research article optimal design and acoustic assessment of...

Research ArticleOptimal Design and Acoustic Assessment ofLow-Vibration Rotor Blades

G Bernardini1 E Piccione1 A Anobile2 J Serafini1 and M Gennaretti1

1Department of Engineering Roma Tre University 00146 Rome Italy2Department of Mechanical Materials and Manufacturing Engineering The University of Nottingham Nottingham NG7 2RD UK

Correspondence should be addressed to J Serafini serafiniuniroma3it

Received 28 December 2015 Accepted 2 March 2016

Academic Editor Jiun-Jih Miau

Copyright copy 2016 G Bernardini et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An optimal procedure for the design of rotor blade that generates low vibratory hub loads in nonaxial flow conditions is presentedand applied to a helicopter rotor in forward flight a conditionwhere vibrations and noise become severe Blade shape and structuralproperties are the design parameters to be identified within a binary genetic optimization algorithm under aeroelastic stabilityconstraintTheprocess exploits an aeroelastic solver that is based on a nonlinear beam-likemodel suited for the analysis of arbitrarycurved-elastic-axis blades with the introduction of a surrogate wake inflow model for the analysis of sectional aerodynamic loadsNumerical results are presented to demonstrate the capability of the proposed approach to identify low vibratory hub loads rotorblades as well as to assess the robustness of solution at off-design operating conditions Further the aeroacoustic assessment of therotor configurations determined is carried out in order to examine the impact of low-vibration blade design on the emitted noisefield

1 Introduction

Main rotors play a fundamental role in helicopter dynamicsproviding both lifting force and thrust but as a by-productthey are sources of vibrations and noise primarily dueto the nonaxial flow condition of most of their typicalmission profiles The reduction of these annoying effectsis of primary interest for rotors designers since vibrationsstrongly affect fatigue life of structures maintenance costsonboard instrumentation efficiency and passengers and pilotcomfort as well as acoustic disturbance inside the cabinsimilar problems concern wind turbines due to wind sheareffect At the same time the external noise emission causescommunity disturbance thus limiting the public acceptanceof helicopters for operations nearby populated areas

In the last years several innovative blade shapes havebeen investigated by manufacturers and researchers in orderto alleviate vibrations and noise caused by main rotor Inparticular advanced geometries characterized by curvedelastic axis (with tip sweep and anhedral angles included)have shown to be effective to this purpose (see eg [1 2])

Such blades reduce normal tip Mach number and conse-quently drag due to compressibility alleviate the interac-tions occurring between the blades and the vortices of therotor wake (and hence the blade-vortex interaction- (BVI-)noise) and may significantly affect the rotor aeroelasticbehavior introducing strong bending-torsion coupling

The objective of this paper is the presentation and appli-cation of an optimal design procedure aimed at identifyingrotor blades that generate reduced vibratory hub loads It is avery challenging goal in that it deals with an inherently mul-tidisciplinary multidimensional constrained minimizationproblem characterized by nonlinear multimodal objectivefunctions (ie functions with several local minima in thedesign domain) Because of this genetic algorithms (GAs)seem to be an appropriate approach to achieving the optimalsolution Indeed they are able to escape local minima andsearch for the global optimumeven in very complex problemsallowing at the same time the implementation of veryefficient computational algorithms due to their intrinsicallyparallel nature

Hindawi Publishing CorporationInternational Journal of Rotating MachineryVolume 2016 Article ID 1302564 17 pageshttpdxdoiorg10115520161302564

2 International Journal of Rotating Machinery

In the past literature on similar topics several approachesbased on deterministic optimization methods can be found[3ndash7] whereas only a limited number of more recent worksaddress the use of nondeterministic methods such as GAsfor the same aim of [8ndash10] For the sake of computational effi-ciency in both optimizationmethodologies often a surrogatemodel for the direct evaluation of the objective function isadopted [4 8ndash10] Extensive reviews of works dealing withrotorcraft optimization are given in [11ndash13]

In this work a binary-based GA developed by the authorsis adopted where blade tip sweep angle blade tip anhedralangle and the distribution of the mechanicalstructuralproperties are considered as design variables to be identifiedunder aeroelastic stability constraint Recently this binary-based GA has been successfully applied for the optimaldesign of aircraft cabin noise control and marine propellerblades [14 15] The interested readers may find details onGA optimization procedures in [16 17] while a thoroughsurvey of multidisciplinary design optimization techniques ispresented in [18]

In the optimal design process a fundamental role isplayed by the simulation of the aeroelastic behavior of therotor since vibratory loads are a by-product of the interactionbetween the blades and the complex aerodynamic field inwhich they operate In the past years several authors havedeveloped structural models for curvedswept tip bladesusually solved through the finite element method approachOne of the earliest models suited for swept tip blades hasbeen presented and successfully applied to hingeless rotorsin [19 20] Among the others a formulation for blades withvarying sweep droop twist angles and platform has beenintroduced in [21] while more recently a curvilinear-axisblade formulation has been applied to horizontal-axis windturbines [22] In the optimization procedure proposed herethe rotor aeroelastic tool consists of an enhanced versionof that recently developed by the authors presented andvalidated in [23 24] It is based on a nonlinear beam-like model spatially integrated through a Galerkin approachsuited for the analysis of blades having arbitrarily curvedelastic axis (including geometrical discontinuities as sweepand anhedral tip angles) A good trade-off between accuracyand computational efficiency is achieved by the identificationand application of a surrogate model of the wake inflowused for sectional aerodynamic loads prediction relying ona boundary element approach for the analysis of unsteadypotential flows [25]

The numerical investigation will assess the capabilityof the proposed optimal design approach to identify rotorblades generating reduced vibratory hub loads as well as therobustness of the solution at off-design operating conditionsAn analysis of the influence of wake inflow model used inthe aeroelastic tool on the optimization process will be alsopresented Furthermore observing that vibratory hub loadsand noise emission are usually strictly related (alleviation ofvibrations often corresponds to noise increase and vice versa)the acoustic performances of the identified optimal rotor willbe examined To this purpose a boundary element approachbased on Farassatrsquos Formulation 1A [26] for the solution ofFfowcs Williams and Hawkingsrsquo equation [27] is applied

2 Materials and Methods

In the following an outline of the aeroacoustoelastic andoptimization tools developed by the authors is presented

21 Rotor Aeroelastic Solver For the purposes of this workthe availability of an efficient and accurate computational toolfor the aeroelastic analysis of rotor blades having arbitraryshape is of paramount importance The one applied here hasbeen developed by the authors as a combination of a suitedstructural dynamics model with the aerodynamic loads givenby a quasi-steady sectional formulation corrected with inflowcontribution in order to take into account the effect of wakevortices

The rotor blade structural dynamics is described througha beam-like model obtained as an enhanced version of theformulation presented by some of the authors in [23] It isvalid for slender homogeneous and isotropic rotating bladeswith curved elastic axis and includes spanwise variationof mass and stiffness properties as well as variable built-in pretwist precone sweep and anhedral angles Nonlinearstrain-displacement relations are considered with the appli-cation of a second-order approximation scheme in orderto take into account the moderate displacements usuallyexperienced by rotor blades A detailed description of thisblade structural model is presented in Appendix

The distributed aerodynamic loads are modeled by thequasi-steady approximation of the sectional Greenberg the-ory [28] (see Appendix) Three-dimensional unsteady effectsderiving from the wake vorticity are taken into accountthrough the influence of the wake inflow on the relativevelocity at the rear aerodynamic center of the blade crosssectionsThe evaluation of the inflow is obtained by a bound-ary element method (BEM) for the solution of the boundaryintegral equation approach presented in [25] suited for theanalysis of potential flows around rotors in arbitrary motionSimple analytical wake inflow models might also be applied

The aeroelastic integrodifferential model derived by cou-pling these structural dynamics and aerodynamic formula-tions has been spatially integrated through application of theGalerkin approach In particular as outlined in Appendixa spectral description of the curvature components in theundeformed-axis frame is considered in order to develop asolver with good convergence properties even in presenceof sweep and anhedral tip angles (ie for elastic axis shapeswith discontinuous first-order derivatives) The projectionsare applied to the bending and torsion moment equilibriumequations with coinciding sets of trial and test functions(note that in addition to shear undeformable assumption theassumption of inextensible elastic axis has been adopted akinto the approach in [29]) Because of the test functions chosenthe present approach yields equations that are strongly relatedto those that could be derived from application of theRayleigh-Ritz approach (see Appendix)

The aeroelastic response to steady flight conditions isevaluated by integrating the set of ordinary time-differentialequations through a harmonic balance approach [30 31]Aeroelastic stability about the equilibrium solution is instead

International Journal of Rotating Machinery 3

examined by eigenanalysis of the (numerically) linearizedsystem [31]

22 Rotor Aeroacoustics Solver The aeroacoustic solver usedto evaluate the noise radiated by rotor blades is based onthe boundary integral Farassatrsquos Formulation 1A [26] for thesolution of the Ffowcs Williams and Hawkings equation [27]that represents a rearrangement of the mass and momentumconservation laws into an inhomogeneous wave equation Itis composed of three separate integral time-retarded contri-butions known as thickness loading and quadrupole noiseeach related to a specific mechanism of noise generationThethickness term depends on blade geometry and kinematicsof the problem the loading term is related to blade airloadswhereas the quadrupole source contribution accounts for thepossible nonlinear effects taking place in the flow field Thequadrupole field term has been neglected since the bladevelocity is far from the transonicsupersonic regimes (asit is in the rotor configurations examined here) Thicknessand loading noise terms are solved by a simple zeroth-orderformulation applied to the blade surface discretized intopanels with the integrand functions assumed uniform ineach panel and equal to the values at the panel centroid

The aeroacoustic formulation is based on the knowledgeof the aerodynamic loads distributed over the blade surfaceHere they are obtained from the same aerodynamic toolapplied for validating the vibratory performance of the bladeidentified in the optimal design process (see Section 3)

23 The Optimal Blade Design Process The blade opti-mization procedure applied in this work is driven by abinary-based genetic algorithm developed by the authors[14 15] Genetic algorithms are probabilistic programmingtechniques that mimic the natural evolution in finding theoptimal solution of a given problem [16] In this processpotential solutions are called individuals and the whole set ofindividuals is called population Each individual is identifiedby a string (chromosome) of binary digits (genes) ordered ina given sequence The optimization procedure starts from acompletely random-generated population and at each step ofthe evolution process individuals are quantitatively evaluatedin terms of the corresponding value of the objective functionThe population size in genetic algorithms is a crucial issue toconsider when dealing with specific optimization problemsas it can seriously affect their efficiency Indeed a very smallpopulation (ie composed of few individuals) may lead toan unsatisfactory coverage of the problem domain as well asto sampling errors [32] while a large population can lead tohigh computational time due to the number of the objectivefunctions to be evaluated Here following [17] an estimateof the population size based on the variance of the objectivefunctions is used

Constraints are included in the optimization processthrough a quadratic extended interior penalty-functionapproach [33] which enhances the breeding possibility ofindividuals potentially able to generate good offspring Inthis sense constraints are taken into account indirectlyturning the constrained optimization process into a sequenceof unconstrained minimization procedures To build a new

generation the best individuals are selected on the basis ofa fitness measure evaluated from the objective function andconstraints For the present analysis a tournament operatoris used based on a random selection of four parents whichare compared one versus one in two pairs and the couple ofldquowinnersrdquo are selected to be parents of two children with twoindependent crossover operations A single random-pointcrossover operator is used

Once the mate is performed a binary uniform mutationoperation is applied to avoid premature convergence tolocal optima This operator alters one or more binary digits(genes) in the chromosome by flipping it with a givenprobability The amount of chromosome variations duringthe evolutionary process is controlled through a user-definedmutation probability factor which is decreased during theoptimization to reduce the impact of random mutations asthe solution converges to an optimum In order to preventpossible negative aspects of the evolution process and hencedrive the solutions to get better over time at each step ofthe optimization process the best individuals (a given user-defined percentage of the population size) are selected tobecome part of an elite group which is unchanged in thenext generation This technique in addition to avoiding thepossibility to obtain worse generation during the processenhances its convergence properties [34 35] The optimiza-tion procedure is iterated until either the chromosomessimilarity (bit-string affinity) achieves a user-defined value[36] or the maximum number of iterations is reached

Here this optimization process is applied to reduce thevibratory hub loads generated by a helicopter rotor in forwardflight Following past works [5 37 38] the goal is pursued bytailoring the structural inertial and aerodynamic propertiesof the rotor blade In particular the following design variablesare considered bending and torsional stiffnesses mass perunit length (assumed to be uniformly distributed spanwise)and sweep and anhedral angles (defined in the 15 longblade tip region) Simple distributions of the structural designvariables are considered in that the main objective of thiswork consists in the assessment of the effectiveness of theproposed optimization methodology to helicopter rotorsdesign Anyway more complex variables distributions mightbe introduced at the cost of increasing the computationaleffort required Given that 119873-bladed rotor transmits to thehub periodic forces and moments of fundamental frequency119873rev the objective function to be minimized in the opti-mization is a linear combination of the scalar norm of the119873rev harmonics of hub forces and moments

The tailoring of inertial structural and geometricalproperties of the blades for low-vibration purposesmay affectthe aeroelastic stability of the rotor and at the same time thehelicopter trim controls setting Therefore the optimal bladedesign process includes constraints regarding the equilibriumtrim conditions and the rotor stability Specifically at eachiteration trim control settings are reevaluated together withthe vibratory loads through an aeroelastic trim procedurewhereas the stability is imposed by setting a minimumacceptable value of the resulting critical damping Finally inorder to have a dynamic behavior of the optimal blade similarto that of the reference one (in terms of eigenfrequencies


Table 1 Blade design variables

Baseline Single-point opt (Drees inflow) Single-point opt (LIN surrogate) Multipoint opt (LIN surrogate)1198641198681205781198980Ω21198774 001060 001283 000857 000898

1198641198681205771198980Ω21198774 003010 003514 002948 002570

1198661198691198980Ω21198774 000147 000104 000113 000188

1198981198980

10 1053 1188 1033Λ119860[deg] 00 424 minus44 minus41

Λ119878[deg] 00 1929 237 182

and blade deflections amplitude) upper and lower bounds onthe design variables are imposed In particular with respectto the baseline values the largest acceptable variations ofbending and torsional stiffnesses are selected to be equal to30 and the largest acceptable variation of distributed massis 20 while the tip sweep angle Λ 119878 (positive backwards)and the tip anhedral angle Λ119860 (positive downwards) areconstrained to be minus20

∘le Λ 119878 le 30

∘ and minus10∘le Λ119860 le 15

∘respectively

All the numerical results that are presented in the nextsection have been obtained by considering populations withindividuals identified through a chromosome string of 24

digits (which allows a very fine resolution of the designvariable range) The optimization process is iterated untila bit-string affinity of 85 or a maximum number of 50populations is reached

3 Results and Discussion

In this section application strategies and effectiveness ofthe proposed optimal approach for the design of rotorblades generating low vibratory hub loads are examined Inparticular four main issues are investigated (i) single-pointand multipoint optimization algorithms performance (ii)effect of aerodynamic modeling on the optimization process(iii) robustness of optimal blade design in off-design flightconditions and (iv) impact of low-vibration blade design onthe emitted noise

The optimal design processes have been applied to a Bo-105-like rotor with four hingeless blades and solidity120590 = 007operating at Lock number 120574 = 55 and thrust coefficient119862119879 =

0005 The baseline values of the blade design variables aregiven in Table 1

Akin to the baseline rotor blade the optimized bladehas been assumed to have uniform structural propertiesAll computations have been carried out using nine shapefunctions in the modal description of each structural dofwhereas five harmonics have been included in the harmonicbalance solution (these discretization parameters guaranteeaeroelastic converged results)

The optimization process includes a constraint imposingaeroelastic stability in the design advancing flight condition(in order to cover the whole flight envelope more than onecondition may be considered) However noting that hoverflight is usually critical in terms of aeroelastic stability thisconstraint has been imposed in hovering as well Controlsettings have been determined as those corresponding to therotor momentum trim

31 Single-Point Optimization First a single-point optimiza-tion algorithm has been applied assuming the design flightcondition at advance ratio 120583 = 03 The analytical Dreesformula has been considered as the first candidate for wakeinflow model in the aeroelastic tool in that yielding anextremely computationally efficient solution process fullysuitable for use in GAs

In this case the optimization led to a reduction of about84 of the objective function given by the following linearcombination of the scalar normof the 4rev harmonics of hubforces and moments

= (1198652

119909+ 1198652

119910+ 1198652

119911)12

+ (1198722

119909+ 1198722

119910+ 1198722

119911)12

(1)

This is achieved with the identified optimal blade design vari-ables given in Table 1 With respect to the baseline values theoptimal design shows an increase of blade mass and bendingstiffnesses a reduction of torsional stiffness a rearward tipsweep angle of 193 deg and a downward tip anhedral angle of42 deg Figure 1(a) depicts the 4rev vibratory hub loads frombaseline and optimal rotors evaluated through the aeroelasticmodel used in the optimization procedure demonstratingthat very good reductions ranging from 60 to 85 areachieved

