Nonlinear Resonance Phenomenon in Cylindrical Helical Springs

In this paper, the non-linear resonance phenomenon of helical springs is studied. The mathematical formulation describing this phenomenon is constituted of a system of four non-linear partial differential equations of first order of hyperbolic type. The coefficients of this system are functions of the dependent variables of the problem. Those are the axial and rotational strains and velocities at any section of the spring. Since the governing equations are non-linear, the solution of the dynamic behavior of the spring can only be obtained by approximate numerical techniques. The non-linear characteristics method is applied to calculate the oscillating strains and velocities at equidistant sections of the spring. When the strains are small, the motion equations are rendered linear. In this case the impedance method is applied to calculate the natural frequency spectrum of the spring. The linear resonance of the spring may be produced by applying, at its extremity, a sinusoidal excitation of frequency coinciding with one of the natural frequencies. In these conditions, significant and amplified in time axial and rotational oscillations occur in the spring. A condition of resonance is then established. When the non linear system is excited with these frequencies the resonance phenomenon occurs at the beginning of the dynamic behavior. But, as the natural frequencies are function of strains, the results show that, the frequencies can deviate and after some time the variables starts to decrease and evolutes in a different manner of those in linear case.