ahs09

Effect of Uncertainty on Hub Vibration Response of Composite Helicopter Rotor BladesYung Hoon Yu11

Prashant M. Pawar2

Sung Nam Jung1

Department of Aerospace Information Engineering Konkuk University, Seoul 143-701, Korea2

Department of Mechanical Engineering, S.V. E. R. Is College of Engineering, Pandharpur, Maharashtra, India

Abstract Main focus of this study is to understand the effects of uncertainty in the composite material properties on the helicopter hub vibratory loads. The stochastic behaviors of composite materials properties obtained from previous experimental studies are used to evaluate the stochastic behaviors of the cross-sectional stiffness properties of composite rotor blades. The stochastic behavior of cross-sectional stiffness of composite blades introduces dissimilarity in the rotor system. A comprehensive aeroelastic code suitable for dissimilar rotor analysis, which is based on the governing equations of motion for composite helicopter rotor blades, obtained using the Hamiltons principle is used for vibratory load analysis. The stochastic behaviors of hub vibratory loads are obtained using Monte-Carlo simulation along with the aeroelastic analysis code. The baseline blade is modeled as a one dimensional thin-walled box-section beam with stiffness properties similar to a stiff-inplane rotor blade.

I. I NTRODUCTION Composite materials are the most preferred materials in the aircraft industry because of their superior fatigue characteristics, damage tolerance and stiffness to weight ratio as compared to that of metals (Ref. 1, 2, 3, 4). These materials are the most preferred materials for rotor blades in the modern helicopters. Composite materials for rotor system bring classical advantages along with reduction in bulkiness of rotor hub inturn advantage of reducing the prole drag. However, materials uncertainty is the major problem of composite materials that of isotropic materials. Uncertainty in the basic material properties of composites could affect the behavior of the composite rotor system through the behavior of individual rotor blade. Uncertainty analysis of aeroelastic response is a complex process which involves various nonlinearties and interactions between structural and aerodynamic disciplines. A review paper by Pettit (Ref. 5) gives a comprehensive survey on the inuence of the uncertainties in the aeroelastic analysis. Uncertainties in the aeroelastic analysis could arise from the structural, aerodynamic and control dynamic factors. Various studies have been conducted to understand the inuence of the structural uncertainties such as Youngs modulus, boundary conditions, geometrical congurations, and loads on the aeroelastic response of the structure. Recent applications of uncertainty quantication to various aeroelastic problems suchPresented at the 65th American Helicopter Society Annual Forum, May 27May 29, 2009, Grapevine, Texas, USA. Copyright c 2009 by the American Helicopter Society International, Inc. All rights reserved.

as utter ight-testing, prediction of limit-cycle oscillations, and design optimization with aeroelastic constraints gives a new physical insights and promising path towards design improvement of the structure. However, almost all the uncertainty analysis studies were focused on the aeroelastic response of the xed wing aircraft (Ref. 6, 7, 8). Rotorcraft uncertainty analysis is even more complicated than that of the xed wing aircraft analysis due to unsymmetrical nature of the lift pattern and the rotating components. Murguan et al. (Ref. 9) initiated the rotorcraft uncertainty analysis by investigating the effects of material uncertainties of the composites on the cross-sectional stiffness properties, natural frequencies, and aeroelastic responses of the composite helicopter rotor blades. Stochastic behaviors composite material properties available in the literature (Ref. 10, 11, 12, 13) in the form experimental data along with Monte-Carlo simulation methods are used to estimate the stochastic behaviors of the cross-sectional stiffness properties and aeroelastic response of a box beam model of composite rotor blade. The numerical results show about 6 % coefcient of variation for crosssectional stiffness properties and about 3% coefcient of variation for non-rotating rotor blade natural frequencies. This study demonstrates that uncertainty in the composite material properties get propagated into aeroelastic response, which causes large deviations, particularly in the higher-harmonic components that are critical for the accurate prediction of helicopter blade loads and vibration. However, this study was focused for blade level analysis. The blade level effects may get transmitted through rotor hub to the helicopter fuselage, which could change the rotor system behavior affecting the helicopter vibratory load patterns. Main rotor system is the principle source of the helicopter vibrations, which restricts them from achieving higher speed, maneuverability, agility, and crew effectiveness. These vibrations are of the two types: vibrations inherent to the asymmetric nature of a rotor in forward ight and are present even in case of balanced rotor (tracked rotor) system; vibrations due to the blade-to-blade dissimilarity, which results from manufacturing uncertainties, highly vibratory operating conditions and environmental effects (Ref. 14). The current study is intended to predict the hub vibratory loads arising from the disimilarity caused due to composite material uncertainties. Few researchers (Ref. 15, 16, 17, 18) have focused on predicting the rotor vibrations for assuming the dissimilarity in the rotor system due to various fault for structural health

