V.L. Litvinov. Solution of boundary value problems with moving boundaries by an approximate method for constructing solutions of integro-differential equations ... P. 188-199

The problem of oscillations of objects with moving boundaries formulated as a differential equation with boundary and initial conditions is a nonclassical generalization of the hyperbolic type problem. To facilitate the construction of a solution to this problem and justify the choice of the solution form, we construct equivalent integro-differential equations with symmetric time-dependent kernels and time-varying integration limits. The advantages of the method of integro-differential equations are found in the transition to more complex dynamic systems that carry concentrated masses oscillating under mobile loads. The method is extended to a broader class of model boundary value problems that take into account the bending stiffness, environmental resistance, and stiffness of the base of the oscillating object. Special attention is paid to the analysis of the most common applied case when the boundaries are subject to external perturbations. The problem is solved in dimensionless variables up to the values of the second order of smallness relative to the small parameters that characterize the speed of the boundary movement. We find an approximate solution of a problem on transverse vibrations of a rope with bending stiffness in a lifting device; one end of the rope is wound on a drum and the other is fixed to a load. The results obtained for the oscillation amplitude corresponding to the nth dynamic mode are presented. The phenomena of steady-state resonance and passage through the resonance are studied by numerical methods.

Keywords: resonance properties, oscillations in systems with moving boundaries, laws of motion of the boundaries, integro-differential equations, amplitude of oscillations

Received March 10, 2020

Revised  May 11, 2020

Accepted  May 18, 2020

Vladislav L’vovich Litvinov, Cand. Sci. (Tech.), Syzran Branch of Samara State Technical University, Syzran, 446001 Russia; Lomonosov Moscow State University, Moscow, 119991 Russia,
e-mail: vladlitvinov@rambler.ru

REFERENCES

1.   Kolosov L.B., Zhigula T.I. Longitudinal-transverse vibrations of a string of a rope of a lifting system. Izv. Vyssh. Uchebn. Zaved. Gorn. Zh., 1981, no. 3, pp. 83–86 (in Russian).

2.   Zhu W.D., Chen Y. Theoretical and experimental investigation of elevator cable dynamics and control. J. Vibr. Acoust., 2006, vol. 128, no. 1, pp. 66–78. doi: 10.1115/1.2128640 

3.   Shi Y., Wu L., Wang Y. Analysis on nonlinear natural frequencies of cable net. J. Vibr. Eng., 2006, vol. 19, no. 2, pp. 173–178.

4.   Goroshko O.A., Savin G.N. Vvedenie v mekhaniku deformiruemykh odnomernykh tel peremennoi dliny [Introduction to the mechanics of deformable one-dimensional bodies of variable length]. Kiev: Naukova dumka Publ., 1971, 224 p.

5.   Litvinov V.L., Anisimov V.N. Transverse vibrations rope moving in longitudinal direction. Izvestiya Samarskogo Nauchnogo Tsentra Rossiiskoi Akademii Nauk, 2017, vol. 19, no. 4, pp. 161–165 (in Russian).

6.   Savin G.N., Goroshko O.A. Dinamika niti peremennoi dliny [Dynamics of a thread of variable length]. Kiev: Nauk. Dumka Publ., 1962, 332 p.

7.   Liu Z., Chen G. Analysis of flat non-linear free oscillations of the supporting rope taking into account the effect of flexural rigidity. J. Vibr. Eng., 2007, no. 1, pp. 57–60.

8.   J. Palm et al. Simulation of mooring cable dynamics using a discontinuous Galerkin method. In: V Internat. Conf. on Computational Methods in Marine Engineering, 2013, pp. 455–466.
ISBN: 978-84-941407-4-7 .

9.   Litvinov V.L. The study of free vibrations of mechanical objects with moving boundaries using the asymptotic method. Zhurn. Srednevolzh. Mat. Obshchestva., 2014, vol. 16, no. 1, pp. 83–88 (in Russian).

10.   Litvinov V.L., Anisimov V.N. Matematicheskoe modelirovanie i issledovanie kolebanii odnomernykh mekhanicheskikh sistem s dvizhushchimisya granitsami [Mathematical modeling and study of oscillations of one-dimensional mechanical systems with moving boundaries]. Samara: Samar. Gos. Tekhn. Univ. Publ., 2017, 149 p. ISBN: 978-5-7964-4984-7 .

11.   Lezhneva A.A. Free bending vibrations of a beam of variable length. In: Uchenye zapiski Permskogo gosudarstvennogo universiteta. Mekhanika [Scientific notes of Perm state University. Mechanics], no. 156. Perm: Perm State Univ. Publ., 1966, pp. 143–150.

12.   Wang L., Zhao Y. Multiple internal resonances and non-planar dynamics of shallow suspended cables to the harmonic excitations. J. Sound Vibr., 2009, vol. 319, no. 1–2, pp. 1–14. doi: 10.1016/j.jsv.2008.08.020 

13.   Zhao Y., Wang L. On the symmetric modal interaction of the suspended cable: three-to one internal resonance. J. Sound Vibr., 2006, vol. 294, no. 4–5, pp. 1073–1093. doi: 10.1016/j.jsv.2006.01.004 

14.   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. Mech. Solids., 2018, vol. 53, no. 2, pp. 177–183. doi: 10.3103/S0025654418020085 

15.   Berlioz A., Lamarque C.-H. A non-linear model for the dynamics of an inclined cable. J. Sound Vibr., 2005, vol. 279, no. 3, pp. 619–639. doi: 10.1016/j.jsv.2003.11.069 

16.   Sandilo S.H., van Horssen W.T. On variable length induced vibrations of a vertical string. J. Sound Vibr., 2014, vol. 333, no. 11, pp. 2432–2449. doi: 10.1016/j.jsv.2014.01.011 

17.   Zhang W., Tang Y. Global dynamics of the cable under combined parametrical and external excitations. Internat. J. of Non-Linear Mechanics, 2002, vol. 37, no. 3, pp. 505–526. doi: 10.1016/S0020-7462(01)00026-9 

18.   Faravelli L., Fuggini C., Ubertini F. Toward a hybrid control solution for cable dynamics: Theoretical prediction and experimental validation. Struct. Control Health Monit., 2010, vol. 17, no. 4, pp. 386–403. doi: 10.1002/stc.313 

19.   Vesnitskii A.I. Volny v sistemakh s dvizhushimisya granitsami i nagruzkami (Waves in Systems with Moving Boundaries and Loads). Moscow: Fizmatlit Publ., 2001, 320 p. ISBN: 5-9221-0172-2 .

20.   Anisimov V.N., Litvinov V.L., Korpen I.V. On a method of analytical solution of wave equation describing the oscillations of systems with moving boundaries. Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki, 2012, vol. 3(28), pp. 145–151 (in Russian). doi: 10.14498/vsgtu1079 

21.   Vesnitskii A.I. The inverse problem for a one-dimensional resonator the dimensions of which vary with time. Radiophys. Quantum Electron., 1971, vol. 14, no. 10, pp. 1209–1215. doi: 10.1007/BF01035071 

22.   Barsukov K.A., Grigoryan G.A. Theory of a waveguide with moving boundaries. Radiophys. Quantum Electron., 1976, vol. 19, no. 2, pp. 194–200. doi: 10.1007/BF01038526 

Cite this article as: V.L. Litvinov. 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, vol. 26, no. 2, pp. 188–199.