Tamás Insperger

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.

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.

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.

Act-and-wait concept for continuous-time control systems with feedback delay

The act-and-wait control concept is introduced for continuous-time control systems with feedback delay associated with infinite poles. The point of the method is that the feedback is periodically switched on (act) and off (wait) during the control. It is shown that if the duration of waiting (when the control is switched off) is larger than the feedback delay, then the system can be represented by a finite dimensional monodromy matrix, and a finite number of eigenvalues describe stability. This way, the infinite dimensional pole placement problem is reduced to a finite dimensional one. The efficiency of the method is demonstrated on a case study

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

Machine Tool Chatter and Surface Location Error in Milling Processes

A two degree of freddom model of the milling process is investigated. The governing equation of motion is decomposed into two parts: an ordinary differential equation describing the periodic chatter-free motion of the tool and a delay-differential equation describing chatter. The stability chart is derived by using the semi-discretization method for the delay-differential equation corresponding to the chatter motion. The periodic chatter-free motion of the tool and the associated surface location error (SLE) are obtained by a conventional solution technique of ordinary differential equations. It is shown that the SLE is large at the spindle speeds where the ratio of the dominant frequency of the tool and the tooth passing frequency is an integer. This phenomenon is explained by the large amplitude of the periodic chatter-free motion of the tool at these resonant spindle speeds. It is shown that large stable depths of cut with a small SLE can still be attained close to the resonant spindle speeds by using the SLE diagrams associated with stability charts. The results are confirmed experimentally on a high-speed milling center.

DELAY, PARAMETRIC EXCITATION, AND THE NONLINEAR DYNAMICS OF CUTTING PROCESSES

It is a rule of thumb that time delay tends to destabilize any dynamical system. This is not true, however, in the case of delayed oscillators, which serve as mechanical models for several surprising physical phenomena. Parametric excitation of oscillatory systems also exhibits stability properties sometimes defying our physical sense. The combination of the two effects leads to challenging tasks when nonlinear dynamic behaviors in these systems are to be predicted or explained as well. This paper gives a brief historical review of the development of stability analysis in these systems, induced by newer and newer models in several fields of engineering. Local and global nonlinear behavior is also discussed in the case of the most typical parametrically excited delayed oscillator, a recent model of cutting applied to the study of high-speed milling processes.