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Experimental Measurement of Mode Shapes and Frequencies for Vibration of Plates by Optical Interferometry Method

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Journal of

Vibration

and Acoustics

Technical Briefs

Experimental Measurement of Mode

Shapes and Frequencies for

Vibration of Plates by Optical

Interferometry Method

Chi-Hung Huang

Associate Professor, Department of Mechanical

Engineering, Ching Yun Institute of Technology, Chung-Li, Taiwan 320, Republic of China

Chien-Ching Ma

Professor, Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan 106, Republic of China

Most of the published literature for vibration mode shapes of plates is concerned with analytical and numerical results. There are only very few experimental results available for the full field configuration of mode shapes for vibrating plates. In this study, an optical system called the AF-ESPI method with the out-of-plane displacement measurement is employed to investigate experimentally the vibration behavior of square isotropic plates with different boundary conditions. The edges of the plates may either be clamped or free. As compared with the film recording and optical reconstruction procedures used for holographic inter-ferometry, the interferometric fringes of AF-ESPI are produced instantly by a video recording system. Based on the fact that clear fringe patterns will appear only at resonant frequencies, both resonant frequencies and corresponding mode shapes can be obtained experimentally at the same time by the proposed AF-ESPI method. Excellent quality of the interferometric fringe pat-terns for the mode shapes is demonstrated.

关DOI: 10.1115/1.1352018兴

1 Introduction

Holographic interferometry opened new worlds of research by making possible accurate, global measurement of small dynamic surface displacements in a two-step process for a wide variety of objects. For this purpose, different methods of holographic inter-ferometry have been developed for vibration analysis, which have made possible the gathering of a large amount of practical and theoretical information. Unfortunately, the slow and cumbersome process of film development limits the application of holographic vibration analysis in industry. Electronic speckle pattern interfer-Contributed by the Technical Committee on Vibration and Sound for publication in the JOURNAL OFVIBRATION ANDACOUSTICS. Manuscript received Nov. 1999; revised Oct. 2000. Associate Editor: J. Q. Sun.

Fig. 1 Schematic layout of the experimental ESPI setup for out-of-plane displacement measurement

Fig. 2 Geometric dimension and configuration of isotropic square plates

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ometry共ESPI兲, which was first proposed by Butters and Leendertz

关1兴 to investigate the out-of-plane vibration behavior, is a

full-field, noncontact, and real-time measurement technique of defor-mation for structures subjected to various kinds of loadings. As compared with the traditional holographic interferometry共Rastogi

关2兴兲, the interferometric fringe patterns of ESPI are recorded by

video camera, which speeds up the process by eliminating time-consuming chemical development. Since the interferometric im-age is recorded and updated by the video camera every 1/30 sec-ond, ESPI is faster in operation and more insensitive to

Fig. 3 The first 17 mode shapes obtained by using the experimental AF-ESPI system and the finite element analysis for the FCFF plate

Table 1 Comparison of theoretical predicted resonant fre-quencies with experimental results for the FCFF plate

Table 2 Comparison of theoretical predicted resonant fre-quencies with experimental results for the CCFF plate

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environment than holography. Because of the reasons mentioned above, ESPI has become a powerful technique in many academic research and engineering applications. In order to increase the visibility of the fringe pattern and reduce the environmental noise simultaneously, an amplitude-fluctuation ESPI共AF-ESPI兲 method was proposed by Wang et al. 关3兴 for out-of-plane vibration measurement. In the amplitude-fluctuation ESPI method, the ref-erence frame is recorded in a vibrating state and subtracted from the incoming frame. Ma and Huang 关4,5兴 used the AF-ESPI method to investigate the three-dimensional vibrations of piezo-electric rectangular parallelepipeds and cylinders; both the reso-nant frequencies and mode shapes were presented and discussed in detail.

The study of the vibration behavior of a plate is a problem of great practical interest. However, very few experimental results, especially for the full field measurement of mode shapes, are available in the literature. In this paper, the optical method based on the amplitude-fluctuation ESPI 共AF-ESPI兲 is employed to study experimentally the resonant characteristics of free vibration for isotropic square plates with different boundary conditions. The optical arrangement of the AF-ESPI method for the out-of-plane vibrating measurement is show schematically in Fig. 1. The

ad-vantage of using the AF-ESPI method is that both resonant fre-quencies and the corresponding mode shapes can be obtained si-multaneously from the experimental investigation. The fringe patterns shown in the experimental results are correspondent to the vibrating mode shapes. Two cases are studied which involve the possible combinations of free and clamped edge conditions; they are Free-Clamped-Free-Free共FCFF兲 and Clamped-Clamped-Free-Free共CCFF兲. In addition to the AF-ESPI experimental tech-nique, numerical computations based on a finite element package are also presented and good agreements of resonant frequencies and mode shapes are found for both results.

