Nonlinear vibration analysis of steam turbine bladed disk ...€¦ · shroud tie-boss Demand from...

Josef Voldřich

Ukraine, November 2018

Nonlinear vibration analysis

of steam turbine bladed disk

with friction contact between

adjacent blades

Проект „Развитие международного сотрудничества с украинскими ВУЗами

в областях качества, энергетики и транспорта“г. Харьков, 11/2018

Степан Прокопович Тимошенко1878 - 1972

- A grand native of Ukraine, the father ofmodern engineering mechanics.

- His portrait, as the only one, has displayedwith reverence in my workroom for 20 years.

- He was a giant for mathematical modelling of strength problems in mechanicalengineering.

Encouragement:- Let’s not be afraid to use MATHEMATICS

for solution of our actual problems.- And it is not inevitable to look up only to

commercial FEM software.

Content

Motivation and introduction

Methodology

Calculations of nonlinear vibration for bladed disks

Contact stiffnesses

Conclusions


Pilsen (170 000 inhabitants):

Steam turbines

produced by Doosan Škoda Power Ltd.Nuclear reactors WWER for NPPsproduced by Škoda JS, Ltd. (25 so far).

Well-known for its brewingThe world’s first-ever pilsner type blond lager, making it the inspiration for much

of the beer produced in the world today, many of which are named pils, pilsnerand pilsener.

The Škoda company (established 1859) was one of the biggest European armsfactories up to the WW II.

Pilsen


shroud tie-boss

Demand from Doosan Škoda Power Ltd. :

Vibration analysis of bladed disk with 66 blades of type LSB48 (1220 mm)

277,269 DOFs


Reactor heatpower

[%]

Beforemodernization

LP[MWe]

Power risegaranted[MWe]

Power risemeasured

[MWe]

100 1016,9 22 28,8

renovation of steam turbines’ low-pressure

stages in NPP Temelín by Doosan Škoda

Power

NPP Temelín in south Bohemia

Turbine room

Methodology – Modal analysis

turbin room

297k DOFs

𝓜=

𝑴 00 𝑴

⋯00

⋮ ⋱ ⋮0 0 ⋯ 𝑴

𝓚 =

𝑲0 𝑲1 𝟎𝑲𝑁−1 𝑲0 𝑲1𝟎 𝑲𝑁−1 𝑲0

⋯𝟎 𝑲𝑁−1𝟎 𝟎𝟎 𝟎

⋮ ⋱ ⋮𝟎 𝟎 𝟎𝑲1 𝟎 𝟎

⋯𝑲0 𝑲1𝑲𝑁−1 𝑲0

−𝜔2𝓜+𝓚 𝒘 = 𝟎

−𝜔2𝑴+𝑲0 +𝑲1ei𝑘𝜎 +𝑲𝑁−1e

−i𝑘𝜎 𝒗 = 0

It is sufficient to consider only a sector with 1 blade –the reference sector

Eigenvalue problém for the reference sector:

Eigenvalue problem for the whole bladed disk

𝑴, 𝑲0 : mass and stiffness matrices𝑲1, 𝑲𝑁−1 : sector-to-sector coupling𝑁 : number of blades, 𝜎 = 2π 𝑁, 𝑘 : harmoni index (nodal diameter)

Methodology

𝑴d2𝒖 𝑡

d𝑡 2+ 𝑩d𝒖 𝑡

d𝑡+ 𝑲𝒖 𝑡 + 𝒇L 𝒖 𝑡 − ∆𝑡 , 𝒖 𝑡 + 𝒇R 𝒖 𝑡 , 𝒖 𝑡 + ∆𝑡 = 𝒑 𝑡

Vibration of the reference sector

Linear partNonlinear forces between blades(friction contact)

