a new approach to identifying the dynamic behavior of cnc machine

International Journal of Machine Tools & Manufacture 72 (2013) 73–84

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International Journal of Machine Tools & Manufacture

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A new approach to identifying the dynamic behavior of CNC machinetools with respect to different worktable feed speeds

Bin Li a,b, Bo Luo b, Xinyong Mao b,n, Hui Cai b, Fangyu Peng b, Hongqi Liu b

a State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, PR Chinab National NC System Engineering Research Center, Huazhong University of Science and Technology, Wuhan 430074, PR China

a r t i c l e i n f o

Article history:Received 24 January 2013Received in revised form12 June 2013Accepted 13 June 2013Available online 27 June 2013

Keywords:Natural frequencyDamping ratioModal shapeThe dynamics of worktableMachine tools

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espondence to: National NC System Engineeriical Science and Engineering, Huazhong UniveST), 1037 Luoyu Road, Hongshan District, Wua. Tel.: +86 27 87542613 8428, mobile: +86 15027 87540024.ail addresses: [email protected], maoxy

a b s t r a c t

The dynamics of the machine tool structure are important in high precision machining. Some researchershave studied that the dynamics are expected to change under different machining conditions. However,the dynamic behaviors of the machine tool at different worktable feed speeds are rarely studied. In thispaper, an output-only modal identification available to predict the dynamics of the machine tool atdifferent feed speeds is proposed. The excitation of this method uses the inertia force sequence caused byrandom idle running of the worktable. The first six modes of the entire machine tool structure areestimated using the proposed method. The results indicate that the running state of the worktable caninfluence the modes in which the worktable vibrates. The estimated natural frequencies and dampingratios decrease obviously as the feed speed increases. Furthermore, because this method enable todetermine modal parameters by measuring the response of machine tool structure without using anyartificial excitation, it can be used to predict the dynamic behaviors of the machine tool in entire workingspace effectively.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

In terms of geometrical and dimensional accuracy, the qualityof machined parts depends largely on the dynamics of themachine tool, and an analysis of the tools' dynamic behavior isan important method for optimizing the product design andmanufacturing process. Several researchers have predicted thedynamic parameters of the spindle nose or the tools [1–4].However, other researchers have proven that the structure of themachine tool also contributes significantly to the dynamic stiffnessof the spindle nose. Schmitz et al. [5,6] divided the dynamicmachine tool into two components: the overhang portion of thetool, which is modeled analytically, and the remainder of theassembly or the spindle holder, which is modeled experimentally.Kolar et al. [7] performed an experimental analysis of the impact ofthe machine frame on the tool's dynamic properties, as evaluatedat the spindle nose. To capture the influence of the machine toolframe, the dynamic properties at the spindle nose were evaluatedfor both a spindle-free state and for a spindle mounted in themachine tool. The results indicated that the natural frequencies

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ng Research Center, School ofrsity of Science and Technol-han, 430074 Hubei Province,007120546;

[email protected] (X. Mao).

of the spindle nose were shifted. Therefore, the analysis of thedynamic behavior of the entire machine tool structure is highlysignificant.

In general, two types of methods are used to estimate thedynamic parameters of the entire machine tool structure. Onemethod is based on computer-aided engineering. Altintas et al. [8]summarized the use of finite element analysis of the dynamicbehavior of a machine tool structure in a review. However, it isdifficult to calculate the dynamic parameters with high accuracyusing this method because approximately 60% of the total dynamicstiffness and approximately 90% of the total damping of the entiremachine tool structure originates in the joints [9]. Another type ismodal analysis based on an experiment, and this method can befurther divided into experimental modal analysis (EMA) andoperational modal analysis (OMA).

Experimental modal analysis (EMA) is commonly performedusing artificial excitation of a structure and measurement of theinput forces and output responses. The excitation consists of eitherimpact hammer testing or shaker testing (random, burst-randomand sinusoidal, etc.). Nevertheless, the EMA can be applied onlywhen the machine tool is inactive. Thus, there are two disadvan-tages to EMA when estimating the modal parameters of a machinetool. First, because the dynamics of the machine tool structurevary as the slide positions change during processing, the modalparameters must be considered for the entire working space[10,11]. Altintas also suggested that the structural dynamics ofthe machine tools change as a function of the tool position or

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Fig. 1. Schematic diagram of the excitation force. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web versionof this article.)

