International Journal of Machine Tools & Manufacture 71 (2013) 26–40
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International Journal of Machine Tools & Manufacture
journal homepage: www.elsevier.com/locate/ijmactool
Estimation of CNC machine–tool dynamic parameters based on
random cutting excitation through operational modal analysis
Bin Li a,b, Hui Cai b, Xinyong Mao b,n, Junbin Huang b, Bo Luo b
a
b
State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China
National NC System Engineering Research Center, Huazhong University of Science and Technology, Wuhan 430074, China
art ic l e i nf o
a b s t r a c t
Article history:
Received 10 January 2013
Received in revised form
7 April 2013
Accepted 9 April 2013
Available online 17 April 2013
Dynamic properties of the whole machine tool structure including tool, spindle, and machine tool frame
contribute greatly to the reliability of the machine tool in service and machining quality. However, they
will change during operation compared with the results from static frequency response function
measurements of classic experimental modal analysis. Therefore, an accurate estimation of the dynamic
modal parameters of the whole structure is of great value in real time monitoring, active maintenance,
and precise prediction of a stability lobes diagram.
Operational modal analysis (OMA) developed from civil engineering works quite efficiently in modal
parameters estimation of structure in operation under an intrinsic assumption of white noise excitation.
This paper proposes a new methodology for applying this technique in the case of computer numerically
controlled (CNC) machine tools during machining operations. A novel random excitation technique based
on cutting is presented to meet the white noise excitation requirement. This technique is realized by
interrupted cutting of a narrow workpiece step while spindle rotating randomly. The spindle rotation
speed is automatically controlled by G-code part program, which contains a series of random speed
values produced by MATLAB software following uniform distribution. The resulting cutting produces
random pulses and excites the structure in all three directions. The effect of cutting parameters on the
excitation frequency and energy was analyzed and simulated. The proposed technique was experimentally validated with two different OMA methods: the Stochastic Subspace Identification (SSI) method and
the poly-reference least square complex frequency domain (pLSCF or PolyMAX) method, both of which
came up with similar results. It was shown that the proposed excitation technique combined successfully
with OMA methods to extract dynamic modal parameters of the machine tool structure.
& 2013 Elsevier Ltd. All rights reserved.
Keywords:
CNC Machine tools
Dynamics
Random cutting technique
Operational modal analysis
1. Introduction
The objective of high performance cutting is to machine the
parts in the shortest time, while respecting the physical constraints
of the process, such as torque, power, vibrations, tool wear and
failure, surface quality, and tolerances [1].This puts high demands
on good dynamic behavior of the machine tool during machining
when the power of the spindle is settled. Since Tobias et al. [2]
recognized that the tool or spindle should not be treated in isolation
from the whole mechanical system which includes the machine–
tool structure and the foundation the characteristics of which partly
determine its dynamic behavior. Kolar et al. [3] studied the impact
of the machine frame on the dynamic properties at the tool end
n
Correspondence to: National NC System Engineering Research Center, School of
Mechanical Science and Engineering, Huazhong University of Science and Technology (HUST), 1037 Luoyu Road, Hongshan District, Wuhan 430074, Hubei Province,
PR China. Tel.: +86 27 87542613 8428, mobile: +86 15007120546;
fax: +86 27 87540024.
E-mail addresses: maoxyhust@hust.edu.cn, maoxyhust@163.com (X. Mao).
0890-6955/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.ijmachtools.2013.04.001
point experimentally with a representative stiff face milling cutter
and with a slim compliant shank cutter. It was recognized that the
machine tool frame causes new critical compliances in the low
frequency range in the former case as well as a shift of existing
eigenfrequencies and an increased level of damping in both cases.
Thus, an accurate estimation of the dynamic modal parameters of
the whole mechanical system, including the tool, the spindle and
the frame, is of great value in online/real time monitoring, active
maintenance and precise prediction of stability lobes diagrams in
order to achieve high performance cutting.
This is generally done by experimental modal analysis (EMA)
approaches, an impact test or a shaker test, which calculate
frequency transfer functions (FRF) from measurements of both
input excitations and corresponding responses at the tool tip when
the machine is under rest [4]. However, impact testing usually
cannot be repeated unless a special device such as a swing for the
impact hammer is used. The frequency bandwidth of the excitation
depends on what hammer tip is chosen, and its energy depends
more on the operator's force. All these put high demands on the
operator's experience. Also, some parts of the machine tools, for
B. Li et al. / International Journal of Machine Tools & Manufacture 71 (2013) 26–40
example, the high speed spindle, are too delicate to be hammered
heavily upon while weak input causes poor data quality. In addition,
the slight nonlinearities of the machine structure appear regularly
in impact testing resulting in less reliable results of the linear modal
analysis.
Moreover, significant changes in modal parameters are expected
to occur due to spindle rotation and changes of machine–tool–
workpiece boundary conditions between the inactive state and the
machining operation state [5,6]. However, impact testing at the tool
tip is not feasible during high-speed machining because the operation
presents high risk of injury both to machine tools and to the hammer,
and thus violates health and safety regulations. Shaker testing is also
impossible because the shaker stinger has to be in contact with the
rotating tool. All these lead to a failure of applying EMA to identify the
dynamic modal parameters of machine tools during machining
operation. Fortunately, the cutting force resulting from machining
has long been a natural source to excite the structure. Kwiatkowski
and Al-Samarai [7] determined the dynamics of a milling machine
measuring its structural responses to the random component of the
cutting force during normal milling. However, the random component of cutting force is too weak compared with the harmonic ones
making it quite difficult to distinguish between natural frequencies
and tooth passing frequency and its harmonics.
A lot of methods have been developed to create strong and
broadband excitation. Bonzanigo and Tsudi [8] used an unequally
spaced milling cutter to generate a uniform, broadband cutting
force that excited all the modes of a milling machine in the
bandwidth of interest. However, their results are valid only for a
particular direction of the cutting force. Opitz and Weck [9] excited
the machine tool's structure by the random cutting force generated
from the continuous cutting of a “random” workpiece. They derived
the transfer functions of the structure in a limited frequency range
of 40–80 Hz using spectral density measurements of both the force
and the displacement signals. Minis [10] developed an improved
technique, similar to that of Opitz and Weck, which provides a
strong, broadband excitation by interrupted cutting of a specially
designed workpiece, where the surface is modulated with pseudorandomly distributed teeth and channels. They identified the
dynamics of machine tool structure from input–output measurements. This also proved to be an effective method to excite a 5-axis
machining center in Budak's case [11]. However, the interrupted
cutting of such a workpiece generated a pseudorandom periodic
force signal resulting in a lower limit of the valid excitation
frequency band. What's more, this lower limit could be much
higher in high-speed machining which will cut off the properties
in lower frequency range due to machine tool frame, for example, in
Kolar's case. Besides, new workpieces have to be designed and
machined if the frequency bandwidth of interest changes according
to different machine tools. Also, the machining of the complicated
workpiece is time and effort consuming resulting in high cost.
