшить ошибки моделирования коэффициента лобового сопротивления при незначительном усложнении процесса его расчета.

To provide effective performance of the aerospace defence (ASD) strategic system, its integrated equipment and systems shall be tested on a regular basis. Since testing with full-scale prototypes is very expensive and technically complicated, and sometimes even infeasible, alternative methods of research in this field need to be found. One of such methods is the experimental and theoretical method combining full-scale tests of individual system components and simulation of the whole system.

The methodology based on the principles of systematic approach and integrated goaloriented planning is the foundation for solving the problem of ASD system integration verification. This methodology implies the use of experimental and theoretical research methods. The integrated simulation system is widely used for theoretical studies.

The main component of the integrated aerospace environment simulation system is the software module of airspace attack weapons (ASAW) motion dynamics, intended for implementing the following system of differential equations of ballistic target (BT) motion:

(1)

where VX, VY, VZ, RX, RY, RZ, PX, PY, PZ, gX, gY, gZ – components of velocity, total aerodynamic force, thrust force and gravity acceleration, respectively, in projections on the axes of the Earth-fixed launch site coordinate system, m – ballistic missile (BM) weight.

To estimate the influence of aerodynamic coefficients on the velocity, the following relationship may be used:

(2)

Let us analyse the non-powered (atmospheric) segment of the trajectory. Based on the above, the weight is assumed to be constant m = const. According to [

(3) where – density at the ground level, Hk – current altitude.

In its turn, the altitude – time dependence can be expressed as follows:

Hk = Hk – 1 – Vв(k – 1)t, (4) (4) where Hk – 1 – initial BT altitude at a given time t, Vв(k – 1) = Vk – 1sin θвх – vertical velocity at a given time t, θвх = 15° – atmospheric entry angle.

In this respect, the following assumption is taken: we suppose that during the 30-second flight in the non-powered atmospheric segment velocity Vв for a short period of time Δt = 1 s and angle θвх changes but slightly within 30 s.

Taking into account (3) and (4), equation (2) has the following form:

(5)

We shall use the following designations:

(6)

(7)

By integrating equation (5), we have:

(8)

(9) where Δt – increment, k ∈ N – discrete value of flight time, s.

Thus, we have determined the dependence of the current velocity on the aerodynamic drag coefficient. Dependence (9) is the recurrence formula, where the input element – velocity Vk – 1, will be substituted with velocity Vk calculated at the previous step, with relevant altitude recalculation as per formula (4).

Fig. 1 shows the dependence of BT velocity variation on time with the increment equal to Δt = 1, at different values of cx.

According to the diagram shown in Fig. 1, when the drag coefficient changes by 0.1, the difference in velocity at the 30th second at an altitude of about 30 km may exceed 200 m/s. Thus, to obtain the BT current velocity estimation error of not more than 10 m/s, the approximation meansquare error shall be less than 0.005.

According to equation (1), aerodynamic forces and moments affect trajectory parameters in a longer flight segment.

There are some well-known methods of approximation of the dependence of aerodynamic characteristics (ADC) on the angle of attack and Mach number, including: − method based on representation of the specified dependence in the form of Taylor’s series [

(10)− methods based on the use of simplified dependences of ADC on the angle of attack [

Cx = Cx0 + Cxi(α); Cy = Cyα α; Cz = Czβ β. (11)

According to (10), using the method based on the expansion of aerodynamic coefficients into Taylor’s series requires a large amount of discrete values, including those for determining partial derivatives, which also include wellknown dependence on the rotation speed, angle of control surface deviation (if any), etc. This makes the method labour-consuming for realtime simulation of missile and aerospace environment.

Besides, the method is applied only in case of low transient disturbances [

Fig. 2 shows mean-square deviations within the interval of Mach numbers (0; 8) when using the method based on the use of simplified ADC – angle of attack dependences.

Fig. 2. Mean square deviations between approximated and calculated values of lift force coefficient

According to the figure, the use of simplified dependence leads to large errors (from 15 to 30 %) within a large range of Mach numbers. Therefore, errors in BM deceleration estimation may be equal to fractions of g, affecting the accuracy of BM trajectory simulation.

Another drawback associated with the use of simplified dependences is that only one parameter (angle of attack) is considered, while one of the requirements for the dependence is that it shall be a combined two-parameter dependence on angle of attack and Mach number.

As a result, to ensure the required performance of the integrated ASD simulation system, we need to find an approximating function which complies with the requirements listed below:

− continuous approximation of coefficient Cxa depending on the angle of attack and Mach number; − coefficient Cxa approximation error shall not exceed 5 %; − approximation dependence parameters shall be determined based on available data on BT coefficient Cxa at certain values of angle of attack and Mach number.