Excitation forces

Steady-state vibration responce can be represented by Fourier series

𝒁𝑘 𝜔 𝑼𝑘 + 𝑭𝑘 𝑼 − 𝑷𝑘 = 𝟎, 𝑘 = 0,… , 𝑛

𝒁𝑘 𝜔 = 𝑲+ i𝑘𝜔𝑩 − 𝑘𝜔2𝑴, Matrix of dynamic stiffnesses for the system without couplings

Substituing into the equation we obtain

𝒑 𝑡 =

𝑘=1

𝑛

𝑷𝑘c cos 𝑘𝜔𝑡 + 𝑷𝑘

s sin 𝑘𝜔𝑡

𝒖 𝑡 = 𝑼0 +

𝑘=1

𝑛

𝑼𝑘c cos 𝑘𝜔𝑡 + 𝑼𝑘

s sin 𝑘𝜔𝑡 = 𝑼0 +

𝑘=1

𝑛

𝑼𝑘 e−i𝑘𝜔𝑡 + 𝑼𝑘e

i𝑘𝜔𝑡

Methodology –Nonlinear vibration, Multiharmonic Balance Method

1) Calculate a multiharmonic FRF matrix of the linear part of the system

4 important steps for the solution of equation

2) Separate linear and nonlinear degrees of freedom

𝑨𝑘 =𝑨𝑘l n ln 𝑨𝑘

l n nln

𝑨𝑘nl n ln 𝑨𝑘

nl n nln𝑼𝑘 = 𝑼𝑘ln, 𝑼𝑘nln

3) Split the equation into linear and nonlinear part

4) Accomplish effective calculation

𝑼𝑘ln + 𝑨𝑘

l n nln𝑭𝑘nln 𝑼nln − 𝑨𝑘𝑷𝑘

ln = 𝟎ln

𝑼𝑘nln + 𝑨𝑘

nl n nln𝑭𝑘nln 𝑼nln − 𝑨𝑘𝑷𝑘

nln = 𝟎nln

𝒁𝑘 𝜔−1= 𝑨𝑘 ≈

𝑟=1

𝑚1

1 + i𝜂𝑘,𝑟 𝛺𝑘,𝑟2 − 𝑘𝜔 2

𝝓𝑘,𝑟 𝝓𝑘,𝑟H

𝑭𝑘nln 𝑼nln

using 𝑭𝑘 𝑼 = 0, 𝑭𝑘nln 𝑼nln

𝒁𝑘 𝜔 𝑼𝑘 + 𝑭𝑘 𝑼 − 𝑷𝑘 = 𝟎,𝑘 = 0,… , 𝑛

𝑼nln

Dry Coulomb friction model for contactforces is used

friction coefficient𝜇

𝑁0 normal preloading force of the contact

𝑥, 𝑦

𝑘t, 𝑘n tečné a normálové kontaktní tuhosti

𝑼nln → 𝑭𝑘nln

Calculation of harmonic coefficients fornonlinear forces

[Petrov E.P. and Ewins D:Analytical formulation of friction interface elements for analysis of nonlinear multiharmonic vibrations of bladed disks. ASME Journal of Turbomachinery 125:364-371, 2003]

relative displacement of nodes of the nonlinear contactelement in the tangential and the normal directions

Methodology – Mathematical model of contacts

Methodology – mathematical model of contacts

Concept of „contact stiffnesses“

Finite number𝒎 of mode shapes

Notion of „contact stiffness“ is misleading, because it does not label only physicalproperties of contact.Contact stiffnesses compensate higher mode shapes that are not included in theapproximation of 𝑨𝒌 .

