It is seen that the oscillations are not isochronous. This agrees well with previous results reported in reference [ 9 ] Fig.

However, our approach allows for a significant reduction of the computation time in [ 9 ] to obtain similar results the approximation of the fourth order is taken. Amplitude of the initial disturbance versus oscillation frequency of stringer shell 1 — according the proposed method, 2 — data [9].

In solving various kinds of problems of modern development and improvement of thin-walled machine elements operating under the conditions of intensive manufacturing process, the use of a holographic interferometry method should be emphasized [ 10 , 11 , 14 ]. It allows for a more accurate and complete investigation of shell structures under complex stress-strain state. The accuracy of interpretation of holographic interferograms is mainly determined by the number of support points of the design used for the construction regarding displacements and stresses.

Improvement of the accuracy requires a large amount of routine preparations for writing the coordinates of points and their corresponding numbers of lines when developing data on a computer, which is particularly important in the case of an experiment. The existing methods of automated data entry and processing of interferograms yield, as a rule, the specific configuration of the optical system and the types of strain state flat, one-dimensional, etc.

In addition, although several authors proposed methods of interpretation [ 10 ], they did not fully take into account the statistical nature of input data.

### Theory and Applications

For cylindrical shells a method for automated processing of the results of the holographic research has been proposed, which eliminates the above drawbacks [ 12 ]. Next, we have extended this technique to study the motion of shell structures of zero Gaussian curvature which is based on modern means of an interactive data processing. The surface of zero Gaussian curvature can be approximated with sufficient accuracy with respect to the system of flat rectangular panels whose sides are segments close to the case which occurred during the analysis of generators. To determine all components points of the displacement vector, three holograms of a circuit design interferometer based on a reference beam is used.

The interferometer is shown schematically in Fig 7. The scheme of the interferometer 1 — laser generator, 2 — mirror, 3 — expanding lens, 4 — studied object, 5 — camera. After registering the two exposures, i. Let us enter the order line using a computer in the following manner. The photos of interferograms are scanned and entered into the computer memory in the form of graphic files with the extension, for example, jpg, which is the most popular choice of compression of graphic information on all platforms, or equivalently in other file formats.

Next, the file is displayed on the screen in a specially designed box on the toolbar image processing. The information produced is removed by a successive mouse click on the corresponding image points at the request of a specially created database. Algorithms for further processing of the data are widely described in [ 12 ]. Further calculations are performed in the X O Y system in which the entered coordinates of the points of lines of equal order are transformed by the formulas.

While computing the physical coordinates, the approximation of the mentioned shell surface by the system of folds is applied Fig. Therefore, the so far obtained arrays of point coordinates of the lines of equal order corresponding to the three noncoplanar directions of observation allow us to approximate the surface bands. The most appropriate method to do this is the structural analysis of extrapolation MSEA [ 6 ] using step by step the best choice of the model.

Indeed, the formalization of the input source data for inhomogeneous stress-strain state requires a large number of points. The above method allows us to determine the coordinates of the centers of bands up to 0. This procedure is used to significantly increase the number of input points up to for each direction of observation. The use of spline functions for smoothing requires the enumeration of all coordinates of control points for each calculation of the order of the band.

### 1. Introduction

This slows down the calculation and requires a significant memory space. In addition, these disadvantages are compounded by the increasing number of control points. The use of MSEA allows each step to obtain unbiased estimates of the effective coefficients of the model to ensure a maximum plausible value of the order of the reference points [ 11 ]. This eliminates the problem of choosing a smoothing parameter, with the number needed to calculate the coefficients one order of magnitude smaller than the number of coordinates of reference points. In addition, the incremental method allows us to formalize the process of selecting the optimal order of approximating polynomial based on the assessment of the significance of the model and the adequacy of its source data.

Note that in this case the number of points is much larger than the number of estimated parameters, which suggests a considerable power of the statistical tests like those of Student's, Fisher and Durbin-Watson , and indicates the validity of hypotheses taken in selecting the best model. An increase of the number of points improves a regression model, and a loss of accuracy in the summation can be successfully overcome by standardizing the original data according to the known methods.

Displacements are defined by the equation [ 4 ]:. Further transformation of movements, according to the Cauchy relations and equations of state of the environment, can also yield the stress state at the point. Performing the calculation of the stress-strain state parameters to form and direct the shell with a certain step, it is possible to obtain data for plotting the distribution of displacements and stresses.

Loading capacity of cylindrical shells is significantly affected by the unevenness of deformation caused by the ovality ends of the shell [ 13 ]. Imperfections in face of shells usually occur as a result of their deformation either under their own weight or during the mechanical handling, storage, as well as installation and assembly of the shells as individual elements. In all cases the shape of each end should be within the required tolerances, and roundness introduced by the collection process should not be reduced by more than 0. These studies have shown the need for a more correct approach in establishing the correspondence between the magnitude of these abnormalities and the level of carrying capacity.

As a consequence, it is necessary to study the nature of deformation of shells with different ratios of the parameters of roundness and taper. The use of a multi-factor approach allows one to solve correctly the problem of the nonlinear joint influence of defects on the loading capacity of the shell. Tests on the stability of prototypes carried out on a UMETM machine showed that the exhaustion of loading capacity of the shell took place at one stage by reaching a limit point. The loss of stability of a conical shell with the same low ovality ends Fig. On one side of the shell there are two or three belt dents located at the larger end.

