The framework for uncertainty and global sensitivity analysis . 60

The uncertainty in the crystal orientation of the blades will result in uncertainty in the static and dynamic response of a bladed disk. When the probability density function of anisotropy angles, defining the crystal orientation of blades are available, uncertainty and sensitivity analysis can be performed to quantify the scattering in the response of the bladed disk. In order to perform uncertainty and sensitivity analysis, a Python code namely ChaoStat was developed the author. The code allows seamless integration of different tools used for FE analysis, forced response analysis, and for building polynomial chaos approximation.

A schematic diagram illustrating different tools used by ChaoStat, the specific func-tion of each of those tools, and the flow of control within ChaoStat is shown in Fig.

3.6(a). A brief description of the different tools used in ChaoStat is given below:

1. ChaosPy [103] – An open source tool used for designing methods for uncer-tainty analysis using polynomial chaos expansion and Monte Carlo methods.

The tool is available as a Python module for installation. Within ChaoStat code, the tool is used for sampling the input parameter space based on the user-defined probability distribution of individual blade anisotropy angles, for generating orthogonal basis functions for PCE, and to calculate the statisti-cal characteristics of blade response as well as the global sensitivity indices of blade anisotropy angles. As part of this work, the capability to use gra-dient values of functions for calculating the coefficients in polynomial chaos approximation was developed using the standard ChaosPy module.

2. CalculiX [102] – An open source FE package used for building, solving and post–processing finite element models. CalculiX package includes the FE solver, CalculiX CrunchiX, and the graphical interface CalculiX GraphiX.

Within ChaoStat, CalculiX is used for linear and non-linear static analysis,

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modal analysis, and local sensitivity analysis of static deformation and modal properties of the bladed disk with respect to blade anisotropy angles [105].

3. ContaDyn – An in–house code developed as part of the current project [110].

The code is used for forced response analysis of complex structures with con-tact non-linearities using harmonic balance method and a novel approach for condensation of DOF for high-fidelity FE models [111]. Within ChaoStat, ContaDyn is used for forced response analysis and to obtain the local sensi-tivity of forced response amplitude [95] of the bladed disk with respect to the crystal orientation of blades.

4. SciPy – SciPy is a collection of mathematical algorithms, arranged as different sub–packages, available to a Python programmer [112]. Within ChaoStat, the SciPy package for linear algebra named scipy-linalg is used for solving a system of linear equations.

3.3.2 The framework for optimisation

The amplification of forced response of a mistuned bladed disk, in comparison to that of a tuned bladed disk with same blade crystal orientations, can be reduced by optimising the crystal orientation of blades in the bladed disk. In order to perform the optimisation of the blade anisotropy angles, a Python code was developed to integrate an open source optimisation framework namely OpenMDAO [113] to the FE solver CalculiX and the forced response analysis tool ContaDyn. In order to convert the modal analysis results of CalculiX to the input format of ContaDyn, a code developed by Adam Koscso [105] was used. A schematic diagram illustrating different tools used in the code for optimisation of blade anisotropy angles, the specific function of each of those tools, and the flow of control within the code is shown in Fig. 3.6(b).

(a)

(b)

Figure 3.6: Schematic diagram showing the framework for (a) uncertainty and global sensitivity analysis, and for (b) optimisation of the blade anisotropy angles.

Chapter 4

Sensitivity of static deformation to blade anisotropy orientations

In this chapter, the methods discussed in Chapter 2 for obtaining local sensitivity of displacements (Eqn. 2.13) and stresses (Eqn. 2.18, 2.20) and for global sensitivity (Eqn. 2.55) are applied to static deformation of anisotropy mistuned bladed disk under centrifugal loading. To study the effects of blade material anisotropy mis-tuning on static displacements and stresses, bladed disk with linear and non-linear contacts at fir–tree root and shroud interfaces are analysed. Mistuning is introduced by sampling the material anisotropy angles, α, β and ζ, of each blade of the bladed disk randomly from the realistic statistical distribution for these angles provided by the manufacturer. An example of the realistic distribution of blade anisotropy angles for all blades in the mistuned bladed disk is shown in Fig. 4.1. In the figure, the angles for α, β and ζ are normalized by the maximum value of those angles in the considered pattern.

4.1 Deformation of a mistuned bladed disk

In order to investigate the effects of blade anisotropy mistuning on static defor-mation of the bladed disk under centrifugal loading, corresponding to a realistic rotational speed specified by the manufacturer, both linear and non–linear mod-els are analysed. While non–linear model has friction contacts at fir–tree root and

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Figure 4.1: Distribution of the normalised anisotropy angles of all blades for an example mistuning pattern.

shroud interfaces, for a linear model these interfaces are modelled as bonded con-tact. For the mistuning pattern shown in Fig. 4.1 the variation in normalised axial, circumferential and radial displacements at blade tip node of all blades in the bladed disk is shown in Fig. 4.2(a), (b) and (c) respectively for linear and non-linear bladed disk. The value of displacement and the local sensitivity of displacement presented in this chapter are normalised with respect to the maximum radial displacement value for the non-linear tuned bladed disk with crystal orientation aligned with blade geometry. For the mistuning pattern studied, the variation in displacements along axial, circumferential and radial direction are 12%, 14% and 2.5%. In com-parison with the variation of displacement for non-linear bladed disk, it is observed that the scatter in displacement for the linear bladed disk is larger with respective values for axial, circumferential and radial displacement being 18%, 33% and 5%.

Note that for the linear bladed disk model, each blade is connected to its adjacent blade through bonded contact at shroud and therefore, results in a stiffer connection between blades. Therefore, in the case of linear bladed disk, the structure is com-paratively more sensitive to variation in blade anisotropy orientation of blades. The magnitude of blade displacements for the non-linear bladed disk is higher compared to that of the linear bladed disk given the same mistuning pattern which

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tiates the argument that bonded contacts at interfaces increases the stiffness of the structure. In the case of a bladed disk with friction contact interfaces at shrouds, the friction interface reduces the effect of variation in anisotropy angles of a blade on the deformation of other blades.

(a)

(b)

(c)

Figure 4.2: Displacements at blade tip node of all blades in (a) radial (b) circum-ferential and (c) axial direction in a mistuned bladed disk

For a sample mistuning pattern, the normalized displacements of the bladed disk along axial, tangential and radial directions are shown in Figs. 4.3(a), (b) and (c) respectively for a blade tip node. The displacement values are normalized with respect to the value of radial displacement at the considered node for the case of a tuned bladed disk with the blade anisotropy axis aligned with the blade geometry axis. By comparing the axial displacement of different blades, it can be observed that, due to mistuning, there is a difference in displacements from blade to blade.

In order to highlight the small variation in axial displacements between blades, a zoomed view of several blades in the bladed disk is shown in the inset in Fig.

4.3(a). Compared to the differences in blade displacements along the axial direction between different blades in the bladed disk, the difference in displacements along the tangential direction and radial direction is not easily discernible from Fig. 4.3(b) and (c).

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(a)

(b)

(c)

Figure 4.3: Figure showing (a) axial, (b) circumferential and (c) radial displacement of a mistuned bladed disk.