On the validity of several common assumptions in the design of machine tool spindle-bearing systems

Int. J. Mach. Tools Manufact. Vol. 31, No. 2, pp.235--248,1991. Printed in Great Britain

0890--6955/9153.00 + .00 Pergamon Press plc

ON THE VALIDITY OF SEVERAL COMMON ASSUMPTIONS IN THE DESIGN OF MACHINE TOOL SPINDLE-BEARING SYSTEMS J. A. B~NDON* and K. J. H. AL-SnARF.EF* (Received 24 July 1990) Al~traet--The paper examines a number of common assumptions which have been used for the modelling of spindle-bearing systems. It is shown that the particular functional geometrical configuration of some categories of machine tools may lead to an apparent validation of the more simplistic models whilst significant structural behaviour, predicted by the Timoshenko model but neglected in the Euler-Bernoulli model, is extremely difficult to measure experimentally, leading to specious acceptance of the erroneous analysis.

1. INTRODUCTION

THE REVIEWERSof recent papers by the authors [1, 2] have, quite properly, and with some justification, questioned a number of assumptions used by the authors for the modelling of spindle-bearing systems. The authors have valued the criticism and advice received and believe that there is considerable merit in the wider discussion of the issues raised. The modelling assumptions discussed in the current paper are: the neglect of shear deformations of the shaft; neglect of rotary inertia; neglect of gyroscopic effects. In asssessing the validity of these assumptions, it is, firstly, appropriate to ask fundamental questions concerning the nature of mathematical models in design. If these considerations are met, on a prima facie basis, then the value of the specific assumptions may then be addressed. Having established the criteria of assessment, the specific modelling assumptions under consideration may then be measured against criteria of necessity and/or desirability under the constraints of available resources. As will be seen, the more simplistic models may be capable of providing close approximations to some of the required structural properties whilst completely failing to predict other dynamic behaviour. 2. GENERAL CONSIDERATIONS RELATING TO THE VALIDITY OF DESIGN MODELS

The validity of a mathematical model may be judged, both axiomatically and pragmatically, in terms of its conformance to specification under prescribed operating conditions. Specifically, if the predictions of a structure's behaviour, under all conceivable operating conditions, match those observed by an experiment spanning those operating conditions, then the mathematical model may be regarded as valid. Brandon [3] has characterized the interactions and relationships between design specification, mathematical model and physical realization in terms of a reanalysis triangle, as shown in Fig. 1. As has been suggested by Brandon and AI-Shareef [4], in a commercial design

*Division of Mechanical Engineering and Energy Studies, School of Engineering, University of Wales College of Cardiff, P.O. Box 917, Cardiff CF2 1XH, U.K. 235

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environment competing models must be justified in cost-benefit terms, i.e. comparison between ~the amount and quality of information derived and the effort involved in gaining that information. 3. VALIDITY CRITERIA FOR MACHINE TOOL SPINDLE BEARING MODELS

Machining is generally classified into one of two categories: firstly, "roughing cuts", where the principal objective is high metal removal rates; secondly, "finishing cuts", where dimensional accuracy and surface finish are the main priorities. In the first category the primary source of concern for the dynamicist is the limitation on width of cut determined by chatter sensitivity. This is a stability criterion based on a regenerative relationship between the cutting force and the resulting deformation at the same point (the tool point), though not necessarily with the same orientation. Under normal cutting conditions, in single point cutting machines, it is usual for the spindle speed to be set substantially greater than the most sensitive chatter frequency, i.e. that corresponding to the first mode. As has been suggested by the authors, using a Bernoulli-Euler model [2, 4], in a typical lathe spindle bearing assembly the higher modes may be expected to be near-nodal in the region of the cutting zone. Although, as a consequence, the higher modes may have minimal effect on the cutting process itself, long term performance may be prejudiced due to significant loads at the bearings in higher modes of vibration, particularly when these coincide with spindle speed or other excitations, such as rolling element passing frequencies. The remainder of the current paper provides a critical examination of the modelling assumptions used by the authors previously [1, 2, 4], following earlier work by a number of authors. 4. MODELLING ASSUMPTIONS

4.1. Euler-Bernoulli and Timoshenko beams The (Euler-Bernoulli) model of a vibrating beam, used by the authors previously, is described by the equation

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(1)

(see for example Rao [5]). In the Euler-Bernoulli model deformations due to shear effects are discounted. In contrast the Timoshenko beam model incorporates these effects. As has been suggested above, the criterion for demanding the more sophisticated model depends on the

Machine Tool Spindle-bearing Systems

237

relative contributions of these effects. 4.1.1. Shear deformations. Shear deformations give rise to the extra term

