HELICOPTER GROUND RESONANCE ANALYSIS USING MULTIBODY DYNAMICS

Transcription

1 ERF00- PAPER 9 HELICOPTER GROUND RESONANCE ANALYSIS USING MULTIBODY DYNAMICS A. Rezaean, German Aerospace Center, Insttute of Aeroelastcty, Bunsenstraße 0, Göttngen, Germany alreza.rezaean@dlr.de Keywords: Aeroelastcty, Helcopter Ground Resonance, Multbody Smulaton, Tme-perodc System, Multblade Coordnates Abstract Helcopter dynamc response s one of the most mportant crterons of the helcopter desgn. For an accurate predcton and calculaton of the vbratory responses of the helcopter structure n dfferent flght condtons, an exact modellng s essental. The accuracy of the results returns back to the accuraces of the model and calculaton procedures. The multbody dynamc smulaton tool allows usng modular models to buld a complete detaled system. Ths could be used to model a complex helcopter. SIMPACK (SImulaton of Multbody systems PACKage) as a multbody smulaton tool has shown consderable ablty to model and analyse lnear and non-lnear systems durng dfferent DLR-projects, however use of ths tool for modellng the helcopter dynamcs s somethng whch s currently beng nvestgated [], []. Ths tool was orgnally developed by DLR (German Aerospace Center) and s now further developed and commercally dstrbuted by SIMPACK AG. In ths work the multbody smulaton of the ground resonance effect s nvestgated. For ths purpose, the nterface of SIMPACK wth a FEM code for modellng the elastc rotor blades s consdered. Ths nterface creates a Standatrd Input Data, SID, based on blade geometry and egenvalue analyss. The SID-fle s then used for the calculaton of elastc deformaton. SIMPACK provdes also an nterface wth MATLAB, whch allows performng a part of postprocessng wthn ths program. Fg. shows the general vew of the ground resonance analyss usng multbody dynamcs smulaton. Fg. General vew of the ground resonance analyss procedure Helcopter ground resonance s a self exted dynamc nstablty, whch may occur when helcopter s n contact wth the ground. Elastc deformaton and out of phase laggng moton of the rotor blades durng rotor rotaton lead to the oscllaton of the rotor centre of gravty around the rotor rotaton axs. Due to the nteracton between rotor and fuselage, ths oscllaton exctes the flexble fuselage and the rotor support and may cause a volent damage of the helcopter structure. Ths phenomenon s known as ground resonance. Modellng ths phenomenon needs consderaton of the followng ponts: Lnearsaton, Geometrc stffenng effect, Tme-perodc system, Multblade coordnates transformaton. The work descrbed wthn ths paper s generally dvded nto three man parts. The frst part s related to the lnearzaton and geometrc stffenng

2 effect, whch s a prelmnary step for the dynamc stablty analyss of a rotatng system. Here, the smulaton results are compared wth analytcal results. Creatng the Fan-Dagram of an solated rotor and comparson of the results wth those produced wth CAMRAD II belongs also to ths part. The second part deals wth the helcopter ground resonance analyss. Modellng the ground resonance starts wth a smplfed model (rgd blades wth mechancally equvalent elastc laggng deformaton and fuselage wth two translatonal degrees of freedom) and s contnued wth more complex models comprsng elastc blades and fuselage. Data used n ths part belongs to DLR scaled (:.5) BO05 research rotor model (confguraton K0) [3] and CAMRAD II model of full scale BO05 wth modfed skd gear provded by Eurocopter Germany. One of the features of the ground resonance model s ts tme-perodc characterstc, whch leads to the nvaldty of the drect usage of the classcal egenvalue method for the dynamc stablty analyss. To perform the ground resonance analyss, the model s frst lnearzed about the equlbrum state. Then the lnear system matrx of the model s extracted and transformed from the rotatng coordnates to the non-rotatng multblade coordnates. Ths coordnate transformaton changes the system from a tme-perodc to a tme-nvarant one. Fnally, a classcal egenvalue analyss of the transformed system matrx results n the frequences and dampng of the system. To evaluate the smulaton procedures, the SIMPACK results are compared wth results produced wth CAMRAD II model. To valdate the model and calculaton procedures n addton to a comparson wth the CAMRAD II-results an analytcal valdaton of the dynamc response of the system s also performed. The thrd part of the work deals wth the parametrc studes of the ground resonance effect and analysng the results and tendences of the changes of the dynamc response.. INTRODUCTION Testng