Then the optimal blade design has been validated againstapplication of an aerodynamic model more accurate thanthat used in the synthesis process To this purpose a high-fidelity aeroelastic model based on the numerical free-wakeinflow evaluated through the BEM formulation mentionedin Section 21 has been considered This analysis confirmsthe aeroelastic stability of the optimal rotor configurationbut a reduction of the objective function with respect tothe baseline configuration of only 5 is obtained To betterunderstand the results deterioration the vibratory hub loadsof both baseline and optimal rotor from the high-fidelityaeroelastic model are presented in Figure 1(b) From thecomparison of Figures 1(a) and 1(b) it is apparent thatthe vibratory loads are very sensitive to the aerodynamicmodel used for their evaluation with considerably highervalues predicted by the high-fidelity aerodynamic modelFurthermore the out-of-plane component of the hub forces119865119911 is significantly increased from its baseline value whereasthe in-plane force119865119909 and the torquemoment119872119911 areweaklyaffected by the blade redesign These results show that thesensitivity of the vibratory loads to the design variables aspredicted by the two inflow models is very different (forsome load components even opposite) thus suggesting theneed of using the more accurate aerodynamic model in rotor


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(b) Validation (BEM wake inflow)

Figure 1 Vibratory 4rev hub loads optimal versus baseline configuration at 120583 = 03

aeroelastic optimizationThus the process has been repeatedreplacing the Drees model with the inflow numericallyevaluated through the free-wake BEM in the aeroelastic toolThis allows the introduction of aerodynamic phenomena likeblade-vortex interactions that may play an important rolein the generation of vibratory loads and that are completelyneglected in using simple analytical wake inflow modelsHowever noting that a direct use of the BEM solver in theoptimization process would significantly decrease its compu-tational efficiency a surrogate model of the BEMwake inflowhas been synthesized and used in the optimizer This modelis based upon a linear interpolation of a database of the BEMwake inflow previously evaluated for a limited number ofblade operating conditions falling in the domain of definitionof the optimization problem in terms of both design variablesand flight conditions In order to limit the number of rotordisk wake inflow computations to be performed to definethe surrogate model (rapidly increasing with the numberof considered parameters) and considering that only one-point and two-point optimizations have been performedfor the purposes of this work a different surrogate modelhas been synthesized for each considered flight conditionFurthermore for the database definition nine blades havebeen considered differing only in the values of the sweep andanhedral blade tip angles Indeed a preliminary sensitivityanalysis has shown that the wake inflow is weakly affected bymechanicalstructural blade properties

Using this linear (LIN) surrogate inflow model in theoptimization process the optimal blade variables given incolumn 3 of Table 1 have been identified showing decreasedblade mass and bending stiffnesses with respect to thebaseline values and an upward tip anhedral angle Thisblade model yields a reduction of the objective function of74 in the synthesis phase that is slightly lower than thatobtained with the Drees inflow model However in thiscase validating the optimal design against application of the

high-fidelity aeroelastic solver based on the (nonsurrogate)BEM free-wake inflow model has given positive resultsIndeed the optimal blade has confirmed both a stableaeroelastic behavior and a significant (63) reduction of theexamined objective function

Figure 2(a) presents the comparison among vibratoryhub loads given by (i) the baseline rotor (ii) the optimalblade rotor within the optimal design process (synthesis)and (iii) the optimal blade rotor in the validation analysisIt demonstrates the effectiveness of the proposed design theresults obtained in the synthesis and validation phases areindeed quite similar as only small discrepancies appear inthe prediction of the in-plane force 119865119910 of the out-of-planeforce 119865119911 and of the torque moment 119872119911 The sensitivity ofvibratory loads to variations of the design variables predictedby the surrogate wake inflowmodel is similar to that from thehigh-fidelity aerodynamic model and hence it has proven tobe well suited for rotor blade optimization applications Thevalidation of the optimal design against aerodynamic model-ing variation has been performed also in terms of aeroelasticstability the high-fidelity aeroelastic solver predicts a stablebehavior of the optimal blade rotor both at 120583 = 03 and inhovering that are the two flight conditions considered for thestability constraint Further the 1rev and 2rev blade loads inthe rotating frame which do not contribute to the vibratoryhub loads but stress blade root are monitored in Figure 2(b)It shows that these loads although not taken into accountin the minimization process at least in this case are eitherpractically unaffected or decreased with the exception of the2rev normal shear force that is subject to an increase of about25 However this can be considered as an acceptable minordrawback of the optimal design configuration

Next in order to assess the robustness of the designwith respect to off-design flight conditions vibratory hubloads and aeroelastic stability of the optimal rotor have beenexamined at advance ratio 120583 = 015 Akin to the case with


BaselineOptimal (synthesis)Optimal (validation)

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s (times10minus4)

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(a) Vibratory 4rev hub loads

1rev baseline1rev optimal


MzMyMxFzFyFx

1r

ev2

rev

bla

de lo

ads (times10minus4)

0

05

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15

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(b) Vibratory loads at blade root (validation)

Figure 2 Vibratory loads from single-point surrogate wake inflow optimization 120583 = 03

BaselineOptimal

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load

s (times10minus4)

MzMyMxFzFyFx0

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Figure 3 Optimized versus baseline rotor 4rev hub loads at off-design condition 120583 = 015

120583 = 03 the critical eigenvalues are not appreciably affectedby the blade redesign and thus a stable behavior of the rotor ismaintained With regard to the vibratory loads although theobjective function is reduced by about 32 Figure 3 showsthat the vibratory lateral shear force and torque moment areconsiderably increased with respect to those at the baselineconfiguration In addition it is worth noting that at 120583 = 015

the vibratory hub loads are higher than those at the 120583 =

03 design flight condition The reason for this is explainedby Figures 4(a) and 4(b) which depict the time historiesof the blade lift spanwise distribution for the baseline rotorin the design and off-design flight conditions respectively

Indeed these figures show a more irregular distribution(in space and time) of the airloads in the off-design flightcondition because of the occurrence of strong blade-wakeinteraction effects as revealed in Figures 5(a) and 5(b) whichdepict an isometric view of the computed wake geometryat the two flight conditions Figure 5(b) clearly shows thatduring the low-speed flight (120583 = 015) the wake remainsclose to the rotor disk thus inducing severe blade-wakeimpingement at both the advancing and retreating sides ofthe rotorThe strong blade-wake interactions inducing highervibratory loads at lower advance ratio may be captured byfree-wake aerodynamic simulation andwould remain hiddenif simplified wake inflows were used (like eg those basedon prescribed wake shape or semianalytic ones) [39 40]Note also that neither advancing-side blade tip transoniceffects nor retreating-side dynamic-stallreverse flow effectsare considered here although these might significantly affectthe vibratory loads in high-speed flights

The observation that the low-speed flight condition (120583 =

015) is more severe in terms of vibratory hub loads than thedesign flight condition combined with the fact that the off-design behavior of the optimal blade configuration has beenproven to be unsatisfactory suggests to apply a multipointoptimization approach in order to take into account severalflight conditions in the optimization process so as to broadenthe range of the flight envelope where the optimal bladedesign might be effective

32 Multipoint Optimization For the multipoint optimiza-tion the objective function has been defined as a combi-nation of the vibratory hub loads arising at 120583 = 015 and120583 = 03 The attempt is to develop a blade design processtaking into account aerodynamic effects that characterizeboth high-speed and low-speed flight conditions Specificallythe following combination of the scalar norm of the 4rev


minus002

0

002

004

006

008

01

012

Non

dim

ensio

nal s

ectio

nal l

ift

(a) Design flight condition 120583 = 03

minus002

0

002

004

006

008

01

012

014

Non

dim

ensio

nal s

ectio

nal l

ift

(b) Off-design flight condition 120583 = 015

Figure 4 Rotor disk distribution of blade sectional lift

(a) Design flight condition 120583 = 03 (b) Off-design flight condition 120583 = 015

Figure 5 View of BEM free-wake geometry

harmonics of hub forces and moments at the two flightconditions has been considered

=

2

sum

119894=1

[(1198652

119909+ 1198652

119910+ 1198652

119911)12

]119894

+ [(1198722

119909+ 1198722

119910+ 1198722

119911)12

]119894

(2)

As in the single-point optimization process the designvariables are the blade mass per unit length the bending andtorsional stiffnesses and the sweep and anhedral angles

The result of the optimization process has been a reduc-tion of the objective function of about 64 with theoptimal design variables given in the last column of Table 1These show a reduction of bending stiffnesses with respectto the baseline values whereas torsional stiffness and blademass per unit length are increased Furthermore a rearwardtip sweep angle of about 18 deg and an upward tip anhedralangle of about 4 deg have been identified

The corresponding 4rev vibratory hub loads and 1revand 2rev blade root rotating loads are presented in Figures6(a) 6(b) 7(a) and 7(b) for the two design flight conditionsSignificant reductions of vibratory loads are evident at bothdesign conditions although small spillover on the in-planeshear force 119865119910 and on the torque moment 119872119911 is presentat 120583 = 015 (see Figure 7(a)) However with these being

the lowest loads the overall quality of the results may beconsidered very good In addition these figures present thevalidation of the identified optimal blade against the high-fidelity aerodynamic model Indeed vibratory loads from thesurrogate inflow model used in the optimal process are quitesimilar to those from themore accurate BEM solutionmodelwith the only exception of the out-of-plane shear force at120583 = 015 in Figure 7(a) which is thoroughly overestimated bythe solver in the optimizer (anyway the effect of the changesof the design variables on this load seems to bewell captured)The multipoint optimization has produced vibratory loadsalleviation quite uniformly distributed between the twodesign conditions although these alleviations for120583 = 03 arelower than those obtained by the single-point optimizationprocedure

For the 1rev and 2rev rotating blade root loads con-clusions in line with those made in the case of single-pointoptimization can be drawn Indeed Figures 6(b) and 7(b)show that although not considered in the objective functionalso these loads are reduced with the only exception of the2rev normal shear force that is subject to some amplificationboth at 120583 = 03 and at 120583 = 015 Again the increase is suchthat it can be considered as an acceptable minor drawback ofthe optimal blade configuration

Then the robustness of the optimal rotor configura-tion has been assessed by application to off-design flight



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Figure 6 Vibratory loads from multipoint surrogate wake inflow optimization 120583 = 03


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conditions at advancing ratios 120583 = 01 02 and 025 Astable aeroelastic response has been observed in two of thesethree flight conditions with a slightly unstable eigenvalueappearing at 120583 = 010 However this is not a critical issuein that just the inclusion of a realistic structural damping inthe analysis (not considered here) would have avoided theonset of such a weak instability (more generally slight insta-bilities might be simply overcome by including structuraldampers)

Concerning the assessment of the off-design vibratoryloads generated by the optimal rotor a hub loads magnitudeindex is introduced as the sumof the scalar normof 4rev hubforces and moments The values of this index computed by

the high-fidelity aerodynamicmodel at design and off-designflight conditions are depicted in Figure 8(a) These showthat although never increased with respect to those relatedto the baseline blade very small reductions are obtained at120583 = 010 and 120583 = 025 This is essentially due to spillovereffects which typically appear at off-design applications andthat here are of particular strength at 120583 = 010 and 120583 =

025 As an example Figure 8(b) compares the optimal rotor4rev vibratory hub loads with those from the baseline rotorevaluated at 120583 = 025 Indeed significant increases of in-plane 119865119910 and out-of-plane 119865119911 shear forces as well as oftorque moment 119872119911 are observed in contrast to the reducedcorresponding hub loads magnitude index in Figure 8(a)


BaselineOptimal

Hub

load

s mag

nitu

de in

dex

(times103)

05

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015 02 025 0301Advance ratio 120583

(a) Hub loads magnitude index

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BaselineOptimal

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(b) Vibratory 4rev hub loads at 120583 = 025

Figure 8 Multipoint optimization off-design conditions

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Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

1

2

3

4

5

6

7

8

9

MzMyMxFzFyFx


(b) Advance ratio 120583 = 030

Figure 9 Vibratory loads from multipoint TPS surrogate wake inflow optimization

33 Effects of Surrogate Models on Optimal Design Herethe sensitivity of the multipoint optimization results tothe surrogate wake inflow model is assessed In particulartwo additional techniques are investigated thin-plate splines(TPS) and multilayer feed-forward neural networks (NN)

First for both advance ratios considered in themultipointoptimization the vibratory loads reduction obtained througha thin-plate spline surrogate model is presented in Figures9(a) and 9(b) The optimal process has led to a reductionof the objective function of about 62 with the designvariables listed in the third column of Table 2 labeled as TPSsurrogate

The optimal configuration is similar to that obtained byusing the linear wake inflow model with a reduction of thebending stiffnesses and an increase in torsional stiffness andblade mass Also in this case the vibratory loads are quitesimilar to those estimated by the high-fidelity BEM solverwith the exception of the out-of-plane shear force at 120583 = 015

and the torque moment 119872119911 at 120583 = 030Then the results obtained by using a NN surrogatemodel

are presented in Figures 10(a) and 10(b) in terms of vibratoryhub loads Also in this case a reduction of the objective func-tion of about 60 is achieved with the optimal configurationgiven in the fourth column of Table 2 Again the optimal


Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

2

4

6

8

10

12

14

16

18

20

MzMyMxFzFyFx




Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

1

2

3

4

5

6

7

8

9

MzMyMxFzFyFx


Figure 10 Vibratory loads from multipoint NN surrogate wake inflow optimization

Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

2

4

6

8

10

12

14

16

18

20



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

1

2

3

4

5

6

7

8

9



Figure 11 Vibratory loads from multipoint LIN-NN surrogate wake inflow optimization

blade properties are similar to those related with the linearsurrogate optimal configuration except for the sweep anglewhich in this case is about 20 deg The verification with thehigh-fidelity BEM solver has shown a satisfactory level ofaccuracy in the evaluation of the vibratory hub loads duringthe synthesis phase with the better correlation achieved inthe slower flight condition However some differences arepresent in the shear forces evaluation at 120583 = 030 with loadsoverestimated in the synthesis phase

These results indicate that the linear surrogate model isaccurate enough for the wake inflow interpolation at 120583 =

030 whereas the neural networks interpolation is requiredat 120583 = 015 This is due to the fact that at higher advance

ratios the wake is far from the rotor disk and the wakeinflow seems to be weakly influenced by the geometricaldesign variables while at lower advance ratios the wakeremains nearby the rotor disk with the rotor operating in amore complex aerodynamic field thus producing strongervariations in the wake inflow when the geometrical designvariables change These observations suggest to perform anew multipoint optimization procedure combining the twodifferent approaches the linear interpolation at 120583 = 030 andthe neural networks at 120583 = 015

The reduction of the objective function obtained is ofabout 62 with the design variables listed in Table 2 columnLIN-NN Also in this case they seem to remain similar to the


Table 2 Blade design variables multipoint optimization

Baseline LINsurrogate

TPSsurrogate

NNsurrogate

LIN-NNsurrogate

1198641198681205781198980Ω21198774 001060 000898 000856 000910 000857

1198641198681205771198980Ω21198774 003010 002570 002940 002557 002664

1198661198691198980Ω21198774 000147 000188 000187 000180 000188

1198981198980

10 1033 1164 1101 1051Λ119860[deg] 00 minus41 minus39 minus40 minus42

Λ119878[deg] 00 182 187 201 177

OASPL (dB)

minus10

minus5

0

5

10

y(m

)

minus10 minus5 0 5 10minus15

x (m)

85

88

91

94

97

100

103

106

109

112

115

Figure 12 OASPL contour plot at 120583 = 03 baseline configuration

other cases except again for the sweep angle which seemsto be the variables mainly affected by the new analysis Theperformances of this combined optimization in terms ofvibratory hub loads are depicted in Figures 11(a) and 11(b)the comparison of the loads predicted through the surrogatemodels with those from the high-fidelity solver reveals thatthe accuracy of the combined surrogate model is satisfactoryalthough of quality similar to the results of completely LINanalysis It is worth reminding that in all analyses discussedthe satisfaction of the stability constraints is confirmed in thehigh-fidelity verification

34 Assessment of the Emitted Noise Noting that designstrategies leading to reduced vibratory hub loads oftenresult in more acoustically annoying rotors the comparisonbetween baseline and optimal rotor configurations is com-pleted by the assessment of the emitted noise To this aimthe acoustic field predicted for the optimal rotor consideringthe linear surrogate inflow model is considered as the rep-resentative one The results are shown on a horizontal planelocated 55m below the rotor hub in terms of the OverallSound Pressure Level (OASPL) that is a parameter typicallyconsidered to measure the acoustic disturbance

Firstly the flight condition at 120583 = 03 is examinedFigure 12 shows the OASPL contour plot related to thebaseline rotor whereas Figures 13(a) and 13(b) show OASPLcontour plots concerning the optimal blade configurationsidentified through single-point and multipoint optimizationprocedures respectivelyThe sound radiated by both optimal

rotor configurations shows higher OASPL peaks (the onefrom the single-point optimization in particular) along witha more pronounced directional propagation pattern Thenthe additional flight condition (120583 = 015) considered in themultipoint optimization procedure is examined Figures 14(a)and 14(b) depict the OASPL contour plots from baseline andthe optimal rotor respectively In this case uniformly highernoise levels are produced by the optimal rotor throughoutthe entire domain considered with slight modification of thenoise radiation pattern

These results confirm the more acoustically annoyingnature of low-vibration rotors although it is worth highlight-ing that in high-speed flight the observed impact may beconsidered low

