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Free vibrations of a cylindrical shell by holographic interferometry
Journal of Sound and Vibration (1988) 121 ( I ), 169-180
FREE VIBRATIONS OF A CYLINDRICAL SHELL BY H O L O G R A P H I C INTERFEROMETRY A. G. KllADAKKAR, R. NARAYANAN, K. RAVI SANKAR, N. R. IYER AND T. V. S. R. APPA RAO Structural Engineering Research Centre, CSIR Campus, Taramani, Madras 600 i 13, India (Received 8 January 1987, and in revised form 7 Afay 1987) Holographic interferometry has been successfully used to record the various mode shapes of a cylindrical shell model with clamped-free boundaries. Experimental results in some cases are compared with those obtained by using the finite element method. Effects of an opening on the natural frequencies and on the associated mode shapes have also been studied experimentally on models with openings of different aspect ratios and are presented in this paper.
I. INTRODUCTION Holographic interferometry is a powerful technique used in experimental studies on various structural forms to measure their surface displacements in either static or dynamic state. The technique has the merit of being highly sensitive, precise, whole-field and non-contacting. Its unique capability to record the m o d e shapes of a resonantly vibrating object showing nodal and antinodal loci in a visual form makes it the most suitable technique for vibration studies on structures of even complex geometry. The holographic technique of vibration analysis appeals greatly, because one can see the object covered by dark and bright interference fringes which represent the ampliiudes of vibration of the object resonating in one of its natural frequencies. This paper gives an account of the application of holographic interferometry to the vibration study of clamped-free cylindrical shell models with emphasis on mode shapes at various resonating frequencies. Studies were also made on shell models which had openings of various sizes and cgnfigurations, to find the influence of an opening on the mode shapes and frequencies. Theoretical analysis of natural frequencies and the m o d e shapes of shells is generally very complex [1]. The generality of the shell equations representing all the degrees of freedom permits a shell to have a wide variety of mode shapes. These could be indicative of different types o f behaviour. For example, in a cylindrical shell, there can be transverse vibrations, longitudinal vibrations and torsional vibrations of tubular beam type, as well as flexural in-plane and extensional vibrations of shell ring type. Because of the complexity of the shell equations, closed form solutions for free vibration analysis are available only for some simple cases. Generally, it is difficult and time-consuming to obtain analytically mode shapes over the entire surface of the shell. Most o f the theoretical methods [2-4] are based on one or more simplifying assumptions so as to render the complex problem comparatively simple and to obtain amenable solutions, in this paper, some natural frequencies and the associated mode shapes obtained by the finite element method for the case of a fixed-free cylindrical shell without an opening are presented. These results are then compared with those found experimentally. 169 0022-460x/88/030169 -i-12 s03.00/0 O 1988 Academic Press Limited
170
a.G. 2. T H E O R Y
KIIADAKKAR
OF IIOLOGRAPHIC
I'T AL.
INTERFEROMETRY
Holographic interferometry can be considered as the interferometric comparison of two or more coherent light wavefronts [5]. The composite reconstruction of these two or more waves is called a holographic interferogram or, simply, a "hologram". Two or more light waves emanating from the same point on an object with an induced deformation causing a change in the phase (or in the optical path length) interfere with each other in the presence of a reference light wave, creating an image of the object superposed with fringes. In a case of mechanical vibration of an opaque object, holographic exposure produces an image of its surface modulated by a system of interference fringes. The brightest of these fringes coincides with the nodal region; that is, the portion of the surface that remains stationary during the vibration. In addition to the bright nodal fringe, a number of black and white fringes may be observed in the holographic image, representing variation of vibration amplitude. If the vibrational motion has a simple time dependence, as in the case of free vibrations of an object, the vibrational amplitude of each point on the surface can be determined from these fringes. In time-averaged holographic interferometry, a hologram of a freely vibrating obiect is recorded by exposing a film to the object and the reference beams simultaneously over a period of time. During such periodic motion, the object spends most of its periodic time near the two extreme positions of maximum amplitude where the direction of vibration reverses and the velocity is zero. An object vibrating sinusoidally with an amplitude of vibration [A(x,y, z)sin tot] will cause optical phase changes in the complex amplitude of light scattered by the object. This phase change A4, at any instant of time is given by
Acl,(x, .t.', z, t) = (2rr/X)2A(x, ); z) sin tot,
( 1)
where A(x,)', z) is the amplitude of vibration at location (x,y, z), A is the wavelength of the light and w is the radian frequency of the vibration. The corresponding complex amplitude of light (object wave) in the plane of the hologram will then be modified due to this phase change from [ a ( x , y ) e i~] to [a(x, y) e~6+'J*~], where a(x, y) is the real amplitude of light wave. A time-averaged hologram is recorded by simultaneously exposing a film to the object wave emanating from the vibrating object and to an otI-axis reference wave for a period of time T. When this hologram is developed and illuminated by the reference beam for reconstruction, the complex anaplitude of the reconstructed wave will be given by [a(x, y) ele'Mr ],
(2)
where Mr is called the characteristic function. Irradiance or intensity of reconstructed hologram image is then given by the square of the above term. When the exposure time T is large compared to ( l / w ) , the characteristic function Mr for sinusoidal vibration becomes
Mr = Jo[(4~/X )A(x, ), z)]
(3)
where Jo is the Bessei function of the first kind and of order zero. The corresponding irradiance of the time-averaged hologram is given by
! = a'-(x, y)J~[(4,-r/A )a(x, y, z)].