monitoring or vibration analysis. The track and balance and rotor smoothening are the most popular techniques used for used for correcting or alleviating the dissimilarities of the rotor system (Ref. 14). These methods use adjusting weights, the pitch link and trim tabs are adjusted to minimize the vibrations using a predetermined relation between vibrations and corrective adjustments. Few researchers have devoted efforts to develop an in-ight tracking correction based on the electro-mechanical actuator, shape memory alloy tracking tabs (Ref. 19) and a trailing-edge ap (Ref. 20). Roget and Chopra (Ref. 21) developed trailing edge ap based approach for simultaneous reduction of the regular vibrations along with vibrations due to rotor dissimilarities. Recently, Pawar and Jung (Ref. 22) demonstrated use of active twist control for vibration reduction in dissimilar composite rotor blades. Main aim of the current study is to understand the inuence of the composite material uncertainties on the rotor hub vibratory loads. Uncertainties in the mechanical properties of the composite materials E1 , E2 G12 and 12 are used to evaluate the stochastic behaviors of the cross-sectional stiffness values of the composite rotor blades. Using the Monte-Carlo simulation along with the aeroelastic code, the hub vibratory loads are obtained under the uncertainties in the cross-sectional properties of composite blades. The baseline rotor blade is modeled as a one dimensional thin-walled box-section beam with stiffness properties similar to a stiff-inplane rotor blade. First sensitivity analysis is carried out to understand inuence of uncertainty in each cross-sectional stiffness properties. Finally, stochastic behaviors of rst few modes hub vibratory loads are studied to understand the inuence of composite material uncertainty on the hub vibratory loads. II. A EROELASTIC ANALYSISOF DISSIMILAR ROTOR SYSTEM

developed by Leishman and Beddoes (Ref. 23) along with a free wake model developed by Bagai and Leishman (Ref. 24) The effects of composite material are included in the aeroelastic analysis through the strain energy expression. The strain energy expression of the system can be written in symbolic form as U = UI + UC (2)

where UI is the contribution from isotropic materials, UC is the contribution from the composite elastic coupling (Ref. 1). The nite element method is used to solve the governing equations of the motion and allows non-uniform blade properties along the length. A 15-degree of freedom beam nite element is used to describe the ap-lag-torsion coupled behavior of rotating blades. Unlike the identical blade analysis, in the dissimilar rotor system, the blade response of each blade is obtained individually. Steady and vibratory components of blade loads are calculated using the force summation method. In this method, blade inertia and aerodynamic forces are integrated directly over the length of the blade. The xed frame hub loads are obtained by summing the contributions from the individual blades. III. N UMERICAL R ESULTS Uncertainties in the composite material properties are used to evaluate the stochastic behaviors of the cross-sectional stiffness properties of the rotor blades. Stochastic behaviors of cross-sectional stiffness properties of the blades are used to evaluate stochastic behaviors of hub vibratory loads using Monte-Carlo simulation along with aeroelastic analysis code. The baseline rotor blade is modeled as a uniform singlecell thin walled composite box beam that matches with the realistic magnitudes of cross-section stiffness, inertia and rotating frequencies of the stiff-inplane rotor blade (Ref. 25). Respective dimensions of the box section are: outer width 203.2 mm, outer depth 38.1 mm and wall thickness 3.556 mm. The mechanical properties are of AS4/3501-6 graphite/epoxy lamina. The layups of each wall of the box section are given as [04 /(+15/ 15)3/(+30/ 30)2]s . Effects of uncertainties on hub vibratory loads are studied at in a forward ight with an advance ratio of 0.3 with a moderate thrust condition CT / = 0.07 and for a Lock number of 6.34. A. Stochastic behavior of cross-sectional stiffness properties Experimental studies in the literature (Ref. 10, 11, 12, 13) on the stochastic behaviors of composite materials show coefcient of variation (CoV) values of about 5-10 %. Experimental scattering of E1 , E2 , 12 of a graphite/epoxy material given in reference (Ref. 26) have mean values of the material properties equivalent to that of the baseline blade. Hence, the stochastic behaviors of E1 , E2 and 12 given in (Ref. 26) are considered for the current analysis. Variations in G12 are obtained from the CoV of the E2 distribution as considered in (Ref. 27). Table 1 show the stochastic behavior of the material properties used for the design of composite blade.

Uncertainties in the cross-sectional stiffness properties of the composite rotor blade introduce dissimilarity in the composite rotor system. A comprehensive aeroelastic analysis system has been used to obtain the vibrations of a helicopter with dissimilar rotor system. For the aeroelastic analysis, the helicopter is represented by a non-linear model of rotating elastic blades dynamically coupled to a six-degree-of-freedom rigid fuselage. Each blade undergoes ap (out-of-plane) bending, lag (in-plane) bending, elastic twist and axial displacement. The governing equations are derived using a generalized Hamiltons principle applicable to non-conservative systems: =2 1

(U T W )d = 0

(1)

where U, W and T are the virtual variations of strain energy, kinetic energy and virtual work done, respectively, and represents the total potential of the system. The U and T include energy contributions from components that are attached to the blades, e.g., pitch link and lag damper. The aerodynamic forces acting on the blades contribute to the virtual work variational, W . The aerodynamic forces and moments are calculated using the unsteady aerodynamic model

