Dynamic analysis and crack observability of combined-material spur gear pairs: a model-based study

The use of lightweight materials is increasingly demanded to support energy-saving and weight-reduction trends. In this work, the dynamic behavior and crack detection of a spur gear system are studied for four material combinations of the engaging gears: steel–steel, polyamide–polyamide, steel–polyamide and polyamide–steel. The analytically computed time-varying mesh stiffness (TVMS) drives a six-degree-of-freedom (6DOF) dynamic model, from which the residual response and the corresponding frequency response functions of each configuration are obtained over a range of pinion root-crack depths. The analysis yields a closed-form crack observability coefficient Λ showing that the normalized sensitivity of the equivalent mesh stiffness to a crack-induced loss of tooth stiffness is governed by the ratio of the two elastic moduli alone, independently of crack depth, transmitted load, rotational speed, and damping. Λ = 0.985 when the crack lies in the compliant polyamide member of a dissimilar pair, against 0.0148 when it lies in the steel member, a factor of about seventy. Combined with the crack-induced modal shift, Λ yields a dimensionless detectability index D that spans nearly two orders of magnitude across the four pairings, from 2.50 for steel–steel and 1.85 for polyamide–steel, through a marginal 0.375 for polyamide–polyamide, to 0.028 for steel–polyamide, and that classifies a proposed pairing as monitorable, marginal, or masked before any hardware exists. Material combination therefore carries a cost that is not only dynamic but diagnostic. A crack in the stiffer member of a dissimilar pair is predicted to be effectively invisible to vibration-based monitoring, in both the time and frequency domains. The paper presents different diagnostic results analysis and concludes with several findings, which are model-based and await experimental confirmation on a dedicated mixed-material test rig.