Abstract
The dynamic behavior of rod systems under the action of external force factors described by multivalued (subdifferential) relations is studied. The mathematical formulation of the problem is given in the form of a dynamic quasivariational inequality. With the use of the Newmark difference scheme, successive approximations, and finite-element discretization, the problem is reduced to minimization of a convex nonsmooth finite-dimensional functional with respect to velocities at each time step. Introduction of auxiliary variables by the method of a modified Lagrangian reduces the problem of minimization of this functional to a sequence of smooth problems of nonlinear programming. The algorithm is verified using the numerical solution for a problem with one degree of freedom. The algorithm proposed is used to calculate the rods of deep-well pumps.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
V. L. Biderman, Applied Theory of Mechanical Oscillations [in Russian], Vysshaya Shkola, Moscow (1972).
N. V. Butenin, Yu. I. Neimark, and N. A. Fufaev, Introduction into the Theory of Nonlinear Oscillations [in Russian], Nauka, Moscow (1976).
V. I. Gulyaev, V. A. Bazhenov, and S. L. Popov, Applied Problems of the Theory of Nonlinear Oscillations of Mechanical Systems [in Russian], Vysshaya Shkola, Moscow (1989).
A. A. Andronov, A. A. Vitt, and S. É. Khaikin, Theory of Oscillations, Pergamon Press, Oxford (1966).
A. F. Filippov, Diffierential Equations with a Discontinuous Right Side [in Russian], Nauka, Moscow (1985).
M. I. Feigin, Forced Oscillations of Systems with Discontinuous Nonlinearities [in Russian], Nauka, Moscow (1994).
E. N. Rozenvasser, “General equations of discontinuous-system sensitivity,” Avtomat. Telemekh., No. 3, 52-57 (1967).
G. Duvaut and J.-L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris (1972).
R. Glowinski, J. L. Lions, and R. Tremolieres, Analyse Numerique des Inequations Variationnelles, Dunond, Paris (1976).
P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions, Birkhäuser Verlag, Boston (1985).
C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems, John Wiley, Chichester (1984).
I. Hlavacek, J. Haslinger, J. Necas, and J. Lovisek, Solution of Variational Inequalities in Mechanics, Springer (1982).
A. S. Kravchuk, “To the theory of contact problems with friction on the contact surface,” Prikl. Mat. Mekh., 44, No. 1, 122-129 (1980).
A. V. Vovkushevskii, “Variational formulation and solution methods of a contact problem with friction with allowance for surface roughness,” Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 3, 56-62 (1991).
P. Alart and A. Curnier, “A mixed formulation for frictional contact problems proned to Newton-like solution method,” Comput. Meth. Appl. Mech. Eng., 92, 353-375 (1991).
D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York (1982).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Vasserman, I.N., Shardakov, I.N. Formulation and Solution of Dynamic Problems of Elastic Rod Systems Subjected to Boundary Conditions Described by Multivalued Relations. Journal of Applied Mechanics and Technical Physics 44, 406–414 (2003). https://doi.org/10.1023/A:1023493425848
Issue Date:
DOI: https://doi.org/10.1023/A:1023493425848