Tounsi and Otho [12] developed a pulse-like cutting force excitation
as a result of the interrupted cutting of a narrow workpiece width
through single tooth milling operations. In the above cases, all
the cutting forces and output responses must be measured by
an expensive dynamometer in order to identify the dynamics of the
structure. This results in high cost to do real time monitoring in a
factory background and is sometimes even impossible.
Operational modal analysis (OMA), also known as output-only
modal analysis, works quite efficiently in modal parameters estimation of a structure on duty under an intrinsic assumption of white
noise excitation. The response is the only information to identify the
modal parameters of the structure making it quite simple and
relatively inexpensive. The theoretical assumption of white noise
excitation turns out to be too strict in practical applications. Fortunately, as long as (unknown) input spectral is reasonably flat, OMA
methods will work fine [13]. Some of the popular methods are the
27
numerically robust Stochastic Subspace Identification (SSI) method
[14–19], the user friendly Frequency Domain Decomposition (FDD)
method [20], the industry standard, Least Square Complex Exponential (LSCE) method [5,21,22] and the poly-reference Least Square
Complex Frequency domain (pLSCF or PolyMAX) method [13,23,24].
As the model order increases, the SSI method requires a large
expenditure of memory and is not suitable for cases that need to
handle large amounts of data or require high computational efficiency [25], such as real time monitoring. Also, poles arising from
noise and redundant model orders can be so scattered as to mess up
the physical poles. FDD is under the assumption that the structure is
lightly damped and the modal shapes of close modes are geometrically orthogonal [20]. This may not be the case for machine tools
whose damping ratios can be much higher because of many joint
interfaces between different parts, and the modal shapes can have
arbitrary directions. The PolyMAX method which is a polyreference
version of the LSCF method can also be applied to operational data
(referred to as Op. PolyMAX) when appropriate preprocessing and
post-processing is applied [13]. It proceeds similarly and as fast as the
polyreference LSCE method. Moreover, it can identify closely spaced
poles quite well, and produce extremely clear stabilization diagrams
making the automatic parameter identification process rather
straightforward. This enables continuous monitoring of the dynamic
properties of machine tools.
So far, the only complete methodology to apply OMA under
machining operations was detailed by Zaghbani and Songmene [5].
They tried to estimate modal parameters through SSI and the
autoregressive moving average (ARMA) method during normal milling
operations. However, it is quite difficult to distinguish between natural
frequencies and tooth-passing frequencies and their harmonics.
Although some criteria were presented to eliminate these harmonic
frequencies, the methods were complex and rather experiencedependent. The reason for this problem is that the cutting force
generated by normal machining is rather periodic, which does not
fulfill the assumption of white noise excitation for OMA.
In the present study, the main goals are: (1) to propose a new
random excitation technique in accord with the excitation requirement of OMA, (2) to develop a new complete method to apply
OMA during machining operations based on the proposed technique, (3) to verify the effectiveness of the excitation through
spectral analysis, and (4) to estimate the dynamic modal parameters of the machine tool structure during machining through
OMA methods and compare them with the results of conventional
impact test and normal cutting tests. The employed OMA methods
were SSI and Op.PolyMAX methods.
This paper is organized as follows: Section 2 presents the
theoretical background of OMA and describes the proposed excitation technique originating from random impulses signal. This signal
is modeled and simulated and then the technique to realize this
signal with cutting force is proposed. After an analysis of the effect of
cutting parameters on excitation bandwidth and energy, the steps to
estimate CNC machine-tool dynamic parameters through OMA are
summed up. Section 3 presents the experimental verification of the
method. The characteristics and feasibility of the proposed technique
are discussed and compared with the impact testing. The work is
summarized and future works are presented in the conclusions.
2. Operational modal analysis based on random cutting
2.1. Background of OMA
The relationship between the inputs x(t) and the responses y(t)
[26] can be expressed as
Gyy ðωÞ ¼ HðωÞGxx ðωÞHðωÞH
ð1Þ
28
B. Li et al. / International Journal of Machine Tools & Manufacture 71 (2013) 26–40
where Gxx(ω) is the (n n) power spectral density (PSD) matrix of
the input, Gyy(ω) is the (r r) PSD matrix of the responses, H(ω) is
the (r n) frequency response function (FRF) matrix and superscript H denotes the hermitian of a matrix. The FRF matrix can be
written in the following well-known partial fraction, i.e. pole/
residue form:
N
Q r Ψ r Ψ Tr Q nr Ψ nr Ψ H
r
ð2Þ
þ
HðωÞ ¼ ∑
n
jω−λr
jω−λr
r¼1
where λr, Ψr and Qr are respectively, the pole, the mode shape
vector and a scalar constant of mode r. Superscripts T and n denote
the transpose and complex conjugate of a matrix, respectively.
The poles occur in complex-conjugated pairs and are related to the
natural frequencies ωr and damping ratios ξr as follows:
qffiffiffiffiffiffiffiffiffiffiffi
ð3Þ
λr ; λnr ¼ −ξr ωr 7jωr 1−ξ2r
Operational modal analysis (OMA) is developed under an
intrinsic assumption of white noise excitation, i.e. the Gxx (ω)¼
const., at least in the frequency band of interest, then Eq. (1)
becomes
n
N
ar Ψ r Ψ Tr anr Ψ nr Ψ H
br Ψ r Ψ Tr br Ψ nr Ψ H
r
r
þ
þ
þ
ð4Þ
Gyy ðωÞ ¼ ∑
jω−λr
−jω−λr
jω−λnr
jω−λnr
r¼1
n
where ar, anr , br and br all are scalar constant coefficients. The goal
of operational modal analysis is to identify the right hand side four
terms of Eq. (4) based on measured output data pre-processed into
output spectral. However, it is obvious that this PSD model of
outputs has four poles (λr,−λr, λnr , and −λnr ) for each mode r, which
means its order is twice the order of the FRF model as shown in Eq.
