aeroelasticity · 2018. 5. 24. · aeroelasticity 2014 prof. sangjoonshin. active aeroelasticityand...

Active Aeroelasticity and Rotorcraft Lab.

Aeroelasticity

2014

Prof. SangJoon Shin

Active Aeroelasticity and Rotorcraft Lab., Seoul National University

Turbomachinery


Turbomachinery

5 10

bending torsion flutter

single D.O.F flutterstall flutter

- Structural Formulation; blade-to-blade coupling (structurally, aero)

- Aero Formulation; cascade

“static” flutterforced response(vibration)

fatigue on blade

*Ref: AGARD manual on Aeroelasticity of Axial Flow Turbulence Vol II, Chap.19 “Aeroelastic Formulation for Tuned and Mistuned Rotors”, E.F.Crawley


Modeling Problems

. Modeling problems

i) Varying starting points

- Mode shapes of individual blades- Mode shapes of entire blade-disk assembly- Properties of typical section

ii) Varying objectives

- Simple stability assessment- Full forced vibration response- Completely coupled, time-accurate dynamic-aero. Analysis

iii) Analytical tools not available

- ex) 3-D large shock motion aero.


Modeling Problems

General flowchart

i) Single D.O.F, typical section of the i-th blade: motion dependent: aero. disturbance force

: augmented state variable

time history to represent lag effect of


Ni 2,1=

Ni 2,1=

Ni 2,1=

nqβ 0=n

1=n

1−= Nn

General flow chart


General flow chart

― Matrix formwhere

― forces acting on zero-th blade undergoing the n-th travelling wave

: amplitude of the n-th travelling wave

: complex force coefficient due to

[ ]

n

j ti

j ti

q E q e

q e

ωβ

ω

=

=

=

−−−

−

1,10,1

1,00,0

][

NNN

N

EE

EEE

Nklj

lk eEπ2

, =

nlβ nβ

2 20 n n

m j tf b l q e ωβ βπρ ω=

Travellingwave

― forces acting on i-th blade due to superposition of all the intermediate phase angle waves

1( )2 2

0

n

n n

Nj t im

in

f b l q e ω ββ βπρ ω

−+

=

= ∑2 2 [ ]

n n

m j tif b E l q e ω

β βπρ ω

=

nqβ


3 Fundamental Relationships

1) Dynamic governing eqn of motion

2) Kinematic relationship between individual and travelling wave blade

3) Relationship between travelling wave motion and unsteady aerodynamics

[ ] n

j t j ti iq E q e q eω ω

β= =

2 (1 ) i i i i i i im q m jg q fω + + =

2 2 [ ] n n


β βπρ ω

=


Travelling wave formulation

• Description of aeroelastic eigenvalue problem

i) Travelling wave formulation

2 2

2 2

[ ] (1 ) [ ]

[ ]

n n

n n

j t ji i i i

j t

m E q e m jg E q e

b E l q e

ω ωβ β

ωβ β

ω ω

πρ ω

− + +

=


• Description of aeroelastic eigenvalue problem― Single D.O.F. uniform mass and stiffness

[ ],im m I =

0

2 2(1 ) (1 )[ ]ii i im jg m jg Iω ω

+ = +

2 2 2 20[ ] (1 )[ ]

n n n nim I q m jg I q b l qβ β β βω ω πρ ω

⇒ − + + =



• Description of aeroelastic eigenvalue problemi) Travelling wave formulation

― Will be completely uncoupled , travelling wave coordinate are the normal aeroelastic eigenmodes for a tuned rotor.

, eigenvalues are directly relatedto (unsteady aero coeff.))/1(

)1(

)1(

2

202

2220

2

mlbjg

lbjgmm

n

n

β

β

πρωω

ωπρωω

++

=

=++−

nlβ




― AdvantageConstant inter-blade phase angle -> the eigenmodesof the aeroelastic problem for a tuned rotor

― Disadvantage1. Requires a transformation of the structural modes to

travelling wave coord.2. Very difficult to interpret the aeroelastic response of

the mistuned rotor with non-uniform blades3. Difficult to explicitly include the effect of shroud and dis

k elastic coupling4. Representation of the aero. forces in this form obscures

the real physical dependence of forces on specific blade motion -> much more insight into these aspect is gained by individual blade coord.