(4)
This expression can be interpreted as follows: "the holographic virtual image is modulated by a system of fringes described by the Jo~ term, which can be shown analytically to represent the ettect of free vibration of the object".
StlEI.L V I I | R A T I O N STU DY BY H O L O G R A P I I Y 3. E X P E R I M E N T A L
SET-UP
AND
TEST
171
PROCEDURE
Five cylindrical shell models with clamped-Free boundaries and identical overall dimensions were machined from a solid rod of mild steel. Each model was made by first welding a base plate to the solid rod. The thin cylinder was machined out of the solid rod by boring. This sequence eliminated any possible distortion in geometry of shell models which would have been caused due to welding. The details and geometry of the five models are shown in Figure I. Precision machining of these models gave a dimensional accuracy better than 0.5%. For the holographic recording, the shell model was rigidly clamped through its base plate to the table of the vibrating isolation system. The shell model was excited to resonance by means of a piezoelectric shaker which was connected to an audio oscillator. This oscillator could provide an input signal to the piezoelectric shaker in the frequency range of 20 to 20 000 Hz with adequate force to excite a shell model over the lower half o f its frequency range: i.e., up to 10 kHz. An aluminium exponential horn, mounted on the piezoelectric shaker, was kept in physical contact with the shell model to impart optimum excitation energy. Different shaker locations were used to excite the various modes of
ill
0D=66
I
I] II
Model A
(without opening]
:1 II
l"
Centreline
of openlnc
25
t
i
1
Model B (with squore opening)
i21 ,:, 170 ElevoUon
','i i
Model C (wlth elllptlcol opening]
c~
5
Model D (w~th reclon, ruler open;nq| I
2
,~I
S
170
Plon
1
t Model
(with recton alor open;rig]
Figure I. Geometry o f cylindrical shell models. All dimensions are in mm.
172
E T AL.
A. G. K H A D A K K A R
vibrations. It was verified experimentally from the recorded holograms that the physical contact o f t h e horn with the model does not alter its natural frequencies and mode shapes except in causing a slight change in fringe pattern around the contact point in a very localized manner. An electronic frequency counter was used for determining the frequencies o f the resonantly vibrating models. A standard holographic interferometry set -up was used for recording and reconstruction o f the holograms as shown in Figure 2. The positions of the spatial filter on the object beam path and the holographic plate were kept in the same horizontal plane symmetrically about the shell model at the smallest angle (about 4~ This enabled recording of predominantly out-of-plane deflections to be made.
Piezoelectric shoker
M
Shell mode
It SF.-
L H ~B,5,
M
'"'M
Figure 2. Schematic of experimental set-up. L, lie-Ne laser; M, mirror; S.F., spatial filter; B.S., beam spllttcr; H, holographic plate; P.S., piezoelectric shaker; S.M., shell model.
To identify the resonant frequencies, real-time holography was used. By observing the vibrating model in real time through the hologram (recorded earlier with the model stationary), the resonances at various frequencies were identified whenever the fringe patterns were seen to be frozen. The corresponding frequencies and the locations of the piezoelectric shaker were noted down. The interference fringes, created as a result of the sinusoidal vibrations of a model, were recorded on a holographic plate by a time-averaged holography technique. For further analysis, the fringe patterns corresponding to different resonant frequencies were photographed during reconstruction of the holograms. Identification of resonant frequencies, recording of holograms at these frequencies and photographing the fringe patterns were carried out for all the five models by using the same procedure. As per the definition of modes given in reference [ 1], m is the number ofaxial half-waves along the shell length and n is the number of circumferential waves considered over the
S H E L L V I B R A T I O N S T U D Y IIY I I O L O G R A I ' I I Y
173
entire periphery of the shell. The latter is the same as the number of half-waves of the mode shapes contained in the (front) half o f the shell periphery for which holographic records are made. From a photograph of the fringe pattern, the axial m o d e number could be identified easily by counting the lobes or half-waves. Counting of circumferential half-waves could be tricky, mainly because the curvature of the shell surface in the horizontal plane foreshortens the projected diametric distance. This makes the number of lobes a p p e a r less than the actual number. To remove this ambiguity, the number of half-waves contained in the front quarter sector of the circumference, was counted accurately and extrapolated for the entire shell surface. This procedure, as shown in Figure 3, simplified the determination o f the mode number and also made it accurate. The procedure was found to be necessary particularly for the case of the fourth circumferential mode (n = 4) in the present study.
I
LApporentmodenumber n=3
I
i Figure 3. Counting of circumferential mode number n.
