Uncertainty Quantification in Helicopter Performance ... - AIAA ARC

JOURNAL OF AIRCRAFT Vol. 48, No. 5, September–October 2011

Uncertainty Quantification in Helicopter Performance Using Monte Carlo Simulations C. Siva∗ Indian Institute of Science, Bangalore, Bangalore 560 012, India M. S. Murugan† Swansea University, Swansea, Wales SA2 8PP, United Kingdom and Ranjan Ganguli‡ Indian Institute of Science, Bangalore, Bangalore 560 012, India DOI: 10.2514/1.C000288 The effect of structural and aerodynamic uncertainties on the performance predictions of a helicopter is investigated. An aerodynamic model based on blade element and momentum theory is used to predict the helicopter performance. The aeroelastic parameters, such as blade chord, rotor radius, two-dimensional lift-curve slope, blade profile drag coefficient, rotor angular velocity, blade pitch angle, and blade twist rate per radius of the rotor, are considered as random variables. The propagation of these uncertainties to the performance parameters, such as thrust coefficient and power coefficient, are studied using Monte Carlo Simulations. The simulations are performed with 100,000 samples of structural and aerodynamic uncertain variables with a coefficient of variation ranging from 1 to 5%. The scatter in power predictions in hover, axial climb, and forward flight for the untwisted and linearly twisted blades is studied. It is found that about 20–25% excess power can be required by the helicopter relative to the determination predictions due to uncertainties.

modeling, the manufacturing process, and operational environments. However, modern systems require more critical and complex designs and need to operate at high performance, narrow margins of safety, and high reliability. Consequently, the design approaches to assess uncertainties have gained importance and are finding application in the multidisciplinary design environment [1,2]. An accurate representation of uncertainties for a given system is crucial in a stochastic analysis. Uncertainties can be classified into two categories: aleatory and epistemic [3]. Aleatory uncertainty is an inherent variation associated with the physical system or the environment. It is also referred to as variability, irreducible uncertainty, random uncertainty, and stochastic uncertainty. In structural mechanics, the uncertainties in design values and other parameters involving geometric and material properties [4,5] can be classified as aleatory uncertainties. Besides structural uncertainty, the randomness in properties such as flow velocity, lift and drag coefficients, and variation in atmospheric conditions can also be termed as aleatory uncertainties [6,7]. Epistemic uncertainty is an uncertainty that is due to a lack of knowledge of quantities or processes of the system or the environment. It is also referred to as reducible uncertainty, subjective uncertainty, and model form uncertainty. Lack of experimental data to characterize new materials and processes, human error, and poor understanding of coupled physics phenomena can be referred to as epistemic uncertainty. The modeling uncertainty, such as structural damping, is difficult to handle; hence, it is defined in terms of variability among experimental estimates [8]. Despite the complexities, different theories have been proposed to model and quantify this uncertainty effect [9]. A better understanding in modeling the physics of the problem and the availability of sufficient data can reduce the adverse effects of epistemic uncertainty. In an aircraft design process, uncertainties can often arise from both structural and aerodynamic parameters. The quantification of a pure structural uncertainty effect is well established in the literature. Oh and Librescu [10] investigated the free vibration and reliability of a cantilever composite beam considering randomness in material properties, lamina thickness, and fiber orientation of the individual constituent lamina. The effect of deviations in mechanical properties, lamina thickness, ply angle, and applied loads on the output response of a composite structure was studied by Antonio and Hoffbauer [11].

Nomenclature B Cd0 Cp Ct Cl k o tw 1s c f r

= = = = = = = = = = = = = = = =

tip-loss factor blade profile drag coefficient power coefficient thrust coefficient two-dimensional lift-curve slope induced power correction factor rotor disk angle of attack blade pitch angle blade twist rate per radius of the rotor longitudinal cyclic pitch angle inflow ratio climb inflow ratio mean value advance ratio standard deviation rotor solidity

I. Introduction

U

NDERSTANDING and management of uncertainties have become increasingly important for aircraft industries to advance in the design process, make better field development decisions, optimize the manufacturing techniques, and improve dayto-day technical operations. Uncertainty exists everywhere, such as in raw data measurements, data processing and interpretation, system