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In machine tool spindles, shear deformations are likely to be of increased significance under certain specific conditions: (i) when the spindle is short in comparison to other machine dimensions, for example in grinding machines; (ii) where bearings impart a significant reaction moment, for example roller bearings. 4.1.2. Rotary inertia. Inclusion of rotary inertia to the Euler-Bernoulli model leads

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It should be noted that Beards [6] quotes this equation correctly, although his previous discussion is confusing and/or erroneous. In machine tool spindle-bearing systems significant local concentrations of rotary inertia are implicit in the function of the machine, for example the chuck of a lathe and bearing inner race elements in all types machines. 4.1.3. Combination of shear and rotary inertia. When rotary inertia and shear effects are combined, an additional fourth order derivative term is necessary to take account of the coupling between these effects O4y OZy E1 ~x 4 + pA Otz

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For beams of regular geometry the arguments presented by Beards [6] are generally valid: Solutions to these equations are available, which generally lead to a frequency a few percent more accurate than the solution to the simple beam equation. However, in most cases the modelling error exceeds this. In general, the correction due to shear is larger than the correction due to rotary inertia. 4.2. Gyroscopic effects It is essential to realize that the stability problems encountered in machine tool dynamics are overwhelmingly due to the dynamics of the cutting process itself, rather than due to shaft whirl or gyroscopic effects. Inclusion of the rotational effects tends to increase the critical speeds. The consensus of experience with dynamic modelling of machine tools suggests that such models are unnecessarily complicated. For example, Allemang et al. [7] carried out experimental testing of a drilling machine, a lathe and a grinding machine within their operational speed range. In the first two cases little difference was observed between non-rotational natural frequencies and rotational critical speeds, but there was a significant decrease in the effective damping under rotational conditions. Results for the grinding machine were inconclusive, due to limitations of the equipment, particularly the difficulty of providing excitation under rotating conditions.

238

J . A . BRANDON and K. J. H. AL-SHAREEF

Ozgiiven, discounts rotary inertia, shear deformation and gyroscopic effects, noting that: "These approximations limit the applicability of the presented curves to low speeds" [8]. Obviously, within the context of the current problem, this is acceptable if "low speed" implies within the operating range of the machine, as discussed in Section 3. 4.3. Axial loading It is common for machine tool spindles to undergo significant axial loading, for example in lathes when undertaking facing operations. In this case equation (1) becomes

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O2y + T~x2 = 0

(5)

where T is the axial thrust (assumed compressive), in the absence of shear and rotary effects. The inclusion of this additional term tends to reduce the natural frequencies (see Beards [6]). 5. COMPARISON OF MODELS FOR TYPICAL MACHINE TOOL CONFIGURATIONS

5.1. Test configurations The authors have examined two spindle bearing systems in detail. Firstly, a lathe type spindle having a large diameter disc at its extremity, to simulate a lathe shaft, has been used by a number of authors previously. In particular Bollinger and Geiger [9] used an Euler-Bernoulli model, using a finite difference method to solve the resulting equations. Computed and measured deflection curves compared well but the first and second natural frequencies were approximately 8% less than the measured values. The design configuration is shown in Fig. 2 and the assumed properties are as shown in Table 1. In previous papers [1, 2, 4], the authors have presented results for this system based on the Euler-Bernoulli model. The second example used by the authors was that of a boring spindle, previously used by a number of authors (Neves [10, 11], Sadeghipour et al. [12], Brandon et al. [13], and Sadeghipour and Cowley [14]). Whereas, in the first example, the authors have undertaken experimental investigations to examine the applicability of the simpler beam model, the test results previously presented by Neves [11] have been used to test the validity of the simple model. 5.2. Theoretical predictions With uniform beams the mode shapes approximate to the familiar deformed shapes of the transverse vibrations of a stretched string (although the actual shapes involve both trigonometric and hyperbolic functions). As will be seen, the mode shapes of 138

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TABLE 1. GEOMETRIC AND MATERIALPROPERTIES FOR LATHE TYPE SPINDLE External diameter Internal diameter Front overhang Rear overhang Bearing span Modulus of elasticity Chuck mass Diameter of chuck Rear bearing stiffness E l e m e n t length Mass of elements 1, 2, 4, 5, 6, 7, 8, 10 Mass of elements 3, 9

