Recent technological trends based on miniaturization of mechanical, electromechanical and photonic devices to the microscopic scale have led to the development of microelectromechanical systems (MEMS). Effective development of MEMS components requires the synergy of advanced design, analysis, and fabrication methodologies as well as application of quantitative metrology techniques for characterizing their performance, reliability and integrity during design stage. In this paper, we describe analysis of holographic interferograms of MEMS links which includes interferometric fringes related to deformation appearing because of the non-harmonic excitation of MEMS link

A. Palevicius

R. Vasiliauskas

English

Vasiliauskas, R.

Palevicius, A.

Journal of Mechatronics and Artificial Intelligence in Engineering

 332. ANALYSIS OF HOLOGRAPHIC INTERFEROGRAM OF MICROELECTROMECHANICAL SYSTEM WITH NON HARMONIC CYCLES EXCITATION.  R. VASILIAUSKAS1,A, A. PALEVICIUS2,B   © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2008 JANUARY/MARCH,   VOLUME  10,  ISSUE  1,  ISSN 1392-8716  42 332. Analysis of Holographic Interferogram of Microelectromechanical          System with Non Harmonic Cycles Excitation  R. Vasiliauskas1,a, A. Palevicius2,b 1 Mykolo Romerio University,    V. Putvinskio 70, LT-44211 Kaunas, Lithuania 2Kaunas University of Technology,    A.Mickeviciaus 37-204, LT-44244, Kaunas, Lithuania   E-mail: ar.vasiliauskas@kpf.mruni.lt, barvydas.palevicius@ktu.lt  (Received 19  January 2008; accepted 17  March  2008)         Abstract. Recent technological trends based on miniaturization of mechanical, electromechanical and photonic devices to the microscopic scale have led to the development of microelectromechanical systems (MEMS). Effective development of MEMS components requires the synergy of advanced design, analysis, and fabrication methodologies as well as application of quantitative metrology techniques for characterizing their performance, reliability and integrity during design stage. In this paper, we describe analysis of holographic interferograms of MEMS links which includes interferometric fringes related to deformation appearing because of the non-harmonic excitation of MEMS links.  Keywords: microelectromechanical system, holographic interferometry, mechanical vibration.  Introduction  Holographic interferometry is the powerful tool for analysis of dynamics of microelectromechanical systems [1-6]. It is a non-destructive full-field measurement technique capable of registering micro-scale oscillations of MEMS components [2]. There exist numerous methods [3] used for interpretation of patterns of fringes in the holograms of analyzed objects. Unfortunately, sometimes straightforward application of these motion reconstruction methods (fringe counting technique, etc.) does not produce acceptable results. A typical example is holographic analysis of a micro-electromechanical switch which is described in [7]. The vibration of cantilever of micro-electromechanical switch produces nonlinear effects in a microstructure. Holographic interferogram of vibrating surface of such cantilever cannot be evaluated directly using characteristic function for distribution of interference fringes in the case when time-averaging method for recording hologram is used. The characteristic function of distribution of interference fringes of the vibrating surface of the object should be chosen taking into account the nature of nonlinearities taking place in working regimes of microelectromechical system. In the paper [5] it was demonstrated that the specific dynamical properties of a microswitch cantilever need to adopt the existing numerical simulation techniques. First of all it is necessary to evaluate the fact that the period of motion is generally not a harmonic function as cantilever tips touch the drain. In a high frequency repetitive mode of operation the deflection from the status of equilibrium can be characterized as the time function η(t) (1) that is schematically presented in Fig. 1:  ( ) ( ) ( )( )⎩⎨⎧−≤+−−>++=ctcctttϕωϕωϕωηsin if sin if sin       (1)  where c is a constant satisfying inequalities − ≤ − ≤1 0c  and characterizing the clearance between the cantilever and the drain.   Fig. 1. Displacement of a cantilever tip in time with a limiter Because of this cantilever produces non-harmonic cycles of fluctuations and the system of dark interference fringes of the recorded holographic interferogram of   332. ANALYSIS OF HOLOGRAPHIC INTERFEROGRAM OF MICROELECTROMECHANICAL SYSTEM WITH NON HARMONIC CYCLES EXCITATION.  