Received 1 February 2010; revision received 18 April 2011; accepted for publication 2 May 2011. Copyright © 2011 by C. Siva, M. S. Murugan, and Ranjan Ganguli. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0021-8669/11 and $10.00 in correspondence with the CCC. ∗ Graduate Student, Department of Aerospace Engineering; mitaerosiva@ yahoo.co.in. † Postdoctoral Research Assistant, School of Engineering; s.m.masanam@ swansea.ac.uk. Senior Member AIAA. ‡ Professor, Department of Aerospace Engineering; [email protected]. ernet.in. Associate Fellow AIAA. 1503

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SIVA, MURUGAN, AND GANGULI

A health monitoring system was developed for civil structures modeled with randomness in loads and elastic modulus [12]. Mahadevan et al. [13] have developed a probabilistic analysis framework to predict the fatigue reliability of laminated composites featuring uncertainties in elastic constants, bending angle, axial load, and strain energy release rate under centrifugal and oscillatory bending loads. In the literature, the studies focusing on aerodynamic uncertainty are much less when compared with those focusing on structural uncertainties. Loeven et al. [14] considered the freestream velocity over the airfoil as uncertain input data and studied its effect on aerodynamic parameters like lift and drag. In general, several stochastic fluid mechanics problems have been addressed, combining uncertainties in fluid properties and geometry of the structure [15,16]. However, the aerodynamic uncertainty has a substantially greater effect and is the source of some of the key nonlinear effects in aeroelasticity. Recently, there has been some research oriented toward the study of the effects of various uncertainties in aeroelasticity. Pettit [17] discusses the uncertainty sources and quantification in various aeroelastic problems such as flutter and limit-cycle oscillations. Murugan et al. [18,19] have investigated the effect of material uncertainty on blade cross-sectional stiffness, rotating natural frequencies, aeroelastic response, and vibratory loads of a composite helicopter rotor. Hosder et al. [20] studied the transient response of a transonic wing featuring uncertainty in Mach number and angle of attack. Most research on helicopter aeroelastic analysis and design has focused on improving the physical and mathematical models in order to improve predictions. Sitaraman and Roget [21] coupled a computational fluid dynamics (CFD) framework with computational structural dynamics to predict helicopter maneuver loads. Yeo and Johnson [22] used the comprehensive rotorcraft analysis software CAMRAD II for the performance, stability, and control analysis of a compound helicopter. Bain et al. [23] investigated the variable droop leading-edge concept using a modified version of the Navier–Stokes solver OVERFLOW. Such variable droop leading-edge airfoils reduce the increase in drag and moment that occurs due to dynamic stall. The CFD code OVERFLOW was coupled with the structural dynamic code DYMORE, and the numerical predictions were compared with test data. Gagliardi and Barakos [24] studied a flapequipped low twist rotor in hover using a blade element method combined with two-dimensional (2-D) and three-dimensional (3-D) CFD. Yee et al. [25] studied the aerodynamic performance of the Gurney flap by using a 2-D Navier–Stokes equation. The advancements in computational power, in the last decades, have led to the use of sophisticated methods in helicopter analysis and design. However, these methods attempt to reduce epistemic uncertainty through better modeling. The effect of aleatory uncertainty can have a significant impact on helicopter performance and needs to the addressed in the literature. Research in applied mechanics has shown that aleatory uncertainty can significantly impact the design process. Predicting aircraft performance is a crucial part of the aircraft design process. Such predictions based on deterministic computational methods can differ from the measured system performance because of the unaccounted randomness in the data or model used in the analysis. In general, the field performance cannot be exactly predicted by today’s computational methods or by a finite number of experiments [26,27]. Therefore, the use of probability and statistics in guiding engineering design and maintenance decisions is needed. The present work attempts to identify the structural and aerodynamic uncertain variables, apply the probability techniques, and assess the impact of uncertainties on helicopter performance using the classical Monte Carlo simulation (MCS). The MCS is one of the most popular nonintrusive uncertainty analysis techniques that can be performed without any modification or approximation of the governing equations. The performance parameters are formulated using blade element theory in conjunction with momentum theory for three flight conditions of the helicopter: hover, axial climb, and forward flight. The uncertainty effect on thrust and power predictions of the untwisted (UT) and linearly twisted (LT) blades are studied. To the best of the authors’ knowledge, helicopter performance has not

been analyzed to account for uncertainty effects throughout the flight envelope. This study aims to address this gap in the literature. A simple aerodynamic model assuming constant coefficients is used to establish benchmark results and allow a very large number of MCSs to ensure convergence.

