MINIMIZING SENSITIVITY OF BLADED DISKS TO MISTUNING

Joseph C. Slater and Andrew J. Blair

Department of Mechanical and Materials Engineering Wright State University Colonel Glenn Highway

Dayton, OH 45435

ABSTRACT

High cycle fatigue due to vibration continues to be a leading cause of blade failure in turbomachinery. Designing for the alleviation of high cycle fatigue is one of the most complex problems facing engineers. The complexity of the problem has given rise to many design propositions and many techniques for evaluating their merit. In this work, the Finite Element Method is used to analyze minor hub design modifications intended to minimize the propensity for mode localization in turbomachinery blades, thus alleviating high cycle fatigue.

NOMENCLATURE

[M]  mass matrix
{Φ}  normalized mode shapes
E  Young’s modulus
M  mass matrix
R  stress ratio
V M  Von Mises stress
ν  Poisson’s Ratio
ρ  density
i  mode shape #
m  mistuned case
t  tuned case

1 INVESTIGATION

In bladed disk assemblies, the disk acts as a coupling device between the blades. As the stiffness of the disk increases, blade coupling decreases. It has been shown that weak interblade coupling leads to high levels of mode localization when blades are mistuned^{[1−6]}. Mode localization also occurs in bladed disks as a result of their symmetry. Sets of axisymmetric modes combine to form a basis set from which

There once was a figure here

Figure 1: Finite element model.

drastically localized mode shapes can be generated. Bladed disk assemblies are traditionally designed to be symmetric for the sake of balance. This leads to two hypotheses to be investigated in this work: 1) Decreasing the stiffness of the disk by varying the geometry and/or material composition may reduce mode localization due to mistuning, 2) Destroying the symmetry of the disk, yet maintaining balance, may reduce mode localization due to mistuning (The idea here being to split repeated modes such that they have unique natural frequencies). Each of these was investigated separately, as well as combinations of the two.

A model of an eight bladed disk based on an experimental testbed at Wright State University was constructed in AN-SYS\begin{matrix}r \\ ◯ \end{matrix} using eight noded “brick” elements (Figure (sorry, it’s not in this doc)). The model was designed to exhibit a propensity for mode localization similar to that of turbomachinery bladed disk assemblies. The bladed disk model was adjusted to provide weak coupling between the blades resulting in tightly packed sets of natural frequencies, eight modes in each. The blade deformation in the first set of modes is predominately a first beam bending mode, in the second set of modes it is predominately a first beam torsional mode, and in the third set of modes it is predominately a second beam bending mode.

Three types of mistuning were chosen for the investigation. The first two were: one percent of the mass of one blade added to the tip of one blade, and one percent blade mass removed from the tip of a single blade. This results in two cases of mistuning for the symmetric models. For non-symmetric models, however, each type (addition and subtraction) must be investigated on each of four different blades. This results in eight different cases of mistuning for the non-symmetric models. The preceding models, representing minor damage cases to an individual blade or all but one blade, are in agreement with accepted practice in prior studies^{[1, 2]}. The final type of mistuning cases is random mistuning. Three random patterns were chosen such that the mass added to the tip of each blade varied between plus or minus one percent of the mass of one blade.

1.1 Models

All models used in this study were variations of the symmetric, constant stiffness system referred to as the baseline model (Figure 1). The dimensions of the disk are as follows: OD = 188 mm, ID = 87.5 mm, and thickness = 14 mm. The dimensions of each blade are: length = 62 mm, width = 32 mm, and thickness = 1.5 mm. This model was constructed entirely of eight noded brick elements (ANSYS\begin{matrix}r \\ ◯ \end{matrix} element— Solid 45), having three degrees of freedom per node. The material properties used were that of mild steel: Young’s Modulus E = 200 GPa, density ρ = 7800 kg/m^{3}, and Poisson’s Ratio ν = 0.3. Lines were also included in the geometry that allowed meshing to be performed in ANSYS\begin{matrix}r \\ ◯ \end{matrix}. A proper mesh in this study had two important properties. Yada yada.

1.2 Mistuning

Three basic types of mistuning were chosen to best represent realistic challenges to the robustness of a bladed disk design. The first two were addition and subtraction of mass to a single blade. These were chosen to represent foreign object damage (FOD) to either a single blade (subtraction) or all but one blade (addition). The final type of mistuning chosen was the random addition and subtraction of mass from each blade, representative of the small variance between blades due to manufacturing variations or more distributed wear/damage.

Yada yada.

Three random patterns of mistuning were also investigated. As mentioned before, the addition of one percent blade mass to each blade was chosen as the tuned, or nominal case. In determining the random patterns to be used, the mass to be added to any location was allowed to vary from zero to two percent of a single blade mass. The patterns generated follow in Table 1 and the blade numbering scheme is shown in Figure (not in sample).

ACKNOWLEDGEMENTS

Thanks to all those who’ve built LaTeX into what it is today.

Random 1 Random 2 Random 3

Blade Number

10^{−3} Kg  10^{−3} Kg  10^{−3} Kg
1  0.0168  0.3343  0.3413
2  0.0260  0.2867  0.4431
3  0.2579  0.4529  0.3710
4  0.3267  0.4119  0.1278
5  0.0037  0.2565  0.0231
6  0.1866  0.0448  0.3583
7  0.0325  0.3183  0.1598
8  0.2032  0.2025  0.3080

TABLE 1: Random mistuning patterns used in the investigation.

REFERENCES

[1] Pierre, C. and Murthy, D. V., Aeroelastic Modal Characteristics of Mistuned Blade Assemblies: Mode Localization and Loss of Eigenstructure, AIAA Journal, Vol. 30, No. 10, pp. 2483–2496, October 1992.

[2] Pierre, C., Smith, T. E. and Murthy, D. V., Localization of Aeroelastic Modes in Mistuned High-Energy Turbines, Journal of Propulsion and Power, Vol. 10, No. 3, pp. 318– 328, May-June 1994.

[3] Wei., S. T. and Pierre, C., Localization Phenomena in Mistuned Assemblies with Cyclic Symmetry Part 1: Free Vibrations, Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 110, pp. 429–438, October 1988.

[4] Ewins, D. J., Vibration Characteristics of Bladed Disk Assemblies, Journal of Mechanical Engineering Science, Vol. 15, No. 3, pp. 165–186, 1973.

[5] Griffin, J. H., On Predicting the Resonant Response of Bladed Disk Assemblies, Journal of Engineering for Gas Turbines and Power, Vol. 110, pp. 45–50, Jan 1988.

[6] Kaza, K. R. V. and Kielb, R. E., Effects of Mistuning on Bending-Torsion Flutter and Response of a Mistuned Cascade in Incompressible Flow, AIAA Journal, Vol. 20, No. 8, pp. 1120–1127, August 1982.