R. VASILIAUSKAS1,A, A. PALEVICIUS2,B   © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2008 JANUARY/MARCH,   VOLUME  10,  ISSUE  1,  ISSN 1392-8716  43 cantilever should be analyzed using characteristic function which have to be derived for this case of periodical excitation of cantilever. To illustrate this case the holographic interferograms of vibrating cantilever were recorded using Denisiuk hologram recording method [7] (He-Ne laser with the wavelength 0,6328 µm was employed). Fig. 2 a and b represent time- averaged holographic images of the microcantilever beams. The image of the harmonically excited cantilever beam of the micro-mechanical switch without limiters is presented in Fig 2 a. The holographic image of cantilever beam with limiters at c = 0,5 (Eq. 1) is presented in Fig 2 b. The left white zones in both pictures correspond to the clamped part of the cantilever beams. The determination of the centre of interference fringes is straightforward and does not require special phase unwrapping techniques basically due to the fact that the analyzed shape of the vibrating beam is very simple – in both cases it is the first eigenshape.      a      b  Fig. 2. Time-averaged holograms of the cantilever beams: a - representing cantilever without limiter (no drain contact); b - cantilever with limiter (drain contact representing c = 0,5) The problem is that for analysis of holographic interferogram presented in the Fig. 2b characteristic function for the interpretation of holographic interferogram is different than is used usually in time-averaging holography in case of sinusoidal vibration of object and characteristic function should be derived taking into account nonlinearities of vibrating cantilever. The derivation of the characteristic function of distribution interference fringe in case of non-harmonic oscillation is presented in this paper.  Characteristic Functions for Interpretation of Holographic Interferogram        In full-field nondestructive testing using holographic interferometry methods it is necessary to describe the physical origin of the interference bands, the function describing the intensity of these bands and the methodology of their interpretation.       The applied method for development of the interference patterns on the surface of a vibrating body is based on the time-averaging methodology which presumes that the exposition of the vibrating surface to the laser beam is many times longer than the period of vibrations.       In the case when each point of the surface performs harmonic oscillations, the averaged lightening intensity distributed on the vibrating surface may be described by the following formula [5]:  20 0210 )()cos(cos2exp∫⎭⎬⎫⎩⎨⎧⎥⎦⎤⎢⎣⎡∫ ++−≈T tdtdttFiTaI θθλπψ ,   (2)  where I − intensity; T − time of exposition; λ − length of the laser beam light; R − the vector of vibration; t − time; θ1 − angle between the lightening and the vibration vectors; θ2 − angle between the observation and the vibration vectors; λθπψ sin20 = ; θ − angle between the observation and the normal vector of the hologram plane; a − the coefficient of the light reflection of the surface; F(t) – harmonic function related to vibrating surface of the object.  Let us consider that for excitation of links of microelectromechanical system is used the signal with the period T which is described as presented in Fig. 3.   F(t)      A    t  0  T/2 T  2/)1(2/,2/,)(TktTkTATkTtkTAtF+≤≤+−+≤≤=     a) F(t)      A    t   0   T/2 T  b)    2/)1(2/,02/,)(TktTkTTkTtkTAtF+≤≤++≤≤=    Fig. 3. Signal for external excitation of the links of the microelectromechanical system          The time-averaged holographic interferogram analysis of microelectromechanical systems in case when their parts perform non-harmonic oscillations is based on non- harmonic periodic cycles decomposition using Fourier transformation.        In general case signal F(t) is expressed as tkTTkTTkTTATATtF ik ii0110101010 cossin2)( ωππ∑∞=+=  (3) where T0 – period of rectangular signal; 0ω = 02Tπ - frequency of the signal; A – amplitude of vibrations normal  332. ANALYSIS OF HOLOGRAPHIC INTERFEROGRAM OF MICROELECTROMECHANICAL SYSTEM WITH NON HARMONIC CYCLES EXCITATION.  R. VASILIAUSKAS1,A, A. PALEVICIUS2,B   © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2008 JANUARY/MARCH,   VOLUME  10,  ISSUE  1,  ISSN 1392-8716  44 to the surface of the analyzed part of the MEMS; ki 0ω  - ki – harmonic components (ki = 1,2,3,…..n); T1 – number less than 5 – 7.        Characteristic function of distribution of interference fringes taking into account (3) is expressed as follow  { } [ ]dttkTkTTkTTATTATiTdttFiTtFMtkt∫⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡∑+==∫=∞=0 101010101000)cossin2(4exp1/)(4exp1)(ωππλπλπ (4) It is shown in [8] that in the case when T1 is less than 5 – 7, T0/T1= 1η  and number of harmonics is 7 the formula (3) could be transformed as follow:  { }dttkkkATidttkkkATAitFMktkt)cossin8sin()cossin8cos(1)4exp()(071110007111001ωπηπηλωπηπηλλπη∑∫++∑∫++===(5)      The results of calculation by formula (5) in case η1 = 0,1; 0,25; 0,5; 0,75 are presented in Table 1 and Fig. 4 a,b. Table 1.   