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Probabilistic Modal Response Analysis for Turbine Engine Blade Design Using MSC.Nastran

(#2001-26)

Jeffrey M Brown Lead Structural Analyst Turbine Engine Division

Air Force Research Laboratory Wright-Patterson Air Force Base, OH, 45433 937-255-2734, jeffrey.brown@wpafb.af.mil

Recent increases in blade failures have lead to a US Air Force initiative to enhance its

blade design process. Part of this initiative is replacing the deterministic design margins with a physics-based resonance avoidance process that relies on analytical modeling, characterization of the as-manufactured geometry, and probabilistic methods. The design system will also account for variation in modal stresses caused by geometry variations. This paper presents a probabilistic methodology that uses MSC.Nastran to predict the probability distributions of blade frequency and modal stress. The method uses Monte Carlo sampling from the population of random blades. Each Monte Carlo iteration obtains a random blade geometry, modifies a MSC.Nastran data file, runs the solution, and stores the new frequency and modal stress values. The distribution of response gives the engineer a broader understanding what input variations are influential on response variation and the margin their systems have. This approach represents aspects of the new Air Force approach for designing blades to avoid high cycle fatigue failure. INTRODUCTION

A major technical challenge of the US Air Force is to reduce turbine engine blade high cycle fatigue failures. These failures are driven, in part, by variations from nominal design geometry that cause variation in the blade natural frequency and modal stress. Improved reliability will be achieved by more accurately accounting for variation in response with probabilistic methods. This paper introduces three approaches to account for the distributions of response that are relevant to turbine engine blade design: a probabilistic resonance avoidance criteria, a probabilistic gauge factor, and probabilistic modal stress. These are extensions of the current, simplistic, blade design process based on blade resonance frequency avoidance, empirically demonstrated stresses below the material endurance limit, and the assumption of a Kt=3 at critical stress locations. These three approaches are being replaced with probabilistic frequency margin, probabilistic gauge factor, and probabilistic modal stress. This paper shows how MSC.Nastran can be used to take part in this process.

Probabilistic Frequency Margin

Traditional deterministic blade design attempts to avoid resonance failures with design margin and the Campbell Diagram, Figure 1. The Campbell Diagram is used to compare a blades natural frequency and the excitation frequencies produced in an operating engine. The excitation frequency is a function of engine RPM and the number of upstream disturbances caused by stators, struts, etc. On this diagram, the natural mode frequency is superimposed on

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the excitation frequency and where the two cross, resonance and the inevitable resonance failure will occur. The sloped excitation frequency curve, 4E, signifies the upstream distortion from four components. Campbell Diagrams have many excitation curves and many modes and it becomes very difficult to avoid resonance across all operation speeds. Still, resonance must be avoided and a standard has been set in the Air Force ENSIP, Engine Structural Integrity Program. The document calls for a 10% frequency margin on all resonance crossings, Figure 2. This margin has no technical basis other than being used with some success with previous engine designs. Current failure rates indicate that this margin may be inadequate.

A physics-based avoidance process relies on an analytical model representing response, a characterization of the as-manufactured geometry, and probabilistic methods. With these three components, the probability distribution of blade frequency can be predicted. The distribution of blade frequency gives the engineer a broader understanding of what input variations are influential on response variation and the margin their systems have. Specifically, use of this

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Figure 1: Campbell Diagram

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Figure 2: Campbell Diagram Design Margin

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Figure 3: Probabilistic Campbell Diagram

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process will allow designers to predict the probability of resonance of a given blade. Ultimately, the Campbell Diagram will still be used but it will become a Probabilistic Campbell Diagram, Figure 3. The Air Force has set standards for acceptable probability of failures for its future engines, and the probabilistic process described in this paper are a part of the overall probabilistic blade design system that will allow the industry to design to this guidance.

Probabilistic Gauge Factor

The intent is to avoid resonance, but in practice blade resonance does occur for brief

periods and at low amplitudes. Current design practice is to ensure that the magnitude of this stress is below the material endurance limit through instrumented engine tests. Limited instrumentation is used and, as an example, three gauges are used to measure the stress across the entire blade. Stresses at non-instrumented locations are predicted based off of finite element models that predict modal stress. Traditionally, these modal stresses are predicted with the nominal geometry. It is important to assess the impact of geometry variations on the modal stress to ensure a proper prediction of the alternating stress in the blade. The traditional gauge factor will be replaced with a probabilistic gauge factor that use the distribution of gauge factors caused by variations in modal stress.