(2). Fortunately, it is sufficient to compute the so-called half
spectral, Gþ
yy ðωÞ, which only consists of the first two terms in Eq.
(4) [24]:
N
ar Ψ r Ψ Tr anr Ψ nr Ψ H
r
Gþ
þ
ð5Þ
yy ðωÞ ¼ ∑
n
jω−λr
jω−λr
r¼1
This expression of Gþ
yy ðωÞ is almost the same as the expression
of H(ω) identified by Eq. (2) except for the scalar constant Qr and
ar. Each element of matrix Gþ
yy ðωÞ is a spectral density function. The
diagonal elements of the matrix are the so-called auto-power
spectral density (Auto PSD) functions which are the magnitudes of
the spectral densities between a response and itself. The offdiagonal elements are the cross-power spectral densities (CSDs)
between different responses. The Auto PSDs are all real-valued
elements while the CSDs take complex values, carrying the phase
information between the measured and the reference degree of
freedom. The matrix is symmetric with complex conjugate elements around the diagonal, namely a Hermitian. Any column or
row of the matrix carries enough information to extract the modal
parameters like the H(ω) matrix. Then the natural frequency ωr,
damping ratio ξr and unscaled mode shape Ψr can be estimated
based on Gþ
yy ðωÞ with classical frequency domain identification
methods based on FRF in EMA. Of course, there are some time
domain identification methods of OMA based on the correlation
function model similar to the impulse response function (IRF)
in EMA.
It should be noted that when OMA is applied in the estimation
of machine–tool dynamics, there are two critical requirements for
excitation from the analysis above. First, it needs white noise
excitation intrinsically, namely the PSD of the excitation should be
reasonable flat over the frequency bandwidth of interests. Second,
the corresponding frequency range and energy of the excitation
should be adjustable according to different machine tools and
actual situations so that all the structure modes in the frequency
range of interest are excited. The following section presents the
proposed random excitation technique based on cutting that
meets the needs mentioned above.
2.2. Structure excitation with random cutting force
2.2.1. The random impulses excitation
In EMA, random and impulse signals are two of the most
popular excitations used by the shaker test and the impact test
respectively. The force signal for random excitation is an ergodic,
stationary random signal containing all frequencies within the
frequency range. It is nondeterministic which cannot be described
by a mathematical function but rather by its statistical characteristics. The white noise excitation needed in OMA is actually a pure
random signal whose energy is equally distributed through the
whole frequency range (−∞∼+∞); namely, the PSD of the signal is a
flat line over the entire frequency band. However, ideal white
noise excitation cannot be obtained in reality while the most
common one has a reasonable flat PSD in a limited frequency
range. Fig. 1a shows the PSD of such a typical random signal
generated by MATLAB which has the uniform distribution over
[0,100]. Of course, whatever distribution having a flat PSD is fine.
Random excitation has the tendency to linearize the behavior of a
structure from the measurement data even though it behaves
nonlinearly. However, the fact that neither the force signal nor the
response is periodic with infinite time history gives rise to an error
called leakage.
The ideal impulse signal is a Dirac function δ(t) and its spectral
is a straight line over the entire frequency range. The practical
impulse signal for impact testing is deterministic with limited
amplitude and duration of time which can be described by a
mathematical function as
(
A0 ; t∈½0; τ
f si ðt Þ ¼
ð6Þ
0; t∉½0; τ
where fsi(t) is the time function of
duration of the impulse, A0 is its
spectrum Fsi(f) ¼(A0e−iπft sin πft)/πf,
signal processing book. Then the
Gss(f) of fsi(t) is defined as [21]
2
sin πτf
Gss ðf Þ ¼ F si ðf ÞF nsi ðf Þ ¼ A20
πf
single impulse input, τ is the
amplitude, and its frequency
which can be found in any
auto-power spectral density
ð7Þ
Its zeros are f¼ k/τ (k ¼ 71, 72 …). It is clear that the
frequency bandwidth of the first power spectral lobe (k¼ 1)
increases as τ decreases, as shown in Fig. 1b.
Though impact excitation is convenient to use and very
portable for field and laboratory tests, it is difficult to control
either the force level or the frequency range of the impact. This
could affect the signal-to-noise ratio in the measurement, thus
resulting in poor quality data. Fortunately, distributing many
impulses randomly in time domain gives a new random excitation
fri(t):
(
Ai ; t∈½t i ; t i þ τ
f ri ðt Þ ¼
ð8Þ
0; t∉½t i ; t i þ τ
where n is the number of impulses the excitation contained,
Ai (i¼1, 2, …, n) is the amplitude of each impulse, and ti (ti≥0,
i¼1, 2, …, n) is a random variable representing the start moment
of the ith impulse. The spectral of fri(t) is
Z
F si ðf Þ ¼
∞
−∞
n
f ri ðtÞe−j2πf t dt ¼ ∑
n
∑ Ai e
i¼1
i¼1
−j2πf ðτ=2þt i Þ
Z
t i þτ
ti
Ai e−j2πf t dt ¼
sinπτf
πf
ð9Þ
B. Li et al. / International Journal of Machine Tools & Manufacture 71 (2013) 26–40
29
Fig. 1. (a) Random signal with reasonable flat PSD in a limited frequency band and (b) Time signal and PSD of impulses with different durations.
yielded:
The PSD of the signal is
Grr ðf Þ ¼ F ri ðf ÞF nri ðf Þ ¼
2
sin πτf
πf
2
Grr ðf Þ ¼ nðAτÞ2
3
6 n
7
n
n
6
7
6 ∑ A2i þ ∑ ∑ Ai Aj cos 2πf ðt i −t j Þ7
4i ¼ 1
5
i¼1 j¼1
i≠j
ð10Þ
j≠i
where tj (j¼ 1, 2,…, n; j≠i) following the same distribution as ti. It is
clear that Eq. (10) contains three independent parameters, namely
τ, Ai (or Aj), and ti (or tj), which originate from the excitation signal
fri(t) and has the same zeros as Eq. (7). Then the bandwidth of the
first spectral lobe (referred to as BW1st) is the inverse of the pulse
duration:
BW 1st ¼
1
τ
ð11Þ
tj (or tj; i, j ¼1, 2,…, n; j≠i) is a very important variable characterizing the statistical properties of the random impulses excitation.