The values of the characteristic function in the center of the dark interference fringes at the various intervals of the period  Dark interference fringes. 1η   0,1 0,25 0,5 0,75  0,1 0,25 0,5 0,75 1 0,25 0,25 0,25 0,25 11 2,45 2,75 2,8 2,8 2 0,45 0,5 0,5 0,5 12 2,7 3,0 3,05 3,05 3 0,69 0,75 0,75 0,75 13 2,95 3,3 3,35 3,35 4 0,9 1,0 1,0 1,0 14 3,2 3,55 3,6 3,6 5 1,14 1,25 1,24 1,24 15 3,6 3,8 3,8 3,8 6 1,29 1,49 1,49 1,49 16 4,0 4,05 4,15 4,15 7 1,49 1,7 1,7 1,7 17 4,3 4,3 4,45 4,45 8 1,69 1,95 1,99 1,99 18 4,55 4,6 4,75 4,75 9 1,999 2,249 2,249 2,249 19 4,8 4,85 5,0 5,0 10 2,24 2,5 2,5 2,5 20 5,0 5,05 5,35 5,35                   Fig. 4. Distribution of the intensity of interference fringes versus amplitude of vibrating surface of the MEMS: a) η1 =  0,25;      b) 1 -η1 =  0,1; 2, 3 η1 =  0,5; 0,75    332. ANALYSIS OF HOLOGRAPHIC INTERFEROGRAM OF MICROELECTROMECHANICAL SYSTEM WITH NON HARMONIC CYCLES EXCITATION.  R. VASILIAUSKAS1,A, A. PALEVICIUS2,B   © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2008 JANUARY/MARCH,   VOLUME  10,  ISSUE  1,  ISSN 1392-8716  45 Characteristic function of interference fringes distribution in case when excitation is expressed as periodic triangle signal is expressed as follow  tkkkTATTATtFk0211711010 cos)22(sin4)( ωπηπη∑+==                                       (6) The results of calculation using formula (5) are performed in case η1 = 0,1; 0,25; 0,5; 0,75 (Table 2 and Fig. 5).   Table 2. The values of the characteristic function in the center of the dark interference fringes at the different intervals of the period  1η  Dark interference fringes 0,1 0,25 0,5 0,75  0,1 0,25 0,5 0,75 1 0,4 0,49 0,4 0,4 11 4,0 3,1 2,9 2,9 2 0,7 0,6 0,6 0,65 12 4,1 3,45 3,2 3,15 3 1,04 0,9 0,85 0,9 13 4,65 4,05 3,45 3,4 4 1,24 1,15 1,1 1,15 14 4,95 4,45 3,7 3,65 5 1,65 1,5 1,35 1,4 15 5,4 4,75 3,95 3,9 6 1,5 1,7 1,6 1,6 16 5,7 5,0 4,2 4,15 7 2,6 2,0 1,9 1,85 17 6,0 5,35 4,5 4,45 8 2,7 2,45 2,15 2,1 18 6,25 5,6 4,75 4,7 9 3,5 2,55 2,4 2,4 19 6,65 5,9 5,0 4,95 10 3,6 2,8 2,65 2,65 20 6,95 6,20 5,25 5,20                          Fig. 5. Distribution of the intensity of interference fringes versus amplitude of vibrating surface of the MEMS when triangle signal is used for excitation    332. ANALYSIS OF HOLOGRAPHIC INTERFEROGRAM OF MICROELECTROMECHANICAL SYSTEM WITH NON HARMONIC CYCLES EXCITATION.  R. VASILIAUSKAS1,A, A. PALEVICIUS2,B   © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2008 JANUARY/MARCH,   VOLUME  10,  ISSUE  1,  ISSN 1392-8716  46 3. Concluding Remarks            Though time-averaging holography is a powerful experimental technique for analysis of MEMS dynamics, the interpretation of produced patterns of fringes in holograms must be performed with care. Complex physical and nonlinear dynamical processes taking place in MEMS systems may influence the results produced by optical methods and require additional analysis of the vibrational processes and the derivation of the characteristic function of distribution of interference fringes which must correspond to the character of vibration that appear in real design of microelectromechanical systems.  Reference  [1] Brown G. C., and Pryputniewicz R. J. Holographic microscope for measuring displacements of vibrating microbeams using time-average electro-optic holography,  Opt. Eng., 37:1398-1405, 1998. [2] Pryputniewicz R. J., Furlong C., Brown G. C. and Pryputniewicz E. J. Optical methodology for static and dynamic measurements of nanodisplacements, Proc. Internat. Congress on Experimental and Applied Mechanics in Emerging Technologies, Portland, OR, 2001, pp. 826-831.                                      [3] Pryputniewicz R. J., Stetson K. A. Measurement of vibration patterns using electro-optic holography, Proc. SPIE, v.1162, 1989. [4] Ostasevicius V., Palevicius A. Daugela A, Ragulskis M., Palevicius R. Holographic imaging technique for characterization of MEMS switch dynamics, Proc. SPIE Vol. 5389 , Smart Structures and Materials 2004: Smart Electronics, MEMS, BioMEMS, and Nanotechnology; Vijay K. Varadan; Ed., 2004, pp. 73-84. [5] Ostaševičius Vytautas, Palevičius Arvydas, Ragulskis Minvydas Kazys et al. Hybrid numerical – experimental optical investigation of non-linearity in MEMS dynamics. Iš: Unification of Analytical, Computational, and Experimental Solution Methodologies in MEMS and Nanotechnology, Springfield, Massachusetts, 27-29 October, 2004: proceedings of the International Invitational UACEM Symposium 15th in the Series. Bethel, 2004, pp. 347-353. [6] Ragulskis M., Palevicius A. and Ragulskis L. Plotting Holographic Interferograms for Visualization of Dynamic Results from Finite-Element Calculations, International Journal for Numerical Methods in Engineering, Vol. 56, 2003, pp.1647-1659. [7] Denisiuk J. N. The status quo and perspectives of development of holography. Nauka, Moscow, 1974, 214p. [8] Бансявичюс Р., Рагульскис К. Вибродвигатели.- Вильнюс, Мокслас. 1981.- 194 с..   