40 m m 24 m m 100 m m 100 m m 240 m m 1.96 x 10 H N m -~ 9.66 kg 200 m m 6.73 x 10s N M - l 40 m m 0.25 kg 0.7 kg

spindles with a large rotary inertia at one extremity (typical of lathe spindles) differ markedly from this conceptual template. In particular, the general shape of the first two modes is broadly similar, with the portion of the mode shape which enables discrimination between the modes located in the overhang region, between the front bearing and the chuck jaws, where access to attach instrumentation is particularly limited. Reliance on the general rule, quoted from Beards above, that the effects of shear deformation are likely to be more influential than those due to rotary inertia has only limited validity in the present case. Where the modes of the two systems coincide, the effect of shear deformations is larger, consistent with the guidance of Beards. As can be seen, however, the Euler-Bernoulli model fails to identify the second mode completely, because of the neglect of the (extremely large) rotary inertia of the chuck. There is, consequently, a substantial discrepancy in frequency of the first mode, but the mode shape is predicted accurately. The mode shapes for these modes are shown in Fig. 3. As can be seen there is virtually no difference in mode shape between comparable modes. 6. E X P E R I M E N T A L

RESULTS

Two sets of tests were undertaken. In the first instance the dynamics of the spindle were investigated when set into taper roller bearings in a stiff frame. These tests used traditional instrumentation, i.e. frequency response analysers, spectrum analysers, electrodynamic shakers and accelerometers. Because of the effects of bearing compliance, neglected in Tables 2 and 3, the measured frequencies were compared with revised theoretical predictions based on applying the measured static compliance at the bearings. Estimates of modal characteristics were made under both free and forced vibration conditions, although only the forced vibration tests will be discussed here. (The differences in measured properties were marginal, for example the first natural frequency under free vibration conditions was 171 Hz whilst that in the forced vibration test was 173 Hz.) In the second series of tests, the spindle was tested in its free-free condition using a scanning laser Doppler vibration pattern imager, using sinusoidal excitation. 6.1. Forced vibration tests--spindle constrained The test rig was designed to simulate the constraint conditions commonly encountered in machine tools. Consequently, accessibility to attach transducers was severely restricted. In particular, the region of most interest, the chuck overhang, could not be instrumented. Furthermore, the discriminator between the first and second modes may be characterized as an acute local bending at the spindle-chuck interface which is measurable primarily as a rotational displacement of the chuck. As described by Ewins [15, p. 145], the measurement of rotational displacements remains a considerable challenge:

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Distance measured from rear free end (cm) FIG. 3. (a) First analytical mode Euler-Bernoulli model. (b) First analytical mode Timoshenko model. (c) Second analytical mode Timoshenko model. (d) Second analytical mode Euler-Bernoulli model. (e) Third analytical mode Timoshenko model. (f) Third analytical mode Euler-Bernoulli model. (g) Fourth analytical mode Timoshenko model. (h) Fourth analytical mode Euler-Bernoulli model. (i) Fifth analytical mode Timoshenko model.

TABLE 2. EFFECT OF INCORPORATINGSHEAR EFFECTS

Mode

Without shear

With shear

Error %

272 1396 2369 4197

267 1356 2294 4059

1.8 2.86 3.16 3.28

1 2 3 4

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With rotary inertia

Error %

272 -1396 2370 4197

225 1007 1373 2359 4318

17.3 1.65 0.42 2.86

• . . it is extremely rare to find any reference to methods for the measurement of rotational mobilities and this reflects the fact that virtually none are made. This situation arises from a considerable difficulty which is encountered when trying to measure either rotational responses or excitations and also when trying to apply rotational e x c i t a t i o n . . . The first three mode shapes identified are shown in Fig. 4. Apparent discrepancies in measured and predicted amplitudes, particularly in the second mode, are due, at least in part, to the modal normalization chosen, that of unity displacement at the left end of the spindle• As can be seen, the point response measurements would not provide any evidence that would indicate to an analyst that any mode other than a suitable linear combination of the first and second mode was being measured. 6.2. Forced vibration tests--free-free configuration The authors have been fortunate, during the course of the study to have access to a laser Doppler scanning laser system, which allows full field area scanning of vibration responses. Because the authors wished to examine certain substructuring methods, these tests were undertaken with the shaft in its free-free configuration.

