Projected fringe pattern analysis by the Fourier method: Application to
vibration studies
R. Rodriguez-Vera, C. Meneses-Fabian,a J.A. Rayas, and F. Mendoza-Santoyo
Centro de Investigaciones en Optica
Loma del Bosque 115, Col. Lomas del Campestre, León, Mexico
[email protected]
a
Benemérita Universidad Autónoma de Puebla, Facultad de Ciencias Físico-Matemáticas
Av. San Claudio y 18 Sur, C.U. San Manuel, Puebla, México, [email protected]
ABSTRACT
3-D displacement field measuring by using optical methods is an attractive task, mainly in engineering structures subjected to
vibration. This paper is concern on describe a new method of phase extraction with two fringe pattern and without carried
frequency focused for an application to study of vibrations. The method is based in the analysis of the Fourier method and with
an approach of the first derivative phase function to removed ambiguity sign due to absent carried frequency. The fringe
pattern is obtained from fringe projection on a homogenous fixed-free cantilever beam undergoing to harmonic vibration. With
this technique, we can measure amplitude and frequency of the first two vibration modes of the cantilever at each camera
frame rate (33.3 ms). Theoretical analysis and experimental results are shown.
1. Introduction
Several optical techniques to study vibrations have been used amply. In general, some have been oriented to study the
vibrations of low amplitudes and high frequencies and other on the contrary, to study high amplitudes and low frequencies.[1]
For example, in middle-aged studies, it has been synchronized the acquisition the object’s image on vibratory movement by a
high-speed camera and it has been developed associated electronics to generate triggers for image of capture.[2] Also, it has
been used techniques of fringe projection time-averaged and stroboscopic illumination to study the modal patterns on plates
and complex surfaces under vibration.[2,3] As well as, it has been improved techniques to analyze and evaluate 3-D
deformation fields for different kinds of surfaces by fringe projection.[4] Recently, it has been used the digital time-averaged by
fiber optic fringe projection[5] and the stroboscopic scanning moiré projection[6] techniques to determine the amplitude and
frequency of the vibration modes. Some applications have been discussed to aeolic vibrations of aerial electricity transmission
cables.[7] Other works have employed the phase shifting of nine steps to study modal parameters of a homogeneous
cantilever beam[8]. Last techniques have been used employing triggering wardware.
This paper is focused to measure high amplitude, low frequency, and the 3-D spatial contour of natural vibration for the two
first modes on a fixed-free homogenous cantilever beam, without use electronics, neither stroboscopic techniques, and
inclusively, without requirement to synchronize the vibrating cantilever beam image. Instead, fringe projection technique is
employed in combination with of phase extraction techniques, without to use spatial phase shifting. Both Fourier method and
temporal phase shifting are used. This technique is based on the application of consecutives differences of the two camera
frame adjacent patterns projected on the cantilever beam, when it is vibrating in one of its natural modes.
2. Projected fringe pattern and the Fourier method
The intensity of a Ronchi ruling when is projected on a homogenous cantilever beam, which is subjected to a forced vibration
(Fig. 1) can be considered as an interference pattern with a carrier frequency in the horizontal direction, this can be modelled
as
I k (x, y ) = a ( x, y ) + b( x, y ) cos[φk ( x, y ) + 2πµ0 x ] ,
(1)
where the subscript k, is a positive integer that indicates the k-th captured pattern,
should be interpreted as the background intensity;
b( x, y )
I k (x, y ) , by the CCD camera; a( x, y )
is the modulation intensity;
proportional to the ruling deformation on the vibrating cantilever surface; and
µ0
φk ( x, y )
is the optical phase
is the carrier frequency, correspondent to the
spatial Ronchi ruling frequency. The carrier frequency is chosen in such away that term
2πµ0 x
will be not bigger
(x, y ) , b(x, y ) , and φk (x, y ) . This is so, because the pattern of Eq. (1) does not have a monotonous behaviour, it is
that means the comporting depend on the natural mode of the cantilever beam. Note that φk ( x, y ) is directly proportional to
than a
the ruling deformation on surface target, being major when its vibration reach to its bigger amplitude and it is annulled when
the ruling is in stationary state, or when is in its equilibrium position. In such a way, the amplitude of the vibration can be
calculated interpreting the information of the measured phase properly (see section 3 below).
y
L
x
z
α
Ronchi
Ruling
β
Homogenous
Cantilever
Projector
CCD Camera
(a)
(b)
Figure 1. (a) Experimental setup, a Ronchi ruling is projected on a homogenous fixed-free cantilever beam of length L and the image is
captured by a CCD camera. Both, projector and CDD form an angle α and an angle β with respect to the normal line to the cantilever,
respectively. (b) Top view of the cantilever beam when it is vibrating at the mode 3.
