MEASUREMENT OF FREQUENCY RESPONSE FUNCTIONSOF ROTATING SPINDLES

Yum Ji Chan 1, and Kuang-Yu Chen 2

1,2 Department of Mechanical Engineering, National Chung Hsing University, Taiwan ROC, yjchan@nchu.edu.tw.

ABSTRACTTo obtain accurate FRFs of rotating spindles thus mode shapes, both the response and the excitation forcehave to be measured accurately. A non-contact excitation system with real-time force level measurementis designed based on two U-shaped electromagnets arranged face-to-face. The ends of the electromagnetsform a hole for a dummy tool to be inserted with the radial clearance of 0.5 mm. The total attractive forcelevel is calculated and effects such as magnetic saturation and the difference between the force transducerreading and the actual attractive force, are considered. Preliminary results show that while aligning thedummy tool may be difficult, the effect of misalignment between the two arms of the electromagnet islow. On top of measuring the FRF of a rotating spindle, the system is also a handy way to measure theODS of machine tools.

Keywords: rotating spindles, electromagnetic excitation, experiments

1. MOTIVATION

Frequency response functions (FRFs) of spindle heads determine the precision of a machine tool, and itcan also be used to predict chatter stability (Abele et al. [1]). The FRF of a spindle is a function of therotor material, rotor geometry and bearing stiffness. The stiffness of bearings is a function of the exertedpreload (Gunduz et al. [5]) and its properties changes upon rotation due to centrifugal force of bearingballs and the thermal deformation of the spindle. For example, Matsubara et al. [7] found that the naturalfrequency of a spindle can be lower due to rotation.

Vibration analysis on spindles can be carried out by obtaining waterfall diagrams during a free run orby exciting the tool tip. Electromagnetism has been utilized in past research to excite spindles, suchas Matsubara et al. [7], Rantatalo et al. [8], Yamazaki et al. [9]. Rantatalo et al. [8] developed a non-contact excitation system consisted of 8 electromagnets surrounding a dummy tool, and Matsubara et al.[7], Yamazaki et al. [9] measured spindle stiffness with non-contact excitation system.

The past research focused on obtaining natural frequencies and operational deflection shapes (ODS) ofspindles. To obtain accurate FRFs of spindles thus mode shapes, both the response and the excitation

force have to be measured accurately. On the one hand, this is not a problem as the response of rotatingspindles are routinely measured using Laser Doppler Vibrometers (LDVs). On the other hand, it isdifficult to excite rotating structures using predefined waveforms, and the forced excitation level onrotating shafts are seldom measured directly in past research. Due to the difficulties in exciting rotatingstructures, FRFs were only sought on non-rotating structures (Gunduz et al. [5]).

Based on devices designed to excite bladed disks (Firrone and Berruti [4]), a system that can obtainaccurate FRFs of rotating spindles is proposed in the current work. In this article, the design and the pre-liminary results are presented. The current device consists of electromagnets with real-time excitationmeasurement using force gauges. Due to time constraint, the experimental results are presented usingone electromagnet, and accelerometers are used to measure the vibration amplitude in this study. Never-theless, it will show the capability of such an excitation system, with potential improvements discussedat the end of the article.

2. THEORY

2.1. Generation of electromagnetic force

In our excitation system, the tool tip is deflected by a magnetic field generated by an electromagnet. Thegap between the magnet and the tool tip is assumed to be constant and denoted δ. This is justified sincethe vibration amplitude of the tool tip should be within 0.05 mm. The derivation below is similar to theresults of Firrone and Berruti [4] but adapted to a cylindrical gap between the tool and the electromagnet.The magnetic attractive force generated over a small area dA is is given by Feynman [3]:

dF =(Ni)2µ0dA

2δ2=

(Ni)2µ0rdθ

2δ2(1)

where N , i, µ0 and δ refer to number of turns in electromagnet, electrical current, permeability of airand the air gap between electromagnet and tool, respectively. Also, r denotes radius of the tool. We canobserve from Eq. (1) that the magnetic attractive force is inversely proportional to δ2, the square of thegap.