Calculation of contact stiffnesses:

Model of nonlinear systém –resonant frequency

Linearized contactstick or 𝝁=0 ≈

Measurement – resonantfrequenciesFEM calculations – natural frequencies

𝑨𝑘 ≈

𝑟=1

𝒎1

1 + i𝜂𝑘,𝑟 𝛺𝑘,𝑟2 − 𝑘𝜔 2

𝝓𝑘,𝑟 𝝓𝑘,𝑟H

Methodology – Computational diagram

NPP Temelín in souh Bohemia

Analysis of nonlinear vibration for cyclically symmetric system with contacts

NONVIBTW_V4

Modal anallysis of thesector without binding

ANSYS

Calculation of contactslocalization and preload

ANSYS

Operating FEM mesh

Input file– specificationof excitation

Input file withdescription of analysis

Modal analysis of thesystem with stuck contacts

ANSYS

Input files NONVIB_VIEW

Stress field

ANSYSOutput files

Pictures

Graphs

Numbers

Results ofmodal analysis

Quiescent FEM mesh

Input file– specificationof binding

NONVIBCS

In-house software

In-house software

In-house software

Calculation of nonlinear vibration

Shr

oud

TieB

oss

xy

z𝝋 Low

High

Low High

HighHigh

LSB 48″1220 mm, steel

LSB 54″1375 mm, titanium alloy

Each blade has two integral coupling:Tie-boss – middle partShroud (bandage) – top

Blade geometries of LSB48 and LSB54 are moderately different.

Contact surfaces are plane.


0.9 0.95 1 1.0510

-2

10-1

100

Normalizovaná budící frekvence

No

rma

lizo

van

á v

ých

ylk

a k

mitá

ní

šp

ičky lo

pa

tky

kT = 0,510

5 N/mm

= 0,35

= 0,2

kT = 210

5 N/mm

= 0,35

= 0,2

St-St systém

s2

s3

s1

Limited number (10) of mode shapes for approximation of blade dynamicbehavior → necessity to fit contact stiffnesses (influenceing resonantfrequencies of computational model)

Normalized excitation frequency

No

rmal

ize

dm

axvi

bra

tio

nam

plit

ud

e,

shro

ud

con

tact

LSB 48″, excitation by traveling waves with 2 ND


- Higher friction coeficient, µ → more friction energy dissipated in contact.

- Higher energy dissipated in contacts does not lead to drop of displacementaplitudes.

-0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Normalizovaný relativní prokluz, shroudN

orm

alizo

van

á s

myko

vá s

íla

f T /

fN

0,

sh

rou

d

= 0,35

= 0,2

0.9 0.95 1 1.0510

-2

10-1

100


Norm

alizovaná v

ýchylk

a k

mitání

špič

ky lopa

tky

kT = 0,510

5 N/mm

= 0,35

= 0,2

kT = 210

5 N/mm

= 0,35

= 0,2

St-St systém

s2

s3

s1


No

rmal

ize

dm

axvi

bra

tio

nam

plit

ud

e,

shro

ud

con

tact

Normalized relative displacement, shroud

No

rmal

ize

dta

nge

nti

alfo

rce

, sh

rou

d

friction damper ??

0 1 2 3 4 5 6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Fáze kmitu [rad]

Norm

alizované k

onta

tní

síly,

shro

ud

fT/ f

N0

fN/ f

N0

- fN/ f

N0

0 1 2 3 4 5 6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Fáze kmitu [rad]N

orm

aliz

ova

né k

onta

ktní

síly

, sh

roud

fT/ f

N0

fN/ f

N0

- fN/ f

N0

0.9 0.95 1 1.0510

-2

10-1

100


Norm

alizovaná v

ýchylk

a k

mitání

špič

ky lopa

tky

kT = 0,510

5 N/mm

= 0,35

= 0,2

kT = 210

5 N/mm

= 0,35

= 0,2

St-St systém

s2

s3

s1

- Frictional sliding occurs in contact.

- Excitation by travelling wave of 2ND → two cycleswithin the period (0,2π).

Phase of vibration [rad]Phase of vibration [rad]N

orm

aliz

ed

con

tact

forc

e, s

hro

ud

No

rmal

ize

dco

nta

ctfo

rce

, sh

rou

d



0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

10-2

10-1

100


No

rma

lizo

van

á v

ých

ylk

a k

mitá

ní

šp

ičky lo

pa

tky

Sl-Sl system Sl-St system St-St system

= 0.2

Sl-Sl systém Sl-St systém St-St systém

= 0,2

- Linearized problems: Contacts in durable ideal sliding („Slide“) orsticking („Stick“).