174
A.G. KIIADAKKAR ET ,,~L. 4. RESULTS
The experimental results were obtained for the five cylindrical shell models in the form o f resonant frequencies of various modes and their associated mode shapes. The mode shapes are clearly discerned from the fringe patterns obtained from the time-averaged holograms. A few of the resonant frequencies and the mode shapes for model A are shown in Figure 4. The mode shapes (m-axial, n-circumferential) for these mode numbers can be clearly visualized from observation of the respective fringe patterns. Figure 5 shows the fringe patterns obtained for the five shell models when these were resonantly vibrating in a particular mode (m = 4, n = 3). For a typical circumferential mode (n = 3) and various axial modes (m = 2, 3, 4) o f m o d e l A, the amplitudes ofvibrations were calculated from the fringe patterns. Figure 6 shows deformation variations of the shell model A in the circumferential and axial directions for a few mode numbers. The deformations are normalized and are shown in dimensionless form in the plots. Appendix A shows a calculation for a typical circumferential mode and its comparison with theoretical results. 5. COMPARISON OF RESULTS AND DISCUSSION For obtaining natural frequencies and the corresponding mode profiles of the shell model A, an axisymmetric shell element was used in the finite element analysis. The analysis was carried out with two different sizes of elements and the results were found to be almost identical. The natural frequencies corresponding to the various mode shapes were obtained experimentally for the five models and were compared with the finite element results obtained for model A. The experimental and theoretical values of resonant frequencies of vibration for various modes of the shell model A are given in Table 1 for comparison. Table 2 gives the experimentally determined frequencies o f the five shell models for a few modes. It is possible to obtain theoretical values of resonant frequencies for any mode number o f the shell model. However, due to overlapping of the modes, it becomes increasingly difficult to record holographic fringe patterns when a combination of higher mode number n along tile circumferential direction and lower mode number m along the axial direction or uice rersa is to be rccorded. Another limitation on recording the fringes at high frequency is thai, at higher frequencies, the energy required to excite a model increases considerably. It can be seen from Table 1 that the values of the frequencies of vibration obtained experimentally are, in general, slightly less than those obtained by theory. The amplitudes of the vibrations also compare well, both along the axial and the circumferential directions in case of the shell model A, as can be seen from a few typical plots (Figure 6). From the comparison of tile natural frequencies for different models (five models), it is seen that there is some amount of scatter in most o f the modes (see Table 2). Also, it is evident, that an opening of size up to L / 5 • L / 8 (L being length of the cylindrical shell model) does not attect either the natural frequencies or the mode shapes appreciably. Only in the case of model E, which has a relatively large opening (size 7.5 x 2.5 cm), has the mode shape changed appreciably. However, its natural frequencies at various modes have not changed much compared to those of model A. The reliability of the experimental results depends mainly on the correct simulation o f the boundary conditions and the sensitivity and precision of the measurement techniques. Holographic interferometry has very high sensitivity and precision for measurement o f
SIIELL VIBRATION STUDY BY I'tOLOGRAPIIY
f
.=i. Mode 2 X 2 f = 1965-7 Hz
Mode I X 3 f = 2 6 3 0 . 0 Hz
Mode 2 • f = 2911.6 Hz
Mode :5 X 3 f=3825-OHz
Mode 2 X 4 f =5187.0Hz
Mode 4 X 4 f = 6 3 8 5 . 0 Hz
Figure 4. ttolographic fringe patterns for model A vibrating in different modes.
Model A f = 5 5 6 2 - 8 Hz
Model B f = 5 5 8 3 . 0 Hz
Model D f = 5 5 6 9 . 0 Hz
Model C f = 5 5 9 2 . 0 Hz
Model E f = 5 4 0 0 . 0 Hz
Figure 5. Holographic fringe patterns for the five different models vibrating in the mode m = 3 , n =4.
175
176
A . G . KIIAI)AKKAR E T ,,~L.
-08
-08
t i /,
~
/
-06
-02
Nodes
I
I
,
-I
Mode 2 x 3
=3 -3
i( -0-2 I
-0.2
Amplitude
--,
=2
-04 /
\
0
3
-06
-04
I -I
=
0 Amplitude Mode3 x 3 Along meridion
1 -1
0 Amplitude Mode 4x3
f Along circumference
Figure 6. Comparison o f normalized amplitudes ofvibratlon of model A for various modes. O, Experimental; theoretical.
TABLE 1
Comparison of experimental and theoretical t,alues of vibration frequencies of shell model A for different modes (values in Hz) Circumferential mode no., n
i
2
Axial mode number, m ^ 3
4
5
!
(a) (b)
916"0 923.3
2
(a) (b)
989-6 993-3
1965.7 2030.9
4393.0 4417.8
3
(a) (b)
2630.0 2646.0
2911.6 2899.5
3825.0' 3760.6
5357.8 5266.4
7180"0 7187.5
4
(a) (b)
5077.9 5030.0
5187.0 5239.0
5562.8 5574.0
6385"0 6314-0
7358.0 7342.9
(a) Experimental, time-averaged holograph)'. (b) Theoretical, finite element method. deformations. Also the technique has a unique self-checking capability to ensure the desired fixity at the b o u n d a r y o f a model. When the shell model was fixed perfectly t h r o u g h its base to the table top, no interference fringes were seen on its base. The high sensitivities o f the other instrumentation, including the frequency counter and oscillator, also ensured highly reliable results from the experiments.
S I I E L L V I B R A T I O N STU DY BY H O L O G R A P I I Y
177
TABLE 2
Comparison of experimental values of frequencies of vibrations offive shell models for different modes (values in Hz) Circumferential mode no. n
Axial mode number, m " 2 3 4
Model
I
A B C D E
916.0
2
A B C D E
989"6 996-0 981"0 -981.0
1965"7 2089"0 1974"0 2079.0 1911-2
4393"0 4422.0 4289"0 -4390.0
3
A B C D E
2630.0 -2650.0 -2637.0
2911.6 3005.0 2887.0 2905.0 2866.0
3825-0 3827-0 3781-0 3894.0 3964-5
5357.8 5328-0 5324.0 5400-0
7040.0
A B C D E
5077.9 4967.0 5028.0 -4974.0
5187.0 5109.0 5148.0 5129-0 5114.0
5562.8 5583.0 5592.0 5569.0 5592.0
6385.0 6368.0 6295.0 6343.0 622 !.0
7358.0
I
4
887.0 885-4
7180.0 7118.0
7384.0 --
It can be c o n c l u d e d from this study that h o l o g r a p h i c i n t e r f e r o m e t r y is an e x t r e m e l y useful a n d p o w e r f u l t e c h n i q u e in v i b r a t i o n studies, p a r t i c u l a r l y for d e t e r m i n a t i o n o f m o d e s h a p e s o f even c o m p l e x structures such as shells when the a c t u a l b o u n d a r y c o n d i t i o n s are properly simulated.
