All cylinders are measured prior to testing using the ATOS-system at DLR. From the measured data points an equidistant mesh of surface point coordinates is extracted.
This can be used as an input for a finite element analysis. Also, deviations of the node positions to a best fit cylinder can be computed and the resulting imperfection pattern plotted. The radii of the best fit cylinders for all 12 cylinders are presented in Table 5-8.
Table 5-8 Radii of best fit cylinders from ATOS measurements Radius [mm]
Cylinder .1 .2 .3 .4 .5 .6
Z1.x 114.852 114.828 114.856 114.870 114.852 114.865 Z2.x 114.873 114.857 114.863 114.858 114.883 114.889
5.4.2 Comparison of imperfections prior and after mounting
The cylinders 2.1-2.6 are measured using the ATOS system before they are glued into the mountings and after. This was done to quantify the influence of the mounting on the shape of the cylinders.
The resulting measurements are similar for all 6 cylinders. Exemplary, the deviations from the best-fit cylinder of Z2.1 are shown in Figure 5-20. The equidistant peak-lines visible in circumferential direction result from peel ply fibres that could not be re-moved.
The cylinder shows a slight ovalized shape whereat ovalization at top and bottom are offset by 90° with respect to the cylinder axis.
With deflections of up to 0.70 mm from the nominal radius, the deformation is not visible to the naked eye. However, it is more than three times the maximum deflection measured after mounting. A direct comparison with adjusted legend is depicted in Figure 5-20, showing significant higher amplitudes of deviation from the nominal radius for the unmounted cylinder.
The large deflections due to the tendency of the unrestrained cylinder to ovalize give no hint towards the imperfection pattern that will be present after mounting and during the tests. Only at a small circumferential section at mid-height deviations are in the order of magnitude as observed after mounting.
Figure 5-20 Deviations from best-fit cylinder for Z2.1 5.4.3 Surface plots
The winding-up of the mounted cylinder surface with its shell-wall mid-surface imper-fection on the z-axis are plotted for cylinders Z 1.1 and Z 1.2 in Figure 5-21 [Sch15].
Prior to mounting After mounting
Figure 5-21 Shell-wall mid-surface imperfections
The tendency for long wave modes in axial and circumferential direction is clearly visible on all plots. Furthermore, there is an influence of the mounting sequence visi-ble: at the lower edge with z=0 mm, which has been put on the steel mounting and glued first, the mid-surface imperfections are significantly lower than at the upper edge. This is due to the fact that the cylinder could deform more freely as compared to the second mounting step when the lower edge is already constrained. These patterns are hence indicating a certain pre-stress state.
An overview of occurring deviations can be found in the histograms given in Appendix A6 for each cylinder.
5.4.4 Power Spectral Density (PSD)
The characteristics of the shell-wall mid-surface imperfection can be considered as signal information with varying amplitude over the cylinders circumference or length. A common way to transform this information from a length-domain to a frequency do-main (with unit 1/mm) is to use a Fourier transformation and decompose the imperfec-tion pattern into its sine or cosine contents. In the context of geometric imperfecimperfec-tions, the frequency axis is commonly scaled to represent the number of half-waves fitting into the considered length of the cylinder (compare e.g. Refs. [Arb02], [Arb05], [Kri11]).
Since the Fourier transform assumes next to stationary also ergodic signals, a more general way to analyse a random signal is to consider its power spectral density (PSD), which has been done by e.g. [Eli82], [Sch10], [Kep13]. A truncated Fourier transform F(x) is used, which decomposes a zero-mean random field f(x) of finite length L into its sine and cosine contributions as a function of frequency.
L
x l dx e x f F
0
) 2 ( ) 1
(
(5-2)
Z1.1 Z1.2
The PSD is then described through [Sch10]
dx x S F
E
L
∫
=
0
2 ( , )
)
(ω ω (5-3)
In this case, the well-known Welch method [Wel67] is used to estimate the PSD. It consists of the following steps
1. Dividing the data sequence into segments
2. Multiplying a segment by a window function (here: turkey window) 3. Taking the Fourier transform of the product
4. Multiplying procedure 3 by its conjugate to obtain the spectral density of the segment
5. Repeat procedures 2 through 4 for each segment so that the average of these periodogram estimates produce the power spectral density estimate The resulting plots are depicted in Figure 5-22 for cylinders Z1.1 (left) and Z1.2 (right).
They illustrate the strong narrow-bandedness of the information analysed, which is represented solely through very low frequencies with their power increasing along the z-coordinate of the cylinder.
Figure 5-22 Power spectral density
For an easier read of the amplitudes, the following representation is chosen (Figure 5-23). From the PSD the y-axis of the plots is scaled down to represent the amplitude of the mid-surface imperfection. On the x-axis, instead of frequencies, the more intuitive unit of number of half-waves fitting in axial direction into the considered direction of the cylinder is depicted. For each half wave number occurring, the median (red bar), standard deviation (blue box) and 5 % to 95 % percentile (black bars) are depicted.
Appearances of outliers are marked with red crosses. In contrast to the dominance of
Z1.1 Z1.2
the ovalization mode (4 half waves) of the unconstrained cylinder, it is now the mode consisting of two half waves that show the largest amplitude contribution, with the mean lying below 5% of the cylinder thickness.
Figure 5-23 Geometrical imperfections represented through half waves in axial direction
5.4.5 Root mean square
From the PSD the root mean square (rms) of the amplitude can directly be calculated as the square root of the area under the spectrum vs. frequency curve corresponding to certain height of the cylinders. So for each z-position on the cylinder the rms-value is computed and depicted for cylinders of set 1 and set 2 in Figure 5-24.
Figure 5-24 Root mean square values over cylinder height
Set 1 Set 2
Z1.1 Z1.2
Although some information about stochastic properties is lost, this representation offers a very direct way to compare an important characteristic regarding the ampli-tude of the geometric imperfection. However, the strong deviation between the rms values at the bottom of the cylinders at z=0 mm and the top at z=215 mm indicate that in the case of these cylinders, taking only the mean rms over the whole cylinder surface (refer to Table 5-9) may be misleading in terms the geometric quality achieved.
Table 5-9 Mean root mean square values
Cylinder no. .1 .2 .3 .4 .5 .6
mean rms [mm]
set Z1. 0.06 0.07 0.07 0.1 0.07 0.08 set Z2. 0.07 0.06 0.08 0.05 0.10 0.12