- „Sl-Sl system“ = sliding without frictionfor Shroud and Tie-Boss contact.

- Dangerous situation: the excitationfrequency of variation of torquemoment is close to the natural frequency of Sl-Sl system.

- Contact couplings could activatea „sleeping resonance regime“.

LSB 48″, excitation by harmonic variation of rotor torque moment


No

rmal

ize

dm

axvi

bra

tio

nam

plit

ud

e,

shro

ud

con

tact


- Analogous character as in the case of LSB 48″ excited with 2ND: higher values of µ → higher amplitudes of displacements of forcedvibrations

- loss of numerical konvergency for µ > cca 0.6 .


No

rmal

ize

dm

axvi

bra

tio

nam

plit

ud

e,

shro

ud

con

tact



Analysis of nonlinear vibration results:- Dissipated power in level tens W for each blade,

if slipping.- Higher displacement amplitude rise for higher

friction coefficient.

µ

Normalized excitation frequencyD

issi

pat

ed

po

we

r, s

hro

ud

[W

]

Relative displacement of coupled nodes, shroud [mm]

Tan

gen

tial

forc

e f

T, s

hro

ud

[kN

]


- Excitation by traveling wave with 8 ND → 8 cycles within the period (0,2π).

- Higher friction coefficient → higher amplitude of tangential and normal contact forces

Phase of vibration [rad] Phase of vibration [rad]Phase of vibration [rad]

No

rmal

ize

dco

nta

ctfo

rce

s, s

hro

ud



0 1 2 3 4 5 6

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Fáze kmitu [rad]N

orm

alizované k

onta

ktn

í síly,

shro

ud

fT/ f

N0

fN/ f

N0

- fN/ f

N0

µ = 0.6 : big amplitude of normal forces forexcitat. frequencies close to resonance→ intersection of curves for µFN and -µFN

We can interpreted the intersection as contactseparation.

0 1 2 3 4 5 6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Fáze kmitu [rad]

Norm

alizované k

onta

ktn

í síly,

shro

ud

fT/ f

N0

fN/ f

N0

- fN/ f

N0

0,988 𝝎𝟖,𝟏,𝑺𝒕 0,994 𝝎𝟖,𝟏,𝑺𝒕

Contact stiffnesses

NPP Temelín insouth Bohemia

turbineroomTo our knowledge, up to now, no general method for finding the values of “contact stiffnesses” has been introduced.

We developed a (computational) method that fits the contact stiffnesses effectively, at least in our case of bladed disks (mentioned also in this lecture).

1

𝑘t=1

𝑘x+

1

𝑘t,comp

1

𝑘n=1

𝑘y+

1

𝑘n,comp

The compliance of interface contact elements

Related to the deformations of the asperities of the contactingsurfaces. May be omitted (an order of magnitude higher).

𝑘t ≈ 𝑘t,comp 𝑘n ≈ 𝑘n,comp

Related to the compensation of omitted higher modes.

Conclusions

Harmonic Balance Method (HBM) - It is possible to use to calculations of steady state of nonlinear vibration forbladed- It is possible to perform the calculations on PC

„Contact stiffnesses“ - They compensate higher mode shapes omitted in the approximation ofdynamic compliance of a calculated system- It is necessary to fitted these contact stiffnesses

Ne vždy lze třecí vazby považovat pouze za třecí tlumiče:- Větší disipovaný výkon nevede pokaždé k většímu poklesu amplitud výchylek.- Prokluz ve třecí vazbě může být příčinou nárůstu amplitud výchylek.- Separation of contact surfaces can occur

Notion of „sleeping resonant regime“

Thank You very much

Nonlinear vibration analysis of steam turbine bladed disk ...€¦ · shroud tie-boss Demand from...

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