II.

Formulation

The Rankine–Froude momentum theory and blade element theory are classical tools in the helicopter rotor design and performance analysis. The Rankine–Froude momentum theory is a simple aerodynamic model that considers the main rotor to be a thin actuator disk with pressure difference, and uniform inflow assumptions as well. In contrast, blade element theory accounts for individual blades, and the rotor performance is obtained by integrating the sectional airloads at each blade element over the length of the blade and averaging the result over a rotor revolution. This theory provides the designer with radial and azimuthal distributions of the blade aerodynamic loading over the rotor disk. Therefore, the blade element theory can be used as a basis to assist the design of rotor blades in terms of blade twist, planform distribution, and perhaps the airfoil shape to provide a specified overall rotor performance. The blade element theory assumes no mutual influence of adjacent blade sections, and each blade section is assumed to be a quasi-2-D airfoil in producing aerodynamic loads and moments. The real inflow at the rotor is highly nonuniform in nature because of the vortical wake trailed from each blade and flow interactions with airframe components. However, in this study, the induced velocity is assumed to be of uniform distribution over the rotor disk for simplicity. Thus, momentum theory and blade element theory are combined for the results. The tip-loss factor is incorporated in the formulation to account for the adverse effects on rotor thrust and induced power because of the locally high induced velocities produced at the blade tips by the trailed tip vortices. The Prandtl tip-loss factor (B) is used to represent the loss of blade lift, and its typical value is between 0.95 and 0.98 for most helicopter rotors. For a constant thrust, there is an increase in induced inflow by a factor of B1 . Because of tip-loss effects, a real rotor will always have a higher overall average induced velocity compared with that given by momentum theory, and so the induced power will be higher relative to the simple momentum theory result. A summary of the key mathematical expressions of blade element theory used in the performance analysis is given in the following sections. Detailed derivations of those equations can be found in books [28–31]. A.

Hover

Hover is vertical flight with zero climb velocity. In hover, the thrust produced should be equal to weight of the helicopter. The rotor blade is taken to be of rectangular shape for both UT and LT cases considered in this study. Although the 2-D lift-curve slope of the airfoil section is a function of local incident Mach number and Reynolds number, an average value for the rotor is assumed without any serious loss of accuracy. The tip-loss factor is applied to account for 3-D effects. A measure of the thrust produced by the rotor is expressed by a nondimensional number called thrust coefficient, which is obtained by integrating the incremental thrust coefficient along the blade from root to tip. All helicopter rotor blades use some amount of spanwise twist in their shape, in different forms and different amounts. The use of blade twist provides the rotor with several advantages. Many rotor blades are designed with a linear twist of the form r o rtw , where tw is the blade twist rate per radius of the rotor. So, for a LT blade in hover, the thrust coefficient is given by 1 o tw (1) Ct r Cl 2 2B 3 4 The thrust coefficient given by Eq. (1) is used for a blade with zero twist by considering (tw 0).

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Hover is a simple flight regime to analyze mathematically because of its axisymmetric flowfield. As a result, assuming the inflow to be of uniform distribution and from a simple momentum theory, the inflow ratio is given as r Ct (2) 2 The power coefficient, a nondimensional number, has high significance in power calculations of a helicopter. In calculating the power coefficient, the induced power is corrected for both the tip loss and nonuniform inflow by incorporating induced power correction factor k in the equation. Taking into account induced power and profile power, the power coefficient is given by 1 Cp kCt r Cdo 8 B.

(8)

where i Ct =2f is Glauert’s high-speed approximation of the induced inflow ratio for advance ratios greater than 0.1. The power coefficient comprising induced power and profile power is given by Cp

kCt2 1 C 1 K2f 2f 8 r do

(9)

where the numerical value of K varies from 4.5 in hover to 5 at an advance ratio of 0.5 in forward flight, depending on the various assumptions and approximations that are made.