of rotorcraft dynamc stablty s a necessary step for the successful development of a new desgn. Testng a prototype at full scale or model scale s typcally very expensve. Therefore usage of a smulaton code for nvestgaton of dynamc stablty s a consderable alternatve soluton. Each smulaton code has ts own computatonal methods and modellng approaches, whch results n dfferent accuraces. Therefore usage of a smulaton code for a specal analyss demands prenvestgatons and valdatons. The multbody dynamc smulaton tool allows usng modular models to buld a complete detaled system. Ths could be used to model a complex helcopter. SIMPACK (SImulaton of Multbody systems PACKage) as a multbody smulaton tool has shown consderable ablty to model and analyse lnear and non-lnear systems durng dfferent DLR-projects, however use of ths tool for modellng the helcopter dynamcs s somethng whch s currently beng nvestgated [], []. In general helcopter dynamc nstablty happens less frequently but ends more dramatcally. A vbratory mode of a rotor can be stable over a wde range of operatng condtons and wth a small change n speed or torque may become volently unstable wth a rapd growth n ampltude leadng ether to falure or to a lmt cycle f nonlneartes lmt the growth. Havng deep knowledge of the physcs of the aeroelastc nstablty of a system makes t easer to fnd an effectve techncal soluton to prevent t. To acheve ths, parametrc studes, usng smulaton code, for nvestgaton of the effects of dfferent system parameters on the nstablty wll be benefcal. Insde each smulaton code frst physcal model s estmated wth a mathematcal model. Ths mathematcal model s generally n form of a system of nonlnear dfferental equatons: () x & = f ( x, x,, x ) K n For stablty analyss of ths system, the response of the physcal system after a small perturbaton from ts equlbrum state s studed. In general the behavour of ths nonlnear system could be approxmated wth the behavour of a dynamcally equvalent lnear system. Applyng the Jacoban lnearzaton method for equaton () wth consderaton of the lnearzaton pont LP results the followng lnear equaton: f f f L X& x x LP x LP n LP X () = X & f f f = X X& L x x x M LP LP n LP M X& M M M M n f n f n f n L x x x LP LP n LP Ths equaton can be rewrtten n the followng form: (3) X& = [ A] X X The matrx [ A] s called system matrx. For the stablty analyss of the lnear equaton (3) followng methods can be utlzed: If the system matrx s constant n tme, then egenvalue analyss leads to stablty/nstablty predcton If the system matrx s tme dependent and perodc, then applyng the Floquet theory or an approxmaton method (for example usage of multblade coordnates transformaton wth tme average approxmaton) s common... Helcopter Ground Resonance The theory of helcopter ground resonance was orgnally developed by Coleman and Fengold [4]. Ths phenomenon s a self-excted dynamc nstablty caused by the nteracton of the laggng moton of the rotor blades wth other modes of the moton of the helcopter. Elastc deformaton and out of phase laggng moton of the rotor blade durng rotor rotaton causes the oscllaton of the rotor centre of gravty around the rotor rotaton axs. Due to the nteracton between rotor and fuselage, ths oscllaton exctes the flexble fuselage and rotor support and may cause a volent damage of the helcopter structure. Ths dynamc nstablty happens when helcopter landng gear or skd gear s n contact wth the ground. Therefore ths nstablty s called ground resonance. Some of the characterstc parameters of ths nstablty are: n

3 Blade lag frequency Blade lag dampng Frequences of the structure supportng the rotor Damng of the structure supportng the rotor To perform ground resonance analyss, degrees of freedom of the modelled helcopter are reduced. An Elmnaton of a DOF s based on ts effect on the ground resonance. For a classcal ground resonance model just the laggng of the rotor blades and longtudnal and lateral n-plane moton of the fuselage are consdered. For ths model aerodynamc forces have lttle nfluence on the nstablty effect n compare to structural and nertal forces, therefore aerodynamc forces can be neglected [5]. Wth consderaton of the flappng moton of the rotor blades aerodynamc forces can be neglected... Multblade Coordnates Generally the equaton of moton of a rotor s derved n the rotatng frame. In ths frame each rotor blade s consdered separately but mostly the rotor responds as a whole to an exctaton. Changng the coordnate system from rotatng to the non-rotatng allows to analyse the rotor as a whole system. In the non-rotatng frame the so