Equation (1) can be written in a more convenient way as:
I k ( x, y ) = a( x, y ) + ck ( x, y )ei 2πµ0 x + ck * ( x, y )e −i 2πµ0 x ,
where i is the imaginary unit given by
(2)
− 1 , and the symbol * denoted the complex conjugated of c k ( x, y ) , which is given
by
1
ck ( x, y ) = b( x, y )e iφk ( x , y ) .
2
(3)
From the principle of the Fourier method [9] and with the idea of phase extraction introduced by Kreis [10] and the others
workers [11-16], the term
ck ( x, y )ei 2πµ0 x
can be separated from
I k ( x, y )
propose two consecutive patterns, k and k+1, calculating the quotient ∆ck
in Eq. (2). Considering this possible fact, we
(x, y ) = ck +1 (x, y )ei 2πµ x / ck (x, y )ei 2πµ x ,
0
0
∆ck ( x, y ) = ei∆φk ( x , y ) .
(4)
(
)
Eliminating in this way, the corresponding term to the modulation intensity b x, y and the corresponding term to the carrier
frequency or, in other words, to the corresponding information to the frequency of the used Ronchi ruling, giving as a result a
∆φk ( x, y ) = φk +1 ( x, y ) − φk ( x, y ) of the pattern
k+1 and the pattern k, it is to say, to the difference of deformation ruling at the instant t k +1 and t k , where t k is the capture
complex exponential function with relative exponent to phase difference
time of the pattern k-th.
Note that, it is chose the term
ck ( x, y )ei 2πµ0 x
instead of the term c k
(x, y ) , as is usually used, due to that is easiest to realize
the filtering in the frequency space of the fringe pattern. Moreover, with this election, the operation indicated in Eq. (4),
possible ambiguity when locating the filtering component in the centre coordinates is removed. Additionally, implicitly is
removed the carrier frequency, and also, with this operation is possible removed problems as the detector non-linearity and
the non-uniformed illumination. Then, the phase difference can be calculated choosing the imaginary part of the logarithm of
the equation (4)
∆φk ( x, y ) = Im{log[∆ck ( x, y )]}.
Equation (5) shows the principal value wrapping between
2π
(5)
radians. Though, due to that it can be chose properly, without
missing generality, the Ronchi ruling period, such that Eq. (5) should be minor or equal than 2π radians, then Eq. (5) can be
considered as the equivalent measure to the 3-D spatial contour field of the vibration mode without necessity of the
unwrapping phase, in this way can be determined the vibration mode on the fixed-free homogenous cantilever beam.
3. Amplitude and frequency measuring theory
In order to measure the vibration amplitude and frequency, we assume that the temporal part of the vibration mode has a
harmonic behaviour, in such a way that a point (x, y) on the cantilever surface oscillates according to,
where
z k ( x, y )
zk ( x, y ) = φk ( x, y ) = z (x, y, t k ) = z A ( x, y )sin (2πµ s t k + φ0 ) ,
is the observable amplitude in the instant
t k , z A ( x, y ) is the maximum amplitude, µ s is the oscillation
frequency of the temporary part of the stationary wave on the cantilever, and
instant
(6)
φ 0 is the initial phase accumulated in the
t 0 . If we calculate the difference between the patterns in the consecutive capture instants t k and t k +1 , assuming
that the period of CCD camera capture is τ c
= t k +1 − t k
and therefore t k
= kτ c , we have
φ ⎞
⎛φ ⎞ ⎛
∆z k ( x, y ) = z k +1 ( x, y ) − zk ( x, y ) = ∆φk ( x, y ) = 2 z A (x, y )sin ⎜ 1 ⎟ cos⎜ φ0 + φk + 1 ⎟ ,
2⎠
⎝2⎠ ⎝
(7)
where φ k
= 2πµ s t k = 2kπµ sτ c ,
and
φ1 = 2πµ s t1 = 2πµ sτ c .
It is easy show for induction that
φ k +l = φ k + φl ,
for l
non-negative integer. Applying the previous properties and subtracting two adjacent patterns consecutively for k+l, with
l = 0,1,2,... we have without loss of generality
φ ⎤
⎛φ ⎞ ⎡
∆z k + l ( x, y ) = z k +l +1 ( x, y ) − z k +l ( x, y ) = 2 z A ( x, y )sin ⎜ 1 ⎟ cos ⎢φ 0 + φ k + (2l + 1) 1 ⎥ .
2⎦
⎝2⎠ ⎣
At this point,
φ 0 + φ k can
(8)
be considered as a single variable because remains constant for all permitted value of l. As a
consequence, equation (8) has three variables, the amplitude z A
(x, y ) , the phase constant φ0 + φ k , and the dependent
phase of the vibration frequency φ1 . This way, to solve Eq. (8) is necessary to have a minimum of 3 consecutive values of l to
generate 3 equations and in this way to calculate the variables of interest, this mean, the amplitude and the frequency.
