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International Journal of Engineering and Advanced Research Technology (IJEART)
Volume1, Issue1, July 2015
45 www.ijeart.com
Abstract— For the first time, a method for frequencies and
form determination of hydroelastic natural oscillations of
hydroturbine head covers has been developed. Eigenmodes of
the structure oscillations in fluid were investigated as a series in
terms of vibration eigenmodes in vacuum. Hydroelasticity
problem solving was obtained by making use of singular integral
equations and a finite element method. A numerical analysis was
carried out.
Index Terms— Hydroelastic Vibrations, Turboatom, FEM
I. INTRODUCTION
Head cover of a hydraulic turbine is a stationary annular
part limiting from above the turbine water passageway, and
being used for placement of guide vanes and other assemblies.
The main requirement to it at designing stage consists in that
strength and stiffness are to be provided at a minimum
specific metal content. The head cover structure of hydraulic
turbine represents a combination of thinwalled bodies of
revolution, which are stiffened with a system of
closelyspaced multiply connected meridional plates.
However, structural features of the head cover are determined
by entire layout of a turbine and its type and size. When in
operating condition, the head cover is affected by significant
axissymmetrical loads both from mass forces and from
hydrodynamic pressure acting on its surface in contact with
water, as well as by radial load from the turbine rotor. As
concerns previous design versions, the head covers were
made as iron castings, whereas nowadays they are made as
welded structures of carbon steel Ст3сп. It is to note that
elastic properties of those grey cast iron types as used
previously for casting purposes are dependable on amount of
graphite inclusions: elasticity modulus of these cast iron types
makes up (40…75)% of elasticity modulus for steel qualities,
Poisson's ratio  about 67% [1, 2]. Cast iron density makes up
(90…95)% of steel density.
Recently, the level of requirements to effectiveness and
reliability of power generating plants has been raised
drastically, and significant utilization of power generation
potential in many countries in the world, including Ukraine
[3] resulted in particular in a necessity to modernize and
replace hydroturbine equipment at hydroelectric power plants
that are in operation for a long time.
When taking decision as to a scope of modernization, due
consideration shall be given either to necessary replacement
or service life prolongation of the hydraulic turbine head
cover because it is one of its most metalintensive assemblies.
At Public JointStock Co. “Turboatom”, works are in
T. Medvedovskaya, Cand. of Techn. Sc.;
E. Strelnikova, Doct. of Techn. Sc.,
K.Medvedyeva
progress for normative basis perfection for service live
estimation of hydroturbine head covers [4]. Analyzing of their
structural features and load application has permitted us to
work out firstly an effective estimation methodology for
strength and dynamic characteristics determination in vacuum
by making use of the finite element method (FEM) in
combination with expansion of the unknown quantities of
displacements and loads into a Fourier series [4], [5].
Trustworthiness of results obtained by this methodology is
confirmed in some works [68]. Said approach was further
developed in [9] for strainstressed state determination of a
structurally orthotropic body under nonsymmetrical load
application, and this makes it possible to reduce calculations
of unknown quantities of displacements to solutions of
independent problems for each term of Fourierseries
expansion.
Because of data nonavailability in literature as to
numerical investigations for determination of natural
frequencies and oscillation forms of the hydroturbine head
covers in water, the results given by S.P. Timoshenko for a
radial plate oscillating in fluid [10] were used previously for
estimation of water influencing on their dynamic
characteristics. Specified definition of natural frequencies of
hydroturbine head cover hydroelastic oscillations is
indispensable both at estimation of its residual service life and
at the service life forecasting in case when head covers made
of cast iron shall be substituted by those ones made of steel,
because of a significant difference in elastic characteristics
between them. This problem is dealt with in this paper, in
which in contrast to [10] oscillation forms of the head cover in
fluid are represented in terms of formwise decomposition of
its oscillations in vacuum.