Discrete data on coefficient Cxa can be determined using the existing mathematical apparatus, for example, the mathematical apparatus based on the SolidWorks Flow Simulation software package. For calculation of discrete values of coefficient Cxa, atmosphere parameters are taken as per [

We analysed the dependence of coefficient Cxa on Mach number in the following sequence:

− identification of characteristic points of the function, including its maximum and minimum points and break points; − dividing the function setting interval into segments with constant and monotonically increasing or decreasing function values; − approximation function selection; − calculation of approximation parameters with the sought function bound to characteristic points; − determination of approximation errors.

Fig. 3 shows discrete dependences of drag coefficient on Mach number at different angles of attack.

The figure also shows the following characteristic points for the dependence at zero angle of attack: x0, x1, x2 related to the points of function Cxa. characteristic variation. The analysis of Fig. 3 allows to reveal typical points of inflection of Cxa depending on Mach number and select the appropriate type of approximation for them.

We selected the type of approximation, knowing characteristic points x0, x1, x2, as well as additional conditions for the maximum at point x1 and uniformity in segments x x2.

As a result, function f(x) of the dependence of coefficient Cxa on Mach number can be represented in the following form

(12)where f * (x) – function variable f(x).

Based on the above, the approximation function will be generally formed on the basis of five discrete values which comply with the following system of equations

(13)

According to the analysis, the following function can be used as approximating function f *апп (x*)

(14) where A, B, a, b, c – approximation parameters;

(15) where x* – argument of approximating function f *апп (x*) to be selected based on the convenience of approximation parameters estimations,

x* = x – x0, .

To determine coefficients a, b, c, the following system of equations shall be solved

(16)

Taking into account that the resulting system of equations is a linear system with regard to a, b, c, it can be solved using Cramer’s rule.

After completing the analysis of coefficient Cxa – Mach number dependence, we analysed dependences of approximation parameters on the angle of attack. Characteristic points along with the relevant function values are supposed to be approximation parameters. Values of characteristic points x0, x2 are constant. The following system of equations (17) was obtained for determining the dependences of input approximation parameters on the angle of attack.

(17)

The proposed approach allowed to form the following functional dependence of the drag coefficient on the angle of attack and Mach number.

(18) where M* = M – M0; M0 = x0 – calculated by formula (17); A [f0(α)] = f0(α) – 1; B [f0(α), f2(α)] = f2(α) – f0(α) + 1; φ [M*, x*1(α)] – calculated by formula (15); a [f0(α), f1(α), f2(α), x*1(α), x*2(α)] – calculated by formula (16); b [f0(α), f1(α), f2(α), x*1(α), x*2(α)] – calculated by formula (16); c [f0(α), f1(α), f2(α), x*1(α), x*2(α)] – calculated by formula (16); x*1(α) = x1(α) – x0(α); x*2(α) = x2(α) – x0(α); f0(α), f1(α), f2(α), x0(α), x1(α), x2(α) – calculated by formula (17).

Fig. 4 shows the diagram of comparison of the initial and approximated functions of drag coefficient – Mach number dependence for zero angle of attack, plotted with the help of function (18).

Points in Fig. 4 designate discrete calculated values of aerodynamic coefficients, and the solid line represents approximation with the help of the approach described above.

The approximation error is taken as value

Fig. 5 shows the error Δ – Mach number dependence diagram at different angles of attack.

The analysis of errors of approximation curves in Fig. 5 proved the following. The maximum discrepancy between drag coefficient values calculated in the SolidWorks Flow Simulation environment and values obtained by approximation is observed in segment x0 ≤ x ≤ x1. This is caused by a steep increase in the function within a short interval. This segments needs further in-depth analysis. The error value in the segment may reach 29 %, while it does not exceed 2.5 % in the remaining part.

Therefore, according to the analysis of existing methods of ADC calculation, available approximation methods have some drawbacks such as labour intensity if used for real-time simulation of missile and aerospace environment, restrictions on the use or a large error of up to 30 %. A five-parameter function was generated for approximating the dependence of ballistic targets’ drag coefficient Cxa on Mach number and angle of attack. This function allows to obtain an approximation error of 2.5 % maximum at supersonic velocities with Mach numbers over 1.5, thus giving the accuracy 10 times greater than the simplified method. In further studies, it is reasonable to consider the possibility to apply the proposed approach to other classes of ballistic missiles represented in various assembly variants. This will enable further ADS simulation using a unified approach.

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