In this paper, an unbalanced Jeffcott rotor supported by the roller bearings is modelled, and the dynamic differential equations are obtained by the Lagrange's equations. The fourth order Runge-Kutta method is employed to simulate this system in two cases. Simulation results show the combined effect of the coupled lateral and torsional vibrations and the variable compliance vibration of the rotor. In the case of that only a large and constant radial force is applied, the sum and difference between variable compliance and rotational frequency components of the torsional vibrations of the disk are induced because of the roller bearings' rotation. And when they are near to the natural frequency of the torsional vibration, the resonance peaks appear. In the case of that a large and constant radial force and a harmonic torque are simultaneously applied, the two components, the sum and difference between rotational frequency and the frequency of the external torque, of the lateral vibrations of the disk are induced. When the frequency of the external harmonic torque approximatively equal to a special value, the two frequency band lines, variable compliance and the sum of rotational frequency and the frequency of external torque, intersect near the natural frequency of the lateral vibration, and the vibration may aggravate.

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The multiple solution problem of nonlinear energy sinks (NESs) is of great interest in academic research and engineering applications; however, there exists only limited work on the multiple solution branches of piecewise linear NESs. Piecewise linear NESs can give rise to the appearance of isolated solution branches (ISBs), which are harmful to the performance of dynamical systems. It is difficult to detect the ISBs of a dynamical system with a piecewise linear NES. Hence, in this study, the multiple periodic solution branch distribution of a four-degree-of-freedom dynamical system with a piecewise linear NES is investigated. A method based on polynomial homotopy that can capture ISBs is developed to search multiple solution branches. Spurious periodic solutions can exist owing to the use of the implicit polynomial approach; therefore, the harmonic balance method with high harmonic orders and alternating frequency/time-domain technique are employed to distinguish true periodic solutions. Numerical experiments demonstrate that the proposed method can search for multiple periodic solution branches. A parameter analysis is conducted on the multiple solution branches, and the inner and outer ISBs are observed.