Probabilistic Modal Stress

With the probabilistic gauge factor we will have an improved representation of stress

magnitude. The variation of the location of this peak stress is also critical. The location of peak stress must be well characterized when evaluating the foreign object damage tolerance. Current design practice requires blades to tolerate a kt=3 at critical stress locations to have this tolerance. An improved method would use the statistical distributions of FOD impact locations, imparted stress concentration, modal stress. The integration of these would give the probability of failure from FOD incidents. PROBLEM DEFINITION Low Aspect Ratio Blade Model An Air Force low-aspect ratio blade is used for this demonstration because it is closely related in dynamic response to many of the advanced designs that are under development within the turbine engine industry. It’s span is approximately 14, chord length is 12 inches, and is made from titanium. Industry experience has shown that the response of low-aspect ratio blades are more sensitive to geometry variations, particularly at higher modes, than past blades with high aspect ratio. The MSC.Nastran finite element model of this blade has 1200 elements, 1764 nodes, for a total

Figure 4: Low Aspect Ratio Blade

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of 5,292 degrees-of-freedom. The blade is fixed in all transnational degrees of freedom at the base of the blade Blade Geometry Variations

Because machining processes are not perfectly repeatable, blade geometry has random variations. These variations are quite small, on the order of mils, and yet can have significant effects on the modal response of the engine. Blades are inspected with automated measuring devices to ensure that they meet stringent geometric tolerances and the blade geometry variation can be captured from these inspections. The population of these blade measurements is required for the probabilistic prediction.

Because of the proprietary nature of actual blade geometry and the differences in blade-to-blade variations caused by various manufacturing techniques, the data used for this analysis was simulated. The simulation process was based on a “Manufacturing Modes” concept which has been observed with actual manufacturing processes. Manufacturing Modes are based on the fact that blade variations are not truly random, but instead are controlled by variations and tolerances of the manufacturing process. For instance, often in the machining process, misalignment of the raw ingots from the vertical axis of the machine tool causes a, for lack of a better word, slant on the blade. Therefore, there is a variation in the blade geometry, but it is spatially correlated because it follows a defined pattern. In this slant case, the manufacturing mode is close to the familiar bending mode of flat plates. Other manufacturing modes include twist, bows, and waviness. In practice, these modes occur simultaneously with each other.

Spatial functions were created to replicate potential manufacturing modes. These functions were then linearly summed with random weighting functions to generate a simulated population of 500 blades. The data show in Figure 5 represents the variation of blade thickness across the surface at 400 locations. Figure 6 shows the geometry variation across the blade for a single blade. This plot show increase variation at the leading and trailing edge which is expected because of difficulties in keeping tolerance at these locations. There is also twist, slant, and waviness visible.

Figure 5: Blade Geometry “Simulated” Measurements

Figure 6: A sample blade variation

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Probabilistic Method

The remaining component needed for the probabilistic prediction is the actual probabilistic method. There many probabilistic methods available that are all intended to do the same thing, integrate the following equation:

∫ dxxf )( (1) where f(x) is a function describing a probability density function, PDF, such as a normal distribution. You integrate the probability over a given part of the function. The limits on this integration start at the value of the PDF that causes a failure and extend to infinity. In reality, equation one is multi-dimensional and the instead of a PDF, you must integrate over a joint probability density function. This joint PDF is in almost all cases, no longer an analytical equation, and must be determined numerically. Methods to calculate the response probability include Monte Carlo, response surface, and mean value approximations. Each has its strengths and weaknesses, but only one is undeniably robust, Monte Carlo. Monte Carlo analysis could also be the brute force method. With Monte Carlo analysis you randomly select values from the probability density functions that define your models input parameters, calculate the model response, and store this response in a histogram. Continue this iteration for a large number of trials and you, on most cases, will converge to a probability density function. The major complaint against Monte Carlo is that it takes too much computational time. All other methods are potentially inaccurate. With the increasing speed of computers and the fact the Monte Carlo Analysis can practically be linearly scaled with multi-processor computers, Monte Carlo analysis is the preferred method in this process until robustness can be shown in other methods. This paper uses the Monte Carlo approach to show that realistic problems can be solved with Monte Carlo in a reasonable amount of time. ANALYSIS

Implementation of the probabilistic blade modal response was enable through the use of both MSC.Patran and MSC.Nastran. A software routine written in Matlab was also used to automate the iterative solution process. MSC.Patran

