Bifurcation Control Of Some Kinds Of Vibro-Impact Systems

Impact and vibration are very common phenomena in mechanical engineering field.On the one hand, the bifurcation and chaos dynamical behaviors caused by the discontinuous property of impacts lead to loss of stability and subsequent failure of the system. These bifurcation phenomena are usually avoided or delayed by control in engineering. On the other hand, people are starting to be concerned about how to actively use nonlinear characteristics of bifurcation for some production purposes and realize the bifurcations with desired properties through design or control. This dissertation takes several vibro-impact systems as the research object, develops some control methods to study co-dimension-one bifurcation 、 co-dimension-two bifurcation and grazing non-smooth bifurcation in vibro-impact systems and degenerate Neimark-Sacker bifurcation in a high dimensional map, and then investigates nonlinear dynamical behaviors of a two-degree-of-freedom vibro-impact system through experiment. The main research works and results of this dissertation are listed as follows:1. Design of quasi-periodic impact motion and anti-controlling Period-doubling bifurcation in an impact shaker system are studied. When the difficulties given by the classical Neimark-Sacker bifurcation critical criterion described by the properties of eigenvalues are considered, an explicit critical criterion without using eigenvalues is given to obtain the Neimark-Sacker bifurcation diagram of two parameters. A stable quasi-periodic impact motion is designed by utilizing the center manifold and normal formal theory. Aimed at the discontinuity caused by impact and the implicit Poincarémap of inertial impact shaker system, a linear feedback control method is developed in the premise of no change of periodic solutions of the original system. The explicit critical criterion without using eigenvalues calculation about period-doubling bifurcation is applied to obtain the controlling parameters area with good robustness.The stability of the period-doubling bifurcation solution is further analyzed by utilizing the center manifold and normal formal theory. The ultimate numerical experiments verify that the stable quasi-periodic impact motion is realized and the stable period-doubling bifurcation solutions can be generated at an specified parameter point by controlling.2. Anti-control of Neimark-Sacker bifurcation 、 Pitchfork bifurcation and HopfHopf interaction bifurcation of periodic motion are studied in a three-degree-of-freedom vibration system with clearance. Impact periodic solution is obtained and a six-dimensional Poincaré map of the close-loop control system is established. As the analytical expressions of all eigenvalues of Jacobi matrix for six-dimensional map are unavailable, the classical bifurcation critical criteria described by the properties of eigenvalues have a great limitation in obtaining control gains. Aiming at the limitation,some explicit bifurcation criteria including eigenvalue assignment, transversality condition and non-resonance condition are established. The established criteria are equivalent to the classical critical criteria, but they do not depend on eigenvalue computations of Jacobi matrix. The numerical simulations show that Neimark-Sacker bifurcation 、 Pitchfork bifurcation and Hopf-Hopf interaction bifurcation of the vibro-impact system are created in a desired parameter location and verify theoretical analysis.3. The near-grazing dynamics and experimental investigation of dynamical behaviors are studied in the two-degree-of-freedom vibratory system with a clearance.The discontinuity mapping is used to derive a normal form map near the grazing point.The stability criterion of grazing motion is formulated through the normal form map.Numerical simulations reveal a discontinuous grazing bifurcation with a jump between the transitions of periodic motions and validate the stability of grazing motion on basis of the stability criterion. An experimental apparatus is designed and built to investigate the dynamical behaviors of a two-degree-of-freedom vibro-impact system. The dynamic responses of periodic motion、grazing impact and chaotic motion of the vibro-impact system are investigated experimentally by changing the excitation frequencies and gap sizes.4. Anti-control of degenerate Neimark-Sacker bifurcation is studied in a four-dimensional generalized Hénon map. An explicit critical criterion without using eigenvalues is used to obtain the area of control gains, the four-dimensional Hénon map is reduced into a two-dimensional map by utilizing the center manifold and normal formal theory. On basis of degenerate Neimark-Sacker bifurcation theory proposed by Chenciner, a feedback controller of polynomial functions is designed to realize the degenerate Neimark-Sacker bifurcation of the map and numerical simulations demonstrate the theoretical analysis.