Analysis of holographic interferogram of microelectromechanical system with non harmonic cycles excitation

JVE International

Journal of Vibroengineering

 332. ANALYSIS OF HOLOGRAPHIC INTERFEROGRAM OF MICROELECTROMECHANICAL SYSTEM WITH NON HARMONIC CYCLES EXCITATION.  R. VASILIAUSKAS1,A, A. PALEVICIUS2,B 
 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2008 JANUARY/MARCH,   VOLUME  10,  ISSUE  1,  ISSN 1392-8716 
 
42 
332. Analysis of Holographic Interferogram of Microelectromechanical  
        System with Non Harmonic Cycles Excitation 
 
R. Vasiliauskas1,a, A. Palevicius2,b 
1
 Mykolo Romerio University,  
  V. Putvinskio 70, LT-44211 Kaunas, Lithuania 
2Kaunas University of Technology,  
  A.Mickeviciaus 37-204, LT-44244, Kaunas, Lithuania 
  E-mail: ar.vasiliauskas@kpf.mruni.lt, barvydas.palevicius@ktu.lt 
 
(Received 19  January 2008; accepted 17  March  2008) 
 
 
 
 
 
 
 
 
Abstract. Recent technological trends based on miniaturization of mechanical, electromechanical and photonic devices to 
the microscopic scale have led to the development of microelectromechanical systems (MEMS). Effective development of 
MEMS components requires the synergy of advanced design, analysis, and fabrication methodologies as well as application 
of quantitative metrology techniques for characterizing their performance, reliability and integrity during design stage. In 
this paper, we describe analysis of holographic interferograms of MEMS links which includes interferometric fringes 
related to deformation appearing because of the non-harmonic excitation of MEMS links. 
 
Keywords: microelectromechanical system, holographic interferometry, mechanical vibration. 
 
