534.131.2 Identification of operational parameters ensuring the stability of a monopropellant-fueled thermocatalytic liquid rocket engine
Grigor’ev V. G. (Moscow Aviation Institute (National Research University)), Kurakin V. V. (Moscow Aviation Institute (National Research University))
EIGENOSCILLATIONS, THREE-DIMENSIONAL PROBLEM, FINITE ELEMENT METHOD, MIXED VARIATIONAL PRINCIPLE, HYDROELASTIC STRUCTURE, NUMERICAL INTEGRATION, GAUSS-LEGENDRE QUADRATURE
doi: 10.18698/2309-3684-2025-4-418
The application of the finite element method based on the mixed variational principle for solving a three-dimensional eigenvalue problem using the example of a cavity with an elastic bottom is considered. A brief review of the literature on the subject of the study is provided. A rigorous mathematical formulation of the problem for the described mechanical system is presented. A hexagonal finite element of the fluid is introduced, the techniques of numerical integration of the kinetic energy of the liquid volume is described in using the Gauss-Legendre quadrature. A quadrangular finite element of the free surface is introduced, the techniques of numerical integration of the potential energy of wave formation using the Gauss-Legendre quadrature is described. An analytical expression for integration of the potential of the bottom contact interaction forces with a liquid is obtained as applied to a finite element of the bottom. The techniques of integrating the terms of the functional of the total mechanical energy of the system, providing the conditions for conjugation of the introduced degrees of freedom is described. The conjugation condition of the free surface displacement and the displacement potential of the fluid volume is integrated numerically using the Gauss-Legendre quadrature. The conjugation condition of the elastic bottom bending displacement and the medium displacement potential is integrated analytically. The description of the algorithm of numerical solution of the frequency-modal problem is given. The calculation results for the case of a rigid bottom are presented. The analysis of convergence of finite element solutions to analytical ones for different variants of partitions by finite element mesh is performed. The calculation results for the case of an elastic bottom are presented. The convergence analysis of the solution results for the case of an elastic bottom to the case of a rigid bottom with an increase in its thickness is performed. The analysis of the first form of oscillations for the case of an elastic bottom is presented. Conclusions are made about the applicability of the implemented algorithms to mechanical engineering problems.
[1] Kirilin A.N., Ahmetov R.N., Sollogub A.V. Proektirovanie, dinamika i ustoichivost’ dvizheniya raket-nositelej: metody, modeli, algoritmy, programmy v srede MathCad [Design, dynamics and stability of the motion of launch vehicles: methods, models, algorithms, programs in the MathCad environment]. Moscow, Mashinostroenie Publ., Mashinostroenie-Polet Publ., 2013, 294 p.
[2] Anisimov A.V., Zabudkin V.V., Lihoded A.I., Ponomarev D.A. Dynamicheskoe nagruzhenie pilotiruemyh kosmicheskih stanciy slozhnoi prostranstvennoy konstrukcij iz kompozicionnyh materialov [Dynamic loading of manned space stations with complex spatial layout]. Kosmonavtika i raketostroyeniye [Cosmonautics and rocket science], 1988, no. 13, pp. 130–140.
[3] Balakirev U.G. Issledovaniye ustoychivosti sistemy uprugiy korpus – toplivnyye magistrali – dvigateli dlya zhidkostnykh raket paketnoy komponovki. [Study of the stability of the system elastic body – fuel lines – engines for liquid rockets of package layout]. Izv. RAN. MTT – Proc. Of the Russ. Acad. Sci. Mech. Rigid Body, 1994, no. 2, pp. 129–137.
[4] Park S., Grigor’ev V.G. Ustoychivost' tonkostennykh osesimmetrichnykh soosnykh konstruktsiy, soderzhashchikh zhidkost', pri mnogofaktornykh nagruzkakh [Stability of thin-walled axisymmetric coaxial structures containing liquid under multifactor loads]. Trudy MAI, 2021, no. 119. DOI: 10.34759/trd-2021-119-08.
