Experimental measurements on transverse vibration characteristics of piezoceramic rectangular plates by optical methods

This study provides two non-contact optical techniques to investigate the transverse vibration characteristics of piezoceramic rectangular plates in resonance. These methods, including the amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) and laser Doppler vibrometer (LDV), are full-field measurement for AF-ESPI and point-wise displacement measurement for LDV, respectively. The edges of these piezoceramic rectangular plates may either be fixed or free. Both resonant frequencies and mode shapes of vibrating piezoceramic plates can be obtained simultaneously by AF-ESPI. Excellent quality of the interferometric fringe patterns for the mode shapes is obtained. In the LDV system, a built-in dynamic signal analyzer (DSA) composed of DSA software and a plug-in waveform generator board can provide the piezoceramic plates with the swept-sine excitation signal, whose gain at corresponding frequencies is analyzed by the DSA software. The peaks appeared in the frequency response curve are resonant frequencies. In addition to these optical methods, the numerical computation based on the finite element analysis is used to verify the experimental results. Good agreements of the mode shapes and resonant frequencies are obtained for experimental and numerical results.     

EXPERIMENTAL MEASUREMENTS BY AN OPTICAL METHOD OF RESONANT FREQUENCES AND MODE SHAPES FOR SQUARE PLATES WITH ROUNDED CORNERS AND CHAMFERS

Most of the work done on vibration of plates published in the literature includes analytical and numerical studies with few experimental results available. In this paper, an optical system called the amplitude-fluctuation electronic speckle pattern interferometry for the out-of-plane displacement measurement is employed to investigate the vibration behavior of plates with rounded corners and with chamfers. The boundary conditions are traction free along the circumference of the plate. 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 using the present method. Numerical calculations by finite element method are also performed and the results are compared with the experimental measurements. Good agreements are obtained for both results. It is interesting to note that the mode number sequences for some resonant modes are changed. The transition of mode shapes from the square plate to the circular plate is also discussed.     

The separation of out-of-plane displacement from in-plane components by fringe carrier method based on large image-shearing ESPI

A fringe carrier method for separating out-of-plane displacement from in-plane components based on large image-shearing electronic speckle pattern interferometry (ESPI) is presented. If the test object is respectively illuminated by two expanded symmetric illuminations in large image-shearing ESPI, two interferometers are formed. Carrier fringe patterns can be introduced by tilting reference surface a small angle. The carrier fringe patterns are demodulated after deformation of the object. Two phase maps, which include out-of-plane and in-plane displacement, can be obtained by using Fourier transform. Then out-of-plane displacement can be easily separated from in-plane displacement by simple operation between two unwrapped phase distributions. The principle of spatial carrier frequency modulation in large image-shearing ESPI is discussed. A typical three-point-bending experiment is completed. Experimental results are offered. The results show that the method offers high visibility of carrier fringes. And the system presented does not need a special beam as a reference light and has simple optical setup.     

LINEAR VIBRATION CHARACTERISTICS OF CABLE-BUOY SYSTEMS

A theoretical model for the linear vibration of a cable tensioned by a subsurface buoy is developed. The equilibrium of the cable-buoy system subject to drag is evaluated using an approximate closed-form solution whose range of validity is confirmed through comparison with numerical solutions. The three-dimensional equations of cable-buoy motion are linearized about this equilibrium and then used to assess vibration characteristics. The characteristic equations for the natural frequencies of both in-plane and out-of-plane vibration modes are derived. The in-plane natural frequency spectrum exhibits the curve veering phenomena due to asymmetry of the associated mode shapes. Parameter studies reveal the dependencies of the in-plane and out-of-plane vibration modes on the cable tension, the buoy mass, and the current velocity.     

Application of ESPI to three-dimensional vibration measurements

There are many situation in vibration analysis where it is necessary or desirable to make three-dimensional measurements. By using three different illumination geometries around a single imaging system, electronic speckle pattern interferometry (ESPI) can be used to measure the orthogonal components of vibration amplitude independently. These can be combined to determine the three-dimensional amplitude and mode shape. Examples of experimental results are presented for volume vibrations of a thick cylinder, identification of vibration modes of a turbocharger blade, and observation of in-plane modes in a thin plate.     

