Three Models of Shimmy in Rolling Tasks of a Towed Pneumatic Wheel

Abstract

Rolling without sliding of the pneumowheel, towed by means of a rigid device of length L, is considered. The angles of rotation of this device during vibrations are studied. Cases L ≈ L_* and L > L_* are distinguished, where L_* is the length at which the system is located at the boundary of stability. To describe the interaction of the running surface of the tire with the supporting plane, three well-known linear rolling models are used — the nonholonomic one, as well as the string and slip models. The case L > L_* is studied with the help of reduced mathematical models — modeling oscillators. The main results of the study: all three models indicate the same length L_* ≈ D/2 (D is the outer diameter of the free tire), in the parameter range studied the first two models actually coincide, with an imbalance of the pneumatic wheel and L ≈ L_*–which is often found in practice–parametric resonances occur if the Clarke–Dodge–Naibekken numbers DΩ/V_A (V_A is the speed of the hinge-hitch, Ω is the natural frequency of oscillations when the pneumatic wheel does not roll) belong to the natural series. In addition, the effect of the speed V_A on the natural frequency of oscillation is quantitatively described, and comments are made on the verification of shimming models and the relationship of mechanical permanent tires.