Table 1: Stochastic material properties of graphite/epoxy Material properties E1 , MPa E2 , MPa G12 , MPa 12 Mean (N/m2 ) 141.96e3 9.79e3 6.00e3 0.42 CoV, % 3.39 4.27 4.27 3.65 Distribution Normal Normal Normal Normal

Table 2: Statistics of cross-sectional stiffness properties C/S Stiffness EIy EIz GJ Mean, N.m2 47,811.09 761,304.16 22,800.96 CoV, % 3.053 3.052 2.678 Distribution Normal Normal Normal

Using the Monte Carlo Simulation, stochastic behaviors of composite material properties are transmitted to the crosssectional stiffness properties of the composite blade. Mixed beam theory (Ref. 28) is used to obtain the cross-sectional stiffness properties of the composite blade with geometric properties of the baseline blade and the stochastic composite material properties given in Table 1. For this stochastic analysis 6000 samples will lead to the convergence of the standard deviation (Ref. 9). Figure 1 shows the histograms of the crosssectional stiffness properties of the composite blades. It can be noted that the normal distribution of composite material properties gets transmitted to cross-sectional stiffness properties as normal distribution. Table 2 shows the mean values and CoV values for all the cross sectional stiffness properties. The mean values, CoV and type distribution of cross-sectional stiffness properties are used for aeroelastic analysis to understand the effect of composite material uncertainties on the vibratory hub load behavior. B. Sensitivity Analysis of Stochastic behavior of Vibratory Loads Uncertainties of the cross-sectional stiffness properties of composite rotor blade are propagated to the nonlinear aeroelastic response of the helicopter rotor blade. First, sensitivities of the hub vibratory loads to the uncertainties in each crosssectional property of composite rotor blade are studied by assuming the stochastic value of one cross-section stiffness and the deterministic values of other two cross-sectional stiffness properties. Baseline vibratory loads of the rotor obtained using the mean values of cross-sectional stiffness are shown in Figure 2 and tip responses for this analysis are shown in Figure 3. It should be noted that these values are equivalent to deterministic values in absence of the uncertainties in composite material property. The baseline values of vibratory loads for deterministic cross-section stiffness values are used to evaluate the inuence of uncertainties on the hub vibratory loads and tip responses of the blades. 1) Flap cross-sectional stiffness: Sensitivities of hub vibratory loads for uncertainty in the ap cross-sectional stiffness properties are studied by assuming the stochastic behavior of the ap cross-sectional stiffness and deterministic values of other two cross-sectional stiffness properties. A set of rotor system with composite blades having stochastic values of apwise cross-section stiffness properties and deterministic values for other two cross-sectional stiffness properties is

Fig. 1.

Section stiffness distribution

Fig. 2.

Baseline Vibratory Hub Loads

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considered to obtain the sensitivity. The mean values of lag and torsion cross-section stiffness properties are considered as their deterministic values. As the stochastic values of ap crosssectional stiffness are considered, cross-section properties of all the blades in rotor system for each analysis will not be same which brings the dissimilarity to the rotor system. Therefore, dissimilar rotor analysis of 6000 rotor system sets is performed to obtain the stochastic behavior of the vibratory loads transferred through rotor hubs. As 4/rev harmonics are not much affected by the stiffness uncertainties, variations in 1/rev, 2/rev, 3/rev and 5/rev harmonics of loads are considered for analysis. Figure 4 shows histograms whereas Figures 5 and 6 show probability plots of the hub loads normalized with 4/rev loads. From the histograms and probability plots, it can be observed that even though the cross-section properties show the normal distribution, the vibratory loads show skewness in the histograms and nonlinear behavior in the probability distribution. As the baseline rotor is considered a tracked and symmetric, the values of 1/rev, 2/rev, 3/rev and 5/rev harmonics of hub loads are zero. However, uncertainties in the cross-sectional properties of the rotor system bring dissimilarity in the rotor

