NASA‘ Technical MemoranIiiIm'8l6l3
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`5
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`(
`3
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`(HA5A-TH-81613)
`HIGH 59330 ra:s guésa)
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`SUPEBSOBIIC STALL FLUTTBR OF
`15 p ac A02/HF A01
`CSCL 011
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`rm-1u91a""’
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`3"
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`5.3/01
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`Unclaa
`2969“
`
`Supersonic Stall Flutter
`of High Speed Fans
`
`~
`
`J. J. Adamczyk and W. Stevans
`Lewis Research Center
`Cleveland, Ohio
`
`and
`
`R. Jutras
`General Electric Company
`Evendale. Ohio
`
`
`
`Prepared for the
`Twenty-sixth Annual International Gas Turbine Conference
`sponsored by the American Society of Mechanical Engineers
`Houston, Texas. March 8-12, 1981
`
`NASA
`
`UTC—2005.00l
`
`GE v. UTC
`
`Trial IPR2016-00534
`
`

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`SIIPERSGIIIC STALL FLUTTER or HIGH SPEED FANS
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`J. J. Admlczyk and ii. Stevans
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`llational Aeronautics and Space Administration
`Lewis Research Center
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`Cleveland. Ohio
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`R. Jutras
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`General Electric Company
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`
`Evendale. Ohio
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`INTRODIKIT ION
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`Flutter can result in costly (both in time and
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`money) overruns in turbofan-engine development pro-
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`grams. Solving the problem of flutter (at the
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`engine development stage) may well mean major engine
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`redesign and retesting.
`for this reason. engine
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`manufacturers and goverrmient agencies are currently
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`supporting numerous research programs in an attemt
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`to develop flutter prediction systems that can be
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`used to design flutter-free engines.
`To date, these
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`research programs have identified five regions of
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`the conpressor performance on
`where flutter is
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`generally encountered (fig. 1 .
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`Of the five regions shown in figure 1. the
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`supersonic low back-pressure flutter region has been
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`the most thoroughly investigated analytically
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`(refs.
`1 to 3).
`In general. these analyses have
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`considered the flow field through a cascade of two-
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`dimensional air-foils undergoing simple harmonic
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`pitching or plunging. The gas stream was assumed to
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`be an inviscid. nonconducting. perfect gas.
`Shock
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`waves that originated in the flow field were assumed
`to be weak so that supersonic small-disturbance the-
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`ory could be used. Although these assumptions ap-
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`pear to oversimplify the flow conditions encountered
`by a rotor at the onset of flutter. the flutter
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`boundaries predicted by these analyses correlate
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`well with experimental data.
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`A recent analysis (ref. 4) has atteapted to
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`apply the supersonic.
`linearized. small-disturbance
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`theory to higher backpressure operating conditions
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`(i.e.. region xv of fig. 1) by including a finite-
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`strength shock wave within the cascade passage.
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`Results from this analysis show that the unsteady
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`motion of the shock wave tends to induce bending
`flutter. The existence of this flutter mode is
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`documented in reference 5.
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`The remaining operating region where supersonic
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`flutter occurs (region V of fig. 1) lies close to
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`the stall line of a stage. Analyses of this region
`have not qipeared in the open literature. Litera-
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`ture on this subject (e.g., refs. 5 to 7) generally
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`presents experimental data to document the extent of
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`the flutter region. These data provide a very
`limited base from which an empirical correlation can
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`be derived for predicting the onset of this flutter
`mode.
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`The objective of the present analysis is to
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`develop a model for predicting the onset of super-
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`sonic stall flutter encountered by rotors that do
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`not have part—span dampers or tip shrouds.
`l-.xperi-
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`mental data reveal that the flutter mode is gen-
`erally the first flexural mode at a reduced fre-
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`ouency (based on tip relative velocity and tip semi-
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`chord) of about 0.2.
`The vibratory pattern around
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`the rotor tends to be very regular: All blades
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`vibrate at the same frequency but are shifted in
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`phgse by 1; positive interblade phase angle of about
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`to 50
`(ref. 7). This positive phase shift
`20
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`implies that the vibratory pattern is traveling
`around the wheel
`in the direction of rotation (i.e..
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`a forward-traveling wave).
