Piecewise approximate analytical solutions for a Jeffcott rotor with a snubber ring

In the paper two approximate analytical methods for calculating nonlinear dynamic responses of an idealised model of a rotor system are devised in order to obtain robust analytical solutions, and consequently speed up the computations maintaining high computational accuracy. The physical model, which is similar to a Jeffcott rotor, assumes a situation where gyroscopic forces can be neglected and concentrates on the dynamic responses caused by interactions between a whirling rotor and a massless snubber ring, which has much higher stiffness than the rotor. The system is modelled by two second-order differential equations, which are linear for non-contact and strongly nonlinear for contact scenarios. The first and the simpler method has been named one point approximation (1PA) and uses only one point in the first-order Taylor expansion of the nonlinear term. It is suitable for soft impacts and gives a reasonable prediction of responses ranging from period one to period four motion. The second and more accurate method of multiple point approximation (MPA) expands the nonlinear term many times when the rotor and the snubber ring are in contact and it can even be used for calculating chaotic responses. The methods are evaluated by a comparison with direct numerical integration showing an excellent level of accuracy.