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A new physical pattern – consistency of the factor of dynamic relations between velocity vector modulus and longitudinal acceleration of missile warheads


The paper describes the existing problems in determination of all scheduled evaluations of missile warhead performance during flight tests and puts forward one of the possible methods of problem solving. Besides, the paper gives the results of investigation of the properties of the factor of dynamic relations between the velocity vector modulus and longitudinal acceleration of missile warheads within the atmospheric passive flight leg – the dynamic relation factor is constant in different flight test conditions. The notion of the reference dynamic relation factor is reasonably introduced for both parameters under study in order to provide reliable determination of parameter estimates, and hence, to conduct a complete analysis of experimental launch results.

For citations:

Ustinov A.S. A new physical pattern – consistency of the factor of dynamic relations between velocity vector modulus and longitudinal acceleration of missile warheads. Journal of «Almaz – Antey» Air and Space Defence Corporation. 2021;(2):72-75.

Development of new missile warhead (hereinafter – MW) prototypes for strategic missile systems is currently growing at a quick pace. Flight tests are supposed to be a necessary phase to prove that MWs meet the set performance specifications and to make a decision on large-scale production launch. Evaluation of combat capabilities of MWs to be developed, as well as the efficiency of their intended use directly depend on the quality of flight tests (hereinafter – FT).

The importance of solving problems related to MW flight tests, development of new MW prototypes and high precision weapons, as well as reduction of a total number of tests imply stricter requirements to the reliability of missile warhead performance evaluation.

To process measurement results and evaluate MW performance, the personnel working at the test range apply a set of methods developed decades ago and proved efficient during multiple test launches.

A major drawback of such methods is that they give good results only if the entire volume of ground-based and telemetric measurement data is acquired. If measurement results are partially or completely unavailable, this causes serious problems of method application.

A lack of measurement results regarding at least one of the parameters at the final leg of the MW trajectory does not allow to apply the existing methods of integrated measurement data processing to the full extent, and hence, to run a subsequent analysis of test launch results as a whole. With a total amount of tests reduced, this is not acceptable.

Many top experts in the field paid special attention to the problems related to the methodology of aircraft performance evaluation based on measurement results [1–5 et al.].

At the same time, according to research data, contemporary theoretical studies along with existing methods of analysis and evaluation of experimental launch results fail to cover the problem of estimated parameter value determination with no relevant measurement results available.

Therefore, it is evident that we need to find new approaches to determination of MW motion parameters estimates if there is no direct measurement data or if acquired measurement data is not suitable for processing by applying the existing methods.

According to the analysis of existing new approaches to solving problems of evaluation and analysis of MW flight tests, it is possible to solve the problem in question using the theoretical basis of the dynamic relation method [1], which has successfully been tested and proved its operability and efficiency (in terms of accuracy, reliability and certainty of MW performance evaluation improvements) in different test launch conditions. Unlike other methods, this one allows to evaluate the performance of new missile prototypes with higher accuracy, reliability and certainty using low-quality measurement data, as well as to solve a bunch of problems of FT result analysis.

The method of dynamic relations is based on the dynamic relation factor (hereinafter – DRF), i.e. a mathematical model of the direct functional relation between elements of a dynamic system under study, namely, the relation between two parameters normalized by the mean:

where qij – dynamic relation factor;
xi , xj – estimated parameters;
xicp , xjcp – mean parameter values in the analysed range.

This coefficient has some unique properties described in detail in [1]. Moreover, the behaviour of many DRFs remains unchanged during multiple launches, and values of some factors are sufficiently close to each other.

Based on real data acquired after over 20 test launches, the author investigated the properties of the DRF of MW velocity vector modulus and longitudinal acceleration:

where qVNxi – dynamic relation factor of MW velocity vector modulus and longitudinal acceleration;
Vi – MW velocity vector modulus value;
Viср – mean value of MW velocity vector modulus in the analysed interval;
Nxi – MW longitudinal acceleration values;
Nxiср – mean value of MW longitudinal acceleration in the analysed interval.

Calculated values of DRF of the specified pair of parameters are shown in Fig. 1.


Fig. 1. DRF values of velocity vector modulus and longitudinal acceleration


The fact that there is a wide range of conditions for test launches, the results of which are selected for processing, requires special consideration. Test launches under consideration were conducted using different trajectories (atmospheric entry conditions: entry velocity of 3 000 to 7 000 m/s, entry angle of minus 20º to minus 60º); in different meteorological conditions (in different seasons and light conditions); with different configuration and combat formation of elements of a complex ballistic target; with different configuration of measuring instruments applied for testing; different types of MW with different mass, inertial, aerodynamic and other characteristics. Based on results of every single test launch, we acquired different volumes of measurement data of different quality.

According to the data specified herein, despite absolutely different launch conditions, DRF values for the pair of parameters being analysed are quite close to each other in the atmospheric non-powered flight segment.

Therefore, we can make a reasonable conclusion that there is a physical pattern proving the consistency of the DRF of MW velocity vector modulus and longitudinal acceleration. According to research data, this fact can be used for eliminating the existing problems related to total lack of measurement data on one of the specified parameters during FT or to acquisition of measurement data not suitable for processing by applying existing methods.

The proposed solution is to determine the mean DRF value, take it as the reference value (Fig. 2) for all completed and subsequent launches and use it to determine or correct one of the parameters using available values of the other one if the relevant measurement data are not acquired, which is not infrequent during test launches, because velocity vector modulus values are obtained by processing ground-based measurement data, and longitudinal acceleration values – by processing telemetric data.


Fig. 2. Reference DRF of velocity vector modulus and longitudinal acceleration


Further studies have proven that the normalized reference value of the DRF can be used for solving the problems of FT result analysis. For now, the relevant set of methods is currently in development and will be presented in further publications.

Hence, the paper:
– describes the existing problems of determining all the scheduled evaluations of MW performance during FT;
– justifies the selection of the method of dynamic relations as one of possible methods of problem solving;
– introduces a new physical pattern – the consistency of the dynamic relation factor of missile warheads’ velocity vector modulus and longitudinal acceleration, identified during processing a large volume of statistic data based on real warhead flight tests;
– introduces a new notion – the reference DRF value of warhead’s velocity vector modulus and longitudinal acceleration. In the author’s opinion, this notion will be important for further theoretical studies of dependencies of different warhead performance parameters related to a dynamic system with a variety of random disturbing factors, as well as to their practical implementation.



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About the Author

A. S. Ustinov
4th State Central Interbranch Test Range of the Ministry of Defence of Russia
Russian Federation

Ustinov Andrey Sergeevich – Head of Laboratory, Research and Testing Department.
Science research interests: missile launch information and ballistics support.

Znamensk, Astrakhan Region


For citations:

Ustinov A.S. A new physical pattern – consistency of the factor of dynamic relations between velocity vector modulus and longitudinal acceleration of missile warheads. Journal of «Almaz – Antey» Air and Space Defence Corporation. 2021;(2):72-75.

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ISSN 2542-0542 (Print)