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文獻、資料題目：A new analytical–experimental method for the identification of stability lobes in high-speed milling
文獻、資料來源：國外金屬加工2OO5年第26卷
第3期
文獻、資料發(fā)表（出版）日期：2005.3
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外文文獻
A new analytical–experimental method for the identification of stability lobes in high-speed milling
There is a pressing economic need for efficient machine-tool operation. A wide range of research has been conducted to determine the optimal parameters for machining (i.e. feed of rate, depth of cut, and spindle speed). Studies have focused on cost minimization, machining time minimization and metal-removal rate (MRR) maximization. Most optimization methods seek to increase the MRR and are oriented towards optimizing cutting speeds and accelerations. Even so, thesemethods do not guarantee the optimal solution, since they are developed in a space conspicuously free fro mall the restrictions entailed in real machining.
Some attempts at finding the optimal values of cutting parameters consider different objective functions , including production-cost minimisation [1], production time minimisation [2] and a combination thereof [3,4].
However, the limiting factor for most optimization methods is the instability involved in milling operations. Instability is a vibratory phenomenon and can be measured and described in a quantitative form. Diverse studies of the vibratory phenomenon have been made for machine tools and with the help of technology, it is now in a certain way scientifically possible to quantify the characteristics of the vibration in machining processes, to predict chatter and to make pertinent recommendations to avoid it.
Tobias, in ‘‘Vibrations in Machine-Tools’’ [5], discusses the principles of vibration theory applicable to machine tools and offers a critical overview of the main theoretical and experimental results in the investigation of chatter. Nevertheless, since 1961, the year when the original manuscript of the book was finished, machine-tool research has made great strides. New techniques for predicting machine-tool instability have been developed and have been applied successfully. Other aspects of chatter theory, however, have not changed; the way the cutting process under chatter conditions is conceptualised has not altered, nor has the exceptional importance of the regenerative effect as the main physical cause of instability been questioned.
In 1983, Tlusty et al. [6] examined machining stability, especially in high-speed machining (HSM). They applied an approach using a time-domain simulation in order to analyze chatter and thus to enhance the knowledge of the effect of cutting speed on milling operations. In this paper, stability lobe diagrams, based on the axial depth of cut and speed (Fig. 1), are obtained.
Another study of Ismail and Soliman [8] introduces a method for identifying stability lobes in milling operations. This method depends on ramping the spindle speed while monitoring the behavior of a chatter indicator. Based on patterns shown by this indicator, stability lobes can be identified on line. The proposed technique makes it viable to locate stable regions during practical tests while avoiding chatter.
In 1998, Abrari et al. [7] presented a dynamic-force model and a stability analysis for ball-end milling. The concept of equivalent orthogonal cutting conditions, which they applied to modeling the mechanics of ball-end milling, can be extended to include the dynamics of cutting forces. The model thus developed can generate forces very similar to the data from the experiments.
A more recent study from Naterwalla [9] has been published on how to perform machining operations without chatter and maximise MRR for the metal industry.
As a result of technological advances in tools and machine tools, operations are taking place at increasingly higher speeds and accelerations. The terms ‘‘high-speed machining’’, ‘‘high speed of cut’’ (HSC) and ‘‘high-speed performance’’ (HSP) have become more common over the last few years for describing the process of machining at high speeds.
Because machining operations are taking place at increasingly higher speeds, studies in the area of vibrations have branched off into researching the stability of HSM. One of Tlusty’s studies [10] describes HSM applications in facing for airplane structures and introduces a case of ‘‘high-speed grilling’’.
Fig.1 Example of a stability diagram
In the year 2000, Altintas [12] published a paper focusing on the foundations of metal-cutting mechanisms, static and dynamic deformations, principles for the design of CNC, sensor-assisted machining and technology for programming numerical-control machines. He proposes a method for determining the chatter vibrations in orthogonal cuts and introduced a technique for considering chatter in complex milling operations. In the article, Altintas explained the technique with the help of the results of machining simulations and tests.
Insperger and Ste′pe′n [11] proposed and applied a new criterion for delaying the parametric excitation of the milling model wherein stability diagrams are constructed and the non-conventional regions of instability are identified together with the vibration frequencies.
1. A method to identify the stability lobes
1.1. The chatter problem
In the milling process, material is removed from a work piece by a rotating cutting tool. While the tool rotates, it translates in the feed direction at a certain speed. A schematic representation of the milling process is shown in Fig. 2.
One of the most common problems in machining is dynamic deformations, which are structural vibrations between the cutting tool and the work piece. The most common vibrations are the self-excited vibrations of chatter, which grow until the tool leaves its cutting zone due to the exponential increase of the dynamic displacements between the tool and the work piece (regenerative chatter).
Chatter occurs in machining operations due to the interaction between the tool-work piece structure and the force process. Regenerative chatter is so named because of the closed-loop nature of this interaction (Fig. 3). Each tooth pass leaves a modulated surface on the work piece due to the vibrations of the tool and work piece structures, causing a variation in the expected chip thickness.
Under certain cutting conditions (i.e. feed of rate, depth of cut, and spindle speed), large chip thickness variations and hence force and displacement variations occur and chatter is present. The results of chatter include a poor surface finish due to the chatter marks, excessive tool wear, reduce dimensional accuracy, and tool damage. Machine-tool operators often select conservative cutting conditions to avoid chatter, thus, decreasing productivity.