• Description of aeroelastic eigenvalue problemii) Individual blade coordinate

― aero. forces transformed into individual blade coordinate

,

where

― Flutter equation

2 2 [ ] m j ti if b L q e ωπρ ω=

1[ ] [ ] [ ]n

L E l Eβ−

=

Individual blade formulation

2 2 2 2 (1 ) [ ] i i i i i i im q m jg q b L qω ω πρ ω − + + =



ii) Individual blade coordinate- Aerodynamic Influence matrix ---- circulant form

:force acting on any given blade due to its own motion

: the k-th coeff. of discrete Fourier series representation of

0L

2exp,2exp1 1

0

1

0 NknjLl

Nknjl

NL

N

nk

N

nk nn

ππββ −== ∑∑

−

=

−

=

kL


nlβ

=

−−

−

021

201

110

][

LLL

LLLLLL

L

NN

N



ii) Individual blade formulation― Graph of vs.

dominantly first harmonic and an average offset

is almost tridiagonal, only the two adjacent blades andthe blade itself have distinct effect

Higher harmonics --- influence of more distant blades

][L


nlβ β

nlβ

a) average offset represents blade’s influence on itself, L0

Relation between aero. forces in Inter-blade Phase and Complex Influence Coefficient Form



nlβ

b) first harmonic represents neighboring blade influence, L1 and LN-1

c) second harmonicrepresents influence of blades two stations away, L2 and LN-2

Relation between aero. forces in Inter-blade Phase and Complex Influence Coefficient Form



ii) Individual blade formulation― Advantage: natural for structure

• Easy to add a complicating feature (disk elastic coupling, shroud, blade non-uniformity, misturning, multiple blade D.O.F.)

• matrix gives tremendous insight into the unsteady aero. interaction in cascade

][L



0 cos( ) sin( )i n i n in n

q b b n a nθ θ= + +∑ ∑

0 cos( ) sin( ) [ ]cn

i c cn i sn in n

sn

qq q q n q n P

qθ θ

= + + =

∑ ∑

Ni

Nn

Nn

iπθ 2

22,1

212,1

=

=

−=

For N odd

For N even

• Description of aeroelastic eigenvalue problemii) Standing wave formulation

• Starting point: a set of calculated/measured standing structural eigenmodes of the blade-disk assembly

• Two natural ways to represent eigenvectors corresponding to pairs of repeated structural eigenvaluesforward/backward travelling wavessine/cosine standing waves (twin orthogonal modes/

multi-blade coordinate)

- Blade oscillatory motion

Standing wave formulation



- Aeroelastic eqn.

- By comparison

2 1 1

2 2 1

[ ] [ ] [ ] (1 ) [ ]

[ ] [ ][ ]

cn cn

i i i i

sn sn

cn

sn

q qP m P P m jg P

q q

qb P L P

q

ω ω

πρ ω

− −

−

− + + =

1 [ ] [ ]n

cn

sn

qq E P

qβ

−

=




― Mistuned, coupled bending torsion motion --- no longer simple to relate standing and travelling waves

― Still straightforward to relate the standing blade-disk modes to the individual blade deflections

, : special case of normal mode vector matrix

1( )

0

Nn

i i nn

q qϕ−

=

=∑ [ ]

[ ]

i n

cn

i

sn

q qq

q Pq

φ=

=

[ ]P




― Aeroelastic formulation in terms of arbitrary blade-disk modalcoord.

L.H.S.: decoupled, but will coupled by aero. in R.H.S.

