1962 - Dynamic Acceptance Tests

Int. J. Math. Tool Des. Res. Vol. 2, pp. 267-280. Pergamon Press 1962. Printed in Great Britain

DYNAMIC ACCEPTANCE TESTS FOR MACHINE TOOLS

S. A. TOBIAS*

Summary--It is proposed that the dynamic quality of machine tools, in particular of radial drilling machines, be measured by the coefficient of merit. This coefficient is a product of the static stiffness between tool and workpiece, in the direction of the cutting thrust, and a non-dimensional ratio determined from the harmonic response locus of the machine structure.

The determination of the coefficient of merit in the case of radial drilling machines is discussed in details and its relationship with the stability chart of the drilling process is elucidated. It is shown that the coefficient of merit is proportional to the maximum drill diameter which is stable at all speeds and that its increase signifies a contraction of the speed ranges over which the machine chatters.

INTRODUCTION

A Ct;STOtaER buying a machine tool which is up to standard as far as the presently used acceptance tests are concerned has a guarantee that the machine has been properly manu- factured. What guarantee does he have that this machine has been designed properly in the first place ? None whatsoever!

The acceptance tests due to G. Schlesinger [1] deal solely with the machine alignment and accuracy under test conditions. They do not give a picture as to how the machine will behave under working conditions, that is, whether surface finish and machining accuracy will in actual fact come up to expectation. Nor do these tests give any indication whether the whole nominal horsepower capacity of the machine will be available for metal removal, or whether only part of it can be utilized for this purpose, the remainder being used for setting the machine into self-induced vibration, that is, for producing chatter patterns on the machined surface and cutting down tool life.

It might perhaps be assumed that the currently used acceptance tests, even if inadequate as far as the working performance of the machine is concerned, at least give a complete picture of the behaviour of the machine under test conditions. This, however, is not so. There is, for instance, no standard as to the allowable forced vibration between spindle and workpiece in a particular type of machine, an omission which is not altogether surprising. The tests used in 1961 were originally drawn up in 1927 and have survived with only minor modifications. Vibration measurements which to-day are carried out as a matter of routine, were thirty years ago beyond the boldest dream of the machine tool engineer.

STATIC AND DYNAMIC ACCEPTANCE TESTS

The tests presently used can be considered to be static tests since they describe the behaviour of the machine under constant (or zero) loads. Tests which give an indication of the behaviour of the machine under cutting conditions are of the dynamic variety, since they describe machine performance under varying (pulsating) loads. In fact, dynamic acceptance tests describe essentially the resistance of a machine to vibration arising as a result of the cutting process. The vibration might be generated by unbalanced rotating masses or by interrupted cutting (milling) and in that case it is of the forced vibration type.

* Professor of Mechanical Engineering and Head of Department, University of Birmingham. s 267

268 S .A . TOBIAS

Alternatively, it might be a self-induced vibration, due to dynamic instability of the cutting process, commonly known as machine tool chatter.

The following discussion summarizes work in progress which aims at the establishment of dynamic acceptance tests giving a comparative and absolute measure of a machine in its resistance to chatter. It will become clear that such tests constitute also a measure of the dynamic behaviour of a machine structure from the forced vibralion point of view.

The tests discussed apply primarily to cases in which the re-generative effect is the chatter generating and dominating cause. The consideration will be confined to face milling and drilling, these being two machining processes which have been sufficiently explored from the point of view of their dynamic stability.

STABIL ITY CHARTS AS COMPARATIVE DYNAMIC TEST CERT IF ICATES

For a comparative assessment of machine tools of the same type but of differing design, stability charts describing their chatter behaviour can be used. A typical stability chart is shown in Fig. 1. It refers to a vertical milling machine on which a face milling operation is carried out. It was experimentally determined by setting up a certain depth of cut on the machine and machining with a series of speeds covering a wide range. If at a certain depth of cut, and speed, chatter is encountered then the appropriate point in the chart is marked with a dot (or a half filled circle or a circle, depending on the chatter amplitudes). As can

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FIG. 1. Experimentally determined stability chart of a face milling process. Workpiece material: cast iron, number of cutter blades 32. After Tobias [2].

be seen, at a small depth of cut (below 0.045 in.), the cutting process was chatter-free at all speeds. Above this minimum depth of cut, chatter arose in certain speed bands which were separated from each other by stable (chatter-free) speeds. The unstable speeds form a number of lobes in the chart which overlap at large values of the depth of cut, indicating that under these conditions all speeds are unstable. At the lower values of the depth of cut, these lobes touch a hyperbolic line, the stability band envelope, denoted by hm in the figure. Cutting conditions lying below this line are stable.