B. Li et al. / International Journal of Machine Tools & Manufacture 72 (2013) 73–8474

spindle speed, which renders the frequency response function(FRF) of the machines nonlinear [12]. Therefore, the analysisbecomes rather burdensome because a large number of EMA testsare required to measure the dynamic parameters of the entireworking space. Second, the natural frequencies and dampingratios measured in the static state might be different from thoseobtained during the actual machining operation. Minis et al. [13]proposed a method based on interrupted cutting to identify thestructural dynamics of a lathe from input–output measurements.The study showed that the damping ratios of the machine toolstructure increased during cutting by 20–40% percent of thatmeasured via hammer tests. Budak et al. [14] measured the FRFsof a milling machine under cutting conditions using the input–output relationships of the cutting forces and the vibrationresponses. The results showed that the bearing stiffness anddamping changed under rotating conditions. Therefore, the work-ing conditions of the tool must be considered. Nevertheless, theEMA method is able only to identify the dynamic behavior of theinactive machine tool structure.

Operational modal analysis has been proven useful as a methodin civil engineering for situations in which it is difficult andexpensive to excite such structures as bridges and buildings witha hammer or shaker and to obtain artificially induced vibrationlevels that exceed the natural vibrations due to traffic or wind.With this method, it is possible to estimate the system modalparameters from an output signal only. Therefore, two previouslymentioned disadvantages exist in EMA but not in OMA. Fewresearchers have estimated the machine tool dynamic parametersusing OMA methods. Burney et al. determined the dynamicparameters of a machine tool under working conditions using atime-series technique (ARMA) and studied the static stability of amachine tool system under operating conditions. Kushnir [15]estimated the modal parameters of a machine tool structure undercutting conditions using the coherence function of the vibrationsat different measurement points. However, this method can onlybe applied in cases in which the dynamic stiffness of the structureis weak, and moreover, the damping ratios cannot be estimated.Zaghbani and Songmene [16] estimated the machine tool dynamicparameters during machining operations via operational modalanalysis. The results showed that the natural frequencies and thedamping ratios under working condition were smaller than thosein the hammer tests. Although these methods can accuratelyestimate the modal parameters, the feed movement of the work-table was not considered. However, the dynamics of the worktableplays an important role in the cutting process and thus must beconsidered. According to Timoshenko beam models, the dynamicstiffness of the worktable can be reduced by the feed speed due tothe rotation of the ball screw [17]. Unfortunately, almost all of themodal analysis experiments found in the literature were con-ducted with the worktable inactive. Therefore, the dynamics of thefeeding worktable require further study.

The main goals of the research in this paper are: (1) to developa method for estimating the dynamic modal parameters of amachine tool structure with random idle running of the worktable,(2) to study the influence of the movement of the worktable onthe dynamics of the machine tool and (3) to study the dynamics ofthe worktable at different speeds.

The remainder of the paper is organized as follows. In Section 2,the principles of the proposed method are presented. Two work-table movements are designed in Section 3 to predict thedynamics of the machine tool at different feed speeds. In Section4, the modal parameters of the entire machine tool structure areestimated using the proposed method and are compared withthose estimated by EMA. The dynamics of the worktable atdifferent feed speeds are studied in Section 5. Finally, Section 6provides conclusions from the current research study.

2. Theoretical background

2.1. Modal model of the machine tool

The OMA method is an output-only modal analysis methodbased on the assumption that the input excitation can be treatedas white noise. In the OMA method proposed in this paper, theexcitation is caused by the inertial forces of the worktable (asshown in Fig. 1): If the table accelerates or decelerates randomlyand continuously, the corresponding stochastic inertial forcessequence is generated. Using the ball-screw driver system, theinertial forces are applied to the machine tool (as shown by theblue marks in Fig. 1). Because the inertial forces have beencharacterized as square-wave pulses [18], the frequency spectrumof the sequence is flat in a certain frequency bandwidth and hasthe same characteristics as white noise. To eliminate the effects ofstructural changes on the dynamics of the machine tool, theworktable movements are restricted in a small fixed area. There-fore, the dynamics are approximately invariant and the dynamicequation of the machine tool can be expressed as follows:

