A Stochastic Approach to Characterization of Machine Tool System Dynamics Under Actual Working Conditions

The machine tool dynamics is evaluated under actual working conditions by using a time series technique. This technique develops mathematical models from only one signal, viz., the relative displacement between the cutter and the workpiece. Analysis of the experimental data collected on a vertical milling machine indicates that the new methodology is capable of characterizing the machine tool structure and the cutting process dynamics separately. Furthermore, it can also detect and quantify the interaction between these two subsystems.

Computer Design of a Vibration-Free Face-Milling Cutter

A method is developed to choose the blade spacing for a face-milling cutter that will minimize vibration. When the dynamic frequency response of a machine-tool-workpiece system is known, a special-purpose cutter can be designed to minimize the relative cutter-workpiece vibration for a particular cutting speed. When the dynamic system response is unknown the design method is limited to a general-purpose cutter to avoid resonant excitation over a broad frequency range. A nonlinear least-squares method is used to choose the blade angles for both cases. Experimental results using a general-purpose cutter with unequal blade spacing showed an appreciable reduction in noise and vibration compared to a similar cutter with equal blade spacing.

Stability of Random Vibrations With Special Reference to Machine Tool Chatter

Static and dynamic stabilities of self-excited random vibrations are investigated in terms of the differential equation and time series model for the vibrational signal. Various instabilities are demarcated in the parameter space of the time series model, so that the stability of random vibrations can be ascertained by locating the parameters obtained from the vibration data. These results are applied to machine tool chatter by analyzing tool point vibrations in a turning operation under different degrees of chatter. This analysis substantiates the theoretical investigation, which is further confirmed by resonance curves obtained for the workpiece and cutting tool.

Modeling Machine Tool Chatter by Time Series

Machine tool chatter is formulated as self-excited random vibration with white noise forcing function. The formulation takes into account the unknown factors and random disturbances present in the cutting process when chatter occurs. Based on this formulation, a procedure for modeling chatter using the time series of sampled observations on vibration signals is developed. Feasibility of this procedure is established by modeling data obtained from a turning operation under conditions of severe chatter.

Determination of Geometric Properties of Coated Abrasive Cutting Edges

The surface topography of a coated abrasive was measured by a special designed profilometer with an oscillating stylus, revealing very detailed geometric features of the peaks. The criterion for a peak to be a dynamic active cutting edge is analyzed and the results are applied for the identification of active cutting edges of the measured profiles. The distributions of some geometric properties of the active cutting edges as heights, distances, rake-angles and wear-lands are evaluated for six grades of coated abrasives.

On Building Models for the Grinding Process

A new technique of building models for the grinding process is introduced. The grinding model is postulated in terms of grain distributions on the abrasive wheel and kinematic grinding conditions. Transverse workpiece profiles are conjectured as sample realizations of the stochastic grinding process. The sample spectrum of experimentally ground workpiece profiles is proposed as a means to estimate parameters in the postulated model through a nonlinear regression method. A model for a specified grinding process was developed based upon ten workpieces ground within a preselected practical region. The model is limited to single-pass surface grinding without crossfeed. The model was evaluated in regard to its validity, versatility, and sensitivity.

Relationship Between the Parameters of an Autoregressive Model and Grinding Wheel Constituents

Autoregressive models are developed for eight wheels differing in grit size, hardness, and structure. Using the parameters as responses of a factorially designed experiment, the relative contribution of the wheel constituents to the total wheel topography is related and quantified. Utilizing the effects observed, the individual and interactive roles of the wheel constituents are discussed. From the experimental observations a formation mechanism is suggested.

Simulation Study of the Grinding Process

A model in terms of grain distributions and kinematic conditions has been recently developed [8] for a specified grinding process. Using this model, an exploratory attempt was made to study the grinding process with respect to the surface topography and nature of surface interactions by simulation on a digital computer. Fundamental grinding parameters, such as the effective grain density, grain spacings, chip depths, and chip areas, are employed to describe the grain-workpiece interactions. The grinding process is studied under a systematic variation of table speed, wheel depth of cut, and grain apex angle. The physical significance of the Z-distribution and stochastic nature of the grinding process is also discussed.

Characterization of Abrasive Tools by Continuous Time Series

A succinct characterization of abrasive tools using continuous time series is developed. It is based on three parameters: natural frequency ωn, damping factor ζ, and variance γ0. These parameters can be readily interpreted in terms of the visual records. The interpretation, along with the effects of the physical constituents on the parameters, is discussed and illustrated for profiles of 8 grinding wheels, with different grit size, hardness, and structure.