MSC.Patran was used to build the finite element model as well as provide surface normals at the blade nodes. Blade geometry variations are typically measured normal to the surface of the blade, therefore, surface normals directions are required in the geometry perturbation. Modifications to the MSC.Nastran bulk data file are made by adding the product of the normal vector and the measured geometry variation to the nominal grid locations. Obtaining the surface normals at each node can be done in MSC.PATRAN a number of different ways. It was found to be relatively, under the Finite Elements tools, to use Sweep-Element-Normal capability to create a set of nodes offset from the surface nodes a 1.0 distance

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away. The Show-Node-Distance, tool was then used with the first node list containing the blade surface nodes and the second node list containing the offset nodes. The distance can be output as a node distance report that acts as the surface normal vector for each node. This vector is similar to shape basis vectors, used in MSC.Nastran shape optimization. Figure 7 shows the

MSC.Patran model of the blade with swept elements for surface normal generation. MSC.Nastran

The solution process is a standard Monte Carlo approach. Random blade variations are created by generating random values from probability distributions associated with manufacturing modes. These variations are multiplied by the surface normal vectors to gives a vector describing the shape variation of a random blade from nominal. These variations are added to the nominal geometry and MSC.Nastran is used to solve the model.

These steps are automated by using a Matlab routine. The nominal blade geometry is defined with a file of GRID cards. The Matlab code outputs a new set of GRID cards for the random blade. MSC.Nastran is executed to solve an existing bulk data file. This bulk data file has all the required cards to conduct a SOL 103, except the GRID cards. An INCLUDE card is added in the bulk data file that references the INCLUDE file generated by MATLAB that contains the random blade geometry. On completion of the analysis run, the Matlab code reads in the frequencies from the *.f06 file. These are stored in a vector on successive iterations to form a histogram. RESULTS Probabilistic Frequency Margin

The histogram in Figure 8 shows the Mode 1 frequency distribution from 500 iterations of Monte Carlo Analysis. The result can be approximated with a normal distribution with a

Figure 7: Surface Normal Generation

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mean of 375 hz. with a standard deviation of 4.2 corresponding to a to a 1% coefficient of variation. Figure 9 shows the Mode 10 frequency distribution with a mean of 3400 hz with standard deviation of 140 hz. corresponding to a 4% coefficient of variation. These results show increasing variation at higher modes. A probability density such as this would be used in the Probabilistic Campbell diagram described in Figure 3.

Figure 8: Histogram of Mode 1 Frequency Variation

Figure 9: Histogram of Mode 10 Frequency Variation

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Probabilistic Gauge Factor Figure 10 shows the histogram of 100 gauge factors obtained from 100 Monte Carlo simulations of blade geometry variations. The distribution is not normally distributed and shows that a high percentage of the factors clustered around 0.4. The range of factors is between 0.09 and 0.4. The inverse of these factors is a range between 2.38 and 11 multiplication factor applied to the strain gauge reading to predict maximum stress. This variation must be accounted for

when predicting the maximum blade stress. Probabilistic Modal Stress

Figure 11 shows the nominal modal stress of the blade at Mode 10. Figure 12 is the not a modal stress plot, but a plot of the maximum modal stress across the blade surface seen over 100 modal stress predictions. The comparison of these plots show that the maximum modal stress can be located over a larger location than indicated by the nominal plot. Figure 13 is a histogram of the modal stresses at a node on the leading edge over the 100 Monte Carlo simulations. This variation in location should be used in conjunction with the variation in modal stress, the distribution of stress concentrations caused by FOD, and the material capacity to predict probability of failure. DISCUSSIONS These results demonstrate the application of probabilistic methods and MSC.Nastran to aspects of blade design. It is difficult to make concrete conclusions from the results because the

Figure 10: Probability Distribution of Gauge Factor

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geometry variations used were not taken from a true blade population. It is important to note the increased understanding engineers get of the component response. CONCLUSIONS

A process was demonstrated to conduct probabilistic modal analysis and demonstrated on a relevant model. Aspects of this process will replace the existing design margin practices for frequency avoidance, gage factors, and modal stress prediction. The use of probabilistic methods give engineers a more thorough understanding of how the parameter variations affect the

Figure 11: Nominal Modal Stress

Figure 12: 99th Percentile of Modal Stress

Figure 13: Probability Distribution of Modal Stress

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distribution in response. MSC.Patran and MSC.Nastran were both critical to enabling this process.

Even though many MSC users may not be conducting turbine engine blade design, the probabilistic approach used in this report could be of great significance. Probabilistics is rapidly being introduced into the design and analysis field and its demonstrations in this paper may further increase probabilistic use in the MSC community. Adding probabilistic capabilities directly into MSC.Nastran and MSC.Patran would greatly enhance their already excellent capabilities.