Dynamics of an impact-forming machine

An impact-forming machine is considered. Dynamics of the impact-forming system are studied with special attention to stability of period n single-impact motion, Hopf bifurcations in non-resonance and weak resonance cases, subharmonic and Hopf bifurcations in 1:4 strong resonance case, codimension two bifurcations and chaotic motions, etc. Period n single-impact motion is derived analytically. The Poincaré section associated with the state of the impact-forming system, just immediately after the impact, is chosen, and then the disturbed map of period n single-impact motion is established. Stability and local bifurcations of period one single-impact motion are analyzed by using the Poincaré map. Local bifurcation analyses and numerical simulation show that period one single-impact motion, in most cases, undergoes period doubling bifurcation or Hopf bifurcation with change of control parameters. Period one single-impact motion undergoes either subharmonic or Hopf bifurcation in 1:4 strong resonance case. The grazing instability occurs in the strong resonance case. Generally on the grazing boundary of periodic-impact motion a new impact in the motion period appears or an impact in the motion period vanishes. Near the points of codimension two bifurcations there exists not only Hopf bifurcation of period one single-impact motion, but also Hopf bifurcation of period two double-impact motion. Finally, the influences of system parameters on periodic motions and global bifurcations of the impact-forming system are discussed. In designing impact tools it is of great interest to achieve the desired periodic impact velocities. In order to facilitate such design, the global bifurcation diagrams for the relative impact velocities of the impact-forming system versus the forcing frequency are plotted, which enable the practicing engineer to select excitation frequency ranges in which stable period one single-impact response can be expected to occur, and to predict the larger impact velocities and shorter impact period of such response. In designing and remaking the impact-forming machine, if some system parameters have been given or limited, the other parameters and forcing frequency can be optimized by analyses of stability and bifurcation of periodic impact motion so that the impact-forming system exhibits stable period one single-impact motion with larger impact velocity and shorter impact period.