Assume now that a comparative assessment of two vertical milling machines has to be made and that the stability chart of each of these is given by Fig. 2(a) and (b), respectively. In actual fact, both figures have been determined by calculation, Fig. 2(a) being the calculated version of Fig. 1 and Fig. 2(b) having been determined with the view of investigating

Dynamic Acceptance Tests for Machine Tools 269

the effect of a certain design alteration of the machine in question on its stability chart, i.e. its chatter behaviour. It is quite clear that the machine with the stability chart of Fig. 2(b) is superior to that which yields Fig. 2(a). This impression is by no means misleading. Fig. 2(b) corresponds to a machine which is in all respect identical with that which produced Fig. 2(a), except for a hypothetical design modification which increased by twenty-five

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FIG. 2. Stability charts of face milling process determined by calculation. Fig. 2(a) is the calculated version of Fig. 1. Fig. 2(b) shows effect of a design modification which increased

stiffness by 25 per cent. After Tobias [2].

per cent the dynamic stiffness of the mode becoming unstable, this being assumed to be brought about without increasing the vibrating mass of the machine.

On the basis of the example shown in Fig. 2, it will be agreed that the usage of stability charts as a graphical means for presenting the comparative dynamic quality of machines, does recommend itself by a number of attractive features. They present the essential information in an easily understood way, which is closely related to the actual utilization of

270 S.A. TOBIAS

the machines. The charts can be determined by experiment and they can also be found by calculation. In the latter case, they afford a means with which the designer is able to estimate the practical effect of design alterations. However, this type of dynamic test has also a number of disadvantages which make it unsuitable for general application. It is quite obvious that the reasonably accurate determination of these stability charts is by no means an easy matter. A large number of cutting tests have to be performed for this purpose and the cutting conditions must be carefully standardized. The large number of cutting tests are time consuming and make a standardization of the conditions difficult. A change of workpiece or relatively small changes of tool geometry might introduce errors of con- siderable magnitude.* Owing to the wide variation of the machining properties of materials of the same nominal specification, the repeatability of these tests would not be very reliable either. To this must be added also that most machines do not have a sufficient number of speeds to make the reasonably accurate determination of the stability charts possible.

From the enumeration of these difficulties, it is clear that they arise by virtue of the fact that the stability charts embody not only the essential dynamic characteristics of the machine, but also certain aspects of the cutting condition, that is, certain properties of the workpiece material, tool geometry, coolants etc. As far as a dynamic acceptance test is concerned, the essential dynamic characteristics of the machine which determine (with the cutting conditions) its chatter behaviour, are quite sufficient. At the same time, however one is reluctant to drop the idea of using stability charts as a graphical representation of dynamic performance, simply because they give a clear picture of the practical implications involved and are experimentally testable. This dilemma is resolved by considering those features of the stability chart which can be regarded as containing the practically important information, and the relation of these with the dynamic characteristics of the machine in question.

THE STABILITY BAND ENVELOPE AND ITS HORIZONTAL ASYMPTOTE

From the practical point of view, the stability band envelope (the hm line in Fig. 1 and Fig. 2) and the right-hand boundary of the highest unstable band are of greatest importance. This is due to the fact that cutting conditions lying below the envelope and at speeds above the right-hand boundary of that band are unconditionally stable.

The position of the stability band envelope in the chart is determined by its horizontal asymptote, which cuts the ordinate axis at hmo in Fig. 2. hmo is the maximum value of the depth of cut which is stable at all speeds. As a matter of fact, it can be seen from Fig. 2 that a depth of cut slightly larger than hmo is still stable and that consequently hmo errs on the safe side.

The value of hmo is determined partly by the dynamic behaviour of the structure and partly by the cutting conditions. It is equal to a product consisting of a term found from forced vibration tests, the static stiffness As between tool and work piece and a proportion- ality factor derived from cutting tests. Provided that the same cutting conditions are considered, and machines of the same type (but differing design) are compared, the propo- tionality factor will also be the same. It follows that under these conditions the hmo values give an indication of the relative dynamic and static quality of the machines and for these

* A typical example of the type of errors which can thus be introduced will be discussed in connection with Fig. 3.


reasons, they could be used, in certain cases, as the first approximation of a measure of their dynamic performance.

