Bifurcation diagrams for two cases determined by two methods in Fig. Amplitudes of steady motion, obtained by numerical integration are compared with the results of asymptotic method. Examples of resulting displacement time histories and its Fourier spectrum are shown in Figs.
(8) using implicit scheme with iterations. To verify results obtained by asymptotic method we perform numerical time integration of Eq. Amplitudes of limit cycle vs parameter T in Fig. 3, regions A, B) have been analyzed by means of asymptotic method. Here we vary delay parameter, which corresponds to rotational velocity of gundrill.īifurcation diagram for two regions in the vicinity of the stability border.
The system behavior in the vicinity of the bifurcation point has been studied in order to analyze the nature of Andronov-Hopf bifurcation with respect to system parameters variation. All algebraic manipulations were performed in Matlab Symbolic toolbox. Obtained functions f and g are quite cumbersome and are not presented here. The nonlinear behavior of system in vicinity of stability borders was analyzed by using numerical integration of nonlinear equation. Stability diagram was constructed and bifurcation diagrams were obtained by multi-scale expansion. The multiple scale method is applied for nonlinear vibrations analysis. The new approach allows considering the influence of lateral vibrations on the dynamics of the gun drilling system. In this paper gun drill is considered as flexible continuous beam loaded with eccentrically applied cutting force. Most authors modeled drill shank using the reduced single degree of freedom system. The comprehensive review of present state of deep hole drilling modeling was given in. The same mechanism emerges not only in drilling, but in milling, boring etc. And it requires that time-delayed terms in model equations should be taken into account. The regenerative mechanism is the main source of chatter vibrations. Therefore, it is important to predict in advance regimes with chatter vibrations. Excessive vibrations are detrimental to finish surface quality and may damage the tool. Due to low flexural stiffness of gun drill shank lateral vibrations of high magnitude could be excited during the machining. Optimal drill performance in gundrilling is achieved when the combination of the cutting speed, feed rate, tool geometry, carbide grade, and coolant parameters are selected properly depending upon the work material, deep-hole tool machine conditions, and the quality requirements to the drilled holes. The method is widely applied in machining small deep holes as it provides a good straightness and high quality of machined surface due to its self-guiding action. Gun drill has asymmetrical single edge tool design with long straight tube of asymmetrical cross-section with a typical reachable diameter range of 0.5 mm up to 40 mm and length-to-diameter- ratios up to L/ D = 400 (in special applications even L/ D = 900 ). Gundrilling is the mostly used method of deep small hole machining. This efficient process is widely used in the automotive industry to drill deep holes in cylinder heads, crankshafts, fuel pump housings, turbine blades and etc. Long straight holes are usually produced by means of a special drilling tool.
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