Hopf Bifurcation And Semi-discretization Algorithm Study On Machine Tool Chatter

During cutting process, because the interaction between system internalexcitation and feedback, the workpiece and tool often occur strongly vibrations, thesevibrations, known as “chatter”. Chatter not only make machining process becomesunstable, but also the noise generated by vibration can stimulate operation workersfatigue. In addition, chatter can result in lessened productivity, poorer product surfacefinish, and decreased cutting life of the tool. To reduce the adverse effects of chatter,the users are forced to change temporarily the cutting parameters in the machiningprocess, such as reducing the depth of cut and discarding high spindle speed.However, these have hindered the full use of the machine rated power, resulting inmanufacturing cost rise and processing time delay. Therefore, the chatter is a mainobstacle for improving machining capacity. On the premise of ensuring highproductivity, how to use effective methods to analize and predict chatter are urgent toresolve problems. Based on machine tool chatter theory, in depth study of thenonlinear dynamics of the machine tool and the stability prediction algorithm are doneby computer simulation. The main contents include the following aspects:1. We propose a weakly nonlinear model with square and cubic terms in bothstructural stiffness and regenerative terms, to represent self excited vibrations inmachining. It is proved that Hopf bifurcation exists when bifurcation parameter equalsa critical value, a formula for determining the direction of the Hopf bifurcation andthe stability of bifurcating periodic solutions are given by using the normal formmethod and center manifold theorem.2. We propose an improved semi discretization method, based on the Floquettheory, to predict the sinusoidal spindle speed modulation of milling processes.Compared with the non improved semi discretization method, the accuracy andefficiency of the proposed algorithm are verified. In addition, stability is predictedusing benchmark examples. The stability charts for variable spindle speed milling arecompared with those for constant spindle speed milling. The results show that chattercan be effectively suppressed by varying the spindle speed.3. An improved semi discretization method is proposed to predict variablespindle speed milling with helix angle. Based on tool geometry and machining theory,the cutting region is divided into five different cases to calculate the cutting force. Theinfluences of radial immersion rate and modulation parameters relative to variable spindle speed milling are explored. By comparison with constant spindle speed, thesimulation results show that the variable spindle speed scheme can obtain a largerrange of stability. In short, the helix angle and variable spindle speed play animportant role in the stability of milling process.4. An improved semi discretization method for modeling and simulation withvariable pitch and variable helix milling is proposed. Due to the delay between eachflute varies along the axial depth of the tool in milling; the cutting tool is discrete intosome axial layers to simplify the calculation. A comparison of the predicted andobserved performance of variable pitch and variable helix against uniform pitch anduniform helix milling are presented. It is shown that variable pitch and variable helixmilling can obtain larger stable cutting area than uniform pitch and uniform helixmilling. Thus, it is concluded that variable pitch and variable helix milling are aneffective way for suppressing chatter.