II. MATRIX CONSTRUCTION OF ASSOCIATED MASSES OF A
STRUCTURE INTERACTING WITH FLUID
Let us write the free oscillation equation for a structure,
some surfaces of which are in contact with water, in form of a
matrix as follows: [K–2(Me + Ml)]W=0, (1)
where: K, Me, Ml – matrices of stiffness, structure masses
and associated masses of fluid;  natural frequency; W –
matrix, columns of which are eigenvectors of structure
oscillation in water. When applied to the finite element
method, components of vectors W are amplitude
displacements of finiteelement lattice nodes of the structure.
In order to determine matrix elements Ml, it is necessary to
calculate pressure that acts on structure surfaces being in
contact with fluid. Let us assume that fluid is ideal and
noncompressible, fluid motion is considered to be without
vortices. Fluid velocity can be represented in the form of:
Free Hydroelastic Vibrations of Hydroturbine Head
Covers
T. Medvedovskaya, E. Strelnikova, K. Medvedyeva
Free Hydroelastic Vibrations of Hydroturbine Head Covers
46 www.ijeart.com
),,,(),,(),,,( 0tzyxgradzyxvtzyxv
(2)
where:
zyxv ,,
0
– velocity vector of nonturbulent
fluid flow; (x,y,z,t)– potential of velocities induced by free
oscillations of the structure. CauchyLagrange integral [11]
serves for determination of fluid pressure on wetted surfaces
of the structure:
2
,,,
),,,(),,(
2
0
v
t
tzyx
tzyxpzyxp l
, (3)
where: l – fluid density. By substituting (2) into (3) and
keeping only the terms of the first order of smallness, we
obtain:
vtzyxgrad
t
tzyx
pl
,,,
,,,
, (4)
where the point means a scalar product. As indicated in the
work [12], an incidence flow velocity up to 30 m/s affects
insignificantly the frequencies of structure’s natural
oscillations in fluid, therefore the second component in the
formula (4) can be ignored, hence:
t
tzyx
pl
,,,
. (5)
Thus, to find out pressure of fluid onto the structure
surfaces, it is necessary to define the function
(x,y,z,t) by
solving Laplacian equation:
0
2
2
2
2
2
2
zyx
under following boundary conditions:
0;
21
s
ngrad
t
w
s
ngrad
,
where: S1 – population of wetted elastic surfaces of the
structure; S2 – population of wetted rigid surfaces of the
structure;
n
– outer normal to the structure.
Based on [13], we will find
(x,y,z,t) out in the form of a
simple layer potential over surface S limiting the fluid volume
under consideration (S= S1 S2)
)(
,
1
4
1
0
0XdS
XXr
XX
S
. (6)
Here: X0 – point of observation; X – moving point on the
surface; r = r(X,X0) – Cartesian distance from point X0 to
point X; (X) – unknown density.
As follows from [14], we obtain a singular integral
equation relatively to (X):
St
Xw
XdSXXLXX 0
00 )(,
4
1
(7)
Here: w – displacement normally to the wetted surface.
The integral equation nucleus shall be defined by the formula
3
0
0
,r
eXn
XXL r
,
where:
r
e
– unit vector
r
, directed from point X0 to point
X. The right part of equation (7) represents displacement
velocity of strained walls (structure surfaces); the zero
rightside part corresponds to stationary walls. It is to note
that for a case when points X and X0 belong to the same
surface of a plate or a flatshaped shell, then L(X,X0) nucleus
numerator is close to or equals to zero. If these points lay on
sufficiently distant surfaces, then L(X,X0) nucleus
denominator is large. This explains why in a series of works it
came out to obtain good results under assumption that
t
Xw
X
0
0
. Having solved the equation (7) and
calculated (x,y,z,t) by the formula (6), we will define fluid
pressure on the structure walls by using (5).
