Spatial and temporal study of a mechanical and harmonic vibration by high speed optical interferometry

International Journal of JOURNAL Engineering and Technology (IJMET), ISSN AND –
INTERNATIONAL Mechanical OF MECHANICAL ENGINEERING 0976
6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue(IJMET) (2012) © IAEME
TECHNOLOGY 3, Sep- Dec

ISSN 0976 – 6340 (Print)
ISSN 0976 – 6359 (Online) IJMET
Volume 3, Issue 3, Septmebr - December (2012), pp. 96-106
© IAEME: www.iaeme.com/ijmet.html
Journal Impact Factor (2012): 3.8071 (Calculated by GISI) ©IAEME
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SPATIAL AND TEMPORAL STUDY OF A MECHANICAL AND
HARMONIC VIBRATION BY HIGH SPEED OPTICAL
INTERFEROMETRY

David Asael Gutiérrez Hernández*1, Carlos Pérez López1, Fernando Mendoza Santoyo1, Juan
Arturo Aranda Ruiz2

1
Centro de Investigaciones en Óptica A.C. Lomas del Bosque 115. Col. Lomas del Campestre.
León, Guanajuato, 37150, México.
2
Unidad Profesional Interdisciplinaria de Ingeniería campus Guanajuato. Instituto Politécnico
Nacional. Av. Mineral de Valenciana No. 200 Col. Fracc. Industrial Puerto Interior, Silao de la
Victoria, Guanajuato, 36275, México.
*
davidGH@cio.mx

ABSTRACT

Optical techniques for vibration measurements have been of a big importance in last year’s for
industrial applications. By the apparition of high speed cameras, these techniques have been
improved. This work presents a vibration study by means of high speed optical interferometer
which consists, mainly, in an electronic speckle pattern interferometry system improved with a
continuous laser beam and a CMOS high speed camera that allows the analysis of a complete
evolution of the vibration on time and space. The camera is set up at 4000 frames per second to
record the evolution of the free vibration of the object under study which is a metallic rectangular
membrane clamped in its four sides. An experimental analysis of the temporal and spatial
vibration evolution of the membrane is reported. An algorithm for phase extraction of closed
fringes patterns is also used to process and get the results. The resulted graphics shown the
spatial evolution from the minimum to the maximum deformation according to the mechanical
properties of the object and also shows the recovered time function of the vibration induced to
the plate.

96

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEME

1. INTRODUCTION

The study of vibrations has been of great interest in many applications. Many techniques for
measuring vibrations have been developed along the years and some of them have given good
results in industrial applications; this is the case of full-field vibration analysis. For some cases it
is necessary to measure dynamic effects on real time, non-contact optical techniques for
vibration measurement have very well supported the solution to this requirement [1, 2, 3, 4, 5].

Optical interferometry, such as speckle interferometry or normally called electronic speckle
pattern interferometry (ESPI) has been used in dynamic measurements because it can easily
realize a noncontact, full-field and high sensitive real-time measurements [6, 7]. However, this
technique requires of the evaluation of the correlation fringes by a consuming time image
processing in order to quantitative obtain the deformation value.

As a solution to this requirement, phase-shifting speckle pattern interferometry technique was
introduced to acquire the qualitative displacement measurement [8, 9, 10]. This technique
requires either of some frames interferograms taken at the same time or of a well-controlled
phase shifter, the use of this technique in difficult environments results complex.

A modification of the ESPI is done by introducing a carrier pattern into the interference field.
This technique requires only one frame interferograms processed by the Fourier transform
technique to calculate the whole-field phase map.

All these modifications to the ESPI technique require of taking information before and after
deformation to ensure a measure of the displacement done because of the vibration.

In this work a high speed CMOS camera working at 4000 fps modifies a traditional ESPI system
for measuring, in a temporal and spatial way, the absolute evolution of a harmonic vibration over
a four side rectangular clamped plate. To do this experiment it was not needed of any kind of
synchronization of the system, it was not necessary to introduce a phase shifter and was not
necessary to introduce a carrier pattern into the interference field. By doing a computer
processing, the information can be manipulated to obtain the evolution of an unknown vibration.
It is important to mention that a continuous laser beam is used.

2. HIGH SPEED ESPI THEORETICAL ANALYSIS

In this technique, the addition of the reference beam, I R and the object beam, I O can be written
as

I 1 = I O + I R + 2 I O I R cos(Ψ ) (1)

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where Ψ is the subtraction of the phase of the wavefront of the object beam, φ O , and the
reference beam, φ R .

