Articles and Publication  Physics Physical mechanics THE METHOD OF THERMO-MECHANICAL PROCESSES IN A SET OF BARS ASSUMING ELASTIC AND FRICTIONAL JOINTS
THE METHOD OF
THERMO-MECHANICAL PROCESSES IN A SET OF BARS ASSUMING ELASTIC AND FRICTIONAL
JOINTS
© Eugenij
P. Firsanov, Ph.D.
Contact to the author:
mail-fire@yandex.ru
In the article we consider the set rectilinear
bars lying in a plane that are fixed at one side. The bars are coupled
mechanically by inter-bar elastic joints. Two adjacent bars are frictionally
joined assuming the frictional sliding possibility. Thermo-mechanical processes
mean changes of axial shifts, forces and deformations (strains) of bars
occurring due to bar temperature alterations. We study the solution for
“heating-cooling” cycle problem given aforementioned elastic and frictional
joints and bar frictional sliding. The suggested method of research has been
used to investigate thermo-mechanical processes on the stator of turbo-generator
happening under temperature changes of main components.
__________________________________________________________________________________________
The system of differential equations is used to
research thermo-mechanical processes happening under temperature changes of a
set of rectilinear mechanically coupled bars lying in a plane. We consider n-bar
model shown on figure 1.
Figure 1. The n-bar
model
Initial assumptions:
- The bars are fixed at one base (x =
0); at the other base they are tied by the elastic joints (
 )
to the shared plate A which can move along x without turning and
deformation.
Each bar is given by heating temperature (θj),
the (linear expansion) thermal extension coefficient (αj),
the cross sectional area (Fj) , the longitudinal rigidity (Gj)
and the elastic joint rigidity ().
The bars are coupled by inter-bar elastic
joints ( )
evenly distributed along the bar.
The bar with j = 1 and
j = 2 are evenly lengthwise frictionally joined with the linear
friction coefficient τ.
The mechanical parameters (Gj,
,
τ) and heating
temperatures (θj)
are constant (over x axis) for each bar; Gj = EFj,
where E is the elastic modulus.
The right move is stated positive.
Let us denote:
l
+  –
the bar length; m;
l
= const;
–the vector of bars shifts at coordinate x; m;
– the vector of elastic joint shifts between bars at coordinate x;
m;
w
– the axial shift of the plate A; m;
– the vector of axial forces in the bars at coordinate x; N;
 ,
when the bar is stretched;
– the vector of axial strains in the bars at coordinate x;
,  ;
N;
 ;
1/° C
 ;
° C;
τ
– the linear friction coefficient; N/m
– the coefficient of evenly distributed elastic joint between bars j and
j+1; N/m2;
 –
the coefficient of elastic joint of bar j to the plate A; N/m
 ,
 ;
 –
the additional force of the special spring element if installed between the bar
end and the plate A; N.
The relation between force changes and shifts of
the bars under temperature changes are given by two differential
equations describing the relative lengthenings of elementary bar portions and
the inter-bar balance condition for these portions.
j = 1, n;
(1)
The following boundary conditions must be
satisfied.
The absence of bar shifts at x = 0
is given by
. (2)
The condition of deformation of elastic elements
at bar ends is given by
;  . (3)
Assuming the possibility of rigid bar joint to
the plate A, this boundary condition is rewritten as
(4)
where the 
if 
and 
in the rigid joint case. We set 
in the latter case as well.
Finally, the balance condition for the plate A is
given by
 .
(5)
Now let's exclude the vector v(x) from
the equation (1). The elastic joint shifts v(x) are a consequent to the
bar frictional sliding.
(6)
As only the j=1 bar can slide, we get:
.
So it is enough to exclude the v1(x)
function.
The function v1(x) differs
from the function u1(x) at the points where the
frictional sliding took place, namely:
a) If the frictional sliding happens
under given heating conditions in some area x, then  ;
(7)
where τ
> 0 when the
first bar the slides to the right i.e. v1(x)>
u2(x).
b) Otherwise, when there
is no frictional sliding under given heating conditions, the function
representing the frictional sliding value under previous heating conditions
remains constant
 ;
(8)
So a) and b) imply the differential
equation 
can be written as:
or 
(9)
and the latter form doesn't contain unknown
function v1(x).
Thus the problem is defined up to the
initial function 
provided that following values are given: the frictional sliding area, i.e. the
sliding area start coordinate γ
( ),
and the sign of the friction force τ
in the
equations (in other words the slide direction). Later we demonstrate how to
determine these values.
The detailed analysis of shifts and
forces under heating and cooling shows that if initially ≡
0, then the most important case, the
“heating-cooling” cycle calculation given the initial state, results in two
problems with the initial condition ≡
0 and the heatings vectors  .