called multblade coordnates are defned. Transformaton of the dfferental equatons of moton from rotatng frame to non-rotatng frame has the followng advantages: It smplfes the analyss and helps to understand the behavour of the rotatng system, especally, when they are n nteracton wth non-rotatng parts. In the case of tme-perodc system, ths transformaton reduces the number of tmeperodc elements n the dfferental equatons. For a tme-perodc system, a combnaton of multblade transformaton and tme average approxmaton leads to a system of dfferental equatons wth constant coeffcents. Consder a rotor wth N equally spaced blades. The azmuth of the th blade s gven by: π ψ = ψ + N (4) ( ) N ψ s the azmuth of the frst blade. The arbtrary degree of freedom of the rotor blade β s then expressed n the multblade coordnates as follows [6]: n (5) β = β + ( β cos nψ + β sn nψ ) + β ( ) wth n = to or n = to d 0 nc ns d N when N even N when N odd β appears only when N s even. The multblade coordnates from (5) are then calculated by followng equatons: N β = β collectve coordnate (6) 0 ( ) N = β d dfferental coordnate N N ( ) (7) = β ( ) = N β cos cyclc coordnate N (8) nc = ( β nψ ) = N (9) β ns = ( β sn nψ ) cyclc coordnate N = These coordnates defnes the new degrees of freedom, whch descrbe the moton of the rotor as a whole n a non-rotatng frame. Fg. shows as an example the mode shapes of a four bladed rotor n multblade coordnates consderng the lag degree of freedom. FIG. Multblade modes of laggng rotor blade There are smlartes between the multblade coordnates and coeffcents of the Fourer seres but they are not the same. If the moton n the rotatng frame represented by a Fourer seres, then the coeffcents of seres are constant n tme but nfnte n number and f by Multblade coordnates then the coeffcents or not constant n tme and the number of coeffcents depends to the number of rotor blades..3. Smulaton tool SIMPACK SIMPACK (SImulaton of Multbody systems PACKage) software package s used to smulate, analyse and desgn all types of mechancal system. It can analyse the vbratonal behavour of multbody systems and allows predctng and descrbng the moton of a complex machne or mechansms [7]. Ths tool was orgnally developed by DLR (German Aerospace Center) and s now further developed and commercally dstrbuted by SIMPACK AG. The software has a comprehensve range of modellng and calculaton features. Therefore t s appled wthn ndustry, unversty and research nsttutons. There are dfferent modellng elements, whch are used to smulate a complex system. Data related to these elements can be entered n to SIMPACK va the graphcal user nterface. Most mportant modellng elements and optons are: Reference Frames Bodes

4 Jonts, Constrants Force elements Sensors, Control elements Substructures, Substtutons varables SIMPACK creates frst of all the equatons of moton for a modelled mechancal system and then solves t wth dfferent mathematcal procedures. A wde range of analyses features are avalable to analyse dynamc systems. These are: Statc Analyss Knematcs Analyss Non-lnear Dynamc Analyss Lnear System Analyss Symbolc Code Generaton Egenvalue Analyss. GROUND RESONANCE MODELLING APPROACHES There are dfferent modellng approaches of the ground resonance effect. The dfferences of these approaches returns back to the accuraces of the modellng and to the methods of analyss. The structure of a helcopter s n general dvded n to two man groups. The frst group ncludes rotor and the second group fuselage. For each group there are dfferent modellng methods. For reducton of the complexty of the problem, the degrees of freedom of each group can be reduced to the ground resonance domnant degrees of freedom... Modellng of the Rotor Blades Two methods for modellng a rotor blade are used. In the frst method the rotor blade s modelled as rgd body. The elastcty and structural dampng of the blade are modelled wth equvalent sprng and damper. For a hngeless rotor besdes the equvalent sprng and damper an equvalent laggng hnge s also consdered. The calculatons of the equvalent hnge, sprng and dampng coeffcents are done under assumpton of the equvalency of the frst laggng harmonc of both hngeless and equvalent rotor. Fg.3 shows a general form of a rotor blade created wth the frst method. In the second method the blade s modelled as an elastc body usng drectly a FEM code or SIMPACK beam elements (SIMBEAM). In both cases an egenvalue analyss of the structure should be frst performed. Usage of these results n SIMPACK nterface wth FEM code (FEMBS) allows creatng a standard nput data fle (SID), whch s used by SIMPACK to calculate the elastc deformaton of the structure.... Frequency Analyss of the Rotor Blade Generally for the stablty analyss of a nonlnear system n frequency doman, ts lnearzed form s consdered. Therefore for blades created wth the frst or second method the lnearzaton method of SIMPACK should be evaluated. FIG. 3 Rgd blade wth equvalent elastcty Usng the frst method of the blade modellng, a blade of a wndmll-powered plant, shown on Fg.4 was created. I gven on Fg.4 defnes the blade mass moment of nerta at the rotor centre and about the Z axs and K gves the rotatonal sprng stffness. Ths model was lnearzed for dfferent rotatonal veloctes and azmuth angles. FIG. 4 Table SIMPACK wndmll-powered plant model Comparson of Egenfrequences: Analytcal, SIMPACK and MSC.ADAMS The lead-lag frequences obtaned after the lnearzaton were compared wth analytcal results and MSC.ADAMS results publshed on paper [8]. Table gves and compares these frequences for four dfferent azmuth angle and two rotor rotatonal veloctes. For frequency analyss of a rotor blade created wth the second method of the modellng, an elastc rotatng blade wth 3 elements, created wth SIMBEAM opton of SIMPACK, was analyzed. Fg.5 shows ths model. To

5 evaluate ths model the same beam was modelled usng MSC.NASTRAN. FIG. 5 Elastc beam created wth SIMBEAM Egenfrequences of these rotatng beams were compared together for dfferent rotatonal veloctes. Fg.6 shows as an example the egenfrequences of dfferent egenforms of these two beams for rotatonal frequency 6 Hz and compares them together. Ths comparson shows, that the elastc beam created drectly wth SIMPACK consders the rotatonal effects (centrfugal effect, whch changes the stffness of the blade and gyroscopc effect, whch leads to the couplng of DOFs) as t s consdered for NASTRAN model. FIG. 7 Oscllaton of the rotor blade after a small perturbaton FFT analyss of ths result, shown on Fg.8, gves as expected the lag frequency of the blade, whch depends to rotor rotatonal velocty. FIG. 6 Egenfrequences of dfferent Egenforms of the rotatng beam shown on Fg.5 FIG. 8 Frequency spectrum of the blade oscllaton shown on Fg.7 Because of the out of phase oscllaton of the rotor blades, the rotor centre of gravty oscllates too. The Frequences of ths oscllaton can be obtaned wth FFT analyss of the movement of the rotor centre of gravty, whch s captured by a sensor. Fg.9 shows these frequences. It can be seen that, between the frequences of the rotor centre of gravty, rotor rotatonal frequency and blade laggng frequency the followng relaton exsts [9]: (0) f CG = f Blade, Laggng ± f Rotor, Ω... Frequency Analyss of an Isolated Rotor Wth blades created wth the frst method of the blade modellng, an solated rotor was created. The data related to ths rotor belongs to the scaled BO05 rotor [3] and wll be gven later for descrpton of the ground resonance model. For ths solated rotor a created sensor measures the locaton of the rotor centre of gravty. The blades are frst dsturbed out of phase n lag drecton and then tme ntegraton s performed. Fg.7 shows the result of the tme ntegraton of the frst rotor blade. FIG. 9 Frequency spectrum of the oscllaton of the rotor centre of gravty. Performng of ths frequency analyss for dfferent rotor rotatonal veloctes (rotatonal frequences) and plottng all

6 the frequences results n the lnes shown on Fg.0. A Smlar frequency dagram wll be seen later n ground resonance analyss of a whole helcopter. FIG. Dsplacement of the rotor centre of gravty FIG. 0 Frequences of the rotor centre of gravty and blade lag mode As a further analyss for ths solated rotor, Fg., and 3 show graphcally the planer dsplacement of the centre of gravty of ths rotor for three dfferent rotatonal veloctes. Comparson of these fgures shows the rregularty of the patterns of these dsplacements. For ths solated rotor the multblade coordnates were defned as outputs. These Outputs were used to calculate the poston of the rotor centre of gravty wth the followng equatons [9], see Fg.3: () X CG = ξ ( a + R ) C CG FIG. 3 Dsplacement of the rotor centre of gravty As a next step, usng the SIMBEAM opton of SIMPACK the elastc blade of BO05 helcopter was modelled (structure data was provded by Eurocopter). After modellng the blade, an solated rotor as shown on Fg.4 was created. () Y CG = ξ ( a + R ) S CG The results obtaned from these equatons were plotted on Fg., and 3 and compared wth the results obtaned drectly from the sensor measurements. A Concluson n respect to equatons and and Fgures to 3 would be that, nstead of frequences of the oscllaton of the rotor centre of gravty the frequences of the multblade coordnates can be analysed. FIG. 4 Isolated BO05 rotor For ths rotor frequency dagram (Fg. 5) was created. FIG. Dsplacement of the rotor centre of gravty FIG. 5 Cambell dagram of BO05 rotor

