A General Formulation of the Milling Process Equation: Contribution to Machine Tool Chatter Research—5

1968 ◽  
Vol 90 (2) ◽  
pp. 317-324 ◽  
Author(s):  
R. Sridhar ◽  
R. E. Hohn ◽  
G. W. Long
Keyword(s):  
Machine Tool ◽  
Cutting Forces ◽  
Single Point ◽  
Milling Process ◽  
Cutting Process ◽  
Varying Coefficients ◽  
Machine Tool Structure ◽  
Time Varying Coefficients ◽  
Linear Differential ◽  
Vector Matrix

In the cutting process, forces are induced at each of the cutter teeth in contact with the workpiece, and these forces in turn excite the machine tool structure. Due to the inherent feedback which exists between the cutting forces and the structure deflection, there are conditions under which this system becomes unstable. When this occurs, a condition of self-excited chatter exists. In the theory of self-excited chatter for single point tools wherein the tool is continuously in contact with the workpiece, the system can be described by a time-invariant equation, and has been rather fully developed. However, the milling process cannot be described in this manner. It is characterized by a multitooth cutter, and the cutting process itself is interrupted. Also, the direction of the cutting forces generated by each tooth does not remain constant with respect to the machine tool structure as for turning operations, but changes direction as a function of cutter position. In this paper, more complete description of the milling process is formulated. The resulting equation is a general nth order vector-matrix linear equation with periodic coefficients and a transport lag. Or equally, it is a linear differential-difference equation with time-varying coefficients. This equation is then expressed as n-first order equations (state variable form), consistent with current literature and in a form compatible for digital computer analysis.

Uniform global asymptotic stability for half-linear differential systems with time-varying coefficients

2011 ◽  
Vol 141 (5) ◽  
pp. 1083-1101 ◽  
Author(s):  
Masakazu Onitsuka ◽  
Jitsuro Sugie
Keyword(s):  
Linear System ◽  
Zero Solution ◽  
Integrable Case ◽  
Time Varying ◽  
Globally Asymptotically Stable ◽  
Varying Coefficients ◽  
Linear Differential Systems ◽  
Time Varying Coefficients ◽  
Linear Differential ◽  
Image Position

The present paper deals with the following system:where p and p* are positive numbers satisfying 1/p + 1/p* = 1, and ϕq(z) = |z|q−2z for q = p or q = p*. This system is referred to as a half-linear system. We herein establish conditions on time-varying coefficients e(t), f(t), g(t) and h(t) for the zero solution to be uniformly globally asymptotically stable. If (e(t), f(t)) ≡ (h(t), g(t)), then the half-linear system is integrable. We consider two cases: the integrable case (e(t), f(t)) ≡ (h(t), g(t)) and the non-integrable case (e(t), f(t)) ≢ (h(t), g(t)). Finally, some simple examples are presented to illustrate our results.

A Wavelet Based Procedure to Study Vibrations and Stability

1999 ◽  
Author(s):  
S. Pernot ◽  
C. H. Lamarque
Keyword(s):  
Stability Analysis ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Periodic Solutions ◽  
Time Varying ◽  
Differential Systems ◽  
Varying Coefficients ◽  
Linear Differential Systems ◽  
Time Varying Coefficients ◽  
Linear Differential

Abstract A Wavelet-Galerkin procedure is introduced in order to obtain periodic solutions of multidegrees-of-freedom dynamical systems with periodic time-varying coefficients. The procedure is then used to study the vibrations of parametrically excited mechanical systems. As problems of stability analysis of nonlinear systems are often reduced after linearization to problems involving linear differential systems with time-varying coefficients, we demonstrate the method provides efficient practical computations of Floquet exponents and consequently allows to give estimators for stability/instability levels. A few academic examples illustrate the relevance of the method.