½M�f€xg þ ½C�f_xg þ ½K�fxg ¼ ff g ð1Þ

where ½M�, ½C� and ½K� are the mass, damping and stiffness matricesof the machine tool structure; f€xg, f_xg, fxg are time functionsorganized in column vectors that characterize the evolution ofthe acceleration, velocity and displacement, respectively; and fFg isa column vector with the excitation force applied to the machinetool generated by the movement of worktable. By taking theFourier transform of Eq. (1), the excitation force vector andvibration vector can be expressed as follows:

fXðωÞg ¼ ½HðωÞ�ff ðωÞg ð2Þ

with

HðωÞ½ � ¼ ∑n

r ¼ 1

fϕrgfLrgTiω−λr

þ fϕrgfLrgHiω−λnr

ð3Þ

and

λr ¼−ζrωr þ iffiffiffiffiffiffiffiffiffiffiffi1−ζ2r

qωr ð4Þ

The modal model (3) expresses the dynamic behavior of themachine tool as a linear combination of n resonant modes. Eachmode is defined by a natural frequency f r ¼ ωr=2π, a damping ratioζr , a modal shape vector fϕrg and a modal participation vector fLrg.Therefore, if the excitation force vector and vibration vector aremeasured simultaneously, the modal parameters can be predictedfrom the modal model ½HðωÞ�. However, it is difficult to measurethe random forces applied to the machine tool due to thecomplexity of the feed drive system.

B. Li et al. / International Journal of Machine Tools & Manufacture 72 (2013) 73–84 75

2.2. The basic theory of the OMA algorithm

The OMA methods available to identify the modal parametersof a system using only the measured responses are usuallyclassified as frequency domain or time domain methods. ThePolyMAX@ modal parameter estimation is an advanced frequencydomain estimator, which starts from power spectra [19,20].For this method, the inertial forces of the worktable are designedas white noise in a certain frequency bandwidth and the half-spectrum matrix of the measured responses can be expressedusing the modal model as follows:

½SXðωÞ�þ ¼ ∑n

r ¼ 1

fϕrgfKrgTiω−λr

þ fϕrgnfKrgHiω−λrn

ð5Þ

where fKrg represents the operational participation vector, whichdepends on the modal participation vector fLrg and the powerspectrum matrix of the unknown operational force.

The half-spectrum matrix also can be further modeled in thediscrete-time frequency domain in a fractional form as follows:

½SXðωjÞ�þ ¼ BA−1 ¼ ∑p

r ¼ 0Breiωj rΔt

� �∑p

r ¼ 0AreiωjrΔt

� �−1ð6Þ

where Br and Ar are matrices with the model parameters, p is theorder of the polynomials, Δt is the sampling time used to measurethe structural responses and ωj is the discrete frequency. If themeasured half-spectrum matrix of the output time series isrepresented by SþX ð̂ωjÞ, the difference between the measuredhalf-spectrum matrix and the theoretical half-spectrum matrixcan be written as:

E¼ ∑p

r ¼ 0Breiωj rΔt

� �∑p

r ¼ 0AreiωjrΔt

� �−1−SþX ð̂ωjÞ ð7Þ

Next, the matrices Br and Ar can be determined by minimizingthe least squares cost function of Eq. (7), which is obtained byadding all of the squared elements of the error matrix E evaluatedat all of the discrete frequency values from ω1 to ωnf [21].The denominator matrix polynomial of Eq. (6) in the continuous-time frequency domain can be subsequently expressed as follows:

∑2p

r ¼ 0Areiωr ¼ ∏

p

s ¼ 1ðiω−λrÞðiω−λnr Þ ¼ 0 ð8Þ

The complex conjugate roots with λk and λnk , which representthe poles of the half-spectrum matrix, can be calculated fromEq. (8). Hence, according to Eq. (4), the modal natural frequenciesand damping ratios can be obtained as follows:

ωk ¼ffiffiffiffiffiffiffiffiffiλkλ

n

k

q

ζk ¼λk þ λnk2ωk

ð9Þ

Fig. 2. Schematic drawing of the velocities of the worktable and the generated impactforces at excitation A, (c) the velocities of the worktable at excitation B and (d) the generafigure legend, the reader is referred to the web version of this article.)

3. Two different excitation force sequences

In this paper, excitations A and B are proposed to estimate themodal parameters of the machine tool at different worktable feedspeeds. For excitation A, the worktable is accelerated to a specifiedspeed and subsequently decelerated to zero speed immediatelyfollowing. Between every two adjacent accelerations or decelera-tions, the worktable remains stationary for a random time tsi, andthe velocities of the table are shown in Fig. 2a. The stochasticexcitation force sequence caused by these movements is shown inFig. 2b, and the green curves represent the responses of machinetool under a static state. For excitation B, the worktable firstaccelerates to the specified speed and maintains this speed for arandom time toi, and subsequently accelerates to the same speedin the opposite direction. The velocities and excitation forcesequence are presented in Fig. 2c and d, respectively. Becausethe worktable continues to move at a constant speed, the greencurves represent the responses of the machine tool at that speed.

