First International Symposium on Flutter and its Application, 2016
+1luca.pigolotti@dicea.unifi.it, +2claudio.mannini@dicea.unifi.it, +3gianni.bartoli@unifi.it
EFFECTS OF MASS UNBALANCE AND HIGH LEVELS OF EXTERNAL
DAMPING ON THE POST-CRITICAL FLUTTER OF A FLAT PLATE
Luca Pigolotti
+1, Claudio Mannini
+2and Gianni Bartoli
+3+1,2,3
CRIACIV/Department of Civil and Environmental Engineering, University of Florence, Italy
The post-critical flutter regime is still an open issue, despite the interest for its applications in the field of energy harvesting, where the knowledge of the self-sustained motion at large amplitudes is of crucial importance. In the paper, the experimental approach has been followed to investigate this issue. The role of some governing parameters, mainly the mass centre position and the damping level in the translational degree of freedom, were investigated through wind tunnel tests, evaluating their effects on the critical condition and subsequent oscillatory regime. Linear analyses supported the identification of the instability threshold. Downstream mass eccentricity of about 0.05 times the section chord significantly anticipated the critical onset velocity and modified the motion characteristics, in terms of ratio between translational and rotation amplitudes and their phase difference. External damping was introduced, up to a ratio-to-critical value of 18%, and the consequent reduction of the motion amplitudes was clearly shown. In particular, even if the amplitude-velocity diagrams always maintained the same qualitative features, a non-linear dependence on the damping level of its slopes was apparent. In addition, a theoretically stable configuration with large damping was found to perform self-sustained motion if triggered by large initial conditions.
Keyword: Flutter, Wind Tunnel Tests, Post-critical Behaviour, Sub-critical Bifurcation.
1. INTRODUCTION
In the common practice of wind and aeronautical engineering, dynamic fluid-structure interaction is a dangerous phenomenon and the design of structures prone to flow-induced vibrations usually aims at limiting any flow-induced vibration. Nevertheless, the aeroelastic phenomena are characterized by important nonlinear effects that produce Limit Cycles of Oscillation (LCO). In some cases, LCO is restricted to a limited range of flow velocities, as in the case of vortex-induced vibrations, or flow-velocity-unrestricted oscillations can be observed after a critical threshold, as in the case of galloping and flutter. From a different perspective, recent studies on alternative energy sources showed the possibility of exploiting fluid-elastic instabilities and the consequent large steady-state oscillations to capture energy from the flow and generate electricity through
suitable energy conversion apparatus1). Some authors have preliminary explored aero-/hydro-elastic generators
based on: vortex-induced vibrations2); transverse3) and torsional4),5) galloping; wake-galloping6); flapping7),8),9)
and fluttering10) wings.
From the literature analysis one can conclude that flutter-based solutions are the most promising ones8),10) and the actual capability of performing self-sustained large-amplitude motion in the post-critical
regime is a fundamental requirement for any flutter-based generator. Nevertheless, only very few scientific works have been developed so far on the flutter post-critical regime, due to the limited interest for conventional civil/aeronautical structures. Reliable predictive models for the post-critical behaviour are still missing and CFD investigations are hardly applicable due to the very large-amplitude oscillations. Following
an experimental approach, aeroelastic setups can be developed to observe the post-critical response11), but they
require specific and complex design solutions.
In the classical flutter12) involving two Degrees of Freedom (DoFs), the system is excited by
aerodynamic lift and moment forces depending on its motion. The energy transfer between airstream and structure relies on the elastic and/or aerodynamic coupling between two modes, generally with components on
both DoFs, as well as on the phase lag between the displacement and its aerodynamic reaction. This interaction generates phase adjustment and loss of damping in one of the modes, which leads to the instability and drives the growth of the motion. Then, when the cross-flow displacements and rotations are no longer small, nonlinearities occur in the self-excited loads. Particularly important are those due to the massive flow separation encountered for large angles of attack, beyond the dynamic stall angle of the section. The subsequent amplitude-velocity path manifests large amplitudes in both degrees of freedom that generally increase with the flow velocity, after a steep initial jump.