system which brings extra harmonics of loads. These loads are summarized as maximum values at 95 % probability of these loads. Table 3 shows 95% probability of maximum value of vibratory loads due to uncertainty in the ap stiffness values. These values are normalized with respective 4/rev loads. 2/rev Fx and Fy shear forces are the most affected shear forces whose maximum values at 95/In case of moments, Mz moments are the most affected by the uncertainties as compared to other moments. However, it should be noted the baseline value of 4/rev Mz is quite low as compared 4/rev Mx and My . Subsequently, 1/rev Mx and My moments are the most affected whose values at 95/ Figure 7 shows the tip responses of all the blades considered for stochastic analysis due uncertainty in ap cross-sectional stiffness values. These gures show that the ap response is more sensitive to the uncertainty in ap cross-sectional stiffness value. The peak-to-peak ap, lag and torsion response variation from their baseline values are about 6%, 2.25 % and 1 %. 2) Lag Cross-Section Stiffness: Sensitivity to uncertainty in lag cross-sectional stiffness (EIz ) is carried out by aeroelastic analysis of rotor with stochastic values of EIz and deterministic values of other two stiffness properties. Figure 8 shows the histograms whereas Figures 9 and 10 show probability distribution plots of hub vibratory loads due to uncertainty in the EIz . From these gures it can be observed that the vibratory loads are more sensitive to uncertainty in EIz as compared to uncertainty in EIy . Variations in the vibratory loads for stochastic values of EIz are summarized as maximum values at 95 % percent probability in Table 4. 2/rev Fx and Fy are the most affected shear forces whose maximum values at 95/ Figure 11 shows the blade tip responses of rotor systems used for stochastic analysis of hub vibratory loads for uncertainty in EIz . Blade tip responses of various blades used in the stochastic analysis have more inuence of the uncertainty in EIz . Lag response have more inuence of uncertainty in EIz whose peak-to-peak values varies by about 16.5% from its baseline values whereas ap and torsion shows about 5.4 % and 2.32 %. 3) Torsion Cross-section Stiffness: Sensitivities of hub vibratory loads to uncertainty in torsion cross-sectional stiffness (GJ) are studied using aeroelastic analysis of the rotor blades with the stochastic behavior of GJ and deterministic values of other two stiffness properties EIy and EIz . Figure 12 shows histograms whereas Figures 13 and 14 show probability distribution plots of hub vibratory loads. These results are summarized in Table 5 as the maximum values of hub vibratory loads at 95 % probability. It can be observed that 1/rev Fz and 2/rev Fx and Fy are the most affected shear forces whose maximum values at 95% probability are about 0.93, 0.92 and 0.89 times respective 4/rev shear forces, respectively. Subsequently affected shear forces are 3/rev Fx and Fy whose maximum values at 95% probability are about 0.20 times of respective 4/rev shear forces. All other harmonics of shear forces show their maximum values at 95% probability less 0.11 times that of 4/rev loads. Similar to other sensitivity analysis, all harmonics Mz are the most affected moments. Subsequently, 2/rev Mx and My are the most affected

Torsion ()

Flap (w/r)

Fig. 4.

Histograms of hub vibratory load for stochastic values of ap cross-sectional stiffness

Normal Probability Plot




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

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

0.999 0.997 0.99 0.98 0.95 0.90 Probability 0 x 104

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0.999 0.997 0.99 0.98 0.95 0.90 Probability 04

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

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

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Probability distribution plots of shear forces for stochastic values of ap cross-sectional stiffness






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0.999 0.997 0.99 0.98 0.95 0.90 Probability 0 2 4 6 8 10 Mz 1/rev Harmonic Normal Probability Plot 12 14 x 105

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Probability distribution plots of moments for stochastic values of ap cross-sectional stiffness

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C. Stochastic Behaviors of Hub Vibratory Loads Stochastic behaviors of hub vibratory loads are obtained using the stochastic behavior of all the cross-sectional stiffness properties together. Aeroelastic analyses of 6000 rotors with composite blades having stochastic behaviors of EIy , EIz and GJ stiffness properties are performed to obtain the stochastic behaviors of hub vibratory loads. Figure 16 shows histograms whereas Figures 17 and 18 show the probability distribution plots of hub vibratory loads. From these gure it can be observed that the maximum variations in the loads due to uncertainty in each cross-sectional stiffness property are getting reected in the stochastic behaviors of the hub vibratory loads for when uncertainty in all the cross-sectional stiffness properties considered together. Stochastic behaviors of the hub vibratory loads are summarized in Table 6 as maximum values of hub vibratory loads at 95% probability. From the summary of shear forces it can be observed that 1/rev Fx , 2/rev Fx and Fy are the most affected shear forces whose maximum values at 95 % probability are about 1.05, 2.07 and 1.96 times that of respective 4/rev loads, respectively. Subsequently, affected shear forces 2/rev Fz , 3/rev Fx and Fy whose maximum values at 95 % probability are about 0.25, 0.42 and 0.36 times respective 4/rev forces, respectively. Other considerably affected shear forces 1/rev Fx and Fy and 3/rev Fz whose maximum values at 95% probability are about 0.17, 0.15 and 0.12 times respective 4/rev shear forces. In case of hub vibratory moments, similar to sensitivity analysis, maximum variations are observed in all the harmonics in Mz . Even though, the baseline values of 4/rev harmonics Mz is quite low which are about 22 times smaller than 4/rev Mx and My , 1/rev Mz shows quite high variation which is about 1.4 times that of 4/rev Mx and My . Subsequently, inuenced moments are 1/rev Mx and My whose maximum values at 95% probability are about 0.52 and 0.41 times that of 4/rev moments, respectively. Values of 2/rev Mx and My show considerable inuence with their maximum values at 95% probability are about 0.27 and 0.24 times of 4/rev values. All other harmonics shows their values less than 0.1 times that of 4/rev moments. Figure 19 shows variations in tip responses used for the stochastic analysis of vibratory loads due stochastic behavior of ap, lag and torsion cross-sectional stiffness properties. Uncertainty in all the cross-sectional stiffness shows significant inuence on the lag response and subsequently on ap and torsion responses. The peak-to-peak response variations of ap, lag and torsion responses are about 15.2%, 20.27% and 10.2% with respect to their baseline values, respectively. IV. C ONCLUSION In this study the stochastic behaviors of hub vibratory loads due to uncertainty in composite materials are studied. The stochastic behaviors of composite material properties obtained from the experimental data in the literature are used to calculate the stochastic behavior of the cross-section stiffness properties. Following Conclusions are drawn from this study.

