Dispersion relationship extraction using a smooth
complex orthogonal decomposition Rayleigh quotient
R A Caldwell Jr1 , B F Feeny2
1
2
Merrimack College, Department of Mechanical Engineering, North Andover, MA, USA
Michigan State University, Department of Mechanical Engineering, East Lansing, MI, USA
E-mail: [email protected]
Abstract. The geometric dispersion relationships of uniform structures, specifically a
simulated beam, experimental beam, and a simulated mass chain, were extracted using assumed
modes and Rayleigh quotients. The Rayleigh quotients were set up based on the eigenvalue
problem of the smooth complex orthogonal decomposition (SCOD). SCOD is in the family
of decomposition methods born from proper orthogonal decomposition, and is well suited for
extracting frequencies and shapes of underlying characteristic wave patterns. The frequencies of
the traveling waves are the square roots of the eigenvalues of the generalized SCOD eigenvalue
problem of matrices R and S, where R and S are complex Hermitian correlation matrices created
from measured displacements and velocities, or velocities and accelerations. The mode shape
matrix is obtained from the inverse complex conjugate transpose of the eigenvector matrix. The
assumed modes were chosen as harmonic traveling wave modes with assumed wavenumbers, k,
which was then discretized to match the sensor arrangement in the experiment or simulation.
These assumed-mode shapes, along with the correlation matrices R and S from the smooth
orthogonal decomposition, were used to compute the corresponding Rayleigh quotient for each
mode shape. The frequencies and wavenumbers were fitted to the theoretical dispersion curve for
each system, namely for the beams and for the mass chain. The use of assumed modes in the
Rayleigh quotient was shown to accurately extract the dispersion relation in each example.
Additionally, it was found that the initial measurement state offered the best result. For
example, if sensing the beam with accelerometers, it was best to apply SCOD on accelerations
and velocities rather than velocities and displacements.
1. Introduction
During the middle of the 20th century various researchers independently developed proper orthogonal decomposition (POD) [1, 2, 3, 4] which was applied to statistics and turbulent fluid
flows. POD later caught the interest of structural engineers and was used to extract mode shapes
from vibrating structures [5, 6, 7, 8]. Other generalizations of POD have been developed, such
as the smooth orthogonal decomposition (SOD) for finding the natural frequencies and linear
normal modes [9, 10], the Ibrahim time domain method [11, 12] for modal analysis, and the
state variable modal decomposition (SVMD) [13] for finding natural frequencies, normal modes,
and (in theory) modal damping.
Soon after POD was expanded for extracting traveling wave modes using complex orthogonal
decomposition [14]. Complex orthogonal decomposition (COD) [15] can extract nonsynchronous
and standing waves of vibrating structures, and when waves are traveling through an elastic
medium COD can be used to extract the wavenumbers, frequencies, and the dispersion relationship of the waves [16]. In the application of COD the wavenumber is extracted from the
eigenvector, and the frequency is extracted from the modal coordinate.
A new generalization of COD and SOD called the smooth complex orthogonal decomposition
(SCOD) was outlined in [17] . It was shown that with SCOD the wavenumber and frequency
can be extracted from the eigenvector and eigenvalue of the SCOD eigenvalue problem (EVP).
A mathematical overview of SCOD is covered in next section. This paper will focus on
the extraction of the geometric dispersion relationship using Rayleigh quotients and SCOD
assumed eigenvectors. This method will be applied to a simulated infinite beam in Section 2,
experimentally to a rectangular beam in Section 3, and to a simulated mass chain Section 4.
1.1. Mathematical Overview of Smooth Complex Orthogonal Decomposition
In this section we review the framework for SCOD in the time domain for distributed parameter
systems [17]. SCOD is based on SOD, which has been tied to modal analysis of discrete and
continuous systems with real modal properties [9, 10, 16]. Here we will review the connection
between the modal properties of a linear self-adjoint partial differential equation (PDE) for a
one-dimensional medium, and those of the SCOD eigenvalue problem [17].
