USAGE OF DISK MODAL CONDENSATION TO MODELLING OF BLADED DISK VIBRATIONS WITH FRICTION ELEMENTS

USAGE OF DISK MODAL CONDENSATION TO MODELLING OF BLADED DISK

VIBRATIONS

WITH FRICTION ELEMENTS

Vladimír Zeman

^1a

, Jakub Šašek

^1b

, Josef Kellner

^1c

1Department of Mechanics, University of West Bohemia inPilsen e-mail: ^azemanv@kme.zcu.cz, ^bjsasek@kme.zcu.cz, ^ckenny@kme.zcu.cz

Summary

The paper presents the method of the mathematical modelling of rotating bladed disk vibrations with friction el- ements. A modelling is based on bladed disk decomposition into a three-dimensional elastic disk subsystem and a blading subsystem. Modal condensation applied to disk enables reduction of DOF number of the whole system.

The influence of different condensation levels on eigenfrequencies and natural modes of the imperfect experimental bladed disk with friction elements placed between shroud in the end of several symmetrically arranged blades is discussed.

VYUŽITÍ MODÁLNÍ KONDENZACE DISKU

PRO MODELOVÁNÍ KMITÁNÍ OLOPATKOVANÉHO DISKU SE TŘECÍMI ČLENY

Summary

V článku je uvedena metoda matematického modelování kmitání rotujícího olopatkovaného disku se třecími členy.

Modelování je založeno na dekompozici olopatkovaného disku na třírozměrný elastický disk a na subsystém lopatek provázaných třecími členy. Modální kondenzace aplikovaná na disk umožňuje redukci počtu stupňů volnosti celého systému. Vliv rozdílné úrovně kondenzace na vlastní frekvence, vlastní tvary kmitání a na Campbellův diagram je vyšetřován na imperfektním zkušebním olopatkovaném disku se třecími členy umístěnými mezi bandáží několika souměrně umístěných lopatek.

1. INTRODUCTION

Many rotating systems with bladed disks assume rigid disks and flexible blades. This approach can be usually applied on high-pressure stages of steam turbines with relative small disk diameters. A modelling of low- pressure stage bladed disk vibration requires to respect a disk flexibility. The accuracy of finite element disk mathematical models is directly bounded with higher number of degrees of freedom (DOF) of the whole bladed disk which affects a computation.

The aim of this article is to present a generally accepted method of the mathematical modelling of rotating bladed disk vibrations based on the disk modal conden- sation. The paper follows in problems presented in [1], [3].

2. MATHEMATICAL MODEL OF THE BLADED DISK SUBSYSTEMS

We assume that the bladed disk is centrally clamped into turbine shaft rotating with constant angular veloci- ty

ω

around y-axis and blades are harmonically excit-

ed by hydrodynamical forces caused by vapour through fixed nozzles with nozzle frequency

The system (Fig. 1) can be decomposed into disk (su system D) and blades with friction elements placed between shroud (subsystem B). The disk is modelled by 3D hexahedral elements [1] and blades are modelled using 1D elements [2].

Fig.2. Physical model of the bladed disk with friction elements Equations of motion of the subsystems

derived in the coordinate system x, y, z

constant angular velocity

ω

. The vector of disk gene alized coordinates is partitioned with respect to the couplings between disk and blades in the form

( ) ( ) ( ) ( ) ( ) ( )

[

j^C

] [

^T

C j C j F j F j F j

D= u v w u v w =

q L L

,

where displacements marked by superscript correspond to free (coupled) nodes. The vector of bla ing subsystem generalized coordinates is partitioned in the form

[

^{i i i i i i} i

E i B j B j B j B j B j B j

B=_Lu v w ϕ ϑ ψ _Lu v

q ,

ed by hydrodynamical forces caused by vapour flow through fixed nozzles with nozzle frequency

ω

_k

= ^k ω

. The system (Fig. 1) can be decomposed into disk (sub-

) and blades with friction elements placed ). The disk is modelled by ] and blades are modelled

Fig.2. Physical model of the bladed disk with friction elements Equations of motion of the subsystems D and B were

x, y, z rotating with . The vector of disk gener- alized coordinates is partitioned with respect to the couplings between disk and blades in the form

(

( )

) (

^{( )}

) [

D^{C T}

]