Once the surface of the object is deformed, the phase of I 1 changes by ∆φ , and thus the sum of
two beams after deformation, I 2 , can be written as

I 2 = I O + I R + 2 I O I R cos(Ψ + ∆φ ) (2)

The subtraction of intensities of the input images I 1 and I 2 can be given by

 1  1 
I 1 − I 2 = 4 I O I R sin ψ + ∆φ  sin  ∆φ  (3)
 2  2 

Eq. (3) represents a fringe pattern that can be numerically processed to obtain a phase map for
measuring the deformation of the object.

When a sinusoidal signal produces an oscillated vibration on the surface of the object, the phase
change, ∆φ , can be expressed as

4π
∆φ = A sin ω t (4)
Λ

where Λ is the wavelength of a laser, A is the amplitude and ω is the natural frequency.

Eq. (4) indicates that the vibration on the object’s surface will change cyclically from a minimum
to maximum amplitude. If, for this case, it is considered that in a high speed ESPI system, there
are more than two intensities that can be stored by a sensor while the object is deformed,
I 1 , I 2 , I 3 , . . ., I n , the phase of I 1 will change from ∆φ = 0 to 2π . The number of intensities
stored in a complete vibration cycle will depend on the natural frequency, ω , and on the
exposure time of the sensor, τ .

In this case, the intensity stored by the sensor can be written as

 2(n − 1)π 
I n = I O + I R + 2 I O I R cosψ +  (5)
 m 

[ ]
where m ≈ int (ω τ )−1 and n = 1, 2, 3, 4, 5, 6, . .. , m

The subtraction of intensities of the input images I 1 and I 2 to I n can be given by

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
I1 − I n = 4 I O I R sin ψ +
(n − 1)π  sin  (n − 1)π  (6)
  
 m   m 

Each of these subtractions represents a fringe pattern and each of these fringe patterns can be
processed for extracting the corresponding phase map along the deformation of the object. With
this information, the reconstruction of the deformation along a complete vibration cycle of the
object can be done.

One of the most important considerations of this technique is that the measurement can be started
at any time by only selecting an intensity and make a subtraction of the following consecutive
intensities by itself. It is not need of any kind of synchronization or any kind of carrier to do the
measurement.

3. PHASE EXTRACTION OF A SINGLE FRINGE PATTERN

Due to the not existence of a carrier, each fringe pattern must be treated with independency to
obtain the deformation phase map. Phase demodulation from a single fringe pattern has been of a
very high interest for several industrial applications and many algorithms have been published in
last year’s [11-17].

For this work, a function called DemIQT for the demodulation of fringe patterns with closed or
open fringes was used. This function is part of the software called XtremeFringe which was
developed to be compatible with matlab [18].

The function is based in the Isotropic Quadrature Transform [19, 20]. Given the normalized
version of a fringe pattern, the corresponding quadrature term can be get as

Q{I N } = − sin φ (7)

where Q{ } is the isotropic quadrature operator. The wrapped phase of the modulation phase can
be obtained from

 Q{I N }
W {φ } = arctan −
  (8)
 IN  

The demodulated phase obtained for each fringe pattern will give very useful information in
order to analysis the complete evolution of the object along a vibrating cycle.

4. SIMULATING AND EXPERIMENTAL RESULTS

The optical arrangement of the high speed ESPI for vibration measurement is shown in Figure 1.
The beam coming from a Nd:YAG laser with wavelength Λ = 532 nm, is divided into object
and reference beams by a beam splitter. The object beam is projected to the object by L2 and the

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reflection of this is recombined by a second beam splitter through L1. The reference beam is re-
directed by an optical fiber to the second beam splitter and then to the sensor of the CMOS high
speed camera. The laser was set up at 5.5 Watts for the experiment described in this paper.

Laser L1

BS 1

L2
CMOS BS 2 Rectangular plate,
High Speed 1mm thickness
camera

Figure 1. Diagram of high speed ESPI set-up for out-of-plane displacement measurement

A rectangular aluminum membrane clamped in its four sides is used in this study for
experimental investigation. The geometric dimensions of the membrane are Length L = 190 mm,
width W = 140 mm and thickness h = 1 mm.