SCOD generalizes SOD, which is defined as the eigenvalue problem λRr ψ = Sr ψ, where
Rr = Xr XTr /N and Sr = Vr VrT /N , and where Xr and Vr are ensemble matrices of sampled
displacements and velocities, wherein each row of the ensemble matrix is a sensed real quantity,
and each column is a time sample. It has been shown [9, 10] that for undamped (or lightly
damped) free (or randomly excited) vibrations of a symmetric system represented by
Mẍ + Kx = 0
(1)
the natural modal matrix is approximated by Φ = Ψ−T , where columns of Ψ are the eigenvectors
ψ. In real applications, some of the modal vectors produced by SOD are spurious [13, 16].
SCOD [17] is applied to ensemble matrices generated from complex analytic signals, and
can produce complex modes representing normal traveling wave modes. The SCOD generalized
eigenvalue problem is
λRψ = Sψ.
(2)
where R and S are
R = XX̄T /N
S = VV̄T /N ;
(3)
(4)
X and V are analytic measurement ensembles and the overbar represents the complex conjugate.
It was shown in [17] that SCOD can be applied to discretized continua modeled by a partial
differential equation (PDE) such as
m(x)ü + Lu = 0
(5)
where m(x) is the mass per unit length, u = u(x, t) is a function of space x and time t, and L
is a linear operator. The associated complex modes can be considered if we assume u(x, t) is
complex and analytic with the form
u(x, t) = eiωt (c(x) + id(x)) = eiωt φ(x).
Taking the partial derivative of u(x, t) with respect to t twice,
∂2
{u(x, t)} = −ω 2 e iωt φ(x ),
∂t2
and substituting it into Eqn. (5), leads to the continuous eigenvalue problem
Lφ(x) = ω 2 m(x)φ(x),
(6)
with appropriate boundary conditions, where the eigenfunction φ(x) is a complex modal function.
The discretized SCOD problem is applied to the continuous vibration problem as follows.
Let u(x, t) and v(x, t) be complex analytic displacement and velocity time signals evaluated at
point x on the beam and sampled at times tk = k∆t for k = 1, · · · , N . We define
N
1 X
u(xi , tk )ū(xj , tk )
Rij = R(xi , yj ) =
N
Sij = S(xi , yj ) =
1
N
k=1
N
X
v(xi , tk )v̄(xj , tk )
k=1
The matrices of elements Rij and Sij can be expressed in matrix form, in terms of the ensemble
matrices U and V with elements u(xi , tk ) and v(xi , tk ), as
1
UŪT
N
1
S∼
= VV̄T
N
R∼
=
(7a)
(7b)
In application, the discretized SCOD eigenvectors from equation (2) can be used to
approximate discretizations of continuous complex functions ψ(x), and then related to the
complex modal functions φ(x) through the mass distibution via
ψ(x) = m(x)φ(x).
(8)
Thus, the SCOD produces eigenvector ψ from which discretized modal functions can be
approximated as φ = M−1 ψ, where M is an appropriate discretization of the mass distribution
m(x). For uniform structures, we can write φ = ψ, disregarding normalization conventions.
The connection between SCOD an a lumped parameter model carries over directly in terms
of eigenvectors and modal vectors, without the discretization approximations.
Here is a useful tidbit. Taking the time derivative of equation (1) and (5) yields
Mv̈ + Kv = 0
(9)
m(x)v̈ + Lv = 0
(10)
and
where v = ẋ and v(x, t) = u̇(x, t). Thus, in principle, defining SCOD based on
R = VV̄T /N
S = AĀT /N,
(11)
(12)
where A is an acceleration ensemble, will tie SCOD (and similarly SOD) to the modal properties
of systems modeled by equation (9) or (10), which are the same modal properties of systems (1)
or (5) that are sought! This can be useful when dealing with experimental measurement data.
1.2. SCOD Rayleigh Quotient
A challenge mentioned with applying SCOD or SOD is that it produces spurious eigenvectors,
and some insight may be needed to distinguish them from true eigenvectors. Here we use
the SCOD Rayleigh quotient as a means for estimating frequencies when we have meaningful
approximations to modal functions.
The SCOD Rayleigh quotient is based on the SCOD EVP, and is defined as
RQ(λ) =
Rψ a
ψH
a
ψH
Sψ a
a
.
(13)
R and S are the correlation matrices from SCOD, ψ a are columns of a square matrix Ψ = Φ−H ,
where columns of Φ are φa , and are assumed modes with elements φaj , for this work to be
chosen in the form φa j = e ikxj . The value of wave number k can be defined by the user.
