Bifurcation of Periodic Solutions in the Two-Degree-of-Freedom System With Clearances

Clearances exist in many mechanical systems either by design or due to manufacturing tolerances and wear. The characteristics of such systems include abrupt variation of stiffness usually approximated as piecewise linear. It is well-known that the stiffness discontinuity can be a source of the instabilities in the dynamic behavior of the system.

In this paper, periodic solutions of the two-degree-of-freedom mechanical system with clearances subjected to periodic excitations are studied. The periodic solution may lose its stability via a static bifurcation (cyclic-fold or flip), or via a Neimark bifurcation. The bifurcation depends on the eigenvalues of the Jacobian matrix of the nonlinear vector field. By applying Hurwitz criterion on the Jacobian matrix, the bifurcation can be classified. For the analyzed dynamical system with clearances, a Neimark bifurcation occurs.

The analytical results are compared with the numerical solutions obtained by the finite element in time method. The bifurcation analysis in time finite element procedure is performed by using Poincaré map. The system is assumed to be controlled by the excitation frequency (codimension-one bifurcation). Imposing small increments in the excitation frequency, the critical point is found from which the Neimark bifurcation takes place. The qualitatively different phase portraits, prior to and after the critical point, confirm the Neimark bifurcation.