Natural small vibrations of a flat viscoelastic spiral spring

3. Results and analysis
Based on the considered method for determining the frequencies of natural oscillations, an algorithm and a computer program were compiled [6], with the help of which calculations of the frequencies of natural oscillations were carried out. Based on the calculation results, graphs were constructed of the dependence of natural vibration frequencies on the geometric parameters of tubular springs (Fig. 1): bending radius of the tube ( ), tube wall thickness ( ), radius of the workpiece tube ( ), rotation angle ( ) and the ratio of the semi-axes of the elliptical tube-workpiece ( ).
Based on the results of calculations, graphs of the dependence of natural oscillation frequencies on the geometric parameters of tubular springs are constructed (Fig.1): the bending radius of the tube ( ), the wall thickness of the tube ( ), the radius of the tube-blank ( ), the angle of rotation ( ) and the ratio of the semi-axes of the elliptical tube-blank ( ).

4. Conclusions
Thus, a resolving system of differential equations (15) and the corresponding boundary conditions are obtained. Also, based on the methods of differential geometry, the main relationships are obtained. To solve (15), asymptotic or numerical methods can be used.
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