They cover the smaller curvature of the plate and are shifted to the side of panel larger curvature. Local dents have a relatively large size and do not form a regular closed form buckling. The increase of taper and roundness of the ends leads to a shift of the zone of wave generation into the longitudinal direction to a lower end Fig. Forms of supercritical wave generation of the shell with a small taper and the same low ovality of ends a ; with a large taper, and the same large oval ends b ; with a large taper, and a large oval of the lower extremity c ; with a large taper, and a large oval upper end d.

At high cone within a given experiment increased roundness of the lower end, while maintaining the shape of the upper longitudinal, increases the localization of buckling Fig. Conversely, the prevalence of high cone-roundness of the upper end leads to a significant shift of dents to the end of a large oval Fig. The presence of significant second-order terms indicates a significant non-linearity of the relationship between the parameters, and, therefore, incorrect to separate consideration of the parameters and the placement of single-factor experiments. Let us investigate the derived mathematical models.

The corresponding surface of the pair interactions are shown in Fig. They demonstrate good agreement between calculation results of MMPC and experimental data.

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## Pade and Rational Approximation - 1st Edition

Analysis of the surface in Fig. In addition, in these limits roundness has a greater impact on the setting than the taper. This is consistent with the single-factor experiments reported in [ 13 , 14 ]. But the analysis of Fig.

This is essentially a nonlinear effect, which could not be found by single-factor experiments. Further study of the nonlinear interaction of defects Fig. The joint increase in roundness of ends leads to an increase in carrying capacity, which is also an essentially nonlinear effect and is in good agreement with the results shown in Fig.

Subcritical deformation has been studied in thin-walled shells with an oval on the lower and upper end being equal to 0. A qualitative analysis of the effect of displacement fields on the results of the holographic experiment suggests an important role played by the strain state of shells under non-uniform roundness in the district and in the longitudinal direction Fig.

A comparison of interferograms obtained at different load levels shows that an increase in the last number of fringes decreases with equal values of the additional load, and this indicates the hardening of structures, possibly caused by high deformability of the shell at the beginning of loading. The deformation of the shell in the experiment depends on the character of ends, and imperfections differ significantly on the panels of varying curvature. Interferogram envelope with the taper and ovality of the small curvature of the panel a , the joint panel zone b , and the larger curvature of the panel c.

The deformation of the shell with ovality and taper. Solid curves correspond to the middle of panels: black — large curvature, gray — small curvature, dash — forming at the junction of the panels; positive direction goes toward the center of curvature. The field of displacements was explained semi-automatically on the basis of the above algorithm.

The forms of the radial deflection of some shell generatrixes are shown in Fig. A modified method of the parameter continuation MMPC is proposed. This method enables simplification of the calculations both at the stage of constructing the model, and also within its continued use due to precise values of the Taylor coefficients for the solution of the degree not exceeding the number of approximation. The expression to calculate approximations by the MMPC in the general case and with the nonlinearity type of products and squares of the desired functions is presented.

The application of fractional-rational transformation for the polynomial approximation in the form of the 1-D and 2-D PAs used for increasing the degree of convergence and for the analytical continuation of the approximation in the region of its meromorphy was analyzed. It was concluded that such a transformation is justified if it is applied to polynomials which depend on the variable of integration.

We used 2-D PAs for the independent variable and for the artificial parameter applying the scheme proposed by V. In this paper it is shown that this transformation provides a satisfactory quality for the approximation behavior and minimizes its error, in spite of the fact that the use of 2-D PAs requires a further theoretical justification.

The estimation of stability using MMPC approximation is also proposed. A study of numerical results was conducted by applying the methods for three model examples which were perturbed with a natural small parameter. It is shown that the application of PAs provides them with sufficient accuracy in the studied area. This paper shows the advantage of approximations which were obtained based on the MMPC. E-mail address: dirk.

## Pade and Rational Approximation

Use the link below to share a full-text version of this article with your friends and colleagues. Learn more. In the same way the characterization of the object's contour by its Fourier descriptors, and the reconstruction of its region from the object's multidimensional moments, are also dual problems. In Section 2 we discuss how the latest techniques allow to reconstruct an object's shape from the knowledge of its moments. For 2D significantly different techniques must be used, compared to the general 3D case. In Section 3, the parameterization of a 2D contour onto a unit circle and a 3D surface onto a unit sphere is described.

Furthermore, the theory of Fourier descriptors for 2D shape representation and the extension to 3D shape analysis are discussed.

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The reader familiar with the use of either Fourier descriptors or moments as shape descriptors of physical objects may find the comparative discussion in the concluding section interesting. KGaA, Weinheim. The full text of this article hosted at iucr. If you do not receive an email within 10 minutes, your email address may not be registered, and you may need to create a new Wiley Online Library account. If the address matches an existing account you will receive an email with instructions to retrieve your username.

Annie Cuyt E-mail address: annie. Jan Sijbers E-mail address: jan. Brigitte Verdonk E-mail address: brigitte. Dirk Van Dyck E-mail address: dirk. Tools Request permission Export citation Add to favorites Track citation. Share Give access Share full text access.