4 Concluding Remarks

Optimization procedures based on genetic binary-basedsingle-point and multipoint algorithms have been developedand successfully applied for the design of rotor blades gener-ating reduced vibratory hub loads Aeroelastic stability con-straints have been considered during the search of minimumobjective function with structuralmechanical propertiesblade tip sweep angles and anhedral angles considered asdesign variables A computational tool developed by theauthors for arbitrarily shaped elastic axis blades has beenused to predict the aeroelastic behavior of the configurationsexamined in the optimization processes Accuracy and com-putational efficiency has been guaranteed by application ofseveral surrogate wake inflow models The effects of thesemodels on the optimal results have been investigated Thefollowingmain outcomes have been derived from the numer-ical investigation presented (i) both single-point and two-point optimizations successfully identify rotor blades suitedfor vibratory hub loads alleviation at the flight(s) condition(s)considered in the objective function (design conditions) (ii)increasing the number of flight conditions considered in theobjective function does not reduce drastically the level ofalleviation attained at each flight condition (iii) the surrogatewake inflow models introduced are suited to be appliedwithin the optimization process their accuracy depends onthe specific configuration examined (iv) when operating atoff-design conditions the performance of the identified bladegets worse and spillover effect may occur (ie the alleviationof each vibratory load component is not guaranteed) thelatter depending on the sensitivity of aerodynamic phenom-ena to changes of the flight configuration (v) the aeroelasticstability of the optimal rotor is guaranteed at the designflight condition but is substantially maintained also at off-design flights including the critical hovering condition (vi)the optimal design variables seem to be slightly affected bythe surrogate inflow model applied with the sweep anglebeing the variable most affected by it and (vii) a minorcorrelation with the validation solver is shown when ananalytical poorly accurate wake inflow model is used in theoptimization procedure the anhedral angle being the mostsensitive parameter (it has opposite sign with respect to thosepredicted by other wake inflow models)


OASPL (dB)


x (m)

828588919497100103106109112115118

minus10

minus5

0

5

10

y(m

)

(a) Single-point optimization

OASPL (dB)

85

88

91

94

97

100

103

106

109

112

115


x (m)

minus10

minus5

0

5

10

y(m

)

(b) Multipoint optimization

Figure 13 OASPL contour plot at 120583 = 03 optimal configurations

OASPL (dB)

828588919497100103106109112115118


x (m)

minus10

minus5

0

5

10

y(m

)

(a) Baseline

OASPL (dB)

minus10

minus5

0

5

10

y(m

)


x (m)

87

90

93

96

99

102

105

108

111

114

117


Figure 14 OASPL contour plot at 120583 = 015 from baseline and optimal configurations

Since even the two-point optimization has been unable toguarantee significant vibration alleviation in off-design con-ditions it is expected that a reduction uniformly distributedthroughout the flight envelope might be achieved by eitherincluding in the objective function several flight conditionsor introducing active controls to reduce vibrations in off-design flights The first strategy is easier to be applied inthat it does not require the implementation of additionaldevices but it has the drawback of yielding lower alleviationThe second one is more complex but better performance ispossibly achieved Finally the acoustic assessment has shownthat the optimal blade configuration yields higher noiseemissions thus confirming the more acoustically annoyingnature of low vibratory rotors At high speed the increaseis not relevant while the directivity of noise pattern issignificantly altered by blade redesign Conversely at lowspeed the directivity is not significantly modified but theacoustic peaks are subject to higher increase These resultssuggest the inclusion of an acoustic annoyance measurein the objective function using multiobjective optimizationtechniques so as to define a tool capable of identifyingblades combining low-vibration levels with low acousticemissions

Appendix

A Aeroelastic Blade Model

In the following an outline of the mathematical formulationimplemented to predict the aeroelastic behavior of rotorblades within the optimal design process is presented First adetailed description of the structural modeling for arbitrarilycurved blades is given followed by a brief outline of theformulation applied for the prediction of sectional loads Aconcluding section presents the spectral approach applied forthe spatial integration of the differential aeroelastic model

A1 Displacement Variables and Coordinate Systems Severalcoordinate systems are introduced to derive the equations ofmotion of the blade The main ones illustrated in Figure 15are the following

(i) A global blade orthogonal system of unit vectors ( 11989411198942 1198943) which is centered at the rotor hub and rotateswith the blade with 1198941 tangent to the elastic axis at theroot section vectors 1198942 and 1198943 identify the principalaxes of the blade root section note that the orientation


Deformed elastic axis

Undeformed elastic axis

e3e2 e1

i3

r0 r

r

r1

u

r0

r1

u0

Ω

s

i2

i1

1e

e2e3

Figure 15 Sketch of beam representation

of 1198942 and 1198943 depends on both collective and cyclic bladepitch commands

(ii) Local rotating orthogonal systems of unit vectors( 1198901 1198902 1198903) with 1198901 aligned to the undeformed bladeelastic axis and 1198902 1198903 aligned with the blade sectionprincipal axes note that the orientation of 1198902 and 1198903

depends also on blade pretwist

(iii) Local rotating orthogonal systems related to the bladedeformed configuration having base unit vectors(997888rarr 1

997888rarr 2

997888rarr 3) with

997888rarr 1 tangent to the deformed elastic

axis and997888rarr 2

997888rarr 3 aligned with the principal axes of

the elastically twisted blade sections (this assumptionmeans that the sections of the deformed beam remainorthogonal to the deformed elastic axis and hence ashear undeformable beam model is considered)

Deformations are described in terms of displacementsof the elastic axis and rotation of beam sections The dis-placements 119906 V 119908 are defined in the local frame fixedwith the undeformed blade respectively along the directionsidentified by 1198901 1198902 and 1198903 the blade twist 120601 is defined as therotation of blade sections about

997888rarr 1-direction (ie about the

deformed elastic axis)In the development of the blade dynamics formulation

the definition of transformation matrices relating the framesof reference defined above is also convenient

Global to Undeformed Blade References The unit base vectorsof the global rotating frame are related to those of the local-undeformed blade frames through the following expression

119890119894 = A (119904) 119894119894 (A1)

where for 119890119894 = 1198901 1198902 1198903 and 119904 denoting the curvilinearcoordinate defined along the undeformed elastic axis A(119904)

is the transformation matrix yielding the local frame unitvectors in terms of superposition of the global-frame base(ie it collects the components of each 119890119894 in the global frame)

Blade-Undeformed to Blade Deformed References Local-undeformed blade and deformed blade references are relatedby the following expression

997888rarr 119894 = T (119904) 119890119894 (A2)

where T is the transformation matrix depending on thedeformation of the blade The matrix T is obtained as asequence of rotations about 1198903 (rotated) 1198902 and (rotated) 1198901its expression in terms of 119906 V 119908 120601 may be derived from thatgiven in [41]

Global to Blade Deformed References From the transforma-tions mentioned above it is possible to determine the relationbetween global and deformed blade references as

997888rarr 119894 = Λ (119904) 119894119894 (A3)

where Λ = TA

A2 Equilibrium Relations Considering a deformed beamelement of length d119904 the equilibrium of forces and momentsacting on it yields

dVd119904

+ = 0

dd119904

+997888rarr 1 times V + = 0

(A4)

where V and are the internal structural forces andmomentsat the elastic axis while and are the external distributedforces and moments

In order to integrate the above differential equilibriumequations it is convenient towrite them in terms of forces andmoments components v119897 m119897 p119897 and q119897 in the local blade-undeformed frames Observing that for v119892 m119892 p119892 and q119892denoting forces and moments components in the globalframe of reference one has v119892 = A119879v119897 m119892 = A119879m119897 p119892 =


A119879p119897 and q119892 = A119879q119897 the equilibrium equations projectedonto the frame 119894119894 yield

dd119904

(A119879k119897) + A119879p119897 = 0 (A5)

dd119904

(A119879m119897) minus A119879H119897k119897 + A119879q119897 = 0 (A6)

where H119897 is the matrix of the components in the local-undeformed frame of the axial tensor associated to vector

997888rarr 1

Next integration of (A5) yields the following distributionof the internal shear loads

k119897 (119904) = Aint

119877

119904

A119879p119897d (A7)

while the integration of (A6) yields the following distributionof the internal moments

m119897 (119904)

= Aint

119877

119904

A119879q119897d minus int

119877

119904

(A119879H119897Aint

119877

A119879p119897d) d

(A8)

where 119877 denotes the length of the undeformed elastic axis(under assumption of negligible second-order terms relatedto local slope)

Equations (A7) and (A8) are the general solutions for theinternal shear loads and moments arising in a beam fromwhich the equations governing the blade elastic displace-ment may be derived once strain-displacement and load-displacement relations are identified (see next subsections)Note that because of shear undeformable assumption theequations governing the blade motion variables 119906 V 119908 and120601 are derived from the first scalar equation in (A7) andthe three scalar equations in (A8) (the second and thirdscalar equations in (A7) are used to determine the shear loadcomponents lying in the plane of beam sections)

A3 Strain-Displacement Relations In order to express theinternal loads in terms of the (shear undeformable) beamdeformation variables 119906 V 119908 and 120601 the strain tensor hasto be derived from position vectors describing undeformedand deformed beam The position vector 119903 of a point ofthe undeformed beam of coordinates (119904 120578 120577) with 120578 and120577 denoting the coordinates along the principal axes of thesections is given by 119903(119904 120578 120577) = 1199030 + 120578 1198902 + 120577 1198903 while afterdeformation under the assumption of negligible warping itsposition is identified by the following vector

997888rarr (119904 120578 120577) =

1199030 + 119906 1198901 + V 1198902 + 119908 1198903 + 120578997888rarr 2 + 120577

997888rarr 3 where 1199030 denotes the

position of the points along the undeformed elastic axis Fromthe above equations the strain tensor is derived from

(119904 120578 120577) =1

2(

120597997888rarr

120597120585119894sdot120597997888rarr

120597120585119895minus

120597 119903

120597120585119894sdot120597 119903

120597120585119895)

120597 119903

120597120585119894otimes

120597 119903

120597120585119895 (A9)

where (1205851 1205852 1205853) equiv (119904 120578 120577) observing that the local coordi-

nate systems considered are orthogonal

A4 Internal Load-Displacement Relations From the stress-strain relations given by the theory of elasticity combinedwith the strain tensor expression it is possible to derivethe stress-displacement relations and in turn those betweeninternal structural loads and displacement variables afterintegration over the beam sections Under the assumptionsof rigid beam sections and shear undeformable beam thefollowing four loads are given in terms of the four variablesdescribing the beammotion (the evaluation of the remainingtwo loads comes from balancing of the external loads)

119881119909 = V sdot997888rarr 1 = int

119860

120590119909119909d120578 d120577

= 119864119860(1015840+V10158402

2+

10158402

2+ 1198962

1198601198962

1)

119872 = sdot997888rarr 2 = minusint

119860

120590119909119909120577 d120578 d120577 = 1198641198681205781198962

119872= sdot

997888rarr 3 = int

119860

120590119909119909120578 d120578 d120577 = 1198641198681205771198963

119872 = sdot997888rarr 1 = int

119860

(120590119909120578120578 minus 120590119909120577120577) d120578 d120577

= 1198661198691198961 + 1198811199091198962

1198601198961

(A10)

where 119860 is the blade cross-section area 119864 is the Youngmodulus and 119866 is the shear modulus while

119868120578 = int119860

1205772d120578 d120577

119869 = int119860

(1205782+ 1205772) d120578 d120577

119868120577 = int119860

1205782d120578 d120577

1198962

119860=

1

119860int119860

(1205782+ 1205772) d120578 d120577

(A11)

In addition 1198961 1198962 1198963 are such that for = 1198961

997888rarr 1 + 1198962

997888rarr 2 +

1198963

997888rarr 3 the skew-symmetric matrix K = (dTd119904)T119879 is the

matrix of the components in the local-deformed frame ofthe axial tensor associated with vector (specifically 1198962

and 1198963 are the bending curvatures of the deformed elasticaxis whereas 1198961 is the twist of the blade sections afterdeformation) Finally following a second-order geometricdescription the extensional deformation of the elastic axis isgiven by 120598119909119909 =

1015840+ V101584022+

101584022 where

1015840 V1015840 and 1015840 are the

components of dud119904 = dud119904 minusK0u for u = 119906 V 119908119879 and

K0 = (dAd119904)A119879

A5 Inertial Loads Blade rotation and unsteady deforma-tions make inertial loads arise These contribute to theequilibrium equations as external distributed loads and com-bined with the internal loads yield the equations governing


blade structural dynamicsThe acceleration of a generic pointof a rotating blade is given by

= 119903 + 119867 + Ω times Ω times997888rarr + 2Ω times ] (A12)

where 119903 is the acceleration of the point with respect toa frame rigidly connected to the undeformed blade It isderived from the rigid motion of the beam cross sectionsexpressed in terms of displacements of the elastic axis androtations about it In addition 119867 is the rotor hub acceler-ation and ] is the velocity of the examined point with respectto the rotating frame fixed with the undeformed blade whileΩ is the blade angular velocity Then the resulting inertialdistributed loads appearing in (A4) are expressed as

= minusint119860

120588 d120578 d120577

= minusint119860

120588 119903119904 times d120578 d120577(A13)

where 120588 is the material density and 119903119904 =997888rarr minus ( 1199030 + 119906 1198901 + V 1198902 +

119908 1198903)

A6 Equations for Deformation Variables In order to applythe structural formulation outlined above it is necessaryto derive the relationship between the blade displacementunknowns 119906 V 119908 and 120601 appearing in the definition ofthe inertial loads and the deformation variables 120598119909119909 1 2and 3 appearing in the expressions of the internal loads

To this purpose the vector of the cross-section rotationsabout the axes of the local-deformed frame of reference isintroduced observing that the derivative of its global-framecomponents with respect to the curvilinear abscissa 119904 givesthe components in the same frame of vector of SectionA4Reminding the shear undeformable beam assumption thisyields

A120597 (A119879120579)

120597119904= k (A14)

and hence

120579 (119904) = Aint

119904

0

A119879k d119904 (A15)

where 120579 denotes the vector of the local-undeformed framecomponents of the cross-section rotations while k denotesthe components of in the same frame Note that the firstcomponent of 120579 corresponds to the elastic torsion 120601 whereasthe second and third ones are related to the displacementcomponents 119906 V and 119908 through the following second-ordernonlinear kinematic relation

A120597A119879u120597119904

=

120598119909119909 minus 119891nl (119906 V 119908)

1205793

minus1205792

(A16)

where the nonlinear term 119891nl takes into account the exten-sion of the elastic axis due to bending

Thus the deformation variables 120598119909119909 1 2 and 3

appearing in the expression of the internal elastic loads arerelated to 119906 V 119908 and 120601 through combination of (A15) and(A16)

Note that from the combination of the above kinematicrelations with the internal loads the inertial loads the firstscalar equation in (A5) and the three scalar equations in(A6) it is possible to derive a set of four integrodifferentialequations in the deformation unknowns 119906 V 119908 120601 governingthe structural dynamics of a blade with arbitrarily curvedelastic axis

A7 Aerodynamic Loads As already mentioned in the mainbody text the aerodynamic loads are derived from a quasi-steady approximation of the Greenberg theory [28] forairfoils Aerodynamic three-dimensional effects are takeninto account by including wake inflow (either through ananalytical model or by an aerodynamic solution tool) Thussection force119879 orthogonal to the chord and 119878 parallel to thechord are given by

119879 =

984858119862119897120572

119888

2[minus119880119875119880119879 +

119888

2120596119880119879 minus

119888

4119875 + (

119888

4)

2

]

119878 =

984858119862119897120572

119888

2[1198802

119875minus

119888

2120596119880119875 minus

1198621198890

119862119897120572

1198802

119879]

(A17)

while the section pitching moment with respect to thequarter-chord point reads

119872120601 = minus

984858119862119897120572

1198883

32(120596119880119879 minus 119875 +

3119888

8) (A18)

In the above equations 119880119875 and 119880119879 are respectively thequarter-chord velocity components normal and parallel tothe chord after deformation 120596 is the out-of-section compo-nent of the angular velocity of the blade section 119888 denotesthe chord length 984858 is the air density119862119897

120572

is the lift curve slopecoefficient while 119862119889

0

is the drag coefficientFor including these equations within an aeroelastic

model 119880119875 119880119879 and 120596 are expressed in terms of 119906 V 119908 and120601 and the aerodynamic forces 119879 and 119878 are projected onto thelocal blade-undeformed frame of reference

A8 Spectral Solution In the numerical solver developed theblade elastic axis is considered inextensible In this case theaxial degree of freedom 119906 is derived as a consequence ofbeam bending (akin to the approach followed in [29]) Thecorresponding set of governing equations is that in (A8) withunknown variables V 119908 and 120601

To explain the procedure adopted for the numericalsolution of these equations it is convenient to recast themformally as

m119897 = min + maer (A19)

where min and maer denote the inertial and aerodynamiccontributions to the RHS of (A8) respectively The spatialintegration of (A19) is obtained through a spectral approach


The first step consists of the following description of theundeformed-axis frame components of the curvatures

1198961 (119904 119905) =

1198731

sum

119899=1

120572120601

119899(119905) 1205951206011015840

119899(119904) (A20)

1198962 (119904 119905) =

1198732

sum

119899=1

120572119908

119899(119905) 12059511990810158401015840

119899(119904) (A21)

1198963 (119904 119905) =

1198733

sum

119899=1

120572V119899(119905) 120595

V10158401015840119899

(119904) (A22)

where120595120601

119899 120595V119899 and120595

119908

119899may be conveniently chosen as the tor-

sion and bending natural modes of vibration of a nonrotatingbeam [29] Note that in (A20) the first-order derivative ofthe torsion shape function 120595120601

119899 is used in that related to the

elastic twist of the blade 1198961 whereas in (A21) and (A22) thesecond-order derivatives of the bending shape functions 120595V

119899

and 120595119908

119899 have been introduced in that related to the bending

curvatures 1198962 and 1198963Then the equations in (A19) are projected onto the same

set of functions applied in (A20)ndash(A22)

int

119877

0

m119879119897Ψ d119904 = int

119877

0

(m119879in + m119879aer)Ψ d119904 (A23)

where

Ψ =

1205951206011015840

119899

12059511990810158401015840

119899

120595V10158401015840119899

(A24)

