On the approximation of delayed systems by Taylor series expansion

It is known that stability properties of delay-differential equations are not preserved by Taylor series expansion of the delayed term. Still, this technique is often used to approximate delayed systems by ordinary differential equations in different engineering and biological applications. In this brief, it is demonstrated through some simple second-order scalar systems that low-order Taylor series expansion of the delayed term approximates the asymptotic behavior of the original delayed system only for certain parameter regions, while for high-order expansions, the approximate system is unstable independently of the system parameters

On the effect of distributed regenerative delay on the stability lobe diagrams of milling processes

Regenerative machine tool chatter is investigated for milling operations with helical tools. The stability of a two-degrees-of-freedom milling model is analyzed, where the cutting-force is modeled as a force system distributed along the rake face of the tool. Introducing a distributed force system instead of a concentrated cutting-force results in an additional short, peri-odically varying distributed delay in the governing equations of the system. It is shown that the additional delay significantly affects the stability of the machining operation, especially at low spindle speeds. This phenomenon is referred to as the short regenerative effect, and is studied by computing the stability lobe diagrams of milling operations via the semi-discretization technique. The sensitivity of the stability charts to the shape of the force distribution and the contact length between the chip and tool is investigated. Keywords milling, stability, machine tool chatter, regenerative delay, cutting-force distribution

Chatter Control by Spindle Speed Variation in High-Speed Milling

High-speed milling operations are often limited by regenerative vibrations. The aim of this paper is to analyze the effect of spindle speed variation on machine tool chatter in high-speed milling. The stability analysis of triangular and sinusoidal shape variations is made numerically with the semi-discretization method. Parametric studies show also the influence of the frequency and amplitude variation parameters. This modeling is validated experimentally by variable spindle speed cutting tests with a triangular shape. Stable and unstable tests are analyzed in term of amplitude vibration and surface roughness degradation. This work reveals that stability must be considered at period variation scale. It is also shown that spindle speed variation can be efficiently used to suppress chatter in the flip lobe area

Estimation of safe chatter-free technological parameter regions for machining operations

AbstractExperimental cutting tests often show different dynamic behavior from the one predicted by means of theoretical mechanical models. One reason behind this observation is that stability lobe diagrams determined by standard linear stability analysis often under- or overestimate the region of chatter-free parameters. Therefore, reliable selection of technological parameters associated with optimal material removal rate is difficult in practice. In this paper, a simple formula is given to estimate the safe parameter region, where the machining operation is globally stable even for large disturbances. The analytical results are confirmed by numerical simulations

STABILITY TRANSITION BETWEEN 1 AND 2 DEGREE-OF-FREEDOM MODELS OF MILLING

Chatter prediction for 2 degree of freedom (DOF) milling model is presented. The workpiece is assumed to be flexible and the tool to be stiff. Non-linear cutting force model is used, and the linearized equation of motion is derived. Stability charts are constructed for different stiffness values in the directions x and y. The charts for the 1 DOF models associated with the x and the y directions are also given. It is shown that the 2-DOF case can not be given via the pure overlaying of the charts of the two single DOF cases

Az emberi egyensúlyozás mechanikai modellezése PIDA szabályozó segítségével

Ebben a cikkben két egyszerű problémán keresztül vizsgáljuk az emberi egyensúlyozás folyamatát. Vizsgáljuk az ujjhegyen történő rúdegyensúlyozást és az egy helyben állás egyensúlyozási folyamatát, a posturalis kilengést. Az egyensúlyozási problémákat egyszerű mechanikai modellekkel írjuk le, majd egy, az iparban is gyakran használt, PIDA szabályozó segítségével modellezzük az emberi agy szabályozási mechanizmusát egyensúlyozás közben. A mozgást leíró differenciál-egyenletben konstans időkéséssel figyelembe vesszük a reflex-késés hatását és a leíró egyenletek stabilitási vizsgálatával ellenőrizzük a felírt modell stabilizálhatóságát. Végül a kapott számítási eredményeket összevetjük a szakirodalomban található kísérleti eredményekkel

Stability of the milling process

A new technique for determining the stability conditions of delayed differential equations with time-periodic coefficient is presented. The method is based on a special kind of approximation of the delayed term. As a practical application, the stability of the milling process with respect to the technological parameters is analysed, and an unstable zone in the domain of high cutting speed is shown