Longitudinal resonance vibrations of the viscoelastic rod with a variable length
Fluctuations of the core which is burning at one end are investigated in the paper. The research object belongs to a wide range of fluctuating one-dimensional objects with moving boundaries and loadings. The classical mathematical model considering viscoelasticity on the basis of the structural model of Foigt is used for the description of fluctuations. By introducing the dimensionless variables it became possible to reduce the number of parameters (which affect the process of fluctuations) to two. The parameters characterize the speed of the border’s motion and viscoelastic properties of the core. The method of Kantorovich-Galerkin is applied for the solution. Eigen functions of the boundary problem with a motionless border are taken as dynamic modes. The solution when the speed of the border’s motion is equal to zero is precise. The error of the solution increases, when the speed of the border’s motion increases. By neglecting the small values, it became possible to obtain quite a simple expression for the amplitude of resonant fluctuations. The expression obtained for the amplitude of fluctuations contains the integrals having no analytical solution, therefore they were obtained numerically. The solution has a mode structure that allows analyzing the resonant properties of the core. The obtained solution made it possible to analyze the phenomena of the established resonance and the process of passing through the resonance. The analytical expression describing the increase in amplitude of fluctuations is obtained for the established resonance. The passing through the resonance is analyzed quantitatively. The graphs of the amplitude of fluctuations changing in the resonant area for the first dynamic mode at various values of the parameter characterizing the viscoelasticity are presented. Also the graphs of the maximum amplitude of fluctuations are provided when passing through the resonance at the first dynamic mode, depending on the parameters which characterize the viscoelasticity and speed of the border’s motion. The results of numerical calculations are processed by means of the least squares method. The expressions for the maximum deformations of the core when passing through the resonance at the first and second dynamic mode are obtained. The gained expression allows depending on the speed of the border’s motion, the coefficient of viscoelasticity and strength of the core’s material which make it possible to estimate the core durability.