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`A further characteristic of the flow regime
`associated with stall flutter is shown by the
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`steady-state pressure distribution in figure 2.
`This pressure distribution was produced from mea-
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`surements taken across a rotor tip while the rotor
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`was operating near the flutter boundary. This fig-
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`ure clearly indicates a detached,
`leading-edge bow
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`shock wave inoinging on the suction surface of the
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`adjacent blade.
`The large extent of the compression
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`region at the base of the shock wave seems to sug-
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`gest a separated flow region that would increase in
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`size as the fan operating point moved toward the
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`supersonic stall bending flutter boundarv.
`
`UTC-2005.002
`
`UTC-2005.002
`
`

`
`The present analysis develops a flutter Iodel
`for the highly conlicated flow field illustrated in
`figure 2 by using two-dinansional actuator disk the-
`ory.
`The effects of flow separation are included in
`the nodal
`through rotor-loss and deviation-angle
`correlations.
`or low-speed flows. actuator dist
`flutter Q1303“ have bd:n able
`onset
`of a si
`e-deoree-of- reedoa
`r lode
`ng
`u
`(e.g.. 3n. 8 and 9). The success of the aodels at
`low speeds suggests that a coqiressible actuator
`dist nodal night be capable of predicting the onset
`of bending flutter at supersonic speeds. This hy-
`pothesis has been confined by a number of calcula-
`tions. These calculations. however. did require as
`irout the interblada phase angle at the onset of
`flutter.
`So that this requirennt could be avoided.
`the actuator dist nodal was aodifieo to allow for
`uoderate values of inter-blade phase angle. This
`codification results in a flutter nodal that can be
`developed into a flutter prediction systu. The
`validity of this flutter prediction systu is daun-
`strated by covering the predicted flutter boundary
`of a high-speed fan stage with its Ieasured boundary.
`
`where the tilde denotes a tine-dependant variable
`and the bar si
`ifies a steady-state varinle. The
`-:":’:':..........~'- ‘- :2. *:..-~ ‘l'.""'....."i'...
`s
`ve oc
`s.
`was a
`n
`respectively.
`see fig. 5 for definition of coordi-
`nate systaa.)
`n addition the use of repeated
`indices denotes sination with respect to the re-
`peated indea. The linearised nonantu equations for
`the flow fields are
`
`-
`1
`-
`'5
`51-
`._..£.‘...u‘.“ .1; “arr. 1.1.2
`
`(2)
`
`The
`is the nondinensional pressure.
`where 5..
`ruaining field equations are the energy equation
`for an inviscid. aonconducting gas
`
`Q.-.uh_.;;:_'.o
`
`(3)
`
`FORIIILATION
`
`and the equation of state
`
`The present analysis for stall bending flutter
`of an isolated rotor eodels the rotor as a two-
`dinansional cascade of airfoils.
`The cascade is
`defined by the blade-elaent geoaetry on a cylindri-
`cal surface at a distance it
`fro: the axis of rota-
`tion.
`The flow field in the cascade plane is
`assuaed to be two dinensional. cowpressible. and
`tiae dependent. Viscous forces are considered only
`within the blade channel.
`The unsteady flow vari-
`ables associated with the rotor vibratory option are
`assumed to be snaller than their steady-state coun-
`terparts. These variables at an instant in tires are
`required to be periodic around the wheel at a period
`equal to a fractional part of the circuaferential
`distance d - Zail.
`The notion of the ai-foils in the cascade plane
`is restricted to sinle harnonic plu
`ing and edge-
`wise notion at a cyclic frequency -nffig. 3).
`n
`addition the raotion of each airfoil at an instant in
`tile is assuned to be shifted froo that of its
`neighbor by an interblade phase angle of
`o - Zunlh,
`where
`n
`is an integer and
`I
`is the humor of
`blades in the rotor.
`In the present analysis the
`reduced-frequency paraneter
`k
`and the inter-
`blade phase angle are assumed o be saall.
`(Reduced
`frequency In, - ab/3.... where
`b
`is the semi-
`chord of an airfoil and ‘C...
`is the aagnitude of
`the relative inlet velocity in the cascade plane.)
`
`ations
`Field E
`TF3 governing linearized equations for the flow
`field upstream and downstream of the cascade are
`written with respect to the relative coordinate sys-
`tea.
