About determination of parameters of elastic fixing of the non-uniform beam
UDC
539.3Abstract
Natural oscillations of an elastic rod with variable stiffness defined by an arbitrary positive function are considered. Natural oscillations in cases of arbitrary and constant stiffness are studied. Using the shooting method for the Cauchy problem, the differential equation of 4th order is reduced to the canonical system of differential equations, by solving the system and building a linear combination of the solutions, the algebraic system was obtained. To analyze the zeros of the system determinant, a program carrying out the construction of surface solutions is obtained. The three surface sections of solutions were analyzed. An asymptotic representation of the solutions for small and large fixing elastic parameters was constructed. Inverse problem on determination of the parameters of elastic fixing of the inhomogeneous beam was formulated. A new approach to solve this problem was proposed; based on the (priori known) asymptotic and numerical analysis in case of intermediate values of the parameters, the two methods of reconstruction of the described parameters characterizing the elastic fixing at one end of the beam at two known resonance frequencies. The influence of the impact of the input data noising on the reconstruction of the unknown parameters is estimated, the effect of deviations on the results is evaluated. The proposed methods allow restoring the parameters of the noise no more than 3% with high accuracy.
Keywords:
equation of 4th order with variable coefficients, boundary conditions, natural vibrations, reconstruction, asymptotic behavior, elastic fixationReferences
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