The mechanical effects of short-circuit currents in - Montefiore

Table 2.3: Factor γ for different supports
type of beam and support γ
single span beam continuous beam with equidistant supports
A and B:
supported
A: fixed
B: supported
A and B:
fixed
two spans
3 or more spans
a) b)
l
k = 1
k = 2
k = 3
l s
l s
l s
l s
k = 4 ls ≈ 0,2l
22
ls l 0,5 ≈
ls ≈ 0,33l
to 0,5l
ls ≈ 0,25l
perpendicular to the surface
in direction of surface
Figure 2.9 Main conductor consisting of sub-conductors
a) Arrangement of connecting pieces within the span b) Direction of oscillation
The connecting pieces are taken into account by
additional masses mz which are distributed over the n
sub-conductors and weighted by the influence factor ξm:
(2.39)
ω
2
=
m′
s
c
+ ξ
F
m
1
=
ω
m
1+
ξm
nm
l
1 cF
=
mz
l mz
m
1
s′
+ ξm
n nm′
l
z
s′
with ω= 2πf0. Hence the frequency f follows with the
factor cm:
(2.40) 0 m 0
2π
mz
1+
ξm
nms′
l
2
0
ω 1
f = =
f = c
s
f
1,57
2,45
3,56
ξm depends on the number k and the position ls/l of
connecting pieces; mz is the total mass of one piece. ξm
is now calculated from the actual frequency of the main
conductor. An upper limit can be gained analytically by
use of the Rayleigh-quotient [Ref 29, Ref 30], which
compares the maximum kinetic energy Umax when
passing the rest position with the maximum energy Emax
in the reversal of direction of movement during the
undamped oscillation [Ref 31]
2
(2.41) U max = Emax
= ω E max
Hence the Rayleigh-quotient is :
(2.42)
where
2 U
R = ω =
E
max
max