ACKNOWLEDGMENTS T h e a u t h o r s wish to t h a n k their c o l l e a g u e s D r N. L a k s h m a n a n , Messrs G. J a y a r a m a Rao, T. N a r a y a n a n a n d K. Z. Z a c h a r i a for the h e l p a n d s u g g e s t i o n s given by them d u r i n g the c o u r s e o f the investigations. T h e y a r e also grateful to D r M. R a m a i a h , D i r e c t o r , S t r u c t u r a l E n g i n e e r i n g R e s e a r c h Centre, M a d r a s , for his e n c o u r a g e m e n t in c a r r y i n g out this study. This p a p e r is p u b l i s h e d with the p e r m i s s i o n o f the D i r e c t o r , Structural E n g i n e e r i n g R e s e a r c h Centre, M a d r a s .
REFERENCES I. R. D. BLEVINS 1979 Formulasfor Natural Frequeno' and Mode Shape New York: Van Nostrand Reinhold. 2. C. B. SHARMA 1973 Journal of Sound and Vibration 30, 525-528. Frequencies of claraped-free circular cylindrical shells. 3. C. B. SHARMA 1984 77~in-walledStructures 2, 175-193. Free vibrations of clamped-free circular cylinders.
178
A.G. KIIADAKKAR ET AL.
4. G. B. WARBURTON and J. HIGGS 1970 Journal o f S o u n d a n d Vibration II, 335-358. Natural frequencies of thin cantilever cylindrical shell. 5. C. M. VEST 1979 Holographic lnterferometry. New York: John Wiley, first edition.
APPENDIX A: CALCULATION OF DEFLECTIONS ALONG CIRCUMFERENCE FROM HOLOGRAPHIC OBSERVATIONS AND COMPARISON WITH THEORETICAL VALUES A m p l i t u d e s o f v i b r a t i o n c o r r e s p o n d i n g to the 4 x 4 m o d e o f the shell model A are c a l c u l a t e d for the ease when one half-wave a p p e a r s in the front q u a d r a n t of the shell circumference. Figure AI shows the i l l u m i n a t i o n a n d observation directions a n d other geometrical p a r a m e t e r s in the optical set-up used d u r i n g h o l o g r a p h i c recording. For c o p l a n a r displacem e n t , the following g o v e r n i n g e q u a t i o n is valid for a n y p o i n t on the object surface: A~b = 2~-N = ( 4 ~ ' / A ) cos 0 cos ~ ~5(x, y, z),
(AI)
Here A ~ is the c h a n g e in phase, 0 is half the angle b e t w e e n the i l l u m i n a t i o n a n d o b s e r v a t i o n directions, A is the wavelength o f t h e light, ~, is the angle between the direction
b--a
:I=
a
'1
Figure AI. Geometric parameters of the optical set-up used for hologram recordings and reconstruction, tI, holographic plate; S, light source: P, any point on shell surface; R, point on shell surface on axis of symmetry; ~v. ~R, normal vectors at P and R. /~,v, /~R, illumination direction vectors at P and R, ,~l.. o~R,sensitivity vectors at P and R; /~:e./~.~R, observation direction vectors at P and R; ,5~,,fin, amplitudes of vibrations at P and R; a, angle of position vector of P; 0~,. 0~, angles between illumination vector and normal vectors at P and R; O~,+0~, angles bet~veen observation vectors and normaJ vectors at P and R; Op, 0to. angles bet~een sensitivity ~ector and direction of vibration and illumination vector at P. Also, Z >>9 and Z >>d.
SttELL VIBRATION STUDY fly IIOLOGRAPHY
179
of vibrational motion and the direction of the sensitivity vector, and cS(x,); z) is the amplitude of vibration. For point R, lying on the symmetry axis, equation (AI) reduces to
~ = NRA/2 cos OR,
(A2)
where
~Sv = NpA/2 cos Ov cos Or,
(A3)
where
Ov=~(O~,+O~)
and
O,,=a+Ov-O~p.
(A4)
By using the holographic fringe pattern, the positions of the fringe orders (in multiples of a full fringe) are identified on the circumference (as shown in Figure A2). From the optical set-up used during the experimentation, the values of the angles a, 0 n, 02 and are determined for all these marked points. Amplitudes of vibration (absolute values) are computed by using equations (A3) and (A4). These are normalized with respect to the maximum amplitude and are plotted in Figure A2. (Also see Table A1.)
surfoce
Figure A2. Normalized vibration amplitudes over half-wave along circumference of shell model A vibrating in the 4 x 4 mode. x, Experimental; O, theoretical; I, 20..., 9, points on circumference having integer fringe order (from experiment); a, b , . . . , i , points on circumference corresponding to multiples of I/8th angular fraction of half-wave (from theory).
As the basic assumptions in axisymmetric finite element analysis of shell are that the axial and circumferential modes are uncoupled and that the vibration amplitude varies sinusoidally along circumference, the theoretical values of the vibration amplitudes are obtained in a form normalized with respect to the maximum value in the half.wave. Theoretical values are then computed and plotted for comparison (see Table A1 and Figure A2).
180
A.
G.
E T AL.
KIIADAKKAR
i c5r
c5
c~
,:.
