Simulation of beating modes during the rotary-oscillating movement of complex aerodynamic engineering with determination of the conditions of their occurrence




dynamic system, beating modes, nonlinear oscillations


The aim of the article is to determine the conditions of occurrence of beating modes in a nonlinear high-order dynamic system with subsequent computer simulation of these modes. Methods of research of nonlinear oscillatory systems are applied with consideration of two cases of interrelation on rotation (weak and strong) between oscillatory circuits.

In the first of them, the conditions for the existence of beating modes are approximately the same values of the partial frequencies of longitudinal and lateral oscillating motions with a constant increase in the modulus of the phase shift between these oscillations (phase motion is unstable).

At strong forces of interconnection (coefficients of interrelation of various signs) modes of beating arise at close values of sizes of the main frequencies of the interconnected fluctuations lying in a range between partial frequencies. Such modes (in the absence of parametric interaction between the circuits) are possible when the conditions of stability of the biharmonic process are observed.

The study of the complex form of rotational-oscillating motion of an aerodynamic object at the initial stage includes the selection from the complete dynamic system of two interconnected self-oscillating contours of longitudinal and lateral motions as a basis (necessary conditions) for the existence of beating modes.

In cases of observance of existence conditions of single-frequency self-oscillating processes at occurrence of parametric interaction these processes can also pass to beating modes.

In practice (outside the resonant region of the main frequencies) this is often realized when the functional frequency of the contour of lateral motion includes components proportional to the parameters of longitudinal motion.

All these cases are supported by numerous model experiments.

Author Biographies

Volodymyr Kotliarov, Central Research Institute of the Armed Forces of Ukraine

Doctor of Technical Sciences, Professor, Chief Researcher

Oleksandr Voloshchenko, Central Research Institute of the Armed Forces of Ukraine

Candidate of Military Science, Senior Researcher, Leading Researcher

Oleksandr Kuznetsov, Central Research Institute of the Armed Forces of Ukraine

Candidate of Military Science, Senior Research Fellow

Mykola Kushnirenko, Kyiv National University of Construction and Architecture

Candidate of Technical Sciences, Associate Professor, Associate Professor of the Department of Structural Mechanics


Goman M.G., Khramczovskij A.V. Bifurkaczii ustanovivshikhsya rezhimov shtopora samoleta. (Bifurcations of steady-state airplane spin modes). - Issledovanie po dinamike poleta letatel`ny`kh apparatov: mezhduved. sb. - Moskva: MFTI, 1986. - Р. 17–25.

Bukov V.N. Adaptivny`e prognoziruyushhie sistemy`upravleniya poletom. (Adaptive predictive flight control systems). - Moskva: Nauka, 1987. - Р. 97–232.

Kotliarov V.P. Pobudova struktury dynamichnoi systemy, shcho opysuie avtokolyvalnyi rukh obiekta, za materialamy naturnykh vyprobuvan (Construction of the structure of a dynamic system describing the self-oscillating motion of an object, based on field tests). - Zb. nauk. pr. TsNDI ZS Ukrainy no. 3(8). – Kyiv: TsNDI ZS Ukrainy, 1999. - Р. 165–169.

Nedin V.O. Parametrychni kolyvannia sterzhniv, shcho obertaiutsia pid diieiu pozdovzhnoho udarnoho navantazhennia (Parametric oscillations of rods rotating under the action of longitudinal shock load). - Opir materialiv i teoriia sporud no. 104. - Kyiv, 2020. - Р. 309–320.

Bazhenov V.A., Pohorelova O.S., Postnikova T.H. Stvorennia matematychnoi modeli udarno-vibratsiinoho maidanchyka dlia ushchilnennia ta formuvannia betonnykh vyrobiv (Creation of a mathematical model of the shock-vibration platform for compaction and formation of concrete products). - Opir materialiv i teoriia sporud no. 104. - Kyiv, 2020. - Р. 103–116.

Baiev S.V., Volchok D.L. Neliniini kolyvannia poperedno napruzhenoi zalizobetonnoi mostovoi balky pry harmoniinomu obureni v umovakh nechitkykh parametriv (Nonlinear oscillations of prestressed reinforced concrete bridge beam with harmonic perturbation under fuzzy parameters). - Opir materialiv i teoriia sporud no. 104. - Kyiv, 2020. - Р. 147–163.

Kotlyarov V.P., Seryogin G.N., Shihaleyev V.N. Simulation of aircraft autooscillatory motion in high-angel-of attack. Book of Abstracts in Mosaeroshow 92 (11-16 August 1992, Zhukovsky, Russia. - Moskow: Central atrohydrodynamic insntitute, 1992. - Р. 59–60.

Rubanik V.P. Kolebaniya kvazilinejny`kh sistem s zapazdy`vaniem (Oscillations of quasilinear systems with delay). - Moskva: Nauka, 1969. - Р. 17–290 р.

Butenin N.V. Vvedenie v teoriyu nelinejny`kh kolebanij (Introduction to the theory of nonlinear oscillations). - Moskva: Nauka, 1987. - Р. 22–381.

Byushgens G.S, Studnev R.V. Dinamika samoleta. Prostranstvennoe dvizhenie (Aircraft dynamics. Spatial movement). - Moskva: Mashinostroenie, 1983. - Р. 223–320.