The excitation forces will affect the dynamics of the machinetool; however, the dynamics of the system can be identifiedwithout considering the influence of the excitation forcesacting on the system if two conditions are satisfied [22]: (1) thesystem is linear and controllable and (2) there exist no constantsK1, K2,…, Kn such that the system's input can be expressed for alltime by the expression:

F ¼ K1X1þ K2X2þ⋯þ KnXn ð10Þ

where X1, X2,…, Kn are the state variables of the system. Usually,the machine tool structure can be considered as linear andcontrollable, as shown in reference. Therefore, the first conditionis satisfied. The second condition of the theorem is true for bothexcitation A and excitation B. In this case, because Xj≠0 (becausefree vibrations and forced vibrations occur during the excitationand Xj is the state variable of vibrations), Eq. (10) is satisfied only ifKj ¼ 0. However, if all constants are zero, then the excitation forcesare always zero, which is obviously not true. For this reason, thereexists no unique set of constants that satisfies Eq. (10) for any time.Therefore, according to Fisher [23], the dynamics of the machinetool in the absence of the influence of excitation forces can beexpressed as follows:

HFreeðωÞ ¼XðωÞ

FrandomðωÞð11Þ

where HFreeðωÞ represents the dynamics of the machine tool withuncoupling of the influence of the excitation force, FrandomðωÞ is theexcitation force for excitation A or B and XðωÞ is the vibrationcaused by the excitation. For the operational modal analysis, thepower spectrum matrix or covariance matrix of the machine tool

forces: (a) the velocities of the worktable at excitation A, (b) the generated impactted impact forces at excitation B. (For interpretation of the references to color in this

Fig. 3. Experimental system for modal analysis of the running state of the worktable: (a) the sensing system for acquiring the vibrations and (b) the schematic drawing of thepositions for accelerometer attachment.


vibration can be expressed by HFreeðωÞ:½SXðωÞ� ¼ ½HFreeðωÞ�½SF ðωÞ�½HFreeðωÞ�H ð12Þwhere ½SF ðωÞ� is the spectral density matrix of the excitation forces.Because the frequency spectrum of the random excitation forcesequence of both excitation A and B is flat in a certain frequencybandwidth, the ½SF ðωÞ� for both excitations are a constant matrix,and Eq. (12) can be expressed as follows:

½SXðωÞ�∝½HFreeðωÞ�½HFreeðωÞ�H ð13ÞTherefore, the modal parameters of the machine tool for

different motion states of the worktable can be estimated viaoutput-only modal analysis without considering the influence ofthe dynamic excitation forces. Using excitation A and B, the modalparameters of the machine tool can be estimated if the worktableis stationary or moving.

4. Modal analysis of the entire machine tool structure

In this section, both OMA and EMA were conducted to estimatethe modal parameters of the machine tool. The proposed methodwas validated, and the influence of the worktable movement onthe dynamics of the entire machine tool structure was studied. Alltests in the current paper were conducted on the same machinetool: a CNC milling machine built by the National EngineeringResearch Center of Numerical Control System of China, modelnumber XHK-5140.

4.1. Experimental setup

4.1.1. OMA of the machine tool structureAs mentioned in Section 3, the operational modal parameters

of the machine tool with worktable motion can be estimated byexcitation B. In this experiment, the speed of the worktable wasset to 1000 mm/min, and the mean value of toi was 0.05 s. Themodal parameters of the machine tool were estimated from thecross-power spectra using the operational PolyMAX@ modal ana-lysis. Fig. 3a shows the experimental system of the machine toolfor the running state of the worktable:

(i)
To acquire the modal shapes of the machine toolstructure accurately and carefully, 198 measurement pointswere densely selected (shown by the nodes of the grid inFig. 3b).
(ii)
Seven low-mass, wide-bandwidth and three-axis acceler-ometers were mounted at the measurement points to moni-tor the vibrations in three directions (accelerometerreference: PCB-356-A15). Because the accelerometers werefewer in number than the measurement points, six acceler-ometers were shifted 33 times to measure the vibrations, andthe remainder was attached to fixed measurement point asreference. The reference point selection has a significant effecton the measurement results. Typically, points that are non-nodal or peak deflection points are good choices [24]. In thisexperiment, the reference point was the black node at thebase shown in Fig. 3b.
(iii)
All accelerometers were connected to an acquisition system(LMS SCADAS Mobile SCM05), and the sampling rate of thesystem was 1000 Hz.
(iv)
The B excitations were conducted in both the X-direction andY-direction to estimate the operational modal parameters ofthe machine tool in orthogonal directions.
The total travel distance of the X-axis and Y-axis shown inFig. 4a are 1100 mm and 350 mm, respectively. To eliminate theeffects of the table locations on the dynamics of the machinetool, the worktable movements of excitation B are restricted in asmall fixed area, as shown in Fig. 4b. The maximum distancesfor table movements (distances between location A and B) in boththe X- and Y-directions are 10 mm. The FRFs of the worktableat location A and B are shown in Fig. 4c, with the red curvesindicating the FRFs at location A and the blue curves indicating theFRFs of location B. Because the blue curves are nearly coincidentwith the red curves, it can be concluded that the dynamics of themachine tool do not change for excitation B.

