International Conference
"Modern Problems of Applied Mathematics and Mechanics: Theory, Experiment and Applications", devoted to the 90th anniversary of professor Nikolai N. Yanenko

Alexandrova N.I.  

Propagation of resonance waves in a square lattice. An antiplane problem

      The dynamics of a block medium is studied in the pendulum approximation, where it is assumed that the blocks are incompressible, and all the deformations and displacements are due to the compressibility of the interlayers. In this case, a lattice of masses connected together by springs may be considered as a computational model. Within the frames of this model, an antiplane deformation of a two-dimensional square lattice is studied which consists of masses connected by springs with equal stiffness in the both directions. The unsteady propagation of disturbances is studied under the action of concentrated sinusoidal load.
      For this system, there exist two critical frequencies of the load such that a resonance growth of perturbations occurs in the system. The resonant mode of wave propagation in a square lattice under the action of a sinusoidal load with one of the critical frequencies is studied both numerically and analytically. A finite-difference solution of the problem is obtained numerically for the entire time interval. Asymptotic estimates of the behavior of the perturbation are derived analytically for an infinitely long time since the beginning of the excitation. Numerical and analytical results are compared with each other. It is shown that they are in a good agreement on a finite interval of time since the beginning of the excitation.

The research was supported by the Russian Foundation for Basic Research (grant No.08-05-00509) and by the Siberian Branch of the Russian Academy of Sciences (integration project No. 74).