A Stability Algorithm for a Special Case of the Milling Process: Contribution to Machine Tool Chatter Research—6

1968 ◽  
Vol 90 (2) ◽  
pp. 325-329 ◽  
Author(s):  
R. E. Hohn ◽  
R. Sridhar ◽  
G. W. Long
Keyword(s):  
Differential Equation ◽  
Experimental Data ◽  
Machine Tool ◽  
Linear Differential Equation ◽  
Mathieu Equation ◽  
Milling Process ◽  
Machine Tool Chatter ◽  
Linear Differential ◽  
The Stability ◽  
Special Case

In an effort to determine the stability of the milling process, and due to the complexity of its describing equation, a special case of this equation is considered. In this way, it is possible to isolate and study its salient characteristics. Moreover, the simplified equation is representative of a machining operation on which experimental data can be obtained. This special case is described by a linear differential equation with periodic coefficients. A computer algorithm is developed for determining the stability of this equation. To demonstrate the use of the algorithm on an example whose solution is known, the classical Mathieu equation is studied. Also, experimental results on an actual machining operation described by this type of equation are compared to the results found using the stability algorithm. As a result of this work, some knowledge about the stability solution of the general milling process is obtained.

Research on Cutting Stability of High-Efficiency Micro Turn-Milling Compound Machine Tool Based on Lobes

2019 ◽  
Vol 295 ◽  
pp. 59-65
Author(s):  
Zhong Peng Zheng ◽  
Xin Jin ◽  
Ye Wang Sun ◽  
Xin Yang Jiang ◽  
Zhi Jing Zhang ◽  
...  
Keyword(s):  
Machine Tool ◽  
High Efficiency ◽  
Cutting Parameters ◽  
Milling Process ◽  
Test Method ◽  
Cutting Process ◽  
Milling Machine ◽  
Cutting Stability ◽  
Milling Chatter ◽  
Turn Milling

In order to improve the cutting stability of high-efficiency micro turn-milling machine tools, avoid the chattering problem during the cutting process. In this paper, the chatter problem in the cutting process is studied based on the stable lobes. By analyzing the high-efficiency turn-milling machine tool mechanism and the turn-milling model, the micro turn-milling dynamic dynamic vibration model and the mathematical model of turn-milling chatter are obtained. Then, based on the hammer test method, the transfer function of the tool-workpiece system is obtained, and the turn-milling stable lobes of the high-efficiency micro turn-milling machine tool is constructed. Finally, the research on the stable zone of the turning main spindle parts, the turning back spindle parts and the high-frequency milling part are completed. The experimental research results guide and optimize the selection of cutting parameters for turn-milling process.

Numerical Kinematical Analysis of the Shaping Mechanism

2019 ◽  
Vol 17 (17) ◽  
pp. 37-49
Author(s):  
Vladimir Dragoş Tătaru ◽  
Mircea Bogdan Tătaru
Keyword(s):  
Differential Equations ◽  
Machine Tool ◽  
Numerical Integration ◽  
Reference System ◽  
First Order ◽  
Kinematical Analysis ◽  
Machine Tool Structure ◽  
Numerical Integration Methods ◽  
Linear Differential ◽  
Kinematical Parameters

AbstractThe paper deals with the complete kinematical analysis of the mechanism that enters the machine tool structure designed to generate, in particular, plane surfaces. A machine tool of this kind is called shaping machine. For this purpose, Euler’s relations concerning the velocities distribution, written in projections on the fix reference system axes will be used. Starting from these relations we will get to a system of the first order linear differential equations whose unknowns are the kinematical parameters of the mechanism elements. The variation in time of these parameters will be obtained by solving the differential equations system the differential equations system using numerical integration methods.

An Analysis and Modeling of the Dynamic Stability of the Cutting Process Against Self-Excited Vibration

2020 ◽  
Vol 22 (4) ◽  
pp. 1287-1300
Author(s):  
A. Motallebia ◽  
A. Doniavi ◽  
Y. Sahebi
Keyword(s):  
Machine Tool ◽  
Removal Rate ◽  
Dimensional Accuracy ◽  
Modal Testing ◽  
Cutting Process ◽  
Work Piece ◽  
Stability Diagrams ◽  
Machine Tool Structure ◽  
Excited Vibration ◽  
The Stability

AbstractChatter is a self-excited vibration which depends on several parameters such as the dynamic characteristics of the machine tool structure, the material of the work piece, the material removal rate, and the geometry of tools. Chatter has an undesirable effect on dimensional accuracy, smoothness of the work piece surface, and the lifetime of tools and the machine tool. Thus, it is useful to understand this phenomenon in order to improve the economic aspect of machining. In the present article, first the theoretical study and mathematical modeling of chatter in the cutting process were carried out, and then by performing modal testing on a milling machine and drawing chatter stability diagrams, we determined the stability regions of the machine tool operation and recognized that witch parameter has a most important effect on chatter.

