Numerical Study of Longitudinal-Transverse Vibrations of Objects with Moving Boundaries in the Developed Software Package TB-Analisys
Abstract
Difference numerical schemes for solutions of problems describing the longitudinal-cross oscillations of object with moving boundaries are noted. The scheme allows us to solve the Cauchy problem for a system of nonlinear differential equations with nonlinear boundary conditions and to take into account the energy exchange between the parts of the vibrating object on the left and right of the moving boundary. The grid is divided into equally spaced time layers by the time variable. The grid is divided into a fixed number of parts, equidistant nodes by the space variable in each time step to the left and right of the moving boundary. Partitioning step in temporary layers are different in connection with the movement of the boundary. Such a partition avoids the transition moving boundary through the nodes of the grid. To find the functions and their derivatives are used finite difference approximation. Approximation error is of second order of smallness relative to the grid spacing on the space and time variables. The solution obtained by successive transition from one time to another layer. The accuracy of the numerical solution is confirmed by the coincidence of the solutions of the linear and nonlinear models at low vibration amplitudes. The solution is made in dimensionless variables using the TB-Analisys software package developed in the Matlab environment, which allows using the results obtained for calculating a wide range of technical objects.
References
2. Brake M.R., Wickert J.A. Frictional vibration transmission from a laterally moving surface to a traveling beam. Journal of Sound and Vibration. 2008;310(3):663-675. doi: https://doi.org/10.1016/j.jsv.2007.04.029
3. Cho Y.H. Numerical simulation of the dynamic responses of railway overhead contact lines to a moving pantograph, considering a nonlinear dropper. Journal of Sound and Vibration. 2008;315(3):433-454. doi: https://doi.org/10.1016/j.jsv.2008.02.024
4. Ding H., Chen L.-Q. Galerkin methods for natural frequencies of high-speed axially moving beams. Journal of Sound and Vibration. 2010;329(17):3484-3494. doi: https://doi.org/10.1016/j.jsv.2010.03.005
5. Inácio O., Antunes J., Wright M.C.M. Computational modelling of string-body interaction for the violin family and simulation of wolf notes. Journal of Sound and Vibration. 2008;310(1-2):260-286. doi: https://doi.org/10.1016/j.jsv.2007.07.079
6. Nakagawa C., Shimamune R., Watanabe K., Suzuki E. Fundamental Study on the Effect of High-frequency Vibration in the Vertical and Lateral Directions on Ride Comfort. Quarterly Report of RTRI. 2010;51(2):101-104. doi: https://doi.org/10.2219/rtriqr.51.101
7. Ryue J., Thompson D.J., White P. R., Thomspon D.R. Decay rates of propagating waves in railway tracks at high frequencies. Journal of Sound and Vibration. 2009;320(4-5):955-976. doi: https://doi.org/10.1016/j.jsv.2008.09.025
8. Sahebkar S.M., Ghazavi M.R., Khadem S.E., Ghayesh M.H. Nonlinear vibration analysis of an axially moving drillstring system with time dependent axial load and axial velocity in inclined well. Mechanism and Machine Theory. 2011;46(5):743-760. doi: https://doi.org/10.1016/j.mechmachtheory.2010.12.003
9. Shi Y.-j., Wu L.-l., Wang Y.-q. Analysis on nonlinear natural frequencies of cable net. Journal of Vibration Engineering. 2006;(2):173-178. Available at: http://caod.oriprobe.com/order.htm?id=10291215 (accessed 13.09.2022).
10. Sun L., Luo F. Steady-State Dynamic Response of a Bernoulli-Euler Beam on a Viscoelastic Foundation Subject to a Platoon of Moving Dynamic Loads. Journal of Vibration and Acoustics. 2008;130(5):051002. doi: https://doi.org/10.1115/1.2948376
11. Teng Y.-f., Teng N.-g., Kou X.-j. Vibration analysis of maglev three–span continuous guideway considering control system. Journal of Zhejiang University-SCIENCE A. 2008;9(1):1-14. doi: https://doi.org/10.1631/jzus.A071214
12. Wang L., Zhao Y. Multiple internal resonances and non–planar dynamics of shallow suspended cables to the harmonic excitations. Journal of Sound and Vibration. 2008;319(1-2):1-14. doi: https://doi.org/10.1016/j.jsv.2008.08.020
13. Yagci B., Filiz S., Romero L.L., Ozdoganlar O.B. A spectral-Tchebychev technique for solving linear and nonlinear beam equations. Journal of Sound and Vibration. 2009;321(1-2):375-404. doi: https://doi.org/10.1016/j.jsv.2008.09.040
14. Zhao Y., Wang L. On the symmetric modal interaction of the suspended cable: three-to-one internal resonance. Journal of Sound and Vibration. 2006;294(4-5):1073-1093. doi: https://doi.org/10.1016/j.jsv.2006.01.004
15. Zhu W.D., Zheng N.A. Exact Response of a Translating String With Arbitrarily Varying Length Under General Excitation. Journal of Applied Mechanics. 2008;75(3):031003. doi: https://doi.org/10.1115/1.2839903
16. Anisimov V.N., Litvinov V.L. Investigation of resonance characteristics of mechanical objects with moving borders by application of the Kantorovich-Galyorkin method. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2009;(1):149-158. Available at: https://www.elibrary.ru/item.asp?id=13069346 (accessed 13.09.2022). (In Russ., abstract in Eng.)