7 On Fg.5 frequences of the dfferent egenforms calculated wth SIMPACK are compared wth results produced wth CAMRADII. Ths comparson shows a good agreement of the both SIMPACK and CAMRADII models. Because the results of ground resonance analyss produced wth SIMPACK wll be compared wth CAMRADII results, therefore t s necessary both models to be dynamcally equvalent. For classcal ground resonance analyss the frst lag frequency of the blade s most mportant and s often consdered as only model degree of freedom of the blade, therefore a modfcaton (calbraton) s necessary to match these lag frequences wth those obtaned from CAMRADII. Ths ensures the dynamcally equvalence of both models. Fg.6 shows the frst lag frequences of the rotor blade after and before modfcaton and compares them wth CAMRADII results. frequency on ts landng gear vares dependng on ts percent arborne condton. Because of ths a helcopter may be stable when fully on the ground but go n to ground resonance n a partally arborne condton. It s assumed that the helcopter on ts landng gear can be represented by effectve parameters appled at the fuselage model. FIG. 7 Modellng optons of the Fuselage FIG. 6 Lag frequences of the BO05 rotor blade For the modfcaton of the lag frequences a reducton n centrfugal stffness was consdered... Modellng Approaches of the Fuselage For modellng the helcopter fuselage three methods can be appled. Fg.7 shows schematcally these methods. In the frst method the fuselage s modelled drectly usng SIMPACK modellng elements (mass, sprng, damper). For the defnton of the sprng or dampng coeffcents, we consder ths fact that, the fuselage model and real fuselage should be dynamcally equvalent. The calculatons of these values are normally based on the results of the expermental ground vbraton test or modal values. These equvalent values of the fuselage are normally determned expermentally by ground vbraton test. The second method apples the SIMPACK nterface wth FEM code. The fuselage s frst modelled usng a FEM code and then the nformaton related to ths elastc model s sent to SIMPACK. The SIMPACK nterface wth FEM code (FEMBS) creates a standard nput data fle (SID), whch s used to defne the structure and elastcty of a body. All matrces nsde ths fle are n modal coordnates. In the thrd method one can edt or create drectly a SID fle usng the modal values (modal mass matrx, modal stffness matrx, modal dampng matrx and etc.) of the fuselage. Modal values can be obtaned for example after a ground vbraton test. For ground resonance calculaton, t s suffcent to evaluate fuselage frequences not hgher than the rotor frequency mnus laggng frequency. The Fuselage natural 3. GROUND RESONANCE ANALYSIS (G.R.A) In ths part two types of ground resonance model are consdered. In the frst modellng type, blades are rgd and ther elastcty are defned wth equvalent sprngs (frst method of the blade modellng) and n the second model the blades are elastc and are created by fnte element method. To perform a ground resonance analyss after creatng the model the model s frst lnearzed. For ths lnear system then a multblade transformaton s performed to convert the rotor coordnates from a rotatng frame n to a fxed frame. Ths transformaton averages out the perodc terms n the moton equatons over tme, whch leads to a system of equatons wth constant coeffcents. Fnally an egenvalue analyss leads to the frequences and dampng of the system. Ths procedure s repeated for dfferent rotor rotatonal veloctes and at the end the obtaned frequences and dampng are plotted together. 3.. G.R.A: Rgd Blade wth Equvalent Elastcty Wth the data gven on the next page, a whole helcopter model was created to perform the ground resonance analyss. Fg.8 shows ths model. In ths model the fuselage has two translatonal degrees of freedom and rotor blade has only lag degree of freedom. Blades are jonted to the lag hnge wth rotatonal sprngs and dampers. The data related to the rotor are based on the scaled research rotor BO05, DNW, confguraton K0 [3]. Ths SIMPACK blade model s n fact a mathematcally equvalent model of the hngless scaled BO05 rotor blade. Ths hngless rotor blade s modelled wth an equvalent laggng hnge and equvalent sprng and damper to model the stffness and structural dampng of the hngless blade.