It determines the moment of when each impulse occurs and
the number of impulses in unit time (referred to as density ρ).
However, the variable ρ, which is analyzed here, is much more
controllable than the moment, which is rather random. Because ti
and tj are both random variables obeying the same distribution,
the term Ai Aj cos ω(ti−tj) in Eq. (10) becomes zero after many times
of averaging. If the amplitude of each impulse is almost the same,
namely Ai ¼A (i¼ 1, 2, …, n), the averaged PSD of this signal is
sin πτf
πτf
2
ð12Þ
Grr ðf Þ represents the distribution of the signal energy, referred
to as Ee, as a function of frequency. Eq. (11) indicates that the
frequency range BW1st of the excitation is inversely proportional to
the duration of each impulse τ. Eq. (12) indicates that the
excitation energy Ee is proportional to the number of impulses
contained in one sample, which is determined by ρ, and the square
of the impulse amplitude A. Fig. 2a presents three different
random impulse signals of which the amplitude A is 1 N, the
duration of each impulse τ is 2 ms and the total length of time
(referred to as T) is 1 s while the density ρ is 1, 8, and 32 impulses
per second respectively. Fig. 2b presents the PSD of the three
signals. It is clear that BW1st does not change while the amplitude
of the PSD increases almost eightfold (9 dB) as ρ octuples and
increases almost fourfold (6 dB) as ρ quadruples. Fig. 2c presents
the PSD of signal2 in Fig. 2a, while A takes up 1 N, 4 N and 16 N.
The PSD of the three signals have the same BW1st while the PSD
amplitude increases almost 16 times (12 dB) as the impulse
amplitude quadruples. Fig. 2d presents the PSD of signal2 while
τ takes up 2 ms, 4 ms and 8 ms and the corresponding BW1st of
each signal is 500 Hz, 250 Hz and 125 Hz respectively. The PSD
spectral is reasonably flat over almost the entire first lobe with
15 dB roll-off which is acceptable [27]. Moreover, as τ doubles the
amplitude of the first lobe increases almost fourfold (6 dB). This is
30
B. Li et al. / International Journal of Machine Tools & Manufacture 71 (2013) 26–40
Fig. 2. (a) Three random impulse time signals with different ρ (1, 8, and 32 impulses per second); (b) PSD of the three signals; (c) PSD of signal2 while A changes to 1 N, 4 N,
and 16 N; (d) PSD of signal2 while τ changes to 2 ms, 4 ms, and 8 ms.
because when τ is quite small, sin πτf≈πτf and then Grr ðf Þ∝τ2
according to Eq. (12).
To sum up, the frequency bandwidth and energy of this
random excitation can be adjusted easily by its parameters,
namely the duration of impulse τ, the density of the impulses ρ,
and the amplitude of the impulses A. BW1st of the excitation is
inversely proportional to τ, while the energy Ee is proportional to ρ
and the square of both A and τ. Its PSD is reasonably flat over half
the first lobe BW1st/2 with 10 dB variations which is acceptable.
So it is a fairly good input for OMA.
2.2.2. Realization of random impulses excitation with cutting force
Minis [10] developed an improved method to provide a strong,
broadband excitation by interrupted cutting of a specially designed
workpiece the surface of which modulated with pseudorandomly
distributed teeth and channels. The interrupted cutting of that
workpiece generated a pseudorandom impulses force signal. This
also proved to be an effective method in Budak's case [11] to excite a
5-axis machining center. However, new workpieces have to be
designed and machined if the frequency bandwidth of interest
changes according to different machine tools. The machining of the
complicated workpiece are time and effort consuming resulting in
high cost. Moreover, the limited dimension of the workpiece results
in a pseudorandom impulses force signal instead of a pure random
one. In order to generate random impulses excitation with cutting
force and to simplify the machining of the workpiece at the same
time, a new excitation technique specifically suitable for widely
used CNC machine tools is proposed.
Thanks to technological progress, both the main motion and
feed motion of CNC machine tools can be controlled automatically
and accurately. It is common that the main motion and feed
motion are stable in a single pass. However, when the speed of the
main motion is random in a single pass, random cutting force will
be generated. One step further, if what is being machined is a
narrow tooth of a workpiece, cutting force like random impulses
finally occurs.
As the impulse duration decreases, the frequency bandwidth of
the first spectral lobe increases; this is analyzed in the above
section. In order to make sure that the random impulses cutting
force excites all the structure modes in the bandwidth of interest,
the frequency range of the excitation must be wide enough and
the impulse energy should be powerful enough. Therefore, the
duration of the impulses should be short enough, namely mill cuts
engagement should not last much time, but not too short, which
will result in too week impulses. On the other hand, it is known
that the interaction between the machine tool and the cutting
process can be represented by a closed loop system that incorporates two fundamental blocks [10]. One is the machine tool block
GMT describes the relationship between the dynamic cutting force,
which is applied on the tool, and the response of the structure.
The other is the cutting process block GCP which describes the
effect of the relative displacement of the tool and workpiece on
the variation of the cutting force. In order to estimate the
B. Li et al. / International Journal of Machine Tools & Manufacture 71 (2013) 26–40
dynamics of the machine-tool structure GMT during machining, GCP
should be uncoupled from the structure dynamics. Fortunately, the
cutting process block is valid only in a period of time τ when the
cutting is performed. As this tends to zero, the cutting force
becomes more pulse-like and then is uncoupled from the structure
dynamics. Thus, a narrow radial depth was chosen to reduce the
time of engagement and eliminate the effect of the cutting process
GCP [12]. Fig. 3 shows the schematization of the proposed technique applied in a milling machine. In the figure, a workpiece with a
single tooth is machined while the face milling cutter rotates
randomly (the revolution speed N or n(t) is a random variable).
The cutting forces are assumed to be proportional to the uncut
chip area Ac [28]. In the above case, Ac consists of two faces, the
flank and the button, as Fig. 6 shows. So Ac is calculated as
Ac ¼
aw ad
þ aw T w
sin α
ð13Þ
where ad is the axial depth, aw is the width of the tooth, Tw is
the width of the blade, and α is the angle of tool cutting edge.