4.1.2. EMA of the machine tool structureConsidering the machine tool is a complex structure and

the estimated results may be influenced by the position of theexcitation force, two excitation positions were selected. Next, themodal parameters of the machine tool with a stationary worktablewere estimated from the FRFs using the experimental PolyMAX@

modal analysis:

(i)
Hammer excitations were conducted in the X-direction and Y-direction at excitation points A and B, as shown in Fig. 5b andc, respectively. The hammer used in the experiments was a

Fig. 4. The dynamics of the machine tool at different worktable locations: (a) total travel distances of the worktable in the X- and Y-directions, (b) area for the worktablemovements and (c) FRFs of the worktable at locations A and B. (For interpretation of the references to color in this figure legend, the reader is referred to the web version ofthis article.)

Fig. 5. Photographs of the conventional FRF experiments and different excitation points.


Fig. 6. Frequency stability diagram. (For interpretation of the references to color inthis figure legend, the reader is referred to the web version of this article.) Fig. 7. Damping ratio stability diagram.


powerful hammer (reference: HDFC-DFC-1) supplied by theChina Orient Institute of Noise and Vibration.

(ii)
The measurement points and sampling rate were the same asthose in the operational modal analysis experiment describedabove.
4.2. Results and discussions

Figs. 6 and 7 present the stability diagrams of the OMA of themachine tool. In Fig. 6, the green curve represents the summation ofthe cross-power spectra of all measurement points and the referencepoint in both the X-direction and Y-direction. The potential naturalfrequencies are shown on the abscissa axis, and the model orders ofthe algorithm are displayed on the ordinate axis. The red and bluesymbols indicate the relationship between the model order and thesolutions found for the order. Theoretically, if a natural frequencyappeared at any order, there would be a high probability that it is anatural frequency. Therefore, according to the figure, six modes existbetween 0 Hz and 70 Hz. The blue clusters of symbols indicate themodes excited by excitation B in the X-direction, and the red clustersof symbols denote the modes in the Y-direction. The blue clusterbetween 25 Hz and 35 Hz is bifurcated and does not appear at any ofthe same orders as the others, which illustrates a calculated modecaused by the noise. In Fig. 7, the damping ratios are presented on theabscissa axis, and the model orders are shown on the ordinate axis. Itis clear that the estimates of the damping ratio converge when themodel order increases. If the model order is sufficiently high, the staticresults of the damping ratios can be acquired. The clusters in bothFigs. 6 and 7 are fairly clear, which indicates that the results of theoperational modal analysis are believable and the experiment is valid.

Table 1 summarizes the modal parameters of the machine toolestimated by OMA and EMA. The OMA column shows the opera-tional modal parameters from the OMA test when the worktable isoperated at a speed of 1000 mm/s. The results of EMA tests atexcitation positions A and B are shown in the hammer A andhammer B columns, respectively. The natural frequencies of thefirst two modes estimated from all experiments are quite similar.From the third to fifth natural frequencies, the values estimatedfrom OMA are nearly 2 Hz larger than those of the EMA. For thesixth mode, the natural frequency estimated by the OMA is smallerthan that of EMA by approximately 2–3 Hz. However, all of thedamping ratios estimated by OMA are smaller than those of theEMA. These differences are due to many interrelated factors, i.e.,

the boundary conditions and the friction conditions, amongothers. In addition, from the comparison of the results shown inthe hammer A and hammer B columns, it can be observed that thenatural frequencies and damping ratios are not impacted by theposition of the input force.