8 FIG. 8 SIMPACK ground resonance model The data related to the model shown on Fg.8 are : Rotor data 0.35 [m] Dstance between the laggng hnge and the axs of rotaton..39 [Kg] Blade mass.66 [m] Blade length.8[kgm²] Blade mass moment of nerta about laggng hnge 8.6 [Nm/grad] Laggng stffness.5 [% crt.] Dampng rato Fuselage data [N/m] Sprng stffness n Y drecton [Ns/m] Dampng coeffcents n Y drecton [N/m] Sprng stffness n X drecton 65 [Ns/m] Dampng coeffcents n X drecton 3.5 [Hz] Frequency of egenform n Y drecton for Ω = [Hz] Frequency of efenform n X drecton for Ω = [% crt.] Dampng rato n Ydrecton for Ω = [% crt.] Dampng rato n X drecton for Ω = 0 3 [Kg] Fuselage mass To perform the stablty analyss the rotor rotatonal velocty s changed stepwse. For each value of the rotatonal velocty the system s lnearzed about ts equlbrum state. Then the lnear system matrx of the model s extracted and saved n MATLAB m-fle format. A MATLAB program, mplemented by wrter, s used to perform the coordnates transformaton. Ths program A from rotatng coordnates to non-rotatng coordnates. The new matrx s transforms the lnear system matrx [ ] T. Fg. 9 shows the lnear dfferental equaton of the system n rotatng and non-rotatng coordnates and llustrates the defnton of the system matrces A and T. After performng the transformaton, the egenvalues of called [ ] T are determned and then wth respect to the egenforms are sorted. Fnally the frequences and dampng are calculated and evaluated. the transformed lnear system matrx [ ] FIG. 9 Multblade coordnates transformaton A CAMRAD II model of the descrbed ground resonance model was created to valdate the performed procedures. From the data used for SIMPACK model, the data needed for the CAMRAD II model were calculated. Ths s necessary due to the dfferences between the modellng approaches of CAMRADII [0] and SIMPACK. Fg.0 shows schematcally the CAMRAD II rotor blade. FIG. 0 CAMRADII rotor blade model From the mass moment of nerta of the SIMPACK model and the rotor length one can calculate the rotor chord length (see, Fg.0 and Equ.3) as follows: (3) r. [ m] ( + c ).3896( c ) b = r4 = 658 m b I CG = c = [ m] = = Then densty of the blade s calculated from ts geometry and mass: M (4) ρ = blade = = 3.[ Kg / m ] b c d.658 c 0.c Equatons 5 and 6 gve the values of the two other parameters needed n the blade nput fle of CAMRAD II. (5) I THETA = ( x + z ) ρ da = [ kgm]

9 (6) I POLAR = ( x z ) ρ da = [ Kgm] Fg. shows graphcally the frequences and dampng of the both SIMPACK and COMRAD II models n multblade coordnates (frst model). Comparson of the results shows the agreement of the both models and valdates the mplemented MATLAB program for multblade coordnates transformaton and also approved the consdered procedure of ground resonance analyss. Accordng to the Fg. the ground resonance of the model happens by rotatonal frequency ~7 [Hz]. Ths nstablty happens when the regressng frequency of the rotor nterfaces the fuselage frequency. However ths happens n two dfferent ponts. Investgatng the dampng at both ntersecton ponts shows that just n one ntersecton pont the dampng of the fuselage domnant egenform reduces and leads to nstablty. The nstablty of ths ground resonance model depends to the followng parameters: natural frequences of the helcopter and the blades, the dstance between the rotor hub and the lead-lag hnge, the moment of nerta of the blade about the lead-lag hnge, the rotor speed, the mass of blade, and the effectve mass of the fuselage. To see the effect of each of these parameters on the stablty, one can perform parameter studes usng the parameter varaton opton of SIMACK ParVaraton. modfcatons) to prevent any couplng effect. The data used for ths model are: Rotor data Same data used for the frst model Fuselage data 335 [N/m] Sprng stffness n Y drecton 43 [Ns/m] Dampng coeffcents n Y drecton [N/m] Sprng stffness n X drecton 6500 [Ns/m] Dampng coeffcents n X drecton 3.48 [Hz] Frequency of egenform n Y drecton for Ω = [Hz] Frequency of egenform n X drecton for Ω = 0 5 [% crt.] Dampng rato n Y