[5] Grigor’ev V.G. Metodologiya issledovaniya dinamicheskih svojstv slozhnyh uprugih i gidrouprugih system [Methodology for the study of dynamic properties of complex elastic and hydroelastic systems]. Diss. Dr. Sci. (Eng), Moscow, 2000, 328 p.
[6] Gorshkov A.G., Morozov V.I., Ponomarev A.T. Aerogidrouprugost' konstruktsiy [Aerohydroelasticity if structures]. Moscow, Nauka. PML. Publ., 2000, 591 p.
[7] Grigor’ev V.G., Kurakin V.V. Resheniye zadachi chastotno-modal'nogo analiza toplivnogo baka vertoleta Mi-171A3 [Solution of the problem of frequency-modal analysis of the fuel tank of the Mi-171A3 helicopter]. Materialy XXX Mezhdunarodnogo simpoziuma «Dinamicheskiye i tekhnologicheskiye problemy mekhaniki konstruktsiy i sploshnykh sred» im. A.G. Gorshkova T. 1 – Proceedings of the XXX International Symposium “Dynamic and Technological Problems of Structures and Continuous Media” named after A.G. Gorshkov P. 1. Moscow, 2024, pp. 84–86.
[8] Grachev S.V., Smagin D.S., Savel’ev R.S., Napreenko K.S., Zinina A.I. Kontseptsiya modelirovaniya toplivnoy sistemy s uchetom trebovaniy sertifikatsii [Concept of fuel system modeling taking into account certification requirements]. Computational nanotechnology, 2020, vol. 7, no. 3, pp. 45–51.
[9] Sidorenko A.S. Dinamicheskoye sostoyaniye konstruktsii vertoleta pri soudarenii s pregradoy [Dynamic state of the helicopter structure upon impact with an obstacle]. Trudy MAI, 2009, no. 36, pp. 1–14.
[10] Mihailov D.A., Pyhalov A.A., Zenkov E.V., Artunin A.I. Metodika ispytaniya obraztsov iz elastomerov dlya polucheniya mekhanicheskikh kharakteristik ikh gidrouprugosti i konechno-elementnogo modelirovaniya deformiruyemykh sbornykh konstruktsiy [Metodology for testing elastomer samples to obtain mechanical characteristics of their hedroelasticity and finite element modeling of deformable prefabricated structures]. Sovremennyye tekhnologii. Sistemnyy analiz. Modelirovaniye – Modern technologies. System analysis. Simulation, 2021, no. 4 (72), pp. 38–47.
[11] Grigor’ev V.G., Grigor’eva E.V. Contact interaction of bounded liquid volume with a solid under the action of gravity forces. Mechanics of Solids, 2011, vol. 46, no. 2, pp. 280-290.
[12] Grigor’ev V.G. Primenenie metoda konechnyh elementov k raschetu kolebanij uprugih obolochechnyh konstrukcij, soderzhashchih zhidkost’ [Application of the finite element method to the calculation of vibrations of elastic shell structures containing liquid]. Trudy nauchnogo seminara TGU. Dinamika uprugih I tverdyh tel, vzaimodejstvuyushih s zhidkost’yu – Proceedings of the TSU scientific seminar. Dynamics of elastic and solid bodies interacting with a liquid. Tomsk, TSU, 1978, pp. 55–60.
[13] Zienkiewicz O.C., Taylor R.L., Fox D.D. The finite element method for solid and structural mechanics. Butterworth-Heinemann, Elsevier, 2014, 624 p.
[14] Zienkiewicz O.C., Taylor R.L. The finite element method: its basic and fundamentals. Butterworth-Heinemann, Elsevier, 2013, 756 p.
[15] Gupta K.K. Eigenproblem solution by a combined Sturm sequence and inverse iteration technique. International Journal for Numerical Methods in Engineering, 1973, vol.7, no. 1, pp. 17–42.
Григорьев В.Г., Куракин В.В. Моделирование собственных колебаний пространственной гидроупругой конструкции методом конечных элементов на основе смешанного вариационного принципа. Математическое моделирование и численные методы, 2025, № 4, с. 4–18.
Download article
Количество скачиваний: 97