Unlike the availability of semi-empirical predictive models about the dynamic-stall flutter
mechanism13),14),15), due to the importance for wings operating at high angles of attack16),17),18),19),20), the
research on post-critical oscillations due to classical flutter is not so extensive. In addition, the physical sources of nonlinearity still represent an open issue, since the literature studies often focused on classical
flutter LCOs under specific nonlinear mechanical boundary conditions21),22),23). Hence, further research is
required on reliable mathematical models and experimental setups allowing for large oscillations to deal with the classical-flutter post-critical behaviour.
This work concentrates on wind tunnel tests on a sectional model with elongated rectangular cross section (width-to-depth ratio of 25:1, with the smaller dimension of 4 mm facing the flow). The developed
setup allowed motion in two DoFs), namely vertical (heaving, η, maximum ±100 mm) and rotational (pitching,
α, maximum ±150°) oscillations, with linear mechanical features in the tested range. Magnetic dampers were
used to introduce linear viscous damping into the heaving degree of freedom, up to a ratio-to-critical value of about 20%. Linear aeroelastic models were preliminary used to parametrically explore the flutter boundaries.
The main objective of the paper is to improve the understanding of the influence of some of the
governing parameters on the system response24), such as the position of the mass centre, the ratio of pitching
to heaving frequency and the still-air heaving damping. The latter, in the context of flutter-based energy
generators, can be assumed to simulate the energy extraction process25), as it pumps out energy from the
fluid-structure system.
2. METHODOLOGY
(1) Analytical linear modelThe stability of the system is investigated by means of linearized models, arranged for the 2-DoF problem, within the assumption of small perturbations around the equilibrium position. The self-excited loads
are given by Theodorsen’s model12), which derives from potential flow theory applied to a theoretical flat plate
in conjunction with the Kutta condition. The investigated width-to-depth ratio of the cross section is 25:1 and
it is large enough for the oscillation body to be idealized as a flat plate26).
The 2-DoF flutter problem (Figure 1) and the governing equations are reported in a non-dimensional
form in Eqs. 1-2, assuming the heaving and pitching DoFs in the form ( ) = ∗ ( ) and ( ) =
∗ ( ∗)
, where n and φ* are respectively the frequency of oscillation and phase difference at flutter onset: 1 + − 1 ( )− ( ) = , ( , , , , ) (1 + ) − 1 ( ) − ( ) = , ( , , , , ) (1) , ( ) = − −2+ ( ) ( ) − 2 + ( ) + 2 + ( ) 14− ( ) ; , ( ) = − 2 + ( ) ( ) + 1 64+ 2 + ( ) + − 2 1 4− + ( ) 1 4− ( ). (2)
First Internati of rate-indep of the mech damping stands for electromagn aerodynamic is Theodorse playing a ro elasticity an moved air; , unknown = ; In Eq. 3: is the sta Figu (2)
Experim
The CRIACIV, in (width×heig speed up to 3 The a linear mech (1004 mm) o arranged to ends of the b a stiffness o ends of each heaving stiff Moreover, th order to dec allowed to c devised to v dissipation g magnets (ne the heaving ional Symposi pendent dam hanically un = , + the structur netic damper c coefficients en’s circulato ole in the flut nd of the cen, non-dime n non-dimens
= =
is the flow d atic mass unb
ure 1: Sketch
mental cam
experimenta n Prato, Italy ght). The 156 30 m/s throu aeroelastic s hanical beha of two pairs provide a fr beams, while f 195 N/m an h frame thro fness. Each f he model axi couple the tw control the p vary the masgenerated on odymium N4 DoF. ium on Flutter mping12) for th ncoupled sys , and ral damping rs. The para s, which for ory function tter problem ntre of mass; ensional radi sional flutter ; = 2 density; a balance; d of the flutte