To study the influences on the modal shapes of machine tooldue to table movement, the mode shapes estimated by OMA arecompared with those obtained from EMA using the modal assur-ance criterion (MAC) [25]. The MAC is defined as the squaredcorrelation coefficient between two modal vectors vi and vj, andcan be expressed as follows:

MACij ¼jvHi vjj2vHi vivjv

Hj

ð14Þ

The MAC value is between 0 and 1, where 1 means that the twovectors are estimates of the same physical mode shape and 0 meansthat they are orthogonal to each other. There are two types of MAC:the Auto-MAC and Cross-MAC. The Auto-MAC, which compares a setof modes with themselves, is used as a powerful tool for assessing thequality of the extracted mode shapes. If the off-diagonal terms of theAuto-MAC are near 0, this means that the extracted mode shapes arecorrect and accurate; otherwise, they are not accurate [25]. The Cross-MAC is used to compare the mode shapes of a system estimated fromdifferent methods or under different working conditions [26–30].Fig. 8a and c presents the Auto-MACs of the mode shapes identifiedfrom the EMA test at excitation position A and the proposed OMA test,respectively. The results indicate that the mode shapes from the OMAand EMA tests are accurate. Fig. 8b presents the Cross-MAC of themode shapes derived from the EMA tests at excitation positions A andB. As can be seen, the mode shapes estimated from the EMA tests atdifferent excitation positions are the same. The mode shapes derivedfrom the OMA test and the EMA test at excitation position A arecompared in Fig. 8d. From the figure, it can be observed that the firsttwo mode shapes are similar, but the last four mode shapes aredifferent. These phenomena are due to the different working condi-tions of the table. Fig. 9 presents the operational mode shapes of theentire machine tool acquired from the excitation B test. The blackarrows in the picture indicate the directions of the modal vibrations,and their lengths represent the amplitudes. From the figure, it can benoted that no obvious vibration of the worktable exists for the firsttwo modes. However, for the last four modes, the worktable showsobvious vibration. Compared with Fig. 8d, it can be concluded that the

Table 1Comparison of the machine-tool modal parameters estimated by the excitation B test and the conventional impulse tests.

Modes Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6

Natural frequency, Damping ratios ζ (%)

f1 ζ1 f2 ζ2 f3 ζ3 f4 ζ4 f5 ζ5 f6 ζ6

OMA 17.34 2.12 19.36 5.7 41.23 5.08 50.43 2.83 61.78 1.71 66.04 3.25Hammer A 17.09 2.54 19.52 6.91 39.48 6.67 47.88 6.97 59.14 5.55 68.42 5.7Hammer B 17.12 2.36 19.67 6.12 39.62 6.36 48.24 7.43 59.75 6.00 68.812 6.02

Fig. 8. MAC values of different mode shapes: (a) Auto-MAC of modes identified from EMA test A, (b) Cross-MAC comparing mode shapes derived from EMA tests A and B,(c) Auto-MAC of modes identified from the OMA test and (d) Cross-MAC comparing mode shapes derived from the OMA test and EMA test A.


running state of the worktable can influence the modes in which theworktable vibrates.

Because the running state of the worktable can influence thedynamics of the machine tool, the following section will discussthe dynamics of the worktable at different speeds.

5. Dynamic analysis of the worktable at different feed speeds

5.1. Dynamic analysis of the worktable in the X-direction

In this section, the natural frequencies and damping ratios ofthe table were estimated at different feed speeds from theexcitation A and excitation B tests. Two measurement points wereselected. One measurement point was located on the table asshown by the black node in Fig. 3b. Another measurement point

was selected as the reference point on the base. To study theinfluence of the speed on the dynamics of the table, four speeds(500 mm/min, 750 mm/min, 1000 mm/min and 1250 mm/min)were chosen. The sensing system was configured to acquireacceleration signals with a sampling frequency of 1024 Hz andproduce time segments with a length of 300 s, as shown in Fig. 10.To eliminate the effects of the table locations on the dynamics ofthe table, the worktable movements of excitations A and B wererestricted in a small fixed area, which were illustrated in Fig. 4 ofSection 3.

Fig. 11 shows the cross-correlation functions of the worktableat different feed speeds. According to the natural excitationtechnique [31], the cross-correlation function is the sum of thedecaying sinusoids and has the same character as the impulseresponse function. The blue curves in Fig. 11 represent the cross-correlation functions of the worktable under excitation A at

Fig. 9. Mode shapes of the machine tool estimated by the OMA test. (For interpretation of the references to color in this figure legend, the reader is referred to the webversion of this article.)

Fig. 10. Vibrations of the table caused by excitations A and B at a speed of 1000 mm/min: (a) acceleration signal for the entire acquisition time caused by excitation A and anenlarged view of the signal profile for the first 5 s and (b) acceleration signal for the entire acquisition time caused by excitation B and an enlarged view of the signal profilefor the first 5 s.