The cutting speed v (mm/s) is
v¼
πnD
60
ð14Þ
where n is the mean of revolution speed N, and D is the diameter
of the cutter. Then the duration τ of each pulse is
aw
60aw
τ¼
¼
v
πnD
ð15Þ
Substituting Eq. (11) into (15) gives
BW 1st ¼
πnD
60aw
ð16Þ
Because the PSD spectral is reasonably flat over half the first
lobe, which has been discussed in the above section, BW1st/2 can
be set as the upper limit fh of the effective bandwidth of the
excitation. While the lower frequency limit of the excitation fl
given by Minis [10] is
fl ¼
15
Δf
π
ð17Þ
31
where Δf is the inverse of the period of revolution T (¼60/n, and n
is the constant spindle speed). So Eq. (17) becomes
fl ¼
15
n
n
¼
π
60
4π
ð18Þ
There exists a lower frequency limit in Minis' method because
both force components of orthogonal cutting are pseudorandom
periodic signals. In contrast, the proposed excitation in this paper
is truly random resulting in no lower limit. From the above two
sections, the relationships between BW1st (and Ee) of the excitation
and impulse parameters and the relationships between impulse
parameters and cutting parameters are summed up in Table 1.
It can be seen from Table 1 that BW1st is only related to one
impulse parameter τ while Ee is far more adjustable through all three
impulse parameters. The task of realizing random impulses excitation
with cutting force is how to choose the cutting parameters according
to actual needs. What is known before the work is just the frequency
range of interest (referred to as BW1st/2). Before the milling operation,
the tools and workpieces at hand are surely known, so the diameter D
of the cutter and the width aw of the tooth for excitation are actually
determinate. In practice, aw should be chosen carefully in order to
provide strong impulses as well as to eliminate the effect of the cutting
process. Then Eq. (16) can be rewritten as
n¼
60BW 1st aw
πD
ð19Þ
and the mean n of the revolution speed N is obtained. The random
variable N is the key to fulfill white noise excitation. Taking into
account the rotational inertia and response speed of the motor,
the spindle speed cannot change immediately and dramatically.
A parameter Δn is defined to present the maximum difference
between adjacent speed value ni and ni+1 (i¼ 1, 2, …, k) where k is
the desired number of the speed values. Define a continuous random
Table 1
Summary of the relationships between cutting parameters and excitation signal.
Energy Ee
Bandwidth BW1st
Impulse parameters
Cutting parameters
BW1st ¼1/τ
A∝aw, ad
τ∝aw, ∝−n
ρ∝n, af
aw, ad
a w, n
n, af
2
Ee∝A
Ee∝τ2
Ee∝ρ
where ∝ denotes proportional relationship and ∝− denotes inverse relationship.
Fig. 3. Schematization of the proposed excitation technique.
32
B. Li et al. / International Journal of Machine Tools & Manufacture 71 (2013) 26–40
Fig. 4. Random signal and histogram of N and Nu.
variable Nu to be the difference between adjacent speeds which has
the uniform distribution over [−Δn, Δn] and then the random variable
N denoting the spindle speed is defined as
n
N i −N i−1 ¼ N ui or Ni ¼ N 0 þ ∑ N ui ði ¼ 1; 2; ⋯; k; N 0 ¼ nÞ
i¼1
ii.
ð20Þ
According to the definition of the Wiener process (or Brownian
motion) [29], the stochastic process {Ni: i≥0} is a Wiener process if the
increments Nui (i¼1, 2,…, k) are normally distributed with mean μ¼ 0
and variance s2 ¼1. However, they are uniformly distributed over
[−Δn, Δn] in this case. It can be supposed that {Ni: i≥0} are also
uniformly distributed with mean μ¼n which will be checked by
Matlab later.
It should be noted that the spindle speed N should be present in a
limited range [nl, nh] rather than too low or too high during machining
(N should be smaller than the maximum speed and larger than zero),
and its PSD should be reasonable flat. Such a random variable is easily
generated by MATLAB software. Compared with the feed af, the axial
depth ad is more effective to adjust Ee, so only changing ad while
choosing a constant value for af is a good choice to increase the
excitation energy. So far all the cutting parameters are determined.
In summary, the complete steps to apply the proposed random
impulses excitation technique in a CNC machine tool to do OMA
are as follows:
i. Design and manufacture the workpiece. Only one tooth is used
for excitation and the width aw of this tooth should be chosen
iii.
iv.
v.
carefully in order to provide strong impulses as well as to
eliminate the effect of the cutting process.
Calculate average spindle speed n from Eq. (19) according to
the frequency band of interest BW1st/2, the tool diameter D and
the width aw;
Generate the values of random variable N through MATLAB.
A sequence of values of N limited in certain range [nl, nh] can be
easily obtained from Eq. (20) through MATLAB;
Manually prepare the NC part program. The part program
contains a sequence of commands and each command contains
a spindle speed value from sequence obtained in step iii.
Meanwhile, only change the axial depth ad to adjust the
excitation energy while choose a normal value for feed af in
the program;
Machine the workpiece to excite the structure and pick up
vibrations of different points to do OMA.
3. Experimental verification
3.1. Realization of random cutting excitation
The proposed excitation technique was conducted on a 3-axis
vertical milling center XHK5140 made by Huazhong University of
Science & Technology shown in Fig. 5. It is equipped with a
spindle, the speed of which is up to 3000 rpm and the maximum
B. Li et al. / International Journal of Machine Tools & Manufacture 71 (2013) 26–40
power of which is 11 kW, and with a numerical control system
HNC-22M from Wuhan Huazhong Numerical Control Co., Ltd
(HNC). The machine structure represents a very common type of
machine tool design with a moving spindle box, a worktable and a
slide. The primary motion is the spindle rotation and the feed
motion is completed by the spindle box moving along the z-axis,
the worktable moving along the x-axis and the slide along the
y-axis.
According to Minis's method, the lower limit fl is 31.8 Hz [see
Eq. (18)] if the rotational speed is 400 rpm which is common in
milling operations. However, it is not acceptable because the
dynamics of lower frequency range (0–50 Hz) corresponding to
the spindle speed range (0–3000 rpm) is of great significance to
machining behavior. So Minis's method fails while the proposed
excitation technique is a good choice. A representative stiff face
milling cutter with the diameter of 80 mm and total length of
92 mm was used and only one tool tooth was engaged in the
milling operation. The width aw of the workpiece tooth for
excitation was 4 mm and the frequency band of interest BW1st/2
was chosen to be 250 Hz. The feed af was kept at 0.1 mm/tooth and
the axial depth ad of cut could vary from 0.5 up to 4 mm in order
33
to adjust the excitation energy with a depth of 1.2 mm originally
chosen. According to Eq. (19) the average rotation speed n was
477 rpm and it was first made 500 rpm in order to set the speed
limits [300,700] rpm. The maximum difference Δn between
adjacent spindle speeds is set to be 20 rpm. 800 values of a
random variable Nu having the uniform distribution over [−20,
20] is first generated by MATLAB. The mean of the 800 values is
0.365. Then a sequence of values between 300 and 700 rpm for
random variable N denoting the spindle speed was generated
according to Eq. (20). The final average rotation speed n is 490 rpm
resulting in a final BW1st 513.14 Hz. The minimum speed is
311 rpm, the maximum is 676 rpm and the standard deviation is
91.4 rpm. Plotting Nu and N as a function of time in 60 s results in
two random signals shown in Fig. 4a and b.