Based on equation (2), the Rayleigh quotient will approximate an eigenvalue in the eigenspace
of R and S when φa is used as an approximate modal vector (or discretized modal function).
SCOD correlation matrices were used because the square root of the SCOD eigenvalues are the
frequencies of the traveling waves. The SOD Rayleigh quotient was considered in a slightly
different context by Przekop et al. [18].
2. Simulated Infinite Euler-Bernoulli Beam
The response of an infinite Euler-Bernoulli beam was written about in the works of [19, 14].
The initial conditions for the beam are
2
−x
y(x, 0) = f0 exp
4b20
ẏ(x, 0) = 0
which is a Gaussian distribution on the initial displacement, where f0 = 1 mm and b0 = 0.01 m.
Beam parameters are listed in Table 1. The analytical solution to the Euler-Bernoulli beam
with the initial conditions given above is [19]
x 2 b2
− 4 02 2
atx2
1
f0
at
4(b0 +a t )
y(x, t) =
e
cos
− arctan 2
,
(14)
4
4
2
2
1/4
2
2
4(b0 + a t ) 2
b0
(1 + a t /b0 )
where a = EI/ρA, E is Young’s modulus, I is the second moment of area of the cross section,
of area A, about the neutral axis, and ρ is the mass density.
Simulated Beam Dimensions
Length
Cross section
Density
Modulus of elasticity
∞
0.069 m × 0.0045 m
7870 kg/m3
200 GPa
Table 1. Simulated Beam Dimensions
The sensor network on the simulated beam was designed to be the same as the beam
experiment, with identical sensor spacing of ∆x = 0.0461 m over a length of L = 1.42875 m,
where sensor 1 is located at the origin such that point p on the beam is located at p∆x. However,
the sample rate in the simulation was 100,000 Hz, whereas in the beam experiment the sampling
rate was 25,000 Hz.
The displacements and velocities from the simulated beam are shown in figure 1 and figure 2
respectively for selected points on the simulated beam.
Figure 1. Displacement y(x, t) of the
simulated beam at measurement points 1
(——), 16 (- - - -), and 31 (light · · · · · ·).
Figure 2.
Velocity ẏ(x, t) of the
simulated beam at measurement points 1
(——), 16 (- - - -), and 31 ( light · · · · · ·).
2.1. Creating Analytic Ensemble Matrices
After the simulation is completed an ensemble matrix of displacements is formed and used
to derive the velocity ensemble V. The displacement time history for each simulated sensor
is recorded in a row vector, yi such that the first entry is the first time step and the last
entry is the last time step. Therefore yi ∈ R1×N , where N is the number of time steps and
yi = [y(xi , 0), y(xi , ∆t), · · · , y(xi , (N − 1)∆t)]. Y is formed by stacking each row in sensor
order to create Y ∈ RM ×N , where M is the number of sensors. The mean of each row was
subtracted from each element of that row to create Y. Next, the central difference was applied
to the rows of this matrix to compute the velocity ensemble, V̂. Once again the mean of each
row was computed and then subtracted from the element of its respective row, to create V. It is
worthy to mention that notation for the displacement and velocity ensembles matrix Y and V
are used to differentiate displacement and velocity, and not that V is computed from Y. Next,
these two matrices are converted into complex analytic ensemble matrices.
An analytic signal is a signal where the negative frequency content is equal to zero. There
several ways to create an analytic data, one such way is to use the Hilbert transform. Another
way, is to take the Fourier transform of the time signal, zero out the negative frequency
components and multiply the positive components by two, and then take the inverse Fourier
transform. Here the latter is used to created two analytic ensemble matrices Y :→ Z and
V :→ Zv , where Z, Zv ∈ CM×N . For example, to compute Z, first take the fast Fourier
e The
transform (FFT) of each row of Y from the time domain to frequency domain to get Y.
FFT ensemble in the discrete frequency domain can be defined to roughly cover the spectrum
from approximately −ωny to ωny , as indexed from 1 to N/2 (for example in the case that N is
e is the FFT of a sensor signal and each column is a frequency sample, and
even). Each row of Y
e are Yeij with i = 1, · · · , M and j = −( N − 1), · · · , N/2. Then the negative
the elements of Y
2
e
spectrum is nullified, and the positive spectrum is doubled, such that the elements of Zeij of Z
are
(
if j < 0
eij = 0
Z
(15)
e
2Yij if j ≥ 0.
e
The complex analytic ensemble is obtained, using the inverse FFT (IFFT), as Z = IFFT(Z).
