Gábor Stépán

Stability of up-milling and down-milling, part 1: alternative analytical methods

Abstract The dynamic stability of the milling process is investigated through a single degree of freedom mechanical model. Two alternative analytical methods are introduced, both based on finite dimensional discrete map representations of the governing time periodic delay-differential equation. Stability charts and chatter frequencies are determined for partial immersion up- and down-milling, and for full immersion milling operations. A special duality property of stability regions for up- and down-milling is shown and explained.

Semi-Discretization for Time-Delay Systems

Introducing delay.- Basic delay differential equations.- Newtonian examples.- Engineering applications.- Summary.- References.

Subcritical Hopf Bifurcation in the Delay Equation Model for Machine Tool Vibrations

We show the existence of a subcritical Hopf bifurcation in thedelay-differential equation model of the so-called regenerative machine toolvibration. The calculation is based on the reduction of the infinite-dimensional problem to a two-dimensional center manifold. Due to the specialalgebraic structure of the delayed terms in the nonlinear part of the equation,the computation results in simple analytical formulas. Numerical simulationsgave excellent agreement with the results.

Multiple chatter frequencies in milling processes

Analytical and experimental identifications of the chatter frequencies in milling processes are presented. In the case of milling, there are several frequency sets arising from the vibration signals, as opposed to the single well-defined chatter frequency of the unstable turning process. Frequency diagrams are constructed analytically and attached to the stability charts of mechanical models of high-speed milling. The corresponding quasiperiodic solutions of the governing time-periodic delay-differential equations are also identified with some milling experiments in the case of highly intermittent cutting.

Modelling nonlinear regenerative effects in metal cutting

This paper deals with the nonlinear models of regenerative chatter on machine tools. The limitations and actual problems of the existing theory are summarized. These cause difficulties in industrial applications, like in the design of optimum technological parameters, adaptive control and/or vibration–suppression strategies. An industrial machine–tool vibration case is analysed experimentally and conclusions are drawn with respect to the likely mechanism of the nonlinear behaviour in a four–dimensional phase–space representation embedded in the infinite–dimensional phase space of regenerative chatter. In some respects, the whole phenomenon seems to be analogous to the turbulence in fluid mechanics.

Stability of up-milling and down-milling, part 2: experimental verification

The stability of interrupted cutting in a single degree of freedom milling process was studied experimentally. An instrumented flexure was used to provide a flexible workpiece with a natural frequency comparable to the tooth pass frequency, mimicking high speed milling dynamics. The displacement of the system was sampled continuously and periodically once per cutter revolution. These data samples were used to asses the stability of the system. Results confirm the theoretical predictions obtained in Part 1.

Stability chart for the delayed Mathieu equation

In the space of system parameters, the closed–form stability chart is determined for the delayed Mathieu equation defined as ä(t)+(δ+ϵcost)x(t) = bx(t−2&pgr;). This stability chart makes the connection between the Strutt–Ince chart of the Mathieu equation and the Hsu–Bhatt–Vyshnegradskii chart of the second–order delay–differential equation. The combined chart describes the intriguing stability properties of a class of delayed oscillatory systems subjected to parametric excitation.

Stability Analysis of Turning With Periodic Spindle Speed Modulation Via Semidiscretization

We investigate a single-degree-of-freedom model of turning with sinusoidal spindle speed modulation and the corresponding delay-differential equation with time-varying delay. The equation is analyzed by the numerical semidiscretization method. Stability charts and chatter frequencies are constructed. Improvement in the efficiency of machining is found for high modulation frequency and for low spindle speed domain. Period-one, period-two (flip), and secondary Hopf bifurcations were detected by eigenvalue analysis.

Delay effects in the human sensory system during balancing

Mechanical models of human self-balancing often use the Newtonian equations of inverted pendula. While these mathematical models are precise enough on the mechanical side, the ways humans balance themselves are still quite unexplored on the control side. Time delays in the sensory and motoric neural pathways give essential limitations to the stabilization of the human body as a multiple inverted pendulum. The sensory systems supporting each other provide the necessary signals for these control tasks; but the more complicated the system is, the larger delay is introduced. Human ageing as well as our actual physical and mental state affects the time delays in the neural system, and the mechanical structure of the human body also changes in a large range during our lives. The human balancing organ, the labyrinth, and the vision system essentially adapted to these relatively large time delays and parameter regions occurring during balancing. The analytical study of the simplified large-scale time-delayed models of balancing provides a Newtonian insight into the functioning of these organs that may also serve as a basis to support theories and hypotheses on balancing and vision.

Effects of Radial Immersion and Cutting Direction on Chatter Instability in End-Milling

Low radial immersion end-milling involves intermittent cutting. If the tool is flexible, its motion in both the x- and y-directions affects the chip load and cutting forces, leading to chatter instability under certain conditions. Interrupted cutting complicates stability analysis by imposing sharp periodic variations in the dynamic model. Stability predictions for the 2-DOF model differ significantly from prior 1-DOF models of interrupted cutting. In this paper stability boundaries of the 2-DOF milling process are determined by three techniques and compared: (1) a frequency-domain technique developed by Altintas and Budak (1995); (2) a method based on time finite element analysis; and (3) the statistical variance of periodic 1/tooth samples in a time-marching simulation. Each method has advantages in different situations. The frequency-domain technique is fastest, and is accurate except at very low radial immersions. The temporal FEA method is significantly more efficient than time-marching simulation, and provides accurate stability predictions at small radial immersions. The variance estimate is a robust and versatile measure of stability for experimental tests as well as simulation. Experimental up-milling and down-milling tests, in a simple model with varying cutting directions, agree well with theory.Copyright