A common 45# carbon steel workpiece with two teeth was
machined as Fig. 5c shows. One tooth was used here for excitation
and the other was used for another experiment. The length of the
teeth was 80 mm and the height was 4 mm. Finally a NC part
program was manually prepared which contained the spindle
speed values obtained above. The theoretical cutting period
calculated according to NC part program is 101.489 s.
Fig. 5. Experimental setup: (a) XHK5140 milling center and the numbered red squares show the locations of 6 PCB 356A15 accelerometers: 1-Column:L1, 2-Column:L18,
3-Headstock:10, 4-worktable:31, 5-slide:21, 6-base:25; (b) the face cutter used and (c) the workpiece for excitation (For interpretation of the references to color in this figure
legend, the reader is referred to the web version of this article.).
34
B. Li et al. / International Journal of Machine Tools & Manufacture 71 (2013) 26–40
3.2. Experimental setup and measurements
Fig. 5 shows the experimental setup. During the machining, the
cutting forces in the x, y, and z directions were measured using a
three-axis 9253B23 Kistler table dynamometer (Measurement
range: 712 kN; Frequency range (−3 dB):0–45 kHz; Resonant
frequency: fn(x)≈610 Hz, fn(y)≈570 Hz, fn(z)≈570 Hz). The FRF of
the dynamometer was measured when it was mounted on the
machine worktable shown in Fig. 8b. It is clear that the frequency
band of interest BW1st/2 (250 Hz) is still within the bandwidth
of the dynamometer, while the bandwidth BW1st exceeds a little
which is acceptable. A steel plate was faced and drilled to adjust
the table dynamometer holes to the T-slots of the machine worktable and firmly bolted to the worktable ensuring rigid mounting.
Then the dynamometer was bolted to the plate and the workpiece
was bolted to the dynamometer. The tooth of the cutter was an
indexical carbide insert (SEHT1204) from EGO Machine Tools
Co., Ltd.
Six three-axis accelerometers of type PCB 356A15 (Measurement range: 750 g; Frequency range (75%): 2–5000 Hz;
Resonant Frequency: ≥25 kHz) were mounted to measure the
vibrations of the machine structure, the slide, the worktable, the
base, the headstock, and the top and bottom of the column, in all
three directions due to the cutting force. And a one-axis accelerometer of type PCB 352C34 (Measurement range: 7 50 g;
Frequency range: 0.5–10,000 Hz; Resonant frequency: ≥50 kHz)
was used to measure the vibration of the workpiece in the x
direction during machining. The locations of all the accelerometers
are shown in Fig. 5.
In order to make sure that the generated cutting force can be
controlled as expected, the axial depth of all the cutting tests were
chosen according to previous cutting results so that the cutting
Table 2
Summary of the cutting conditions for the experimental tests. Case A is symmetric
tooth milling of the workpiece; Case B is symmetric face milling of the workpiece.
Test #2, #4 and #6 were later used to extract modal parameters.
Case
A
B
Test #
Feed
(mm/tooth)
Axial depth
(mm)
Spindle speeds (rpm)
1
0.1
1.2
2
0.1
1.5
3
4
5
6
7
8
9
0.05
0.1
0.2
0.1
0.1
0.1
0.1
1
1
1
1
1
0.5
0.7
800 random
values over [300,700]
500 random
values over [300,700]
400
400
400
600
800
400
400
process is stable. Table 2 summarizes all the cutting tests during
the experimental study. During all the cutting tests, the tool cuts
symmetrically in end-milling instead of peripheral milling (upmilling or down-milling) while the slide feeding along the +y
direction. Of course, the proposed excitation technique can be
applied easily in other conditions according to the machine tool
considered. The measured boundary condition is that the spindle
is rotating, the machine table is moving and the tool is moved at
the exact position without contact with the workpiece while
machining. Because the face milling cutter is an example of very
compact and stiff tool and both the axial depth and radial depth
are small, the deflection of the tool during the cutting is negligible.
The total 22 signals were collected by the acquisition system
LMS SCADAS Mobile SCM05 simultaneously at a sampling rate of
1024 Hz. The cutting forces were only measured to analysis the
effectiveness of the excitation.
Table 3
Comparison of the machine–tool modal parameters estimated using the Op.
PolyMAX method for test #2, #4 and #6 and the PolyMAX method for tap test.
The SSI method was also employed for test #2.
Modes
Test #2 [300,700]
rpm
SSI
1 ωn
(Hz)
ζ (%)
2 ωn
(Hz)
ζ (%)
3 ωn
(Hz)
ζ (%)
4 ωn
(Hz)
ζ (%)
5 ωn
(Hz)
ζ (%)
6 ωn
(Hz)
ζ (%)
7 ωn
(Hz)
ζ (%)
8 ωn
(Hz)
ζ (%)
9 ωn
(Hz)
ζ (%)
Tap test
0 rpm
Test #4
400 rpm
Test #6
600 rpm
Op.
PolyMAX
17.611
17.473
17.436
17.073
16.839
6.843
19.593
3.687
19.416
1.078
18.976
1.358
20.037
3.86
20.185
1.83
22.351
1.381
22.844
2.207
21.687
0.55
22.912
0.136
22.594
4.702
69.563
1.55
69.469
3.387
71.066
1.145
73.237
1.937
72.035
4.988
91.677
2.261
91.124
4.819
91.098
0.237
93.396
0.468
93.944
5.024
1.19
120.936 122.266
4.764
122.293
0.169
121.432
0.122
123.397
3.552
0.608
130.46 130.577
3.114
133.027
0.071
134.997
0.274
132.304
10.607
1.937
175.055 175.022
2.548
175.153
0.062
173.32
0.164
173.156
3.819
2.739
243.131 242.788
3.915
239.392
0.083
238.872
0.052
241.675
4.933
0.276
3.323
0.808
Fig. 6. Impact tests with different hammers: (a) Tool tip impact test with HPCB and (b) Column impact test with HDFC.