`The variables in these equations are non-
`dinensionalized by the steady inlet speed 3...
`the circumferential distance around the wheel a.
`the inlet steady-state fluid density ’ . and the
`inlet steady-state fluid temperature
`In
`addition the subscripts (u,
`.a) signify oomstrean
`and upstreaw flow variables.
`The linearized continuity equation for the up-
`stream and downstream flow fields is
`
`'
`a- _
`:5
`_
`'.‘:t:’ui.--a1.E'';t-‘a§‘‘'.‘'°
`
`”’
`
`'-‘Er.
`" i3'°‘r..
`The variables wipearing in equations (3) and (4) are
`the entropy s; the ratio of specific heats
`; and
`the inlet steady-state relative liach nufier L.
`+The’ént;-opy was nondiunsionalized with respect to
`-O 0
`The present analysis assumes that there are no
`sources of entropy or vorticity upstrea of the cas-
`cade.
`The upstren. unsteady velocity field lust
`therefore be irrotational and hence is equal to the
`gradient of a potential function e... where
`
`.:__
`-
`101. 2
`U‘.__ car’
`The unsteady velocity field downstreaa of the cas-
`cada is expressed as
`
`(5)
`
`(5)
`
`-
`03.
`1- 1. 2
`U“. --’xT’r§'_
`:3 al
`represen s the irrotational cowo-
`where
`nent of t
`f eld and
`..
`its rotational
`cononent.
`The source o
`the rotation field is the
`vorticity shed by the oscillating air-foils.
`The solution to the field equations is obtained
`by assuoing a simle hanaonic spacial and teworial
`dependency for the potential,
`the rotational veloc-
`ity fields and the dowrstreaa entropy field. The
`details of the solution procedure can be found in
`reference 10.
`
`Condition
`Iounda
`It is shown in reference 10 that four boundary
`conditions Iust be specified at the actuator disk to
`relate the unsteady flow field to the cascade defor-
`Iation and the steady inlet flow conditions.
`The first of these boundary condition requires
`the flow to be continuous across the deforming
`disk.
`The analytical for-Ia of this boundary condi-
`tion is
`
`UTC—2005.003
`
`

`
`_..r.u..........-....
`
`I
`
`V
`
`7
`
`,
`5
`
`u o :__;u',‘;}_ - (:_ « 3_)z:',‘,f,‘,
`U‘
`represents the noreal velocity
`:..'-:..';i.:e'*-
`i relative to the deforming dist. These
`velocity consonants to first order (i.e.. shall
`cascade-plane deflection) are equal to
`
`(7)
`
`a3__
`‘U1,-0.-IxT-I
`an
`o
`
`3
`
`‘U2.-TF2
`
`(8)
`
`‘
`U.-
`U20“?!-2‘. . fit‘
`“:f.'U1.-‘sift?’
`where
`is the displacement of the dis.’ from
`its mean position in the axial
`(X1) direction
`(fig. 4).
`The next boundary condition requires the total
`enthalpy with respect to the deforning CCIIMIOP (I 1 SK
`to be locally conserved across the disk.
`The equiv-
`alent mathematical statement is
`
`(9)
`
`1
`1
`1
`2 Rel
`"-1,- 32' L. ’ 2 ‘LI.
`
`1 2.iiel
`1
`1
`'7‘-Ti!’ ‘- ' 2 ‘L
`-0
`
`OKXIIO
`
`(10)
`
`where T...
`is the u streui and doivnstrearn gas
`teaperature and
`«:3:
`is the magnitude of the
`relative velocity appzafining and leaving the disk.
`The expressions for
`are
`
`2
`
`2 1/2
`
`a-ll.-O’-C.
`
`1
`
`(14)
`
`in this equation is the angle
`The variable a.
`between the chord line and the tangent to the cuber
`line at the leading
`e. The corresponding rela-
`tion for the local dmati n angle is
`
`s-a'_-e‘-s
`
`"2
`
`(15)
`
`relative exit flow angle
`is the local
`a:.
`where
`measured from the nonial to the trailing-edge plane
`and
`a
`is the angle between the chord line
`and the angent to the caper line at the trailing
`N9‘!