~
~
c5c~
~
o ~ I~ I~
~ I~ I~ I ~
.< N
OO
..J ,....
<
F-
?
.< r
OO
N
r
t..
~ It.
O
r~
oN
Q
r rE oM
r-
~
r
FREE VIBRATIONS OF A CYLINDRICAL SHELL BY H O L O G R A P H I C INTERFEROMETRY A. G. KllADAKKAR, R. NARAYANAN, K. RAVI SANKAR, N. R. IYER AND T. V. S. R. APPA RAO Structural Engineering Research Centre, CSIR Campus, Taramani, Madras 600 i 13, India (Received 8 January 1987, and in revised form 7 Afay 1987) Holographic interferometry has been successfully used to record the various mode shapes of a cylindrical shell model with clamped-free boundaries. Experimental results in some cases are compared with those obtained by using the finite element method. Effects of an opening on the natural frequencies and on the associated mode shapes have also been studied experimentally on models with openings of different aspect ratios and are presented in this paper.
I. INTRODUCTION Holographic interferometry is a powerful technique used in experimental studies on various structural forms to measure their surface displacements in either static or dynamic state. The technique has the merit of being highly sensitive, precise, whole-field and non-contacting. Its unique capability to record the m o d e shapes of a resonantly vibrating object showing nodal and antinodal loci in a visual form makes it the most suitable technique for vibration studies on structures of even complex geometry. The holographic technique of vibration analysis appeals greatly, because one can see the object covered by dark and bright interference fringes which represent the ampliiudes of vibration of the object resonating in one of its natural frequencies. This paper gives an account of the application of holographic interferometry to the vibration study of clamped-free cylindrical shell models with emphasis on mode shapes at various resonating frequencies. Studies were also made on shell models which had openings of various sizes and cgnfigurations, to find the influence of an opening on the mode shapes and frequencies. Theoretical analysis of natural frequencies and the m o d e shapes of shells is generally very complex [1]. The generality of the shell equations representing all the degrees of freedom permits a shell to have a wide variety of mode shapes. These could be indicative of different types o f behaviour. For example, in a cylindrical shell, there can be transverse vibrations, longitudinal vibrations and torsional vibrations of tubular beam type, as well as flexural in-plane and extensional vibrations of shell ring type. Because of the complexity of the shell equations, closed form solutions for free vibration analysis are available only for some simple cases. Generally, it is difficult and time-consuming to obtain analytically mode shapes over the entire surface of the shell. Most o f the theoretical methods [2-4] are based on one or more simplifying assumptions so as to render the complex problem comparatively simple and to obtain amenable solutions, in this paper, some natural frequencies and the associated mode shapes obtained by the finite element method for the case of a fixed-free cylindrical shell without an opening are presented. These results are then compared with those found experimentally. 169 0022-460x/88/030169 -i-12 s03.00/0 O 1988 Academic Press Limited
170
a.G. 2. T H E O R Y
KIIADAKKAR
OF IIOLOGRAPHIC
I'T AL.
INTERFEROMETRY
Holographic interferometry can be considered as the interferometric comparison of two or more coherent light wavefronts [5]. The composite reconstruction of these two or more waves is called a holographic interferogram or, simply, a "hologram". Two or more light waves emanating from the same point on an object with an induced deformation causing a change in the phase (or in the optical path length) interfere with each other in the presence of a reference light wave, creating an image of the object superposed with fringes. In a case of mechanical vibration of an opaque object, holographic exposure produces an image of its surface modulated by a system of interference fringes. The brightest of these fringes coincides with the nodal region; that is, the portion of the surface that remains stationary during the vibration. In addition to the bright nodal fringe, a number of black and white fringes may be observed in the holographic image, representing variation of vibration amplitude. If the vibrational motion has a simple time dependence, as in the case of free vibrations of an object, the vibrational amplitude of each point on the surface can be determined from these fringes. In time-averaged holographic interferometry, a hologram of a freely vibrating obiect is recorded by exposing a film to the object and the reference beams simultaneously over a period of time. During such periodic motion, the object spends most of its periodic time near the two extreme positions of maximum amplitude where the direction of vibration reverses and the velocity is zero. An object vibrating sinusoidally with an amplitude of vibration [A(x,y, z)sin tot] will cause optical phase changes in the complex amplitude of light scattered by the object. This phase change A4, at any instant of time is given by
Acl,(x, .t.', z, t) = (2rr/X)2A(x, ); z) sin tot,
( 1)
where A(x,)', z) is the amplitude of vibration at location (x,y, z), A is the wavelength of the light and w is the radian frequency of the vibration. The corresponding complex amplitude of light (object wave) in the plane of the hologram will then be modified due to this phase change from [ a ( x , y ) e i~] to [a(x, y) e~6+'J*~], where a(x, y) is the real amplitude of light wave. A time-averaged hologram is recorded by simultaneously exposing a film to the object wave emanating from the vibrating object and to an otI-axis reference wave for a period of time T. When this hologram is developed and illuminated by the reference beam for reconstruction, the complex anaplitude of the reconstructed wave will be given by [a(x, y) ele'Mr ],
(2)
where Mr is called the characteristic function. Irradiance or intensity of reconstructed hologram image is then given by the square of the above term. When the exposure time T is large compared to ( l / w ) , the characteristic function Mr for sinusoidal vibration becomes
Mr = Jo[(4~/X )A(x, ), z)]
(3)
where Jo is the Bessei function of the first kind and of order zero. The corresponding irradiance of the time-averaged hologram is given by
! = a'-(x, y)J~[(4,-r/A )a(x, y, z)].