2 2

2

[ ] [ ] [ ] (1 ) [ ]

[ ] [ ][ ]

T Ti n i i i n

Tn

m q m jg q

b L q

ω φ φ φ ω φ

πρ φ φ

− + + =




ii) Standing wave formulation

― Advantage: all forms of blade, disk and shroud elastic coupling can be easily included

― Disadvantage: aerodynamics and the resulting flutter eigenvectors may be difficult to interpret physically



• Two types of aerodynamic forces formulation― Generalized forces acting on i-th blade due to n-th travelling

displacement wave pattern and the wake forced vibration

i) formulation

ii) formulation

Generalized Aerodynamic Forces

l

( )2 2[ ] n n

n n n

j t ii w

wf b E l q l e

Uβ ω β

β βπρ ω +

= +

lc

( )22

2 2 2/

/ 4 4 4

nq w n n

q wn

F F F j t ii

M M M

hjkC C C wL bf U ebM b jkC C C U

α

α

ββ ω β

β

πρα

+

− = = −


Aeroelastic Solution

• Aeroelastic eqn in individual blade coord.

- Divide by nominal blade mass non-dimensionalize

: fractional mass/stiffness mistuning of the i-th blade: non-dimensional eigen-frequency: sectional mass density ratio

2 2 2 2 (1 ) [ ] i i i i i i im q m jg q b L qω ω πρ ω − + + =

11 [ ] (1 )(1 )i i i i iL q jg qε δµ

Ω + + = + +

,i iε δ

Ω Rω ω=µ 2m bπρ=


Aeroelastic Solution

•- V-g method

( )( ) ( ) ( )

2

12

11 [ ] (1 ) ,

1 ,

Re , Im Re

i i i iL q q

jg

g

ε δµ

−

+ + = +

+=

Ω

Ω = =

L

L

L L L


Expansion to Multiple Blade DOF

• Multiple section D.O.F. formulation― each section has transformational and pitching D.O.F.

i ii

h bq q

α

→ =

2i ii

m S bm m

S b I b

→ =

2

2 2

2

(1 ) 0(1 )

0 (1 )

B B

i i i i TT

i

m jgm jg m I jg

b

ω ωω ω

+ + → = +

2i ii

L bf f

M b−

→ ≡


• Multiple section D.O.F. formulation― Modifying kinematic relationship

― E matrix: now fully populated one with sub-matrix blocks

Expansion to Multiple Blade DOF

n

n n

n

h bq q ββ β

βα

→ ≡

,, ,

,

00k l

k l k lk l

EE E

E

→ ≡

=

000

00

][1,0

1,00,0

1,00,0

EEE

EE

E


Expansion to Multiple Blade DOF• Multiple section D.O.F. formulation

― three fundamental relations for single d.o.f.i) dynamic equilibrium equation

ii) kinematic relationship between standing and travelling waves

iii) aero. force dependence on motion

2 (1 )i i i i i i im q m jg q fω + + =

[ ] n

j t j tiq e E q eω ω

β=

[ ] 2 2m n


β βπ ρ ω =


Turbomachinery Aeroelasticity- more accuracy --- two approaches available

i) p-k method ---- iterative

ii) expand the explicit function response of L on k

---> new eigenvalue problem (std. eigenvalue problem)RR k

kLLLL ==ΩΩ

+Ω

+=Ωωω,11)( 2210

iiiii qjgqLLL

++=

+Ω

+

+

+Ω

)1)(1(][1][][11 210 δµµµ

ε


Turbomachinery Aeroelasticity• Multiple spanwise blade modes formulation

- integrate the unsteady aero. forces over the entire span.

• Solution for sinusoidal temporal representations.

- key simplification for gas turbines

--- mass ratio usually large (μ>>10),

aero. force very small compared to inertial and elastic forces

oscillatory component of aeroelastic eigenvalue ≈ ref. frequency

- Relative weakness of aero. forces

reference k is calculated based on the structural freq. at the

same speed, but in vacuum. Little difference between

aeroelastic and in vacuum frequency.