(3)

III. Uncertainty Analysis in Hover

Axial Climb

Axial climb is vertical flight with nonzero climb velocity, and the inflow velocity at the rotor disk is the summation of climb velocity and induced velocity at the rotor disk. For linear twist, the thrust coefficient is given as 1 o tw (4) Ct r Cl 2 2B 3 4 More interesting, in hover and axial climb, the blade with a linear twist has the same thrust coefficient as one of constant pitch when is set to the pitch of the twisted blade defined at 0.75 radius. Since the flow is axisymmetric, assuming the inflow to be of uniform distribution, we get the inflow ratio from simple momentum theory as s 2 c c C t (5) 2 2 2 The thrust coefficient equations of hover and axial climb are of the same form but differ only in the inflow ratio because of nonzero climb velocity in the axial climb. The power coefficient comprising induced power and profile power is given by 1 Cp kCt r Cdo 8 C.

f tan i

(6)

Forward Flight

Axisymmetry of the flow is lost in forward flight because of forward tilt in the rotor disk to produce both the lift and thrust. The rotor moves almost edgewise through the air, and the blade section encounters a periodic variation in local velocity. This gives rise to a number of complications in the aerodynamics of the rotor, including the effects of blade flapping, significant compressibility effects, unsteady effects, nonlinear aerodynamics, complex induced velocity from rotor wake, possibility of stall, reverse flow, and so on. The flow properties vary radially and azimuthally, making the mathematical model formidable. However, using the blade element theory and momentum theory with certain simplifying assumptions, the main effects of the rotor aerodynamics can be obtained. For a blade with linear twist, integrating the incremental thrust coefficient from root to tip, the thrust coefficient is obtained as f 1 3 (7) Ct r Cl o 1 2f tw 1 2f 1s 2 2 2B 3 4 2 In forward flight, the effect of the individual tip vortices tends to produce a highly nonuniform inflow over the rotor disk, and the calculation of these effects is computationally expensive. The performance of the rotor can be analyzed with the aid of a simple uniform inflow model from momentum theory to represent the basic effects of inflow resulting from the rotor wake. Assuming the inflow to be of uniform distribution and from a simple momentum theory, we get the inflow ratio as

A.

Uncertain Variables

The uncertainty analysis in hover is carried out for UT and LT blades. The performance parameters of interest are thrust and power coefficients of the rotor. From Eqs. (1–3) for the hover case, it is seen that uncertainty can exist in blade chord, rotor radius, lift-curve slope, blade profile drag coefficient, blade pitch angle, and blade twist rate per radius of the rotor. These values are therefore treated as random variables in hover. The uncertainty in blade twist rate becomes known only in the analysis of LT blades. The uncertainty effect is described in terms of the coefficient of variation (COV), which is defined as the ratio of standard deviation (SD) to the mean of the random variable. The mean value, distribution type, and COV chosen for each random variable are given in Table 1. We have taken a low COVof 1% for geometric variables and a relatively high COVof 5% for aerodynamic coefficients, which can change due to erosion, insect damage, etc., changing the blade profile. The lift-curve slope was set to 5:73=rad, which represents a small reduction of the 2-D thin airfoil result of 6:28=rad because of finite airfoil thickness and Reynolds number effects. The mean value of blade the profile drag coefficient is a typical deterministic value, and the pitch angles of the UT blade and LT blade are fixed at 10 and 16 , respectively. The blade twist rate per radius of the rotor is 8 and the rest of the values are taken for a known helicopter configuration of mass 1363.6 kg [30]. The blade pitch angle and blade twist rate of a LT blade are chosen in such a way that they produce the same thrust as that of an UT blade, numerically. The attempt is to find the uncertainty effect on UT and LT blades while keeping the thrust constant. All the random input variables are expected to follow a normal distribution by the central limit theorem, which states that the sum of many arbitrary distributed random variables asymptotically follows a normal distribution when the sample size grows large. In a few engineering applications, arbitrary distribution of randomness may occur. Such distributions can be converted into equivalent normal distribution using the hybrid algorithm proposed by Hajela and Vittal [31]. The COV for each uncertain variable is specified based on literature surveys and domain knowledge. The geometric parameters are controllable to a certain extent; hence, they take the least covariance. The atmospheric and aerodynamic parameters are much more random in nature; hence, they take the highest covariance.