mpaign
al campaign y. The facilit 6 kW fan, pl ugh an inverte etup (Figure avior. The s of aluminiu rame with sh e the other en nd the bottom ough steel ca frame was eq is was conne wo degrees o position of t ss centre by n an alumin 45, 35×20 mr and its Appli
he heaving a stem in still = , . , while ‘E’ ameters ’ a flat plate a 12) depending m are: and , proportio ius of polar i frequency; ; = and are t denotes the s r problem of n was condu ty is about 22 laced downst er and/or con 2) was spec stiffness of m beams (or hear-type de nds are clam m one of 95 ables, in ord quipped with ected to the e of freedom. the elastic c adding calib nium plate ( mm) at close d ication, 2016 and pitching air, and the Concerning ’ for the ex
and ’ in
are equal res
g on = d , non-d onal to the r inertia; , fr , reduce ; = the inertiae in span of the m f a two-degre ucted in the 2 m long and tream the te ntrolling the cifically desig the heaving r blade-sprin eformation th mped by fixed N/m), place der to compe h a clock spr elastic suppo In addition, centre, while brated mass (5 mm thick) distance, wer DoFs, as ob ey are used the notation xternal dam n Eq. 2 are pectively to / = /2. dimensional ratio of the m requency rati d flow veloc ; = n the heaving model. ee-of-freedom e open-circu d the geomet est section, a pitch angle o gned to allow g DoF was ngs, 50 mm w hanks to a V d supports. T ed outside th ensate the st ring (with a orting frame the connect e two rocker es. Damping ) moving be re installed to btained throu here instea n of an mping that c the slopes 2 and 2 . The other d position resp mass of the m io in still air city or r ; g and pitchin m, two-dime uit, boundary
try of the tes allows contin of the rotor b w large ampl controlled wide and 3 m Vierendeel gi Two linear sp e test section tatic deflecti rotational sti s by means o tion of the f r arms fixed g devices, ba etween a pa o introduce l ugh free-vibr ad of the ra nd , the can be vari of the lift (1/4 + ) dimensionles pectively of model to the of the uncou reduced freq = = ng DoFs, resp ensional flat p ry layer win st section is 2 nuously vary blades. litudes of osc through the mm thick). E irder connec prings (the to n, were linke ion and cont tiffness of 2.0 of radial bal flat plate to d to the mod ased on the air of circula linear viscou ration decay tio-to-critica subscript ‘S ied with the
and momen , while ( ) ss parameter the centre o e mass of the upled modes quency: 2 (3) pectively; plate. nd tunnel o 2.41×1.60 m ying the flow
cillation with e free length Each pair wa
cting the free op one having ed to the free tribute to the 057 Nm/rad) l bearings, in
the end tube del axis were eddy-curren ar permanen us damping in s al S’ e nt ) s f e s; f m2 w h h s e g e e ). n e e nt nt n
The range of 200 rolling motio rotation with rocker arms 2000 Hz. Figure 2: Fr Vierendee heaving mo 0 mm), point on. A third l h increasing to record the rontal view o el girder with
otion was rec ting the ends
aser transduc flow speed. e pitching m of the aeroela h clock sprin corded throu s of the shear cer was poin Two miniatu otion up to l
astic setup (to g and hookin
ugh two ana r-type frame nting directly urized accele large amplitu op); close-up ng system fo alogic laser d s, so enablin y the model t erometers we udes. The sam
ps of the dam or the elastic/ displacement ng also the m to check for ere installed mpling frequ mping devices /mass centre nt transducer monitoring of a possible m at the ends uency during s (bottom-le control (bott s (measuring f the possible mean pitching of one of the the tests wa ft) and of the tom-right). g e g e s e
First Internati