Fig. 11. Cross-correlation functions of the worktable acquired by excitations A and B at different feed speeds. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

Fig. 12. Cross-power spectra of the worktable in the X-direction at different feed speeds.


different speeds. It can be observed that the curves are invariantwhen the feed speed is different. The red curves in Fig. 11represent the cross-correlation functions of the worktable inexcitation B under different feed speeds. It is noted that the curveschange significantly when the speeds are different. These observa-tions illustrate that the impulse response function of the movingworktable varieties with the changes in the feed speed.

Fig. 12a shows the real parts of the cross-power spectra of theworktable caused by excitation A at different speeds, and Fig. 12bshows the imaginary parts. Curves with different colors represent thespectra at four different feed speeds. From these results, it can beobserved that the first three peaks of the cross-power spectra atdifferent speeds are coincident. The modal parameters estimated fromthese spectra are listed in Table 2. It is noted that the parameters at

different speeds are quite similar; the reason for this is that the modalparameters estimated by excitation A represent the dynamics of theworktable in a stationary state. Fig. 12c and d shows the real parts andthe imaginary parts, respectively, of the cross-power spectra acquiredfrom excitation B tests. From the figures, we note that the secondpeaks of the cross-power spectra in the pink boxes shift as the speedincreases. The modal parameters estimated from these cross-powerspectra, as summarized in Table 2, illustrate that the natural frequencyand damping ratio of mode 2 decrease as the speed increases. Thereason for these is that the dynamics of the machine tool may changeas a function of running speed, which makes the FRF of the machinenonlinear [12,17].

To verify the validity of the estimated modal parameters, theFRFs of the worktable in different running states were measured

Table 2Comparison of the modal parameters of the worktable in different experiments.

Speeds Excitation A Excitation B

Mode 1 Mode 2 Mode 3 Mode 1 Mode 2 Mode 3

500 mm/min Natural frequency (Hz) 17.72 37.28 48.11 17.61 32.56 39.36Damping ratio (%) 1.22 16.52 3.28 1.44 13.73 1.84




Hammer testsMode 1 Mode 2 Mode 3 Mode 1 Mode 2 Mode 3

0 mm/min Natural frequency (Hz) 17.809 36.595 48.171 500 mm/min 17.598 33.559 38.802Damping ratio (%) 4.45 34.80 10.07 3.98 18.42 6.92

Fig. 13. FRFs of the worktable at different feed speeds: (a) real parts of the FRFs and (b) imaginary parts of the FRFs.


by the hammer tests. One test measured the FRF when the tablewas in a stationary state. Another test measured the FRF when thefeed speed was set to 500 mm/min. Fig. 13 shows the FRFsmeasured in different states. The solid curves presented inFig. 13a show the real parts of the measured FRFs, and Fig. 13bshows the imaginary parts of the FRFs. From the figure, it can beobserved that a large difference exists between the FRFs in thestationary and the moving states. Fig. 14 shows the coherencefunctions of the input force and the vibration. From 0 Hz to 100 Hz, it can be observed that the coherence functions are almost 1,which means that the vibration of the worktable is primarilycaused by the input force of the hammer, and the vibration due tothe movements of worktable can be ignored. Therefore, the FRFstested in the experiments were valid and believable. The modalparameters estimated from the FRFs are summarized in Table 2. Itcan be concluded that the natural frequencies and the dampingratios for modes 2 and 3 change significantly if the running state isdifferent. The natural frequencies and damping ratios of theworktable in the moving state are smaller than those in thestationary state. The dotted curves in Fig. 13a and b show the realparts and the imaginary parts, respectively, of the FRFs synthe-sized by the modal parameters presented in Table 2. From thefigures, it can be observed that the solid curves and dotted curvesmatch quite well, which reveals that the modal parameters listedin Table 2 are well estimated.

From the modal parameters summarized in Table 2, it can beconcluded that the natural frequencies of the modes estimatedfrom the excitation A and B tests exactly coincide with those from

the hammer tests when the table is stationary and moving.Although the natural frequencies are highly matched, the dampingratios are rather different. The damping ratios estimated from theexcitation A and B tests are much smaller than those from thehammer tests.

5.2. Dynamic analysis of the worktable in the Y-direction

Fig. 15 shows the coherence functions of the input force and thevibration of the worktable in the Y-direction. The blue curverepresents the coherence function acquired from the hammer testin the stationary state, and the red curve represents the coherencefunction of the hammer test at 1000 mm/min. From the figure, weobserve that the values of the red curve are much smaller thanone. The reason for these is that the motor that drives theworktable in the Y-direction is more powerful than the one inX-direction, and the movement in the Y-direction affects thevibrations of the worktable to a great degree. Therefore, thehammer test in the Y-direction cannot measure the FRF of themoving worktable if the feed speed is high.