The attractive force is normal to the surface, thus the component of the force parallel to the line ofsymmetry of electromagnet (Figure 1) is equal to dF cos θ. As a result, the attractive force generated bya single electromagnet, with h being its height, is equal to

Fi = 2

∫ π/2

0dF cos θdθ

= 2

∫ π/2

0(Ni)2µ0

rh

2δ2cos θdθ

= (Ni)2µ0rh

δ2(2)

It can be seen that the attractive force is equal to the square of the input current. As a result, if a harmonicforce of frequency f0 is fed into the solenoid, the attractive force contains a DC component and an time-varying component of frequency 2f0:

Fi ∝ i2 = (sin(2πf0t))2 =

1− cos 2πf0t

2

2.2. Flux saturation

The governing equation of an electromagnet with circuit length Lc and total air gap 2δ:

Ni = B

(Lcµs

+2δ

µ0

)≈ B 2δ

µ0(3)

Tool

Electromagnet

Electromagnet

θ

dF

dF cosθ

d A

Figure 1: Determination of total attractive force.

Flux Saturation: Bs

Figure 2: Saturation of magnetic material. Adapted from Corporation [2]

It has to be emphasised that the relationships such as Eq. (2) do not extend indefinitely. When thecurrent is strong enough, the flux is saturated in the silicon steel in the electromagnet (Fig. 2) when theflux density reaches Bs. Saturation is undesirable because (i) extra power is consumed due to electricalresistance without significant increase in force and (ii) the B-H relationship near saturation is nonlinear,which will generate harmonics. To avoid saturation, the current fed to the electromagnet should notexceed Bs.

2.3. Force transducers

Force transducers are used to measure the electromagnetic excitation force in this study. However, theaccuracy of the readings given by the force transducer may be undermined by two factors. The firstfactor arises because each electromagnet weighs 1.5 kg. Therefore, the inertia of the electromagnet maybe significant. The true excitation force (FT ) is sought by considering acceleration of electromagnets toeliminate the effect of electromagnet inertia:

FT = F −mx

however, this effect is found to be small as the inertia is at least an order below the readings given by theforce transducers.

The electromagnet is held horizontally by the leaf springs, force transducer and the force transducerholder. To measure the force accurately, the stiffness of supports in front of the force transducer (that is,the springs), denoted ks, should be lower than that of the force transducer holder and the transducer itself,denoted kb. The relationship between the magnet, ks and kb is shown in Figure 3. A force transmissionratio is defined as kb/(ks + kb): given the deflection at the base is fixed under a known excitation forcelevel, this ratio determines the portion of the excitation force picked up by the force transducer. The forcetransmission ratio is determined using static deflection at the brace, which are calculated using ANSYS.If the deflection at the base with transducer present is denoted xb and that without the transducer is x0,

the following relationship hold:

xb =F

ks + kb(4)

x0 =F

ks(5)

where F is the total attractive force. As a result,

ks + kb =F

xb

ks =F

x0kb

ks + kb= 1− xb

x0(6)

Magnet

Force transducer

kb

ks

Fi

xb

Magnet

ks

Fi

x0

Figure 3: Calculation of force transmission ratio.

3. FINAL DESIGN

The parts involved in the non-contact excitation system is shown in Figure 4. Two U-shaped electro-magnets (1) are arranged face-to-face in order to excite test piece in both directions, thus the currentdevice is capable of generating harmonic excitation. Braces (2) are made of copper in order to avoidmagnetic interference. Force transducers of type B&K 8202 (3) are installed in the back of the bracesand supported by holders (4). The lateral stiffness of the electromagnet supports is minimized as themain supports of the electromagnets are the 3-mm-thick leaf springs (5). The chassis (6) supports allthe components above. The leaf springs are designed by such that the stiffness ks is minimised. Byassuming a static attractive force of 200 N, the deflection of the electromagnet holder in Z direction arexb = 1.8µm (force transducer present) and x0 = 9.7µm (force transducer removed). According to(6), the force transmission ratio is equal to 0.917, thus the force transducers are assumed to measure theattractive force accurately.