The choice of using the first- and second-order derivativesof the shape functions as test functions is mainly motivatedby the equivalence between the resulting equations and thosethat would be derived from application of the Rayleigh-Ritzapproach The resulting aeroelastic system consists of a setof (1198731 + 1198732 + 1198733) nonlinear time-dependent equationswith unknowns 120572 which can be applied for both aeroelasticresponse and stability analysis

Nomenclature

119888 Blade chord119862119879 Rotor thrust coefficient 119879120588120587Ω

21198774 (119879 =

thrust 120588 = air density)119864119868120578 119864119868120577 Bending stiffnesses119865119909 119865119910 119865119911 Hub force components119866119869 Torsional stiffness Objective function119898 Mass distribution1198980 Baseline mass distribution119872119909119872119910119872119911 Hub moment components119877 Rotor radius

119881 Hub-freestream relative velocity120574 Lock number 31205881198861198881198771198980 (119886 = airfoil lift

curve slope coefficient)Λ119860 Blade tip anhedral angle (positive

downwards)Λ 119878 Blade tip sweep angle (positive backwards)120583 Advance ratio 119881Ω119877

120590 Rotor solidityΩ Rotor blade angular speed

Competing Interests

The authors declare that they have no competing interests

References

[1] P Rauch M Gervais P Cranga et al ldquoBlue edge the designdevelopment and testing of a new blade conceptrdquo in Proceedingsof the 67th Annual Forum of the AmericanHelicopter Society pp542ndash555 Virginia Beach Va USA 2011

[2] Y H Yu ldquoRotor blade-vortex interaction noiserdquo Progress inAerospace Sciences vol 36 no 2 pp 97ndash115 2000

[3] K A Yuan and P P Friedmann ldquoAeroelasticity and structuraloptimization of composite helicopter rotor blades with swepttipsrdquo NASA CR-4665 1995

[4] K-A Yuan and P P Friedmann ldquoStructural optimizationfor vibratory loads reduction of composite helicopter rotorblades with advanced geometry tipsrdquo Journal of the AmericanHelicopter Society vol 43 no 3 pp 246ndash256 1998

[5] R Ganguli and I Chopra ldquoAeroelastic optimization of anadvanced geometry helicopter rotorrdquo Journal of the AmericanHelicopter Society vol 41 no 1 pp 18ndash29 1996

[6] R Ganguli and I Chopra ldquoAeroelastic optimization of ahelicopter rotor with two-cell composite bladesrdquo AIAA Journalvol 34 no 4 pp 835ndash841 1996

[7] R Ganguli and I Chopra ldquoAeroelastic optimization of ahelicopter rotor to reduce vibration and dynamic stressesrdquoJournal of Aircraft vol 33 no 4 pp 808ndash815 1996

[8] S Murugan and R Ganguli ldquoInuence of inow models on heli-copter aeroelastic optimizationrdquo Computational Fluid Dynam-ics Journal vol 16 no 4 pp 444ndash453 2008

[9] B Glaz T Goel L Liu P P Friedmann and R T HaftkaldquoMultiple-surrogate approach to helicopter rotor blade vibra-tion reductionrdquo AIAA Journal vol 47 no 1 pp 271ndash282 2009

[10] B Glaz P P Friedmann and L Liu ldquoHelicopter vibrationreduction throughout the entire flight envelope using surrogate-based optimizationrdquo Journal of the American Helicopter Societyvol 54 no 1 pp 1ndash15 2009

[11] P P Friedmann ldquoHelicopter vibration reduction usingstructural optimization with aeroelasticmultidisciplinaryconstraints-a surveyrdquo Journal of Aircraft vol 28 no 1 pp 8ndash211991

[12] R Celi ldquoRecent applications of design optimization torotorcraftmdasha surveyrdquo Journal of Aircraft vol 36 no 1 pp 176ndash189 1999

[13] R Ganguli ldquoA survey of recent developments in rotorcraftdesign optimizationrdquo Journal of Aircraft vol 41 no 3 pp 493ndash510 2004

[14] G Bernardini C Testa and M Gennaretti ldquoOptimal designof tonal noise control inside smart-stiffened cylindrical shellsrdquo


Journal of Vibration and Control vol 18 no 8 pp 1233ndash12462012

[15] DCalcagni G Bernardini and F Salvatore ldquoAutomatedmarinepropeller optimal design combining hydrodynamics modelsand neural networksrdquo in Proceedings of 11th International Con-ference on Computer Applications and Information Technology inthe Maritime Industries Liege Belgium January 2012

[16] J H Holland Adaptation in Nature and Artificial SystemsUniversity of Michigan Press Ann Arbor Mich USA 1975

[17] D E Goldberg ldquoOptimal initial population size for binary-coded genetic algorithmsrdquo TCGA Report 85001 University ofAlabama Tuscaloosa Ala USA 1985

[18] J R R A Martins and A B Lambe ldquoMultidisciplinary designoptimization a survey of architecturesrdquo AIAA Journal vol 51no 9 pp 2049ndash2075 2013

[19] R Celi Aeroelasticity and structural optimization of heli-copter rotor blades with swept tips [PhD thesis] MechanicalAerospace and Nuclear Engineering Department University ofCalifornia Los Angeles Calif USA 1987

[20] R Celi and P P Friedmann ldquoAeroelastic modeling of swepttip rotor blades using finite elementsrdquo Journal of the AmericanHelicopter Society vol 33 no 2 pp 43ndash52 1988

[21] G S Bir and I Chopra ldquoAeromechanical stability of rotorcraftwith advanced geometry bladesrdquo Mathematical and ComputerModelling vol 19 no 3-4 pp 159ndash191 1994

[22] V A Riziotis S G Voutsinas D I Manolas E S Politis andP K Chaviaropoulos ldquoAeroelastic analysis of pre-curved rotorbladesrdquo in Proceedings of the EuropeanWind Energy Conferenceand Exhibition (EWEC rsquo10) Warsaw Poland April 2010

[23] E Piccione G Bernardini M Molica Colella and MGennaretti ldquoStructural and aeroelastic modeling of curvedrotor blades using a galerkin approachrdquo in Proceedings of the 3rdCeas Air amp Space Conference21st AIDAACongress pp 615ndash624Venice Italy 2011

[24] E Piccione G Bernardini M Molica Colella and MGennaretti ldquoA spectral formulation for structuralaeroelasticmodeling of curved-axis rotor bladesrdquo Aerotecnica Missili ampSpazio vol 91 no 1-2 pp 42ndash52 2012

[25] M Gennaretti and G Bernardini ldquoNovel boundary integralformulation for blade-vortex interaction aerodynamics of heli-copter rotorsrdquo AIAA Journal vol 45 no 6 pp 1169ndash1176 2007

[26] F Farassat ldquoDerivation of formulations 1 and 1A of FarassatrdquoNASA TM-2007-214853 2007

[27] J E Ffowcs Williams and D L Hawkings ldquoSound generatedby turbulence and surfaces in arbitrary motionrdquo PhilosophicalTransactions of the Royal Society A vol 264 no 1151 pp 321ndash342 1969

[28] J M Greenberg ldquoAirfoil in sinusoidal motion in pulsatingstreamrdquo NACA TN-1326 1947

[29] D H Hodges and R A Ormiston ldquoStability of elastic bendingand torsion of uniform cantilever rotor blades in hover withvariable structural couplingrdquo NASA TN D-8192 1976

[30] M Gennaretti and G Bernardini ldquoAeroelastic response ofhelicopter rotors using a 3D unsteady aerodynamic solverrdquoTheAeronautical Journal vol 110 no 1114 pp 793ndash801 2006

[31] M Gennaretti and G Bernardini ldquoAeroacousto-elastic mod-eling for response analysis of helicopter rotorsrdquo in VariationalAnalysis and Aerospace Engineering Mathematical Challengesfor Aerospace Design G Buttazzo and A Frediani Eds vol66 of Springer Optimization and Its Applications pp 27ndash50Springer Berlin Germany 2012

[32] R E Smith and E Smuda ldquoAdaptively resizing populationsalgorithm analysis and first resultsrdquo Complex Systems vol 9no 1 pp 47ndash72 1995

[33] R T Haftka and Z Gurdal Elements of Structural OptimizationKluwer Academic Dordrecht The Netherlands 1992

[34] G Rudolph ldquoEvolutionary search under partially orderedfitness setsrdquo in Proceedings of the International Symposiumon Information Science Innovations in Engineering of Naturaland Artificial Intelligent Systems (ISI rsquo01) pp 818ndash822 ICSCAcademic Press 2001

[35] E Zitzler K Deb and LThiele ldquoComparison of multiobjectiveevolutionary algorithms empirical resultsrdquo Evolutionary Com-putation vol 8 no 2 pp 173ndash195 2000

[36] D P Raymer Enhancing aircraft conceptual design using mul-tidisciplinary optimization [PhD thesis] Royal Institute ofTechnology Stockholm Sweden 2002

[37] C Venkatesan P P Friedmann and K-A Yuan ldquoA newsensitivity analysis for structural optimization of compositerotor bladesrdquoMathematical andComputerModelling vol 19 no3-4 pp 1ndash25 1994

[38] R Ganguli ldquoOptimum design of a helicopter rotor for lowvibration using aeroelastic analysis and response surface meth-odsrdquo Journal of Sound andVibration vol 258 no 2 pp 327ndash3442002

[39] J Zhang E C Smith and K W Wang ldquoActive-passive hybridoptimization of rotor blades with trailing edge flapsrdquo Journal ofthe American Helicopter Society vol 49 no 1 pp 54ndash65 2004

[40] A Datta and I Chopra ldquoValidation and understanding ofUH-60A vibratory loads in steady level flightrdquo Journal of theAmerican Helicopter Society vol 49 no 3 pp 271ndash287 2004

[41] DHHodges and EHDowell ldquoNonlinear equations ofmotionfor the elastic bending and torsion of twisted nonuniform rotorbladesrdquo NASA TN D-7818 1974

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



In the past literature on similar topics several approachesbased on deterministic optimization methods can be found[3ndash7] whereas only a limited number of more recent worksaddress the use of nondeterministic methods such as GAsfor the same aim of [8ndash10] For the sake of computational effi-ciency in both optimizationmethodologies often a surrogatemodel for the direct evaluation of the objective function isadopted [4 8ndash10] Extensive reviews of works dealing withrotorcraft optimization are given in [11ndash13]

In this work a binary-based GA developed by the authorsis adopted where blade tip sweep angle blade tip anhedralangle and the distribution of the mechanicalstructuralproperties are considered as design variables to be identifiedunder aeroelastic stability constraint Recently this binary-based GA has been successfully applied for the optimaldesign of aircraft cabin noise control and marine propellerblades [14 15] The interested readers may find details onGA optimization procedures in [16 17] while a thoroughsurvey of multidisciplinary design optimization techniques ispresented in [18]

In the optimal design process a fundamental role isplayed by the simulation of the aeroelastic behavior of therotor since vibratory loads are a by-product of the interactionbetween the blades and the complex aerodynamic field inwhich they operate In the past years several authors havedeveloped structural models for curvedswept tip bladesusually solved through the finite element method approachOne of the earliest models suited for swept tip blades hasbeen presented and successfully applied to hingeless rotorsin [19 20] Among the others a formulation for blades withvarying sweep droop twist angles and platform has beenintroduced in [21] while more recently a curvilinear-axisblade formulation has been applied to horizontal-axis windturbines [22] In the optimization procedure proposed herethe rotor aeroelastic tool consists of an enhanced versionof that recently developed by the authors presented andvalidated in [23 24] It is based on a nonlinear beam-like model spatially integrated through a Galerkin approachsuited for the analysis of blades having arbitrarily curvedelastic axis (including geometrical discontinuities as sweepand anhedral tip angles) A good trade-off between accuracyand computational efficiency is achieved by the identificationand application of a surrogate model of the wake inflowused for sectional aerodynamic loads prediction relying ona boundary element approach for the analysis of unsteadypotential flows [25]

The numerical investigation will assess the capabilityof the proposed optimal design approach to identify rotorblades generating reduced vibratory hub loads as well as therobustness of the solution at off-design operating conditionsAn analysis of the influence of wake inflow model used inthe aeroelastic tool on the optimization process will be alsopresented Furthermore observing that vibratory hub loadsand noise emission are usually strictly related (alleviation ofvibrations often corresponds to noise increase and vice versa)the acoustic performances of the identified optimal rotor willbe examined To this purpose a boundary element approachbased on Farassatrsquos Formulation 1A [26] for the solution ofFfowcs Williams and Hawkingsrsquo equation [27] is applied

2 Materials and Methods

In the following an outline of the aeroacoustoelastic andoptimization tools developed by the authors is presented

21 Rotor Aeroelastic Solver For the purposes of this workthe availability of an efficient and accurate computational toolfor the aeroelastic analysis of rotor blades having arbitraryshape is of paramount importance The one applied here hasbeen developed by the authors as a combination of a suitedstructural dynamics model with the aerodynamic loads givenby a quasi-steady sectional formulation corrected with inflowcontribution in order to take into account the effect of wakevortices

The rotor blade structural dynamics is described througha beam-like model obtained as an enhanced version of theformulation presented by some of the authors in [23] It isvalid for slender homogeneous and isotropic rotating bladeswith curved elastic axis and includes spanwise variationof mass and stiffness properties as well as variable built-in pretwist precone sweep and anhedral angles Nonlinearstrain-displacement relations are considered with the appli-cation of a second-order approximation scheme in orderto take into account the moderate displacements usuallyexperienced by rotor blades A detailed description of thisblade structural model is presented in Appendix

The distributed aerodynamic loads are modeled by thequasi-steady approximation of the sectional Greenberg the-ory [28] (see Appendix) Three-dimensional unsteady effectsderiving from the wake vorticity are taken into accountthrough the influence of the wake inflow on the relativevelocity at the rear aerodynamic center of the blade crosssectionsThe evaluation of the inflow is obtained by a bound-ary element method (BEM) for the solution of the boundaryintegral equation approach presented in [25] suited for theanalysis of potential flows around rotors in arbitrary motionSimple analytical wake inflow models might also be applied

The aeroelastic integrodifferential model derived by cou-pling these structural dynamics and aerodynamic formula-tions has been spatially integrated through application of theGalerkin approach In particular as outlined in Appendixa spectral description of the curvature components in theundeformed-axis frame is considered in order to develop asolver with good convergence properties even in presenceof sweep and anhedral tip angles (ie for elastic axis shapeswith discontinuous first-order derivatives) The projectionsare applied to the bending and torsion moment equilibriumequations with coinciding sets of trial and test functions(note that in addition to shear undeformable assumption theassumption of inextensible elastic axis has been adopted akinto the approach in [29]) Because of the test functions chosenthe present approach yields equations that are strongly relatedto those that could be derived from application of theRayleigh-Ritz approach (see Appendix)

The aeroelastic response to steady flight conditions isevaluated by integrating the set of ordinary time-differentialequations through a harmonic balance approach [30 31]Aeroelastic stability about the equilibrium solution is instead














1198641198681205771198980Ω21198774 003010 003514 002948 002570

1198661198691198980Ω21198774 000147 000104 000113 000188

1198981198980

10 1053 1188 1033Λ119860[deg] 00 424 minus44 minus41

Λ119878[deg] 00 1929 237 182


∘le Λ 119878 le 30


∘respectively











= (1198652

119909+ 1198652

119910+ 1198652

119911)12

+ (1198722

119909+ 1198722

119910+ 1198722

119911)12

(1)




0

1

2

3

4

5

6

BaselineOptimal

Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx


BaselineOptimal

MzMyMxFzFyFx0

2

4

6

8

10

12

14

16

Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)










Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

1

2

3

4

5

6

7

8

9




MzMyMxFzFyFx

1r

ev2

rev

bla

de lo

ads (times10minus4)

0

05

1

15

2

25

3

35

4



BaselineOptimal

Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

2

4

6

8

10

12

14

16

18

20










minus002

0

002

004

006

008

01

012

Non

dim

ensio

nal s

ectio

nal l

ift


minus002

0

002

004

006

008

01

012

014

Non

dim

ensio

nal s

ectio

nal l

ift






=

2

sum

119894=1

[(1198652

119909+ 1198652

119910+ 1198652

119911)12

]119894

+ [(1198722

119909+ 1198722

119910+ 1198722

119911)12

]119894

(2)









Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

1

2

3

4

5

6

7

8

9




MzMyMxFzFyFx

1r

ev2

rev

bla

de lo

ads (times10minus4)

0

05

1

15

2

25

3

35

4




Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

2

4

6

8

10

12

14

16

18

20




MzMyMxFzFyFx

1r

ev2

rev

bla

de lo

ads (times10minus4)

0

05

1

15

2

25

3

35

4








BaselineOptimal

Hub

load

s mag

nitu

de in

dex

(times103)

05

1

15

2

25

3

35

015 02 025 0301Advance ratio 120583


Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

BaselineOptimal

0

2

4

6

8

10

12

14

MzMyMxFzFyFx



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

2

4

6

8

10

12

14

16

18

20

MzMyMxFzFyFx



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

1

2

3

4

5

6

7

8

9

MzMyMxFzFyFx










Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

2

4

6

8

10

12

14

16

18

20

MzMyMxFzFyFx




Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

1

2

3

4

5

6

7

8

9

MzMyMxFzFyFx



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

2

4

6

8

10

12

14

16

18

20



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

1

2

3

4

5

6

7

8

9












TPSsurrogate

NNsurrogate

LIN-NNsurrogate

1198641198681205781198980Ω21198774 001060 000898 000856 000910 000857

1198641198681205771198980Ω21198774 003010 002570 002940 002557 002664

1198661198691198980Ω21198774 000147 000188 000187 000180 000188

1198981198980


Λ119878[deg] 00 182 187 201 177

OASPL (dB)

minus10

minus5

0

5

10

y(m

)


x (m)