Turbomachinery Aeroelasticity

• Explicit time-dependent formulation

- explicit time-dependence of unsteady aero. forcescertain forced vibration (impacting mechanical ribs)time unsteady aero disturbances (rotating stall surge)

- Unsteady aero. operators assuming sinusoidal behaviour

L : circulant matrix, L0 ---- blade’s aero force on itself

- complex inverse Fourier integral is needed very complex

- approximate transform technique Pade approximation

[C]L = k2 [L]

[ ] tjii eqLbf ωωρπ 22=

[ ] tjiL

tjii eqCUeqL

UbUf ωω ρπωρπ ][2

2

222 =

=



[ ] [ ] [ ]

+

+

+

+

+

+

+

=

•••

.......)2()1()0(

0122

22

iii

iiii

yyy

qCqCUbqC

UbUf

ρπ

[G]0 [G]1[G]2

C2 , C1 , C0 : real circulant matrices, inertial, damping, stiffness effectG0 , G1 , G2 … : sparse real circulant matrices with only one entry in column

impact of the relative lags in the aero. Gj …. lagged forces of the (i+j)–th blade on the i-th blade.yi …. augmented states


Turbomachinery Aeroelasticity•

=+ il

ill

i qUbygj

Ub )()( for i = 0,1,……. , N-1

for l = 0,1,……. , N-1

to evaluate the unknown constants C2 , C1 , C0 , G0 , G1 …. forces actingon the zero-th blade is formulated, assuming pure sinusoidal motion

• Trends in Aeroelastic Stability

four parameters ---- stabilizing/destabilizing influences

blade self-damping

loading 2- dimensionality, stall

difference of actual rotor and “rubber” design

i) Stabilizing/Destabilizing influences- single d.o.f flutter model in non-dimensional form



g : structural dampingδ, ε : stiffness, mass non-uniformity

L : aerodynamic influence coefficients- assume uniformity in stiffness and structural damping

- complex eigenvalue plot centroid and eigenvalues distributed- location of the centroid

[ ] iiiii qLgjgqjg

+

+

+−

=

++

Ω µεδ 11

11)1)(1(1

22

[ ] iii qLgjgq

+

+

+−

=Ω µ

ε 11111

22

gLS21Im

21Re 0 −

>=<

µ 2Re

211Im 0 ><

−

−>=<

εµLS



1. In the absence of g, blade must be self-damped.

2. In the presence of g, large μ are relatively more stable.

3. Cascade unsteady aero. influence is destabilizing.

4. Structural mistuning does not change the location of centroid,

but can re-arrange the distribution of eigenvalues.

“ Single- D.O.F. flutter “ : generally true for solid metallic blade

- hollow or composite blades with significant bending torsion

coupling

Im (S)

Re (S)N > H

Centroid

Sin-1 ζ



• Bending-torsion coupling

- two distinct mechanism of bending-torsion coupling

1. single mode coupling ….. kinematic coupling, root not being

supported along a line normal to e.a., offset between e.a. and c.g.,

presence of anisotropic material or fibers, shrouds at the tip or mid-

span.

2. dynamic coupling between two modes ….. classic coalescence flutter

- gas turbine ….. 1 can be very important

aeroelastic instability in gas turbine components ….. not

classical bending-torsion coalescence flutter, instead a cascade-

induced blade-to-blade interaction flutter.



iii) Effect of loading and 3-D

- loading ….. push the blading to a near stall condition L0 coeff.

slight reduction flutter

- swirl …… complicated downstream couples the acoustic,

vorticity and pressure fields


Turbomachinery Aeroelasticity• Effects of mistuning on stability

- Mistuning optimization ….. const. function n =4

constraint

Eigen values of a tuned (original) rotor

Im (Ω)

Re (Ω)

“S”

Sin-1 ζ

1/nni

Nε

ϕ

=

∑

Niii .......2,1,0 =>−= ζζθ



- Errors introduced in optimal mistuning.

- - 1 % RMS scatter ζ = 0.002 -0.00317

Eigenvalues of optimally mistuned rotor ζ= 0

Eigenvalues of optimally mistuned rotor ζ = 0.002

~ ~~i ispecified ieε ε= +


Turbomachinery Aeroelasticity- extremely sensitive to errors in mistuning

- Alternate mistuning – not as cost effective as optimal mistuning

much more robust to errors

ζ = 0.000171 0.000169


Cascade Aerodynamics

Force and Moment Coefficient for vibrating airfoils in

Cascade (Whitehead 1960, AGARD 1987)1. Introduction

- Simplifying assumptions

i. two-dimensional

ii. incompressible, inviscid

iii. un-stalled

iv. neglect camber, thickness flat plate

v. blade operates at zero mean incidence mean deflection is zero

vi. small amplitude of vibraton wakes straight, linear theory

vii. all blades: same amplitude, constant inter-blade phase angle.