(4)
This expression can be interpreted as follows: "the holographic virtual image is modulated by a system of fringes described by the Jo~ term, which can be shown analytically to represent the ettect of free vibration of the object".
StlEI.L V I I | R A T I O N STU DY BY H O L O G R A P I I Y 3. E X P E R I M E N T A L
SET-UP
AND
TEST
171
PROCEDURE
Five cylindrical shell models with clamped-Free boundaries and identical overall dimensions were machined from a solid rod of mild steel. Each model was made by first welding a base plate to the solid rod. The thin cylinder was machined out of the solid rod by boring. This sequence eliminated any possible distortion in geometry of shell models which would have been caused due to welding. The details and geometry of the five models are shown in Figure I. Precision machining of these models gave a dimensional accuracy better than 0.5%. For the holographic recording, the shell model was rigidly clamped through its base plate to the table of the vibrating isolation system. The shell model was excited to resonance by means of a piezoelectric shaker which was connected to an audio oscillator. This oscillator could provide an input signal to the piezoelectric shaker in the frequency range of 20 to 20 000 Hz with adequate force to excite a shell model over the lower half o f its frequency range: i.e., up to 10 kHz. An aluminium exponential horn, mounted on the piezoelectric shaker, was kept in physical contact with the shell model to impart optimum excitation energy. Different shaker locations were used to excite the various modes of
ill
0D=66
I
I] II
Model A
(without opening]
:1 II
l"
Centreline
of openlnc
25
t
i
1
Model B (with squore opening)
i21 ,:, 170 ElevoUon
','i i
Model C (wlth elllptlcol opening]
c~
5
Model D (w~th reclon, ruler open;nq| I
2
,~I
S
170
Plon
1
t Model
(with recton alor open;rig]
Figure I. Geometry o f cylindrical shell models. All dimensions are in mm.
172
E T AL.
A. G. K H A D A K K A R
vibrations. It was verified experimentally from the recorded holograms that the physical contact o f t h e horn with the model does not alter its natural frequencies and mode shapes except in causing a slight change in fringe pattern around the contact point in a very localized manner. An electronic frequency counter was used for determining the frequencies o f the resonantly vibrating models. A standard holographic interferometry set -up was used for recording and reconstruction o f the holograms as shown in Figure 2. The positions of the spatial filter on the object beam path and the holographic plate were kept in the same horizontal plane symmetrically about the shell model at the smallest angle (about 4~ This enabled recording of predominantly out-of-plane deflections to be made.
Piezoelectric shoker
M
Shell mode
It SF.-
L H ~B,5,
M
'"'M
Figure 2. Schematic of experimental set-up. L, lie-Ne laser; M, mirror; S.F., spatial filter; B.S., beam spllttcr; H, holographic plate; P.S., piezoelectric shaker; S.M., shell model.
To identify the resonant frequencies, real-time holography was used. By observing the vibrating model in real time through the hologram (recorded earlier with the model stationary), the resonances at various frequencies were identified whenever the fringe patterns were seen to be frozen. The corresponding frequencies and the locations of the piezoelectric shaker were noted down. The interference fringes, created as a result of the sinusoidal vibrations of a model, were recorded on a holographic plate by a time-averaged holography technique. For further analysis, the fringe patterns corresponding to different resonant frequencies were photographed during reconstruction of the holograms. Identification of resonant frequencies, recording of holograms at these frequencies and photographing the fringe patterns were carried out for all the five models by using the same procedure. As per the definition of modes given in reference [ 1], m is the number ofaxial half-waves along the shell length and n is the number of circumferential waves considered over the
S H E L L V I B R A T I O N S T U D Y IIY I I O L O G R A I ' I I Y
173
entire periphery of the shell. The latter is the same as the number of half-waves of the mode shapes contained in the (front) half o f the shell periphery for which holographic records are made. From a photograph of the fringe pattern, the axial m o d e number could be identified easily by counting the lobes or half-waves. Counting of circumferential half-waves could be tricky, mainly because the curvature of the shell surface in the horizontal plane foreshortens the projected diametric distance. This makes the number of lobes a p p e a r less than the actual number. To remove this ambiguity, the number of half-waves contained in the front quarter sector of the circumference, was counted accurately and extrapolated for the entire shell surface. This procedure, as shown in Figure 3, simplified the determination o f the mode number and also made it accurate. The procedure was found to be necessary particularly for the case of the fourth circumferential mode (n = 4) in the present study.
I
LApporentmodenumber n=3
I
i Figure 3. Counting of circumferential mode number n.
