Civil Engineering Reference
In-Depth Information
coscos
a
b
sincos
a
b
−
sin
b
[
=
R
c
os sinsin
a
b
g ag a
−
sincos
sinsin
b
sin
g ag g
+
coscos
cossin
b
0
l
cossin
a
b
cos in sin in sin
g
+
a g
a
b
cos
g
−
cos sin os cos
a g
b
g
Rotation about the axes from the local to the global system is simply a
reverse operation. It can be shown that the resulting rotation matrices for
this transformation are the transpose of the rotations from the global axis
to the local axis. This is known as a
symmetric transformation
. The fol-
lowing relationships show those basic expressions:
cos sin
sin os
a
−
a
0
0
[]
=
[]
=
T
a
a
a
a
l
0
0
l
0
0
1
0
010
0
10 0
0
cos
b
sin
b
[]
=
[]
=
T
b
b
l
0
0
l
−
sin
b
cos
b
[]
=
[]
=
T
g
g
cos
g
−
sin
g
l
0
0
l
0
sin os
g
g
The rotation matrix from local to global system, [
R
l
0
], can be found from
the individual rotations or directly from [
R
0
l
].
[
=
[]
=
[][][]
T
T
T
T
RR
abg
l
0
0
l
cos sin
sincos
a
−
a
0
0
cos
0
010
0
b
sin
b
10 0
0
0
−
[
=
R
a
a
cos sin
sin o
g
−
g
l
0
0
0
1
sin
b
cos
b
g
s
g
coscos
a
b
cossin sin in cos os sinco
a
b
g
−
a
g
a
b ss in sin
g
+
a
g
[
=
R
sincos
a
b
sinsin
a
b
sin
g
+
coscos
a
g
sinsin
a
b
cos
g
−
cos
a
sin
g
l
0
−
sin
b
cos sin
b
g
cos cos
b
g
T
coscos
a
b
s
in cos
a
b
−
sin
b
[
=
[]
=
T
RR
cossin sin in cos in sin
a
b
g ag a
−
b
sin
g ag
+
cos cos os
b
sin
g
l
0
0
l
cossin cos in sin in sin
a
b
g ag a
+
b
cos
g ag g
−
cos sin os cos
b