( )i tdx e ω βγ +

( )i t mdx e ω βγ +

2. General method of calculation

2.1. Vorticity

- both the blades and their wakes vortex sheets

- element of vorticity on the ref. blade at x from the origin

(l.e.) on the next blade, at (x + s sin ξ, cos ξ)

- the m-th blade

at (x + ms sin ξ, ms cos ξ)

…. bound vorticity

• Free vorticity

….. a vortex sheet shed form the element, its strength ε eiωt

continually washed downstream at U.

stagger angle


i tdxe ωγ


x

y

c

s

x nU

Stagger angle i tdxe ωγ

ξ


(x,0)

Г0 eiβ

Г0 eimβ

(xm , ym)


• Free vorticity….

strength of free vorticity due to , at (x1 , 0)

- bound vorticity changes during a small time interval δt

… equal in magnitude and opposite in sign to the free vorticity

created in δt, free vorticity moves back a distance U δt.

- the strength of free vorticity just behind the bound vorticity at

(x, 0)

1xi ti t Ue const e

ωωε

− = ⋅

i tdx e i tωγ ωδ

i tdx e i tU t

ωγ ωδδ

1( )i x x Uidx eU

ωωε γ −=−

determine the “constant”


i tdxe ωγ


( )γ ε+ 0γ =

• Free vorticity….

- Total free vorticity at (x1,0) by summing up from x =0 ~ x = x1

total vorticity is on the ref. blade, in the wake

Differentiation w.r.t. x1,

x1 x :

1

1( )/

0

xi x x Ui e dx

Uωωε γ −=− ∫

1

1 / /

0

xi x U i x Uie e dx

Uω ωωε γ=− ∫

( )1 11 1// /1

1

i x Ui x U i x Ud i ie e x edx U U

ωω ωε ω ωε γ+ = −

( ) 0d idx Uε ω ε γ+ + =



• Bound vorticity and the pressure difference across the blade

- eqns. of motion in x-direction

just below the blade ….. suffix (-), just above the blade ….(+)

- Subtracting two eqns.

- (total vorticity)

1i t pU uet x x

ω

ρ∂ ∂ ∂ + = − ∂ ∂ ∂

)(1+−+− −

∂∂

−=−

∂∂

+∂∂ pp

xeuu

xU

tti

ρω

u u γ ε− +− = +

1 ( ) ( )

( )

i t

i t i t

p p U ex t x

d d di U U e U edx dx dx

ω

ω ω

γ ερ

γ ε γω γ ε

− +∂ ∂ ∂ − − = + + ∂ ∂ ∂

= + + + =



• Bound vorticity and the pressure difference across the blade

- Integrating

- Aerodynamic force acting upwards,

- Aerodynamic moment acting anti-clockwise about the l.e.

- Kutta condition ….. must be finite at the T.E.

i tp p U e ωρ γ− +− =−

( )0 0

,c c

i tF e p p dx F U dxω ρ γ− += − =−∫ ∫

0

c

M U x dxρ γ=− ∫( )γ ε+



i tdx e ωγ

2.2 Induced Velocity

- normal velocity (v eiωt) induced at (η,0) by a row of vortices with

(ε, ξ, β)

- normal velocity induced at (η,0) by bound vorticity

at (x,0) from the ref. blade and corresponding elements on the

other blades.

- By free velocity,

- special consideration of the case ….. V(-∞) 0, above

integral does not converge, but oscillates.

