Method demonstrates solving differential equations for boundary conditions in structural mechanics, indicating broader applications in engineering.
A method for solving problems in structural mechanics is presented, based on Cauchy's theorems The method is presented using simple examples: beam bending on an elastic foundation and vibrations of a system with one degree of freedom. Differential equations are presented in generalized functions, which allows taking into account boundary and initial conditions in the equations. The righthand sides of the equations written in this way contain parameters that determine both the specified boundary conditions and the unknowns. The integral Fourier transform is used in the solution. To determine the unknown boundary conditions, the conditions for the analyticity of the Fourier images of the displacement function in the upper complex half-plane are used (the Cauchy integral theorem). Thus, a system of equations is obtained for obtaining unknown boundary conditions. When performing the inverse Fourier transform, the Cauchy residue theorem is used. As an example, the solution of oscillations of a system with one degree of freedom with different damping coefficients is given.
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Kurbatskiy et al. (2025) studied this question.
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