Geometrically nonlinear plate bending under the action of moving load

Considerable scientific interest is the development of mathematical models that describe the behavior of materials that are sensitive to deformation rate and can improve the accuracy of analytical calculations of their deformation in the region of noticeable changes of loading rates. Nonetheless, in most works, the problems were solved under the assumption of small displacements (geometrically linear statement of the problem). Meanwhile, in practice, this is not always true and bending of cover can be commensurable with its thickness, this article approximately solves the problem of geometrically nonlinear deformation of a thin elastic plate in aquasistatic setting under the action of an infinite normal uniformly distributed load moving along its surface at a constant speed. In the article, the methods of mathematical modeling, the analytical method, as well as the methods of spatial characteristics and bicharacteristics are used. The problem is solved in the quasistatic formulation and is reduced to a system of two nonlinear differential equations for deflections of the plate and the stress function, which include the speed of the load as a parameter. The results of methodological calculations are presented; based on these solutions of linear and nonlinear problems, they were compared, and the influence of finiteness of displacements on the critical speeds of the forces was determined. Materials of the article can be useful in the study of wave dynamics, aircraft, mechanics, and engineering.</jats:p

Quasi-static stability of a ribbed shell interacting with moving load

Loss of stability of thin-walled structural elements of aircraft is the most common type of destruction. In this work, the definition of the critical velocity of a pressure wave moving over the surface of an acclivous, ribbed cylindrical shell, which is the most common design element of all aviation and rocket systems, is considered. The article takes into account the discreteness of the location of stiffeners (stringers), in contrast to other works, where ribbing is taken into account only within the framework of a constructive orthotropic model. In the quasi-static formulation, the problem of the stability of a shallow shell with a discrete arrangement of stringers under the action of a moving radial load has been solved. The range of critical speeds of movement of the load is defined. An example is considered.</jats:p

Dynamics of a cylindrical shell with a collapsing elastic base under the action of a pressure wave

In the dynamic and quasi-static statements, the issue of non-stationary deformation and stability of the solid propellant rocket engine (SPRE) was approximately solved. It is modeled by a thin, smooth cylindrical shell, inside of which, on a part of its length, there is an elastic base corresponding to a gradually burning powder charge. A pressure wave is moving along the outer surface of the body, simulated by the running load. The deformed state of the shell is considered axisymmetric and is determined on the basis of the moment theory of the shells. For diverse variants of mounting the ends of the shell in a closed form, expressions were obtained for the critical velocity of the load. Examples were considered.</jats:p

Dynamic Behavior of a Cylindrical Shell with a Liquid under the Action of Nonstationary Pressure Wave

The problem of axisymmetric hydroelastic deformation of a thin cylindrical shell containing a liquid under the action of a moving load is approximately solved. It is reduced to the equation of bending of the shell and the condition of incompressibility of the liquid in the cylinder. The deflections of the shell and the level of lowering of the liquid are unknown. For solution, the Galerkin method is used and the problem is reduced to a system of nonlinear algebraic equations. A simpler solution is considered without taking into account the incompressibility condition. Here, in addition to the deformed state of the shell, the critical speeds of the moving load are determined analytically.</jats:p

Dynamics of shell with destructive heat-protective coating under running load

The problem of dynamic deformation of a plate with a two-layer composite shell with a heat-shattering coating collapsing in time under the action of a running load is solved approximately. The problem is solved in a dynamic formulation, considering that the deformed state of the shell depends both on the spatial coordinates and on time. The problem is reduced to solving two differential equations of the shell in partial derivatives with respect to deflections and the stress function. These equations contain discontinuous ratios for unknowns, which are associated with the dynamic destruction of the heat-shielding coating. According to the Bubnov method, the problem is also reduced to a system of differential equations, but already in ordinary derivatives. The solution of these equations is obtained in closed form. In addition, the natural vibration frequencies of the structure and the critical velocities of the load are found depending on the degree of damage to the protective layer. Formulas for oscillation frequencies and critical speeds are obtained in closed form.</jats:p