Civil Engineering Reference
In-Depth Information
Substituting the Enke roots into the original equations yields
x
1
=−
2
.
000,
x
2
=−
2
.
000,
and
x
3
=
1
.
000
.
1.12.3
REAL AND COMPLEX ROOTS
The third possible solution type will occur after many cycles of squaring
the polynomial; the coefficients of the derived polynomial are the squares
of the terms in the preceding cycle, except if one or more terms have a
sign fluctuation, then two of the roots are complex—the one with the sign
fluctuation term and the next one to the right constitutes the complex con-
jugate pair of roots. The roots (Enke roots) will have a relationship similar
to the following assuming
r
3
and
r
4
are the complex conjugate pair of roots
and considering only the dominant terms in each expression:
(
)
=+
x e
=
i
q
=
cos
q
+
isin u iv
q
3
(
)
=−
x e
=
−
i
q
=
cos
q
−
isin u iv
q
4
The values
i
and
R
for the complex form in polar or Cartesian simple
form are:
2
2
i
=− =+
1
and
R uv
The form of the coefficients will become the following:
(
)
brr Re
=++
mmmi m
q
+
e
−
i m
q
1
1
2
+
(
)
+
(
)
+
(
)
+
(
)
+
m
m
m
m
brr
=
mm
r Re
i
q
rRe
−
i
q
r
Re
i
q
rRe
−
i
q
R
2
m
2
1
2
1
1
2
2
=
(
)
+
(
)
+
(
)
+
(
)
m
m
m
m
br rRe
i
q
r rRe
−
i
q
r R
2
rR
2
3
1 2
1 2
1
2
=
(
)
m
2
br rR
4
1 2
These become the following using polar transformations:
(
)
mm m
brr Rcosm
=++
2
q
1
1
2
(
)
+
mm mm m
2
m
brr Rr
=
+ ++
2
r
cosm R
q
2
1
2
1
2
(
)
(
)
m
2
mm m
b
=
2
rrR osmRr
q
+
+
r
3
12
1
2
=
(
)
m
2
br rR
4
1 2
