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Vibrations of a Gao beam subjected to a moving mass. (English) Zbl 07155286
Summary: This paper models, analyzes and simulates the vibrations of a nonlinear Gao beam that is subjected to a moving mass or a massless point-force. Such problems arise naturally in transportation systems such as trains or trams. The dynamics of the system as the mass or the force move on the beam are investigated numerically in the cases when the vibrations are about a buckled state, and in the cases when the mass is positive or vanishes. The simulations are compared to those of the Euler-Bernoulli linear beam and the differences are highlighted. It is seen that the linear beam may be used only when the loads are small, while the Gao beam allows for moderate loads. The simulations are based on a time-marching finite elements algorithm for the model that has been developed and implemented. The results of representative and interesting computer simulations are depicted. The existence of weak solutions of the model is established using a variational formulation of the problem and results about variational set-inclusions.
74H45 Vibrations in dynamical problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74H20 Existence of solutions of dynamical problems in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
Full Text: DOI
[1] Metrikine, A. V.; Dieterman, H. A., Instability of vibrations of a mass moving uniformly along an axially compressed beam on a viscoelastic foundation, J. Sound Vib., 201, 5, 567-576 (1997) · Zbl 1235.74183
[2] Stojanović, Vladimir; Kozić, Predrag; Petković, Marko D., Dynamic instability and critical velocity of a mass moving uniformly along a stabilized infinity beam, Int. J. Solids Struct., 108, 164-174 (2017)
[3] Xiang, Ping; Wang, H. P.; Jiang, L. Z., Simulation and sensitivity analysis of microtubule-based biomechanical mass detector, J. Nanosci. Nanotechnol., 19, 2, 1018-1025 (2019)
[4] Dyniewicz, B.; Konowrocki, R.; Bajer, C. I., Intelligent adaptive control of the vehicle-span/track system, Mech. Syst. Signal Process., 53, 1, 1-14 (2015)
[5] Gao, D. Y., Nonlinear elastic beam theory with application in contact problems and variational approaches, Mech. Res. Commun., 23, 1, 11-17 (1996) · Zbl 0843.73042
[6] M’Bengue, M. F., Analysis of a Nonlinear Dynamic Beam with Material Damage or Contact (2008), Oakland University, (Ph.D. thesis)
[7] Kuttler, K. L.; Purcell, J.; Shillor, M., Analysis and simulations of a contact problem for a nonlinear dynamic beam with a crack, Q. J. Mech. Appl. Math. (2011) · Zbl 1248.74032
[8] Dahlberg, T., Track issues, (Iwnicki, Simon, Handbook of Railway Vehicle Dynamics (2006), CRC Press), 143-180
[9] Bajer, C. I.; Bogacz, R., Propagation of perturbances generated in classic track, and track with Y-type sleepers, Arch. Appl. Mech., 74, 754-761 (2005) · Zbl 1158.74372
[10] Bajer, C. I.; Dyniewicz, B., Numerical modelling of structure vibrations under inertial moving load, Arch. Appl. Mech., 79, 6-7, 499-508 (2009) · Zbl 1264.74086
[11] Matej, J., A new mathematical model of the behaviour of a four-axle freight wagon with UIC single-link suspension, Proc. IMechE F: J. Rail Rapid Transit., 225, 6, 637-647 (2011)
[12] Gao, D. Y.; Russell, D. L., An extended beam theory for smart materials applications: II. static formation problems, Appl. Math. Optim., 38, 1, 69-94 (1998) · Zbl 0911.35039
[13] Gao, D. Y., Finite deformation beam models and triality theory in dynamical post-buckling analysis, Intl. J. Non-Linear Mech., 35, 103-131 (2000) · Zbl 1068.74569