drecton for Ω = [% crt.] Dampng rato n X drecton for Ω = [Kg] Fuselage mass Wth these data two models were created usng SIMPACK and CAMRADII. For both models the ground resonance analyses were performed. Fg. shows the frequences and dampng of both SIMPACK and CAMRADII models n multblade coordnates. Comparson of the results of the both smulaton tools approved agan the procedure defned for ground resonance analyss wth SIMPACK. FIG. Frequency and dampng of the egenvalues of the transformed system matrx, frst model By the frst ground resonance model the frequences of the fuselage domnant egenforms had nearly the same values. For the next model (second model) these frequences were separated from each other (by model FIG. Frequency and dampng of the egenvalues of the transformed system matrx, second model Analysng the dampng on Fg. shows that at the pont of nstablty the energy needed for ncreasng the ampltude of the fuselage oscllaton comes from the rotor. Increase or decrease of the dampng can be better

10 understood, when the exchange of the energy between the fuselage and rotor s analysed. Fg.3 shows the results of ground resonance analyss of two further consderable test cases. In one case the structural dampng of the fuselage s zero whle these values for blades are non zero and n the second case the structural dampng for both blades and fuselage are zero. The upper dagram of Fg.3 shows the frequences for both test cases. Dagram n the mddle shows the dampng for the frst case and lower dagram shows the dampng for the second case. of Fg.3 that, for a rotor wth constant rotatonal velocty the sum of the energy exchange between the rotor and fuselage at the ground resonance pont wll be zero. 3.. G.R.A: BO05 Blade Created wth FEM It was descrbed, that the hngless rotor blade of BO05 was created usng SIMPACK and calbrated to be dynamcally equvalent wth ts smlar CAMRADII model. Consderng the modal values used by CAMRADII model to model the BO05 fuselage, an dynamcally equvalent fuselage wth mass, sprng and damper was nsde SIMPACK created. The dynamc equvalency means here the equvalency of the egenforms and egenvalues of the both models under the assumpton of the equalty of the masses and mass dstrbutons. Fg.4 shows the ground resonance model of BO05 created nsde SIMPACK. Ths model has 4 modal DOFs for the blades (one for each blade) and translatonal DOFs (longtudnal and lateral) for the Fuselage. FIG. 3 Ground resonance analyss, test cases The change of the dampng n the neghbourhood of the frequences ntersecton pont causes the changes n dynamc behavour. It can be also seen on lower dagram FIG. 4 SIMPACK BO05 ground resonance model The procedure of the ground resonance analyss for ths model s dentcal to the procedure performed for prevous model. Fg. 5 shows the results of ths analyss. The upper dagram shows the frequences and the lower dagram the dampng. On ths Fgure the results are compared wth CAMRADII results. It can be seen that, there s a small dfferences between the CAMRADII and SIMPACK results. Ths dfference s frstly due to the dfference between SIMPACK and CAMRADII fuselage models and secondly due to the rotor models. Although the SIMPACK blades were calbrated wth CAMRADII blades, stll the egenfrequences of the blades do not match 00% to each other and ths results n the dfferent ground resonance ponts (offset between the dampng pcks on Fg.5). It was shown n prevous part, f both rotors/fuselages are exactly smlar, then the ground resonance results of both tools wll be the dentcal. an Interestng effect seen on ths model s that, n the neghbourhood of the frequences ntersecton pont, dampng of the fuselage domnant mode ncreases (fuselage gves energy) and dampng of the rotor domnant modes decreases (rotor receves energy). It should be mentoned that, for coupled fuselage and rotor system, fuselage modes refer to fuselage domnant egenmodes and rotor modes refer to rotor domnant egenmodes. For a fuselage domnant egenmode the ampltude of the fuselage oscllaton s larger than the ampltude of the blade oscllaton.