Consider again Fig. 2. It will be remembered that Fig. 2(b) is derived from Fig. 2(a) by increasing by 25 per cent the stiffness of the mode which tended to become unstable. Other- wise the two machines were identical and the cutting conditions were also the same. It can be seen that the hmo value of Fig. 2(b) is 25 per cent above that of Fig. 2(a). Clearly, in a case when it is already certain that two or more machines have essentially similar stability charts, the hmo values corresponding to each machine, corresponding to a standardized cutting condition, represent a perfectly sufficient measure of their dynamic performance.

It is easy to think of cases in which the hmo value is an insufficient criterion of dynamic performance. This is in general the case when for some reasons the maximum depth of cut which is vibration-free at all speeds is considerably greater than hmo. The conditions are the same when it has not been established that the dynamic behaviour of the machines to be compared shows sufficient similarity. In both cases, at least the stability band envelope and the upper boundary of the highest unstable speed band is required, or the stability chart in its complete or condensed form. The condensed form of the stability chart can be obtained in the case of certain machining processes (for instance drilling), for which it contains more information concerning the chatter behaviour of the machine than is of practical interest. An example for this will be given in the following section.

DYNAMIC ACCEPTANCE TEST FOR RADIAL DRILL ING MACHINES

The question which comes to mind is: How do these suggestions work out in practice ? To answer this question, work in progress, concerned with radial drilling machines, will be discussed.

A stability chart of a drilling operation,* carried out on a radial drilling machine, is shown in Fig. 3(a). The chart represents chatter behaviour of the machine under following conditions:

1. The radial arm was in the middle position and the drilling saddle at the free end of the arm.

2. The drilling operation consisted of drilling up holes which had been pre-driUed to the chisel edge width.

3. Mild steel was used as workpiece material. 4. With the exception of one diameter, geometrically similar drills were used, ground

with care on the same machine. Conditions (1) and (2) were chosen to ensure that the machine will chatter in several

speed bands, so as to test the theoretical method used under highly unfavourable conditions. These were caused by the large flexibility of the structure in the position chosen and by the absence of the stabilizing effect of the chisel edge.

Speeds at which a certain drill diameter produced chatter are marked with circles in the chart. Only a limited number of drill speeds were available since the machine used was of the standard type. Nevertheless, these were enough to allow the drawing of the boundaries of the unstable speed bands. An exception to this statement were results obtained with a drill of ~g in. diameter. This drill was of a different make from the others (with a slightly different geometry), and its chatter behaviour showed some inconsistency. Speeds at which

* Figures. 3, 4, 7, 8 and 9 have been taken from the unpublished Thesis ofI. ONtm entitled "Prediction of the Chatter Behaviour of a Radial Drilling Machine."

272 S.A. TOBIAS

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FIG. 3. Stability charts of drilling process. Fig. 3(a) chart determined by experiment. Fig. 3(b) chart determined by calculation. After Onur [3].


this drill chattered are marked with question marks. One of these, that which was obtained at 1140 rev/min, lies outside the drawn boundary of the unstable band. This is considered to be a "freak result", due to lack of standardization of the test procedure. It was included so as to show the care that has to be taken in these tests as far as the experimental conditions are concerned. Owing to this freak result, the boundaries of the highest and next lobe are considered to be unreliable below a drill diameter of about d ---- in., and for this reason this area is drawn in dotted lines.

Figure 3(a) gives a complete picture of the chatter behaviour of the machine under the conditions stated. However, it actually contains more information than is required in practice. This is due to the fact that, assuming the workpiece material to remain unaltered, to each particular drill diameter there belongs only one optimum drilling speed. The relationship between drill diameter and rotational speed for mild steel, as specified by the manufacturers of the machine tested, is represented by the thick hyperbolic line, denoted "diameter-speed function". This line is chain dotted where it intersects the two highest unstable lobes, indicating that the corresponding drill diameters and speeds are unstable. The remainder of the curve is dotted. The speeds on the boundary of stability are given in the figure.

Noting now those speed ranges, and the corresponding drill diameters, which lead to unstable drilling conditions and plotting them on a linear scale, leads to the condensed version of the stability chart, a simple graph giving a clear picture of the chatter behaviour of the machine. Such a chart is shown in Fig. 4. A similar graph can be obtained for all other positions of the arm and the saddle but in practice only a limited number of these

EXPERIMENT

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DRILL DIAMETER

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FIG. 4. Condensed version of stability chart showing unstable drill speeds along the diameter-speed function of Fig. 3(a).

will be required for giving a complete picture of the chatter behaviour of the machine. As a matter of fact, when comparing two or more machines from the chatter point of view, it will probably be quite sufficient to do this with only three condensed stability charts of the type shown in Fig. 4, these corresponding to the configuration when the arm is in the central position of its total range of travel and the saddle is nearest to the column, in the centre of the arm and at its free end.