Analysis of Oblique Cutting Forces

1967 ◽  
Vol 89 (2) ◽  
pp. 347-355 ◽  
Author(s):  
Russell F. Henke
Keyword(s):  
Steady State ◽  
Cutting Force ◽  
Cutting Forces ◽  
Metal Cutting ◽  
Single Point ◽  
Orthogonal Cutting ◽  
Cutting Process ◽  
Chip Area ◽  
Oblique Cutting ◽  
The Stability

This paper is the latest of a continuing series on the subject of self-excited machine tool chatter. The representation of the metal cutting process as required by the previously developed closed-loop chatter theory is extended to oblique cutting with tools of practical shape and geometry. The cutting process parameters essential to proper application of the stability theory are found by an analytical formulation leading to a classical eigenvalue problem. Techniques are developed to determine the steady-state constant of proportionality between resultant cutting force and uncut chip area, the direction of resultant cutting force, and the direction of maximum cutting stiffness for any single-point cutting operation. In the process, a general method to predict steady-state oblique cutting forces is evolved. The method depends on certain experimentally justifiable assumptions and utilizes previously compiled orthogonal cutting data.

Statistical Analysis of the Dynamic Cutting Coefficients and Machine Tool Stability

1993 ◽  
Vol 115 (2) ◽  
pp. 205-215 ◽  
Author(s):  
M. A. El Baradie
Keyword(s):  
Machine Tool ◽  
Confidence Level ◽  
Cutting Process ◽  
Structure Dynamics ◽  
Cutting Coefficients ◽  
Statistical Variations ◽  
Machine Tool Structure ◽  
Machine Tool Chatter ◽  
Machine Structure ◽  
Tool Chatter

Machine tool chatter is a statistical phenomenon since it is dependent on the interaction of two statistical quantities, these being the dynamic characteristics of the machine tool structure and the transfer function of the cutting process. In this paper, a generalized statistical theory of machine tool chatter has been developed. This takes into consideration the scatter of the dynamic data of the machine structure and/or that of the cutting process. The dynamics of the cutting process have been represented by a mathematical model which derives the cutting coefficients from steady state cutting data, based on a nondimensional analysis of the cutting process. The dynamics of the machine tool structure and the cutting process, being the input data to the theory, were determined experimentally. The predicted stability charts were plotted to take into consideration the scatter in the machine structure dynamics and/or the cutting process. At the threshold of stability, the statistical variations due to the dynamic cutting coefficients amount to ±29.5 percent at 99 percent confidence level, while the statistical variations due to the structure dynamics amount to ±4.5 percent only, at the same confidence level. Therefore, the threshold of stability can be specified only in terms of mean values with confidence limits.

Tool stiffness influence on the chosen physical parameters of the milling process

2012 ◽  
Vol 60 (3) ◽  
pp. 597-604 ◽  
Author(s):  
W. Zębala
Keyword(s):  
Titanium Alloy ◽  
Aluminium Alloy ◽  
Chip Formation ◽  
Cutting Forces ◽  
Contact Point ◽  
Milling Process ◽  
Cutting Process ◽  
Physical Parameters ◽  
Titanium Alloy Ti6al4v ◽  
The Impact

Abstract This article presents our own model researches, relating to the down milling process of Aluminium alloy (Al6061) and Titanium alloy (Ti6Al4V), with a tool made of sintered carbides. These investigations pay the special attention to the impact of the tool rigidity on the process of chip formation. The simulation calculations have been carried out for two cases of the cutting process: case 1 - assuming an ideally rigid construction of a milling cutter (length of tool does not impact its deflection under the cutting forces); case 2 - it is possible that the tool can be subjected to deflection under the cutting forces (length of a tool part is counted from the holder end to the contact point of a cutting edge with the machining material).

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