2 Experimental Results and Numerical Analysis for Vibrating Plates

Two isotropic aluminum plates共6061T6兲 are used in this study for experimental investigations and numerical calculations, where the material properties of the plate are mass density ␳⫽2700 kg/m3, Young’s modulus E⫽70 Gpa and Poisson’s ratio␯⫽0.33. By using the combinations of free共F兲 and clamped 共C兲 edges, the resonant frequencies and mode shapes of FCFF and CCFF plates Fig. 3Continued

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are investigated. The geometric dimensions of two plates with different boundary conditions are shown in Fig. 2. The thickness of two isotropic plates are all 1 mm. The schematic layout of a self-arranged AF-ESPI optical system as shown in Fig. 1 is em-ployed to perform the out-of-plane vibration measurement for the resonant frequency and the corresponding mode shape. As shown in Fig. 1, a 30-mW He-Ne laser with wavelength␭⫽632.8 nm is used as the coherent light source. The laser beam is divided into two parts, the reference and object beam, by a beamsplitter. We use a CCD camera 共Pulnix company兲 and a P360F 共Dipix Technologies Inc.兲 frame grabber with DSP on board to re-cord and process the images. The object beam travels to the specimen and then reflects to the CCD camera. The reference beam goes directly to the CCD camera via a mirror and the reference plate. It is important to note that the optical path and the light intensity of these two beams should remain identical in the experimental setup. The CCD camera converts the intensity dis-tribution of the interference pattern of the object into a corre-sponding video signal at 30 frames per second. The signal is electronically processed and converted into an image on the video

monitor. The interpretation of the fringe image is similar to read-ing a contour map. In order to increase the intensity of light re-flection of specimens and the contrast of fringe patterns, the sur-faces of plates are coated with white paint which is mixed with fine seaweed powder. The isotropic plate is excited by a pi-ezostack actuator 共PI company兲 which is attached behind the specimen. To achieve the sinusoidal output, a function generator HP33120A共Hewlett Packard兲 connected to a power amplifier 共NF corporation兲 is used.

Numerical results of resonant frequencies and mode shapes are calculated by using the commercially available software, ABAQUS finite element package. The eight noded two-dimensional quadrilateral thick shell element共S8R5兲 and reduced integration scheme are used to analyze the problem. This element approximates the Midlin-type element that accounts for rotary inertia effects and first order shear deformations through the thick-ness. The results presented in Tables 1–2 show generally good agreement between the numerically predicted and experimentally measured resonant frequencies. Figures 3–4 are the mode shapes for both experimental measurements and numerical Fig. 4 The first 15 mode shapes obtained by using the experimental AF-ESPI system and the finite element analysis for the CCFF plate

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simulations. There are 17 modes and 15 modes presented for FCFF and CCFF plates, respectively. For the finite element cal-culations, the contours of constant displacement for resonant mode shapes are plotted in order to compare with the experimen-tal observation. In Figs. 3–4, we indicate the phase of displace-ment in finite eledisplace-ment results as solid or dashed line, the solid lines are in the opposite direction to the dashed lines. The transi-tion from solid lines to dashed lines corresponds to a zero dis-placement line, or nodal line. The zero-order fringe, which is the brightest on the experimental image, represents the nodal lines of the vibrating square plate at resonant frequencies. The rest of the fringes are contours of constant amplitudes of displacement. Ex-cellent quality of the experimental fringe patterns for vibration mode shapes are presented in Figs. 3–4. The mode shapes ob-tained by experimental results can be checked by the nodal lines and fringe patterns with the numerical finite element calculations in excellent agreement.

3 Conclusions

Optical techniques have been shown to have certain advantages for vibration analysis and ESPI has been applied to many

vibra-tion problems. The advantages of the optical ESPI method include noncontact and full-field measurement, real-time observation, sub-micron sensitivity, validity of both static deformation and dy-namic vibration, and direct digital image output. Because ESPI uses video recording and display, it works in real time to measure dynamic displacement. Its real-time nature makes it possible to implement this technique for vibration measurement. A self-arranged amplitude-fluctuation ESPI optical setup with good vis-ibility and noise reduction has been established in this study to obtain the resonant frequencies and the corresponding mode shapes of free vibrating isotropic square plates at the same time. Two different types of boundary conditions are investigated in this study and more than thirty excellent quality mode shapes are presented by the proposed experimental optical interferometry method. Numerical calculations of resonant frequencies and mode shapes based on a finite element package are also performed and excellent agreements are obtained if compared with experimental measurements.

Acknowledgments

The authors gratefully acknowledge the financial support of this research by the National Science Council 共Republic of China兲 under Grant NSC 87-2218-E002-022.

References

关1兴 Butters, J. N., and Leendertz, J. A., 1971, ‘‘Speckle Pattern and Holographic

Techniques in Engineering Metrology,’’ Opt. Laser Technol., 3, No. 1, pp. 26–30.