Introduction 
 
Holographic interferometry is the powerful tool for 
analysis of dynamics of microelectromechanical systems 
[1-6]. It is a non-destructive full-field measurement 
technique capable of registering micro-scale oscillations of 
MEMS components [2]. There exist numerous methods [3] 
used for interpretation of patterns of fringes in the 
holograms of analyzed objects. Unfortunately, sometimes 
straightforward application of these motion reconstruction 
methods (fringe counting technique, etc.) does not produce 
acceptable results. A typical example is holographic 
analysis of a micro-electromechanical switch which is 
described in [7]. 
The vibration of cantilever of micro-electromechanical 
switch produces nonlinear effects in a microstructure. 
Holographic interferogram of vibrating surface of such 
cantilever cannot be evaluated directly using characteristic 
function for distribution of interference fringes in the case 
when time-averaging method for recording hologram is 
used. The characteristic function of distribution of 
interference fringes of the vibrating surface of the object 
should be chosen taking into account the nature of 
nonlinearities taking place in working regimes of 
microelectromechical system. 
In the paper [5] it was demonstrated that the specific 
dynamical properties of a microswitch cantilever need to 
adopt the existing numerical simulation techniques. First of 
all it is necessary to evaluate the fact that the period of 
motion is generally not a harmonic function as cantilever 
tips touch the drain. In a high frequency repetitive mode of 
operation the deflection from the status of equilibrium can 
be characterized as the time function η(t) (1) that is 
schematically presented in Fig. 1: 
 
( ) ( ) ( )( )⎩⎨
⎧
−≤+−
−>++=
ctc
ctt
t ϕω
ϕωϕωη
sin if 
sin if sin
       
(1) 
 
where c is a constant satisfying inequalities − ≤ − ≤1 0c  
and characterizing the clearance between the cantilever and 
the drain. 
 
 
Fig. 1. Displacement of a cantilever tip in time with a limiter 
Because of this cantilever produces non-harmonic 
cycles of fluctuations and the system of dark interference 
fringes of the recorded holographic interferogram of 
 
 332. ANALYSIS OF HOLOGRAPHIC INTERFEROGRAM OF MICROELECTROMECHANICAL SYSTEM WITH NON HARMONIC CYCLES EXCITATION.  R. VASILIAUSKAS1,A, A. PALEVICIUS2,B 
 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2008 JANUARY/MARCH,   VOLUME  10,  ISSUE  1,  ISSN 1392-8716 
 
43 
cantilever should be analyzed using characteristic function 
which have to be derived for this case of periodical 
excitation of cantilever. 
To illustrate this case the holographic interferograms of 
vibrating cantilever were recorded using Denisiuk 
hologram recording method [7] (He-Ne laser with the 
wavelength 0,6328 µm was employed). Fig. 2 a and b 
represent time- averaged holographic images of the 
microcantilever beams. The image of the harmonically 
excited cantilever beam of the micro-mechanical switch 
without limiters is presented in Fig 2 a. The holographic 
image of cantilever beam with limiters at c = 0,5 (Eq. 1) is 
presented in Fig 2 b. The left white zones in both pictures 
correspond to the clamped part of the cantilever beams. 
The determination of the centre of interference fringes is 
straightforward and does not require special phase 
unwrapping techniques basically due to the fact that the 
analyzed shape of the vibrating beam is very simple – in 
both cases it is the first eigenshape. 
 
 
 
 
 
a 
 
 
 
 
 
b 
 
Fig. 2. Time-averaged holograms of the cantilever beams: a - 
representing cantilever without limiter (no drain contact); b - 
cantilever with limiter (drain contact representing c = 0,5) 
The problem is that for analysis of holographic 
interferogram presented in the Fig. 2b characteristic 
function for the interpretation of holographic interferogram 
is different than is used usually in time-averaging 
holography in case of sinusoidal vibration of object and 
characteristic function should be derived taking into 
account nonlinearities of vibrating cantilever. The 
derivation of the characteristic function of distribution 
interference fringe in case of non-harmonic oscillation is 
presented in this paper. 
 