[14] Russell, D. L.; White, L. W., A nonlinear elastic beam system with inelastic contact constraints, Appl. Math. Optim., 46, 291-312 (2002) · Zbl 1076.74029
[15] Andrews, K. T.; M’Bengue, M. F.; Shillor, M., Vibrations of a nonlinear dynamic beam between two stops, Discrete Contin. Dyn. Syst. (DCDS-B), 12, 1, 23-38 (2009) · Zbl 1167.74019
[16] M’Bengue, M. F.; Shillor, M., Regularity result for the problem of vibrations of a nonlinear beam, Electron. J. Differential Equations, 27, 1-12 (2008) · Zbl 1137.74027
[17] Andrews, K. T.; Dumont, Y.; M’Bemgue, M. F.; Purcell, J.; Shillor, M., Analysis and simulations of a nonlinear dynamic beam, Appl. Math. Phys. (ZAMP), 63, 6, 1005-1019 (2012) · Zbl 1261.35093
[18] Ahn, J.; Kuttler, K. L.; Shillor, M., Dynamic contact of two Gao beams, Electron. J. Differential Equations, 194, 1-42 (2012) · Zbl 1302.74116
[19] Andrews, K. T.; Kuttler, K. L.; Shillor, M., Dynamic Gao Beam in Contact with a Reactive or Rigid Foundation, Vol. 33, 225-248 (2015) · Zbl 1317.74049
[20] Shillor, M.; Sofonea, M.; Telega, J. J., (Models and Analysis of Quasistatic Contact. Models and Analysis of Quasistatic Contact, Lecture Notes in Physics (2004)) · Zbl 1180.74046
[21] Kuttler, K. L.; Shillor, M., Set-valued pseudomonotone maps and degenerate evolution inclusions, Commun. Contemp. Math., 1, 1, 87-123 (1999) · Zbl 0959.34049
[22] Bajer, C. I.; Dyniewicz, B.; Shillor, M., A Gao beam subjected to a moving inertial point load, Math. Mech. Solids, 23, 3, 461-472 (2018) · Zbl 1404.74058
[23] Kuttler, K. L.; Li, J.; Shillor, M., Existence for dynamic contact of a stochastic viscoelastic Gao beam, Nonlinear Anal. RWA, 22, 4, 568-580 (2015) · Zbl 1326.74099
[24] Migórski, S.; Ochal, A.; Sofonea, M., Nonlinear inclusions and hemivariational inequalities, (Models and Analysis of Contact Problems. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, vol. 26 (2013), Springer) · Zbl 1262.49001
[25] (Wriggers, P.; Nackenhorst, U., Analysis and Simulation of Contact Problems. Analysis and Simulation of Contact Problems, Lecture Notes in Applied and Computational Mechanics, vol. 27 (2006), Springer) · Zbl 1089.74008
[26] Gurtin, M. E., Variational principles for linear initial – value problems, Quart. Appl. Math., 22, 252-256 (1964) · Zbl 0173.37602
[27] Herrera, I.; Bielak, J., A simplified version of Gurtin’s variational principles, Arch. Ration. Mech. Anal., 53, 131-149 (1974) · Zbl 0288.49018
[28] Oden, J. T., A generalized theory of finite elements. II. Applications, Internat. J. Numer. Methods Engrg., 1, 247-259 (1969) · Zbl 0263.73048
[29] Argyris, J. H.; Chan, A. S.L., Application of the finite elements in space and time, Ing. Archiv., 41, 235-257 (1972) · Zbl 0241.73096
[30] Bajer, C. I.; Bohatier, C., The soft way method and the velocity formulation, Comput. Struct., 55, 6, 1015-1025 (1995) · Zbl 0918.73067
[31] Dyniewicz, B.; Pisarski, D.; Bajer, C., Vibrations of a Mindlin plate subjected to a pair of inertial loads moving in opposite directions, J. Sound Vib., 386, 265-282 (2017)
[32] Bajer, C. I.; Dyniewicz, B., Virtual functions of the space – time finite element method in moving mass problems, Comput. Struct., 87, 444-455 (2009)
[33] Dyniewicz, B., Space – time finite element approach to general description of a moving inertial load, Finite Elem. Anal. Des., 62, 8-17 (2012)
[34] Dyniewicz, B.; Bajer, C. I., Paradox of the particle’s trajectory moving on a string, Arch. Appl. Mech., 79, 3, 213-223 (2009) · Zbl 1168.74383
[35] Dyniewicz, B.; Bajer, C. I., New feature of the solution of a Timoshenko beam carrying the moving mass particle, Arch. Mech., 62, 5, 327-341 (2010) · Zbl 1269.74135
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