2.2. SCOD Correlation Matrices
The two correlation matrices for the generalized eigenvalue problem are are determined using
R = ZZH
S = Zv ZH
v ,
(16)
(17)
where H is the Hermitian or complex conjugate transpose. From equations (16) and (17) the
generalized eigenvalue problem is posed as
RΨΛ = SΨ.
(18)
The Λ matrix is a diagonal matrix where λii = ωi2 and Ψ is a eigenvector matrix, from which
we can approximate the traveling mode shape φi from the columns of Φ where
Φ = Ψ−H .
(19)
Applying SCOD, we can extract the frequencies, ω, and wavenumbers, k, from Λ and Φ
respectively. The theoretical relationship of an Euler-Bernoulli beam has the form ω = ak 2 ,
where a = 6.55 m2 /rad computed from the beam parameters.
2.3. Dispersion Extraction using Rayleigh Quotients
The modes are assumed to have the form of φ = exp(ikx), where k is initially taken from the
extracted k above and x is the discretized to match the sensor network used in the beam experiment.
When using the SCOD Rayleigh quotient with assumed modes, the wavenumbers were chosen
to be identical to those extracted by SCOD. The extracted dispersion relationship is shown in
Figure 3. The residuals for both SCOD extracted parameters and Rayleigh quotient extracted
parameters were similar. When the data was fitted to the theoretical dispersion relationship
using noise free displacements and velocities atheory = 6.55, aSCOD = 6.57, and aRQ = 6.91
rad/m2 .
Additionally, noise was added to the simulations at two different levels which resulted in SNR
ratio of 43 and 18.4. At SN R = 43, aSCOD = 6.55 and aRQ = 6.81 which had an error of 0.00%
and 3.97%. For SN R = 13.13 aSCOD = 5.28 and aRQ = 7.41 which had an error of −19.39%
and 13.13%. However, the relative error does not indicate that there are zones where SCOD has
a smaller residual in the neighborhood than Rayleigh quotient method and vise versa.
Figure 3. Analytical dispersion relationship for the theoretical Euler-Bernoulli beam (——).
Extracted wavenumbers and frequencies using SCOD ( ◦ ) and SCOD Rayleigh quotient (u
t).
SCOD residual (- - - -) and SCOD Rayleigh quotient residual (♦) .
Figure 4. Experimental setup showing beam. The left end is suspended by a soft spring and
the other end is buried in sand.
3. Experimental Beam
The test specimen was a rectangular steel beam with a constant cross section. In an effort to
emulate a semi-infinite free beam, the beam was suspended with elastic cords, such that one end
was free, and the other was embedded in a sand pit, as done by Önsay and Haddow [20]. The
sand absorbs the wave and prevents reflections off the buried end of the beam. A schematic of
the experiment is shown in Figure 4. In this case, the sandbox was filled with unpacked coarse
sand. The beam had a 0.0045 m × 0.0698 m cross section. The length of the beam was 2.04 m.
The unburied part of the beam measured 1.43 m, such that approximately 0.609 m was buried
in the sand. The density and modulus of elasticity of the beam were 7870 kg/m3 and 200 GPa,
respectively, based on published values for steel. From the geometry and material properties of
the beam, the theoretical equation for the dispersion ratio, a, is a = ω 2 k and is a = 6.55 m2 /s.
The beam was sensed with 31 accelerometers placed at a distance ∆x = 0.0458 m apart over a
distance of L = 1.4198 m and was sampled at f = 25, 000 Hz using a National Instrument’s PXI
data acquisition system. The beam was struck with a PCB model 086C80 mini impact hammer,
lightly such that bending deflections were not visible to the naked eye.
3.1. Data Processing
The data included 100 samples before the hammer impact and 300 samples after the impact
for a total of 400 samples. We aim to integrate the acceleration data to obtain velocity and
displacement. Numerical integration can be problematic because an integration constant is introduced, and low-frequency noise is amplified, which can cause the integrated signal to drift.