174
A.G. KIIADAKKAR ET ,,~L. 4. RESULTS
The experimental results were obtained for the five cylindrical shell models in the form o f resonant frequencies of various modes and their associated mode shapes. The mode shapes are clearly discerned from the fringe patterns obtained from the time-averaged holograms. A few of the resonant frequencies and the mode shapes for model A are shown in Figure 4. The mode shapes (m-axial, n-circumferential) for these mode numbers can be clearly visualized from observation of the respective fringe patterns. Figure 5 shows the fringe patterns obtained for the five shell models when these were resonantly vibrating in a particular mode (m = 4, n = 3). For a typical circumferential mode (n = 3) and various axial modes (m = 2, 3, 4) o f m o d e l A, the amplitudes ofvibrations were calculated from the fringe patterns. Figure 6 shows deformation variations of the shell model A in the circumferential and axial directions for a few mode numbers. The deformations are normalized and are shown in dimensionless form in the plots. Appendix A shows a calculation for a typical circumferential mode and its comparison with theoretical results. 5. COMPARISON OF RESULTS AND DISCUSSION For obtaining natural frequencies and the corresponding mode profiles of the shell model A, an axisymmetric shell element was used in the finite element analysis. The analysis was carried out with two different sizes of elements and the results were found to be almost identical. The natural frequencies corresponding to the various mode shapes were obtained experimentally for the five models and were compared with the finite element results obtained for model A. The experimental and theoretical values of resonant frequencies of vibration for various modes of the shell model A are given in Table 1 for comparison. Table 2 gives the experimentally determined frequencies o f the five shell models for a few modes. It is possible to obtain theoretical values of resonant frequencies for any mode number o f the shell model. However, due to overlapping of the modes, it becomes increasingly difficult to record holographic fringe patterns when a combination of higher mode number n along tile circumferential direction and lower mode number m along the axial direction or uice rersa is to be rccorded. Another limitation on recording the fringes at high frequency is thai, at higher frequencies, the energy required to excite a model increases considerably. It can be seen from Table 1 that the values of the frequencies of vibration obtained experimentally are, in general, slightly less than those obtained by theory. The amplitudes of the vibrations also compare well, both along the axial and the circumferential directions in case of the shell model A, as can be seen from a few typical plots (Figure 6). From the comparison of tile natural frequencies for different models (five models), it is seen that there is some amount of scatter in most o f the modes (see Table 2). Also, it is evident, that an opening of size up to L / 5 • L / 8 (L being length of the cylindrical shell model) does not attect either the natural frequencies or the mode shapes appreciably. Only in the case of model E, which has a relatively large opening (size 7.5 x 2.5 cm), has the mode shape changed appreciably. However, its natural frequencies at various modes have not changed much compared to those of model A. The reliability of the experimental results depends mainly on the correct simulation o f the boundary conditions and the sensitivity and precision of the measurement techniques. Holographic interferometry has very high sensitivity and precision for measurement o f
SIIELL VIBRATION STUDY BY I'tOLOGRAPIIY
f
.=i. Mode 2 X 2 f = 1965-7 Hz
Mode I X 3 f = 2 6 3 0 . 0 Hz
Mode 2 • f = 2911.6 Hz
Mode :5 X 3 f=3825-OHz
Mode 2 X 4 f =5187.0Hz
Mode 4 X 4 f = 6 3 8 5 . 0 Hz
Figure 4. ttolographic fringe patterns for model A vibrating in different modes.
Model A f = 5 5 6 2 - 8 Hz
Model B f = 5 5 8 3 . 0 Hz
Model D f = 5 5 6 9 . 0 Hz
Model C f = 5 5 9 2 . 0 Hz
Model E f = 5 4 0 0 . 0 Hz
Figure 5. Holographic fringe patterns for the five different models vibrating in the mode m = 3 , n =4.
175
176
A . G . KIIAI)AKKAR E T ,,~L.
-08
-08
t i /,
~
/
-06
-02
Nodes
I
I
,
-I
Mode 2 x 3
=3 -3
i( -0-2 I
-0.2
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--,
=2
-04 /
\
0
3
-06
-04
I -I
=
0 Amplitude Mode3 x 3 Along meridion
1 -1
0 Amplitude Mode 4x3
f Along circumference
Figure 6. Comparison o f normalized amplitudes ofvibratlon of model A for various modes. O, Experimental; theoretical.
TABLE 1
Comparison of experimental and theoretical t,alues of vibration frequencies of shell model A for different modes (values in Hz) Circumferential mode no., n
i
2
Axial mode number, m ^ 3
4
5
!
(a) (b)
916"0 923.3
2
(a) (b)
989-6 993-3
1965.7 2030.9
4393.0 4417.8
3
(a) (b)
2630.0 2646.0
2911.6 2899.5
3825.0' 3760.6
5357.8 5266.4
7180"0 7187.5
4
(a) (b)
5077.9 5030.0
5187.0 5239.0
5562.8 5574.0
6385"0 6314-0
7358.0 7342.9
(a) Experimental, time-averaged holograph)'. (b) Theoretical, finite element method. deformations. Also the technique has a unique self-checking capability to ensure the desired fixity at the b o u n d a r y o f a model. When the shell model was fixed perfectly t h r o u g h its base to the table top, no interference fringes were seen on its base. The high sensitivities o f the other instrumentation, including the frequency counter and oscillator, also ensured highly reliable results from the experiments.
S I I E L L V I B R A T I O N STU DY BY H O L O G R A P I I Y
177
TABLE 2
Comparison of experimental values of frequencies of vibrations offive shell models for different modes (values in Hz) Circumferential mode no. n
Axial mode number, m " 2 3 4
Model
I
A B C D E
916.0
2
A B C D E
989"6 996-0 981"0 -981.0
1965"7 2089"0 1974"0 2079.0 1911-2
4393"0 4422.0 4289"0 -4390.0
3
A B C D E
2630.0 -2650.0 -2637.0
2911.6 3005.0 2887.0 2905.0 2866.0
3825-0 3827-0 3781-0 3894.0 3964-5
5357.8 5328-0 5324.0 5400-0
7040.0
A B C D E
5077.9 4967.0 5028.0 -4974.0
5187.0 5109.0 5148.0 5129-0 5114.0
5562.8 5583.0 5592.0 5569.0 5592.0
6385.0 6368.0 6295.0 6343.0 622 !.0
7358.0
I
4
887.0 885-4
7180.0 7118.0
7384.0 --
It can be c o n c l u d e d from this study that h o l o g r a p h i c i n t e r f e r o m e t r y is an e x t r e m e l y useful a n d p o w e r f u l t e c h n i q u e in v i b r a t i o n studies, p a r t i c u l a r l y for d e t e r m i n a t i o n o f m o d e s h a p e s o f even c o m p l e x structures such as shells when the a c t u a l b o u n d a r y c o n d i t i o n s are properly simulated.