The equation for the vibrating rectangular plate is given by

(
∂ 2 u ∂t 2 = c 2 ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 ) (9)

Where u ( x, y, t ) must satisfy the border conditions u = 0 for all t > 0 . Eq. (9) can be solving by
applying the variables separation method as the product of two functions of the form:

u ( x, y, t ) = F (x, y )G (t ) (10)

and again, applying the variables separation method F ( x, y ) can be written as:

F ( x, y ) = H ( x )Q( y ) (11)

After a mathematical procedure, the expression that gives the solution on function of the
dimensions of the plate, a for the length, b for the width and for each horizontal and vertical
vibration modal, w and z respectively is:

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( )
u w z ( x, y, t ) = Bw z cos λ w z t + Bw z senλ w z t sen(w π a ) sen( zπ / b )
*
(12)

where

λ w z = cπ w 2 + z 2 (13)

c is a constant depending of the mechanical properties of the material.

4.1 SIMULATED RESULTS

Consider a full resonant vibration cycle at ω = 320Hz, the CMOS camera works with an
exposure time, τ = 4000 fps. The relation of intensity patterns of interference recorded by the
camera gives a sample of m=12 fringe patterns along the whole vibration event. The intensity
patterns are separated from each other by a constant time. Figure 2 shows the 12 simulated fringe
patterns to be considered.

Figure 2. Simulation of the 12 fringe pattern of interference coming from the relation between
the resonance frequency and the exposure time of the camera

The demodulation of each of the fringe patterns with closed fringes based on an Isotropic
Quadrature Transform is shown in figure 3.

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Figure 3. Profiles of the demodulated fringe patterns of interference

As it is shown in Figure 3, the rectangular plate goes from a minimum to a maximum deformation and
then, from a maximum to a minimum deformation. Each of these deformations adjusts to itself to a four
degree polinomial function, as the theory expects.

4.2 EXPERIMENTAL RESULTS

The CMOS high speed camera is set-up at 4000 frames per second, thus, it works at an exposure time of
1
τ= = 250µ s . The natural modal vibration frequency of the aluminum plate is found at 320 Hz.
4000
The recorded intensity patterns of interference are shown in figure 4.

FIG. 4. Intensity patterns of interference coming from the HS ESPI. Each of these images is separated by
the same amount of phase from each other

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According to eq. (6) taking the first intensity, I , as a reference, the subtraction of the consecutive
1
intensities to it will give a lot of fringe patterns that will content the deformation phase of the semisolid
membrane along the time. Followed subtractions will give to much information about the vibration
under study, as it is shown in figure 5.

Figure 5. The subtractions of fringe patterns of intensity give information along the vibration cycle

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It is shown in figure 5, how the fringe patterns represent an evolution of the vibrating membrane.

The phase demodulation of the fringe patterns in figure 5 is processed. Once the wrapped phase is
obtained, the unwrapped phase has to be calculated. This phase map will have the value of the
deformation of the plate at a particular time, in other words, these maps have the phase information
wanted. From the unwrapped phases, it is possible to plot the profile deformation evolution of the
membrane along the vibration. Figure 6 shows the unwrapped phase and the profiles obtained at the
middle of the membrane across the x axis for a specific subtraction starting at I9 – I10 to I9 – I20.

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Figure 6. Unwrapped phase and profile of the deformation due to the vibration over the metallic plate

Figure 6 shows how the plate changes its phase spatially. If now the maximum deformation of the
membrane over the taken profile is selected, the resulted plot represents the temporal deformation of a
selected point over the plate. Figure 7 shows the temporal evolution of the vibrating plate.

Figure 7. Temporal evolution of the vibrating plate over a selected point. x – axis represents time and y –
axis represents deformation in rads

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An advantage of the method is that there are 4000 frames patterns of interference that can be use for
this study, in this way, if another vibrating cycle is processed, the function sine can be recovered as it is
shown in figure 8.

30

25

20

15

10

5

0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Figure 8. Temporal analysis of the vibration plate. Maximum values of deformation fits a sine function
according to the induced signal

5. CONCLUSIONS

The vibrating plate conserves its mechanical properties along the vibration on space. It was probe that at
any time of the vibration, the mechanical deformation keeps a four degree polynomial function.

Temporally, the bigger deformation point of the profiles represents a maximum mechanical deformation
along the time. All these points recover the signal used to induce vibration on the plate.

High Speed Optical techniques can work perfectly with a lot of algorithm for phase extraction. It will do
any measurement as simple as possible. It can allow giving a proper following of any vibration event.

The introduction of these techniques could have a big impact in industry applications because of the fast
response and evaluation of the system.

6. ACKNOWLEDGMENT

The authors would like to acknowledge partial financial support from Consejo Nacional de Ciencia y
Tecnología, Grant 157594.

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Spatial and temporal study of a mechanical and harmonic vibration by high speed optical interferometry

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