Chatter resistance of non-uniform turning bars with attached dynamic absorbers—Analytical approach

Forced harmonic vibration of a non-uniform elastic beam with attached dynamic vibration absorbers (DVA) is studied. Analytical approximation of the solution is obtained by the functional perturbation method (FPM). The problem has application to cutting tools operations where the resistance of the tool holder against regenerative chatter can be enhanced by optimizing the real part of the frequency response function (FRF). A test case of a beam with step-like heterogeneity and single DVA at the tip shows that the FPM solution is very accurate for up to ∼40 percent deviation in both stiffness and mass density. Using the analytical results and Sims approach, optimal DVA tuning is found for each set of beam heterogeneity parameters by solving a set of nonlinear algebraic equations numerically. It is found that the optimum can be further improved by searching for the best step location. The system optimization is then expanded to a general heterogeneous beam with a DVA at its tip. The mass and stiffness distribution is optimized by applying the Lagrange variation method on the FPM solution yielding Fredholm integral equations. The optimized morphology is found to be approximately linear and far from the “intuitive” step-like one (Rivin and Kang, 1992) and yields better chatter-resistance.