×

zbMATH — the first resource for mathematics

Unilateral dynamic contact of two beams. (English) Zbl 0991.74046
Summary: We investigate the dynamic unilateral contact between two beams. The contact is modeled with Signorini or normal compliance conditions. The model is given in the form of a variational inequality, for which we establish an existence theorem. A numerical algorithm for the problem is obtained by using the method of lines in which the problem is approximated by a system of ordinary differential equations. Simulation results are given when one of the beams is driven periodically. We also investigate numerically the characteristics of vibration transmission across the joint between the beams.

MSC:
74M15 Contact in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74H20 Existence of solutions of dynamical problems in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
Software:
VODE
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] K.L. Kuttler nd M. Shillor, Vibrations of a beam between two stops, Dynamics of Continuous, Discrete and Impulsive Systems (no appear). · Zbl 1013.74033
[2] Y. Dumont, D. Goeleven, M. Rochdi and M. Shillor, Simulations of Beam’s vibrations between stops, (in preparation).
[3] Andrews, K.T.; Shillor, M.; Wright, S., On the dynamic vibrations of an elastic beam in frictional contact with a rigid obstacle, J. elasticity, 42, 1-30, (1996) · Zbl 0860.73028
[4] Andrews, K.T.; Kuttler, K.L.; Shillor, M., On the dynamic behaviour of a thermoviscoelastic body in frictional contact with a rigid obstacle, Euro. J. appl. math., 8, 417-436, (1997) · Zbl 0894.73135
[5] Paoli, L.; Schatzman, M., Resonance in impact problems, Math. comput. modelling, 28, 4-8, 385-406, (1998) · Zbl 1126.74484
[6] Moon, F.C.; Shaw, S.W., Chaotic vibration of a beam with nonlinear boundary conditions, J. nonlinear mech., 18, 465-477, (1983)
[7] Moon, F.C., Chaotic and fractal dynamics, (1992), Wiley
[8] Kikuchi, N.; Oden, J.T., Contact problems in elasticity, (1988), SIAM Philadelphia, PA · Zbl 0685.73002
[9] Klarbring, A.; Mikelic, A.; Shillor, M., Frictional contact problems with normal compliance, Intl. J. engng. sci., 26, 8, 811-832, (1988) · Zbl 0662.73079
[10] Martins, J.A.C.; Oden, J.T., A numerical analysis of a class of problems in elastodynamics with friction, Comput. meth. appl. mech. engnr., 40, 327-360, (1983) · Zbl 0527.73079
[11] Rochdi, M.; Shillor, M.; Sofonea, M., Quasistatic viscoelastic contact with normal compliance and friction, J. elasticity, 51, 105-126, (1998) · Zbl 0921.73231
[12] Kuttler, K.L., Time dependent implicit evolution equations, Nonlin. anal., 10, 5, 447-463, (1986) · Zbl 0603.47038
[13] Kuttler, K.L.; Shillor, M., Set-valued pseudomonotone maps and degenerate evolution equations, Comm. contemp. math., 1, 1, 87-123, (1999) · Zbl 0959.34049
[14] Seidman, T.I., The transient semiconductor problem with generation terms, II, Nonlinear semigroups, partial differential equations and attractors, Springer lecture notes in math., 1394, 185-198, (1989)
[15] Schiesser, W.E., Method of lines—integration of partial differential equations, (1991), Academic Press · Zbl 0763.65076
[16] Sincovec, R.F.; Madsen, N.K., Software for nonlinear partial differential equations, ACM, 1, 232-260, (1975) · Zbl 0311.65057
[17] Brown, P.N.; Byrne, G.D.; Hindmarsh, A.C., VODE—A variable coefficient ODE solver, SIAM J. sci. stat. compt., 10, 1038-1051, (1989) · Zbl 0677.65075
[18] Adams, R., Sobolev spaces, (1975), Academic Press · Zbl 0314.46030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.