To study the dynamics of the moving worktable in Y-direction,the excitation B test and a contrasting cutting experiment wereconducted. Fig. 16a shows the design of the cutting experiment. Inthis experiment, the modal parameters can be identified from thevibration of the worktable caused by the interrupted cutting of anarrow workpiece. This type of cutting provides a pulse-likecutting force. To eliminate the harmonic frequencies, the widthof narrow workpiece was changed gradually (shown in Fig. 16b),


with one end width of 2 mm and the other of 5 mm. Therefore, thefrequency spectrum of the cutting force was flat in a certainfrequency bandwidth and had the same characteristics as white

Fig. 14. Coherence functions between the input force supplied by the hammer andthe vibration of the worktable in the X-direction.

Fig. 15. Coherence functions between the input force supplied by the hammer andthe vibration of the worktable in the Y-direction. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web versionof this article.)

Fig. 16. Experimental system for interrupted cutting of a narrow workpie

noise. The speed of the table in the Y-direction was 1000 mm/min,and the modal parameters of the worktable were acquired atthis speed.

The blue curves shown in Fig. 17a and b display the auto-powerspectra of the table with a speed of 1000 mm/min from theexcitation B test and the cutting test, respectively. The first peakof the auto-power in Fig. 17a is caused by rigid body motion.Because the worktable moved back and forth in the excitation Btest and the mean value of the time between two adjacentmotions (toi) was 0.2 s, the frequency of rigid body motions wereclose to 5 Hz . The red marks in the two figures represent thefrequency stability diagrams, which were calculated using theoperational PolyMAX@ algorithm. The frequencies of all clusters ofthe red marks in the two figures coincide with the exception of the

ce: (a) the test system and (b) the workpiece with a narrow width.

Fig. 17. Frequency stability diagram and auto-power spectrum of the worktable at1000 mm/min in the Y-direction: (a) frequency stability diagram and auto-powerspectrum in the excitation B test and (b) frequency stability diagram and auto-power spectrum in the cutting experiment. (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of this article.)


fourth cluster, which indicates two conclusions: First, the modalparameters of the moving worktable can be estimated from boththe excitation B test and narrow workpiece cutting test, and theboundary conditions of the two experiments are different, whichmay alter certain modes of the worktable.

6. Conclusions

In this paper, a new operational modal analysis method isintroduced for estimating the modal parameters of a machine toolat different feed speeds using the inertial force of the worktable.The proposed method was applied to extract the operationalmodal parameters of the machine tool structure at a feed speedof 1000 mm/min. Next, the extracted modal parameters (the firstsix modes) were compared with the modal parameters of themachine tool in a static state which were estimated by hammertests. The natural frequencies of the machine in the static statewere 17.09 Hz, 19.52 Hz, 39.48 Hz, 47.88 Hz, 59.14 Hz and 68.42 Hz,and the damping ratios were 2.54%, 6.91%, 6.67%, 6.97%, 5.55% and5.70%. Different from the modal parameters in the static state, thenatural frequencies of the machine tool in the operational statewere 17.34 Hz, 19.36 Hz, 41.23 Hz, 50.43 Hz, 61.78 Hz and 66.04 Hz, and the damping ratios were 2.12%, 5.70%, 5.08%, 2.83%, 1.71% and3.25%. The first two mode shapes of the machine tool in the twostates were similar, but the last four mode shapes are different. Allof these results showed that the table movement could influencethe last four modes of the machine tool structure in which theworktable vibrates.

The dynamics of the worktable, which were rarely mentionedin antecedent literature, were further studied at several differentspeeds (0 mm/min, 100 mm/min, 500 mm/min and 1000 mm/min). The results indicated that both the natural frequencies(32.56 Hz, 28.18 Hz, 25.98 Hz, and 24.08 Hz) and the dampingratios (13.73%, 11.48%, 9.97% and 8.36%) of the second mode ofthe table decreased greatly as the feed speeds were increased. Thisobservation illustrated that the influence of the table movementon the dynamics of the machine tool must be considered in high-quality machining. In addition, because no artificial inputs arerequired, the proposed method is able to estimate the operationalmodal parameters of the machine tool efficiently at different feedspeeds for the entire working space.

Acknowledgments

The research is supported by the National Natural ScienceFoundation of China under Grant nos. 51275188, 51121002 andthe Science and Technology Major Special Project of China underGrant no. 2011CB706803.

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