Table 1

Uncertain variables in hover

Random input variable

Mean value

Distribution type

COV, %

Blade chord Rotor radius Lift-curve slope Blade profile drag coefficient Blade pitch angle

0.314 m 4.0 m 5:73=rad 0.011 10 for UT 16 for LT 8

Normal Normal Normal Normal Normal

1 1 5 5 1

Normal

1

Blade twist rate

1506

SIVA, MURUGAN, AND GANGULI x 10 -4

Power coefficient

2.9

2.8 2.75

7 6 5

0

1

2

3

4

5

6

7

8

9

Number of samples

10 x 10 4

6000

2.65 2.6 2.55 0

5

10

4000 2000 0 5.5

15

Number of samples for MCS

x

6

6.5

7

7.5

8

8.5 x 10 -4

Probability distribution of power coefficient

10 4

Fig. 1 Convergence of SD of thrust coefficient of an UT blade in hover. B.

8

2.7

Samples

SD of thrust coefficient

2.85

x 10 -4

9

Fig. 4 Fluctuations and probability distribution of power coefficient of an UT blade in hover.

Number of Samples

The Monte Carlo methods are stochastic techniques based on the use of random numbers to investigate problems of various kinds. In the present work, evaluation of a deterministic model using sets of random numbers as inputs is used to study the uncertainty effects. From the mean and SD of a chosen random variable, the random input values for MCS are generated as follows:

coefficient and power coefficient in hover, it is apparent that 100,000 samples are sufficient for satisfactory MCS results. It is also clear that relatively low levels of samples, such as 5000, often used in MCS of engineering systems may be too low for the helicopter performance problem. C.

xi ri

(10)

where ri is the random number generated with a mean of 0 and an SD of 1 of a normal distribution. The number of samples for MCS is selected based on the convergence of SD of the performance parameters. From the SD convergence results (Figs. 1 and 2) of thrust

Monte Carlo Simulation Results

The MCS is carried out for the performance parameters, such as thrust coefficient and power coefficient, with generated random numbers. The uncertainty effect is given in the form of fluctuations from the mean value, probability distributions, percentage deviations from the baseline, and normal probability plots. Figures 3–6 show the

3.4

Thrust coefficient

x 10 -3 x 10 -5

3.3

7 6 0

1

2

3

3.25

4

5

6

7

8

9

Number of samples

3.2

10 x 10 4

6000

Samples

SD of power coefficient

3.35

8

3.15 3.1

4000 2000

3.05

0 6

6.5

7

7.5

8

8.5

9

9.5 x 10 -3

Probability distribution of thrust coefficient 0

5

10

15 x 10 4

Number of samples for MCS

Fig. 5 Fluctuations and probability distribution of thrust coefficient of a LT blade in hover.

Fig. 2 Convergence of SD of power coefficient of an UT blade in hover.

Power coefficient

Thrust coefficient

x 10 -3

8 7 6

0

1

2

3

4

5

6

7

8

9

Number of samples

10 x 10 4

9

x 10 -4

8 7 6 5

0

1

2

3

4

5

6

7

8

9

Number of samples

10 x 10 4

6000

Samples

Samples

6000 4000 2000 0

6

6.5

7

7.5

8

8.5

Probability distribution of thrust coefficient

9

9.5 x 10 -3

Fig. 3 Fluctuations and probability distribution of thrust coefficient of an UT blade in hover.

4000 2000 0 5.5

6

6.5

7

7.5

8


8.5

9 x 10 -4

Fig. 6 Fluctuations and probability distribution of power coefficient of a LT blade in hover.

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Table 2

Uncertainty effect on performance parameters in hover

Performance parameter

Mean

COV, %

Thrust coefficient Power coefficient

7:572 103 7:201 104

3.52 4.45


7:572 103 7:202 104

4.02 5.03

Uncertain Variables

The uncertainty analysis in axial climb is carried out for UTand LT blades. From Eqs. (4–6) for the axial climb case, it is seen that

0.3

0.3

0.15 0.1

0.25 0.2 0.15 0.1 0.05

0.05 0 -20

-15

-10

-5

0

5

10

0 -20

15

-15

% deviation from baseline - thrust coefficient

-5

0

5

10

15

20

Fig. 9 Probability histogram of thrust coefficient of a LT blade in hover.