For the practical engineers, the condensed stability chart of Fig. 4 is probably the ideal dynamic test certificate which gives him all the information he requires to decide the capa- bilities of a machine or the relative merits of one design over another. However, when it comes to using these graphs for selecting from competitive machines the one which is the best from the dynamic point of view, difficulties are likely to arise. As already stated, unless these charts are all determined under closely controlled identical experimental conditions, it may well be that the scatter caused by different drill geometries and workpiece materials may make a comparison impossible. For this reason a dynamic criterion is sought which depends solely on the dynamic characteristics of the machine structure and from which, if

274 S.A. TomAs

necessary, the information contained in Fig. 4 could be derived by calculations, to give the practical engineer an indication of the type of chatter behaviour he can expect in the workshop.

Such a dynamic criterion is in fact, already available, at least for machines which from the dynamic point of view behave in as simple a fashion as radial drilling machines. For the appreciation of this dynamic criterion, and its relation to the chatter behaviour of the machine, some knowledge of chatter theory, and in particular, of the graphical method for chatter analysis developed by Gurney and Tobias [4] is prerequisite.

The method due to Gurney and Tobias is based on the harmonic response locus of the machine tool structure. This is obtained by setting the machine into forced vibration by a harmonic force imitating the dynamic action of the cutting force and by measuring the amplitude and phase of the ensuing vibration. The experimental set-up used in the case of radial drilling machines is shown in Fig. 5. The vibration generator, which is driven by an oscillator and power amplifier, acts in the direction of the drill thrust, the exciting force

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FIG. 5. Experimental set-up for the measurement of the harmonic response locus of a radial drilling machine.

being measured by a dynamometer. In the figure the relative vibration amplitudes are determined with the capacitative method. The method of measurement used is, of course, immaterial, provided that both the dynamometer and the relative vibration pick-up work on the same principle so as to avoid spurious phase errors. Otherwise these errors must be eliminated by correcting the results appropriately. The result of the amplitude measurements is usually given in the form of a resonance curve, which for radial drilling machines is generally of the type shown in Fig. 6(a).

Figure. 6(b) shows the corresponding phase measurements. The harmonic response locus is obtained by plotting the resonance curve and the phase curve in polar co-ordinates, as represented in Fig. 6(c).

The harmonic response locus describes completely the dynamic (and static) behaviour of the machine with respect to a force acting in the direction of the cutting thrust. Its form constitutes a complete test certificate of the dynamic (and static) characteristics of the machine, as relevant from the chatter point of view, which can be used for comparing the


dynamic quality of machines of the same type but of differing design. From it, with the aid of the method due to Gurney and Tobias, the chatter behaviour of these machines is easily determined, as will be seen presently.

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FIO. 6. The relationship between the resonance and phase curves and the harmonic response locus. Fig. 6(a) resonance curve, Fig. 6(b) phase curve, Fig. 6(c) harmonic responses locus combining amplitude and phase variation in polar co-ordinate form. After Gumey and

Tobias [4].

276 S.A. TOBIAS

To make the implications of these statements clearer, it is convenient to normalize the locus by multiplying the amplitude vectors with the static stiffness ,~8 between tool and workpiece. After normalization the distance OP = xs (static deflection) becomes equal to the force xshs ----- P producing this deflection, where P is the amplitude of the exciting force with which the locus was determined.

Consider now the normalized loci of a number of machines obtained with the same force amplitude P, a typical curve being presented in Fig. 7. It can be shown that from the chatter point of view only that part of the locus is of interest which corresponds to

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phase angles larger than 90 , that is, which lies to the left of the vertical axis in the figure. Let R be that point on the locus which is furthest from the vertical axis, then the machine for which the distance of R from the vertical axis is greatest is the worst and for which it is smallest is the best design.

This somewhat vague measure of the dynamic quality of the machines can be put into more precise terms in the following way: Let OP' - - OP and find the bisector of OP' denoted by line AB in Fig. 7. The intersection of AB with the radius vector OR is denoted by R'. The dynamic quality of the machine is now measured by the product.

