Quantification of Damage Variation of a Blade under Impact Loading due to Manufacturing Tolerance

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ScienceDirect Materials Today: Proceedings 5 (2018) 11838–11844

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ICMMM - 2017

Quantification of Damage Variation of a Blade under Impact Loading due to Manufacturing Tolerance Tirumala Rao Kokaa*, Arun Tom Mathewb a

b

Honeywell Technology Solutions Lab Pvt. Ltd., Bangalore, 560037,INDIA School of Mechanical Engineering, VIT University, Vellore, 632014, INDIA

Abstract Methods to model the dimensional variation into computer based simulations by the use of error propagation, random sampling and principle component analysis (PCA) and quantify the variations in their functionalities are discussed. Quantifying these functionalities is important in case of rotating equipment like fan blades of aircraft engine or compressor blades of an automobile turbocharger. This paper takes the example of a blade under impact loading and the maximum permanent deformation as the measure of damage quantifies the variation in damage due to manufacturing variations. Three approaches are discussed namely, error propagation, random sampling and constructing a maximum and minimum variance models using single value decomposition techniques like principle component analysis(PCA). Ten blades of same design are considered with their dimensional variations due to manufacturing tolerances. Finite element model of these blades are developed along with variations in the thickness of the blade due to manufacturing variations. Simulations are carried out on these ten blades with different thickness values for impact loading. All the three methodologies of damage quantifications namely error propagation, random sampling and PCA are applied and benefits and draw backs of each method are discussed. © 2017 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of International Conference on Materials Manufacturing and Modelling (ICMMM - 2017).

Keywords:Impact loading; Error propagation; Monte carlo simulations; principle component analysis; manufacturing tolerences;

1. Introduction The increase in the computing power has taken the computer aided engineering from simulating the field event or designing a new component or improvement of the existing design from a simplified model, to modelling the actual

* Corresponding author. Tel.: +91-944-995-1113 E-mail address: [email protected] 2214-7853© 2017 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of International Conference on Materials Manufacturing and Modelling (ICMMM - 2017).

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Nomenclature PCA N R f() i n

Principle Component Analysis Nodal thickness matrix Damage response Finite element method representation as general function Node number total number of nodes

or intended geometry with all detailed features. Simplifications of the components for the purpose of simulations using numerical techniques like finite element method, finite volume method etc., have no or less significance. It is easy to model the component as it is, than to apply thought process to simplify it, due to vast increase in the computational power. Current trends in the structural and aero dynamics simulations intend to model the actual geometry with its manufacturing tolerances. This although is an ultimate way of checking the functionality of a particular component, it opens up for a custom analysis for each and individual component. In case of compressor blades of an aircraft engine, each blade manufactured either by investment casting or milling process can have different tolerances. In a full set of blades, each blade can have different dimensional accuracy within the given tolerance. Though it lies within the tolerances of the respective manufacturing process, the differences in each blade can pose a different opportunity to design and estimate their respective functionality. The current paper discusses the damage due to soft body impact on a blade and the damage variation due to its manufacturing tolerances. Estimating damage of blade due to soft body impact is done by a numerical simulation. As per current practices, carrying out simulations to each blade from a batch manufactured is not practical. This issue can be handled by bounding the maximum and minimum limits in the functionalities. In case of soft body impact of blade, by finding the maximum and minimum damage due to dimensional tolerance bounds functional variation due to dimensional tolerance. This is not a single variable problem, where we can find the upper and lower limits. Dimensional tolerance in a blade profile is a multi-dimensional problem and there are interactions between the dimensions at different locations. Three ways of treating this issue are discussed, namely error propagation, Monte Carlo simulations and principle component analysis (PCA). These methods helps us in giving bounds to the damage due to impact on blade. Error propagation gives us the possible damage variation due to the error from the mean design dimension. Monte Carlo simulations generates many possible blade geometries and estimating the damage for each of such blade gives the bounds. PCA helps in generating the blades with maximum and minimum variations due to tolerance and helps in giving the damage bounds. In other words, instead of coming out with a deterministic analysis report for the designed component, application of PCA to the simulations gives a lower and upper bound for analysis results, which cover all the components generated by a particular manufacturing process. 2. Case study: NASA rotor 67 derived blade under soft body impact For the study of damage variation due to soft body impact on blade, a model blade is derived from NASA rotor 67 [4]. It is modelled with a shell elements with variable nodal thickness to model the 3 dimensional blade model precisely. This blade model is assigned with Ti-6Al 4V material properties [3] and Johnson-cook material model is used to represent the constitutive model under different strain rates. Rotor blades manufactured by either milling [1] or by investment casting [2] will have certain dimensional tolerances due to their design thickness and curvatures. For soft body impact, a 100 mm diameter spherical bird projectile impacting the blade at 75% of span is modelled. Bird material [5] is modelled with linear equation of state and 950 Kg/m3 density as fluid material with an initial velocity of 400m/sec. Soft body impact simulation is carried out on a stationary blade and the maximum deviation of leading edge of the blade with respect to the undamaged leading edge is taken as the response. For the study purpose 10 virtual blades with different dimensions are generated within the tolerances by having an error value at 100 locations on blade surface in 10 X 10 two dimensional array. These modeled errors are mapped onto the blade geometry at each nodal location and 10 different blade models with the respective dimensional values are modelled.

