A Two-layer Semi Empirical Model of Nonlinear Bending of the Cantilevered Beam

This paper suggests a semi empirical model of nonlinear bending of the cantilevered beam. The model learns by using the results of measurement of nonlinear bending of the cantilevered metal pipe loaded by sinkers on the end. These studies are of interest for longterm forecasting of the behavior of building beams, various structural elements of load-lifting machines and mechanisms. 1.Introduction Modeling complex technical objects is often hampered by insufficient knowledge of the processes occurring in them. The model of such an object in the form of a differential equation and boundary conditions can cause false ideas about the accuracy of modeling, since the structure and coefficients of the equations are not known accurately, even less accurately the boundary conditions are usually known. All these parameters are usually changing during the operation of the object, which makes the prediction of its state on the basis of the original model even more problematic. To identify the state of an object, one can involve the results of observations of it, but the problem of identifying equations and boundary conditions from these data (the inverse problem) is usually much more complicated than the direct problem of solving a differential equation with boundary conditions. Previously, we solved such problems using our methodology for constructing the neural network model of the object by differential equations and additional data [1-8]. However, the training of neural networks requires a fairly large computational cost. To solve this problem, a new class of multi-layer models [9] was developed, with the help of which it is possible to do it without a complicated training procedure. This approach creates additional opportunities for combining classical and new methods. The essence of the approach is to apply known numerical methods for integrating differential equations to an interval with a variable upper limit. As a result, an approximate solution is obtained not in the form of a set of numerical values, but as a function of this upper limit. The method [9] can be used together with neural networks, but in this paper we managed without using them. In this paper, this approach is tested on the bend of a cantilevered metal tube. The developed methods can be applied for long-term forecasting of the behavior of building beams, various structural elements of load-lifting machines and mechanisms, taking into account the real picture of wear, aging and corrosion of metal. In [10] the necessity of an estimation of influence of deterioration of the 4 Author to whom any correspondence should be addressed building equipment and the hand tool on working conditions of workers of building industry as the most traumatic kind of activity is revealed and justified. From this point of view, solving inverse problems in modeling the state of loaded elements will increase the reliability of forecasting results. 2.Material and methods The measurements were carried out at the next experimental setup. A straight metal tube 1060 mm long was taken. And a mass of 116 g of circular cross-section with an outer diameter of 1 sm and a wall thickness of 2 mm, onto which labels were applied after 50 mm. One end of the tube, at a distance of 60 mm from the left end, was clamped in the vice, and loads with a weight of 100 g to 1300 g were attached alternately to the right with the help of a thread. A screen with a millimeter grid was located behind the experimental setup at a distance of 100 mm in a vertical plane. The horizontal and vertical lines of the grid were controlled by a bubble level with an error of no more than 3 mm per 1000 mm in length. Before the experimental installation at a distance of 1500 mm there was a camera, connected with a computer. In the center of the image was the middle of a tube with no load. To convert the coordinates from the pixel space into the space of a millimeter grid with optical distortion compensation, two calibration functions were constructed from two variables over a sufficiently large number of points. The resulting measurement error should not exceed 5 mm. As a mathematical model we use the equation of a large static deflection of a thin homogeneous physically linear elastic rod under the action of distributed q and concentrated p forces [11].