The the geometry flow and the Circular alu ensure time-The 4 mm deep 541 mm, cal 0.43° to com the whole se limit the unw rest position amplitudes a The 0.7%. The m corrected th (defined as The s verified thro inertia of th previously e the frequenc configuratio order to eval ( , ). H The static u coupled pitc Configuration #1 #2 #2a #2b #3 #3a #4 Figure ional Symposi suspension s y of a NAC e unwanted uminium end -averaged tw steel model (D), the sma lculated as th mpensate the etup as symm wanted rollin n, was about around 90°. tests were c mean flow s hrough know = / stiffness line ough static te he oscillating stimated stif cies were m n, several fr luate the natu Higher levels unbalance, ching-heaving Table 1. Pa n [kg/m3] 1.15 1.17 1.16 1.17 1.16 1.15 1.16 3: Static mea ium on Flutter system was s CA0020 profi aeroelastic e d-plates, 400 wo-dimension had a rectan aller dimens he distance b vertical inci metric as po ng motion. T t 0.25%. Th onducted in speed was m wn flow map , with = 1 earity of the a ests (Figure 3 g system in ffness and the measured for ee decay tes ural frequenc of were , was estim g motion27). arameters of [kg] [kg·m 8.058 0.031 7.867 0.031 ˶ ˶ ˶ ˶ 7.872 0.029 ˶ ˶ ˶ 0.031 asurements t
r and its Appli
sheltered from file, limiting
effects of the 0 mm large a
nal flow cond ngular cross ion being th etween end-p idence of the ssible with r The blockage
his value inc smooth flow measured by ps to infer t 15 mm2/s) d aeroelastic se 3), measuring the heaving e correspondi different ad ts (Figure 4) cies of oscill e reached thr mated by me Table 1 sum the configur m2] [kg·m] 47 0.073 65 0.045 ˶ ˶ 02 0.000 ˶ 65 0.045 to verify the ication, 2016 m the flow b both the dis e oscillating and 1.5 mm ditions. s section wit he one facing plates. The m e flow in the respect to th e ratio, calcu creased up t w conditions y means of a the velocity during the tes etup, both in g static displa g and pitchin ding frequenc dditional ine ) were perfor lation ( ,
rough the ele easuring the mmarises the c rations tested [Hz] [Hz] 1.886 1.820 1.731 1.814 ˶ ˶ ˶ ˶ 1.731 1.895 ˶ ˶ ˶ 1.814 linearity of h by means of sturbances p supporting thick, were th sharp edg g the wind. model was ar e test section he along-win lated in a ve to 6.25% w s with a free-a Prfree-andtl tub at the mode sts was in the the pitching acements for ng motions, cy of oscillati rtias, confirm rmed in still ) and the ectromagneti frequencies characteristic d during the e [%] [% 0.05 1.6 0.04 1.2 9.52 ˶ 18.13 ˶ 0.05 1.2 9.52 ˶ 0.05 1.2 heaving (left two screens roduced by elements (m also provid ges and it wa The free sp rranged with n. Much atten d centreline rtical plane c when the mod
-stream turbu be installed el centreline e range 33,00 and heaving r different kn and , w ion in still air med the resu
air for diffe ratio-to-criti c system, up s of oscillati cs of the test experimental %] [-] [ 67 0.00 0.0 24 ˶ 0.0 ˶ ˶ 23 ˶ 0.0 ˶ 24 ˶ 0.0 ) and pitchin s having a no the setup to mainly the bl ded to the m as 100 mm w pan of the m h a nose-up a ntion was pa of the mode crossing the del experien ulence inten upstream th e. The Reyn 00 to 107,00 g degrees of f nown loads. were calcula r. Dynamic te ults. Moreov erent initial c tical damping p to a value o ion of the sy ted configura l campaign. [-] [-] 091 2708.9 057 2604.7 ˶ 2615.4 ˶ 2605.3 000 2617.1 ˶ 2660.6 057 2627.6 ng (right) stif ose following o the inciden lade-springs) model ends to wide (B) and model (S) wa ngle of abou aid to arrange el in order to model in the nced pitching nsity of abou he model and nolds numbe 0. freedom, wa The effective ated from the ests, in which ver, for each conditions, in g coefficient of about 18% ystem during ations. [-] [-] 0.625 0.965 0.634 1.04 ˶ ˶ ˶ ˶ 0.607 1.095 ˶ ˶ 0.634 1.04 ffnesses. g nt ). o d s ut e o e g ut d er s e e h h n s %. g 5 8 5 8
Figure 4:
3. RESULT