Mechanical characterization of unplasticised polyvinylchloride thick pipes by optical methods

In this work a number of techniques (electronic speckle pattern interferometry, holographic interferometry, strain gauge and finite element method) are brought to bear in order to establish consistency in the results of strain measurement. This is necessary if optical non-destructive testing methods, such as those used here, are to gain acceptance for routine industrial use. The FE model provides a useful check. Furthermore, ESPI fringe data facilitates the extension of FE models, an approach that is of growing importance in component testing.

The use of in-plane and out-of-plane sensitive electronic speckle pattern interferometry (ESPI) for non-destructive material characterization of thick unplasticised polyvinylchloride (uPVC) pipes is presented. A test rig has been designed for stressing pipes by internal pressure. ESPI gives a complete mapping of the displacement field over the area imaged by the video camera. The results for the strain of uPVC obtained from ESPI data and from strain gauges are in good agreement. The value of Young's modulus has been obtained from the fringe data and compared with results obtained using holographic interferometry and from strain gauge measurements. The FE model also produces fringe data that is consistent with the ESPI results.     

Experimental investigation of free vibrations of clamped sector plates

The real time technique of time-averaged holographic interferometry has been applied to determine the natural frequencies and the corresponding mode shapes for the transverse vibrations of clamped wedge-shaped and ring-shaped sector plates. Over 200 resonant modes have been obtained for wedge-shaped sector plates and over 170 for ring-shaped sector plates. The natural frequencies obtained have been expressed in terms of a dimensionless frequency parameter, and the results are shown graphically as a function of the sector angle for the wedge-shaped plates and of the radii ratio for the ring-shaped sector plates, respectively. Some of the present results for wedge-shaped plates are compared with the analytical values obtained by other authors.     

Additive-subtractive phase-modulated speckle interferometry: Fringe visibility under partial decorrelation

Additive-subtractive phase modulated speckle interferometry (ASPMSI) is a technique that minimizes the susceptibility of speckle methods to environmental noise while providing fringes of good visibility. The method requires the acquisition of two consecutive video frames of additive-speckle images of the same two deformed states of an object at a rapid enough rate such that ambient noise is not a problem. The additive-speckle images as expected are of very poor visibility due to the presence of the self-interference term. An interframe phase-modulation is introduced and the two additive-speckle images are digitally subtracted to improve the fringe visibility by removing the self-interference term. The ASPM-SI method works with in-plane and out-of-plane deformation sensitive ESPI as well as with displacement-gradient sensitive speckle-shearing interferometry. It is shown that the ASPM-SI scheme has higher visibility than conventional additive-SI and performs consistently better than subtractive-SI schemes in the presence of partial interframe speckle decorrelating optical noise. Furthermore, it is shown that the fringe visibility of the out-of-plane displacement sensitive interferometer which uses a protected reference beam separate from the object beam can be made to be essentially unity even at complete interframe decorrelation.     

Natural frequencies and mode shapes of flexural vibration of plates: laser-interferometry detection and solutions by Ritz's method

This paper presents an experimental and theoretical study of flexural symmetric vibration modes of a linear elastic plate. A laser interferometer is used as detector of the free vibration of a rectangular parallelepiped-shaped aluminium plate. The vibration spectrum gives the lowest natural frequencies of the sample. Assumption that the vibration of the plates may be described by some approximate theories is proven to be inconsistent. The Ritz method, with products of powers of the co-ordinates as basis functions, is applied to obtain the lowest flexural natural frequencies. Three-dimensional solutions are obtained, unlike those provided by simpler theories. The experimental results are compared with the numerical predictions and a good agreement is obtained. Finally, forced motion is applied to the centre of the plate and the out-of-plane and in-plane displacement components for the first symmetric mode are measured. A good fit of the calculated values to the experimental values is found.     

