MS15.4: Time-periodic systems

Imperfections of cutting tool manufacturing and clamping are regular problems in machining, which affect the chip formation kinematics, forced vibrations, quality of the machined surface, and tool wear. Such effects are often neglected in classical studies due to the modeling difficulties, measurement inaccuracies, and mathematical complexity. Misalignment of the milling cutter axes results in unequal force load on the cutting edges that changes the periodic forcing in the system, and hence changes the ideal cutter-workpiece-engagement. This effect is especially important when machining is performed at small radial immersions or with small feeds, where the cutting forces acting on the cutting edges can be drastically different due to the relatively small uncut chip thickness. In addition to this, the arising vibrations elevate the early fly-over effect leading to missed cuts during the operations. Nonlinear behavior is upraised by the forced vibrations and gives rise to multiple stable and unstable periodic solutions. The presented case study demonstrates that such effects can be strong in a certain domain of technological parameters, where new unexpected solutions and stability boundaries are formed.

Frequency-based determination of stability of periodic solutions is often carried out using Hill's method. The authors recently introduced a novel projection method which uses this Hill matrix to arrive at the Floquet multipliers of the periodic solution in a numerically efficient manner. In this work, we illustrate how a certain nontrivial choice of projection, in conjunction with explicit consideration of subharmonic frequencies in the Hill matrix, can further improve the accuracy while still retaining low computational effort. The Duffing oscillator is presented as a numerical illustration of these properties.

This paper deals with the parametric stiffness excitation of an unconstrained dynamical system with three degrees of freedom. A constant force at the first degree of freedom drives the system, and both mass- and stiffness-proportional damping are considered. The parametrically excited vibrations are studied numerically by using fourth-order Runge-Kutta method. Amplitudes of the stationary (steady state) difference vibrations show resonances at both the second and third eigenfrequency of the structure. Floquet theory is employed for a numerical stability analysis. Large instability regions appear at fundamental parametric resonance frequencies of first order and at a parametric combination frequency of first order.

In the present contribution, the suppression of parametric resonances of mechanical systems by introducing impulsive force excitation is investigated. It is shown that with the proposed approach, impulsive actuation is capable of stabilizing mechanical systems which are parametrically excited at primary parametric resonance frequencies or parametric combination resonance frequencies. A semi-analytical system description is presented allowing to determine the stability of the mechanical system a priori.