0.35

0.4

0.3

0.35

Probability of occurrence


-10


Fig. 7 Probability histogram of thrust coefficient of an UT blade in hover.

0.25 0.2 0.15 0.1 0.05 0 -20

8:960 103 8:990 104

IV. Uncertainty Analysis in Axial Climb A.

0.35

0.2

8:613 103 8:485 104

The thrust coefficient shows a percentage deviation of 15 to 15 from the baseline for an UT blade, whereas in the case of a LT blade, it is 20 to 20 from the baseline. The power coefficient shows a percentage deviation of 20 to 20 from the baseline for an UT blade, whereas in the case of a LT blade, it is 20 to 25 from the baseline. Predominantly, it is seen that a deviation of more than 20% exists in the predicted hovering power. Normal probability plots of MCS results are carried out to formally verify the Gaussian nature of output distributions. In these figures, a linear variation would indicate a Gaussian distribution. Any deviation from linearity is an indication of non-Gaussian distribution. From the normal probability plots (Figs. 11 and 12), it is seen that the performance parameters of UT and LT blades follow a Gaussian distribution.

0.35

0.25

Maximum value

UT blade 6:427 103 5:879 104 LT blade 6:138 103 5:645 104



fluctuations and probability distributions of performance parameters for UT and LT blades because of the uncertainty propagation in a hover case. There is considerable dispersion in the predictions due to randomness, but the distributions appear to be Gaussian, as can be seen from the bell-curve slope. The MCS results of UT and LT blades of a hover case are calculated and summarized in Table 2. It is seen that the performance parameters of a LT blade show a larger COV than the UT blade. In particular, the power coefficient of a LT blade shows a maximum COV of 5.03% and necessitates the appropriate selection of a power plant for the helicopter. For the UT blade, the maximum change in power is about 18%, and for the linear twisted blade, it is about 25%. Thus, considerable excess power capacity should be present in the helicopter to account for uncertainty effects. The stochastic analysis provides quantitative bounds on the power required, which is useful in design. Figures 7–10 show the percentage deviation from the baseline deterministic value of the performance parameters for UT and LT blades because of the uncertainty propagation in a hover case.

Minimum value

0.3 0.25 0.2 0.15 0.1 0.05

-15

-10

-5

0

5

10

15

20

0 -25

% deviation from baseline - power coefficient

Fig. 8 Probability histogram of power coefficient of an UT blade in hover.

-20

-15

-10

-5

0

5

10

15

20

25


Fig. 10 hover.

Probability histogram of power coefficient of a LT blade in

Thrust coefficient


0.999 0.997 0.99 0.98 0.95 0.90 0.75

x 10 -3

7 6

1

2

3

4

5

6

7

8

9

Number of samples

0.25 0.10 0.05 0.02 0.01 0.003 0.001

10 x 10 4

8000

6.5

7

7.5

8

6000 4000 2000

8.5

0 5

x 10 -3

Thrust coefficient

6

6.5

7

7.5

8 x 10 -3

Fig. 13 Fluctuations and probability distribution of thrust coefficient of a LT blade in axial climb.

0.999 0.997 0.99 0.98 0.95 0.90 0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001

5.5


Power coefficient

Fig. 11 Normal probability plot of thrust coefficient of an UT blade in hover.