Multiple modes in the vibration of cantilevered shells

The term multiple modes describes pairs of modes which are similar in shape but occur at different frequencies. This phenomenon has been observed in holographic vibration test results for a turbine blade. Pairs of modes were found, such as two modes which both resembled first torsional modes. In this investigation holographic interferometry was used to verify the earlier results for the turbine blade and to investigate three shell segments simulating blades. The shells ranged in size from moderately to very thick with length to thickness ratios of 16, 8 and 5·6. The blade geometry is characterized by a circumferential angle of 142° and a ratio of length to inner radii arc length near 1·0. In addition, a NASTRAN finite element analysis was performed on these simulated blades. Both mode shapes and frequencies were found to be in good agreement with the results from the experiment. The multiple mode phenomenon was found to be an artifact of the holographic experiment. Pairs of modes were found in the NASTRAN results for the simulated blades in which the out-of-plane displacements (those seen in the hologram) were very similar, but for which the displacements in the plane of the hologram differed significantly. Thus, the two modes which appeared in the experimental results as first torsional modes were seen to include quite different in-plane displacements. The two modes are therefore quite different and do not contradict the normal result, which may be justified from such elementary considerations as a Rayleigh quotient, that similar modes must produce similar frequencies.     

Influence of in-plane displacement and strain components on slope fringe distributions in double-aperture speckle wedge-shearing interferometry

The influence of in-plane displacement and strain components on slope (first-order derivative of out-of-plane displacement component) fringe distributions in double-aperture speckle wedge-shearing interferometry is discussed in detail. The research results show that only the in-plane displacement component parallel to the centre line of double apertures has an influence on the slope fringe distributions. It is also shown that the in-plane strain components have no influence on the slope fringe distributions when utilising normal illumination and an axisymmetric system. A theoretical analysis and an experimental demonstration are given. The experimental results are in good agreement with the quantitative analysis.     

Automatic analysis of three-dimensional displacement field using ESPI

Electronic speckle pattern interferometry (ESPI) has been used to study the in-plane and out-of-plane displacements of the object. In order to improve the accuracy, a Fourier filtration algorithm has been used to remove the speckle noise and get the holographic-quality ESPI fringe pattern. The processing steps of this method are described in detail in this paper. In addition, a phase shifter, which is easily used with simplified structure and high stability, is also presented. It can be applied to various coherent arrangements in experiments to obtain phase shifted fringe patterns. Experiments of determining the 3D displacement field of a circular fixed plate with a uniform load have been carried out using these methods. The results presented in this paper indicate that the accuracy of this method is satisfactory.     

Studies on vibration of some rib-stiffened cantilever plates

An experimental study was carried out to determine the resonant mode shapes and frequencies of some rib-stiffened skew cantilever plates by holographic interferometry. The influences of varying the sweep back angle, the rib stiffness and the aspect ratio, and the effect of varying the boundary conditions at the root chord, on the frequencies and mode shapes were also investigated. Results of the above investigation and also those of a comparative study with the finite element solution obtained for some of the cases studied are presented and discussed.     

Interferometric studies of a piano soundboard

Electronic speckle pattern interferometry has been used to study the deflection shapes of a piano soundboard. A design for an interferometer that can image such an unstable object is introduced, and interferograms of a piano soundboard obtained using this interferometer are presented. Deflection shapes are analyzed and compared to a finite-element model, and it is shown that the force the strings exert on the soundboard is important in determining the mode shapes and resonant frequencies. Measurements of resonance frequencies and driving-point impedance made using the interferometer are also presented.     

Estimation of the 2D measurement error introduced by in-plane and out-of-plane electronic speckle pattern interferometry instruments

Optical interferometric metrology techniques are being increasingly used in industry. These techniques assure a greater accuracy in measuring displacements caused by deformations. One such technique, electronic speckle pattern interferometry (ESPI), has been used successfully to measure in-plane and out-of-plane deformations. The usual model describing ESPI instruments behaviour is only valid in or near the centre of the illuminated surface. In general, this model is used as such for all the points on the surface creating thus an approximate figure of the reality. This study has led to an improved 3D vectorial model, allowing us to assess qualitatively and quantitatively what is actually measured throughout all the inspected surface. Calculations with practical parameters taken from real ESPI instruments designed in the Joint Research Centre of Ispra (Italy) were carried out. The results showing three-dimensional diagrams of the measurement errors are presented and discussed.     