If we represent all the unknown functions in the form of a
product of their amplitude values multiplied by exp(it), then:
;; i
t
wi
t
w
l
,
where amplitude values are kept with their initial
designations and the exponent is eliminated. We will solve the
equation (7) by a projective method, making use of unknown
function representation in the form of eigenmode expansions
of those displacements normally to a wetted surface, which
were obtained in the process of problem solving with regard
to the structure natural oscillations in vacuum. Let be W = Va,
where: V – a rectangular matrix consisting of n1vectors of
values for displacements normally to the wetted surface
calculated through known nodal values of vectors V of the
structure eigenmode oscillations in vacuum; a – vector of
unknown coefficients. Let us represent the function (X) for
points lying on movable walls by the formula
1= Vb, (8)
where: b – columnvector of unknown coefficients. For
points lying on stationary surfaces, we will find out the
function in the form of an expansion using the system of
functions set forth by n2vectors of Unodal values; then
2=Uc , (9)
where: c  columnvector of unknown coefficients. In the
capacity of U it is convenient to adopt normal displacements
having been calculated by oscillation eigenmodes of
freelysupported thin shells, median surface of which
coincides with wetted stationary surfaces of the structure. If
we substitute expansions (8)(9) into equations (7),
premultiply by UT and VT and integrate over movable and
stationary surfaces that are limiting the fluid (T  transposition
sign), we will obtain two coupled systems of algebraic
equations
.0)(),()(
4
1
,
4
1
;
)(),()(
4
1
,
4
1
2 21
1
1 21
022010
01
012010
S SS
T
S
T
S SS
T
XdScXdSXXLXUUbXdSXXLXVU
XVdSVi
XdScXdSXXLXUbXdSXXLXVVV
Let us represent obtained equations in a matrix form
A11b+A12c= – iA11a (10)
A21b+A22c=0 (11)
Square matrices A11 and A22 are symmetric here, in
matrices Aij(i,j=1,2) there are per nirows and njcolumns. If
we, based on the equation (11), shall express the vector of
coefficients c by means of the vector b and substitute it into
(10), we will obtain the relationship between b and a:
International Journal of Engineering and Advanced Research Technology (IJEART)
Volume1, Issue1, July 2015
47 www.ijeart.com
BaiaAAAAAib
11
1
21
1
221211
.
This relation permits to express density at the wetted
strained surface through its normal displacements taking into
account an effect of stationary fluid boundaries. Using (5),
(8), and (9), we find
BaXdS
XXr
Vp
S
l
1
1
0
2
,
1
4
1
and derive the formula for performance of fluid pressure
forces on normal displacements of the wetted structure
surface by premultiplying p by VT and integrating over the
area of this surface:
aMaBaXdSXdS
XXr
XVVa l
T
S S
T
l
T2
011
0
2
1 1
)(
,
1
4
As we can see, the quadratic form obtained has meaning of
kinetic energy of the fluid. In this way, the matrix of
associated masses of fluid Ml is specified.
Natural frequencies of hydroelastic oscillations of the
structure can be found from equation (1). By means of
premultiplying it by matrix WT, and by virtue of vectorial
orthonormalization of oscillation eigenmodes in vacuum
relatively to the structure mass matrix, we will obtain
0
2 aMEa l
T
,
where:  scalar matrix, components of which are
frequency quadrates of the structure oscillations in vacuum, E
 scalar identity matrix. Natural frequencies of hydroelastic
oscillations can be found by Jacobian method by means of
solving the eigenvalue problem
0 ED
,
where matrix components D are
kiikikik Md
/
,
whereat =1. Eigenvectors a (=1,2,…n1) calculated in
such a way are coefficients of free oscillation eigenmodes of
the structure in vacuum. Using them, we obtain vectors of
nodal values for natural hydroelastic oscillations of the
structure W = Ua
by known eigenvectors U of its oscillations in vacuum.
In order to solve singular equations (7), we will apply the
boundary element method [1519]. For this purpose, the range
of integration (streamlined surface of the head cover) was
dissected into a finite number of tetragonal subdomains NS, in
each of them the unknown density having been substituted by
a constant [19].
1. Method for determination of natural
frequencies and modes of hydroturbine head covers
in vacuum
Determination of natural frequencies and modes of
hydroturbine head covers in vacuum can be performed on the
methodology basis stated in [9].
Dynamics problem of the hydroturbine head cover
structure shall be solved based on a matrix equation for free
oscillations
K M( ) ( )U p U
20
, (12)
where: K and M  stiffness matrix and structure mass
matrix, respectively.
