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Abstract: Two vibroimpact systems are considered, which can exhibit symmetrical double-impact periodic motions under suitable system parameter conditions. Dynamics of such systems are studied by use of maps derived from the equations of motion, between impacts, supplemented by transition conditions at the instants of impacts. Two-parameter bifurcations of fixed points in the vibroimpact systems, associated with 1:2 strong resonance, are analyzed. Interesting features like Neimark–Sacker bifurcation of period-1 double-impact symmetrical motion, tangent bifurcation of period-2 four-impact motion, period-doubling bifurcation of period-2 four-impact motion and Neimark–Sacker bifurcation of period-4 eight-impact motion, etc., are found to occur near 1:2 resonance point of a vibroimpact system. The quasi-periodic attractor, associated with the fixed point of period-1 double-impact symmetrical motion, is destroyed as a tangent bifurcation of fixed points of period-2 four-impact motion occurs. However, for the other vibroimpact system the quasi-periodic attractor is restored via the collision of stable and unstable fixed points of period-2 four-impact motion. The results mean that there exist possibly more complicated bifurcation sequences of period-two cycle near 1:2 resonance points of non-linear dynamical systems. [Copyright &y& Elsevier]

Abstract: A three-degree-of-freedom vibratory system impacting symmetrical rigid stops is considered. Dynamics of the vibroimpact system is studied by use of a mapping derived from the equations of motion, supplemented by transition conditions at the instants of impacts. Two-parameter bifurcations of fixed points in the vibroimpact system, associated with strong resonance, are analyzed. Neimark–Sacker bifurcations of period-1 double-impact symmetrical orbits, near resonance point, are found, and the transition of quasi-periodic impact motions to unstable period-3 six-impact motions, are obtained numerically. The results conform to resonance bifurcation theory of maps available. More complicated “tire-like” quasi-periodic attractor and torus bifurcation associated with the transition of the closed invariant curve to torus, near resonance point, are found to exist in the nonlinear system. The results imply that there exist possibly more complicated bifurcation sequences near resonance points of nonlinear dynamical systems. [Copyright &y& Elsevier]

Abstract: A multi-degree-of-freedom vibratory system having symmetrically placed rigid stops is considered. The system consists of linear components, but the maximum displacement of one of the masses is limited to a threshold value by the symmetrical rigid stops. Repeated impacts usually occur in the vibratory system due to the rigid amplitude constraints. Such models play an important role in the studies of mechanical systems with clearances or gaps. Local codimension two bifurcation of maps, involving a real eigenvalue and a complex conjugate pair escaping the unit circle simultaneously, is analyzed by using the center manifold theorem technique and normal form method for maps. Symmetrical double-impact periodic motion and Poincaré map of the system are derived analytically. A center manifold theorem technique is applied to reduce the Poincaré map to a three-dimensional one, and the normal form map associated with the codimension two bifurcation is obtained. Local behaviors of the vibratory systems with symmetrical rigid stops, near the points of codimension two bifurcations, are reported by the presentation of results for a two-degree-of-freedom vibratory system with symmetrical stops. The existence and stability of symmetrical double-impact periodic motion are analyzed explicitly. Also, local bifurcations at the points of change in stability, are analyzed. Near the point of codimension two bifurcation, there exists not only Hopf bifurcation of period-one double-impact motion, but also pitchfork bifurcation of the motion. Pitchfork bifurcation of period-one double-impact symmetrical motion results in the period-one double-impact unsymmetrical motion. The unsymmetrical double-impact motion is of two antisymmetrical forms due to different initial conditions and symmetrical stops. With change of the forcing frequency, the unsymmetrical double-impact periodic motion will undergo Hopf bifurcation. Moreover the period-one double-impact symmetrical motion will undergo Hopf bifurcation directly as the forcing frequency is changed in the contrary direction. The routes of quasi-periodic impact motions to chaos are observed by results from simulation. [Copyright &y& Elsevier]

Abstract: A dual component system with vibro-impact is considered. Local codimension two bifurcation of maps, involving a real eigenvalue and a complex conjugate pair escaping the unit circle simultaneously, is analyzed by using the center manifold and normal form method for maps. The period n single-impact motion and Poincaré map of the vibro-impact system are derived analytically. Stability and local bifurcation of single-impact periodic motion are analyzed by using the Poincaré map. A center manifold theorem technique is applied to reduce the Poincaré map to a three-dimensional one, and the normal form map associated with the codimension two bifurcation is obtained. Local behavior of the dual component system with vibro-impact, near the point of codimension two bifurcation, is analyzed. It is found that near the point of codimension two bifurcation, there exists not only Hopf bifurcation of period one single-impact motion, but also Hopf bifurcation of period two double-impact motion. Period doubling bifurcation of period one single-impact motion commonly exists near the point of codimension two bifurcation. However, no period doubling cascade emerges due to change of the type of period two fixed points and occurrence of Hopf bifurcation associated with period two fixed points. The period two fixed points are symmetrical about the corresponding period one point. The results from simulation show that the attracting invariant circles associated with period two points are symmetrical about the corresponding period one point near the value of Hopf bifurcation of period two points. Two actual examples, the impact-forming machinery and inertial shaker, are chosen to analyze further the phenomena of codimension two bifurcation of maps, and local bifurcation analysis and numerical simulation are carried out to unfold dynamical behavior of these vibro-impact systems near the points of codimension two bifurcations. [Copyright &y& Elsevier]

Abstract: A multidegree-of-freedom system having symmetrically placed rigid stops and subjected to periodic excitation is considered. The system consists of linear components, but the maximum displacement of one of the masses is limited to a threshold value by the symmetrical rigid stops. Repeated impacts usually occur in the vibratory system due to the rigid amplitude constraints. Such models play an important role in the studies of mechanical systems with clearances or gaps. Double Neimark–Sacker bifurcation of the system is analyzed by using the center manifold and normal form method of maps. The period-one double-impact symmetrical motion and homologous disturbed map of the system are derived analytically. A center manifold theorem technique is applied to reduce the Poincaré map to a four-dimensional one, and the normal form map associated with double Neimark–Sacker bifurcation is obtained. The bifurcation sets for the normal-form map are illustrated in detail. Local behavior of the vibratory systems with symmetrical rigid stops, near the points of double Neimark–Sacker bifurcations, is reported by the presentation of results for a three-degree-of-freedom vibratory system with symmetrical stops. The existence and stability of period-one double-impact symmetrical motion are analyzed explicitly. Also, local bifurcations at the points of change in stability are analyzed, thus giving some information on dynamical behavior near the points of double Neimark–Sacker bifurcations. Near the value of double Neimark–Sacker bifurcation there exist period-one double-impact symmetrical motion and quasi-periodic impact motions. The quasi-periodic impact motions are represented by the closed circle and “tire-like” attractor in projected Poincaré sections. With change of system parameters, the quasi-periodic impact motions usually lead to chaos via “tire-like” torus doubling. [Copyright &y& Elsevier]