Different studies have been developed to avoid operations with chatter. One of these studies is the analytical prediction of chatter presented by Altintas and Budak. In the next subsection, a summary of this stability analysis is presented.
Fig.2 Mechanical model for milling
1.2. Analytical prediction of chatter vibrations in milling
In the articles ‘‘Analytical predictions of stability lobes in milling’’ and ‘‘Analytical prediction of chatter stability in milling. Part I: General formulation and Part II: Application of the general formulation to common milling systems’’ published by Altintas and Budak, and described in the book ‘‘Manufacturing Automation. Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design’’ [12], the analytical chatter prediction model was presented and provide practical guidance to machine-tool users for optimal process planning of depth of cuts and spindle speeds in milling operations.
The tool vibrations are assumed to occur at a chatter frequency, when a marginally stable depth of cut is taken. The forces (F) are described as harmonic functions and the closed-loop equation of the face milling operation is written as:
(1)
where is the stiffness of the work piece material, is the axial depth of cut, T is the tooth passing period, is the most simplistic approximation of the average component of the Fourier series expansion. The transfer function matrix [ is defined at the cutter work piece contact zone as
= (2)
where and are the direct transfer functions in the x and y directions, where and are the cross transfer functions, which has a nontrivial solution if its determinant is zero:
(3)
The eigenvalue of the characteristic equation is:
(4)
where N is the number of the tool teeth. The final expression for chatter-free axial depth of cut is found as:
(5)
The spindle speed n is simply calculated by finding the tooth passing period:
(6)
In summary, the transfer functions of the machinetool system are determined, and the dynamic cutting coefficients are evaluated from the derived equations for a specific cutter, workpiece material, and radial immersion of the cut.
(7)
However, the transfer function for multi degree systems can be identified experimentally using a piezoelectric force transducer and it can simplify this analytical method to determine the stability lobes in milling. One methodology to determine experimentally the transfer function is the experimental modal analysis for multi degree-of-freedom systems, which is described in the next subsection.
2.3. Experimental modal analysis for
Multi degree-of-freedom systems
Transfer functions of existing multi degree-of-freedom systems are identified by a structural dynamic test. An impact hammer instrumented with a piezoelectric transducer and a accelerometer can be used (Fig. 4). With this method, we try to excite a range of frequencies that contain the natural modes of the system. In order to obtain our objective, we need to generate an impulse and this can be given with a short impact. To choose the appropriate hammer and sensor, we have to consider the mass, the stiffness and the material of the structure.
Once the response and impulse signals are acquired, an analysis takes place to detect and eliminate signals with double impact and overload; and a ponderation in frequency is made. The modals parameters must be estimated. It is possible to use several systems.
The excitation and the answer contains very few data, but our precision is improved using order analysis methods different as the traditional frequency response function (FRF). Because the order analysis uses models, the coefficients that we needed are adjusted to a similar model of the obtained
signal. A damped sinusoidal signal can be described as a lineal combination of the different damped signals with the following expression:
(8)
where are the amplitudes of the damping factor and indicates the complex amplitudes. Written as a matrix form:
(9)
Baron de Prony discovered that are roots of the polynomial equation:
(10)
that facilitates the resolution using mathematical models. In general, these models solve the coefficients of auto-regression (AR) using diverse regression methods.
Then, the complex roots of the polynomial in (10) are found. The phase of indicates the frequency and the amplitude is the damping factor. Finally, to solve Ck, the values of must be inserted in Eq. (9). Considering that the amplitude and the phase of the sinusoidal component is equal to the amplitude and the phase of Ck, respectively.
The characteristics of damping, natural frequency and stiffness of the present vibration mode in the systemare obtained. These methods are sensible to the noise, reason why the measurement must be made without noise and properly treated.
1.4. The new method to obtain the stability lobes for high-speed milling
The method to generate the stability lobes for a certain vibration mode identifies the transfer function of the system with the experimental analysis. The natural frequency, damping factor and stiffness of the mechanical system are determined experimentally.
Once these values are found, the real and the imaginary components of the transfer function for a certain chatter frequency can be calculated using:
(11)
where and is a chatter frequency.
It is necessary also to consider the average number of teeth during the cut (m):
(12)
where is the tool diameter, N the number of teeth of the tool and the radial depth of cut. The axial depth of cut limit of stability is calculated with the following equation:
1 (13)
where l is directional orientation factor and Km the cutting stiffness. To calculate the spindle speed using Eq. (7) is necessary to identify the phase shift (e) between the inner and outer modulations (present and previous vibrations marks), with the equation:
(14)
The new proposed method to identify the stability lobes for high-speed milling using a combination between the analytical prediction of chatter and the experimental modal analysis for multi degree-off reedom systems is:
Step 1: Obtain the characteristics of the tool, material to mechanise and milling process, and obtain the characteristics of the systemusing the experimental analysis of the system;
Step 2: Calculate the real and imaginary component of the transfer functions (KR, KI), Eq. (11);
Step 3: Select a chatter frequency from the transfer function around a dominant mode;
Step 4: Calculate the critical depth of cut from
Eq. (13);
Step 5: Calculate the spindle speed from n
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