On the basis of a linear and squarelaw representation of
an arbitrary triangular finite element (FE) in the system of
oblique coordinates
l m n
, ,
[4] special stiffness matrix
expressions for a finite element (FE) of body of revolution
have been defined for an arbitrary term of Fourierseries
expansion.
Energy of one triangular finite element (FE) lmn is written
by formula
A L G LT,
where: (L)  complete vector of main parameters of the
triangular element; (G)  its stiffness matrix
( )G
G G
G G
11 12
21 22
, (13)
where:
,
gg
gg
gg
,
g.
gg
gg
nn
mm
ll
12
nn
mnmm
lnlm
11
n
m
lll
g
g
g
G
symmetr
g
G
,
g.
gg
gg
,221221
symmetr
g
GGG T
Here: qij  matrixcells allowing for connection between
nodes i and j
( , , , , , , )i j l m n
, l,m,n  apical nodes of the
triangular element,
, ,
 median nodes.
Block cells G11, G12, G22 (13) for the finiteelements (FE)
of meridional plates are calculated respectively by formulae
such as:
q D N D drdz
lm
l m
lmn
( ) ( )( )
П
T
П П
,
q D N D drdz
l
l l m
lmn
4( ) ( )( )
П
T
П П
,
q D N D drdz
l m m n
lmn
16 ( ) ( )( )
П
T
П П
. (14)
Stiffness matrix cells for the finiteelement body of
revolution
q q q
lm
k
l
k
l
k
, ,
are described in the form of:
q d N d rdrdz
lm
k
l
k k
m
lmn
( ) ( ) ( )
( ) ( )( )
1
2
T
,
q d N d rdrdz
l
k
l
k k
l m
lmn
( ) ( ) ( )
( ) ( )( )
2T
,
q d N d rdrdz
k
l m
k k
m n
lmn
( ) ( ) ( )
( ) ( )( )
8T
. (15)
Free Hydroelastic Vibrations of Hydroturbine Head Covers
48 www.ijeart.com
It seems to be reasonable that in order to reduce
calculations, segregation shall be done of matrices that are
undependable on k (k = 0,1,2,…)
d D k
rd
k
0в
,
where:
DD
D
011
22
0
0
,
dв
0 0 0
0 0 1
0 0 0
0 0 0
1 0 0
0 1 0
.
As a result, the stiffness matrix cells for the finite element
(15) are described by a sum of four summands:
q D N D rdrdz
lm
k
l m
lmn
1
2
0 0
( ) ( )( )
T
drdzdNDk m
lmn
l
BT0 )()(
lmn
ml drdzDNdk )()( 0TB
lmn
ml drdz
r
dNdk 1
)()( BTB2
,
q D N D rdrdz
l
k
llm
lmn
20 0
( ) ( )( )
T
k D N d drdz
l l m
lmn
( ) ( )( )
0 T в
k d N D drdz
llm
lmn
( ) ( )
вT 0
k d N d rdrdz
l l m
lmn
21
( ) ( )
вTв
,
q D N D rdrdz
k
lm mn
lmn
80 0
( ) ( )( )
T
k D N d drdz
lm m n
lmn
( ) ( )( )
0 T в
k d N D drdz
l m mn
lmn
( ) ( )
вT 0
k d N d rdrdz
l m m n
lmn
21
( ) ( )
вTв
, (16)
where:
Dlm
0
 matrixconstants have the following form:
D D D D
lm l m l m m l
0 0 0 0
.
For estimation of integrals (16) onepoint and threepoint
Gauss formulae shall be used.
The mass matrix being a part of (12) shall be calculated at
dynamics problem solving. The finite element mass matrix M
and its blocks m11, m12, m22 shall be described similarly (13).
Cells of the linear block m11 shall be defined by formulae:
m W r E d d
lm
l
lmn
m l m
, (17)
where:
,,2,1,
,0,2
k
k
W
E
1 0 0
0 1 0
0 0 1
.
Here: ρ  density of material;
r z r z
mn mnln ln

doubled area of the triangular finite element (FE) lmn.
Mass matrix blocks m12, m22 bounded with the quadratic
element shall be formed from cells of type
ml
,
m
in the
form of:
m W r E d d
l
l
lmn
m l m
2
,
m W r E d d
l
lmn
m n l m
2
.
