Civil Engineering Reference
In-Depth Information
Z
z
y
z
0
y
l
d
z
l
γ
c
y
0
Y
b
a
Figure 4.4.
Example 4.3 Rotation,
g
.
dysin
yabycos
=
=+=
g
+
z sin
g
l
0
0
zcd
=−
=
zcos
g
−
ysin
g
l
0
0
xx
l
=
0
The equations for
x
,
y
, and
z
can be represented in matrix form as follows:
10 0
0
0
x
y
z
x
y
z
o
l
cos sin
sin os
g
g
=
(4.3)
o
l
−
g
g
o
l
The rotation matrix from the global to the local system, [
R
0
l
], is found
from these rotations about the global Z, Y, and X axes, in that order. The
order of the matrix multiplication goes from right to left. Alpha is first
multiplied by beta and the resultant is multiplied by gamma.
[
=
[][[]
R
l
gba
Substituting Equations 4.1 through 4.3 for [
g
], [
b
], and [
a
], respectively,
results in the following:
10 0
0
0
cos
0
010
0
b
−
sin
b
cos sin
sin os
a
a
0
0
[
=
R
cos sin
sin os
g
g
−
a
a
0
l
−
g
g
sin
b
co
s
b
0
0
1
10 0
0
coscos
a
b
sincos
a
b
−
sin
b
[
=
R
cos
sin
sin os
g
g
−
sin
a
cos
a
0
0
l
0
−
g
g
c
os sinsin sincos
a
b
a
b
b