^{T nD}

T F

D q ∈R

q ,

(1) where displacements marked by superscript

^F ( ) ^C

correspond to free (coupled) nodes. The vector of blad- ing subsystem generalized coordinates is partitioned in

]

^nB

E i E i E i E i E

i w ϕ ϑ ψ ∈R

(2)

where displacements marked by superscript

correspond to blade

B

_i (friction element between shroud of blades i and i + 1). According [3], the equ tions of motion of both uncoupled subsystems can be written in the form

( ) (

D D

)

D

( )

D

Dq t B ωG q t

M && + + & + ,

( ) ( (

(

, ²

2 ,

,B EB B

s

e B B

B t

ω ω

ω K K

K K

q B B q M

− +

+ +

+

& +

&

Symmetric mass

M

_X and static stiffness ces of subsystems (

X = D , B

corresponding to mutually isolated non

= 0

ω

) disk and blade subsystems. Skew symmetric matrices

ω G

_X express gyroscopic effects, symmetric matrix

ω

²

^K

_ω,B represents blade bending centrifugal stiffening and symmetric matrices

dynamic spin softening caused by vibration modelling in the rotating coordinate system. Matrix

contact stiffness between shrouds and friction elements linearized under assumption of the contact normal forces calculated from a quazistatic equilibrium cond tions under rotation. Centrifugal load vectors are constant in time and blade excitation is characte ized by vector

f

_B of complex amplitudes. Matrices express material damping considered as proportional to static stiffness matrices

K

_{s ,X}

ear friction forces, harmonic balance method [4] is used.

Friction forces acting in contact surfaces between fri tion elements and blade shrouds can be approximately expressed by forces transmitted by couple perperdicular viscous dampers in contact surfaces placed in central contact points

A ,

_i

B

_i (see Fig. 1) and rotating dampers in the same contact surfaces. Equiv

coefficients of these viscous dampers are derived assu ing Coulomb friction in following form [2]

where displacements marked by superscript

^B

_i

( ) ^E

(friction element between + 1). According [3], the equa- tions of motion of both uncoupled subsystems can be

(

Ks_,D−ω²Kd_,D

)

qD

( )

t =ω²f_ω,D

+

(3)

) ) ( )

) ( )

^.

,

, 2 ,

t i B B B

B d

B B k B

e k

t

t

ω

ω ω

ω ω

f f q

K

q G q

+

=

+ + &

(4)

and static stiffness

K

_s,X matri-

B

) are standard matrices corresponding to mutually isolated non-rotating (for ) disk and blade subsystems. Skew symmetric express gyroscopic effects, symmetric represents blade bending centrifugal stiffening and symmetric matrices

ω

²

K

_{d ,X} represent dynamic spin softening caused by vibration modelling in tating coordinate system. Matrix

K

_{E ,B} expresses contact stiffness between shrouds and friction elements linearized under assumption of the contact normal forces calculated from a quazistatic equilibrium condi-

gal load vectors

ω

²

f

_ω,X

are constant in time and blade excitation is character- of complex amplitudes. Matrices

B

_X

express material damping considered as proportional to . To linearize the nonlin- ear friction forces, harmonic balance method [4] is used.

Friction forces acting in contact surfaces between fric- and blade shrouds can be approximately expressed by forces transmitted by couple perperdicular viscous dampers in contact surfaces placed in central (see Fig. 1) and rotating dampers in the same contact surfaces. Equivalent damping coefficients of these viscous dampers are derived assum- ing Coulomb friction in following form [2]

( ) ( ) ( )

k X

X k

X e k X

X k X e k X

X k X e

i i

i i

i i

b M b

b T a b

a T

b Φ ω πΦ ω

ω ω π

ω

ω π 4 , , ⁴

, 4 ,

, = = =

, (5)

where terms

Xi

a

and

Xi

b

are amplitudes of steady slip translation motion in directions of dampers and

Xi

Φ

are amplitudes of steady slip rotation motion (X

= A, B). The magnitudes of friction forces (torques) corresponding to the quazistatic equilibrium condition of friction elements under rotation by angular velocity

ω

are indicated

T

X

( M

X

)

.

3. CONDENSED MATHEMATICAL MODEL OF THE BLADED DISK

The displacements of the coupled disk nodes on condi- tion of rigid blade feet modelled as a disk part can be expressed by displacements of the first blade nodes

R

_i

at blade roots (see Fig. 1) in the form [1]

( )

B B D C

D

T q

q =

_, , (6)

where rectangular matrix

T

_D,B

∈ R

^{n( )DC,nB} describes the linkages between disk and blading subsystems.