−

Γ=

cx

cV

cv η0

( ) ( ) ( )v x dxV xη γ η= −

111 )()()( dxxVxvx∫∞

−= ηεη



2.2 Induced Velocity…

- total vorticity on each blade and in its wake is zero

- velocity induced by a bound vorticity and its associated free

vorticity

1 10

( ) 0 ( ) ( ) 0x

dx x dx x dxγ ε γ ε∞ ∞

+ = + =∫ ∫

( ) ( ) 1 1( ) ( ) ( ) ( )x

v V x V rdx x V x V dxη η ε η∞

= − − −∞ + − − −∞∫

( ) ( ) 1( )/1 1( ) ( )i x x U

x

iV x V dx dx e V x V dxU

ωωη γ γ η∞

−= − − −∞ − − − −∞∫



2.2 Induced Velocity…

- let z1= x1-η , λ=ωc/U (c=1)

- Integrating for the total velocity induced by all bound vorticity

i) Pure translational motion of the blades (q eiωt )

…… induced velocity must be equal to the blade velocity

v = q

( ) ( ) 1( )1( ) ( ) ( ) ( )i x z

x

v x dx V x V i e V z V dzλ η

η

η γ η λ∞

− −

−

= − − −∞ − − − −∞∫( ) ( )x dx K xγ η= × −

Kernel function of z = (x-η)

1

0

( ) ( ) ( )v K x x dxη η γ= −∫



ii) Torsional motion about the l.e. … air velocity

normal to the blade = velocity of the blade itself,

iii) blades are operating in the wakes of same kind of periodic

obstruction for upstream ….. weiωt induced normal velocity at the

l.e., at (η, 0)

- All three cases present together

)( tie ωαtieUv ωα )( −

)( tiedtd ωαη )1( ηλα iUv +=

ληωηω itiUti ewvvewe −− −==)/(

(-)

ληηλα iweiUqv −−++= )1(



The strength of the sheet of bound vorticity required to

induce this:

are the solutions of the following integral equations.

subject to that is finite at x = 1

2.3 Solution of the Integral Equations

Appendix I. The velocity induced by a row of vortices

- the velocity normal to the blade surface induced by a row of vortices

for one vortex of strength Γm at (xm , ym), velocity at (η, 0) is

q wq U wαγ γ α γ γ= + −, ,q wαγ γ γ

1

0

, , ( ) [1, (1 ), ]iq w K x dx i e λη

αγ γ γ η λη − − = + ∫

( )( ) 222 mm

mm

yxnxnv+−

−Γ=

π


γ


2.3 Solution of the Integral Equations….

- the center vortex at (x, 0) has strength Γ 0, that on the m-th blade

or in the wake

Γ m = Γ 0 eimβ at (x+ms sinζ, ms+cosζ)

- summing the effects for all the blades

- This will be written

V(z) : non-dimensional function

∑+∞

−∞= +−−−−Γ

=m

im

msmsxmsxev 22

0

)cos()sin()sin(

2 ζζηζη

π

β

−

Γ=

cx

cV

cv η0

2 2

( sin )1( )2 ( sin ) ( cos )

im

m

se z mcV z s sz m m

c c

β ζ

π ζ ζ

+∞

=−∞

−=

− +∑




- This series can be summed analytically, for 0 < β < 2 π

( ) ( )

exp ( )( ) exp ( )( )1 1( )

4 sinh ( ) 4 sinh ( )a ib z a ib z

V z a ib a iba ib z a ib z

π β π βπ π

− − + − −= + + −

+ −

ζcossca = ζsin

scb =




- Induced velocity must be equated to the actual upwash velocity

i) pure translational motion

v = q = q

is the solution of the following integral equation

1

0

( ) 1 , 0 1 ....... (3)q K x dxγ η η− = < <∫


qγγqγ


- Numerical method of integral eqn. (3)

transformation to new independent variables θ and Φ

- specify bound vorticity at (n+1) points : θ = π l/n, l = 0 ~ n

- Trapezoidal rule used to integrate (4)

…… n simultaneous equations for the n unknowns

)cos1(21;)cos1(

21 φηθ −=−=x

0

1 1 1( cos cos ) sin 1 .........(4)2 2 2qK d

π

φ θ γ θ θ− =∫

( )0

1 1 1( cos 2 1/ 2 cos / )[ ] sin / /2 2 2

1 ............... 5

n

ql

K m n l n l n l nπ π γ π π=

+ − •

=

∑


qγ


(5) : matrix notation AГ= B

A : n x n square matrix

Γ : n x 1 matrix, whose lth row [ ] x π/2n sin π l/n

(except for the first row which has half weights)