关2兴 Rastogi, P. K., 1994, Holographic Interferometry, Springer-Verlag. 关3兴 Wang, W. C., Hwang, C. H., and Lin, S. Y., 1996, ‘‘Vibration Measurement

by the Time-Averaged Electronic Speckle Pattern Interferometry Methods,’’ Appl. Opt., 35, No. 22, pp. 4502–4509.

关4兴 Huang, C. H., and Ma, C. C., 1998, ‘‘Vibration Characteristics for

Piezoelec-tric Cylinders Using Amplitude-Fluctuation Electronic Speckle Pattern Inter-ferometry,’’ AIAA J., 36, No. 12, pp. 2262–2268.

关5兴 Ma, C. C., and Huang, C. H., 2001, ‘‘The Investigation of Three-Dimensional

Vibration for Piezoelectric Rectangular Parallelepipeds by Using the AF-ESPI Method,’’ IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 48, No. 1, pp. 142–153.

A Comment on Boundary Conditions

in the Modeling of Beams with

Constrained Layer Damping

Treatments

Peter Y. H. Huang, Per G. Reinhall, and

I. Y. Shen

Mechanical Engineering Department, University of Washington, Seattle, Washington 98195-2600 关DOI: 10.1115/1.1349887兴

Introduction

The most commonly used mathematical formulation for a beam with a constrained layer damping treatment was developed by Mead and Markus关1兴. In their formulation, the base beam and the constraining layer are set to undergo identical transverse deflec-tions and the longitudinal displacements of the base beam and the constraining layer are set to be related via the thickness and Contributed by the Technical Committee on Vibration and Sound for publication in the JOURNAL OFVIBRATION ANDACOUSTICS. Manuscript received Jan. 2000; revised Sept. 2000. Associate Editor: J. Q. Sun.

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Young’s modulus of each layer. As a result, the Mead-Markus formulation only applies to a class of systems with boundary con-ditions described in Markus et al.关2兴, Rao 关3兴, and Lifshitz and Leibowitz关4兴. Trompette et al. 关5兴 were first to investigate the practical implication of this issue by investigating a specific con-strained layer damped cantilevered beam. The purpose of this study was to investigate a broader range of boundary and damping treatment configurations. The error of the Mead-Markus formula-tion was investigated as a funcformula-tion of the thickness of viscoelastic layer and was shown to be large for certain common boundary conditions. A modified Mead-Markus formulation that allows the longitudinal motion of the base beam and constraining layer to be independent from each other is suggested as a remedy.

Cantilevered Beam

Figure 1 shows the displacement variables fields associated with the Mead-Markus model for an Euler-Bernoulli beam. The transverse displacement is represented by w(x,t), and longitudi-nal displacement of the j-th layer is represented by uj(x,t), where

x is the position along the beam and t is time. The subscripts 1, 2 and 3 refer to the base beam, the viscoelastic layer, and the con-straining layer, respectively. All the layers have length L and width b. The j-th layer has thickness hj, and storage moduli Ej

and Gj. In the Mead-Markus model, assuming equilibrium of

axial forces results in E1h1b

⳵u1

⳵x ⫽⫺E3h3b ⳵u3

⳵x (1)

where E1h1b and E3h3b are axial rigidities of the base beam and

the constraining layer, respectively. Integration of Eq. 共1兲 with respect to x leads to

E1h1u1⫽⫺E3h3u3⫹C (2)

where C is an arbitrary function of time. In the original derivation by Mead and Markus关1兴, it is assumed that C⫽0, resulting in

u1⫽

E3h3

E1h1

u3 (3)

Note that u1and u3are no longer independent.

The presence of Eq.共3兲 has a profound implication in that the dependence of u1and u3limits the boundary conditions allowed

in the Mead-Markus formulation. Figure 2 illustrates a simple lab setup consisting of a cantilevered base beam and a free-free con-straining layer. The proper boundary conditions at the left end should be

⳵u1共0,t兲

⳵x ⫽0 and u3共0,t兲⫽0 (4)

which implies that u1(0,t)⫽0 for the free-free constraining layer

and violates Eq. 共3兲. To incorporate these boundary conditions properly, one can modify the Mead-Markus formulation by as-suming that u1and u3are independent as was done by Shen关6兴 共and experimentally validated by Huang et al. 关7兴兲. Figure 3共a兲

Fig. 1 Displacement variable fields of the Mead-Markus model

Fig. 2 Experimental setup

Fig. 3aPredicted and experimental frequency response of a cantilevered beambError produced by a Mead-Markus model as compared to the modified theory for a cantilevered beam

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Fig. 4aFrequency response of a clamped A - clamped A beam. Clamped B boundary conditions were required for the original Mead-Markus formulation.bError produced by a Mead-Markus model as compared to the modified theory for a clamped A - clamped A beam Table 1 Various boundary conditions for the Mead-Markus and the modified Mead-Markus formulation

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shows the comparison between predictions given by the Mead-Markus formulation 共where we are forced to set u1(0,t) ⫽u3(0,t)⫽0 at the left end兲 and the modified Mead-Markus

for-mulation, and experimental results共both with boundary conditions at the left end given by Eq.共4兲兲. Good agreement between the modified theory and experiments can be seen and a significant error in the first natural frequency and corresponding amplitude predicted by the Mead-Markus model can be observed.

