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Exact and approximate equations of static and dynamic web-strip and generalized classical model

Paimushin VN, Polyakov TV

#### Abstract:

We consider the two-dimensional spatial coordinates linear problems of statics and dynamics of the rod-band linear elastic orthotropic material under the action of arbitrary volume and surface loads applied both to the end transverse and longitudinal sections of the facial. The latter are represented in the expansion phase and antiphase components, resulting in the original problem to the problem of in-phase and antiphase kinds of strains, called tasks A and B in the literature of Using trigonometric basis functions corresponding to the allocation of the individual Fourier series of odd (for task A) and even ( As for the problem) harmonics constructed approximating the transverse coordinate movement functions accurately meet the static boundary conditions on the longitudinal edges of the bar. It was found that based on them are absolutely precise equation for problems of free oscillations (in the absence of zero harmonics) and absolutely not suitable for the problems of static bars. At the same time in the limit to zero harmonic motion features built into the kinematic relations are reduced, called relations generalized classical model. Based on them, such as built kinematic relations of the zero approximation, in which the desired unknowns are the two unknown functions of the classical theory. On this basis, we derived one-dimensional spatial coordinates of the equation of static equilibrium and motion, which are different versions of equations of zero approximation and similar equations generalizing the classical theory. By introducing a single simplifying assumptions about the vanishing of the coefficient of Poisson constructed as such updated equation of the first approximation, based on an approximation of the transverse voltage component trigonometric functions at the exact satisfaction of the boundary conditions A and B on the longitudinal edges of the rod, which are valid for the description of both static and dynamic deformation of the rods, as well as allow for the limiting transition to classical equations of the theory of rods.

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