ACKNOWLEDGMENTS T h e a u t h o r s wish to t h a n k their c o l l e a g u e s D r N. L a k s h m a n a n , Messrs G. J a y a r a m a Rao, T. N a r a y a n a n a n d K. Z. Z a c h a r i a for the h e l p a n d s u g g e s t i o n s given by them d u r i n g the c o u r s e o f the investigations. T h e y a r e also grateful to D r M. R a m a i a h , D i r e c t o r , S t r u c t u r a l E n g i n e e r i n g R e s e a r c h Centre, M a d r a s , for his e n c o u r a g e m e n t in c a r r y i n g out this study. This p a p e r is p u b l i s h e d with the p e r m i s s i o n o f the D i r e c t o r , Structural E n g i n e e r i n g R e s e a r c h Centre, M a d r a s .
REFERENCES I. R. D. BLEVINS 1979 Formulasfor Natural Frequeno' and Mode Shape New York: Van Nostrand Reinhold. 2. C. B. SHARMA 1973 Journal of Sound and Vibration 30, 525-528. Frequencies of claraped-free circular cylindrical shells. 3. C. B. SHARMA 1984 77~in-walledStructures 2, 175-193. Free vibrations of clamped-free circular cylinders.
178
A.G. KIIADAKKAR ET AL.
4. G. B. WARBURTON and J. HIGGS 1970 Journal o f S o u n d a n d Vibration II, 335-358. Natural frequencies of thin cantilever cylindrical shell. 5. C. M. VEST 1979 Holographic lnterferometry. New York: John Wiley, first edition.
APPENDIX A: CALCULATION OF DEFLECTIONS ALONG CIRCUMFERENCE FROM HOLOGRAPHIC OBSERVATIONS AND COMPARISON WITH THEORETICAL VALUES A m p l i t u d e s o f v i b r a t i o n c o r r e s p o n d i n g to the 4 x 4 m o d e o f the shell model A are c a l c u l a t e d for the ease when one half-wave a p p e a r s in the front q u a d r a n t of the shell circumference. Figure AI shows the i l l u m i n a t i o n a n d observation directions a n d other geometrical p a r a m e t e r s in the optical set-up used d u r i n g h o l o g r a p h i c recording. For c o p l a n a r displacem e n t , the following g o v e r n i n g e q u a t i o n is valid for a n y p o i n t on the object surface: A~b = 2~-N = ( 4 ~ ' / A ) cos 0 cos ~ ~5(x, y, z),
(AI)
Here A ~ is the c h a n g e in phase, 0 is half the angle b e t w e e n the i l l u m i n a t i o n a n d o b s e r v a t i o n directions, A is the wavelength o f t h e light, ~, is the angle between the direction
b--a
:I=
a
'1
Figure AI. Geometric parameters of the optical set-up used for hologram recordings and reconstruction, tI, holographic plate; S, light source: P, any point on shell surface; R, point on shell surface on axis of symmetry; ~v. ~R, normal vectors at P and R. /~,v, /~R, illumination direction vectors at P and R, ,~l.. o~R,sensitivity vectors at P and R; /~:e./~.~R, observation direction vectors at P and R; ,5~,,fin, amplitudes of vibrations at P and R; a, angle of position vector of P; 0~,. 0~, angles between illumination vector and normal vectors at P and R; O~,+0~, angles bet~veen observation vectors and normaJ vectors at P and R; Op, 0to. angles bet~een sensitivity ~ector and direction of vibration and illumination vector at P. Also, Z >>9 and Z >>d.
SttELL VIBRATION STUDY fly IIOLOGRAPHY
179
of vibrational motion and the direction of the sensitivity vector, and cS(x,); z) is the amplitude of vibration. For point R, lying on the symmetry axis, equation (AI) reduces to
~ = NRA/2 cos OR,
(A2)
where
~Sv = NpA/2 cos Ov cos Or,
(A3)
where
Ov=~(O~,+O~)
and
O,,=a+Ov-O~p.
(A4)
By using the holographic fringe pattern, the positions of the fringe orders (in multiples of a full fringe) are identified on the circumference (as shown in Figure A2). From the optical set-up used during the experimentation, the values of the angles a, 0 n, 02 and are determined for all these marked points. Amplitudes of vibration (absolute values) are computed by using equations (A3) and (A4). These are normalized with respect to the maximum amplitude and are plotted in Figure A2. (Also see Table A1.)
surfoce
Figure A2. Normalized vibration amplitudes over half-wave along circumference of shell model A vibrating in the 4 x 4 mode. x, Experimental; O, theoretical; I, 20..., 9, points on circumference having integer fringe order (from experiment); a, b , . . . , i , points on circumference corresponding to multiples of I/8th angular fraction of half-wave (from theory).
As the basic assumptions in axisymmetric finite element analysis of shell are that the axial and circumferential modes are uncoupled and that the vibration amplitude varies sinusoidally along circumference, the theoretical values of the vibration amplitudes are obtained in a form normalized with respect to the maximum value in the half.wave. Theoretical values are then computed and plotted for comparison (see Table A1 and Figure A2).
180
A.
G.
E T AL.
KIIADAKKAR
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