To reduce these effects, the following steps were taken. First, the data was filtered forward
and backward with a high-pass filter with a cutoff frequency of 100 Hz. Second, the mean of
each sensor’s time history was subtracted from its samples using MATLAB’s "detrend" with
a "constant" modifier. Third, any linear trends were removed using the same command as
above with a "linear" modifier. Fourth, the signals were translated on the time axis such that
the sample before the hammer impact had a time and force value of zero. The first and last
100 samples were truncated leaving the start of the impact plus 200 samples.
Next, to get velocities, the signals were numerically integrated using the "cumtrapz" command. The means were subtracted from the velocities, which were then high pass filtered, and
integrated once more to get displacements. The means were subtracted from displacements and
the displacements were filtered for a final time. These N = 200 samples of displacements were
then used for COD.
2π
= 4.4
L
rad/m. The maximum detectable wavenumber is defined by a spatial Nyquist criterion as
π
kmaxs =
= 68.6 rad/m. Based on the sampling rate f = 25, 000 Hz and time record of
∆x
N/f = 0.008 s, the maximum detectable frequency is fmaxt = 12, 500 Hz and the minimum
detectable frequency is fmint = 125 Hz.
The minimum detectable wavenumber defined by the span of sensors, L, is kmins =
p
Making use of the theoretical dispersion relation ω = ak 2 , or k = 2πf /a, we find that
the temporal sampling parameters correspond to theoretical wavenumber limits of kmaxt =
109 rad/m and kmint = 10.9 rad/m. The approximate total wavenumber limits are thus
kmax = min(kmaxs , kmaxt ) = 68.6 rad/m, and kmin = max(kmins , kmint ) = 10.9 rad/m. Thus,
the upper limit on extractable wavenumbers (and hence frequencies) is determined by the
spatial sampling interval Nyquist criterion, and the lower limit on extractable wavenumbers
is determined by the length of the time record.
3.2. Experimental Results using Rayleigh Quotients
In this application, SCOD involves signals that resulted from the integration of accelerometers.
When we applied the SCOD Rayleigh quotient, results were better when using signals “close”
to the original measurement. As such, we present results for which acceleration and velocity
ensembles A and V were used to build S and R in the spirit of equation (10). Furthermore, we
used a spatial distribution of sensors that was a few sensors away from the impact location,
as the impact produced a local extreme variation from sensor to sensor, and perhaps was
not represented well when assumed harmonic modes were applied. That is, we processed the
correlation matrices S and R while omitting the first two sensors next to the impact location.
Assumed modes were created using wavenumbers that range from 5 to 40 rad/m in increments
of 0.05 rad/m. Assumed modes were computed using
φa = exp(ika x).
(20)
The theoretical dispersion coefficient for an Euler-Bernoulli beam is atheory = 6.548 m2 /rad
the coefficient for the experimental beam was aexperiment = 6.792 m2 /rad and using assumed
modes with Rayleigh quotient aRQ = 6.558 m2 /rad. Figure 3.2 shows the extracted dispersion
relationship using SCOD and Rayleigh quotients.
Figure 5. Analytical dispersion relationship for the theoretical Euler-Bernoulli beam (——).
Extracted wavenumbers and frequencies using SCOD applied to the experimental beam ( ◦ )
and SCOD Rayleigh quotient using assumed modes and SCOD correlation matrices (u
t). SCOD
experimental beam residual (- - - -) and SCOD Rayleigh quotient using assumed modes and
SCOD correlation matrices (♦).
4. Simulated Mass Chain
4.1. Analytical Model for an Infinite Periodic Chain
The wave behavior in an infinite uniform linear mass-spring chain with stiffness α has been
studied in detail along with a nonlinear chain in [21, 22]. Here, we show the derivation for a
linear chain shown in Figure 6.
The mass-spring chain is arranged in a fashion such that each mass is separated by a distance
h from its nearest neighbor. h is also the relaxed length of each spring before any deformation
occurs. The relaxed position and displacement of mass mj are denoted by xj and uj respectively.