The first problem – with the heating
vector 
and the initial boundary conditions (possibly inhomogeneous). The second problem
– with the heating vector ,
homogeneous boundary conditions and doubled friction force. The
“heating-cooling” cycle solution comes as superposition of solutions of
these two problems if the slide area for the second task doesn't exceed that of
the first task.
Let 's
demonstrate this.
Assuming ≡
0 at the initial state, we examine two consecutive thermal states given by the
heatings vectors .
Under the state  the
first bar moves to the right at the slide area coordinate  .
Under the state 
the bar moves to the left with the new slide area coordinate  .
For each state we denote by 
and 
the solution of corresponding problem. Now let's find out which equations are
satisfied by the below functions
 .
Observing the linearity of equations (1), we can
write
j
= 1,…,n ; (10)
 ,
j
= 3,…,n.
And so we need to research
only (
is similar).
Examining the equations separately for each
interval (0, γ1),
(γ1,
γ2),
(γ2, l)
we have:
for (0, γ1) where
the slide didn't happen
 ;
(11)
for ( γ1,
γ2)
thus  .
(12)
While the slide shift function is given by
and writing (1) the heating state
. (13)
We get that (11) holds on (γ1,
γ2)
as well.
For
(γ2, l)
we have
,  , whence
 . (14)
So summing up the above the function Q1(x)
satisfies equation (11) on the interval (0, γ2
) and (14) on (γ2,
l).
The condition (3) takes the form
 .
(15)
Evidently the boundary conditions (2), (3) and
(5) are satisfied. The proof is completed.
Similar reasoning could be employed for the
series of heating states, however it would require the complicated algorithm for
calculating the slide area boundaries for each successive calculation.
Below we try to solve the (1) in the form
(9) assuming ≡0.
By γ we denote the slide area
start coordinate. The elementary bar portions balance condition equation takes
the form of:
Basing on the above the equations (1) can be
rewritten in matrix form:
where  ,
 ,
 ,
 ,
Similarly the boundary conditions are:
(19)
where
, I – is the unitary matrix.
Additionally the solution must be continuous at γ:
 .
(20)
Thus we've got the system of ordinary
linear differential equations with constant coefficients on two adjacent
intervals (0, γ),
(γ, l).
Instead of Q(x) we will examine the
following functions:
 .
The solutions of the LDE system can be
represented as the matrix exponent [1]
if  ;
(21)
if 
;
where
In our case the matrix exponent R and the matrix
P have the following form:
,
(22)
 ,
and
.
where 
are the diagonal matrices of eigenvalues
, 
;
Z, V
are the eigenvector matrices from the generalized eigenvalue problem,
 ,
 ;
;
23)
.
The matrices in (22) are well-defined as the
generalized eigenvalue problem has the complete G – orthonormal system
of eigenvectors.
Usually the matrices are not used in fractions as
they don't commute. However as the diagonal matrices commute, the matrix
fractions are correctly defined and corresponding formulates are easier to read.
If some of the eigenvalues  ,
are zero then the corresponding fractions in the generalized solution (21, 1.22)
can be obtained calculating the limits
,  ,  ,
 . (24)
The formulas (19) and (20) result in the system
of linear algebraic equations in w, u(l), Q(l). The
solvability of the system is proved by the completeness and the consistency of
the boundary conditions (out of the scope of the article).
In order to solve the eigenvalue problem we
observe that M·K·M* is the 3-diagonal symmetric nonnegative-definite
matrix.
The effective algorithms for finding the
eigenvalues of generalized eigenvalue problem for such matrices are well known;
for example the bisection method [2]. The corresponding eigenvectors are
calculated by simple recurrent formulas.
So far we've constructed the solution to
the problem given by equations (17) – (20) assuming the frictional joints
between the first and the second bars and the first bar frictional sliding. Now
let’s show how to determine the frictional sliding area start coordinate
γ
and the sign of the friction force τ
in the
equations (in other words the frictional sliding direction).
The frictional sliding at x happens if
, where 
(25)
and the sign of τ
is defined as sign(τ)
= sign(Δ)
(26)
The coordinate γ
can be determined by the method of successive approximations - changing the γ
from l to 0 until
or γ
= 0 .
As it was mentioned, it is relatively difficult
to find u(x), Q(x) from the differential equations when
there is series of heating conditions. It is also known that differential
equations with variable coefficients (to account the fact the mechanical
parameters are unevenly distributed along the bars) and arbitrary right-hand
sides (temperature, friction) are harder to solve. The finite difference method
is used in practice to overcome the above mentioned difficulties.