0.06
B. Li et al. / International Journal of Machine Tools & Manufacture 71 (2013) 26–40
What's more, multiple-input multiple-output (MIMO) impact
tests with two different impact hammers were conducted on the
machine shown in Fig. 6 under the same experimental setup. One
is a light hammer of type PCB-086C04 (referred to HPCB), and the
other is a much more powerful hammer of type DFC-1 (referred to
as HDFC) which comes from INV (China Orient Institute of Noise &
Vibration). The light hammer tapped the cutter and the heavy
hammer tapped the top of the column both in x and y directions.
35
The resulting modal parameters acted as a comparative reference
and are presented in Table 3 (referred to as EMA).
3.3. Results and discussion
3.3.1. Spectral analysis
Fig. 7 presents the cutting force signals in x, y, and z directions
and the corresponding acceleration signals of point Headstock: 10
Fig. 7. The cutting force and acceleration signals (point Headstock:10) of test #1 in Table 2: (a) Cutting force and acceleration signal for the whole acquisition time together
with an enlarged view of the signal profile for 1 s in x direction; (b) Cutting force and acceleration signal in y direction and (d) Cutting force and acceleration signal in z
direction.
36
B. Li et al. / International Journal of Machine Tools & Manufacture 71 (2013) 26–40
of test #1. It is clear that the cutting forces excited the structure in
all three directions during the cutting process. A good similarity was
observed between the force signals which fulfilled the assumption
of random impulses excitation in the time domain. The effective
cutting period appeared from 8.125 s to 117.148 s and the average
amplitude of the impulses was about 700 N in x, 400 N in y, 380 N
in z directions respectively. Fig. 8a presents the PSDs of the cutting
forces in cutting tests and the PSDs of impulse force in the impact
tests. The spectral estimation was performed using the modified
averaged periodogram method (Welch's technique) with an overlap
of 50% and a Hanning weighting function. This ensures that all data
are equally weighted in the averaging process, minimizing leakage
and picket fence effects. It can be seen that the PSDs of the cutting
forces in three directions present about 15 dB roll-off cross the first
lobe BW1st (¼511.9 Hz, which match the desired 513.14 Hz) with
some fluctuations. They are reasonably flat over half the first lobe,
which closely meets the excitation requirements of OMA. Besides,
the PSD of the x component of the cutting force is higher than the
HDFC hammer impact force in the x direction. Yet the other two
cutting force components are almost the same level as the HDFC
impact force in y direction. In addition, the PSDs of all the cutting
force components are at least 10 dB (10 times) higher than the ones
of the HPCB hammer impact forces although both of which are
inputted at the tool tooth. It means that the designed cutting force
excitation is quite powerful which can greatly improve the energy
level of the corresponding responses.
The real spindle speeds during machining can be estimated from
the measured cutting force impulses. Because the tool cut the
workpiece once per revolution, the duration Δt between two
adjacent force impulses is the time the spindle cost to finish one
revolution. The average rotation speed n(Δti) in this revolution can be
estimated by n(Δti)¼ 60/Δti rpm. Fig. 8c shows the designed and
estimated spindle speed start at the same time 8.125 s. It is clear that
the trend of the estimated speed match quite well with the designed
speed with some delay in time. The maximum of this measured
speed is 682.7 rpm, the minimum is 305.7 rpm, the mean is
492.7 rpm and the standard deviation is 94.3 rpm. Compared with
Fig. 4b, the statistic properties and distribution of the estimated
speed were almost the same as the ones of designed speed N
verifying that the excitation technique was carried out successfully.
Fig. 9 presents the PSD of the acceleration signals at the points
Headstock:10 and Column:L1 under cutting test #1 and HPCB
impact test. Since the vibration modes in all three directions are of
interest while the tool tip cannot be tapped in the z direction, only
the signals of the x and y directions are presented here to do
comparison while all acceleration signals within the effective
cutting period would later be used in modal parameters estimation. It can be observed that the PSDs of the acceleration signals in
both directions under the cutting excitation move almost parallel
to the ones of the hammer impact outputs. The former is at least
10 dB higher than the latter leading to a clearer presentation of the
peaks and stronger ant-jam capability in a factory background.
What's more, the modes in both directions are excited well in the
cutting test compared with impact test. So it can be assumed that
the modes in all three directions in the frequency range of interest
are excited by the cutting forces.
3.3.2. Structure identification
All the acceleration signals of 7 points within the effective cutting
time period were truncated for operational modal analysis. Each of the
signals must contain at least 65,000 data points. Two OMA methods
Fig. 8. (a) Power spectral density of different excitations; (b) Driving point FRF (of Point 1) between impact input and dynamometer outputs in x, y and z direction and
(c) Designed and estimated spindle speed during the effective cutting period.
B. Li et al. / International Journal of Machine Tools & Manufacture 71 (2013) 26–40
(SSI and Op.PolyMAX) were used to estimate the modal parameters of
the machine tool, namely the natural frequencies, damping ratios, and
unscaled modal shapes. Fig. 10a presents the sum of the PSD of all
signals within the mode stabilization diagram for the results of test #2
using the Op.PolyMAX method. It is obvious that 9 modes corresponding to the nine peaks in Fig. 9 fall in the frequency range of interest
(0–250 Hz). Another numerical method (SSI) was used to confirm the
validity of the results and the efficiency of the Op.PolyMAX method.
Fig. 10b presents the stabilization diagram for the results of test #2
using SSI. A tolerance of 1% for frequency stability, 2% for vector
stability and 5% for damping stability is used in both diagrams. It turns
out that the Op.PolyMAX yields an extremely clear stabilization
diagram leading to a quite easy selection of the physical poles. In
contrast, the physical poles calculated by the SSI method are messed
up by non-physical poles which tend to wander in the stabilization
diagram as the model order increases. When it comes to computational efficiency, a lot of research efforts [23,30–33] have investigated.
In this work, the Op.PolyMAX method takes about 2 s to calculate and
display the stabilization diagram at a model order of 40, while the SSI
method takes 4 s on the same PC platform. As the model order
increases, the advantage of the Op.PolyMAX goes further. The modal
parameters of test #2 (referred to as [300,700] rpm) by the two
methods are presented in Table 3.