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Table 3: Maximum values of hub vibratory loads at 95% percent probability for stochastic behavior of ap stiffness values 1 2 3 5 Fx 0.092 0.285 0.030 0.015 Fy 0.085 0.287 0.023 0.013 Fz 0.074 0.078 0.105 0.43e-4 Mx 0.489 0.063 0.043 1.91e-3 My 0.398 0.020 0.046 1.81e-3 Mz 4.343 1.024 0.099 0.030

moments whose maximum values at 95% probability are about 0.24 and 0.23 times that of 4/rev moments, respectively. 1/rev Mx and My moments are 0.099 and 0.088 times of that respective 4/rev harmonics. Variations of all other moment harmonics are less than 0.024 times of that respective 4/rev harmonics. Figure 15 show tip responses of the rotors used for vibratory load sensitivity assessment for uncertainty in GJ. Flap deection shows maximum variation as compared to other responses due to uncertainty in GJ. Peak-to-peak variation in ap response from its baseline value is about 13.71% whereas peak-to-peak variations for torsion and lag response are about 11.5% and about 3.7% from their baseline values, respectively.

Fig. 8.

Histograms of hub loads for stochastic values of lag cross-sectional stiffness






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Probability distribution plots of shear forces for stochastic values of lag cross-sectional stiffness






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

0.999 0.997 0.99 0.98 0.95 0.90 Probability 0 x 105

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

0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001

0

0.5

1

1.5 2 My 1/rev Harmonic Normal Probability Plot

2.5

3 x 105

0

1

2

3 4 5 My 2/rev Harmonic Normal Probability Plot

6

7

1

2

3 4 My 3/rev Harmonic Normal Probability Plot

5

6 x 105

0

2

4

6

8 10 12 My 5/rev Harmonic

14

16

18 x 107



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

0.999 0.997 0.99 0.98 0.95 0.90 Probability 04

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

0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001

0

1

2

3

4 5 6 Mz 1/rev Harmonic

7

8

9 x 104

0

0.5

1

1.5

2 2.5 3 Mz 2/rev Harmonic

3.5

4

4.5 x 10

0.5

1

1.5 2 2.5 Mz 3/rev Harmonic

3

3.5

4 x 105

0

2


8

10 x 107

Fig. 10.

Probability distribution plots of moments for stochastic values of lag cross-sectional stiffness

0.085

0.075

4.

0.065

0.055

0.045

0

90

180

270

360

5.

0.013

0.004

0.005

0.014

6

0.023

0.032

0

90

180

270

360

0.03

0.02

0.01

7

0

0.01

0.02

0.03

0.04

0

90

180

270

360

Fig. 11. Tip responses of rotors for stochastic values of lag cross-sectional stiffness properties

as compared to other moments and 1/rev Mx and My moments are the subsequently affected moments. Uncertainty in lag cross-sectional stiffness show, very high inuence on Second harmonics of Fx and Fy as was observed as compared to that of other stiffness properties and 1/rev Fz and 2/rev Fx and Fy are the subsequently affected moments. All the harmonics Mz have very high inuence of uncertainty in the EIz as compared to uncertainty in the EIy . Uncertainty in torsion wise cross-sectional stiffness properties show high inuence on 1/rev Fz and 2/rev Fx and Fy and subsequently, on 3/rev Fx and Fy shear forces. In case of moments, similar to other sensitivity analysis, most affected moment are all harmonics Mz and subsequently, second harmonics Mx and My whose values are about 0.24 and 0.23 times that of 4/rev moments, respectively. when the all the cross-sectional stiffness properties are considered with stochastic behavior, 1/rev Fx , 2/rev Fx and Fy show quite high inuence whose distribution is about 1-2 times that of respective 4/rev loads whereas 2/rev Fz , 3/rev Fx and Fy show variations upto 0.25-0.40 times that of respective 4/rev loads. Similar to sensitivity analysis, Mz moments is highly inuenced by the uncertainty in all the cross-sectional stiffness properties. Comparison of the sensitivity analysis and the stochastic analysis of hub vibratory loads with uncertainty in all the blade cross-sectional stiffness properties show that lag stiffness uncertainty is responsible for 2/rev, 3/rev Fx and Fy and torsion stiffness uncertainty is responsible for 1/rev Fx , Fy and Fz . Uncertainties in ap stiffness properties affects less as compared to other stiffness uncertainties. Large variations in Mz are combined effect of uncertainty in lag and torsion stiffness properties. This comparison helps in minimizing the hub load variations by minimizing the uncertainties in various cross-sectional stiffness properties. R EFERENCES 1. Smith, E. C. and Chopra, I., Aeroelastic Response and Blade Loads of a Composite Rotor in Forward Flight, AIAA Journal, Vol. 31 (7), 1993, pp. 12651273. Volovoi, V., Hodges, D. H., Cesnik, C., and Popescu, B., Assessment of Beam Modeling for Rotor Blade Application, Mathematical and Computer Modelling, Vol. 33, Nos. 10-11, 2001, pp. 10991112. Friedmann, P. P., Rotary-Wing Aeroelasticity: Current Status and Future Trends, AIAA Journal, Vol. 42, No. 10, 2004, pp. 1953-1972. Ganguli, R., A Survey of Recent Developments in Rotorcraft Design Optimization, Journal of Aircraft, Vol. 41, No. 3, 2004, pp. 493-510. Petit, C. L., Uncertainty Quantication in Aeroelasticity: Recent Results and Research Challenges, Journal of Aircraft, Vol.43, No. 5, 2004, pp. 12171229.

