Abstract
The synthesis of factorization and symmetry methods produced a general analytical solution to the fourth-order differential equation with variable coefficients. The form and structure of the variable coefficients correspond, in this case, to the problem of the oscillations of a concave beam of variable thickness. The solution to this equation makes it possible to study in detail the oscillations of such and similar, for example convex, beams at the different fixation of their ends' sections. A practical confirmation has been obtained that the beam whose thickness changes in line with the concave parabola law H=a2x2+1, where a is the concave factor, demonstrates an increase in the natural frequencies of its free oscillations with an increase in its rigidity. As an example, the object's maximum deflection dependence on the beam rigidity factor has been established. The nature of this dependence confirmed the obvious conclusion that the deflections had decreased while the rigidity had increased. The evidence from the calculation results can be a testament to the correctness of the reported procedure of problem-solving. The considered problem and the analytical solution to it could serve as a practical guide to the optimal design of beam structures. In this case, it is very important to take into consideration the place and nature of the distribution of cyclical extreme operating stresses. The resulting ratios to solve the problem make it possible to simulate the required normal stresses in both the fixation and central zones when the rigidity parameter is changed. Designers could predict such a parabolic profile of the beam, which would ensure the required reduction of maximum stresses in the place of fixing the beam. The considered example of solving the problem of the natural oscillations of the beam with rigid fixation of the ends illustrates the effectiveness of the factoring and symmetry methods used. The developed solution algorithm could be extended to study the natural bending oscillations of the beam at other fixing techniques, not excluding a variant of a completely free beam
Highlights
Beams with various cross-sections are widely used as components of machines and structures
The results of detailed studies into variable cross-section beams under a free oscillation mode are very limited due to known mathematical difficulties arising when trying to analytically solve the problem of oscillations. This is directly emphasized by the authors of article [5] who state that obtaining an accurate analytical solution is a complex procedure due to the presence of variable coefficients in the main resolving equation
Analytical solutions, unlike approximate or numerical ones, make it possible to expand the existing estimation base for beams with variable thickness, supplementing it with new results obtained in the final form
Summary
Beams with various cross-sections are widely used as components of machines and structures. Researchers use approximate numerical methods to find a solution to a problem, which tend to be cumbersome and impossible to generalize for direct, broad practical applications The relevance of this scientific issue, in addition to the extremely diverse technical application of beam structures, is that it is necessary to have an accurate analytical solution to the relevant boundary problem in order to analyze the natural bending oscillations of the beam of the variable cross-section. This requirement is explained by the fact that in order to ensure the desired operational resource of structures with such beams, at a different form of their fixation, it is necessary to have information on the distribution of deflections and stresses lengthwise a beam element. Such a requirement could only be met on the basis of theoretical analysis of the free oscillations of beams with variable cross-sections
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