85

88

91

94

97

100

103

106

109

112

115










OASPL (dB)


x (m)

828588919497100103106109112115118

minus10

minus5

0

5

10

y(m

)


OASPL (dB)

85

88

91

94

97

100

103

106

109

112

115


x (m)

minus10

minus5

0

5

10

y(m

)



OASPL (dB)

828588919497100103106109112115118


x (m)

minus10

minus5

0

5

10

y(m

)

(a) Baseline

OASPL (dB)

minus10

minus5

0

5

10

y(m

)


x (m)

87

90

93

96

99

102

105

108

111

114

117




Appendix








e3e2 e1

i3

r0 r

r

r1

u

r0

r1

u0

Ω

s

i2

i1

1e

e2e3






997888rarr 2

997888rarr 3) with










119890119894 = A (119904) 119894119894 (A1)




997888rarr 119894 = T (119904) 119890119894 (A2)



997888rarr 119894 = Λ (119904) 119894119894 (A3)

where Λ = TA


dVd119904

+ = 0

dd119904

+997888rarr 1 times V + = 0

(A4)





dd119904

(A119879k119897) + A119879p119897 = 0 (A5)

dd119904

(A119879m119897) minus A119879H119897k119897 + A119879q119897 = 0 (A6)


997888rarr 1


k119897 (119904) = Aint

119877

119904

A119879p119897d (A7)


m119897 (119904)

= Aint

119877

119904


119877

119904

(A119879H119897Aint

119877

A119879p119897d) d

(A8)




997888rarr (119904 120578 120577) =

1199030 + 119906 1198901 + V 1198902 + 119908 1198903 + 120578997888rarr 2 + 120577



(119904 120578 120577) =1

2(

120597997888rarr

120597120585119894sdot120597997888rarr

120597120585119895minus

120597 119903

120597120585119894sdot120597 119903

120597120585119895)

120597 119903

120597120585119894otimes

120597 119903

120597120585119895 (A9)




119881119909 = V sdot997888rarr 1 = int

119860

120590119909119909d120578 d120577

= 119864119860(1015840+V10158402

2+

10158402

2+ 1198962

1198601198962

1)


119860

120590119909119909120577 d120578 d120577 = 1198641198681205781198962

119872= sdot

997888rarr 3 = int

119860

120590119909119909120578 d120578 d120577 = 1198641198681205771198963

119872 = sdot997888rarr 1 = int

119860

(120590119909120578120578 minus 120590119909120577120577) d120578 d120577

= 1198661198691198961 + 1198811199091198962

1198601198961

(A10)


119868120578 = int119860

1205772d120578 d120577

119869 = int119860

(1205782+ 1205772) d120578 d120577

119868120577 = int119860

1205782d120578 d120577

1198962

119860=

1

119860int119860

(1205782+ 1205772) d120578 d120577

(A11)


997888rarr 1 + 1198962

997888rarr 2 +

1198963




1015840+ V101584022+

101584022 where

1015840 V1015840 and 1015840 are the


K0 = (dAd119904)A119879






= minusint119860

120588 d120578 d120577

= minusint119860

120588 119903119904 times d120578 d120577(A13)


119908 1198903)



A120597 (A119879120579)

120597119904= k (A14)

and hence

120579 (119904) = Aint

119904

0

A119879k d119904 (A15)


A120597A119879u120597119904

=

120598119909119909 minus 119891nl (119906 V 119908)

1205793

minus1205792

(A16)






119879 =

984858119862119897120572

119888

2[minus119880119875119880119879 +

119888

2120596119880119879 minus

119888

4119875 + (

119888

4)

2

]

119878 =

984858119862119897120572

119888

2[1198802

119875minus

119888

2120596119880119875 minus

1198621198890

119862119897120572

1198802

119879]

(A17)


119872120601 = minus

984858119862119897120572

1198883

32(120596119880119879 minus 119875 +

3119888

8) (A18)


120572


0





m119897 = min + maer (A19)




1198961 (119904 119905) =

1198731

sum

119899=1

120572120601

119899(119905) 1205951206011015840

119899(119904) (A20)

1198962 (119904 119905) =

1198732

sum

119899=1

120572119908

119899(119905) 12059511990810158401015840

119899(119904) (A21)

1198963 (119904 119905) =

1198733

sum

119899=1

120572V119899(119905) 120595

V10158401015840119899

(119904) (A22)

where120595120601

119899 120595V119899 and120595

119908





119899

and 120595119908




int

119877

0

m119879119897Ψ d119904 = int

119877

0

(m119879in + m119879aer)Ψ d119904 (A23)

where

Ψ =

1205951206011015840

119899

12059511990810158401015840

119899

120595V10158401015840119899

(A24)


Nomenclature


21198774 (119879 =






Competing Interests


References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






















1198641198681205771198980Ω21198774 003010 003514 002948 002570

1198661198691198980Ω21198774 000147 000104 000113 000188

1198981198980

10 1053 1188 1033Λ119860[deg] 00 424 minus44 minus41

Λ119878[deg] 00 1929 237 182


∘le Λ 119878 le 30


∘respectively











= (1198652

119909+ 1198652

119910+ 1198652

119911)12

+ (1198722

119909+ 1198722

119910+ 1198722

119911)12

(1)




0

1

2

3

4

5

6

BaselineOptimal

Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx


BaselineOptimal

MzMyMxFzFyFx0

2

4

6

8

10

12

14

16

Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)










Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

1

2

3

4

5

6

7

8

9




MzMyMxFzFyFx

1r

ev2

rev

bla

de lo

ads (times10minus4)

0

05

1

15

2

25

3

35

4



BaselineOptimal

Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

2

4

6

8

10

12

14

16

18

20










minus002

0

002

004

006

008

01

012

Non

dim

ensio

nal s

ectio

nal l

ift


minus002

0

002

004

006

008

01

012

014

Non

dim

ensio

nal s

ectio

nal l

ift






=

2

sum

119894=1

[(1198652

119909+ 1198652

119910+ 1198652

119911)12

]119894

+ [(1198722

119909+ 1198722

119910+ 1198722

119911)12

]119894

(2)









Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

1

2

3

4

5

6

7

8

9




MzMyMxFzFyFx

1r

ev2

rev

bla

de lo

ads (times10minus4)

0

05

1

15

2

25

3

35

4




Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

2

4

6

8

10

12

14

16

18

20




MzMyMxFzFyFx

1r

ev2

rev

bla

de lo

ads (times10minus4)

0

05

1

15

2

25

3

35

4








BaselineOptimal

Hub

load

s mag

nitu

de in

dex

(times103)

05

1

15

2

25

3

35

015 02 025 0301Advance ratio 120583


Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

BaselineOptimal

0

2

4

6

8

10

12

14

MzMyMxFzFyFx



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

2

4

6

8

10

12

14

16

18

20

MzMyMxFzFyFx



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

1

2

3

4

5

6

7

8

9

MzMyMxFzFyFx










Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

2

4

6

8

10

12

14

16

18

20

MzMyMxFzFyFx




Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

1

2

3

4

5

6

7

8

9

MzMyMxFzFyFx



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

2

4

6

8

10

12

14

16

18

20



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

1

2

3

4

5

6

7

8

9












TPSsurrogate

NNsurrogate

LIN-NNsurrogate

1198641198681205781198980Ω21198774 001060 000898 000856 000910 000857

1198641198681205771198980Ω21198774 003010 002570 002940 002557 002664

1198661198691198980Ω21198774 000147 000188 000187 000180 000188

1198981198980


Λ119878[deg] 00 182 187 201 177

OASPL (dB)

minus10

minus5

0

5

10

y(m

)


x (m)

85

88

91

94

97

100

103

106

109

112

115










OASPL (dB)


x (m)

828588919497100103106109112115118

minus10

minus5

0

5

10

y(m

)


OASPL (dB)

85

88

91

94

97

100

103

106

109

112

115


x (m)

minus10

minus5

0

5

10

y(m

)



OASPL (dB)

828588919497100103106109112115118


x (m)

minus10

minus5

0

5

10

y(m

)

(a) Baseline

OASPL (dB)

minus10

minus5

0

5

10

y(m

)


x (m)

87

90

93

96

99

102

105

108

111

114

117




Appendix








e3e2 e1

i3

r0 r

r

r1

u

r0

r1

u0

Ω

s

i2

i1

1e

e2e3






997888rarr 2

997888rarr 3) with










119890119894 = A (119904) 119894119894 (A1)




997888rarr 119894 = T (119904) 119890119894 (A2)



997888rarr 119894 = Λ (119904) 119894119894 (A3)

where Λ = TA


dVd119904

+ = 0

dd119904

+997888rarr 1 times V + = 0

(A4)





dd119904

(A119879k119897) + A119879p119897 = 0 (A5)

dd119904

(A119879m119897) minus A119879H119897k119897 + A119879q119897 = 0 (A6)


997888rarr 1


k119897 (119904) = Aint

119877

119904

A119879p119897d (A7)


m119897 (119904)

= Aint

119877

119904


119877

119904

(A119879H119897Aint

119877

A119879p119897d) d

(A8)




997888rarr (119904 120578 120577) =

1199030 + 119906 1198901 + V 1198902 + 119908 1198903 + 120578997888rarr 2 + 120577



(119904 120578 120577) =1

2(

120597997888rarr

120597120585119894sdot120597997888rarr

120597120585119895minus

120597 119903

120597120585119894sdot120597 119903

120597120585119895)

120597 119903

120597120585119894otimes

120597 119903

120597120585119895 (A9)




119881119909 = V sdot997888rarr 1 = int

119860

120590119909119909d120578 d120577

= 119864119860(1015840+V10158402

2+

10158402

2+ 1198962

1198601198962

1)


119860

120590119909119909120577 d120578 d120577 = 1198641198681205781198962

119872= sdot

997888rarr 3 = int

119860

120590119909119909120578 d120578 d120577 = 1198641198681205771198963

119872 = sdot997888rarr 1 = int

119860

(120590119909120578120578 minus 120590119909120577120577) d120578 d120577

= 1198661198691198961 + 1198811199091198962

1198601198961

(A10)


119868120578 = int119860

1205772d120578 d120577

119869 = int119860

(1205782+ 1205772) d120578 d120577

119868120577 = int119860

1205782d120578 d120577

1198962

119860=

1

119860int119860

(1205782+ 1205772) d120578 d120577

(A11)


997888rarr 1 + 1198962

997888rarr 2 +

1198963




1015840+ V101584022+

101584022 where

1015840 V1015840 and 1015840 are the


K0 = (dAd119904)A119879






= minusint119860

120588 d120578 d120577

= minusint119860

120588 119903119904 times d120578 d120577(A13)


119908 1198903)



A120597 (A119879120579)

120597119904= k (A14)

and hence

120579 (119904) = Aint

119904

0

A119879k d119904 (A15)


A120597A119879u120597119904

=

120598119909119909 minus 119891nl (119906 V 119908)

1205793

minus1205792

(A16)






119879 =

984858119862119897120572

119888

2[minus119880119875119880119879 +

119888

2120596119880119879 minus

119888

4119875 + (

119888

4)

2

]

119878 =

984858119862119897120572

119888

2[1198802

119875minus

119888

2120596119880119875 minus

1198621198890

119862119897120572

1198802

119879]

(A17)


119872120601 = minus

984858119862119897120572

1198883

32(120596119880119879 minus 119875 +

3119888

8) (A18)


120572


0





m119897 = min + maer (A19)




1198961 (119904 119905) =

1198731

sum

119899=1

120572120601

119899(119905) 1205951206011015840

119899(119904) (A20)

1198962 (119904 119905) =

1198732

sum

119899=1

120572119908

119899(119905) 12059511990810158401015840

119899(119904) (A21)

1198963 (119904 119905) =

1198733

sum

119899=1

120572V119899(119905) 120595

V10158401015840119899

(119904) (A22)

where120595120601

119899 120595V119899 and120595

119908





119899

and 120595119908




int

119877

0

m119879119897Ψ d119904 = int

119877

0

(m119879in + m119879aer)Ψ d119904 (A23)

where

Ψ =

1205951206011015840

119899

12059511990810158401015840

119899

120595V10158401015840119899

(A24)


Nomenclature


21198774 (119879 =






Competing Interests


References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

1

2

3

4

5

6

BaselineOptimal

Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx


BaselineOptimal

MzMyMxFzFyFx0

2

4

6

8

10

12

14

16

Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)










Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

1

2

3

4

5

6

7

8

9




MzMyMxFzFyFx

1r

ev2

rev

bla

de lo

ads (times10minus4)

0

05

1

15

2

25

3

35

4



BaselineOptimal

Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

2

4

6

8

10

12

14

16

18

20










minus002

0

002

004

006

008

01

012

Non

dim

ensio

nal s

ectio

nal l

ift


minus002

0

002

004

006

008

01

012

014

Non

dim

ensio

nal s

ectio

nal l

ift






=

2

sum

119894=1

[(1198652

119909+ 1198652

119910+ 1198652

119911)12

]119894

+ [(1198722

119909+ 1198722

119910+ 1198722

119911)12

]119894

(2)









Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

1

2

3

4

5

6

7

8

9




MzMyMxFzFyFx

1r

ev2

rev

bla

de lo

ads (times10minus4)

0

05

1

15

2

25

3

35

4




Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

2

4

6

8

10

12

14

16

18

20




MzMyMxFzFyFx

1r

ev2

rev

bla

de lo

ads (times10minus4)

0

05

1

15

2

25

3

35

4








BaselineOptimal

Hub

load

s mag

nitu

de in

dex

(times103)

05

1

15

2

25

3

35

015 02 025 0301Advance ratio 120583


Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

BaselineOptimal

0

2

4

6

8

10

12

14

MzMyMxFzFyFx



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

2

4

6

8

10

12

14

16

18

20

MzMyMxFzFyFx



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

1

2

3

4

5

6

7

8

9

MzMyMxFzFyFx










Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

2

4

6

8

10

12

14

16

18

20

MzMyMxFzFyFx




Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

1

2

3

4

5

6

7

8

9

MzMyMxFzFyFx



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

2

4

6

8

10

12

14

16

18

20



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

1

2

3

4

5

6

7

8

9












TPSsurrogate

NNsurrogate

LIN-NNsurrogate

1198641198681205781198980Ω21198774 001060 000898 000856 000910 000857

1198641198681205771198980Ω21198774 003010 002570 002940 002557 002664

1198661198691198980Ω21198774 000147 000188 000187 000180 000188

1198981198980


Λ119878[deg] 00 182 187 201 177

OASPL (dB)

minus10

minus5

0

5

10

y(m

)


x (m)

85

88

91

94

97

100

103

106

109

112

115










OASPL (dB)


x (m)

828588919497100103106109112115118

minus10

minus5

0

5

10

y(m

)


OASPL (dB)

85

88

91

94

97

100

103

106

109

112

115


x (m)

minus10

minus5

0

5

10

y(m

)



OASPL (dB)

828588919497100103106109112115118


x (m)

minus10

minus5

0

5

10

y(m

)

(a) Baseline

OASPL (dB)

minus10

minus5

0

5

10

y(m

)


x (m)

87

90

93

96

99

102

105

108

111

114

117




Appendix








e3e2 e1

i3

r0 r

r

r1

u

r0

r1

u0

Ω

s

i2

i1

1e

e2e3






997888rarr 2

997888rarr 3) with










119890119894 = A (119904) 119894119894 (A1)




997888rarr 119894 = T (119904) 119890119894 (A2)



997888rarr 119894 = Λ (119904) 119894119894 (A3)

where Λ = TA


dVd119904

+ = 0

dd119904

+997888rarr 1 times V + = 0

(A4)





dd119904

(A119879k119897) + A119879p119897 = 0 (A5)

dd119904

(A119879m119897) minus A119879H119897k119897 + A119879q119897 = 0 (A6)


997888rarr 1


k119897 (119904) = Aint

119877

119904

A119879p119897d (A7)


m119897 (119904)

= Aint

119877

119904


119877

119904

(A119879H119897Aint

119877

A119879p119897d) d

(A8)




997888rarr (119904 120578 120577) =

1199030 + 119906 1198901 + V 1198902 + 119908 1198903 + 120578997888rarr 2 + 120577



(119904 120578 120577) =1

2(

120597997888rarr

120597120585119894sdot120597997888rarr

120597120585119895minus

120597 119903

120597120585119894sdot120597 119903

120597120585119895)

120597 119903

120597120585119894otimes

120597 119903

120597120585119895 (A9)




119881119909 = V sdot997888rarr 1 = int

119860

120590119909119909d120578 d120577

= 119864119860(1015840+V10158402

2+

10158402

2+ 1198962

1198601198962

1)


119860

120590119909119909120577 d120578 d120577 = 1198641198681205781198962

119872= sdot

997888rarr 3 = int

119860

120590119909119909120578 d120578 d120577 = 1198641198681205771198963

119872 = sdot997888rarr 1 = int

119860

(120590119909120578120578 minus 120590119909120577120577) d120578 d120577

= 1198661198691198961 + 1198811199091198962

1198601198961

(A10)


119868120578 = int119860

1205772d120578 d120577

119869 = int119860

(1205782+ 1205772) d120578 d120577

119868120577 = int119860

1205782d120578 d120577

1198962

119860=

1

119860int119860

(1205782+ 1205772) d120578 d120577

(A11)