Characteristic Functions for Interpretation of 
Holographic Interferogram 
 
      In full-field nondestructive testing using holographic 
interferometry methods it is necessary to describe the 
physical origin of the interference bands, the function 
describing the intensity of these bands and the 
methodology of their interpretation.  
     The applied method for development of the interference 
patterns on the surface of a vibrating body is based on the 
time-averaging methodology which presumes that the 
exposition of the vibrating surface to the laser beam is 
many times longer than the period of vibrations.  
     In the case when each point of the surface performs 
harmonic oscillations, the averaged lightening intensity 
distributed on the vibrating surface may be described by 
the following formula [5]:  
2
0 0
210 )()cos(cos
2
exp∫ ⎭⎬
⎫
⎩⎨
⎧
⎥⎦
⎤⎢⎣
⎡ ∫ ++−≈
T t
dtdttFi
T
aI θθλ
πψ ,   (2) 
 
where I − intensity; T − time of exposition; λ − length of 
the laser beam light; R − the vector of vibration; t − time; 
θ1
 
− angle between the lightening and the vibration vectors; 
θ2
 
− angle between the observation and the vibration 
vectors; λ
θπψ sin20 = ; θ − angle between the observation 
and the normal vector of the hologram plane; a − the 
coefficient of the light reflection of the surface; F(t) – 
harmonic function related to vibrating surface of the object.  
Let us consider that for excitation of links of 
microelectromechanical system is used the signal with the 
period T which is described as presented in Fig. 3.  
 
F(t)     
 A    
t 
 
0 
 
T/2 T  
2/)1(2/,
2/,)(
TktTkTA
TkTtkTA
tF +≤≤+−
+≤≤=  
 
  a) 
F(t)     
 A    
t 
 
 
0 
 
 
T/2 T 
 
b) 
 
 
 
2/)1(2/,0
2/,)(
TktTkT
TkTtkTA
tF +≤≤+
+≤≤=  
 
 
Fig. 3. Signal for external excitation of the links of the 
microelectromechanical system 
   
      The time-averaged holographic interferogram analysis 
of microelectromechanical systems in case when their parts 
perform non-harmonic oscillations is based on non- 
harmonic periodic cycles decomposition using Fourier 
transformation. 
       In general case signal F(t) is expressed as 
tk
T
Tk
T
Tk
T
TA
T
AT
tF i
k i
i
0
1
1
0
1
0
1
0
1
0 cos
sin
2)( ωπ
π
∑∞
=
+=
 (3) 
where T0 – period of rectangular signal; 0ω = 
0
2
T
π
 - 
frequency of the signal; A – amplitude of vibrations normal 
 332. ANALYSIS OF HOLOGRAPHIC INTERFEROGRAM OF MICROELECTROMECHANICAL SYSTEM WITH NON HARMONIC CYCLES EXCITATION.  R. VASILIAUSKAS1,A, A. PALEVICIUS2,B 
 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2008 JANUARY/MARCH,   VOLUME  10,  ISSUE  1,  ISSN 1392-8716 
 
44 
to the surface of the analyzed part of the MEMS; ki 0ω  - ki – 
harmonic components (ki = 1,2,3,…..n); T1 – number less 
than 5 – 7. 
       Characteristic function of distribution of interference 
fringes taking into account (3) is expressed as follow 
 
{ } [ ]
dttk
T
kT
T
kT
T
AT
T
AT
i
T
dttFi
T
tFM
t
k
t
∫
⎥⎥
⎥⎥
⎦
⎤
⎢⎢
⎢⎢
⎣
⎡
∑+=
=∫=
∞
=0 1 0
1
0
1
0
1
0
1
0
0
0
)cos
sin
2(4exp1
/)(4exp1)(
ωπ
π
λ
π
λπ
 (4) 
It is shown in [8] that in the case when T1 is less than 5 – 7, 
T0/T1= 1η  and number of harmonics is 7 the formula (3) 
could be transformed as follow: 
 
{ }
dttk
k
kA
T
i
dttk
k
kA
T
AitFM
k
t
k
t
)cossin8sin(
)cossin8cos(1
)4exp()(
0
7
1
1
1
00
0
7
1
1
1
00
1
ωπη
πη
λ
ωπη
πη
λ
λπη
∑∫+
+∑∫+
+=
=
=
(5) 
     
The results of calculation by formula (5) in case η1 = 0,1; 
0,25; 0,5; 0,75 are presented in Table 1 and Fig. 4 a,b. 
Table 1. 
  The values of the characteristic function in the center of the dark interference fringes at the various intervals of the period 
 