The suggested method of research of
thermo-mechanical processes was used to investigate the effect of
“heating-cooling” cycles on the stator of turbo-generator. The analysis of
turbo-generators operation as well as a number of experimental results showed
that “heating-cooling” cycles may threat the stator winding, core and their
fastening details under certain heating conditions because of the thermo
mechanical processes in stator in question.
It was discovered that axial shifts of the
winding and the core under heating and cooling should be calculated accounting
the mechanical joints between each other and other elements of stator by the
method described above instead of well-known formulas of free thermal expansion
of copper and steel.
The 4-bars model of stator was chosen for
numerical experiment assuming that:
- the bars represent the stator elements (winding,
core teeth and body; keybars) that are mechanically tied with each other by
longitudinally distributed elastic joints (
 )
and tied at their ends to stator pressure plates by elastic joints ( ),
both mechanical parameters and temperatures
are evenly circumferentially distributed,
the longitudinally distributed friction joints
are between the winding and core teeth in the slot
areas,
the stator is symmetrical relative to the
center so that both sides can be examined separately.
The effect of mechanical parameters and
“heating-cooling” cycles on calculation results was analyzed. The
computation results for some turbo-generators were compared with the direct
measurements of certain parameters as shown by diagrams on figures 2 and 3.
Figure 2. The
distribution of thermo-mechanical strains (z) along the tooth
end zone when heating the winding bar; L is distance from the tooth end; [3]
The calculations
results are denoted by ―õ― and ―î―;
the measurements are denoted by ●
Figure 3. The distribution of
relative axial deformation of keybar εp(x)/εp(0)
along the keybar end area (x) for different keybar-to-core elastic joint
coefficients (Kp) [4]; 1 – Êð
= 0,17× 1010 N/m2 (min), 6 – Kp =17×
1010 N/m2 (max), 7 – Kp = 0 (no
joint); x is the distance from keybar end; the measurements are denoted
by î , D with unscrewed tensioning nuts and by ●
with tightened tensioning nuts.
The computations showed that the friction joints
between the winding and core teeth cause the pressure growth in teeth ends when
the stator winding is heating up and the pressure decrease when the stator
winding cools down. The reverse phenomenon is observed at subsequent areas along
the teeth − the pressure decrease when the stator winding is heating up
and the pressure growth when the stator winding gets cools [3].
This was confirmed by the measurements on the
full-scale model of the 200 MW turbo-genera-tor’s stator. The calculated
distribution of thermo-mechanical strains (z) along the tooth
end zone when heating up the winding bar is shown on the figure 2 for the wide
range of mechanical parameters.
The measurement results are presented in the same
diagram. The pressure decrease varied from 0,1 to 0,5 MPa. The local pressure
decrease shows up the most when the winding fastening in slots is the tightest
and winding fastening in the winding ends is the lightest in the axial direction.
The winding copper temperature decrease below 30° C when the core is heated has
the negative effect on the teeth ends pressure.
The effect of the rigidity of elastic joints
between the stator core and the keybars on axial strains of the keybars was
examined on the stator of 220 MW turbo-generator when unscrewing the keybar ends
tensioning nuts. The measured axial strains εp were
non-uniformly distributed along the keybar showing the significant decrease from
end to middle.
The figure 3 presents the calculated values of
relative deformations εp(x)/εp(0) and the
measurements where x is the distance from the keybar end. The
calculated values corresponding to greater of rigidity levels of elastic joints
between the stator core and the keybars are in good concordance with measurement
results [4].
The satisfactory variance between calculated and
experimental results indicates the discussed model could be used on practice for
example for choosing the optimal fastening characteristics of stator elements or
validating the specific heating conditions of turbo-generator.
References
- F.R. Gantmacher. The Theory of Matrices. —
AMS Chelsea Publishing: Reprinted by American Mathematical Society, 2000.
- J.H.Wilkinson, C.Reinsch. Handbook for
Automatic Computation Linear Algebra. Heidelberg New York, Springer-Verlag
Berlin, Germany, 1971.
- G.G.Schastlivy, O.S.Golodnova, G.M.Fedorenko,
E.P.Firsanov/ The development of integrated approach to theoretical analysis
of thermo-mechanical processes in turbo-generator stator components. Kiev.
Institute of Electrodynamics of Ukrainian Academy of Sciences. Preprint
559.1988 (In Russian).
- E.P.Firsanov. The modeling and computations of
thermo-mechanical processes in turbo-generator stator components. Abstract
of Ph.D.thesis. Kiev. Institute of Electrodynamics of Ukrainian Academy of
Sciences. 1992 (In Russian).
The author: E.P. Firsanov
The translator – M.V.Maksimov
Publishing date: February 6, 2012
Source: SciTecLibrary.ru
Back
| 
(17)