17. Anisimov V.N., Litvinov V.L. Mathematical models of nonlinear longitudinal-cross oscillations of object with moving borders. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2015;19(2):382-397. (In Russ., abstract in Eng.) doi: https://doi.org/10.14498/vsgtu1330
18. Anisimov V.N., Korpen I.V., Litvinov V.L. Application of the Kantorovich-Galerkin Method for Solving Boundary Value Problems with Conditions on Moving Borders. Mechanics of Solids. 2018;53(2):177-183. doi: https://doi.org/10.3103/S0025654418020085
19. Berlioz A., Lamarque C.-H. A non-linear model for the dynamics of an inclined cable. Journal of Sound and Vibration. 2005;279(3-5):619-639. doi: https://doi.org/10.1016/j.jsv.2003.11.069
20. Sandilo S.H., van Horssen W.T. On variable length induced vibrations of a vertical string. Journal of Sound and Vibration. 2014;333(11):2432-2449. doi: https://doi.org/10.1016/j.jsv.2014.01.011
21. Liu Z., Chen G.-p. Planar non-linear free vibration analysis of stay cable considering the effects of flexural rigidity. Journal of Vibration Engineering. 2007;(1):57-60. Available at: http://caod.oriprobe.com/order.htm?id=11869569 (accessed 13.09.2022).
22. Zhang W., Tang Y. Global dynamics of the cable under combined parametrical and external excitations. International Journal of Non-Linear Mechanics. 2002; 37(3):505-526. doi: https://doi.org/10.1016/S0020-7462(01)00026-9
23. Faravelli L., Fuggini C., Ubertini F. Toward a hybrid control solution for cable dynamics: Theoretical prediction and experimental validation. Structural Control and Health Monitoring. 2010;17(4):386-403. doi: https://doi.org/10.1002/stc.313
24. Anisimov V.N., Litvinov V.L., Korpen I.V. On a method of analytical solution of wave equation describing the oscillations system with moving boundaries. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2012;(3):145-151. Available at: https://www.elibrary.ru/item.asp?id=19092391 (accessed 13.09.2022). (In Russ., abstract in Eng.)
25. Litvinov V.L. Solution of boundary value problems with moving boundaries by an approximate method for constructing solutions of integro-differential equations. Trudy Instituta Matematiki i Mekhaniki URO RAN. 2020;26(2):188-199. (In Russ., abstract in Eng.) doi: https://doi.org/10.21538/0134-4889-2020-26-2-188-199
26. Litvinov V.L., Anisimov V.N. Transverse vibrations rope moving in longitudinal direction. Izvestia of Samara Scientific Center of the Russian Academy of Sciences. 2017;19(4):161-166. Available at: https://www.elibrary.ru/item.asp?id=32269460 (accessed 13.09.2022). (In Russ., abstract in Eng.)
27. Litvinov V.L., Litvinova K.V. An Approximate Method for Solving Boundary Value Problems with Moving Boundaries by Reduction to Integro-Differential Equations. Computational Mathematics and Mathematical Physics. 2022;62(6):945-954. doi: https://doi.org/10.1134/S0965542522060112
This work is licensed under a Creative Commons Attribution 4.0 International License.
Publication policy of the journal is based on traditional ethical principles of the Russian scientific periodicals and is built in terms of ethical norms of editors and publishers work stated in Code of Conduct and Best Practice Guidelines for Journal Editors and Code of Conduct for Journal Publishers, developed by the Committee on Publication Ethics (COPE). In the course of publishing editorial board of the journal is led by international rules for copyright protection, statutory regulations of the Russian Federation as well as international standards of publishing.
Authors publishing articles in this journal agree to the following: They retain copyright and grant the journal right of first publication of the work, which is automatically licensed under the Creative Commons Attribution License (CC BY license). Users can use, reuse and build upon the material published in this journal provided that such uses are fully attributed.