The critical cond system ident different am get informat and the stab featured an oscillation w response wi amplitude-ve LCO amplitu even at close to the end of As pr positive mas importan region in wh components Considering (Figure 6). In as suggested 1 Figure 5: having low linear th 1 : Example of freLTS
experiments dition was th tification, an mplitudes wer tion on the u le subcritica instability m was registere ith respect to elocity path tudes and to e distance. A f the sub-crit redicted by t ss eccentricit ntly changes hich the motare strongl g also configu n fact, in the d by the incre Build-up of w heaving da eory; 2) tran 1 η f free-decay t e-decay osci s were condu heoretically c nd then comp re released fo unstable bran al branch of t mechanism w d (Figure 5) o the nonlin was recorded achieve a fr Amplitude-ve tical branch w the linear the
ty (Figure 6) s the characte tion centre o ly modified uration #1, t e case of ease of the ph f flutter oscill amping. Thre nsition regime adju = 9.52% = 18.13% 2 #3 α
tests for high illation in the ucted in a sy conducted on pared with th or some flow nch position, the amplitud with sub-crit , identifying near aeroelas d for a time l requency res elocity points when the osc ory, the insta . Moreover, eristics of th of rotation li as well, pr he ratio ∗ 1, the cent hase angle to lations for sy ee phases are e due to the a ustment; 3) s % % 3 h levels of he e case of stru ystematic wa n the basis o hose experim w velocities c separating u de-velocity pa tical bifurcat g a transition stic loads du long enough olution in th s with increas cillation died ability thresh comparing c he motion. Th ies from ups roducing larg ∗⁄ seems ∗ tre of rotation o about 164° ymmetric (le e highlighted activation of steady-state L eaving damp uctural damp
ay. For each f the govern mentally obs lose to the th up to the crit ath. Indeed, tion. After t regime in w ue to the lar (about 200 s he spectra to sing and dec d out. hold is signifi configuration he phase ∗ stream to do ger pitching to be import n lies in a sm , and the hea
eft) and mass d: 1) exponen
f nonlineariti LCO amplitu
1
ing (left) and ing only (rig
h configuratio ning paramet erved. Sever heoretical flu tical threshol as a general the flutter on which the sys
ge-amplitude s) both to allo correctly de reasing flow icantly antici ns #3 and #2 varies from wnstream th g-heaving am tantly affecte mall region do aving compon -unbalanced ntially growin ies of self-ex udes. 2 α # d comparison ght).
on, the estim ters, as obtai
ral initial con utter boundar ld the stable l result, all co nset, the bu stem marked e motion. T ow the stabil etect the freq w speed were ipated in the 2, the sole mo m 12° to 152° he midchord. mplitude rati ed by the fre ownstream t onent is enhan d (right) conf ng motion ac xcited loads a = 0.0 η #2 n with the mation of the ned from the nditions with ry, in order to rest position onfiguration ild-up of the dly adjusts it he following lisation of the quency peak observed, up case of smal odification o °, moving the . The motion ios ∗⁄ ∗ equency ratio he midchord nced. figurations ccording to and phase 05% 3 e e h o n s e s g e s p ll f e n . o d,
First Internati Figure 6: C Fig ional Symposi Comparison o gure 7: Effec ium on Flutter of amplitude difference f cts of high le
r and its Appli
e-velocity dia for the config
evels of heav ication, 2016 L (em agrams for h gurations wit L (em ving damping Legend: #1 = #2 = #3 = critical thr mpty markers eaving and p th low heavin Legend: #2 = #2a = #2b = critical thr mpty markers g on the mass = 0.09 , = = 0.06 , = = 0.00 , = reshold by lin are points wi pitching com ng damping. = 0.06 , = 0.06 , = 0.06 , reshold by lin are points wi s unbalanced = 0.96 , = 1.07 , = 1.07 , near analysis ith decreasing mponents and = 1.07 , = 1.07 , = 1.07 , near analysis ith decreasing d configurati = 0.05% = 0.05% = 0.05% s flow speed) their phase = 0.05% = 9.52% = 18.3% s flow speed) on.