B : n x 1 matrix, value = 1

∴ Γ = A-1 B

- Force calculation

in non-dimensional force coefficients

)/cos212/12cos

21( nlnmKA lm ππ −+=

0

( )c

qF U q dxρ γ=− ∫


qγ


- Force calculation

qFF Uc qCπ ρ= ⋅1

0 0

1 1 sinqF q qC dx d

π

γ γ θ θπ π

= − = −∫ ∫

0

1 1 sin2q

n

F ql

lCn nπ πγ

π =

= − ⋅ ∑




AGARD, 1987

V(z) …… transonic, supersonic, ……

K.C. Hall (1994)

ROM (Reduced Order Model)

Aгn+1 + Bгn = wn+1 (time – domain)time step

г = xc

Compressor performance map

Fan

Compressor ….. stagnation pressure

Turbine ….. windmill


•

m

001

02 , PPP

∆

Real Value

Ωr = constant

Turbomachinery Aeroelasticity• Compressor Performance Map (Dowell Sec 8.2)

- angle of attack of each rotor at radius r

tangential component due to rotor rotation, Ωr

through flow velocity, modified in direction by the

upstream stator row , V

angle of attack will increase inversely with the ratio Φ = Vx/ Ωr

- increase in angle of attack (or ‘loading’)

more work being done on the fluid

greater stagnation pressure increase ΔPo

mass flow rate = ∫Annulus

x dAV ρ



- Unique value of angle of attack (incidence)

Optimal design point

= C100

- Complete Multistage Compressor



• Periodically stalled flow in turbo machine (Dowell Sec. 8.6)

- Rotating stall (or propagating stall) ….. circumferentially

asymmetric flow

- in axial compressor appears at rotationally part-speed conditions

- one or more regions of reduced (or even reversed) through flow

- rotating at a speed less than rotor speed, although in the same

direction

Propagating stall …. integrated mass flow over the entire annulus

remain steady

Surge …. integrated mass flow …. not steady

Periodic loading and unloading of the blades



Stall flutter ….. blade vibrates somewhat sporadically at or

near their individual natural frequencies

- no obvious correlation between the motions of adjacent blades

- motion is often in the fundamental bending mode

- random vibration, nonlinear system

- will occur at part-speed operation, confined to those rotor stages

operating at higher than average incidence

• Choking flutter

….. normally occur at part-speed operation, confined to those rotor

stages operating at lower than average incidence (negative values)



• Axial compressor fan characteristic map (Dowell, Fig. 8.13)

- Region I … ‘System mode instability’ … may not involve flow

separation, local Mach No. > 1, oscillating shock

negative aerodynamic damping

- Region II … oscillating shock waves important, but choke flutter

mechanism still controversial type

stage


• Axial compressor fan characteristic map (Dowell, Fig. 8.13)

- Region III … usually encountered along a normal operating line,

overspeed end

- Region IV, V … higher compressor pressure ratio

- Region V … involve stalling at supersonic blade relative Mach no.

mostly flexural flutter ‘supersonic bending stall

flutter’




Forced Vibration and Flutter Design Methodology

( AGARD - 298 – VOL. 2)

• High cycle fatigue failures forced vibration

flutter

- Forced vibration …. extremely excited oscillating motion where

the forces are independent of the motion

- Flutter …. self-induced oscillation, forces are function of

the displacement, velocity or acceleration,

and these forces feed energy into the system



• Sources of unsteady forces

- Forced vibration ….. Table 1 aerodynamic sources

mechanical sources

- Aerodynamic sources

Upstream vanes/struts (blades)

Downstream vanes/struts (blades)

Asymmetry in flow path geometry

Circumferential inlet flow distortion Rotating stall

(pressure, temperature, velocity) Local bleed extraction

- Mechanical sources

Gear tooth meshes

Rub



• Types of turbomachinery blading (Table 2)