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Initial evaluation of the free-free resonant behaviour was made by suspending the spindle, with its bearing inner races attached, using a compliant suspension system ("bungee" ropes), as described by Ewins [15]. The resonant frequencies were evaluated using a "white noise" signal produced by the Hewlett-Packard 3582A spectrum analyser. Rather than suspend the spindle for the scanning laser tests, it was decided that resting the spindle on sponge pads, on a plinth of expanded polystyrene, with the excitation direction horizontal, would be an adequate approximation to free-free suspension conditions. It was found that the resonant frequencies measured under these conditions were the same as those given by the spectral analysis on the suspended spindle. Analytical predictions of the free-free resonant frequencies, for the first three modes, were 389, 955 and 1892 Hz. The corresponding measured frequencies were 396, 917 NTN 31:2-G

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and 1896 Hz, respectively. Analytical results and measured mode shapes are shown in Fig. 5. 7. DISCUSSION

7.1. Interpretation of observed results As has already been mentioned, early workers reported close correlation between predicted and measured properties using the Euler-Bernoulli model. Similarly, earlier papers by the authors indicated that the behaviour predicted by the simpler model was observed in the laboratory. It should be noted that the resonant behaviour of the spindle in the additional mode was detected in the initial tests but was discounted because of the close resemblance of its mode shape to that of the first mode. The resonance detected was attributed (initially) to failure to de-couple the exciter adequately from the test structure leading to resonances of the support frame. It has been shown that the corresponding modes, predicted by the two models, are almost identical in shape, but differ in frequency. However, the Timoshenko beam model predicts an additional mode between the first and second modes of the

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Euler-Bernoulli model. In this the authors do not claim to present any significant advance over earlier contributions to the literature. For example, Benson [16] presents broadly similar analysis for shafts with similar inertia concentrations at their extremities. The primary insight offered by the current paper is that, once an inappropriate model is chosen, in this case the Euler-Bernoulli beam, the test data are likely to reinforce (specious) confidence in the model. Intuitive reliance on comparison with the mode shapes of uniform beams is likely to give further support to this belief, since the additional second mode, introduced by the more complex model, differs only extremely locally from the general shape of the first mode. Further, the third mode of the Timoshenko model cannot be distinguished from the second mode of the Euler-Bernoulli model. From an experimental point of view, the usual method of using discrete contacting transducers, for example accelerometers, excludes the discrimination between the modes because of lack of accessibility to the region of characteristic deformation. 7.2. Interpretation of predicted mode shapes It is, perhaps, of some value to examine the reasons why the Euler-Bernoulli model should predict mode shapes which are indistinguishable from those given by the more complex model yet fail to identify the second mode completely. The analysis will be summarized briefly here. For more detailed discussion see Brandon [3, pp. 29-33]. The complete eigenvalue problem of the (undamped) system may be written in the form

I2 /Mtt Mtr~ h ~MrtMrr)

Ktt Ktr + ( g r t g r r ) ] (0) = (00)

Machine Tool Spindle-bearingSystems

247

where t denotes translational inertia and stiffness effects and r the rotational inertia and stiffness. The terms Mrt, etc., are usually described as coupling matrices. The eigenvalue problem corresponding to the simpler model may be written

[k2 Mtt + Ktt] x = O. The explanation for both the similarities and the differences between the predictions lies in the manner in which the coupling terms, Mrt, Ktr, etc., act to connect the rotational and translocational coordinates. These can best be understood in terms of specific modal kinetic and translational energies d?~M~biand d?~K~bi, respectively (often described as modal mass and modal stiffness) where ~bi is the ith eigenvector of the model. 7.2.1. First mode. In the first mode the predicted mode shape is virtually indistinguishable in both models. This perception is supported by the experimental evidence. However, the natural frequency prediction is substantially different. This suggests that the rotational coordinates provide very little contribution to the estimate of the strain energy distribution (and hence the modal stiffness) whereas the substantial rotational displacement of the chuck, coupled with its large rotary inertia, provides a significant variation in modal mass. Since the natural frequency may be computed by the (square root of the) Rayleigh quotient, the ratio of modal stiffness to modal mass, the discrepancy in predicted frequency of the Euler-Bernoulli model is understandable. 7.2.2. Second mode. The problem posed by the second mode is not its existence, per se, but why other predictions of the Euler-Bernoulli model are so good, despite its failure to identify this mode at all! Similar problems have been addressed in the literature before, for example Benson [16]. Perhaps the most pertinent idea is that of a local vibration, discussed by Bishop and Mahalingam [17]. They consider two cases: hull vibration at the after end of a ship's hull due to propeller excitation: vibration o f minor components such as handrails. Whilst the latter case is understandable, the localization of response to gross excitations is of considerably more interest. Their analysis considers this in terms of the coupling between sub-structures, examining the conditions under which one substructure oscillates grossly whilst the other is quiescent. In the present context the contributions, of the deformations in this mode, to the kinetic and strain energies of the mode, constitute a local vibration in the sense described by Bishop and Mahalingam [17]. In particular, the mode exhibits gross local rotational deformation in the spindle region adjacent to the chuck. As with the first mode, there is substantial rotational kinetic energy at the chuck itself. Thus the dominant effects of the mode, in terms of kinetic and strain energy, are restricted to a small, localized, set of coordinates which are only present in the Timoshenko model. 7.2.3. Third mode. This mode is of particular interest, in that not only are the mode shapes predicted by the two models virtually indistinguishable, so also are the predicted natural frequencies. The explanation lies in consideration of the deformations in the region of the chuck. Because both the modal displacement and rotation of the chuck are negligible in this region, the gross translational and rotational inertia of the chuck contribute very little to the modal kinetic energy. Thus the problem becomes analogous to that of a partiallyconstrained slender cantilever (built in at the chuck position). 8. CONCLUSIONS