9

x 10 -4

8 7 6 5

0

1

2

3

4

5

6

7

8

Number of samples

9

10 x 10 4

8000

Samples

Probability

8

5 0

0.50

Samples

Probability

1508

6

6.5

7

7.5

8

8.5 x 10 -4

Power coefficient

Fig. 12 Normal probability plot of power coefficient of an UT blade in hover.

uncertainty can exist in blade chord, rotor radius, lift-curve slope, blade profile drag coefficient, rotor angular velocity, blade pitch angle, and blade twist rate per radius of the rotor. These values are therefore treated as random variables in axial climb. The uncertainty in the blade twist rate is used only in the analysis of LT blades. As stated in the hover case, the blade pitch angle and blade twist rate of a LT blade are chosen in such a way that it produces the same thrust as

Table 3

Uncertain variables in axial climb

Random input variable

Mean value

Distribution type

COV, %

Blade chord Rotor radius Lift-curve slope Blade profile drag coefficient Rotor angular velocity Blade pitch angle

0.314 m 4.0 m 5:73=rad 0.011 51:83 rad=s UT 10 LT 16 8

Normal Normal Normal Normal Normal Normal

1 1 5 5 2 1

Normal

1

Blade twist rate

Table 4

6000 4000 2000 0

5

6

7

9 x 10 -4

Fig. 14 Fluctuations and probability distribution of power coefficient of a LT blade in axial climb.

an UT blade. All random variables are expected to follow a normal distribution type by the central limit theorem. The mean value and COV chosen for each random variable are given in Table 3.

B.


The MCS is carried out for the performance parameters, such as thrust coefficient and power coefficient, with 100,000 samples of generated random input values, and the results are obtained. Figures 13 and 14 show the fluctuations and probability distributions of performance parameters for LT blades from the uncertainty propagation in the axial climb case. The MCS results of UT and LT blades of the axial climb case are calculated and summarized in Table 4. The power coefficient of LT blade shows a larger dispersion with a COV of 4.83%. As seen in hover, compared with an UT blade, performance parameters of a LT blade show a larger COV. Figures 15 and 16 show the percentage deviation from the baseline deterministic value of the performance parameters for LT blades because of the uncertainty propagation in axial climb case.

Uncertainty effect on performance parameters in axial climb

Performance parameter

8


Mean

COV, %


6:361 103 6:922 104

3.55 4.16


6:361 103 6:923 104

4.18 4.83

Minimum value UT blade 5:372 103 5:612 104 LT blade 5:119 103 5:346 104

Maximum value 7:475 103 8:356 104 7:668 103 8:702 104

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SIVA, MURUGAN, AND GANGULI 0.4

Thrust coefficient

0.3 0.25

9 8 7 6 5

0.2

0

0.1

0 -20

3

4

5

6

7

8

9

10 x 10 4

-15

-10

-5

0

5

10

15

4000 2000

20

0 5.5


Fig. 15 Probability histogram of thrust coefficient of a LT blade in axial climb.

6

6.5

7

7.5

8

8.5

9

9.5 x 10 -3


Fig. 17 Fluctuations and probability distribution of thrust coefficient of a LT blade in forward flight.

0.4

Power coefficient

x 10 -4

0.35 0.3 0.25 0.2

3

2

0

1

2

3

4

5

6

7

8

9

Number of samples 0.15

8000

0.1

6000

Samples


2

6000

0.15

0.05

0.05 0 -30

-20

-10

0

10

20

30

Fig. 16 Probability histogram of power coefficient of a LT blade in axial climb.

The thrust coefficient shows a percentage deviation of 15 to 15 from the baseline for an UT blade, whereas in the case of a LT blade, it is 20 to 20 from the baseline. The power coefficient shows a percentage deviation of 20 to 20 from the baseline for an UT blade, whereas in the case of a LT blade, it is 20 to 25 from the baseline. As seen in hover, a deviation of more than 20% exists in the predicted climb power. The upper bound on climb power is a design point for the maximum power required (the other extreme being at high-speed forward flight); therefore, this extra power needs to be available to account for the uncertainty effect.

V. Uncertainty Analysis in Forward Flight Uncertain Variables

The uncertainty analysis of forward flight at an advance ratio of 0.3 is carried out for UT and LT blades. From Eqs. (7–9), it is seen that uncertainty can exist in blade chord, rotor radius, lift-curve slope, blade profile drag coefficient, rotor angular velocity, blade pitch angle, and blade twist rate per radius of the rotor. These values are therefore treated as random variables in a forward flight analysis, and

2000 2.6

2.8

3

3.2

3.4

3.6


3.8

4 x 10 4

Fig. 18 Fluctuations and probability distribution of power coefficient of a LT blade in forward flight.

they are same as the values given in Table 3 of the axial climb analysis. B.