3.1. Key parameters of electromagnets

Each 25-mm thick electromagnet is piled up by 50 silicon steel sheets. Each arm of the electromagnethas 140 turns of 1.2 mm-diameter copper wire. As the saturation flux density of silicon steel sheet is1.2 Tesla, by Eq. (3) the maximum allowable current is 3.41 A. The tips of the electromagnets are curvedand form a circle (of diameter 33 mm) to fit the test piece. The tip area of the electromagnet is 560 mm2

on each arm. Both gaps between the tips of the electromagnet and between the two electromagnets areequal to 4 mm. The test piece used in the experiments below is a 32 mm-diameter dummy tool, makingthe radial clearance 0.5 mm. The gap between the tool and electromagnet can be adjusted by changingthe test piece diameter, but it should not be greater than the gap between the electromagnets.

1 2 3 4

56

Figure 4: Exploded view of non-contact excitation system.

4. EXPERIMENTAL RESULTS

Preliminary experimental results are discussed. Due to time constraints, only one electromagnet is usedin the experiments. The excitation system is installed on an X-Y table of a machine tool test bed. Thedummy tool is a bar clamped to the spindle head (Figure 5). The electromagnet is excited by usingeither a DC power source or a power amplifier (B&K 2706). Two tests were carried out. The first test isdedicated to find the sensitivity of attractive force level to alignment error and the FRF is determined inthe second test.

+X axis

(a) (b)

Figure 5: Test setup (a) overview and (b) close-up.

Firstly, the electromagnet is excited using a fixed direct current source and the gap between the tool andthe magnet is adjusted by moving along the x-axis. It can be seen that the attractive force increases withreduced gap if the gap is wider than 0.1 mm (Fig. 6), but the relationship does not follow the trendpredicted in Eq. (3). The trend is reversed if the gap is narrower than 0.1 mm. Such anomaly may be dueto uneven distribution of magnetic flux.

Alignment is found to be a issue in practice. A specific test is carried out to find the sensitivity of excita-tion force to the tool location. In this test, a 10 Hz alternate current signal is fed into the electromagnet,generating an excitation at 20 Hz. By varying the location of the tool in the X-axis, the trend shown in

0 0.1 0.2 0.3 0.4 0.50

20

40

60

80

Fi ∝ 1δ2

Current = 1.18 A

Gap width (mm)

Attr

activ

efo

rce

(N)

Figure 6: Dependence of attractive force on gap width. Direct current of 1.18 A is fed into the electromagnet.

Fig. 7(left) is similar to that shown in Fig. 6). The results in Fig. 7(right) also show that the excitationforce level is insensitive to the location of the bar between the two arms of the electromagnet, and theresultant profile does not necessary be symmetric, probably due to the variation in winding the solenoids.

−0.6 −0.4 −0.2 0 0.2 0.4 0.60

50

100

150

200

250

X-coordinate = 0.1 mm

Geo

met

ric

cent

re

Y axis location related to midpoint (mm)

Max

imum

norm

alis

edFo

rce(

N)

0 0.2 0.4 0.6 0.8 10

50

100

150

200

250

Y-coordinate = −0.02 mm

Gap width (mm)

Nor

mal

ised

max

imum

forc

e(N

)

Excitation voltage = 29.6 Vp–pExcitation voltage = 19.8 Vp–p

Figure 7: Dependence of attractive force on location of tool, along (left) X-axis and (right) Y-axis. The forcereadings are normalised against the excitation voltage of 29.8 V.