11 Comparson of the dfferent ground resonance smulaton results shows that, at the ground resonance pont the egenmdoe wth reduced dampng would be the one wth smaller dampng before frequences ntersecton pont. Ths fact can be seen for example by comparson of Fg. and 5. FIG. 5 Result of BO05 ground resonance analyss: Frequency and Dampng Dfferent parametrc studes were performed for the created ground resonance models and dfferent results were produced. Fg.6 shows as an example one of these results. It shows the effect of the ncrease or decrease of the blade structure dampng on the ground resonance effect. 4. CONCLUSION Durng ths work, SIMPACK was nvestgated as a multbody dynamcs tool for ground resonance analyss. Ths nvestgaton s dvded n to two man parts: frst part deals wth the modellng approaches and the second part deals wth the methods of the calculaton and analyss. In the frst part some modellng optons and features of SIMPACk for creatng a helcopter-model wth elastc blades were evaluated. Some of these optons are: Modellng of the elastc blade and fuselage a-rgd blade wth equvalent elastcty b-elastc blade created usng FEM c-rgd fuselage wth equvalent elastcty Parametrc modellng and studes SIMPACK allows defnng parameters nsde the model and performng dfferent analyss wth parameter varaton. Ths opton can be used for example to produce the cambell dagram. In ths case the rotor rotatonal velocty s defned as a model parameter and egenvalue analyss s performed for dfferent values of ths parameter. Expressons (user defned functons) Usng the expressons, the multblade coordnates can be defned nsde SIMPACK and can be requested as output. Ths allows analysng the multblade coordnates n tme doman. Related to the methods of the calculaton, the followng ponts were analysed: Lnearzaton of a nonlnear system Lnearzaton opton of SIMPACK gves vald results for a rotatng system. These results are used for ground resonance analyss. Rotatonal effects(geometrcal stffenng, gyroscopc effect and etc.) For a rotatng system SIMPACK consders the forces resultng from rotaton. Multblade coordnates transformaton After lnearzaton of a non-lnear system, SIMPACK creates the lnear system matrces. These matrces can be saved n MATLAB m-fle format. An mplemented MATLAB program s then used to perform the multblade coordnates transformaton. Ths coordnates transformaton was valdated before ts usage for ground resonance analyss. SIMPACK post-processng optons (for example coordnate transformaton, FFT analyss) In most cases SIMPACK can be used drectly as a post-processor for the evaluaton of the smulaton results. SIMPACK can perform the Fast Fourer Transformaton (FFT) for the results saved as an output or can flter the results wth user defned or selected flter. FIG. 6 SIMPACK ground resonance analyss: parametrc studes For the evaluaton of the smulaton results, SIMPACK results were compared wth analytcal results (n case of lnearzaton) and results of the other smulaton tool (n case of ground resonance analyss the results were compared wth the results of CAMRADII). In ths work two man dfferent ground resonance models were created. In the frst man model the blade was rgd and ts elastcty was modelled wth an equvalent sprng and n the second model the blade was modelled drectly as an elastc part usng FEM. After performance of the ground resonance analyss for

12 each model the results were compared wth the results of ts smlar/equvalent CAMRADII model. Comparson of the results shows the valdty of the ground resonance modellng approaches and calculaton methods defned wthn ths paper and shows SIMPACK as a convenent multbody smulaton tool for the helcopter ground resonance analyss. However, for the usage of ths tool for a complex and exact helcopter dynamcs analyss stll new methods should be developed. ACKNOWLEDGMENT I would lke to express my grattude to Mr. Olver Deterch from Eurocopter Germany and Dr. Frtz Kesslng from Insttute of Aeroelastcty DLR Göttngen, for all ther techncal and scentfc support. REFERENCES [] J. Arnold, Usng Multbody Dynamcs for the Smulaton of Flexble Rotor Blades- Gettng the Mechancal Couplng Rght?, 35 th European Rotorcraft Forum, Hamburg-Germany, September 009 [] A. Rezaean, Stablty Analyss of Tme-perodc Systems Usng Multbody Smulaton for Applcaton to Helcopters, Deutscher Luft- und Raumfahrtkongress 009, Aachen. [3] H. J. Langer, F. Kesslng, R. Schröder, A Model for Wnd Tunnel Rotorcraft Research Ground Resonance Investgaton, nd European Rotorcraft and Powered Lft Arcraft Forum, 976 [4] Robert P. Coleman, Arnold M. Fengold, Theory of Self-Excted Mechancal Oscllatons of Helcopter Rotors wth Hnged Blades, NACA Techncal Report TR 35, 958. [5] Rchard L. Belawa, Rotary Wng Structural Dynamcs and Aeroelastcty, AIAA, Inc., Washngton, DC, ISBN , 99 [6] Wayne Johnson, Helcopter Theory, Dover publcaton, Inc., New York, ISBN , 994. [7] SIMPACK, SIMPACK Reference Gude, SIMPACK release 8.8, 6 th March 006, SIMDOC v8.800, Copyrght INTEC GmbH006. [8] Jose L. Ortz, Gunjt S. Br, Verfcaton of New MSC.ADAMS Lnearzaton Capablty For Wnd Turbne Applcaton, 44th AIAA Aerospace Scence Meetng and Exhbt, 9- January 006, Reno, Nevada [9] A.R.S. Bramwell, Helcopter Dynamcs, Department of Aeronautcs, The Cty unversty, London, ISBN , 976 [0] CAMRAD II, Comprehensve Analytcal Models of Rotorcraft Aerodynamcs and Dynamcs, Volume VI, Rotorcraft Input, Release 4.6, Wayne Johnson, Johnson Aeronautcs, Palo Alto, Calforna, 007.