Table 4: Maximum values at 95% percent probability for stochastic behavior of lag stiffness values 1 2 3 5 Fx 0.100 1.917 0.367 0.002 Fy 0.016 1.792 0.347 0.003 Fz 0.468 0.232 0.032 0.86e-3 Mx 0.144 0.137 0.085 1.71e-3 My 0.039 0.093 0.080 1.54e-3 Mz 27.322 13.815 1.328 0.027

1. It was noticed that the composite material properties have stochastic behavior in the form of normal distribution lead to same form of stochastic behavior with different COV for cross-sectional stiffness properties. 2. Similar to previous dissimilar rotor analysis studies, major inuence of uncertainty which brings dissimilarity was noticed on the non Nb /rev harmonics. Eventhough the cross-sectional stiffness properties show normal distribution, the histograms and probability distribution of hub vibratory loads shows nonlinearity in the distribution. 3. For the uncertainty in the ap cross-section stiffness the second harmonics of Fx and Fy shear forces are the most affected shear forces and subsequently, all the 1/rev, 2/rev and 3/rev Fz shear forces are affected. In case of moments, Mz moments are the most affected by the uncertainties

2.

3.

4.

5.

Fig. 12.

Histograms of hub loads for stochastic values of torsion cross-sectional stiffness






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

0.999 0.997 0.99 0.98 0.95 0.90 Probability 0 1 Fx 3/rev Harmonic Normal Probability Plot 2 x 10 34

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

0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001

0

2

4


10

12

14 x 105

0

5 Fx 2/rev Harmonic

10 x 10

154

0

1

2


5

6 x 106



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

0.999 0.997 0.99 0.98 0.95 0.90 Probability 0 0.5 1 1.5 Fy 3/rev Harmonic 2 2.5 x 104

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

0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001

0

2

4

6

8 10 12 Fy 1/rev Harmonic Normal Probability Plot

14

16

18 x 105

0

5 Fy 2/rev Harmonic

10

15 x 104

0

1

2

3 4 Fy 5/rev Harmonic

5

6 x 106





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

0.999 0.997 0.99 0.98 0.95 0.90 Probability 0 1 2 3 4 Fz 3/rev Harmonic 5 6 x 10 75

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

0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001

0

2

4

6

8 10 12 Fz 1/rev Harmonic

14

16

18 x 104

0

0.5

1 1.5 Fz 2/rev Harmonic

2

2.5 x 104

0

1


4

5 x 106

Fig. 13.

Probability distribution plots of shear forces for stochastic values of torsion cross-sectional stiffness






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

0.999 0.997 0.99 0.98 0.95 0.90 Probability 0 2 4 6 8 10 Mx 3/rev Harmonic 12 14 16 x 106

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

0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001

0

1

2


5

6 x 105

0

2

4

6

8 10 12 Mx 2/rev Harmonic

14

16

18 x 105

0

2

4


10

12 x 107




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

0.999 0.997 0.99 0.98 0.95 0.90 Probability 0 2 4 6 8 10 My 3/rev Harmonic Normal Probability Plot 12 14 16 x 106

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

0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001

0

1

2


5

6 x 105

0

2

4

6

8 10 12 My 2/rev Harmonic

14

16

18 x 105

0

2

4


10

12 x 107



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

0.999 0.997 0.99 0.98 0.95 0.90 Probability 0 x 104

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

0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001

0

1

2 Mz 1/rev Harmonic

3

4 x 104

0

1 Mz 2/rev Harmonic

2

1

2

3 Mz 3/rev Harmonic

4

5 x 10

66

0

1

2

3


7

8

9 x 107

Fig. 14.

Probability distribution plots of moments for stochastic values of torsion cross-sectional stiffness

Table 5: Maximum values at 95% percent probability behavior of torsion stiffness values Fx Fy Fz Mx My 1 0.100 0.119 0.931 0.100 0.089 2 0.918 0.891 0.111 0.242 0.228 3 0.204 0.189 0.033 0.024 0.022 5 4.13e-3 3.77e-3 2.79e-3 2.03e-3 1.88e-3

for stochastic Mz 13.552 6.594 0.196 0.029

9.