OR' A~. -- Coefficient of Merit*

OR

The greater the value of the coefficient of merit the better the chatter performance of the machine under the conditions investigated. One such factor can be allocated to each posi-

* Readers familiar with the theory of re-generative chatter will have recognized that the coefficient of merit is equal to 2'klmin where z represents the number of cutting edges of the tool (z = 2 for drilling) and klmin is the minimum value of the chip thickness variation coefficient for which chatter arises, klmin is closely related to the Grenz-Tiefenkoeffizient rg used by Tlusty and Polacek [5] in their work. Concerning the relation between klmin and rg see Andrew and Tobias [6].


tion of the radial arm and saddle. As already mentioned it can be expected that in practice only a limited number of positions of these machine components have to be considered, for instance, in all probability only three. At the utmost, the top and the bottom positions of the arm might also be included, a total of nine values of the coefficient merit being required.

THE PHYSICAL SIGNIFICANCE OF THE COEFFICIENT OF MERIT* Practical engineers are likely to be sceptical that only three or at the utmost nine values

of a coefficient be sufficient for forming a measure of the chatter performance of radial drilling machines. They will be quite justified to insist of knowing, (1) what the coefficient of merit actually measures, (2) whether the measured quantity is of practical importance, and (3) that it measures this feature of the machine consistently and reliably.

The physical significance of coefficient of merit is most easily explained by referring to the stability chart of a machining process, for instance, Fig. 2. In that figure, the horizontal asymptote of the stability band envelope hm is denoted by hmo and it represents that depth of cut (of the face milling process to which the chart refers) which is stable to all speeds. It can be shown that the coefficient of merit is proportional to hmo, the factor of propor- tionality being dependent on the tool design (geometry number of blades, number of blades in contact, etc.) and the workpiece characteristics (hardness and general machinability, etc.) and the cutting conditions (cutting speed, feed, coolants, etc.).

Analogous conditions arise in the case of drilling machine, the only difference being that the stability chart the unstable bands lie above the dm curve, this being asymptotic to the dmo line which represents the maximum drill diameter which remains stable at all speeds, dmo is found from:

As OR' coefficient of merit dmo -- -- (2)

x OR x

where X is a constant dependent solely on the drilling conditions. It is clear that from the point of view of using the coefficient as a comparative measure

of the dynamic behaviour of radial drilling machines, the actual value of x need not to be known. A value for x is required solely for attributing an actual physical quantity (the drill diameter dmo) to it.

It now remains to be investigated whether the coefficient of merit represents a sufficiently reliable and consistent measure of the chatter behaviour of the machine. This can be tested either directly, by carrying out a limited number of drilling tests and finding the maximum drill diameter which is stable at all speeds, or indirectly, by verifying the theory on which the concept of the coefficient is based. Direct tests may be impractical, as will be seen presently, and in that case it must be remembered that the coefficient of merit is but a special feature of the graphical method due to Gurney and Tobias. Finding the stability chart of the machining process by that method and comparing it with the experimentally determined chart of the same process, confirms not only the soundness of the method but also that of the coefficient of merit.

Consider now, as a concrete case, the machining process the stability chart of which was shown in Fig. 3(a). Owing to the impossibility of obtaining a series of drills covering a wide range of diameters which are sufficiently similar to each other, the direct determination of the maximum drill diameter which remains stable at all speeds is hardly feasible, at least not with the required accuracy. The only remaining course is thus the determination of the whole stability chart and/or its condensed form, as shown in Fig. 4.

278 S.A. TOBIAS

For the determination of the stability chart the harmonic response locus of the machine is found, this being presented in Fig. 8. Applying the method due to Gurney and Tobias [2] to the locus yields the stability chart o f the drilling process, presented in Fig. 3(b). The chart shows the practical significance of dmo, that is, that it is slightly smaller than the maximum drill diameter which is stable at all speeds. From the theoretical point of view, it is the asymptote of the stability band envelope, denoted by din.

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90* FIG. 8. Experimentally determined harmonic response locus of radial drilling machine used in tests, corresponding to the arm and saddle positions shown in Fig. 5. After Onur [3].

Comparing Fig. 3(a) with Fig. 3(b) shows that the correspondence between the theoretical and experimental results is not altogether satisfactory. The discrepancies are largely due to the shape of the dm curve which determines the ordinate position of the lower unstable speed bands. The reasons for the unsatisfactory correlation between the theoretical and the experimental stability charts will be discussed later. Research is already in progress which has the general aim of improving the correlation, although in the present case no such improvement of the method is required, since the theoretical stability charts yield the practically significant results with surprising accuracy.

EXPERIMENT

THEORY

DRILL SPEED

DRILL DIAMETER

!