Separation of vibration fringe data from rotating object fringes using pulsed ESPI

In industrial and other types of non-controlled environments, an unbalanced rotating object may present characteristic out-of-plane vibration amplitude at a specific frequency. For this type of cases and as a first step towards a complete evaluation, it is only desired to visualize the effect of the vibration on the rotating object, or vice versa, for instance to achieve object balancing. Real time optical non-intrusive measurement techniques such as pulsed electronic speckle pattern interferometry (ESPI), are well suited to study this rotating-vibrating object. The advantage offered by ESPI is that real-time fringe data is qualitatively analyzed while being observed on a TV monitor. The present paper proposes a qualitative method, based on pulsed ESPI, to separate rotation fringes from fringes solely related to vibration. The method relies on a high precision scheme that synchronizes and fixes an object point during rotation, without the use of an optomechanical object derotator.     

The investigation of high frequency vibration modes of PZT-4 transducers using ESPI techniques with reference beam modulation

In this paper the experimental vibration amplitudes together with the vibration modes of the PZT-4 transducer radiating front face are presented for generation in various media. The studies were carried out using an electronic speckle pattern interferometer (ESPI) with reference beam modulation to give resolution down to 20 Å of surface amplitude. Analogies are drawn between these transducers and the resonant vibration modes of thin plates. The relationship between the surface vibration amplitudes and the radiation beam pattern of the transducers is also presented.     

Contrast enhancement of pulsed ESPI addition fringes

In this paper, the contrast enhancement produced by the subtraction of two sets of double pulse electronic speckle pattern interferometry (ESPI) addition fringes is investigated. It is shown that in most practical cases this process will produce a second set of interference fringes in addition to those which can be directly related to the vibration mode of interest. The formation of this second fringe pattern is explained by theory and computer simulation. Finally, the results are verified using an ESPI system with a pulsed Nd: YAG laser.     

Study of a vibrating plate: comparison between experimental (ESPI) and analytical results

Real-time electronic speckle pattern interferometry (ESPI) was used for tuning and visualization of natural frequencies of a trapezoidal plate. The plate was excited to resonant vibration by a sinusoidal acoustical source, which provided a continuous range of audio frequencies. Fringe patterns produced during the time-average recording of the vibrating plate—corresponding to several resonant frequencies—were registered. From these interferograms, calculations of vibrational amplitudes by means of zero-order Bessel functions were performed in some particular cases. The system was also studied analytically. The analytical approach developed is based on the Rayleigh–Ritz method and on the use of non-orthogonal right triangular co-ordinates. The deflection of the plate is approximated by a set of beam characteristic orthogonal polynomials generated by using the Gram–Schmidt procedure. A high degree of correlation between computational analysis and experimental results was observed.     

Free vibrations of spatial Timoshenko arches

This paper addresses the evaluation of the exact natural frequencies and vibration modes of structures obtained by assemblage of plane circular arched Timoshenko beams. The exact dynamic stiffness matrix of the single circular arch, in which both the in-plane and out-of-plane motions are taken into account, is derived in an useful dimensionless form by revisiting the mathematical approach already adopted by Howson and Jemah (1999 [18]), for the in plane and the out-of-plan natural frequencies of curved Timoshenko beams. The knowledge of the exact dynamic stiffness matrix of the single arch makes the direct evaluation of the exact global dynamic stiffness matrix of spatial arch structures possible. Furthermore, it allows the exact evaluation of the frequencies and the corresponding vibration modes, for the distributed parameter model, through the application of the Wittrick and Williams algorithm. Consistently with the dimensionless form proposed in the derivation of the equations of motion and the dynamic stiffness matrix, an original and extensive parametric analysis on the in-plane and out-of-plane dynamic behaviour of the single arch, for a wide range of structural and geometrical dimensionless parameters, has been performed. Moreover, some numerical applications, relative to the evaluation of exact frequencies and the corresponding mode shapes in spatial arched structures, are reported. The exact solution has been numerically validated by comparing the results with those obtained by a refined finite element simulation.