We use the assumption that all the masses are equal (mj = m) and only the nearest neighbors
have direct effects on each other. As such, we consider a linear spring force relative to the
equilibrium state. The equations of motion (EOM) in physical coordinates can then be written
as
müj = α̃ [(uj+1 − uj ) − (uj − uj−1 )]
(21)
for j = · · · , −2, −1, 0, 1, 2, · · · . Letting α =
α̃
m
we get
üj = α [(uj+1 − uj ) − (uj − j − uj−1 )]
(22)
The non-dimensionalization is done by assuming x = hx̃ , which results in xj = j and
xj±1 = j ± 1. The non-dimensional wave number is denoted by µ. We assume a traveling
Figure 6. Infinite mass chain. The relaxed position and displacement of mass mj are denoted
by x̃j and uj respectively. In this case, the springs are linear with relaxed length h.
dispersive wave solution at frequency ω and wave number µ and plug into the equations of
(23)
uj (t) = Aei(µj−ωt) + Āe−i(µj−ωt) .
motion. Let
Substituting uj±1 = e±iµ Aei(µj−ωt) into 2the equations and balancing leads to
ω = 2α(1 − cos µ)
(24)
for the required relationship between ω and µ. This is the dispersion relationship for the linear
chain. Thus for a linear chain, the dispersion is given as
p
(25)
ω = 2α(1 − cos µ).
In our simulation, we used m = 1 and α = 1. The simulated 250 mass mass-spring chain
was excited with a unit impulse on the left hand side. The disturbance propagated toward the
right. The response of the first 100 masses were recorded as the disturbance passed the 100th
mass but before reflections traveled backward to the 100th mass. Essentially, the time record
was truncated at a time such that no reflections were recorded. The means were subtracted
from the displacements of each mass.
The displacements for four of the masses are shown in Figure 7. It can be seen here that the
lower frequency waves travel faster than the higher frequency waves, which is consistent with
the predicted dispersion behavior of a linear uniform chain. Velocities were also captured for
use in SCOD.
Figure 8 shows the theoretical dispersion relationship, and SCOD and Rayleigh quotient
extracted relationships. From the data in the plots, the estimated parameters were αSCOD =
0.81 and αRQ = 0.98 which have errors of 19% and 2%. In this example the Rayleigh quotient
was able to capture the dispersion relationship over a much greater range than the SCOD modal
extractions.
5. Summary
5.1. Simulated Beam
The response of a infinite beam was simulated from rest and a initial displacement profile that
was equal to a Gaussian distribution. The displacements and velocity from the right half of
beam x ≥ 0 were used to created two measurement ensembles, X and V, which were in turn
used to compute the correlation matrices R and S. Next, the assumed modes were computed
using φa = exp (ika x). The frequencies were computed using the assumed modes, the correlation
matrices, and the generalized Rayleigh quotient, where
Figure 7. Displacements for masses 1, 31, 61, and 91.
Figure 8. Analytical dispersion relationship for the linear mass chain (——). Extracted
wavenumbers and frequencies using SCOD applied to the mass chain ( ◦ ) and SCOD Rayleigh
quotient using assumed modes and SCOD correlation matrices (u
t). SCOD residual (♦) and
SCOD Rayleigh quotient using assumed modes and SCOD correlation matrices (- - - -).
ω 2 = R(φa , S, R) =
φH
a Sφa
H
φa Rφa
(26)
The extracted ω and assumed k were fitted to the theoretical dispersion relation ω = ak 2 .
Including the simulated beam, this methodology was applied to an experimental beam and a
simulated discrete mass chain.
When applying this method to the simulated beam atheory = 6.55rad/m2 , aSCOD = 6.57,
and aRQ. = 6.91 using identical wavenumbers, SCOD was computed with the displacement and
velocity ensembles. The effect of added noise was also demonstrated in example calculations.
5.2. Experimental Beam
When these methods were applied to the experimental beam aSCOD = 6.78 and aRQ = 6.56
which had an error of 3.51% and 0.12%. The reason for the performance difference between
the simulated beam and the experimental beam is the choice in wavenumber used in the experiment. Also, it was found, when ensembles closer the measurement source are used the results
are slightly better. For example, since accelerometers were used, then the ensemble matrices
used are V and A.
5.3. Mass Chain
Finally when applied topa simulated mass chain with linear springs, the dispersion relationship
will have the form ω = 2α(1 − cos(µ). The frequencies and wavenumbers were extracted using
SCOD and Rayleigh quotient using assumed modes and applied to the theoretical dispersion
relationship. The range of extracted wavenumbers and frequencies are much larger for Rayleigh
quotient with assumed modes as compared to SCOD. This is because assumed modes will have
better structure than the SCOD derived modes.
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