The number ( ) ( )C D D F

D

n n

n = −

of free node disk dis- placements (dimension of subvector ( )F

q

D ) may be very large for the vibration analysis of the bladed disks with friction elements. Hence, it is desirable to reduce the DOF number by use of the modal disk condensation [1].

Let modal properties of the conservative model of the non-rotating disk (for

ω = 0

) with blade feet isolated from blades in cross-sections passing through referential nodes

R

_i be characterized by spectral

Λ

_D and modal

V

D matrices satisfying the orthogonality conditions

D D D T D D

D T

D

M V E V K V Λ

V = , =

, (7)

where E is unit matrix. The modal matrix of the disk can be rearranged into the block from

 



 



=

_{m C s C}

F D s F D m

D

V V

V

V V

(8)

corresponding to decomposition (1) and eigenvectors separation into frequency lower eigenvectors marked by superscript m (master) and frequency higher eigenvec- tors s (slave). The subvector ( )F

q

D of free node disk displacements can be approximately transformed by modal submatrix of the isolated disk corresponding to free node disk displacements and frequency lower eigen- vectors in the form

( ) ^{( )} D

F

D m

F n D m D F D m F

D

= V x , V ∈ R

^,

q

. (9)

Frequency higher natural modes usually contribute less to the disk deformation and their influence can be neglected.

The motion equations (3) and (4) of the fictive system assembled from uncoupled subsystems D and B can be formally rewritten in the configuration space

( ) [ (

^{( )}

) ( )

^{( ) T}B

]

^T

T C D T F

t q

D

q q

q =

as

( ) ( ) ( ) (

s d

) ( )

^{i t}

e k

t t

t B ωG q K ω K_ω ω K q ω f_ω f ^ω

q

M&& + + & + + ² − ² = ² +

, (10)

where global matrices have the block-diagonal form

(

D B

)

d diag

(

B

)

diag X ,X , X M,G,K , K_ω 0,K_ω,

X= = =

( )

(

D B e B k

)

s diag

(

sD sB CB

)

diag B ,B B q , , K K_, ,K_, K _,

B= + ω = +

and global vectors are

[ ] [

B^T

]

^T

T T

T B T

D

f f 0 f

f

f

_ω

=

_{ω, ω,}

, =

.

The vector

^q ( ) ^t

in consequence of the coupling (6) and the modal transformation (9) can be transformed into new vector

( ) [

^TB

]

^T

T

t x

D

q

q ,

~ =

of dimension

B

D

n

m

m = +

by means of transformation matrix

n m n n

n B D F D m

R

^{D B D}

B

∈

 







 







=

_, ^{+ , +}

E 0

T 0

0 V T

The condensed mathematical model of the coupled bladed disk with friction elements after elimination of the disk coupled displacements takes the form

( )

t

(

B ωG

)

q

( )

t

(

Ks ω K_ω ω K q

M~~&& + ~+ ~ ~& + ~ + ₂~ − ₂~ ,

where condensed mass, damping, gyroscopic, static stiffness, centrifugal stiffening and dynamic spin softe ing matrices

s T

X T X M, B, G, K T

X ~ ,

=

=

order

m

_D

+ n

_B and

f ~

_ω

= T

^T

f

_ω

.

Fig.2. Comparison of the first natural mode of the referential (left) and condensed model for

Table 1. Eigenfrequencies of the bladed disk condensed models

ν _{f [Hz ]}

ν

f

ν

( ²⁰

1 89,233 89,343

2 93,729 93,823

3 110,407 110,444

4 120,384 120,440

5 123,601 124,084

6 137,329 137,616

7 139,865 140,099

8 140,669 140,707

9 140,848 140,931

10 141,073 141,089

D D

n^B

, m < n

.

(11) The condensed mathematical model of the coupled bladed disk with friction elements after elimination of

coupled displacements takes the form

) ( )

^{i t}

d q t ω f_ω fe^ωk K~ ~ = ₂~ +

(12)

where condensed mass, damping, gyroscopic, static stiffness, centrifugal stiffening and dynamic spin soften-

d s

, K ,

_ω

K

are

4. APPLICATION

The presented method is tested on the experimental imperfect bladed disk. This system includes 2 x 25 smaller blades without shroud and 2 x 5 blades with 2 x 4 friction elements placed between theirs shrouds (Fig.