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Once having all the blade models with its corresponding actual dimensions, by carrying out bird impact simulation the damage response can be measured, but in case of actual manufactured batches it is not practical, due to high cost involved in the simulations. Fig. 1. Shows the mapped dimensional errors on the blade surface. Fig. 2. Shows the finite element model of the blade undergoing spherical bird projectile impact.

Fig. 1. Virtual dimensional variations in blade geometry.

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Fig. 2. Finite element model of blades subjected to impact loading.

Next sections discussed the ways to find the bounds of response by building virtual max response model and min response model from these available blade geometries. 3. Error propagation Error propagation [6] is a method of estimating an uncertainty in the response if the uncertainties in the system initial and boundary conditions are given. In our case, the uncertainty in the initial condition is nodal thickness and the uncertainty in the response can be estimated in principle. Considering the finite element impact simulation as general function and the maximum leading edge displacement as response, we can write, at i=1,2,…n

(1)

In such case, the uncertainty in the maximum leading edge displacement can be obtained by, ∑

(2)

To calculate the uncertainty in the response, we need to evaluate partial differential for each nodal thickness, which in the model of very high degrees of freedom is not practical. This method works better for a closed form solutions than a numerical simulation models. 4. Monte Carlo simulations In this method, by collecting the nodal thickness of all the virtual blades at a particular node a normal distribution can be fit by calculating the mean and standard deviation. Similarly we can have normal distributions at each node with the corresponding nodal thickness values from all virtual blades. Fig. 3. Shows the distributions of thickness variations on the blade surface at hub, mid span, shroud and leading edge, mid chord and trailing edge. These nodal thickness distribution models are used to draw random samples of nodal thickness at each node to construct a new blade model. Thus we can randomly [7] select many blade models from within the given nodal thickness distribution models and generate an infinite number of blade models. These modelsare representative of blade model dimensional variations possible from the given manufacturing process. If all these randomly drawn

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Tirumala Rao Koka and Arun Tom Mathew/ Materials Today: Proceedings 5 (2018) 11838–11844

blade models are simulated for bird impact the collective responses of these simulations gives the distribution of damage response. This is very impractical for such high fidelity simulations, unless a simpler empirical model is build.

Fig.3. Nodal thickness error variations at hub, mid span, shroud X le, mid chord, te. 5. Principle component analysis PCA was invented in 1901 by Karl Pearson [9] as an analogue of the principal axes theorem in mechanics; it was later independently developed (and named) by Harold Hotelling in the 1930 [10]. The method is mostly used as a tool in exploratory data analysis [8] and for making predictive models. In PCA a linear transformation is used to change a set of observations possibly correlated into a set of uncorrelated variables called principle components. In PCA, the data is transformed to an eigen space with highest eigen value at the first base. This property gives a maximum variance in the first basis of PCA. To make use of the PCA basis, the nodal thickness errors are arranged into an N-dimensional column vector and these error vectors from all the initial 10 blades are arranged into an NX10 matrix. PCA projects these values onto principle eigen vectors and the first basis have maximum variance. Using this variance upper bound of blade dimensional tolerance is made by adding the first principle component to the mean nodal thickness. Similarly, lower bounds of blade nodal thickness is made by subtracting the first principle component to the mean nodal thickness.

Fig 4. Blade upper and lower bounds built from principle component.

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In this approach, irrespective of number of components manufactured in a batch, by applying the PCA we can build the maximum and minimum bounds of dimensional variations. By conducting the functional requirements on these 2 models gives the functional bounds of the all the manufactured components. In our case, on these 2 upper and lower bound blades bird impact simulation is carried out with the same initial and boundary conditions. 6. Results After the finite element simulation of bird impact on NASA rotor 67 derived blade, the blade is removed of transients due to impact loading and the LE permanent deformation is measured. To compare the deformation from all the blades and PCA bounded blades more quantitatively, deformations at different spans is measured and is plotted as box plot. Leading edge deformation values of PCA bounded blade models are also plotted over the box plot with asterisk Fig. 5. Shows the LE shapes of the mean designed blade, blades with manufacturing tolerances and also the PCA bounded blades plotted with respect to the radius of the blade.