10.0.085

0.075

11.

0.065

0.055

12.0 90 180 270 360

0.045


13.0.004

Lag (v/r)

0.005

0.014

14.

0.023

15.0.032 0 90 180 270 360


0.02

16.

0.01

0

0.01

17.

0.02

0.03

0.04

0

90

180

270

360


18.

Fig. 15. Tip responses of rotors for stochastic values of torsion cross-sectional stiffness

19. 6. Kim, T. K., and Hwang, I. H., Reliability Analysis of Composite Wing Subjected to Gust Loads, Composite Structures, Vol. 66, Nos. 1-4, 2004, pp. 527-531. Pradlwarter, H. J., Pellissetti, M. F., Schenk, C. A., Schuller, G. I., Kreis, A., Fransen, S., Calvi, A., and Klein, M., Realistic and Efcient Reliability Estimation for Aerospace Structures, Computer Methods in Applied Mechanics and Engineering, Vol. 194, Nos. 12-16, 2005, pp. 1597-1617. Koutsourelakis, P. S., Kuntiyawichai, K., and Schuller, G. I., Effect of Material Uncertainties on Fatigue Life Calculations of Aircraft Fuselages: A Cohesive Element Model, Engineering Fracture Mechanics, Vol. 73, No. 9, 2006, pp. 1202-1219.

20.

7.

21.

8.

22.

Murgan, S., Ganguli, R. and Harursampath D., Aeroelastic Response of Composite Helicopter Rotor with Random Material Properties, Journal of Aircraft, Vol. 45, No. 1, January-February, 2008, pp. 306-322. Salim, S., Yadav, D., and Iyengar, N. G. R., Analysis of Composite Plates with Random Material Characteristics, Mechanics Research Communications, Vol. 20, No. 5, 1993, pp. 405-414. Yadav, D., and Verma, N., Buckling of Composite Circular Cylindrical Shells with Random Material Properties, Composite Structures, Vol. 37, Nos. 3-4, 1997, pp. 385-391. Onkar, A. K., and Yadav, D., Non-linear Response Statistics of Composite Laminates with Random Material Properties under Random Loading, Composite Structures, Vol. 60, No. 4, 2003, pp. 375383. Onkar, A. K., and Yadav, D., Forced Nonlinear Vibration ofLaminated Composite Plates with Random Material Properties, Composite Structures, Vol. 70, No. 3, 2005, pp. 334-342. Rosen A and Ben-Ari R 1997 Mathematical modelling of a helicopter rotor track and balance: theory Journal of Sound and Vibration 200 589-603. Wang, J. M. and Chopra, I., Dynamics of Helicopters with Dissimilar Blades in Forward Flight, 17th European Rotorcraft Forum (Berlin, Germany), 1991 Ganguli, R., Chopra, I. and Haas, D. J., Simulation of Helicopter Rotor-System Damage, Blade Mistracking, Friction, and Freeplay, Journal of Aircraft, Vol. 35, 1998, pp. 591-597. Yang, M., Chopra, I. and Haas, D. J., Vibration Prediction for Rotor System with Faults using Coupled Rotor-Fuselage Model, Journal of Aircraft, 2004 Vol. 41, pp. 348-358. Hemant K. Singh, Prashant M. Pawar, Ranjan Ganguli, Sung Nam Jung On the effect of mass and stiffness unbalance on helicopter tail rotor system behavior, Aircraft Engineering and Aerospace Technology 2008, 80, 2, 129 - 138 McKillip, R., Digital SMA-based Tracking Tabs for One-per-rev Vibration Reduction, AHS International 59th Annual Forum Proceedings (Phoenix, AZ, USA) pp.1692-1719, May 2003. Hall, S.R.; Spangler, Jr.; Ronald L., Piezoelectric Helicopter Blade Flap Actuator, Massachusetts Institute of Technology (Cambridge, MA), U.S. patent 5,224,826 , July 1989 Roget, B. and Chopra, I., Individual Blade Control Methodology for a Rotor with Dissimilar Blades, Journal of the American Helicopter Society, Vol. 48, (3), Jul 2003, pp. 176-185. Prashant M Pawar and Sung Nam Jung1 Active twist control methodology for vibration reduction of a helicopter with dissimilar rotor system, Smart Mater. Struct. 18 (2009) 035013 (11pp)

Flap (w/r)

Torsion ()

Fig. 16.