Blades VanesShrouded/shroudless Cantilevered/inner

bandedAxial/circumferential attachmentStiff/flexible disk High/low aspect ratioHigh/low aspect ratio Solid/hollowHigh/low speed Metal/ceramicSolid/hollow Compressor/

turbineFixed/variableMetal/ceramic/composite High/low pressure ratioHigh/low hub-to-tip radius ratio Compressor/turbine

High/low pressure ratio



• Types of turbomachinery blading (Fig. 3)



• Mode shape and frequency (Fig. 5)



• 10 Steps of forced vibration design ….. second stage gasifier

turbine, air-cooled design, shroudless, integrally bonded to the

disk, hollow blade, low aspect ratio, 22 airfoils in a stage, nickel

alloy.

i) Identify possible sources of excitation …. two upstream and

two downstream

ii) Determine operating speed ranges

iii) Calculate natural frequencies …. spacer and disk constant

contact during operation disk flexibility is eliminated from

the assembly modes

only the blade geometry and fixing is modeled in natural

mode calculation



iv) Construct resonance diagram (Campbell diagram)

Fig. 11, dropping of natural frequency with rotor speed

temperature effect dominant over centrifugal stiffening

possible resonant condition ….. intersection of natural frequencies

and order lines



v) Determine response amplitude

….. empirically defined based on experience

Fig. 12 …. response of 1T mode due to an upstream vane source



Fig. 13 …. response of 1T mode due to an downstream vane

source



vi) Calculate stress distribution

….. finite element prediction, measurement through strain gages

vii) Construct modified Goodman diagram ….. Fig. 16



vii) Construct modified Goodman diagram …..

steady stress vs vibratory stress ..... Straight line (Conservative)

affecting : notch factor, data scatter, temperature…. Fig. 17

lowers the mean fatigue strength line ….. Fig. 16



rvxΩ

=φ

Ωr

v vxrΩ

At radius r, AOA

Tangential component Ωr

through flow velocity v

A.O.A inverse ~

• forced vibration

- A.O.A A.O.A ~

- Load Work being

done on the

fluid

Stagnation

pressure

ΔPo

increase of

A.O.A

Stator Rotor

w

v

Ωr



•

m

001

02 , PPP

∆

Real Value

Ωr = constantmass flow rate = dAVx ρ∫

P0 exitSurge or stall limit

Optimal design point

Ωrtip= C100

Choke flutter

Normal operatingC25

C50

C75

0

*0

PrA

RTm

n

n

•


Turbomachinery Aeroelasticity• Surge ------ = steady

• Stall flutter, rotating (propogating) stall ----- = steady

• Choke flutter – negative A.O.A

I. Subsonic flutter transonic

I a. System mode instability

II. Choke flutter

III. Supersonic flutter

IV, V. Supersonic flutter (high compressor ratio)

•

m

•

m

0

*0

PrA

RTm

n

n

•

SurgeNormal Operating

ChokeIII

IaIII

IVV



f(motion, displacement, velocity acceleration)

• Campbell diagram

flutter ------load =

forced vibration ---load ≠ f(motion, disp, vel, accel)

---load = external excitation

AGARD Chap. 22

• 10 steps of forced vibration design

i) id. of possible source of excitation

aerodynamic

mechanical ---------hub gear

ii) operating range (RPM)

iii) natural frequencies ---- F.E.M

blade/disk assembly

Fatigue

Stator Rotor

13

M

14

M/rev.

E.O. – Engine Order


Turbomachinery Aeroelasticityiv) construct resource diagram. possible response

v) response amplitude

vi) stress distribution

strain gauge (vibratory component)

analysis

from previous design experience

WRoom Temp.

2B

1T

1B

19 E.O.

13 E.O.

10 E.O.

Idle DesignΩ

Temperature elevation is considered

Ω the mode shape curves lower


Turbomachinery Aeroelasticityvii) modified Goodman diagram

viii) IⱭ, fatigue life infinite redesign

Vibratory Stress

Steady Stress

Working CurveTemperature notch factor data scatter

Ultimate strength

Endurance limit

aeroelasticity · 2018. 5. 24. · aeroelasticity 2014 prof. sangjoonshin. active aeroelasticityand...

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