`
`Relationships for the angles. inlet liach number
`and cascade solidity in the above equations are
`derived by expanding these variables along with
`The
`equations (13). (14) and (15) to iirst order.
`details of this expansion can be ftund in refer-
`ence
`.
`The last boundary condition specifies the local
`entropy that is generated as the fluid passes
`through the cascade.
`The loss associated with the
`local entropy rise is defined in terms of a loss
`coefficient defined as
`
`Rel
`el
`-9o.
`pg
`03
`'1-‘"—r.r€T
`
`(16)
`
`in this
`The variable 95¢}, - P891
`equation represents the local total-pressure loss
`neasured relative to the deforming actuator disk.
`The entropy rise across the disk is related to this
`quantity by the equation of state
`
`*2)
`0'35.‘ -("1.~.*-ax,—'sr') ’("2,-‘si--T
`
`a;__
`
`3711
`
`a;__
`2
`
`Pilel
`
`0,-
`
`_ 93:1. _ pail. (1_ e'l"E..5..)
`
`:3
`
`- (vi... ~ as o 81..
`
`1
`
`(ll)
`
`Thus
`
`5., must be equal to
`
`-
`
`2,Rel
`I
`1°-Iq--
`X
`S.--Ftn I-ET (17)
`
`:3,
`
`a (U2... 9 ix; + fig" -
`
`1/2
`
`2
`
`_,
`N12
`Ti-
`
`loss co-
`The analysis assumes that the local
`efficient
`x
`is related to the local inlet flow
`field. as observed with respect to the deforming
`disk. by a relationship of the fans
`
`(12)
`
`where
`R
`is the local displacement of the cas-
`g
`cade in he tangential (X2) direction (fig. 4).
`The third boundary condition requires that the
`local inlet and exit ilovr angles relative to the
`deforming disk satisfy a deviation angle correlation
`of the f0TI'|
`
`6 - gin. H__. u. a‘)
`
`(13)
`
`0
`where
`is the local. relative incidence angle.
`H
`is the local. relative inlet liach number,
`p
`_.
`is the local solidity of the deforraed cascade. and
`o‘
`is the local stagg
`er angle measured from the
`The
`normal
`to the leading-edge plane (fig. 4).
`local. relative incidence angle is
`related to the
`local. relative inlet flow angle
`a... measured
`from the normal to the leading-edge plane by the
`EQUIHON
`
`x- Xiu.
`
`li__. u. 0')
`
`(18)
`
`As in equation (13) the parameters appearing in this
`equation can be related to the unsteady flow field
`approaching the cascade and the cascade defc'M-
`tion. Upon combining equation (17) with equation
`(18)
`the entropy rise across the cascade can be
`evaluated in terms of these perturbed quahities.
`
`_A_egod§narnic Force
`av rig described the procedure for r£e°.9ru_ining
`the flow field surrounding the cascade.
`the unsteady
`forces acting on the cascade and hence its stability
`can be assessed.
`The aemdynanic force acting on
`the cascade is obtained by considering the flow
`field through I control value that
`is fixed to a
`cascade passage (fig. 3).
`The present analysis
`assures that the relative motion between neighboring
`airfoils. a measure of which is the interblade phase
`
`UTC-2005.004
`
`3
`
`

`
`
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`a
`la.
`is small and that the reduced frequency of
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`th s notion is also snail.
`The first assumption
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`implies that the spatial variations of the flow
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`variables across a cascade passage are small and
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`thus can be neglected.
`The low-frequency assusution
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`suggests that the rate of cha
`e of mass and momen-
`tum within the control volume
`s smaller than the
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`net mass and momentum flux across the control volume
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`surface and thus will also be neglected. Based on
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`these considerations the linearized momentum equa-
`
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`tion for the control volume illustrated in figun 3
`can be written as
`
`
`
`
`
`
`
`F, - (9, - n__)? * (n_ - n__)*
`
`o i“°‘(U,.
`
`- lIl___) o il(Gl‘_ - GL__)
`
`()9)
`
`
`
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`
`
`£2 ' (B; ‘ 3_,)733
`
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`
`
`(20)
`o i7“"(ll '_ - ll2‘__) + ll(G2‘_ - uZ___)
`
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`
`is the force exerted by the airfoils on
`where F,
`the control volume, r
`is the pitch of
`the cascade.