Histograms of hub loads for stochastic values of all the cross-sectional stiffness properties






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

0.999 0.997 0.99 0.98 0.95 0.90 Probability 0 1 2 3 4 Fx 3/rev Harmonic Normal Probability Plot 5 6 x 104

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

0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001

0

0.5


2

2.5 x 104

0

0.5

1

1.5 2 Fx 2/rev Harmonic Normal Probability Plot

2.5

3

3.5 x 103

0

0.5


2 x 105


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

0.999 0.997 0.99 0.98 0.95 0.90 Probability 0 1 2 3 4 Fy 3/rev Harmonic Normal Probability Plot 5 6 x 104

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

0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001

0

1 Fy 1/rev Harmonic Normal Probability Plot

2 x 104

0

0.5

1

1.5 2 Fy 2/rev Harmonic Normal Probability Plot

2.5

3

3.5 x 103

0

0.5

1 1.5 Fy 5/rev Harmonic Normal Probability Plot

2

2.5 x 105


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

0.999 0.997 0.99 0.98 0.95 0.90 Probability 04

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

0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001

0

0.5

1 1.5 Fz 1/rev Harmonic

2 x 103

0

1


4

5 x 10

1 Fz 3/rev Harmonic

2

3 x 104

0

1

2


5

6

7 x 106

Fig. 17.

Probability distribution plots of shear forces for stochastic values of all the cross-sectional stiffness






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

0.999 0.997 0.99 0.98 0.95 0.90 Probability 0 1 2 3 4 Mx 3/rev Harmonic Normal Probability Plot 5 6 7 x 105

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

0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001

0

0.5

1

1.5 2 Mx 1/rev Harmonic Normal Probability Plot

2.5

3

3.5 x 104

0

0.5

1 1.5 Mx 2/rev Harmonic Normal Probability Plot

2

2.5 x 104

0

0.5

1 1.5 Mx 5/rev Harmonic Normal Probability Plot

2

2.5 x 106


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

0.999 0.997 0.99 0.98 0.95 0.90 Probability 0 x 104

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

0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001

0

0.5

1

1.5 My 1/rev Harmonic

2

2.5 x 104

0

1 My 2/rev Harmonic Normal Probability Plot

2

1

2


5

6

7 x 105

0

0.5

1 1.5 My 5/rev Harmonic Normal Probability Plot

2

2.5 x 106



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

0.999 0.997 0.99 0.98 0.95 0.90 Probability 04

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

0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001

0

2

4

6 Mz 1/rev Harmonic

8

10 x 104

0

1

2


5

6 x 10

0.5

1

1.5 2 2.5 Mz 3/rev Harmonic

3

3.5

4 x 105

0

2

4

6


14

16

18 x 107

Fig. 18.

Probability distribution plots of moments for stochastic values of all the cross-sectional stiffness properties

Table 6: Maximum values at 95% percent probability for stochastic behavior of all the stiffness values Fx Fy Fz Mx My Mz 1 0.172 0.151 1.055 0.523 0.410 31.005 2 2.070 1.963 0.248 0.271 0.240 15.858 3 0.417 0.359 0.118 0.097 0.097 1.343 5 0.016 0.014 3.10e-3 3.40e-3 3.15e-3 0.050

Blade Tip Response (Flap) 0.085 0.08 0.075 0.07 w/r 0.065 0.06 0.055 0.05 0.045

tural Behaviour of Composite Materials Structural Components, Composite Structures, Vol. 32, Nos. 1-4, 1995, pp. 247-253. 27. Singh, B. N., Yadav, D., and Iyengar, N. G. R., Free Vibration of Composite Cylindrical Panels with Random Material Properties, Composite Structures, Vol. 58, No. 4, 2002, pp. 435-442. 28. Jung, S. N., Nagaraj, V. T. and Chopra, I. Rened Structural Dynamics Model for Composite Rotor Blades, AIAA Journal, Vol. 39 (2), 2001, pp. 339348.

0

50

100

150 200 250 Azimuth angle (degree) Blade Tip Response (Lag)

300

350

400

0.015 0.01 0.005 0 0.005 v/r 0.01 0.015 0.02 0.025 0.03 0.035 0 50 100 150 200 250 Azimuth angle (degree) Blade Tip Response (Torsion) 0.03 300 350 400

0.02

0.01

phi (rad)

0

0.01

0.02

0.03

0.04

0

50

100

150 200 250 Azimuth angle (degree)

300

350

400

Fig. 19. properties

Tip responses of rotors for stochastic values all the stiffness

23.

Leishman, J. G. and Beddoes, T. S. A semiempirical model for dynamic stall, Journal of American Helicopter Society, Vol. 34, 1989, pp. 3-17. 24. Bagai, A. and Leishman, J. G., Rotor freewake modeling using a pseudo-implicit technique including comparisons with experiment, Journal of American Helicopter Society, Vol. 40, 1995, pp. 2941. 25. Smith, E. C., Vibration and Flutter of Stiffinplane Elastically Tailored Composite Rotor Blades, Mathematical and Computer Modelling, Vol. 19 (3-4), 1994, pp. 27-45. 26. Vinckenroy, G. V. and Wilde, W. P. De., The Use of Monte Carlo Techniques in Statistical Finite Element Methods for the Determination of the Struc-

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Documents

composite rotor system

dissimilar rotor analysis

composite material properties

materials uncertainty

rotor system behavior

main rotor system

aeroelastic analysis

uncertainty analysis