Reliance on the Euler-Bernoulli model of beam vibration can lead to a paradoxical position where some modes of the spindle-bearing system can be predicted with considerable accuracy whilst others are missed completely. In the case of a lathe-type spindle the missing mode may mimic the behaviour of correctly identified modes so that evidence of its existence could be rejected as spurious. The particular effects which allow the analyst to discriminate between the modes may

248

J . A . BRANDONand K. J. H. AL-SHAREEF

be missed both because of accessibility difficulties and the lack of availability of suitable instrumentation for measuring rotational displacements. Acknowledgements--As was mentioned in the introduction, the comments of referees of earlier papers was the stimulus for the current study. Use of the Ometron Vibration Pattern Imager instrument was provided by the U.K. Science and Engineering Research Council, through the Rutherford-Appleton Laboratory loan pool. The second author is funded by the Republic of Iraq. REFERENCES [1] K. J. H. AL-SHAREEFand J. A. B~NDON, On the quasi-static design of machine tool spindles, Proc. Institution of Mechanical Engineers, Seriesr B, J. Engng Manufact. 204, 91-104 (1990). [2] K. J. H. AL-SHAREEFand J. A. BRANDON,On the effects of variations in the design parameters on the dynamic performance of machine tool spindle-bearing systems, Int. J. Mach. Tools Manufact. 30, 431-445 (1990). [3] J. A. BRA~DON, Strategies for Structural Dynamic Modification. Research Studies Press, Taunton, England (1990). [4] J. A. B~NDON and K. J. H. AL-SHAREEF,On the applicability of modal and response representations for analysis of machine tool spindle bearing systems, Proc. Inst. Mech. Engrs, Series B, J. Engng Manufact. (in press). [5] S. S. RAo, Mechanical Vibrations. Addison-Wesley, Reading, MA (1986). [6] C. F. BEARDS,Structural Vibration Analysis: Modelling, Analysis and Damping of Vibrating Structures. Ellis Horwood, Chichester (1983). [7] R. J. ALLEMANG,H. T. GRAEFand C. D. POWELL,Dynamic characteristics of rotating and nonrotating machine tool spindles, American Society of Mechanical Engineers 73-DET- 29 (1973). [8] H. N. OZ6OVEN, On the critical speed of continuous shaft-disk systems, Trans. Am. Soc. Mech. Engrs J. Vib. Acoust. Stress Reliab. Des. 106, 59-61 (1984). [9] J. G. BOLLINGERand G. GEIGER,Analysis of the static and dynamic behaviour of lathe spindles, Int. J. Mach. Tool Des. Res. 3, 93-209 (1964). [10] F. J. R. NEVES, Investigation into the dynamic performance of a machine tool spindle by the finite element method, M.Sc. Thesis, UMIST (1976). [11] F. J. R. NEVES, A study of the damped vibration of spindle bearing systems, Ph.D. Thesis, UMIST (1978). [12] K. SADE6HIPOtJR,J. A. BRAm~ONand A. COWLEY,The receptance modification strategy of a complex vibrating system, Int. J. Mech. Sci. 27, 841-846 (1985). [13] J. A. BRANDON,K. SADEGmPOtJRand A. COWLEY,Exact reanalysis techniques for predicting the effects of modification on the dynamic behaviour of structures, their potential and limitations, Int. J. Mach. Tool Des. Manufact. 28, 351-357 (1988). [14] K. SADEOHXPOURand A. COWLEY,The effect of viscous damping and mass distribution on the dynamic behaviour of a spindle-bearing system, Int. J. Mach. Tools Manufact. 28, 69-77 (1988). [15] D. J. Ewms, Modal Testing: Theory and Practice. Research Studies Press, Taunton, England (1984). [16] R. C. BENSON,The steady state response of a cantilevered rotor with skew and mass unbalances, Trans. ASME, J. Vib. Acoust. Stress Reliab. Des. 105, 456--460 (1983). [17] R. E. D. BISHOP and S. MAHALINGAM,An elementary investigation of local vibration, J. Sound Vib. 77, 149-163 (1981).