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`is the mass flow exiting the control volume.
`NR9‘
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`
`and hi
`is the unit vector normal to the exit
`plane of
`the control volume.
`(This equation can be
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`derived by expanding the quasi-steady momentum equa-
`tions for the control volume shown in figure 3 to
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`
`
`
`
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`first order in perturbed variables.
`The details can
`
`
`
`
`
`
`
`be found in reference 10.)
`
`
`
`
`
`The work done by the aerodynamic force on an
`
`
`
`
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`
`
`
`
`airfoil over a cycle of motion is equal to
`
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`
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`
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`
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`
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`Zelk
`
`0
`
`.5;
`‘S;
`’1‘iT’‘27t‘'
`
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`‘"
`
`W’
`
`l
`
`*‘xem"29¢
`
`
`
`
`
`
`
`
`the expres-
`where R; [
`] denotes the real part of
`sion and the asterisk superscript signifies the com-
`
`
`
`
`
`
`
`plex conyugate of
`the variable.
`The airfoil's
`
`
`
`
`
`
`
`
`motion can be described by
`
`
`
`
`
`
`H: X.
`_
`"‘_hle 22eikt
`
`-
`h‘ _ h
`4
`
`2
`
`9
`
`-
`ih.X
`2 2 em
`
`
`
`
`
`
`
`
`
`in the X2 direc-
`the wave number
`kg,
`where
`
`
`
`
`
`
`
`
`
`tquation (21) can be
`tion, is equal to kg - Zen.
`
`
`
`
`integrated to yield
`
`
`R2)
`"Aero ' 'j“(Fi;:)
`
`
`
`
`
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`
`
`
`
`
`
`the
`when }.a[ J represents the imaginary part of
`expression within the brackets.
`the onset of
`t
`
`
`
`
`
`
`
`
`flutter the aerodynanic work per cycle is equal to
`
`
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`
`
`
`
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`if
`the mechanical energy dissipated over a cycle.
`
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`the mechanical dissipative force is proportional to
`
`
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`
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`the velocity of the airfoil.
`the mechanical energy
`dissipated over a cycle of motion is
`
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`(83)
`
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`
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`
`
`where the dawning coefficient
`
`
`
`C
`
`
`
`
`
`is defined as
`
`
`
`an
`(24)
`cu,-'—"-uq—‘d-’
`
`
`
`
`
`
`
`
`
`
`in this equation is the log decre-
`The variable a
`
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`
`
`
`
`
`
`ment of the mechanical dating. and
`is the
`
`
`
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`
`
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`mass per unit length of a rotor blade. This vari-
`able is defined as
`
`
`
`
`
`lib - Kphtc
`
`
`
`
`
`
`
`
`is
`K
`is a constant of proportionality, an
`where
`
`
`
`
`
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`
`
`the blade density. 1
`is the average thickness of
`
`
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`
`
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`c
`the blade. and
`is its average chord length.
`
`
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`introducing this equation along with equation (24)
`
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`
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`
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`into the inteeuitad form of equation (23) yields
`9
`. .
`-
`if, (nlh; « nzig)
`
`no - 2eK
`
`(as)
`
`
`(25)
`
`
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`
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`
`
`
`
`kb?(hlh‘1 , "2"? ""'k2["'(F2"2)]
`‘onset ' " 'K(ob> _;_
`
`
`
`
`
`'- I
`
`I
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`
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`Equating equation (26) with equation (22) and solv-
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`
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`a establishes the wininuw level of each-
`ing for
`anical damping required for stability at an operat-
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`
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`
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`
`
`ing point as
`I
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`‘ “
`
`
`
`
`
`
`
`(37)
`
`
`
`
`
`
`
`
`
`
`denotes the minimum value of
`The symbol Mink
`
`respect to k .
`If the available
`the function wit
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`mechanical duping oi’ a rotor b one exceeds this
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`
`
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`critical level. any small bending notion iqiarted to
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`
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`the blade will decay in time. Hence the system is
`
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`stable and flutter will not occur. However.
`if
`
`
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`
`
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`
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`is greater than the available mechanical
`eon“,
`
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`
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`dawning of a rotor blade. any small b;-nding motion
`
`iuparted to the blade will cause the blade to flutter.
