Experimental characterization of the stick/sliding transition in a precision mechanical system using the third order sinusoidal input describing function

In this paper a new, non-parametric frequency domain based measurement technique is introduced that enables capturing the stick to gross sliding transition of a mechanical system with dry friction. The technique is an extension of the Sinusoidal Input Describing Function theory (SIDF) to Higher Order Describing Functions. The resulting Higher Order Sinusoidal Input Describing Functions (HOSIDF) relate the magnitude and phase of the higher harmonics in the periodic response of a non-linear system to the magnitude and phase of the sinusoidal excitation. A non-linear mechanical system with dry friction is analyzed using both the classical Frequency Response Function (FRF) technique and the newly developed HOSIDF technique. Where the FRF technique is not able to identify the stick/sliding transition of the system, the third order SIDF clearly displays this transition. From the third order SIDF the pre-sliding displacement of the system is determined. The first order SIDF is used to generate information about the resonance frequency of the system due to the friction-induced stiffness. From the pre-sliding displacement and the friction-induced stiffness, the friction force is calculated which must be present in the stick-phase. Validation with force measurements shows excellent agreement.