Dark interference 
fringes. 1η  
 0,1 0,25 0,5 0,75  0,1 0,25 0,5 0,75 
1 0,25 0,25 0,25 0,25 11 2,45 2,75 2,8 2,8 
2 0,45 0,5 0,5 0,5 12 2,7 3,0 3,05 3,05 
3 0,69 0,75 0,75 0,75 13 2,95 3,3 3,35 3,35 
4 0,9 1,0 1,0 1,0 14 3,2 3,55 3,6 3,6 
5 1,14 1,25 1,24 1,24 15 3,6 3,8 3,8 3,8 
6 1,29 1,49 1,49 1,49 16 4,0 4,05 4,15 4,15 
7 1,49 1,7 1,7 1,7 17 4,3 4,3 4,45 4,45 
8 1,69 1,95 1,99 1,99 18 4,55 4,6 4,75 4,75 
9 1,999 2,249 2,249 2,249 19 4,8 4,85 5,0 5,0 
10 2,24 2,5 2,5 2,5 20 5,0 5,05 5,35 5,35 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig. 4. Distribution of the intensity of interference fringes versus amplitude of vibrating surface of the MEMS: 
a) η1 =  0,25;      b) 1 -η1 =  0,1; 2, 3 η1 =  0,5; 0,75 
 
 
 332. ANALYSIS OF HOLOGRAPHIC INTERFEROGRAM OF MICROELECTROMECHANICAL SYSTEM WITH NON HARMONIC CYCLES EXCITATION.  R. VASILIAUSKAS1,A, A. PALEVICIUS2,B 
 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2008 JANUARY/MARCH,   VOLUME  10,  ISSUE  1,  ISSN 1392-8716 
 
45 
Characteristic function of interference fringes distribution in case when excitation is expressed as periodic triangle 
signal is expressed as follow 
 
tkk
k
T
AT
T
AT
tF
k
0
2
1
1
7
1
1
0
1
0 cos)
2
2(sin
4
)( ωπη
πη
∑+=
=
                                       (6) 
The results of calculation using formula (5) are performed in case η1 = 0,1; 0,25; 0,5; 0,75 (Table 2 and Fig. 5). 
 
 
Table 2. 
The values of the characteristic function in the center of the dark interference fringes at the different intervals of the period 
 
1η  Dark interference 
fringes 0,1 0,25 0,5 0,75  0,1 0,25 0,5 0,75 
1 0,4 0,49 0,4 0,4 11 4,0 3,1 2,9 2,9 
2 0,7 0,6 0,6 0,65 12 4,1 3,45 3,2 3,15 
3 1,04 0,9 0,85 0,9 13 4,65 4,05 3,45 3,4 
4 1,24 1,15 1,1 1,15 14 4,95 4,45 3,7 3,65 
5 1,65 1,5 1,35 1,4 15 5,4 4,75 3,95 3,9 
6 1,5 1,7 1,6 1,6 16 5,7 5,0 4,2 4,15 
7 2,6 2,0 1,9 1,85 17 6,0 5,35 4,5 4,45 
8 2,7 2,45 2,15 2,1 18 6,25 5,6 4,75 4,7 
9 3,5 2,55 2,4 2,4 19 6,65 5,9 5,0 4,95 
10 3,6 2,8 2,65 2,65 20 6,95 6,20 5,25 5,20 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig. 5. Distribution of the intensity of interference fringes versus amplitude of vibrating surface of the MEMS when triangle signal is 
used for excitation 
 
 
 332. ANALYSIS OF HOLOGRAPHIC INTERFEROGRAM OF MICROELECTROMECHANICAL SYSTEM WITH NON HARMONIC CYCLES EXCITATION.  R. VASILIAUSKAS1,A, A. PALEVICIUS2,B 
 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2008 JANUARY/MARCH,   VOLUME  10,  ISSUE  1,  ISSN 1392-8716 
 
46 
3. Concluding Remarks 
 
          Though time-averaging holography is a powerful 
experimental technique for analysis of MEMS dynamics, 
the interpretation of produced patterns of fringes in 
holograms must be performed with care. Complex physical 
and nonlinear dynamical processes taking place in MEMS 
systems may influence the results produced by optical 
methods and require additional analysis of the vibrational 
processes and the derivation of the characteristic function 
of distribution of interference fringes which must 
correspond to the character of vibration that appear in real 
design of microelectromechanical systems. 
 
Reference 
 
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[3] Pryputniewicz R. J., Stetson K. A. Measurement of 
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Analysis of holographic interferogram of microelectromechanical system with non harmonic cycles excitation

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