**6. Experimental technique**

In solving various kinds of problems of modern development and improvement of thinwalled 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.), making them difficult to use in this case. In addition, although several authors proposed methods of interpretation [10], they did not fully take into account the statistical nature of input data. 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.

16 Nonlinearity, Bifurcation and Chaos – Theory and Applications

nonlinear oscillations in the form

order is taken).

The application of the proposed method of parameter continuation to the Cauchy problem (22) gives approximation of the second order for the artificial parameter for frequency of

2 4

It is seen that the oscillations are not isochronous. This agrees well with previous results reported in reference [9] (Fig. 6). However, our approach allows for a significant reduction of the computation time (in [9] to obtain similar results the approximation of the fourth

**Figure 6.** Amplitude of the initial disturbance versus oscillation frequency of stringer shell

In solving various kinds of problems of modern development and improvement of thinwalled 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.), making them difficult to use in this case. In addition, although several authors proposed methods of interpretation [10], they did not fully take into account the statistical nature of input data. For cylindrical shells a method for

(1 – according the proposed method, 2 – data [9]).

**6. Experimental technique** 

1 *fAA fAA f*

21 31 / / . <sup>1</sup>

.

**Figure 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.e. unloaded and loaded state of the object, we get a flat image of the interference pattern corresponding to the observation of points , , *Mi <sup>н</sup><sup>i</sup> <sup>н</sup><sup>i</sup> <sup>н</sup><sup>i</sup> xyz* , *i* 1,3 . 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]. In the *XOY* coordinate system (Fig. 8) associated with the imaging plate, base point *MBBB x y* , and a segment of the *OY* axis of the *XOY* coordinate system, whose direction coincides with the vertical axis of the projection, are given. Further calculations are performed in the *XOY* system in which the entered coordinates of the points of lines of equal order are transformed by the formulas

$$\alpha = \alpha' \cos \varphi + y' \sin \varphi - \alpha\_{B'} \ y = y' \cos \varphi - \alpha' \sin \varphi - y\_{B\_{\varphi}}$$

where is the angle of rotation of the *XOY* system with respect to *XOY*

cos sin , sin cos . *BB B B B B xx y y x y* 

Applications of 2D Padé Approximants in Nonlinear Shell Theory:

*x y* 1, 1, *i i* ; , *x y* 2, 2, *i i* ; are the coordinates of corner points of the projection of the folds into

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

The above method allows us to determine the coordinates of the centers of bands up to 0.1 mm without any additional devices. This procedure is used to significantly increase the number of input points (up to 200-400 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. 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

the *XOY* system (Fig. 8).

number of points.

original data according to the known methods.

Displacements are defined by the equation [4]:

<sup>1</sup> *<sup>i</sup> <sup>i</sup> N N* – vector of lines order.

observation in the form

wave; <sup>3</sup>

Because the shape of the surface is analytically given by equation *rr z* (,)

where: *M* – optic matrix; *U uvw* , , – vector of displacement;

obtain two-dimensional regression models for the line of the *i* -order for the direction of

( ) ( )

*n i n j <sup>j</sup> <sup>k</sup> i jki j k N bz*

0 0

*MU N* ,

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

 

.

Stability Calculation and Experimental Justification 19

, it is possible to

– length of the laser

**Figure 8.** The scheme of approximation of the surface by the system of folds

While computing the physical coordinates, the approximation of the mentioned shell surface by the system of folds is applied (Fig. 8). In this case, the physical coordinates of point *M*(,, ) *r z* in the *i* fold ( 1, ) *i n* are defined by

$$
\theta = \frac{\theta\_i - \theta\_{i-1}}{\chi\_i - \chi\_{i-1}} (\mathbf{x} - \mathbf{x}\_{i-1}) + \theta\_{i-1}
$$

$$
\mathbf{z} = \mathbf{z}\_1 + \frac{y - \left[y\_{2,i-1} + \left(y\_{2,i} - y\_{2,i-1}\right)\left(\mathbf{x} - \mathbf{x}\_{2,i-1}\right) / \left(\mathbf{x}\_{2,i} - \mathbf{x}\_{2,i-1}\right)\right]}{y\_{1,i} - y\_{2,i}} (\mathbf{z}\_2 - \mathbf{z}\_1).
$$

where *r z* , , are the coordinates of a point in the cylindrical coordinate system associated with the axis of the shell, and the shape of the surface is analytically given by equation *rr z* (,) ; , , *ijk r z* are the physical coordinates of corner points of the considered fold;

 *x y* 1, 1, *i i* ; , *x y* 2, 2, *i i* ; are the coordinates of corner points of the projection of the folds into the *XOY* system (Fig. 8).

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 0.1 mm without any additional devices. This procedure is used to significantly increase the number of input points (up to 200-400 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. 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.

Because the shape of the surface is analytically given by equation *rr z* (,) , it is possible to obtain two-dimensional regression models for the line of the *i* -order for the direction of observation in the form

$$N\_i = \sum\_{j=0}^{n(i)} \sum\_{k=0}^{n(j)} b\_{jki} \theta^j z^k \dots$$

Displacements are defined by the equation [4]:

18 Nonlinearity, Bifurcation and Chaos – Theory and Applications

is the angle of rotation of the *XOY* system with respect to *XOY*

**Figure 8.** The scheme of approximation of the surface by the system of folds

in the *i* fold ( 1, ) *i n* are defined by

While computing the physical coordinates, the approximation of the mentioned shell surface by the system of folds is applied (Fig. 8). In this case, the physical coordinates of point

1

 

*y y y y xx x x z z z z y y* 

*i i*

*i i*

*x x* 

<sup>1</sup>

*x x*

1 2 1 1, 2,

, are the coordinates of a point in the cylindrical coordinate system associated

*i i*

with the axis of the shell, and the shape of the surface is analytically given by equation

1 1

*i i*

2, 1 2, 2, 1 2, 1 2, 2, 1

/ , *i ii i ii*

are the physical coordinates of corner points of the considered fold;

cos sin , sin cos . *BB B B B B xx y y x y*

u

v

where 

*M*(,, ) *r z* 

where *r z* ,

*rr z* (,) 

 ; , , *ijk r z* 

$$M\underline{L}\underline{L} = \mathcal{Z}N\_{\prime\prime}$$

where: *M* – optic matrix; *U uvw* , , – vector of displacement; – length of the laser wave; <sup>3</sup> <sup>1</sup> *<sup>i</sup> <sup>i</sup> N N* – vector of lines order.

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.

Applications of 2D Padé Approximants in Nonlinear Shell Theory:

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. 9b) while maintaining the

**Figure 9.** 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

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. 9c) shifting the dents closer to the lower end, while maintaining the variability in the circumferential direction. Conversely, the prevalence of high cone-roundness of the upper

oval of the lower extremity (c); with a large taper, and a large oval upper end (d)

end leads to a significant shift of dents to the end of a large oval (Fig. 9d).

Results of the experiment allow us to derive mathematical models of the form

overall character of buckling.

Stability Calculation and Experimental Justification 21