Fig.5. Leading edge deformation w.r.t to span PCA bounded blades damage is covering the damage by all the ten virtually manufactured blade damages for majority of the LE span. PCA bounded blades damage values are covering the variations due to manufacturing tolerances in majority of the span. This method gives a way to carryout 2 simulations and predict the damage variations due to manufacturing tolerances in the majority of the spectrum. In rotating components, modal frequencies of all the blades needs to be matched as close as possible. In such scenarios instead of carrying out modal analysis for all the blades including their actual manufactured dimensions, this method provides an easy and efficient way to determine the variations. 7. Conclusions NASA rotor 67 derived blade models are used for bird impact simulations and the maximum leading edge deformation is measured. Virtual blades with dimensional tolerances are built and are simulated for the same loading. A method to capture the damage variation due to dimensional tolerances is explored. Error propagation gives a direct way to estimate the variation in response due to variation in the initial conditions. But evaluating error propagation for finite element codes is very costly and impractical.

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Fig.6. shows the damage in terms of box plots overplayed with the PCA bounded blades damage data Monte Carlo simulations are another way of bounding the variation in response. In this first build the distributions in the initial conditions and then draw random samples and conduct the finite element simulations. This requires carrying out large number of simulations and the accuracy of Monte Carlo simulations depends on the number of random draws. So this method fails due to time and cost involved. Principle component analysis gives a way of projecting the nodal thickness error data onto principle vectors, with highest variance in the first vector. This highest variance model is used to build upper and lower bound blade models. By conducting the simulations on these 2 models majority of the blade manufactured can be bounded. Maximum leading edge deformation plots shows that these bounds are capturing maximum possible variations in the response due to manufacturing variations. Moreover, while carrying out PCA, if the data can be normalized, I.e., instead of taking the thickness errors, if percentage of thickness errors are considered then these bounds becomes more accurate. Application PCA for many other such loading conditions and for different responses needs to be studies further. References [1] Ashwin Polishetty, Moshe Goldberg, Guy Littlefair, Mahesh Puttaraju, Prasad Patil, Akshay Kalra,A PRELIMINARY ASSESSMENT OF MACHINABILITY OF TITANIUM ALLOY TI 6AL 4V DURING THIN WALL MACHINING USING TROCHOIDAL MILLING, Procedia Engineering 97 ( 2014 ) 357 – 364. [2] Parlad Kumar, Rupinder Singh, I.P.S. Ahuja, Investigations on dimensional accuracy of the components prepared by hybrid investment casting, Journal of Manufacturing Processes, Volume 20, Part 3, October 2015, Pages 525–533 [3] A. Tabei, F.H. Abeda, G.Z. Voyiadjis, H. Garmestani, Constitutive modeling of Ti-6Al-4V at a wide range of temperatures and strain rates, European Journal of Mechanics - A/Solids, Volume 63, May–June 2017, Pages 128–135 [4] Anthony J. Strazisar, Jerry R. Wood, Michael D. Hathaway and Kenneth L. Suder, Laser Anemometer Measurements in a Transonic AxialFlow Fan Rotor, NASA Technical Paper 2879, November 1989. [5] Serge Abrate, Soft impacts on aerospace structures, Progress in Aerospace Sciences, Volume 81, February 2016, Pages 1–17 [6] Ku H. H., Notes on the use of propagation of error formulas, Journal of Research of the National Bureau of Standards, National Bureau of Standards, October, 1966, 70C (4): 262. [7] William L. Dunn, J. Kenneth Shultis, Exploring Monte Carlo Methods, ISBN: 978-0-444-51575-9 [8] Farid Ali Mousa, Reda A. El-Khoribi, Mahmoud E. Shoman, A Novel Brain Computer Interface Based on Principle Component Analysis, Procedia Computer Science, Volume 82, 2016, Pages 49–56 [9] Pearson, K. (1901). "On Lines and Planes of Closest Fit to Systems of Points in Space". Philosophical Magazine 2 (11): 559–572 [10] Hotelling, H. (1933). Analysis of a complex of statistical variables into principal components. Journal of Educational Psychology, 24, 417441, and 498-520