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Conservative linear models for the vibrating string and rod free to slide at one end point, on a vertical guide. (English) Zbl 0903.73033

Summary: The motion of a linear vibrating string, or rod, fixed at one point and free to slide on a vertical guide at the other one, is studied. Only transversal vibrations are considered. The motion is simulated by a discrete conservative model. In order to prove that the total energy is conserved, a matrix form of the potential and kinetic energy is used. The numerical method related to the model turns out to be unconditionally stable. Some examples of the motion simulated by the model are given.

MSC:

74H45 Vibrations in dynamical problems in solid mechanics
74S20 Finite difference methods applied to problems in solid mechanics
74K05 Strings
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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References:

[1] Frontini, M.; Gotusso, L., A discrete conservative model for the linear vibrating string and rod, Computers Math. Applic., 33, 10, 53-65 (1997) · Zbl 0890.73032
[2] Greenspan, D., Computer Oriented Mathematical Physics (1981), Pergamon: Pergamon New York · Zbl 0462.70001
[3] Frontini, M.; Gotusso, L., Numerical study of the motion of a string vibrating against an obstacle by physical discretization, Appl. Math. Modeling, 14, 489 (1990) · Zbl 0712.70040
[4] Gotusso, L.; Veneziani, A., Discrete and continuous non linear models for the vibrating rod, Comp. Math. Modeling, 24, 99 (1996) · Zbl 0920.73079
[5] Gotusso, L.; Veneziani, A., Discrete and continuous non linear models for the vibrating string, WSSIAA, 4, 295 (1995) · Zbl 0840.73027
[6] Frontini, M.; Gotusso, L., Some results about boundary conditions for a discrete model in elasticity, (Proceedings Seventh Int. Coll. on Diff. Eq. (1996)) · Zbl 0953.74617
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