997888rarr 1 + 1198962

997888rarr 2 +

1198963




1015840+ V101584022+

101584022 where

1015840 V1015840 and 1015840 are the


K0 = (dAd119904)A119879






= minusint119860

120588 d120578 d120577

= minusint119860

120588 119903119904 times d120578 d120577(A13)


119908 1198903)



A120597 (A119879120579)

120597119904= k (A14)

and hence

120579 (119904) = Aint

119904

0

A119879k d119904 (A15)


A120597A119879u120597119904

=

120598119909119909 minus 119891nl (119906 V 119908)

1205793

minus1205792

(A16)






119879 =

984858119862119897120572

119888

2[minus119880119875119880119879 +

119888

2120596119880119879 minus

119888

4119875 + (

119888

4)

2

]

119878 =

984858119862119897120572

119888

2[1198802

119875minus

119888

2120596119880119875 minus

1198621198890

119862119897120572

1198802

119879]

(A17)


119872120601 = minus

984858119862119897120572

1198883

32(120596119880119879 minus 119875 +

3119888

8) (A18)


120572


0





m119897 = min + maer (A19)




1198961 (119904 119905) =

1198731

sum

119899=1

120572120601

119899(119905) 1205951206011015840

119899(119904) (A20)

1198962 (119904 119905) =

1198732

sum

119899=1

120572119908

119899(119905) 12059511990810158401015840

119899(119904) (A21)

1198963 (119904 119905) =

1198733

sum

119899=1

120572V119899(119905) 120595

V10158401015840119899

(119904) (A22)

where120595120601

119899 120595V119899 and120595

119908





119899

and 120595119908




int

119877

0

m119879119897Ψ d119904 = int

119877

0

(m119879in + m119879aer)Ψ d119904 (A23)

where

Ψ =

1205951206011015840

119899

12059511990810158401015840

119899

120595V10158401015840119899

(A24)


Nomenclature


21198774 (119879 =






Competing Interests


References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

1

2

3

4

5

6

7

8

9




MzMyMxFzFyFx

1r

ev2

rev

bla

de lo

ads (times10minus4)

0

05

1

15

2

25

3

35

4



BaselineOptimal

Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

2

4

6

8

10

12

14

16

18

20










minus002

0

002

004

006

008

01

012

Non

dim

ensio

nal s

ectio

nal l

ift


minus002

0

002

004

006

008

01

012

014

Non

dim

ensio

nal s

ectio

nal l

ift






=

2

sum

119894=1

[(1198652

119909+ 1198652

119910+ 1198652

119911)12

]119894

+ [(1198722

119909+ 1198722

119910+ 1198722

119911)12

]119894

(2)









Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

1

2

3

4

5

6

7

8

9




MzMyMxFzFyFx

1r

ev2

rev

bla

de lo

ads (times10minus4)

0

05

1

15

2

25

3

35

4




Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

2

4

6

8

10

12

14

16

18

20




MzMyMxFzFyFx

1r

ev2

rev

bla

de lo

ads (times10minus4)

0

05

1

15

2

25

3

35

4








BaselineOptimal

Hub

load

s mag

nitu

de in

dex

(times103)

05

1

15

2

25

3

35

015 02 025 0301Advance ratio 120583


Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

BaselineOptimal

0

2

4

6

8

10

12

14

MzMyMxFzFyFx



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

2

4

6

8

10

12

14

16

18

20

MzMyMxFzFyFx



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

1

2

3

4

5

6

7

8

9

MzMyMxFzFyFx










Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

2

4

6

8

10

12

14

16

18

20

MzMyMxFzFyFx




Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

1

2

3

4

5

6

7

8

9

MzMyMxFzFyFx



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

2

4

6

8

10

12

14

16

18

20



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

1

2

3

4

5

6

7

8

9












TPSsurrogate

NNsurrogate

LIN-NNsurrogate

1198641198681205781198980Ω21198774 001060 000898 000856 000910 000857

1198641198681205771198980Ω21198774 003010 002570 002940 002557 002664

1198661198691198980Ω21198774 000147 000188 000187 000180 000188

1198981198980


Λ119878[deg] 00 182 187 201 177

OASPL (dB)

minus10

minus5

0

5

10

y(m

)


x (m)

85

88

91

94

97

100

103

106

109

112

115










OASPL (dB)


x (m)

828588919497100103106109112115118

minus10

minus5

0

5

10

y(m

)


OASPL (dB)

85

88

91

94

97

100

103

106

109

112

115


x (m)

minus10

minus5

0

5

10

y(m

)



OASPL (dB)

828588919497100103106109112115118


x (m)

minus10

minus5

0

5

10

y(m

)

(a) Baseline

OASPL (dB)

minus10

minus5

0

5

10

y(m

)


x (m)

87

90

93

96

99

102

105

108

111

114

117




Appendix








e3e2 e1

i3

r0 r

r

r1

u

r0

r1

u0

Ω

s

i2

i1

1e

e2e3






997888rarr 2

997888rarr 3) with










119890119894 = A (119904) 119894119894 (A1)




997888rarr 119894 = T (119904) 119890119894 (A2)



997888rarr 119894 = Λ (119904) 119894119894 (A3)

where Λ = TA


dVd119904

+ = 0

dd119904

+997888rarr 1 times V + = 0

(A4)





dd119904

(A119879k119897) + A119879p119897 = 0 (A5)

dd119904

(A119879m119897) minus A119879H119897k119897 + A119879q119897 = 0 (A6)


997888rarr 1


k119897 (119904) = Aint

119877

119904

A119879p119897d (A7)


m119897 (119904)

= Aint

119877

119904


119877

119904

(A119879H119897Aint

119877

A119879p119897d) d

(A8)




997888rarr (119904 120578 120577) =

1199030 + 119906 1198901 + V 1198902 + 119908 1198903 + 120578997888rarr 2 + 120577



(119904 120578 120577) =1

2(

120597997888rarr

120597120585119894sdot120597997888rarr

120597120585119895minus

120597 119903

120597120585119894sdot120597 119903

120597120585119895)

120597 119903

120597120585119894otimes

120597 119903

120597120585119895 (A9)




119881119909 = V sdot997888rarr 1 = int

119860

120590119909119909d120578 d120577

= 119864119860(1015840+V10158402

2+

10158402

2+ 1198962

1198601198962

1)


119860

120590119909119909120577 d120578 d120577 = 1198641198681205781198962

119872= sdot

997888rarr 3 = int

119860

120590119909119909120578 d120578 d120577 = 1198641198681205771198963

119872 = sdot997888rarr 1 = int

119860

(120590119909120578120578 minus 120590119909120577120577) d120578 d120577

= 1198661198691198961 + 1198811199091198962

1198601198961

(A10)


119868120578 = int119860

1205772d120578 d120577

119869 = int119860

(1205782+ 1205772) d120578 d120577

119868120577 = int119860

1205782d120578 d120577

1198962

119860=

1

119860int119860

(1205782+ 1205772) d120578 d120577

(A11)


997888rarr 1 + 1198962

997888rarr 2 +

1198963




1015840+ V101584022+

101584022 where

1015840 V1015840 and 1015840 are the


K0 = (dAd119904)A119879






= minusint119860

120588 d120578 d120577

= minusint119860

120588 119903119904 times d120578 d120577(A13)


119908 1198903)



A120597 (A119879120579)

120597119904= k (A14)

and hence

120579 (119904) = Aint

119904

0

A119879k d119904 (A15)


A120597A119879u120597119904

=

120598119909119909 minus 119891nl (119906 V 119908)

1205793

minus1205792

(A16)






119879 =

984858119862119897120572

119888

2[minus119880119875119880119879 +

119888

2120596119880119879 minus

119888

4119875 + (

119888

4)

2

]

119878 =

984858119862119897120572

119888

2[1198802

119875minus

119888

2120596119880119875 minus

1198621198890

119862119897120572

1198802

119879]

(A17)


119872120601 = minus

984858119862119897120572

1198883

32(120596119880119879 minus 119875 +

3119888

8) (A18)


120572


0





m119897 = min + maer (A19)




1198961 (119904 119905) =

1198731

sum

119899=1

120572120601

119899(119905) 1205951206011015840

119899(119904) (A20)

1198962 (119904 119905) =

1198732

sum

119899=1

120572119908

119899(119905) 12059511990810158401015840

119899(119904) (A21)

1198963 (119904 119905) =

1198733

sum

119899=1

120572V119899(119905) 120595

V10158401015840119899

(119904) (A22)

where120595120601

119899 120595V119899 and120595

119908





119899

and 120595119908




int

119877

0

m119879119897Ψ d119904 = int

119877

0

(m119879in + m119879aer)Ψ d119904 (A23)

where

Ψ =

1205951206011015840

119899

12059511990810158401015840

119899

120595V10158401015840119899

(A24)


Nomenclature


21198774 (119879 =






Competing Interests


References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus002

0

002

004

006

008

01

012

Non

dim

ensio

nal s

ectio

nal l

ift


minus002

0

002

004

006

008

01

012

014

Non

dim

ensio

nal s

ectio

nal l

ift






=

2

sum

119894=1

[(1198652

119909+ 1198652

119910+ 1198652

119911)12

]119894

+ [(1198722

119909+ 1198722

119910+ 1198722

119911)12

]119894

(2)









Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

1

2

3

4

5

6

7

8

9




MzMyMxFzFyFx

1r

ev2

rev

bla

de lo

ads (times10minus4)

0

05

1

15

2

25

3

35

4




Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

2

4

6

8

10

12

14

16

18

20




MzMyMxFzFyFx

1r

ev2

rev

bla

de lo

ads (times10minus4)

0

05

1

15

2

25

3

35

4








BaselineOptimal

Hub

load

s mag

nitu

de in

dex

(times103)

05

1

15

2

25

3

35

015 02 025 0301Advance ratio 120583


Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

BaselineOptimal

0

2

4

6

8

10

12

14

MzMyMxFzFyFx



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

2

4

6

8

10

12

14

16

18

20

MzMyMxFzFyFx



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

1

2

3

4

5

6

7

8

9

MzMyMxFzFyFx










Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

2

4

6

8

10

12

14

16

18

20

MzMyMxFzFyFx




Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

1

2

3

4

5

6

7

8

9

MzMyMxFzFyFx



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

2

4

6

8

10

12

14

16

18

20



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

1

2

3

4

5

6

7

8

9












TPSsurrogate

NNsurrogate

LIN-NNsurrogate

1198641198681205781198980Ω21198774 001060 000898 000856 000910 000857

1198641198681205771198980Ω21198774 003010 002570 002940 002557 002664

1198661198691198980Ω21198774 000147 000188 000187 000180 000188

1198981198980


Λ119878[deg] 00 182 187 201 177

OASPL (dB)

minus10

minus5

0

5

10

y(m

)


x (m)

85

88

91

94

97

100

103

106

109

112

115










OASPL (dB)


x (m)

828588919497100103106109112115118

minus10

minus5

0

5

10

y(m

)


OASPL (dB)

85

88

91

94

97

100

103

106

109

112

115


x (m)

minus10

minus5

0

5

10

y(m

)



OASPL (dB)

828588919497100103106109112115118


x (m)

minus10

minus5

0

5

10

y(m

)

(a) Baseline

OASPL (dB)

minus10

minus5

0

5

10

y(m

)


x (m)

87

90

93

96

99

102

105

108

111

114

117




Appendix








e3e2 e1

i3

r0 r

r

r1

u

r0

r1

u0

Ω

s

i2

i1

1e

e2e3






997888rarr 2

997888rarr 3) with










119890119894 = A (119904) 119894119894 (A1)




997888rarr 119894 = T (119904) 119890119894 (A2)



997888rarr 119894 = Λ (119904) 119894119894 (A3)

where Λ = TA


dVd119904

+ = 0

dd119904

+997888rarr 1 times V + = 0

(A4)





dd119904

(A119879k119897) + A119879p119897 = 0 (A5)

dd119904

(A119879m119897) minus A119879H119897k119897 + A119879q119897 = 0 (A6)


997888rarr 1


k119897 (119904) = Aint

119877

119904

A119879p119897d (A7)


m119897 (119904)

= Aint

119877

119904


119877

119904

(A119879H119897Aint

119877

A119879p119897d) d

(A8)




997888rarr (119904 120578 120577) =

1199030 + 119906 1198901 + V 1198902 + 119908 1198903 + 120578997888rarr 2 + 120577



(119904 120578 120577) =1

2(

120597997888rarr

120597120585119894sdot120597997888rarr

120597120585119895minus

120597 119903

120597120585119894sdot120597 119903

120597120585119895)

120597 119903

120597120585119894otimes

120597 119903

120597120585119895 (A9)




119881119909 = V sdot997888rarr 1 = int

119860

120590119909119909d120578 d120577

= 119864119860(1015840+V10158402

2+

10158402

2+ 1198962

1198601198962

1)


119860

120590119909119909120577 d120578 d120577 = 1198641198681205781198962

119872= sdot

997888rarr 3 = int

119860

120590119909119909120578 d120578 d120577 = 1198641198681205771198963

119872 = sdot997888rarr 1 = int

119860

(120590119909120578120578 minus 120590119909120577120577) d120578 d120577

= 1198661198691198961 + 1198811199091198962

1198601198961

(A10)


119868120578 = int119860

1205772d120578 d120577

119869 = int119860

(1205782+ 1205772) d120578 d120577

119868120577 = int119860

1205782d120578 d120577

1198962

119860=

1

119860int119860

(1205782+ 1205772) d120578 d120577

(A11)


997888rarr 1 + 1198962

997888rarr 2 +

1198963




1015840+ V101584022+

101584022 where

1015840 V1015840 and 1015840 are the


K0 = (dAd119904)A119879






= minusint119860

120588 d120578 d120577

= minusint119860

120588 119903119904 times d120578 d120577(A13)


119908 1198903)



A120597 (A119879120579)

120597119904= k (A14)

and hence

120579 (119904) = Aint

119904

0

A119879k d119904 (A15)


A120597A119879u120597119904

=

120598119909119909 minus 119891nl (119906 V 119908)

1205793

minus1205792

(A16)






119879 =

984858119862119897120572

119888

2[minus119880119875119880119879 +

119888

2120596119880119879 minus

119888

4119875 + (

119888

4)

2

]

119878 =

984858119862119897120572

119888

2[1198802

119875minus

119888

2120596119880119875 minus

1198621198890

119862119897120572

1198802

119879]

(A17)


119872120601 = minus

984858119862119897120572

1198883

32(120596119880119879 minus 119875 +

3119888

8) (A18)


120572


0





m119897 = min + maer (A19)




1198961 (119904 119905) =

1198731

sum

119899=1

120572120601

119899(119905) 1205951206011015840

119899(119904) (A20)

1198962 (119904 119905) =

1198732

sum

119899=1

120572119908

119899(119905) 12059511990810158401015840

119899(119904) (A21)

1198963 (119904 119905) =

1198733

sum

119899=1

120572V119899(119905) 120595

V10158401015840119899

(119904) (A22)

where120595120601

119899 120595V119899 and120595

119908





119899

and 120595119908




int

119877

0

m119879119897Ψ d119904 = int

119877

0

(m119879in + m119879aer)Ψ d119904 (A23)

where

Ψ =

1205951206011015840

119899

12059511990810158401015840

119899

120595V10158401015840119899

(A24)


Nomenclature


21198774 (119879 =






Competing Interests


References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

1

2

3

4

5

6

7

8

9




MzMyMxFzFyFx

1r

ev2

rev

bla

de lo

ads (times10minus4)

0

05

1

15

2

25

3

35

4




Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

2

4

6

8

10

12

14

16

18

20




MzMyMxFzFyFx

1r

ev2

rev

bla

de lo

ads (times10minus4)

0

05

1

15

2

25

3

35

4








BaselineOptimal

Hub

load

s mag

nitu

de in

dex

(times103)

05

1

15

2

25

3

35

015 02 025 0301Advance ratio 120583


Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

BaselineOptimal

0

2

4

6

8

10

12

14

MzMyMxFzFyFx



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

2

4

6

8

10

12

14

16

18

20

MzMyMxFzFyFx



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

1

2

3

4

5

6

7

8

9

MzMyMxFzFyFx










Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

2

4

6

8

10

12

14

16

18

20

MzMyMxFzFyFx




Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

1

2

3

4

5

6

7

8

9

MzMyMxFzFyFx



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

2

4

6

8

10

12

14

16

18

20



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

1

2

3

4

5

6

7

8

9












TPSsurrogate

NNsurrogate

LIN-NNsurrogate

1198641198681205781198980Ω21198774 001060 000898 000856 000910 000857

1198641198681205771198980Ω21198774 003010 002570 002940 002557 002664

1198661198691198980Ω21198774 000147 000188 000187 000180 000188

1198981198980


Λ119878[deg] 00 182 187 201 177

OASPL (dB)

minus10

minus5

0

5

10

y(m

)


x (m)

85

88

91

94

97

100

103

106

109

112

115










OASPL (dB)


x (m)

828588919497100103106109112115118

minus10

minus5

0

5

10

y(m

)


OASPL (dB)

85

88

91

94

97

100

103

106

109

112

115


x (m)

minus10

minus5

0

5

10

y(m

)



OASPL (dB)

828588919497100103106109112115118


x (m)

minus10

minus5

0

5

10

y(m

)

(a) Baseline

OASPL (dB)

minus10

minus5

0

5

10

y(m

)


x (m)

