A novel harmonic solution for chatter stability of time periodic systems

Chatter vibrations strongly limit productivity in milling. Due to the presence of rotating parts with asymmetric stiffness and stability enhancement strategies which act through a periodic variation of stiffness, there is growing interest in estimating the stability maps of systems with Linear Time Periodic dynamics together with periodic cutting excitation. Applying Exponentially Periodic Modulated test signals to the dynamic cutting force equation and representing the dynamics of the system through the Harmonic Transfer Function, the innovative Harmonic Solution (HS) and its zero-order approximation were derived in this research. HS is a frequency domain representation of a system described by the combination of two independent periodicities. It is possible to take into account these periodicities together in HS or singularly, resulting in the Zero Order HS or in the well-known Multi-Frequency Solution. This novel formulation can deal efficiently with spindle dependent and independent dynamics and is prone to industrial applications due to its flexibility and efficiency. More specifically, in this work the developed methodologies were used to assess the cutting stability of systems with a periodically modulated stiffness. The accuracy and efficiency of HS were validated by comparison with the results achieved by the use of the semi-discretization method. Results are in agreement with those obtained using semi-discretization. Moreover, admitting a slight precision loss, HS and its zero-order approximation are orders of magnitude faster than semi-discretization, giving reliable stability maps from seconds to a few minutes.