梁的固有弯曲振动具有特殊的宽度变化规律

K. Trapezon, A. Trapezon
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Two examples of the analysis of oscillations of such beam in case of its bilateral rigid fastening and cantilever fastening are resulted. For these cases, the frequency equations are obtained, the natural frequencies and amplitude coefficients are found, which are necessary for the construction of natural forms of oscillations. Originality.The approach presented in this paper is based on the idea of symmetries of differential equations and is characterized by a sim- plified analysis of the solution of the problem of bending oscillations of the beam with a special law of width. The method proposed for solving the boundary value problem is convenient and simple, because the results are found without the use of numerical research methods. Practical value.Examples of the exact analysis of fluctuations which allow to assert about real possibility of expansion of an existing number of configurations of a beam, both on width, and on ways of fastening are resulted. Such beams can, for example, be used as prototype samples for resonant tests of materials for fatigue strength. Сonclusions. The given algorithm for constructing the solution of the problem on eigenvalues for a beam with the given special law of change of width is universal and can be extended to other constructions of beams. 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引用次数: 0

摘要

目的。本文的目的是建立梁的固有振动问题的封闭解析解,其宽度根据exp (αx2)定律变化。方法。该方法基于变系数微分方程对称分析的规定。这种方法可以让你找到一种方法来获得相应微分方程的解析解,并最终得到边值问题。发现。主要成果是构造了该算法并得到了描述梁的横向弯曲振动具有特殊宽度变化规律(梁的厚度为定值)的IV阶微分方程的解。给出了双侧刚性紧固和悬臂紧固两种情况下的振动分析实例。在这些情况下,得到了频率方程,得到了固有频率和幅值系数,这是构造振荡的自然形式所必需的。创意。本文提出的方法基于微分方程的对称性思想,其特点是对具有特殊宽度规律的梁的弯曲振动问题的解进行了模拟化分析。所提出的求解边值问题的方法方便、简单,无需使用数值研究方法即可得到结果。实用价值。给出了对波动进行精确分析的例子,这些例子使我们能够断言梁在宽度和紧固方式上的现有若干构型的扩展的真实可能性。例如,这种梁可以用作材料疲劳强度共振试验的原型样品。Сonclusions。本文给出的构造具有特定宽度变化规律的梁的特征值问题解的算法具有普适性,并可推广到其他梁结构。因为在这种情况下,问题只出现在近似函数的选择上,它允许使用对称方法并获得相应的IV阶微分方程的精确解,这反过来又描述了梁的横向振荡。
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NATURAL BENDING VIBRATIONS OF THE BEAM WITH THE SPECIAL LAW OF CHANGE OF WIDTH
Purpose. The aim of the work is to construct a closed analytical solution of the problem of natural vibrations of the beam, the width of which varies according to the law exp (αx2). Methodology. The approach is based on the provisions of symmetric analysis of differential equations with variable coefficients. This approach allows you to find a way to obtain an analytical solution of the corresponding differential equation and, ultimately, the boundary value problem. Findings. The main result is the construction of the algorithm and obtaining the solution of the differential equation of the IV order, which describes the transverse bending vibrations of the beam with a special law of change of width (the thickness of the beam is a constant value). Two examples of the analysis of oscillations of such beam in case of its bilateral rigid fastening and cantilever fastening are resulted. For these cases, the frequency equations are obtained, the natural frequencies and amplitude coefficients are found, which are necessary for the construction of natural forms of oscillations. Originality.The approach presented in this paper is based on the idea of symmetries of differential equations and is characterized by a sim- plified analysis of the solution of the problem of bending oscillations of the beam with a special law of width. The method proposed for solving the boundary value problem is convenient and simple, because the results are found without the use of numerical research methods. Practical value.Examples of the exact analysis of fluctuations which allow to assert about real possibility of expansion of an existing number of configurations of a beam, both on width, and on ways of fastening are resulted. Such beams can, for example, be used as prototype samples for resonant tests of materials for fatigue strength. Сonclusions. The given algorithm for constructing the solution of the problem on eigenvalues for a beam with the given special law of change of width is universal and can be extended to other constructions of beams. Since in this case the question arises only about the choice of the approximation function, which allows to use the method of symmetries and obtain an exact solution of the corresponding differential equation of the IV order, which in turn describes the transverse oscillations of the beams.
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