87

90

93

96

99

102

105

108

111

114

117




Appendix








e3e2 e1

i3

r0 r

r

r1

u

r0

r1

u0

Ω

s

i2

i1

1e

e2e3






997888rarr 2

997888rarr 3) with










119890119894 = A (119904) 119894119894 (A1)




997888rarr 119894 = T (119904) 119890119894 (A2)



997888rarr 119894 = Λ (119904) 119894119894 (A3)

where Λ = TA


dVd119904

+ = 0

dd119904

+997888rarr 1 times V + = 0

(A4)





dd119904

(A119879k119897) + A119879p119897 = 0 (A5)

dd119904

(A119879m119897) minus A119879H119897k119897 + A119879q119897 = 0 (A6)


997888rarr 1


k119897 (119904) = Aint

119877

119904

A119879p119897d (A7)


m119897 (119904)

= Aint

119877

119904


119877

119904

(A119879H119897Aint

119877

A119879p119897d) d

(A8)




997888rarr (119904 120578 120577) =

1199030 + 119906 1198901 + V 1198902 + 119908 1198903 + 120578997888rarr 2 + 120577



(119904 120578 120577) =1

2(

120597997888rarr

120597120585119894sdot120597997888rarr

120597120585119895minus

120597 119903

120597120585119894sdot120597 119903

120597120585119895)

120597 119903

120597120585119894otimes

120597 119903

120597120585119895 (A9)




119881119909 = V sdot997888rarr 1 = int

119860

120590119909119909d120578 d120577

= 119864119860(1015840+V10158402

2+

10158402

2+ 1198962

1198601198962

1)


119860

120590119909119909120577 d120578 d120577 = 1198641198681205781198962

119872= sdot

997888rarr 3 = int

119860

120590119909119909120578 d120578 d120577 = 1198641198681205771198963

119872 = sdot997888rarr 1 = int

119860

(120590119909120578120578 minus 120590119909120577120577) d120578 d120577

= 1198661198691198961 + 1198811199091198962

1198601198961

(A10)


119868120578 = int119860

1205772d120578 d120577

119869 = int119860

(1205782+ 1205772) d120578 d120577

119868120577 = int119860

1205782d120578 d120577

1198962

119860=

1

119860int119860

(1205782+ 1205772) d120578 d120577

(A11)


997888rarr 1 + 1198962

997888rarr 2 +

1198963




1015840+ V101584022+

101584022 where

1015840 V1015840 and 1015840 are the


K0 = (dAd119904)A119879






= minusint119860

120588 d120578 d120577

= minusint119860

120588 119903119904 times d120578 d120577(A13)


119908 1198903)



A120597 (A119879120579)

120597119904= k (A14)

and hence

120579 (119904) = Aint

119904

0

A119879k d119904 (A15)


A120597A119879u120597119904

=

120598119909119909 minus 119891nl (119906 V 119908)

1205793

minus1205792

(A16)






119879 =

984858119862119897120572

119888

2[minus119880119875119880119879 +

119888

2120596119880119879 minus

119888

4119875 + (

119888

4)

2

]

119878 =

984858119862119897120572

119888

2[1198802

119875minus

119888

2120596119880119875 minus

1198621198890

119862119897120572

1198802

119879]

(A17)


119872120601 = minus

984858119862119897120572

1198883

32(120596119880119879 minus 119875 +

3119888

8) (A18)


120572


0





m119897 = min + maer (A19)




1198961 (119904 119905) =

1198731

sum

119899=1

120572120601

119899(119905) 1205951206011015840

119899(119904) (A20)

1198962 (119904 119905) =

1198732

sum

119899=1

120572119908

119899(119905) 12059511990810158401015840

119899(119904) (A21)

1198963 (119904 119905) =

1198733

sum

119899=1

120572V119899(119905) 120595

V10158401015840119899

(119904) (A22)

where120595120601

119899 120595V119899 and120595

119908





119899

and 120595119908




int

119877

0

m119879119897Ψ d119904 = int

119877

0

(m119879in + m119879aer)Ψ d119904 (A23)

where

Ψ =

1205951206011015840

119899

12059511990810158401015840

119899

120595V10158401015840119899

(A24)


Nomenclature


21198774 (119879 =






Competing Interests


References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










BaselineOptimal

Hub

load

s mag

nitu

de in

dex

(times103)

05

1

15

2

25

3

35

015 02 025 0301Advance ratio 120583


Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

BaselineOptimal

0

2

4

6

8

10

12

14

MzMyMxFzFyFx



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

2

4

6

8

10

12

14

16

18

20

MzMyMxFzFyFx



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

1

2

3

4

5

6

7

8

9

MzMyMxFzFyFx










Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

2

4

6

8

10

12

14

16

18

20

MzMyMxFzFyFx




Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

1

2

3

4

5

6

7

8

9

MzMyMxFzFyFx



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

2

4

6

8

10

12

14

16

18

20



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

1

2

3

4

5

6

7

8

9












TPSsurrogate

NNsurrogate

LIN-NNsurrogate

1198641198681205781198980Ω21198774 001060 000898 000856 000910 000857

1198641198681205771198980Ω21198774 003010 002570 002940 002557 002664

1198661198691198980Ω21198774 000147 000188 000187 000180 000188

1198981198980


Λ119878[deg] 00 182 187 201 177

OASPL (dB)

minus10

minus5

0

5

10

y(m

)


x (m)

85

88

91

94

97

100

103

106

109

112

115










OASPL (dB)


x (m)

828588919497100103106109112115118

minus10

minus5

0

5

10

y(m

)


OASPL (dB)

85

88

91

94

97

100

103

106

109

112

115


x (m)

minus10

minus5

0

5

10

y(m

)



OASPL (dB)

828588919497100103106109112115118


x (m)

minus10

minus5

0

5

10

y(m

)

(a) Baseline

OASPL (dB)

minus10

minus5

0

5

10

y(m

)


x (m)

87

90

93

96

99

102

105

108

111

114

117




Appendix








e3e2 e1

i3

r0 r

r

r1

u

r0

r1

u0

Ω

s

i2

i1

1e

e2e3






997888rarr 2

997888rarr 3) with










119890119894 = A (119904) 119894119894 (A1)




997888rarr 119894 = T (119904) 119890119894 (A2)



997888rarr 119894 = Λ (119904) 119894119894 (A3)

where Λ = TA


dVd119904

+ = 0

dd119904

+997888rarr 1 times V + = 0

(A4)





dd119904

(A119879k119897) + A119879p119897 = 0 (A5)

dd119904

(A119879m119897) minus A119879H119897k119897 + A119879q119897 = 0 (A6)


997888rarr 1


k119897 (119904) = Aint

119877

119904

A119879p119897d (A7)


m119897 (119904)

= Aint

119877

119904


119877

119904

(A119879H119897Aint

119877

A119879p119897d) d

(A8)




997888rarr (119904 120578 120577) =

1199030 + 119906 1198901 + V 1198902 + 119908 1198903 + 120578997888rarr 2 + 120577



(119904 120578 120577) =1

2(

120597997888rarr

120597120585119894sdot120597997888rarr

120597120585119895minus

120597 119903

120597120585119894sdot120597 119903

120597120585119895)

120597 119903

120597120585119894otimes

120597 119903

120597120585119895 (A9)




119881119909 = V sdot997888rarr 1 = int

119860

120590119909119909d120578 d120577

= 119864119860(1015840+V10158402

2+

10158402

2+ 1198962

1198601198962

1)


119860

120590119909119909120577 d120578 d120577 = 1198641198681205781198962

119872= sdot

997888rarr 3 = int

119860

120590119909119909120578 d120578 d120577 = 1198641198681205771198963

119872 = sdot997888rarr 1 = int

119860

(120590119909120578120578 minus 120590119909120577120577) d120578 d120577

= 1198661198691198961 + 1198811199091198962

1198601198961

(A10)


119868120578 = int119860

1205772d120578 d120577

119869 = int119860

(1205782+ 1205772) d120578 d120577

119868120577 = int119860

1205782d120578 d120577

1198962

119860=

1

119860int119860

(1205782+ 1205772) d120578 d120577

(A11)


997888rarr 1 + 1198962

997888rarr 2 +

1198963




1015840+ V101584022+

101584022 where

1015840 V1015840 and 1015840 are the


K0 = (dAd119904)A119879






= minusint119860

120588 d120578 d120577

= minusint119860

120588 119903119904 times d120578 d120577(A13)


119908 1198903)



A120597 (A119879120579)

120597119904= k (A14)

and hence

120579 (119904) = Aint

119904

0

A119879k d119904 (A15)


A120597A119879u120597119904

=

120598119909119909 minus 119891nl (119906 V 119908)

1205793

minus1205792

(A16)






119879 =

984858119862119897120572

119888

2[minus119880119875119880119879 +

119888

2120596119880119879 minus

119888

4119875 + (

119888

4)

2

]

119878 =

984858119862119897120572

119888

2[1198802

119875minus

119888

2120596119880119875 minus

1198621198890

119862119897120572

1198802

119879]

(A17)


119872120601 = minus

984858119862119897120572

1198883

32(120596119880119879 minus 119875 +

3119888

8) (A18)


120572


0





m119897 = min + maer (A19)




1198961 (119904 119905) =

1198731

sum

119899=1

120572120601

119899(119905) 1205951206011015840

119899(119904) (A20)

1198962 (119904 119905) =

1198732

sum

119899=1

120572119908

119899(119905) 12059511990810158401015840

119899(119904) (A21)

1198963 (119904 119905) =

1198733

sum

119899=1

120572V119899(119905) 120595

V10158401015840119899

(119904) (A22)

where120595120601

119899 120595V119899 and120595

119908





119899

and 120595119908




int

119877

0

m119879119897Ψ d119904 = int

119877

0

(m119879in + m119879aer)Ψ d119904 (A23)

where

Ψ =

1205951206011015840

119899

12059511990810158401015840

119899

120595V10158401015840119899

(A24)


Nomenclature


21198774 (119879 =






Competing Interests


References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

2

4

6

8

10

12

14

16

18

20

MzMyMxFzFyFx




Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

0

1

2

3

4

5

6

7

8

9

MzMyMxFzFyFx



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

2

4

6

8

10

12

14

16

18

20



Non

dim

ensio

nal4

rev

hub

load

s (times10minus4)

MzMyMxFzFyFx0

1

2

3

4

5

6

7

8

9












TPSsurrogate

NNsurrogate

LIN-NNsurrogate

1198641198681205781198980Ω21198774 001060 000898 000856 000910 000857

1198641198681205771198980Ω21198774 003010 002570 002940 002557 002664

1198661198691198980Ω21198774 000147 000188 000187 000180 000188

1198981198980


Λ119878[deg] 00 182 187 201 177

OASPL (dB)

minus10

minus5

0

5

10

y(m

)


x (m)

85

88

91

94

97

100

103

106

109

112

115










OASPL (dB)


x (m)

828588919497100103106109112115118

minus10

minus5

0

5

10

y(m

)


OASPL (dB)

85

88

91

94

97

100

103

106

109

112

115


x (m)

minus10

minus5

0

5

10

y(m

)



OASPL (dB)

828588919497100103106109112115118


x (m)

minus10

minus5

0

5

10

y(m

)

(a) Baseline

OASPL (dB)

minus10

minus5

0

5

10

y(m

)


x (m)

87

90

93

96

99

102

105

108

111

114

117




Appendix








e3e2 e1

i3

r0 r

r

r1

u

r0

r1

u0

Ω

s

i2

i1

1e

e2e3






997888rarr 2

997888rarr 3) with










119890119894 = A (119904) 119894119894 (A1)




997888rarr 119894 = T (119904) 119890119894 (A2)



997888rarr 119894 = Λ (119904) 119894119894 (A3)

where Λ = TA


dVd119904

+ = 0

dd119904

+997888rarr 1 times V + = 0

(A4)





dd119904

(A119879k119897) + A119879p119897 = 0 (A5)

dd119904

(A119879m119897) minus A119879H119897k119897 + A119879q119897 = 0 (A6)


997888rarr 1


k119897 (119904) = Aint

119877

119904

A119879p119897d (A7)


m119897 (119904)

= Aint

119877

119904


119877

119904

(A119879H119897Aint

119877

A119879p119897d) d

(A8)




997888rarr (119904 120578 120577) =

1199030 + 119906 1198901 + V 1198902 + 119908 1198903 + 120578997888rarr 2 + 120577



(119904 120578 120577) =1

2(

120597997888rarr

120597120585119894sdot120597997888rarr

120597120585119895minus

120597 119903

120597120585119894sdot120597 119903

120597120585119895)

120597 119903

120597120585119894otimes

120597 119903

120597120585119895 (A9)




119881119909 = V sdot997888rarr 1 = int

119860

120590119909119909d120578 d120577

= 119864119860(1015840+V10158402

2+

10158402

2+ 1198962

1198601198962

1)


119860

120590119909119909120577 d120578 d120577 = 1198641198681205781198962

119872= sdot

997888rarr 3 = int

119860

120590119909119909120578 d120578 d120577 = 1198641198681205771198963

119872 = sdot997888rarr 1 = int

119860

(120590119909120578120578 minus 120590119909120577120577) d120578 d120577

= 1198661198691198961 + 1198811199091198962

1198601198961

(A10)


119868120578 = int119860

1205772d120578 d120577

119869 = int119860

(1205782+ 1205772) d120578 d120577

119868120577 = int119860

1205782d120578 d120577

1198962

119860=

1

119860int119860

(1205782+ 1205772) d120578 d120577

(A11)


997888rarr 1 + 1198962

997888rarr 2 +

1198963




1015840+ V101584022+

101584022 where

1015840 V1015840 and 1015840 are the


K0 = (dAd119904)A119879






= minusint119860

120588 d120578 d120577

= minusint119860

120588 119903119904 times d120578 d120577(A13)


119908 1198903)



A120597 (A119879120579)

120597119904= k (A14)

and hence

120579 (119904) = Aint

119904

0

A119879k d119904 (A15)


A120597A119879u120597119904

=

120598119909119909 minus 119891nl (119906 V 119908)

1205793

minus1205792

(A16)






119879 =

984858119862119897120572

119888

2[minus119880119875119880119879 +

119888

2120596119880119879 minus

119888

4119875 + (

119888

4)

2

]

119878 =

984858119862119897120572

119888

2[1198802

119875minus

119888

2120596119880119875 minus

1198621198890

119862119897120572

1198802

119879]

(A17)


119872120601 = minus

984858119862119897120572

1198883

32(120596119880119879 minus 119875 +

3119888

8) (A18)


120572


0





m119897 = min + maer (A19)




1198961 (119904 119905) =

1198731

sum

119899=1

120572120601

119899(119905) 1205951206011015840

119899(119904) (A20)

1198962 (119904 119905) =

1198732

sum

119899=1

120572119908

119899(119905) 12059511990810158401015840

119899(119904) (A21)

1198963 (119904 119905) =

1198733

sum

119899=1

120572V119899(119905) 120595

V10158401015840119899

(119904) (A22)

where120595120601

119899 120595V119899 and120595

119908





119899

and 120595119908




int

119877

0

m119879119897Ψ d119904 = int

119877

0

(m119879in + m119879aer)Ψ d119904 (A23)

where

Ψ =

1205951206011015840

119899

12059511990810158401015840

119899

120595V10158401015840119899

(A24)


Nomenclature


21198774 (119879 =






Competing Interests


References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












TPSsurrogate

NNsurrogate

LIN-NNsurrogate

1198641198681205781198980Ω21198774 001060 000898 000856 000910 000857

1198641198681205771198980Ω21198774 003010 002570 002940 002557 002664

1198661198691198980Ω21198774 000147 000188 000187 000180 000188

1198981198980


Λ119878[deg] 00 182 187 201 177

OASPL (dB)

minus10

minus5

0

5

10

y(m

)


x (m)

85

88

91

94

97

100

103

106

109

112

115










OASPL (dB)


x (m)

828588919497100103106109112115118

minus10

minus5

0

5

10

y(m

)


OASPL (dB)

85

88

91

94

97

100

103

106

109

112

115


x (m)

minus10

minus5

0

5

10

y(m

)



OASPL (dB)

828588919497100103106109112115118


x (m)

minus10

minus5

0

5

10

y(m

)

(a) Baseline

OASPL (dB)

minus10

minus5

0

5

10

y(m

)


x (m)

87

90

93

96

99

102

105

108

111

114

117




Appendix








e3e2 e1

i3

r0 r

r

r1

u

r0

r1

u0

Ω

s

i2

i1

1e

e2e3






997888rarr 2

997888rarr 3) with










119890119894 = A (119904) 119894119894 (A1)




997888rarr 119894 = T (119904) 119890119894 (A2)



997888rarr 119894 = Λ (119904) 119894119894 (A3)

where Λ = TA


dVd119904

+ = 0

dd119904

+997888rarr 1 times V + = 0

(A4)





dd119904

(A119879k119897) + A119879p119897 = 0 (A5)

dd119904

(A119879m119897) minus A119879H119897k119897 + A119879q119897 = 0 (A6)


997888rarr 1


k119897 (119904) = Aint

119877

119904

A119879p119897d (A7)


m119897 (119904)

= Aint

119877

119904


119877

119904

(A119879H119897Aint

119877

A119879p119897d) d

(A8)




997888rarr (119904 120578 120577) =

1199030 + 119906 1198901 + V 1198902 + 119908 1198903 + 120578997888rarr 2 + 120577



(119904 120578 120577) =1

2(

120597997888rarr

120597120585119894sdot120597997888rarr

120597120585119895minus

120597 119903

120597120585119894sdot120597 119903

120597120585119895)

120597 119903

120597120585119894otimes

120597 119903

120597120585119895 (A9)