2). DOF number of the disk is subsystem is

n

_B

= 2208

.

The first step of the bladed disk modelling using a presented approach is modal analysis of the non rotating

( ^{ω = 0} )

conservative system (for

the different disk condensation levels. As an illustration 10 lowest eigenfrequencies of the tested bladed disk are presented in Table 1 for three levels of the disk conde sation defined by numbers

m

the disk master natural modes. The each eigenfrequency of condensed models is characterized by relative error

( )

[

ν ν

]

ε

ν

= 100 f m

_D

− f / f

eigenfrequency of full (non condensed) model.

Comparison of the first natural mode of the referential (left) and condensed model for

m

_D

= 20

Table 1. Eigenfrequencies of the bladed disk condensed models

Eigenfrequencies of condensed models [Hz]

)

20 ε

_ν

f

ν

( ) ⁸⁰ ε

_ν

f

_ν

89,343 0,123 89,239 0,007 89,233

93,823 0,100 93,738 0,010 93,730

110,444 0,034 110,410 0,003 110,407

120,440 0,047 120,389 0,004 120,384

124,084 0,391 123,610 0,007 123,602

137,616 0,209 137,354 0,018 137,334

140,099 0,167 139,884 0,014 139,869

140,707 0,027 140,691 0,016 140,675

140,931 0,059 140,874 0,018 140,854

141,089 0,011 141,087 0,010 141,080

(11) The presented method is tested on the experimental imperfect bladed disk. This system includes 2 x 25 smaller blades without shroud and 2 x 5 blades with 2 x 4 friction elements placed between theirs shrouds (Fig.

2). DOF number of the disk is

n

_D

= 7200

and blade

The first step of the bladed disk modelling using a presented approach is modal analysis of the non-

conservative system (for

B = 0

) for the different disk condensation levels. As an illustration 10 lowest eigenfrequencies of the tested bladed disk are presented in Table 1 for three levels of the disk conden-

320 , 80 ,

= 20

m

D of

isk master natural modes. The each eigenfrequency of condensed models is characterized by relative error

f

ν, where

f

_ν is eigenfrequency of full (non condensed) model.

20

(right) of bladed disk

Eigenfrequencies of condensed models [Hz]

( ³²⁰ )

ν

ε

ν

89,233 0,000 93,730 0,001 110,407 0,000 120,384 0,000 123,602 0,001 137,334 0,004 139,869 0,003 140,675 0,004 140,854 0,004 141,080 0,005

The first natural mode of bladed disk corresponding to eigenfrequency

f

₁ for full model and condensed model with highest condensation

( m

D

^{= 20} )

are compared in Fig. 2.

5. CONCLUSION

The paper presents an original analytical numerical method of the modelling of rotating bladed disk with friction elements embedded between blade shrouds. The methodology is based on the modal reduction of disk DOF number corresponding to elastic displacements of the free disk nodes and the elimination of the coupled disk node displacements on the blade feet. The mathe-

matical model allows to introduce continuously distrib- uted centrifugal and gyroscopic effects, contact and friction forces in contact surfaces between friction elements and blade shrouds linearized by harmonic balance method. The methodology is tested on the imperfect experimental bladed disk developed in Insti- tute of Thermomechanics, Academy of Science of the Czech Republic.

REFERENCES

1. Zeman V., Šašek J., Byrtus M.: Modelling of rotating disk vibration with fixed blades. Modelling and optimiza- tion of physical systems, 8, 2009, Gliwice, p. 125-130.

2. Zeman V., Byrtus M., Hajžman M.: Harmonic forced vibration of two rotating blades with friction damping.

Engineering Mechanics, Vol. 17, 2010, No ¾, p. 187-200.

3. Šašek J., Zeman V., Kellner J.: Modal analysis of the imperfect bladed disk with friction elements. In: 17^th Int.

Conf. Engineering Mechanics 2011, Institute of Thermomechanics AS CR, 2011, p. 587-590.

4. Sextro W.: Dynamic contact problems with friction. Springer, Berlin, Heidelberg, 2007.

ACKNOWLEDGEMENT

This work is supported by the grant of Grant Agency of the Czech Rep. No. 101/09/1166 “Research of dynamic behaviour and optimization of complex rotating systems with non-linear couplings and high damping materials”.