Analysis and Design of a Drill Grinder and Evaluation of Grinding Parameters

A prototype drill grinder was designed and built based on a computer aided drill point geometry analysis. The new grinder controls all essential drill point grinding parameters. The new grinder was evaluated by grinding drills, measuring their point geometry parameters, and comparing these measurements with their expected values. The effects of five parameters, consisting of three grinding and two cutting condition parameters, on the drill thrust and torque are determined by an experiment using a two-level factorial design.

Surface Profile Characterization by Autoregressive-Moving Average Models

The surface texture of a machined part is in general composed of three topographical components: waviness, roughness, and errors of form. A new technique for surface profile characterization is introduced which employs parametric stochastic models of the autoregressive-moving average (ARMA) class. The method for obtaining these models for surface profiles is shown by an example. The ARMA modeling technique for profile description is evaluated in three parts to determine its validity, workability, and descriptive power. This analysis is developed through the criteria of ergodicity, sensitivity, and describability. The ergodicity criterion tests the ability of models for physically identical profiles to convey equivalent information. The sensitivity criterion measures the level of detection of topographical differences among profiles by the ARMA model parameters. The descriptive ability of the models is examined by interpreting their parameters in light of the physical components of the profile. To implement this evaluation, ARMA models for eight different milled surfaces are determined and used.

Analysis of the Chisel Edge and the Effect of the d-Theta Relationship on Drill Point Geometry

Following the investigation of drill point geometry previously reported [1, 2], the chisel edge profile and the chisel edge angle are analyzed mathematically. The relationship between the cone semiangle, θ, and the x-coordinate of the cone vertex, −d, is derived for a given chisel edge angle and nominal relief angle at the outer corner. This relationship is used to study the effects of grinding cone parameters on drill point geometry. The analysis indicates the existence of an optimum θ value for best drill performance.

Drill Temperature Distributions by Numerical Solutions

The backward finite-difference method is used to determine three-dimensional drill temperature distributions. The geometry of the drill was described by (1) approximating the drill as a one-quarter cone and (2) sectioning a true drill point and measuring its profiles. The three-dimensional temperature distributions provided both drill cutting edge and drill flank temperature profiles which were close to prior experimental data and showed improvement over the previous analytical solutions.

An Analysis of Drill Geometry for Optimum Drill Design by Computer. Part I—Drill Geometry Analysis

A comprehensive analysis of the twist drill point geometry is made in order that the high-speed digital computer can be used as an aid in the design of a drill. This subject is treated in two parts. In Part I, the drill geometry is analyzed with respect to the drill flute and flank contours by considering cross sections of the drill cut by planes perpendicular to its axis. Since several important drill angles are defined in planes inclined to the drill axis, the analysis is extended to cover the general case where the drill is cut by any plane inclined to its axis.

Evaluation of the Effects of Design Variables on Drill Temperature Responses

The effect of five design and three operating variables on three different drill temperature responses is investigated. A total of 512 observations was made using a replicated two-level factorial design. Problems associated with the selection of the temperature responses are studied including the transient nature of the response, the location of the thermocouple, and the interrelationship of the workpiece volume and drill wear. The five drill design variables are web thickness at point, margin width, relative lip height, helix angle, and surface condition, of which the helix angle and web thickness at point had the most significance within the experimental range. The effect of two-factor interactions was also discussed.

An Analysis of Drill Geometry for Optimum Drill Design by Computer. Part II—Computer-Aided Design

In Part II, a computer program that aids in the design of a twist drill is described, based on the analysis presented in Part I. A true drill and a “computer-designed” drill having identical design and grinding parameters are compared in orthogonal cutting planes. The effects of the design parameters on drill geometry are investigated utilizing the computer program.

Building a Mathematical Model to Predict Transient Drilling Temperature Responses

A transient drilling temperature predicting equation is developed using a statistical model-building technique. The functional form of the model is obtained by combining both theoretical considerations and an empirical approach. The parameters of this model are estimated and improved using a sequential procedure in designing the experiments. Confirmatory tests show that the model describes the transient drilling temperature responses remarkably well. The effectiveness of the design procedure for obtaining the “best” estimates of the parameters is also demonstrated.

Investigation of Face-Milling Tool Temperatures by Simulation Techniques

This paper presents simulation techniques to further study the tool temperature distribution curve previously reported. The simulation setup employed a miniature oxyacetylene torch incorporated with a pulse control mechanism. The experimental simulation yielded similar temperature distribution curves to those obtained in the cutting experiments. The temperature distribution curve obtained by digital computer simulation shows a good agreement with that obtained in the cutting experiments. An analysis of the effect of the removed material near the tool tip was made using the numerical method. The result indicates that the material removed up to a tool tip thickness of 0.085 in., which was used in the previous paper, does not show significant effect on the temperature responses.