Zaghbani and Songmene's method [5] was used here to extract
modal parameters during two representative normal machining
tests #4 (referred to as 400 rpm) and #6 (referred to as 600 rpm).
All the results are summarized in Table 3. From the table, it can be
37
seen that the results of the Op.PolyMAX method are quite similar
to their SSI counterparts, for example, the relative variation of
natural frequencies ([ωOp.PolyMAX−ωSSI]/ωSSI) lies between 0.02%
(mode 8) and 2.21% (mode 3). Fig. 11 presents the damping ratios
and relative variation of the natural frequencies of the tap test
(0 rpm), the test at 400 rpm and the test at [300,700] rpm
compared with the reference test at 600 rpm. It can be observed
that the relative variation of natural frequencies between
[300,700] rpm and 600 rpm lies between 0.46% (mode 9) and
3.81% (mode 2) while the variation between 0 rpm and 600 rpm
lies between 0.55% (mode 7) and 5.99% (mode 2). The difference of
damping ratios between [300,700] rpm and 600 rpm lies between
0.17% (mode 1) and 2.69% (mode 8) while the difference between
0 rpm and 600 rpm lies between 1.45% (mode 3) and 4.87% (mode
9). So the results of test #2 ([300,700] rpm) which employed the
proposed random cutting excitation technique are quite close to
the results of the normal machining operation (test #6), much
closer than the tap test. The same conclusion can be obtained
when the reference changes to test #4.
Besides, a powerful tool, the modal assurance criterion (MAC),
was used to evaluate the quality of the mode shapes. MAC is
defined as the squared correlation coefficient between two mode
shape vectors [21], which assesses the correlation between these
two vectors:
MACðΨ r Ψ s Þ ¼
2
jΨ H
r Ψ sj
H
H
ðΨ r Ψ r ÞðΨ s Ψ s Þ
Fig. 9. PSD of acceleration signals of Tool holder: 10 and Column:L1 under different excitations.
ð21Þ
38
B. Li et al. / International Journal of Machine Tools & Manufacture 71 (2013) 26–40
Fig. 10. Mode stabilization diagram: (a) using operational PolyMAX method (Op. PolyMAX); (b) using the stochastic subspace identification (SSI) method. The treated
acceleration signals are from test #2. The symbol ‘o’ indicates a new pole; ‘f’ indicates stable frequency; ‘d’ indicates stable frequency and damping; ‘v’ indicates stable
frequency and eigenvector; ‘s’ indicates that all criteria are stable.
Fig. 11. Comparison of the machine–tool modal parameters generated by the PolyMAX method at different spindle revolution speeds. The former graph is a relative variation
( ¼ ωn−machining/ωn−600 rpm) of natural frequencies and the latter is an absolute presentation.
If two vectors are estimates of the same physical mode shape,
the MAC should approach unity (100%), otherwise the MAC should
be low. A high quality mode set normally contains diagonal
elements which are 100% (by definition) and off-diagonal elements which have a low value (close to 0%). The unscaled mode
shapes of test #2 are rather good and the mode shapes of the two
OMA methods are very similar, which is evidenced by MAC values
represented in Fig. 12a and b. Fig. 12c and d indicated that the
mode shapes of test #2 are reasonably similar to the shapes of test
#6 compared with the shapes of tap test. Similar results can be
observed for test #2 and test #4 according to Fig.12e and f.
To sum up, the results obtained from operational modal
analysis based on the proposed random cutting technique are
extremely close to the results of normal cutting conditions. The
estimated modal parameters can characterize the dynamic properties of machine–tool system during machining within the chosen
revolution speed range, so they are of great value in on-line
monitoring, active maintenance and even precise prediction of
the stability lobes diagram which needs further investigation to
obtain the modal shape scaling factor. Also, many peaks existing in
the lower frequency range of the responses' PSD verifies that the
dynamics of lower frequency range determined by machine frame
is of great significance to machining behavior.
4. Conclusions
The dynamic performance of machine tools will change during
machining, so an accurate estimation of the dynamics is of great
significance in on-line/real time monitoring, active maintenance,
and precise prediction of the stability lobes diagram in order to
reach high performance cutting. This paper presents a new
technique to generate a strong and broadband excitation to meet
the white noise excitation requirements of operational modal
B. Li et al. / International Journal of Machine Tools & Manufacture 71 (2013) 26–40
39
Fig.12. MAC values of different mode shapes: (a) Auto-MAC of mode shapes using Op.PolyMAX of test #2; (b) MAC between mode shapes using the Op.PolyMAX and SSI
method of test #2; (c) MAC between mode shapes of test #2 and test #6; (d) MAC between mode shapes of tap test and test #6; (e) MAC between mode shapes of test #2 and
test #4 and (f) MAC between mode shapes of tap test and test #4.
40
B. Li et al. / International Journal of Machine Tools & Manufacture 71 (2013) 26–40
analysis, which is a powerful tool for dynamic modal parameter
estimations during machining. The mathematical model of this
excitation signal was first proposed, and the effect of the signal
parameters on its energy and effective excitation frequency
bandwidth was analyzed. Then the technique to realize this
excitation on CNC machine tools with cutting force was detailed.
Finally, this technique was experimentally validated and applied
successfully using two OMA methods (SSI and Op.PolyMAX) to
extract the modal parameters of the whole mechanical system,
including the tool, the spindle and the frame under machining
conditions. It was demonstrated in the mode stabilization diagram
that this technique can generate a quite powerful excitation and
avoid the problem of distinguishing between natural frequencies
and the harmonics of tooth passing frequency leading to a much
easier pole selection. Besides, the modal parameters estimated
during the random machining operation are quite close to the
estimated parameters during normal machining operations. Thus,
they can characterize the dynamic properties of a machine–tool
system during machining within the chosen revolution speed
range. Because the modal shapes are not mass normalized, a
scaling factor (or modal mass) must be estimated to synthesize
FRF, which needs further investigation.
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
Acknowledgment
[20]
This work was funded by the National Natural Science Foundation of China (NSFC) under Grant nos. 51275188 and 51121002, and
the Science and Technology Major Special Project of China under
Grant no. 2011CB706803. The authors would like to acknowledge
Mr. and Mrs. Serody for their help to check and revise the
grammatical and spelling errors in the paper.
[21]
[22]
[23]
[24]
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