Modeling of flexible waveguides for ultrasonic vibrations transmission: Longitudinal and flexural vibrations of non-deformed waveguide

The article presents the mathematical model allowing to investigate longitudinal and flexural vibrations of stepped flexible waveguides with transitional section without regard to various vibration modes interaction. The model uses original numerical-analytic calculations based on analytical solutions of the equation of waveguide steps vibrations and their continuous matching with numerical solution of the equation of transitional section vibrations. The proposed model can be considered as an initial approximation to the solution of the problem of flexible waveguides design, which makes it possible to determine and validate effective methods of its addressing. Resonant curves of longitudinal and flexural vibrations of two-step waveguide are traced for the given vibration frequency. Step lengths values providing simultaneous resonance of longitudinal and flexural vibrations for the given frequency are determined. Validity of the proposed model is proved by the results of finite elements method (FEM) modeling using ANSYS® software. Application of Timoshenko’s model instead of Euler–Bernoulli’s model for description of flexural vibrations enabled reduction of relative deviation of resonant frequencies calculated using ANSYS® from the value specified during resonant curves tracing down to negligible value (0.17%).