Temperature Responses and Experimental Errors for Multitooth Milling Cutters

A milling cutter temperature measuring technique was devised using garter spring pickups and standard thermocouples positioned at the back of inserts. Temperature signals as many as eight teeth were recorded simultaneously. It was found that the average cutting temperature of a multitooth cutter was higher than a flycutter at the same cutting conditions. The difference between one and two-tooth cutters was the largest and it tended to decrease when the number of teeth increased. The experimental error was thoroughly analyzed. The standard error for a flycutter was found to be about one half of that for multitooth cutters. Analysis of the temperature responses and the experimental errors indicated that a two-tooth cutter rather than a flycutter should be used to evaluate the performance of a multitooth cutter.

Temperature Distributions in Drilling

An analytical temperature distribution along the cutting edge and on the flank face of a drill is obtained using Tsueda’s and Loewen and Shaw’s equations after modifications. Experimental temperature distributions are also investigated and the effects of a pilot hole and the heat sink are evaluated. Agreement between the analytical and experimental results is fairly close except near the chisel edge and near the drill periphery. The use of workpieces containing a pilot hole provides a method to account for the discrepancy between the analytical solution and the experimental results near the chisel edge while some explanation is offered to describe the discrepancy near the drill periphery.

The Effect of Experimental Error on the Determination of the Optimum Metal-Cutting Conditions

The effect of experimental error in tool life testing on the determination of the minimum cost cutting speed (Vmin) is investigated by using the concept of statistical inference. It is shown that Vmin is not uniquely defined but lies within a probable interval of cutting speeds, and that this interval is affected by the cost and time parameters and the experimental range of feed in tool life testing. The selection of a specific speed from the Vmin confidence interval is illustrated by a decision rule based on the minimax principle.

Maximum Profit as the Criterion in the Determination of the Optimum Cutting Conditions

Maximum profit is an appropriate criterion for the selection of the optimum machining conditions rather than the conventional criteria of minimum cost or maximum production rate. A simple example is presented to illustrate the determination of the maximum-profit cutting speed by application of a fundamental economic principle that maximum profit occurs at the production rate where the marginal revenue equals the marginal cost. The effects of the demand function, feed, and cost and time parameters on the determination of the maximum-profit cutting speed are analyzed. Emphasis is given to the investigation of a range of optimum cutting speeds, instead of the theoretical optimum speed, for practical applications.

An Exploratory Study of Taylor’s Tool-Life Equation by Power Transformations

Transformations of both dependent and independent variables are employed to investigate the linearization of Taylor’s tool-life equation. This exploratory study indicates that a logarithmic transformation, which is a special case of the general class of power transformations, gives the best fit for HSS tool-life data. However, the study does not show that a logarithmic transformation is the best for carbide tool-life data. For a wide cutting range, where Taylor’s tool-life equation does not hold, a linear equation instead of a second-order relationship for the prediction of tool life can be determined by the proper transformation.

Tool-Life Testing by Response Surface Methodology—Part 1

This paper presents a study of tool-life testing by a statistical approach, referred to as response surface methodology, instead of the conventional one-variable-at-a-time method. With this new technique the number of tests required to develop a tool-life predicting equation can be substantially reduced. The reliability of such an equation can also be estimated. Three independent variables, speed, feed, and depth of cut, were investigated in this project. A simple first order equation is “graduated” in Part 1 and a more general second-order equation will be presented in Part 2 of this study.

Tool-Life Testing by Response Surface Methodology—Part 2

This paper is a continuation of a previous paper in which the basic philosophy of response surface methodology has been explained and a first-order tool-life-predicting equation has been developed. This part of the paper illustrates the development of a second-order tool-life-predicting equation in 18 and 24 tests. It was found that the second-order effect did not show statistical significance within the cutting ranges of this project; however, the second-order effect of cutting speed has been found important by the study of residuals. If only one independent variable is investigated, a minimal number of tests can be used to find a second-order equation. Examples of designs in three, five, and six tests are illustrated.

Cutting-Tool Temperature-Predicting Equation by Response-Surface Methodology

Empirical general cutting-tool temperature-predicting equations in terms of speed, feed, and depth of cut are developed by response-surface methodology. The practicability of the first-order model has been shown by confirmatory tests. The importance of the second-order model, particularly at a high-temperature range, is also discussed.