0 100 200 300 400 500 600 700 800 900 1000 I100 R IM I I I i I 3 5 1 3 1 zt 8 2 "8" 4" INCH

FIG. 9. Experimental and theoretical condensed stability charts as derived from Figs. 3(a) and 3(b). After Onur [3].


This will be seen by considering that the discrepancies exist very largely in a region of the stability chart which corresponds to drilling conditions never arising in practice. As already mentioned, the relationship between the drill diameter and drill speed is given by the diameter-speed function and in the vicinity of this line the correlation is perfectly satisfactory, as will be seen from the condensed stability charts of Fig. 9. This figure shows the experimentally determined unstable speed bands and corresponding drill diameter, derived from Fig. 3(a), and already shown in Fig. 4 in addition to the theoretical prediction according to Fig. 3(b). As can be seen, the deviation between the experimental and the theoretical results is better than 5 per cent. A greater accuracy can hardly be expected.

It is obvious that if the coefficient of merit is increased then the unstable speed bands in Fig. 9 contract and the chatter behaviour of the machine improves. This becomes clear by considering that an increase of the coefficient results in an increase of dmo. Owing to this all unstable speed bands shift upwards in Fig. 3(b) and since the diameter-speed function remains unaffected, the speed ranges over which it intersects the shaded (unstable) regions contracts.

CONCLUSIONS

The shape of the stability band envelope, the hm curve in Fig. 2 and the dm curve in Fig. 3, is dependent on the, so-called, penetration rate effect, which represents the resistance encountered of the tool during a dynamic penetration of the workpiece material. The coefficient through which this effect enters the theory of regenerative chatter is a dynamic coefficient which cannot be determined by the customary static cutting thrust measurements. Such experiments yield only approximate values of the penetration rate coefficient and it is quite clear from the lack of correlation between the theoretical and the experimental stability charts that the degree of approximation was insufficient. Special dynamic experiments are already in progress which ought to yield more accurate values and it is expected that with these a greater similarity of the two types of charts will be achieved.

The penetration rate effect is of importance also from the point of view of the general interpretation of the coefficient of merit. In some cases, for instance when drilling into solid material, this coefficient is very large (owing to the presence of the chisel edge) and in that case the stability band envelope, lies far above the dmo line in the speed range which is of practical significance. It follows that the maximum drill diameter which is stable at all (practical) speeds is very much larger than dmo, and with this the simple interpretation of the coefficient of merit given in this paper is lost. However, this introduces only slight complications since once dmo is known and the appropriate penetration rate coefficient is found, the dra curve is easily calculated with the aid of a simple formula [2].

Complications are likely to arise also in cases when more than one mode of vibration of the machine tool structure may become unstable. The stability chart of such a system consists essentially of the superposition of simple charts of the type shown in the paper [4]. These charts will contain two or perhaps more stability band envelopes and corresponding to the horizontal asympote of each of these a coefficient of merit. It is to be expected that the smallest of these will be representative of the chatter behaviour of the machine.

Finally, it ought to be pointed out that although in the example discussed in the paper the unstable drill speeds (and diameters) were predicted with sufficient accuracy and other similarly satisfactory tests could be quoted, cases were observed when the prediction was far off the mark. These deviations are attributed partly to the technique so far used for the determination of the harmonic response loci and partly to the mystery which is still sur-

280 S.A. TOBIAS

rounding the dynamic coefficients of the cutting process. The experimental technique used for the investigation of the vibration of machine tool structure is, however, constantly improved and research relating to the dynamic cutting coefficients is also being actively pursued.

REFERENCES [1] G. SCHLESINGER, Testing Machine Tools. The Machinery Publishing Co., London (1945). [2] S. A. Tomas, Proc. Instn. mech. Engrs, 173, 474 (1959). [3] I. OtqtrR, Prediction of the Chatter Behaviour of a Radial Drilling Machine. Unpublished Thesis, Dept.

of Mech. Eng., University of Birmingham (1961). [4] J. P. GURNEY and S. A. TomAs, Trans. Amer. Soc. mech. Engrs, 84 B, 103 (1962) [5] J. TLUSTY and M. POLACEK, Drittes Forschungs und Konstruktionskolloquim Werkzeugmaschinen,

p. 131, Ill. FoKoMa, Vogel Verlag, Coburg (1957). [6] C. ANDREW and S. A. TOmAS, Int. J. Mach. Tool Des. Res., 1, 325 (1961).

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1962 - Dynamic Acceptance Tests