A methodology for optimizing the fundamental structural designs of rotating turbine disks has been developed to aid in preliminary evaluations of various gas-turbine-engine designs. The basis for this methodology is a combination of turbine-disk and low-cycle-fatigue methodologies that was developed in pioneering work published during the years 1947 through 1965. The present methodology goes beyond the previous methodology in that its structural-analysis component is built on an enhanced mathematical model of a rotating turbine disk and is integrated with an optimization component.

In the previous methodology, the mathematical model of a turbine disk was one of constant thickness. In the present methodology, the thickness of the disk can vary with radius in piecewise-linear fashion; in addition, the temperature gradient in the disk is also modeled as a piecewise-linear function of radius. The differential equations of radial and tangential components of stress and strain in the disk are formulated under some straightforward simplifying assumptions (e.g., thickness << radius at all points of interest, and the disk material obeys Hooke's law and is homogeneous and isotropic). The equations are solved numerically by a finite-difference technique.

The incorporation of an optimization procedure saves much time, inasmuch as iterations that would otherwise have to be initiated manually are initiated and completed automatically by a computer. The time saved can be as much as 3 weeks per rotating disk. In this program, the optimization takes no more than 5 minutes. The optimization is performed by the sequence-of-unconstrained-minimizations (SUMT) technique, which is widely used for solving minimization problems that involve linear and nonlinear constraints or unconstrained functions. In this case, the objective function that one seeks to minimize is the mass of the disk, while the constraints pertain to maximum allowable levels of stress and relationships among the radii where the slope of the thickness-vs.-radius function changes. The optimization procedure can be summarized as follows:

1. Calculate the loads, stresses, and strains for an initial disk geometry that is based on the chord length of the turbine blades and the radii of the shaft and flow path.
2. Given the geometry and loads calculated, evaluate the constraints.
3. Upon violation of any of the constraints, modify the disk geometry until all constraint equations are satisfied and the mass of the disk is a minimum.

The structural-analysis and optimization components of the methodology are implemented in a computer program, along with a low-cycle-fatigue component adopted from the previous methodology. This program is executed in conjunction with an interface computer program that gathers needed data from a flow-path-analysis program called "WATE" (see figure). The combination of programs requires minimal intervention by the user, and can be used as a postprocessor of the WATE output.

In a test case involving a tenth-stage compressor disk, the stress-analysis portion of the methodology and computer program were assessed by comparing the computed stresses and displacements with those predicted in a finite-element analysis; the two sets of predictions agreed within 3 percent. In a test of the optimization component of the methodology, the mass of the rotating compressor disks was reduced by 26 percent from the initial design mass.

This work was done by Sasan C. Armand of Lewis Research Center. Inquiries concerning rights for the commercial use of this invention should be addressed to

###### NASA Lewis Research Center, Commercial Technology Office, Attn: Tech Brief Patent Status, Mail Stop 7-3, 21000 Brookpark Road, Cleveland, Ohio 44135

Refer to LEW-16422. ##### NASA Tech Briefs Magazine

This article first appeared in the September, 1998 issue of NASA Tech Briefs Magazine.

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