`
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`
`RESUL TS
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`For the flutter mode under study the amplitude
`
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`
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`of the motion increases monotonically along the blade
`
`
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`
`
`
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`span from the node line.
`For blades that are rigidly
`
`
`
`
`
`
`
`fixed at their aoot (i.e.. mechanically constrained),
`the node line can be assumed to lie outboard of the
`
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`blade platfonn (i.e.. outboard of the aerodynamic hub
`of the blade).
`The node line of the first flexural
`
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`mode of the rotor to be investigated later in this
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`section is approximately 20 percent of span height
`
`
`
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`
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`outboard of the platfcnh. Thus.
`the vibratory notion
`
`
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`
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`of the blade in the rotor at any spanwise section is
`
`
`
`
`hl . -4; sin 3'“°"
`
`
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`
`
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`
`
`hz - t cos WHO”
`
`
`
`
`
`(23)
`
`
`(29)
`
`
`
`UTC-2005.005
`
`UTC-2005.005
`
`

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`is m aqilitude of the motion at the sec-
`c
`where
`
`
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`
`
`
`
`
`
`tion and T‘(
`l
`is the stagger angle at 20 per-
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`
`
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`cent of span.
`The aerodynaic work per unit of Span at a
`
`
`
`
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`
`
`
`
`
`
`
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`given radial
`location depends on the Qlitude of
`the motion of the section and the relative dynuiic
`
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`pressure of the incoming strenline to the section.
`Thus because both increase with distance from the
`
`
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`
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`the total work done by the airstreae on a blade
`hub.
`
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`
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`is strongly influenced by the unsteady flow field
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`
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`surrounding the tip region.
`A simple calculation
`shows that this influence is concentrated in the
`
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`
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`outer 25 percent of the span.
`If the outer-casing
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`boundary l
`er is assumed to influence the flow
`
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`field over
`percent of the tip region of a blade. a
`
`direct correlation might exist between the flutter
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`boundary of a rotor and a tin.)-dimensional cascade
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`whose geometry and dynamic response coincide with
`
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`that of the rotor at 85 percent of span (i.e.. alge-
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`braic Iiean of 75 and 95 percent of span).
`The re-
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`
`
`sults presented in the ruaainder of this section are
`
`based on this premise.
`
`
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`The objective of the first series of calcula-
`
`tions is to establish the influence of reduced fre-
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`quency it
`and -interblade phase angle on the
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`
`
`aeroelast c stability of a cascade of airfoils. For
`
`
`
`
`
`
`
`this study a normalized darroing parneter defined as
`
`
`
`
`- -
`
`
`a
`
`"
`
`1
`1 7..
`.
`
`
`7 Runs I: c
`
`
`
`
`
`
`-; - sin e‘
`
`- (201),
`
`"-
`
`4 F
`l 1
`
`
`e
`
`
`
`'“‘2"2 an)
`
`
`
`
`
`
`e
`
`
`
`29
`
`(30)
`
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`‘"‘2"2 earn)
`. co, ;.(zosl].. (5
`
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`
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`In and the inter-
`is calculated as a function of
`
`
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`blade phase angle.
`The parameter ‘
`in this
`
`
`
`
`
`
`equation is the inlet stagnation denslty measured in
`
`
`
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`
`
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`the absolute coordinate system.
`a). greater
`for
`
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`
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`than zero the airstreaiii is supplying energy to the
`
`cascade.
`if the mechanical damping of the cascade
`
`
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`system is zero. the operating condition would be
`
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`unstable.
`For values of
`less than zero the
`6"
`
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`
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`cascade is doing worii on the airstream. and hence
`
`the s stem is stable.
`
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`
`
`lving equation (30) requires as input infor-
`
`
`
`
`
`
`
`mation the steady-state inlet flow properties.
`the
`
`
`
`
`
`
`
`
`geometry of the cascade. and the frequency of os-
`cillation.
`A set of loss and deviation-angle cor-
`
`
`
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`
`
`relations must also be specified.
`The present
`
`
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`
`
`
`
`analysis makes use of the correlations derived in
`
`
`
`
`
`
`reference 10.