119881119909 = V sdot997888rarr 1 = int

119860

120590119909119909d120578 d120577

= 119864119860(1015840+V10158402

2+

10158402

2+ 1198962

1198601198962

1)


119860

120590119909119909120577 d120578 d120577 = 1198641198681205781198962

119872= sdot

997888rarr 3 = int

119860

120590119909119909120578 d120578 d120577 = 1198641198681205771198963

119872 = sdot997888rarr 1 = int

119860

(120590119909120578120578 minus 120590119909120577120577) d120578 d120577

= 1198661198691198961 + 1198811199091198962

1198601198961

(A10)


119868120578 = int119860

1205772d120578 d120577

119869 = int119860

(1205782+ 1205772) d120578 d120577

119868120577 = int119860

1205782d120578 d120577

1198962

119860=

1

119860int119860

(1205782+ 1205772) d120578 d120577

(A11)


997888rarr 1 + 1198962

997888rarr 2 +

1198963




1015840+ V101584022+

101584022 where

1015840 V1015840 and 1015840 are the


K0 = (dAd119904)A119879






= minusint119860

120588 d120578 d120577

= minusint119860

120588 119903119904 times d120578 d120577(A13)


119908 1198903)



A120597 (A119879120579)

120597119904= k (A14)

and hence

120579 (119904) = Aint

119904

0

A119879k d119904 (A15)


A120597A119879u120597119904

=

120598119909119909 minus 119891nl (119906 V 119908)

1205793

minus1205792

(A16)






119879 =

984858119862119897120572

119888

2[minus119880119875119880119879 +

119888

2120596119880119879 minus

119888

4119875 + (

119888

4)

2

]

119878 =

984858119862119897120572

119888

2[1198802

119875minus

119888

2120596119880119875 minus

1198621198890

119862119897120572

1198802

119879]

(A17)


119872120601 = minus

984858119862119897120572

1198883

32(120596119880119879 minus 119875 +

3119888

8) (A18)


120572


0





m119897 = min + maer (A19)




1198961 (119904 119905) =

1198731

sum

119899=1

120572120601

119899(119905) 1205951206011015840

119899(119904) (A20)

1198962 (119904 119905) =

1198732

sum

119899=1

120572119908

119899(119905) 12059511990810158401015840

119899(119904) (A21)

1198963 (119904 119905) =

1198733

sum

119899=1

120572V119899(119905) 120595

V10158401015840119899

(119904) (A22)

where120595120601

119899 120595V119899 and120595

119908





119899

and 120595119908




int

119877

0

m119879119897Ψ d119904 = int

119877

0

(m119879in + m119879aer)Ψ d119904 (A23)

where

Ψ =

1205951206011015840

119899

12059511990810158401015840

119899

120595V10158401015840119899

(A24)


Nomenclature


21198774 (119879 =






Competing Interests


References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










OASPL (dB)


x (m)

828588919497100103106109112115118

minus10

minus5

0

5

10

y(m

)


OASPL (dB)

85

88

91

94

97

100

103

106

109

112

115


x (m)

minus10

minus5

0

5

10

y(m

)



OASPL (dB)

828588919497100103106109112115118


x (m)

minus10

minus5

0

5

10

y(m

)

(a) Baseline

OASPL (dB)

minus10

minus5

0

5

10

y(m

)


x (m)

87

90

93

96

99

102

105

108

111

114

117




Appendix








e3e2 e1

i3

r0 r

r

r1

u

r0

r1

u0

Ω

s

i2

i1

1e

e2e3






997888rarr 2

997888rarr 3) with










119890119894 = A (119904) 119894119894 (A1)




997888rarr 119894 = T (119904) 119890119894 (A2)



997888rarr 119894 = Λ (119904) 119894119894 (A3)

where Λ = TA


dVd119904

+ = 0

dd119904

+997888rarr 1 times V + = 0

(A4)





dd119904

(A119879k119897) + A119879p119897 = 0 (A5)

dd119904

(A119879m119897) minus A119879H119897k119897 + A119879q119897 = 0 (A6)


997888rarr 1


k119897 (119904) = Aint

119877

119904

A119879p119897d (A7)


m119897 (119904)

= Aint

119877

119904


119877

119904

(A119879H119897Aint

119877

A119879p119897d) d

(A8)




997888rarr (119904 120578 120577) =

1199030 + 119906 1198901 + V 1198902 + 119908 1198903 + 120578997888rarr 2 + 120577



(119904 120578 120577) =1

2(

120597997888rarr

120597120585119894sdot120597997888rarr

120597120585119895minus

120597 119903

120597120585119894sdot120597 119903

120597120585119895)

120597 119903

120597120585119894otimes

120597 119903

120597120585119895 (A9)




119881119909 = V sdot997888rarr 1 = int

119860

120590119909119909d120578 d120577

= 119864119860(1015840+V10158402

2+

10158402

2+ 1198962

1198601198962

1)


119860

120590119909119909120577 d120578 d120577 = 1198641198681205781198962

119872= sdot

997888rarr 3 = int

119860

120590119909119909120578 d120578 d120577 = 1198641198681205771198963

119872 = sdot997888rarr 1 = int

119860

(120590119909120578120578 minus 120590119909120577120577) d120578 d120577

= 1198661198691198961 + 1198811199091198962

1198601198961

(A10)


119868120578 = int119860

1205772d120578 d120577

119869 = int119860

(1205782+ 1205772) d120578 d120577

119868120577 = int119860

1205782d120578 d120577

1198962

119860=

1

119860int119860

(1205782+ 1205772) d120578 d120577

(A11)


997888rarr 1 + 1198962

997888rarr 2 +

1198963




1015840+ V101584022+

101584022 where

1015840 V1015840 and 1015840 are the


K0 = (dAd119904)A119879






= minusint119860

120588 d120578 d120577

= minusint119860

120588 119903119904 times d120578 d120577(A13)


119908 1198903)



A120597 (A119879120579)

120597119904= k (A14)

and hence

120579 (119904) = Aint

119904

0

A119879k d119904 (A15)


A120597A119879u120597119904

=

120598119909119909 minus 119891nl (119906 V 119908)

1205793

minus1205792

(A16)






119879 =

984858119862119897120572

119888

2[minus119880119875119880119879 +

119888

2120596119880119879 minus

119888

4119875 + (

119888

4)

2

]

119878 =

984858119862119897120572

119888

2[1198802

119875minus

119888

2120596119880119875 minus

1198621198890

119862119897120572

1198802

119879]

(A17)


119872120601 = minus

984858119862119897120572

1198883

32(120596119880119879 minus 119875 +

3119888

8) (A18)


120572


0





m119897 = min + maer (A19)




1198961 (119904 119905) =

1198731

sum

119899=1

120572120601

119899(119905) 1205951206011015840

119899(119904) (A20)

1198962 (119904 119905) =

1198732

sum

119899=1

120572119908

119899(119905) 12059511990810158401015840

119899(119904) (A21)

1198963 (119904 119905) =

1198733

sum

119899=1

120572V119899(119905) 120595

V10158401015840119899

(119904) (A22)

where120595120601

119899 120595V119899 and120595

119908





119899

and 120595119908




int

119877

0

m119879119897Ψ d119904 = int

119877

0

(m119879in + m119879aer)Ψ d119904 (A23)

where

Ψ =

1205951206011015840

119899

12059511990810158401015840

119899

120595V10158401015840119899

(A24)


Nomenclature


21198774 (119879 =






Competing Interests


References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












e3e2 e1

i3

r0 r

r

r1

u

r0

r1

u0

Ω

s

i2

i1

1e

e2e3






997888rarr 2

997888rarr 3) with










119890119894 = A (119904) 119894119894 (A1)




997888rarr 119894 = T (119904) 119890119894 (A2)



997888rarr 119894 = Λ (119904) 119894119894 (A3)

where Λ = TA


dVd119904

+ = 0

dd119904

+997888rarr 1 times V + = 0

(A4)





dd119904

(A119879k119897) + A119879p119897 = 0 (A5)

dd119904

(A119879m119897) minus A119879H119897k119897 + A119879q119897 = 0 (A6)


997888rarr 1


k119897 (119904) = Aint

119877

119904

A119879p119897d (A7)


m119897 (119904)

= Aint

119877

119904


119877

119904

(A119879H119897Aint

119877

A119879p119897d) d

(A8)




997888rarr (119904 120578 120577) =

1199030 + 119906 1198901 + V 1198902 + 119908 1198903 + 120578997888rarr 2 + 120577



(119904 120578 120577) =1

2(

120597997888rarr

120597120585119894sdot120597997888rarr

120597120585119895minus

120597 119903

120597120585119894sdot120597 119903

120597120585119895)

120597 119903

120597120585119894otimes

120597 119903

120597120585119895 (A9)




119881119909 = V sdot997888rarr 1 = int

119860

120590119909119909d120578 d120577

= 119864119860(1015840+V10158402

2+

10158402

2+ 1198962

1198601198962

1)


119860

120590119909119909120577 d120578 d120577 = 1198641198681205781198962

119872= sdot

997888rarr 3 = int

119860

120590119909119909120578 d120578 d120577 = 1198641198681205771198963

119872 = sdot997888rarr 1 = int

119860

(120590119909120578120578 minus 120590119909120577120577) d120578 d120577

= 1198661198691198961 + 1198811199091198962

1198601198961

(A10)


119868120578 = int119860

1205772d120578 d120577

119869 = int119860

(1205782+ 1205772) d120578 d120577

119868120577 = int119860

1205782d120578 d120577

1198962

119860=

1

119860int119860

(1205782+ 1205772) d120578 d120577

(A11)


997888rarr 1 + 1198962

997888rarr 2 +

1198963




1015840+ V101584022+

101584022 where

1015840 V1015840 and 1015840 are the


K0 = (dAd119904)A119879






= minusint119860

120588 d120578 d120577

= minusint119860

120588 119903119904 times d120578 d120577(A13)


119908 1198903)



A120597 (A119879120579)

120597119904= k (A14)

and hence

120579 (119904) = Aint

119904

0

A119879k d119904 (A15)


A120597A119879u120597119904

=

120598119909119909 minus 119891nl (119906 V 119908)

1205793

minus1205792

(A16)






119879 =

984858119862119897120572

119888

2[minus119880119875119880119879 +

119888

2120596119880119879 minus

119888

4119875 + (

119888

4)

2

]

119878 =

984858119862119897120572

119888

2[1198802

119875minus

119888

2120596119880119875 minus

1198621198890

119862119897120572

1198802

119879]

(A17)


119872120601 = minus

984858119862119897120572

1198883

32(120596119880119879 minus 119875 +

3119888

8) (A18)


120572


0





m119897 = min + maer (A19)




1198961 (119904 119905) =

1198731

sum

119899=1

120572120601

119899(119905) 1205951206011015840

119899(119904) (A20)

1198962 (119904 119905) =

1198732

sum

119899=1

120572119908

119899(119905) 12059511990810158401015840

119899(119904) (A21)

1198963 (119904 119905) =

1198733

sum

119899=1

120572V119899(119905) 120595

V10158401015840119899

(119904) (A22)

where120595120601

119899 120595V119899 and120595

119908





119899

and 120595119908




int

119877

0

m119879119897Ψ d119904 = int

119877

0

(m119879in + m119879aer)Ψ d119904 (A23)

where

Ψ =

1205951206011015840

119899

12059511990810158401015840

119899

120595V10158401015840119899

(A24)


Nomenclature


21198774 (119879 =






Competing Interests


References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











dd119904

(A119879k119897) + A119879p119897 = 0 (A5)

dd119904

(A119879m119897) minus A119879H119897k119897 + A119879q119897 = 0 (A6)


997888rarr 1


k119897 (119904) = Aint

119877

119904

A119879p119897d (A7)


m119897 (119904)

= Aint

119877

119904


119877

119904

(A119879H119897Aint

119877

A119879p119897d) d

(A8)




997888rarr (119904 120578 120577) =

1199030 + 119906 1198901 + V 1198902 + 119908 1198903 + 120578997888rarr 2 + 120577



(119904 120578 120577) =1

2(

120597997888rarr

120597120585119894sdot120597997888rarr

120597120585119895minus

120597 119903

120597120585119894sdot120597 119903

120597120585119895)

120597 119903

120597120585119894otimes

120597 119903

120597120585119895 (A9)




119881119909 = V sdot997888rarr 1 = int

119860

120590119909119909d120578 d120577

= 119864119860(1015840+V10158402

2+

10158402

2+ 1198962

1198601198962

1)


119860

120590119909119909120577 d120578 d120577 = 1198641198681205781198962

119872= sdot

997888rarr 3 = int

119860

120590119909119909120578 d120578 d120577 = 1198641198681205771198963

119872 = sdot997888rarr 1 = int

119860

(120590119909120578120578 minus 120590119909120577120577) d120578 d120577

= 1198661198691198961 + 1198811199091198962

1198601198961

(A10)


119868120578 = int119860

1205772d120578 d120577

119869 = int119860

(1205782+ 1205772) d120578 d120577

119868120577 = int119860

1205782d120578 d120577

1198962

119860=

1

119860int119860

(1205782+ 1205772) d120578 d120577

(A11)


997888rarr 1 + 1198962

997888rarr 2 +

1198963




1015840+ V101584022+

101584022 where

1015840 V1015840 and 1015840 are the


K0 = (dAd119904)A119879






= minusint119860

120588 d120578 d120577

= minusint119860

120588 119903119904 times d120578 d120577(A13)


119908 1198903)



A120597 (A119879120579)

120597119904= k (A14)

and hence

120579 (119904) = Aint

119904

0

A119879k d119904 (A15)


A120597A119879u120597119904

=

120598119909119909 minus 119891nl (119906 V 119908)

1205793

minus1205792

(A16)






119879 =

984858119862119897120572

119888

2[minus119880119875119880119879 +

119888

2120596119880119879 minus

119888

4119875 + (

119888

4)

2

]

119878 =

984858119862119897120572

119888

2[1198802

119875minus

119888

2120596119880119875 minus

1198621198890

119862119897120572

1198802

119879]

(A17)


119872120601 = minus

984858119862119897120572

1198883

32(120596119880119879 minus 119875 +

3119888

8) (A18)


120572


0





m119897 = min + maer (A19)




1198961 (119904 119905) =

1198731

sum

119899=1

120572120601

119899(119905) 1205951206011015840

119899(119904) (A20)

1198962 (119904 119905) =

1198732

sum

119899=1

120572119908

119899(119905) 12059511990810158401015840

119899(119904) (A21)

1198963 (119904 119905) =

1198733

sum

119899=1

120572V119899(119905) 120595

V10158401015840119899

(119904) (A22)

where120595120601

119899 120595V119899 and120595

119908





119899

and 120595119908




int

119877

0

m119879119897Ψ d119904 = int

119877

0

(m119879in + m119879aer)Ψ d119904 (A23)

where

Ψ =

1205951206011015840

119899

12059511990810158401015840

119899

120595V10158401015840119899

(A24)


Nomenclature


21198774 (119879 =






Competing Interests


References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













= minusint119860

120588 d120578 d120577

= minusint119860

120588 119903119904 times d120578 d120577(A13)


119908 1198903)



A120597 (A119879120579)

120597119904= k (A14)

and hence

120579 (119904) = Aint

119904

0

A119879k d119904 (A15)


A120597A119879u120597119904

=

120598119909119909 minus 119891nl (119906 V 119908)

1205793

minus1205792

(A16)






119879 =

984858119862119897120572

119888

2[minus119880119875119880119879 +

119888

2120596119880119879 minus

119888

4119875 + (

119888

4)

2

]

119878 =

984858119862119897120572

119888

2[1198802

119875minus

119888

2120596119880119875 minus

1198621198890

119862119897120572

1198802

119879]

(A17)


119872120601 = minus

984858119862119897120572

1198883

32(120596119880119879 minus 119875 +

3119888

8) (A18)


120572


0





m119897 = min + maer (A19)




1198961 (119904 119905) =

1198731

sum

119899=1

120572120601

119899(119905) 1205951206011015840

119899(119904) (A20)

1198962 (119904 119905) =

1198732

sum

119899=1

120572119908

119899(119905) 12059511990810158401015840

119899(119904) (A21)

1198963 (119904 119905) =

1198733

sum

119899=1

120572V119899(119905) 120595

V10158401015840119899

(119904) (A22)

where120595120601

119899 120595V119899 and120595

119908





119899

and 120595119908




int

119877

0

m119879119897Ψ d119904 = int

119877

0

(m119879in + m119879aer)Ψ d119904 (A23)

where

Ψ =

1205951206011015840

119899

12059511990810158401015840

119899

120595V10158401015840119899

(A24)


Nomenclature


21198774 (119879 =






Competing Interests


References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1198961 (119904 119905) =

1198731

sum

119899=1

120572120601

119899(119905) 1205951206011015840

119899(119904) (A20)

1198962 (119904 119905) =

1198732

sum

119899=1

120572119908

119899(119905) 12059511990810158401015840

119899(119904) (A21)

1198963 (119904 119905) =

1198733

sum

119899=1

120572V119899(119905) 120595

V10158401015840119899

(119904) (A22)

where120595120601

119899 120595V119899 and120595

119908





119899

and 120595119908




int

119877

0

m119879119897Ψ d119904 = int

119877

0

(m119879in + m119879aer)Ψ d119904 (A23)

where

Ψ =

1205951206011015840

119899

12059511990810158401015840

119899

120595V10158401015840119899

(A24)


Nomenclature


21198774 (119879 =






Competing Interests


References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article optimal design and acoustic assessment of...

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