For the same boundary conditions, Fig. 3共b兲 shows the error produced by not being able to use correct boundary conditions in Mead-Markus model as compared to the modified theory. As the thickness of the viscoelastic layer was increased, it was found that the error in amplitude of the first mode increased significantly. The errors in the higher modes were found to be insignificant.

Other Boundary Conditions

Many boundary conditions are compatible with the Mead-Markus formulation. Table 1 lists a range of boundary conditions, most of which can be modeled using Mead-Markus. No signifi-cant difference was detected between the predicted frequency re-sponse using Mead-Markus and the modified theory when model-ing boundary conditions compatible with the Mead-Markus formulation.

As soon as the boundary conditions are not compatible, how-ever, significant error in the prediction can occur when using a

Mead-Markus model. Figures 4 and 5 show results for when the base beam was subjected to clamped-clamped and pinned-pinned boundary conditions and the constraining layer was free at both ends. Neither boundary condition is compatible with Mead-Markus and could therefore not be modeled using this theory without modification. The clamped-clamped case was modified such that the constraining layer was also constrained at the ends. For the second case, the pinned ends of the base beam were re-leased such that they were free to move axially. Figures 4 and 5 show the error produced by Mead-Markus model with these modi-fied boundary conditions as compared to using the actual bound-ary conditions with the modified formulation. For both types of boundary conditions, the difference between the two formulations was shown to increase significantly with decreasing thickness of the viscoelastic layer.

The incompatibility of Mead-Markus with these two boundary conditions did not result in significant error in the higher modes. No appreciable difference between the two formulations was seen for mode two and higher.

Conclusions

Careful attention to the boundary conditions must be exercised when predicting the behavior of the first mode using the Mead-Markus formulation. For certain boundary conditions, more accu-Fig. 5aFrequency response of a pinned A - pinned A beam. Pinned B

bound-ary conditions were required for the original Mead-Markus formulation.bError produced by a Mead-Markus model as compared to the modified theory for a pinned A - pinned A beam

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rate results can be obtained by using a formulation that allows independent longitudinal motion of the beam and the constraining layer.

Acknowledgment

This material is based upon work supported by Army Research Office under agreement of No. DAAG 55-98-1-0387.

References

关1兴 Mead, D. J., and Markus, S., 1969, ‘‘The Forced Vibration of a Three-Layer

Damped Sandwich Beam with Arbitrary Boundary Conditions,’’ J. Sound Vib., 10, No. 2, pp. 163–179.

关2兴 Markus, S., Oravsky, V., and Simkova, O., 1974, ‘‘Damping Properties of

Sandwich Beams with Local Shearing Prevention,’’ Acustica, 31, pp. 132– 138.

关3兴 Rao, D. K., 1978, ‘‘Frequency and Loss Factors of Sandwich Beams under

Various Boundary Conditions,’’ J. Mech. Eng. Sci., 20, No. 5, pp. 271–282.

关4兴 Lifshitz, J. M., and Leibowitz, M., 1987, ‘‘Optimal Sandwich Beam Design

for Maximum Viscoelastic Damping,’’ Int. J. Solids Struct., 23, No. 7, pp. 1027–1034.

关5兴 Trompette, P., Boilot, D., and Ravanel, M. A, 1978, ‘‘Effect of Boundary

Conditions on the Vibration of a Viscoelastically Damped Cantilever Beam,’’ J. Sound Vib., 60, No. 3, pp. 345–350.

关6兴 Shen, I. Y., 1994, ‘‘Hybrid Damping Through Intelligent Constrained Layer

Treatments,’’ ASME J. Vibr. Acoust., 116, pp. 341–349.

关7兴 Huang, P. Y., Reinhall, P. G., and Shen, I. Y., 1999, ‘‘A Study of Constrained

Layer Damping Models under Clamped Boundary Conditions,’’ Proceedings of the ASME International Mechanical Engineering Congress and Exposition.

數據

Fig. 2 Geometric dimension and configuration of isotropic square plates
Table 1 Comparison of theoretical predicted resonant fre- fre-quencies with experimental results for the FCFF plate
Fig. 4 „ Continued …
Fig. 1 Displacement variable fields of the Mead-Markus model
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