`The steady-state inlet flow condi-
`tions are derived from the blade-element data. Ilea-
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`sured at 85 percent of design speed. of the second
`
`
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`
`
`
`
`stage of the NASA 1450-ftIsec—tip-speed two-stage
`fan. This blade-elenent data set is reported in
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`reference 5.
`The cascade geometry represents the
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`
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`
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`geometry of the second-stage rotor element at 85
`
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`percent of span height from the hub.
`
`
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`
`
`figure 5 shows a plot of
`a
`as a function
`of interblade phase angle and reduced frequency
`
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`for the second stage rotor operating at the
`in
`flutter-boundary at 85 percent of design speed.
`
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`This figure shows that the bending flutter lode is
`associated with a positive interblade phase angle
`
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`(i.e.. implies that the vibration pattern is travel-
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`
`ing around the rotor in the direction of rotation).
`
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`The figure‘ also shows that increasing the reduced
`
`
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`
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`frequency stabilizes the motion at a constant inter-
`
`
`blade gihase angle.
`
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`as a function
`figure 6 shows a plot of
`an
`
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`
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`
`
`
`of interblade phase angle and percentage of design
`
`
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`
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`weight flow along the 5-percent-speed line.
`for a
`
`
`
`
`
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`given interblade phase angle. decreasing the weight
`
`
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`
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`low at constant wheel speed (i.e.. increasing the
`
`
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`
`
`steady-state aerodynamic loading) tends to
`destabilize the rotor.
`
`
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`The results presented in figures 5 and o are
`also consistent with the observed characteristics of
`
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`supersonic bending flutter reported in reference 7.
`
`
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`A deficiency of the present theory. as shown by
`
`
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`
`
`
`
`the results in figures 5 and b.
`is that it predicts
`
`
`
`
`
`
`
`the daming. an
`to increase monotionically with
`
`
`
`
`
`
`interblade phase angle. This trend is physically
`unrealistic. and is a consequence of the small
`
`
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`
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`interblade phase angle assuuption inherent
`in actua-
`tor disli theory.
`To overcome this deficiency an
`
`
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`additional assumption was incorporated into the the-
`ory which limited the value of the aerodynamic dani-
`
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`
`
`
`
`
`a
`ing.
`The assurrotion is made to replace
`by sin
`
`
`everywhere in the analysis. This assulntion is
`e
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`
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`mathematically constant with the actuator disk
`
`
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`model.
`figure 7 shows the results obtained with
`
`
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`
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`this assunptign incorporated into the analysis.
`
`
`
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`
`
`the modified analysis yields results
`for
`[o] 3 45 .
`that are conparable to those obtained from the
`
`
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`
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`
`
`
`
`
`
`
`
`
`original formulation.
`in addition the modified
`
`
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`
`
`analysis predicts a finite maximum value for
`on
`
`
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`
`
`o . S0 . This maximum value for duping is
`for
`
`
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`
`
`equal to the value computed at
`o - l radian using
`
`
`
`
`
`
`the original formulation. Experimental results of
`
`
`
`reference 7 show that this value for
`o (i.e.. o - 1
`
`
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`
`
`radian) corresponds rather closely to the measured
`
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`interblade phase angle of the least stable flutter
`
`mode.
`
`
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`
`
`An additional validation of the modified analy-
`
`
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`
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`sis as a flutter prediction system for supersonic
`
`
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`
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`stall bending flutter can be established by showing
`
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`that. for a given rotor, flutter will be observed
`
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`
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`whenever the predicted maximum value of
`an ex-
`ceeds the normalized structural oanping of a rotor
`
`
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`
`
`assembly.
`for this demonstration the measured flut-
`
`
`
`
`
`
`ter boundary of a scaled model of fan C of
`the NASA
`
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`
`
`Quiet Engine Program (ref. 6)
`is correlated with the
`predicted boundary.
`The performance map for the
`
`
`
`
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`
`
`
`
`
`
`
`
`
`scale model (fig. 8)
`shows two flutter instability
`The first zone is torsional flutter that
`zones.
`
`
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`occurs at part speed and lies close to the stall
`
`
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`
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`Just above this zone lies a zone of bending
`line.
`
`
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`
`
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`
`
`flutter that appears to extend past 95 percent of
`
`
`
`
`
`
`
`
`design speed. Only the bending flutter mode of this
`rotor is analyzed in this paper.