At last, the tool is excited by feeding a swept sine wave from 5 Hz to 110 Hz, which is the criticalfrequency range for machine tool whole-body dynamics. The swept time is long enough to reducedistortion. It is noted that the excitation system excites both the tool and the X-Y table, in oppositedirections. For comparison, FRFs of impact tests were obtained by exciting the table and the tool one ata time. The inertance plot of Fig. 8 shows that the non-contact excitation system can identify the samethree modes that captured by impact test. Moreover, the excitation system can generate less-noisy FRF atfrequencies below 80 Hz. However, in the current configuration, the FRF related to the excitation systemhas a pair of zero-pole between 100 and 180 Hz. This needs further investigation.

5. DISCUSSION

The results from Fig. 8 offers new applications for the non-contact excitation system beyond spindledynamics. Traditional dynamic analysis of machine tools rely on modal testing using hammers or exter-nal shakers. It is cumbersome to obtain a good estimate of the ODS during operation, because severaloperations are required on the impact test FRFs as action and reaction are exerted on the machine tool atthe same time. It has been attempted to use ball screw drive to excite machine tools, but its accuracy isdifficult to be determined. With the use of the new non-contact excitation system, such procedure can besimplified.

100 101 102

10−6

10−5

10−4

10−3

10−2

10−1

100

Frequency (Hz)

Iner

tan

ce(g

/N)

Excitation systemImpact test

Figure 8: FRFs obtained by using electromagnet and impact tests.

5.1. Further development

The second electromagnet will be installed in due time and the dummy tool is excited from both direc-tions. Kruse and Pierre [6] attempts to separate an excitation signal into “positive” and “negative” partsin order to generate a sinusoidal waveform. Such an approach will be reviewed as it can generate high-order harmonics. Past ideas that propose to refine the experiment, such as grooving the dummy tool toreduce eddy current effect by Matsubara et al. [7], will be included in a refined design.

6. CONCLUSIONS

A non-contact excitation system with real-time force level measurement is presented. The total attractiveforce level is calculated and effects such as magnetic saturation are considered. By supporting the elec-tromagnet on a leaf spring, the difference between the force transducer reading and the actual attractiveforce is minimised. Preliminary results show that while aligning the dummy tool may be difficult, theeffect of misalignment between the two arms of the electromagnet is low. On top of measuring the FRFof a rotating spindle, the system is also a handy way to measure the ODS of machine tools as the FRFgenerated by the system is less noisy than that obtained from impact tests.

ACKNOWLEDGMENTS

This research is supported by Ministry of Science of Technology Project Number MOST 102-2281-E-005-006-MY2. Attendance of Kuang-Yu Chen is supported by a separate grant from Ministry of Scienceof Technology. The testing of the electromagnet was carried out with the generous help from Greg Jonesand Jacob Hanley.

REFERENCES

[1] Abele, E., Altintas, Y., and Brecher, C. (2010). Machine tool spindle units. CIRP Annals - Manu-facturing Technology, 59(2):781 – 802.

[2] Corporation, T. Low loss and high saturation flux density ferrite for high-power transformers andreactors.

[3] Feynman, R. P. (1963). Lectures on Physics, volume 2. Addison-Wesley, New York.

[4] Firrone, C. and Berruti, T. (2012). An electromagnetic system for the non-contact excitation ofbladed disks. Experimental Mechanics, 52(5):447–459.

[5] Gunduz, A., Dreyer, J. T., and Singh, R. (2012). Effect of bearing preloads on the modal char-acteristics of a shaft-bearing assembly: Experiments on double row angular contact ball bearings.Mechanical Systems and Signal Processing, 31(0):176 – 195.

[6] Kruse, M. J. and Pierre, C. (1997). An experimental investigation of vibration localization in bladeddisks, part i: Free response. In Proceedings of the 42nd ASME Gas Turbine & Aeroengine Congress,Users Symposium & Exposition.

[7] Matsubara, A., Yamazaki, T., and Ikenaga, S. (2013). Non-contact measurement of spindle stiffnessby using magnetic loading device. International Journal of Machine Tools and Manufacture, 71(0):20– 25.

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