When solving the dynamics problem of the head cover
structure, the matrices of stiffness (16) and masses (17) shall
be constructed on the basis of the above formulae by a linear
and squarelaw approximation applied to the finite element.
Determination of natural frequencies and oscillation forms
shall be performed by iteration method within a subspace by
solving at each step the system of algebraic equations by
means of LDLTfactorization.
III. FREQUENCIES AND MODES OF NATURAL HYDROELASTIC
VIBRATIONS OF THE HYDROTURBINE HEAD COVER
As an example, we illustrate the head cover of a largesize
Kaplan turbine, which is loaded not only by mass of the guide
vanes turning mechanism but also by a substantially large
mass of the hydraulic unit rotor because the thrust bearing
support is installed on it. The head cover is attached by its
outer flange to the turbine stator with the help of bolts.
Depending on this, on the pitch circle diameter for bolts we
have
u u u
r z
0
. (18)
Since displacements are expanded into Fourier series so in
order to satisfy requirements of (18) the necessary and
sufficiency condition is that amplitude values of
displacements of the circle with given diameter for each of
harmonics would be equal to zero:
u u u
r
k
z
k k( ) ( ) ( )
0
.
Computational scheme and the structure finite element
quantization are shown at Fig. 1.
International Journal of Engineering and Advanced Research Technology (IJEART)
Volume1, Issue1, July 2015
49 www.ijeart.com
Fig. 1: Computational scheme for Head Cover
Fig. 2: The first eigenmode of the hydroturbine head cover
oscillations taking into account associated masses of wicket
gate and turbine rotor parts
Trustworthiness of values of oscillations' natural
frequencies in vacuum as having been calculated on the
methodology elaborated, can be confirmed through their
comparison with results obtained by the finite element method
for a spatial structure on the whole (Fig. 2), for which its
natural frequency makes up 12.2 Hz taking into account
associated masses of guide vanes and turbine rotor parts. This
value concurs with the value obtained by the methodology as
above, with required computational expenditures for the latter
being considerably less.
Investigations were performed on how associated masses
of the above mentioned parts impact the head cover natural
frequencies in vacuum and in water. Computational results
are given in Tables 1, 2 where natural oscillation forms are
characterized by the number of nodal diameters.
Table 1
Natural frequencies of the hydroturbine head cover
oscillations ignoring the mass of wicket gate and turbine rotor
parts
Number of nodal
diameters, KF
Frequency No.
KF = 0
1
2
3
In vacuum
49.8
257.7
289.9
In water
30.5
194.1
287.3
KF = 1
1
2
3
In vacuum
72.3
179.2
238.1
In water
57.1
160.2
233.3
Table 2
Natural frequencies of the hydroturbine head cover
oscillations taking into account the mass of wicket gate and
turbine rotor parts
Number of nodal
diameters, KF
Frequency No.
KF = 0
1
2
3
In vacuum
12.22
55.8
82.8
In water
11.7
55.7
81.7
KF = 1
1
2
3
In vacuum
14.9
38.8
54.5
In water
14.7
38.7
53.9
IV. CONCLUSIONS
1. For the first time, a method for determining natural
frequencies and hydroelastic vibration modes of hydroturbine
head covers has been elaborated, which is based on
combination of the finite element method, Fourierseries
expansions and boundary element method. For this purpose,
unknown eigenmodes of hydroelastic vibrations are expanded
into series in terms of oscillation eigenmodes in vacuum.
2. The method proposed permits substantially to specify
more precisely  as compared with an assessment used
previously in [10]  the hydroturbine head cover dynamics
characteristics and to perform frequency separation away
from dynamic load frequencies, thus increasing structure
reliability as early as at design and modernization stage.
3. For the fullscale head cover structure of Kaplan
hydraulic turbine considered, influence of water on natural
frequencies is insignificant; water influence is lowering with
frequency number increasing.
4. Value of associated masses of the wicket gate parts
and hydraulic unit rotor parts produces a noticeable effect not
only on the natural frequencies of the head cover in vacuum,
but also on their lowering related to the head cover interaction
with water.
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