Research On The Global Bifurcations And Coexistence Of Multiple Solutions In Typical Non-smooth Dynamical Systems

Global dynamics of two types of non-smooth dynamical systems are analyzed,including the bifurcation and coexistence of multiple solutions in this paper.Global analysis methods can be divided into analytical and numerical methods innonlinear dynamical systems. Melnikov method is more famous analytical method,since the flow is not continuous at the switching surface, classical Melnikov methoddoes not apply directly to the non-smooth system. Consider the role of the collisionsurface the subharmonic Melnikov function for periodic orbit is constructed in thequasi-Hamiltonian system in this paper. A quasi-Hamiltonian vibro-impact system isgiven to illustrate the application of the procedures, and the validity of the function isverified by the numerical results. By Melnikov’s criteria, we can derive necessaryconditions of Subharmonic periodic motion. With the change of system parameters theperiodic motion to chaotic motion through the period doubling bifurcation the same ashomoclinic orbits. At the same time global bifurcation diagram is obtained in thequasi-Hamiltonian vibro-impact system. For the first time some special coexistence ofmultiple solutions such as periodic motion and chaotic motion is found.Numerical methods mainly include direct numerical simulation (point mappingmethod) and cell mapping method. There are few people use cell mapping method tonon-smooth dynamical systems. Cell mapping method is improved in this paper andapplied to the quasi-Hamiltonian vibro-impact system. Domain of attraction isobtained with a clear boundary. The number of coexistent attractor changes with thechange of external excitation force, and domains of attraction with complicatedfractal boundary are intertwined. The size of the domain of attraction can predict thestability of this kind of movement to a certain extent. Ensure the system initial valuefar away from the chaotic attractor field and the sensitive region of initial value nearthe saddle point, this may provide leads to the control of chaos and enhance theanti-disturbance capacity of the system.The system bifurcation and coexistence of multiple solutions for a typical airfoilsection with freeplay are analyzed at the same time. The pitch angle at the maximumamplitude is innovatively chosen as the Poincaré sections. Then the bifurcationdiagram of system vibration with the flight speed is obtained in the flutter speedregion. We find the trans-critical flutter area located on (0.71mach,0.75mach) and limit cycle flutter area located on (0.75mach,0.95mach) on basis of the bifurcationdiagram. In trans-critical flutter area, the coexistence of multiple solutions andvarious forms of bifurcation phenomena are first discovered, such as the directtransition from double periodic motion to chaos, the coexistence of multiple periodicmotion and double periodic motion, and the phenomenon of jump on amplitude. Thestudy of the special movement forms in trans-critical flutter area is more engineeringsignificance. It may be the fundamental reason of wing buffeting and other complexvibration. Meanwhile a typical airfoil section with structural and freeplay nonlinearityis analyzed by numerical calculation. Then we know the freeplay nonlinearity playsan important role in the low-speed region, while the structure nonlinearity plays animportant role in the high-speed region.