The MCS is carried out for the performance parameters, such as thrust coefficient and power coefficient, with 100,000 samples of generated random input values, and the results are obtained. Figures 17 and 18 show the fluctuations and probability distributions of performance parameters for LT blades because of the uncertainty propagation in the forward flight case. The MCS results of UT and LT blades are calculated and summarized in Table 5. As seen in hover and axial climb, compared with an UT blade, performance parameters of a LT blade show a larger COV. Figures 19 and 20 show the percentage deviation from the baseline deterministic value of the performance parameters for UT and LT blades in the forward flight case. The thrust coefficient shows a percentage deviation of 20 to 20 from the baseline for an UT blade, whereas in the case of a LT blade, it

Table 5 Uncertainty effect on performance parameters of forward flight Performance parameter

10 x 10 4

4000

0 2.4


A.

1

Number of samples

Samples


0.35

x 10 -3

10

Mean

COV, %


7:168 103 3:024 104

4.68 4.84


7:530 103 3:136 104

5.38 5.32

Minimum value UT blade 5:744 103 2:397 104 LT blade 5:925 103 2:400 104

Maximum value 8:551 103 3:727 104 9:361 103 3:922 104

1510


0.35


0.3 0.25 0.2 0.15 0.1 0.05 0 -25

-20

-15

-10

-5

0

5

10

15

20

25


Fig. 19 Probability histogram of thrust coefficient of a LT blade in forward flight.

flight. The structural and aerodynamic uncertainties result in significant effects on the thrust and power predictions of a helicopter. Uncertainty increases the power required by about 5 to 10%, with a probability of 10 to 12%, relative to the deterministic predictions. It should also be noted that the results in this study are based on the assumed COVs and theoretical models used. However, the outcome of this work clearly shows the need to incorporate randomness of structural and aerodynamic variables in the helicopter rotor design and performance analysis. Understanding the influence of various uncertainties on the performance of a helicopter can lead to a shift from the safety margin concepts to uncertainty quantification by stochastic computational analysis. Further research can focus on the use of sophisticated mathematical models and more realistic probability distributions functions other than normal distribution functions for the uncertainty analysis, with the current results serving as a benchmark solution. Experiments need to be conducted to find the probability distribution functions for different helicopter rotor design parameters.

0.35

References Probability of occurrence

0.3 0.25 0.2 0.15 0.1 0.05 0 -25

-20

-15

-10

-5

0

5

10

15

20

25


Fig. 20 Probability histogram of power coefficient of a LT blade in forward flight.

is 20 to 25 from the baseline. The power coefficient shows a percentage deviation of 20 to 25 from the baseline for an UT blade, whereas in the case of a LT blade, it is 25 to 25 from the baseline. It can be seen that the power required can be underpredicted by as much as 25% relative to the deterministic value. The thrust and power coefficients of UT and LT blades follow a Gaussian distribution.

VI.

Conclusions

The structural and aerodynamic uncertainty effects on helicopter performance predictions are presented. The helicopter aerodynamic model based on blade element theory is used to predict the performance parameters of hover, axial climb, and forward flight. MCSs are performed with 100,000 samples of structural and aerodynamic uncertain variables with a COV ranging from 1 to 5%. The uncertainty effect is given in the form of fluctuations from the mean value, probability distributions, percentage deviations from the baseline, and normal probability plots. The uncertainty effect on power requirements of UT and LT blades is studied. The power coefficient of hover shows a COVof 4.45% for an UT blade and a COVof 5.03% for a LT blade. In an axial climb, the power coefficient shows a COVof 4.16 and 4.83% for the UT and LT blades, respectively. At an advance ratio of 0.3 in forward flight, the power coefficient shows a COVof 4.84 and 5.32% for the UT and LT blades, respectively. The extreme values of scatter in power predictions show an almost 20 to 25% deviation from the baseline for both UT and LT blades, although with a probability of occurrence less than 0.005. The power coefficient of forward flight has a maximum COVof 5.32% and should be considered while choosing a power plant for the helicopter to attain the desired performance. The numerical results of this study provide useful bounds on helicopter power requirements in hover, axial climb, and forward

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