Dynamic model of process of deformation of elastic rack of disk cultivator

The article presents the results of theoretical studies of the dynamic model of the process of deformation of the elastic rack of a disk tool of arbitrary shape, a system of differential equations in general and developed the corresponding program code in Mathematica software package. Taking the form of an elastic discus disk for an Archimedean spiral, when the functions of its boundaries are given in polar coordinates, where the parameters of the geometric shape a (spiral pitch), b (spiral displacement along the radial coordinate), h (elastic column thickness) are determined by its equivalent physical a mathematical model in the form of a rigid mathematical pendulum of length l, to the load of which are attached two springs along the axes Ox and Oz with stiffness coefficients kx and kz, respectively, which deflect it by an angle φ. The dependences of the stiffness coefficients kx and kz, the length l and the angle φ of the equivalent physicomathematical model of the elastic stand of the disc with the parameters of the geometric shape a=0.8 m, b=0 m, h=0.01 m on the values of Fex and Fez, acting on the free end of the rack along the axes Ox and Oz.

Simple Approximate Formulas for Postbuckling Deflection of Heavy Elastic Columns

Columnar buckling is a ubiquitous phenomenon that occurs in both living things and man-made objects, regardless of the length scale ranging from macroscopic to nanometric structures. In general, analyzing the post-buckling behavior of a column requires the application of complex mathematical methods because it involves nonlinear problem solving. To complement these complex methods, this study presents simple analytical formulas for the large deflection of a heavy elastic column under combined loads. The analytical formulas relate the concentrated load acting on the tip of the column, the column’s own weight, and the deflection angle of the column through a simple mathematical expression. This can assist in obtaining an overall picture of the post-buckling behavior of heavy columns from an application point of view.

Optimal Shape and First Integrals for Inverted Compressed Column

We study optimal shape of an inverted elastic column with concentrated force at the end and in the gravitational field. We generalize earlier results on this problem in two directions. First we prove a theorem on the bifurcation of nonlinear equilibrium equations for arbitrary cross-section column. Secondly we determine the cross-sectional area for the compressed column in the optimal way. Variational principle is constructed for the equations determining the optimal shape and two new first integrals are constructed that are used to check numerical integration. Next, we apply the Noether’s theorem and determine transformation groups that leave variational principle Gauge invariant. The classical Lagrange problem follows as a special case. Several numerical examples are presented.