Figure 8: Ef configurati In ge shown in Fig more the hea 9.5%. By co the oscillatio preserve the the lower bo and post-crit Acco configuratio flutter instab artificially d anyway. Thi regime and t the aeroelast adjustment a stall12), the m In the damping ( the symmetr heaving dam seems to ten condition wh ffects of high ion #3a is sta eneral, the in gure 7. In pa aving amplit ontrast, a furt on amplitud main qualita ound of the s tical branche ording to th n #3 fully sta bility, steady disturbed thr is result hig those respon tic coupling and loss of d motion is gov e case of sma = 0.05%) ric configurat mping is equ nd to 90° for v hile exhibitin h level of hea able accordin by ncrease of he articular, for tudes first sig ther increase de. In additio ative features subcritical br es. he linear the abilizes the s y-state oscilla rough large hlights the d nsible for the between stru damping, whe
verned by hy all mass unba
to about 133 tion (Figure ual to 9.52% very high hea ng self-sustai aving dampin ng to the line y the release eaving damp the specific gnificantly re of the damp on, the critic s, such as the ranch, or the eory, the inc
system. Ther ations were a initial cond differences b incipient cla uctural mode en massive f ysteresis loop alance (Figur 3° for = 8) the phase . In general, aving dampin ined oscillati L (em ng on the sym ear theory bu e of large init ping nonlinea case of conf educe for an ping, up to 18 cal condition e sudden jum nearly linear crease of ex refore, althou achieved, as ditions, so th between the assical flutte es and respe flow separati ps in the nonl re 7), the pha 9.52% and angle starts f the phase d ng, suggestin ions in the ca Legend: #3 = #3a = critical thr mpty markers mmetric con ut it exhibits s tial condition arly modifies figuration #2 increment o 8.3%, does n n is postpon mp at the insta r evolution w xternal dam ugh configur shown in Fi hat an ampli self-excited er instability. ective fluid-d on occurs at linear fluid-d ase ∗ decre to about 121 from about 1 difference be ng that the sy ase of very h = 0.00 , = 0.00 , reshold by lin are points wi nfiguration. It steady-state o ns. s the amplitu 2, it is clear th f the heaving not produce a ned but the
ability thresh with the flow mping in the ration #3a is igure 8, prov itude-velocity loads that d In particular dynamic reac high angles dynamic load eases from ab 1° for = 2° and increa etween pitchi ystem adjusts high mechani = 1.07 , = 1.07 , near analysis ith decreasing t is worth rem oscillations, ude-velocity that the pitch g damping fr any dramatic amplitude-v hold and the d w speed of th
e case of th theoretically vided that the ty diagram w dominate the r, while this ctions that le of attack du ds. bout 152° for 18.13%. By ases up to ab ing and heav s the motion ical dissipati = 0.05% = 9.52% s flow speed) marking that if triggered diagrams, a hing and even
rom 0.05% to c reduction o elocity path drop-down a he sub-critica he symmetric y not prone to e system wa was obtained e steady-state last relies on eads to phase ue to dynamic r low heaving y contrast, fo bout 61° if the ving motion to an optima on. t s n o of s at al c o s d e n e c g r e s al
First International Symposium on Flutter and its Application, 2016
In all the tested configurations, the theoretical and experimental critical conditions are in good agreement. Moreover, the instability threshold always lies close to the change of slope between the sub-critical and the post-critical branch, probably suggesting a slightly different excitation mechanism in the two ranges. It is also worth noting that, even though the increase of heaving damping stabilises the system, mainly through a reduction of the motion amplitudes, the unstable branch gets closer to the rest position, so that smaller perturbations can be sufficient to foster the steady-state motion in the sub-critical range.
4. CONCLUSIONS
A systematic experimental approach was used to explore the behaviour of a flat plate model with rectangular 25:1 width-to-depth cross section prone to two-degree-of-freedom classical flutter. Analytical linear models supported the investigation of the flutter critical condition. Wind tunnel tests on a spring-mounted sectional model were conducted to verify the critical condition and to investigate the large amplitude oscillations in the post-critical regime for several dynamic configurations. The key role played by the position of the mass centre and by the damping in the translational degree of freedom was mainly studied.
A small mass unbalance significantly anticipates the instability threshold and modifies the characteristics of the motion in terms of ratio of pitching to heaving components and phase difference. For the investigated set of configurations, the introduction of high levels of heaving damping reduces the motion amplitude and postpones the instability threshold. The amplitude-velocity diagrams are not distorted by increasing the external damping, maintaining a linear evolution with the flow speed of the sub-critical and post-critical branches. By contrast, the slopes of these branches nonlinearly depend on the damping. Steady-state oscillations were found also for a configuration theoretically stable with large damping provided that the motion is artificially triggered through a large enough initial condition
The present results can form the basis for the development of numerical nonlinear models to predict large-amplitude post-critical oscillations that are triggered by the classical-flutter instability but whose evolution to the steady state is mainly driven by the dynamic-stall mechanism.
In the near future, the investigation will be extended to other dynamic parameters governing the flutter problem, in order to achieve a more complete knowledge of their influence on the post-critical flutter regime.
5. ACKNOWLEDGMENT
The authors gratefully acknowledge the contribution of Mikel Ogueta during the experimental tests.
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