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`Figure 9 shows a plot of
`the phase shift be-
`
`
`
`
`
`
`
`
`
`tween the amplitude time history of a rotor blade
`and the reference blade No. 17 at the 95 percent of
`
`
`
`
`
`
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`
`
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`design speed flutter point.
`The data inplies the
`
`
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`
`
`
`
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`
`
`
`
`flutter mode is predominatly a 3 nodal diameter for-
`
`
`
`
`
`
`
`ward traveling wave which corresponds.to an inter-
`blade phase angle of approximately 41 . This value
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`for the interblade phase angle of the flutter mode
`correlates wel’. with the data of reference 7 and
`
`
`
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`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`lies within the unstable range predicted by the
`
`
`
`
`
`
`
`analysis. Figure 10 shows the calculated maximum
`normalized aerodynamic danuing for this rotor as a
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`function of weight flow and wheel speed.
`The re-
`sults presented on this curve were calculated using
`
`
`
`
`
`
`
`
`the blade element data recorded at
`the five operat-
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`ing condition shown on figure 8.
`Thr. shaded region
`
`
`
`
`
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`on figure 10 represents the unstable i‘utter
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`region.
`The boundary of this region 2.-.i~‘. estimated
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`by assuming a log decrement of 0.02:: for the mech-
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`anical daiioing of
`the assembly (ref. 3). Hence all
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`operating conditions which have an aerodynamic dew-
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`5
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`UTC-2005.006
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`UTC-2005.006
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`wi hin the flutter region. while those that have an
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`aerod amic damping level less than 0.026 are stable.
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`he results on figure 10 clearly show the
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`significant influence of wheel speed and aerodynamic
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`loading on the aerodynamic damping.
`The calculated
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`aerodynanic duping at 60 and 70 percent wheel speed
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`lie significantly below the instability boundary.
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`iio bending flutter was observed at these operating
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`conditions. At 90 percent of design speed bending
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`flutter was observed near the stal
`region. The
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`calculated aerodynamic danping for the near stall
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`operating point is seen to lie Just inside the un-
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`stable region. Bending flutter was also observed at
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`95 percent of design speed near the stall boundary.
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`The calculated aerodynamic damping for this operat-
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`ing point lies well within the unstable region of
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`figure 10. At the intermediate operating point on
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`the 95 percent speed line the calculated aerodynunic
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`dairping falls below tfk instability boundary.
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`Therefore the predicted flutter boundary at this
`wheel speed lies between the measured flutter bound-
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`ary and the intermediate operating point. Tran-
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`scribing the stability boundary of figure 10 onto
`the Fan C performance map yields a theoretical flut-
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`ter boundary, which is shown as a dashed line on
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`figure 11.
`The measured operating conditions at
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`flutter occurred are shown on this
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`figure as so id symbols.
`The overall agreement
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`between theory and measurement is very good. with
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`the analysis slightly over predicti
`the extent of
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`the flutter boundary at 95 percent 0 wheel speed.
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`Below 90 percent of wheel speed the analysis pre-
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`dicts the flutter poundary will bend back into the
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`stall region. This result is confirmed by the
`experimental measurements.
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`The results presented in this section clearly
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`show that the current analysis can be used to pre-
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`dict the onset of supersonic bending flutter in un-
`shrouded fans.
`The input variables to the analysis
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`can all be derived from fan structural and aero-
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`dynamic design variables.
`The governing equations
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`are simple algebraic equations. and hence are easily
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`prograrnned.
`hese features provide the designer
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`with a simple and reliable model for analyzing the
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`susceptibility of a fan design to supersonic stall
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`bending flutter.
`In addition it provides him with a
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`tool that can also be used to evaluate proposed
`fixes to an existing flutter problem.
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`The engineering approximation introduced to
`extended the range of validity of actuator disk
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`theory appears reasonable.
`To develop a correct
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`first principles extension of actuator dist theory
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`to finite interblade phase angle for the flow
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`conditions as they exist at the tip of a rotor at
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`the onset of supersonic stall flutter would re-
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`quire a viscous transonic cascade analysis for
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`finite interblade phase angle.
`No such analysis
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`appears to be forth coming in the near future. Any
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