Mathematic Al model of physically nonlinear longitudinal vibrations of a truncated conical rod

Conical elements that make unsteady oscillations are found in the frames of fastening engines, in the columns of construction structures, supporting floors on which unbalanced units are installed, when installing suspensions, etc.Such elements are important structural elements of aerospace vehicles and are widely used in various applications, ranging from beam domes to space launch vehicles with large fuel tanks.Conical sections are found on the details of propeller shafts and shafts of powerful drives.The task of calculating a shaft with a variable cross-section for torsional vibrations is carried out approximately, replacing the shaft with a system of elements with a finite number of degrees of freedom, which leads to complex and cumbersome calculations.In the proposed article, a non-linear mathematical model of unsteady longitudinal vibrations of a truncated conical rod made of elastic material is developed, taking into account the nonlinear relationship between stresses and deformations.Refined, physically nonlinear equations of longitudinal vibrations of a truncated conical rod from a homogeneous and isotropic material are derived, from which, in a particular case, some well-known classical-type rod oscillation equations can be obtained.

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