Therefore, for
and
l = 0,1,2 three
equations ∆z k
z A ( x, y ) , we have respectively,
(x, y ) , ∆z k +1 (x, y ) , and
∆z k + 2 ( x , y )
are generated, and solving for cos φ1
∆z k ( x , y ) + ∆z k + 2 ( x , y )
,
∆z k +1 ( x, y )
(9)
∆z k +1 ( x, y ) − ∆z k ( x, y )∆z k + 2 ( x, y )
1
.
1 − cos(φ1 )
2[1 + cos(φ1 )]
(10)
cos(φ1 ) =
and
z A ( x, y ) =
Solving
φ1
from Eq. (9) and taking in account that φ1
µs =
= 2πµ sτ c , the cantilever beam frequency, µ s , can be obtained,
⎡ ∆z ( x, y ) + ∆z k + 2 ( x, y ) ⎤
φ1
1
arccos ⎢ k
=
⎥.
2πτ c 2πτ c
∆z k +1 ( x, y )
⎣
⎦
(11)
This way, Eqs. (10) and (11) are the expressions that indicate the necessary calculations to obtain vibration amplitude and
frequency present on the cantilever beam.
4. Experimental results
3
A 100 lines/inch Ronchi ruling was projected on a fixed-free plane cantilever beam of dimensions 245 X 40 X 0.68 mm . With a
mechanical shaker a punctual harmonic force placed to 24 mm (and in centre) from the fixed end of the cantilever was used.
Projector and camera optical axes made angles respect to the normal of the cantilever were α = 0º and β = 44º, respectively.
Image of the Fig. 1b depicts a top view of the cantilever beam when it is vibrating at the mode 3. Images observed in Fig. 2 are
those in which a Ronchi ruling is projected on the fixed-free cantilever beam for different condition of moving. For example,
Fig. 2a shows the projection of the ruling when the cantilever is in stationary state. Other images of the same figure 2, (b) and
(c), show the deformation of the ruling when is vibrating in the modes 1 and 2, respectively. We have looked at the Fig. 1b and
Fig. 2 that the configuration of the fringes on the bar changes according to the deformation that suffer when it is vibrating, by
this reason; out-of-plane quantifications give us its 3-D contour geometric field of the cantilever.
(a)
(b)
(c)
Figure 2. Projected images of the Ronchi ruling on a homogenous fixed-free cantilever beam when it is in (a) stationary state, (b) mode 1, and
(c) mode 2, respectively.
(a)
(b)
Figure 3. Geometric form optical phase of the vibrations modes I and 2 on the homogenous cantilever, corresponding to the rulings shown in
the Fig. 2(b) and 2(c), respectively.
x(mm)
y
(mm)
z
(mm)
(a)
(b)
x(mm)
y(mm)
z (mm)
(c)
(d)
Figure 4. Geometric form, in real coordinates, of the two first vibrations modes of the homogenous fixed-free cantilever. The vibration
frequencies were 1.8 Hz and 9.9, respectively, measured by means an oscilloscope.
Figure 3 shows the phase evaluation of the Ronchi ruling pattern projected on surface of the homogenous cantilever,
corresponding to images shown in the Fig. 2(b) and 2(c), respectively, which were captured at the instant t k . As a reference,
the ruling of the Fig. 2(a) captured at the time
t0
was used, in other words, when the cantilever was in state stationary. Each
projected fringe pattern is considered as an interferogram image modelled by the equation (1). The evaluated phase
∆φk ( x, y ) = φk ( x, y )
is directly pertaining to the contour of the cantilever surface at the instant of the image capture by the
camera CCD. The surfaces shown are the out-of-plane deformation measured with respected to the reference. We ca see the
respective optical “unwrapped” phase below of each 3D graph for the modes 1 and 2.
The Fig. 4 shows the surfaces for the same modes of the Fig. 3(a) and 3(b), but in this case the images were used as selfreferences, which were captured at the instant
tk
and these were compared with other images captured at the following
instant t k +1 (images are not shown), for a such mode, respectively, such as is shown in the method exposed in the section 2,
see Eqs. (1)-(5). The frequencies were measured by means an oscilloscope, throwing the values 1.8 Hz and 9.9, respectively.
5. Conclusion and future work
By means of the use of fringe projection was possible to demonstrate contour, amplitude, and frequency for the first two
modes of a homogeneous fixed-free cantilever beam subject to harmonic vibration. The described method has the virtue of to
decrease the noise level of the no uniformity of the illumination, also decrease the no linearity of the detector, and remove the
possible introduction lineal phase when the digital filtering on the fringe patter frequency plane is achieved. Moreover, for our
method, it is not necessary to employ spatial phase shifting. We are working massively in order to measure amplitude and
frequency of the homogenous cantilever when it is vibrating and we have been thought to use the temporal phase shifting
technique. Also, the use of a tuneable high-speed camera will be used in order to determine amplitude and frequency for
modes higher to 2.
Acknowledgments.
This work was partially supported by CONCYTEG (Guanajuato, México) under grant 06-04-K117-36.
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