Page 323 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

310 Introduction to the Finite Elennent Method Chap. 10

Figure 10.3-1.

Consider again a planar structure and examine a local element @ at an
angle a with the global coordinates u, v, which will be assumed in the horizontal
and vertical directions, as shown in Fig. 10.3-l(a).
The displacement of joint ® to must be the same in both the local
and global coordinates. This requirement can be expressed by the equation
Tj - Wji + i; J = WjiT i\S

where i, j, and i, j are unit vectors for the two coordinate systems. Forming the dot
product of the preceding equation with i, we obtain

«i(i • •) + t'lU • 0 = « i ( '- ') + i^i(j • ')
or
j
W -h 0 = i/j cos a sin a
Next, taking the dot product with j, we obtain
0H-r;i= —WjSina + F| cos a
Thus, we can express these results by the matrix equation

cos a sm a (10.3-1)
—sin a cos a

The preceding equation expresses the local coordinates in terms of the
global coordinates These results are readily confirmed geometrically from
Fig. 10.3-l(b).
Similarly, the displacement at joint @ in local coordinates can be expressed
in terms of the global coordinates by the same transformation equation. The
rotation angle for the two coordinate systems must be, of course, the same, so that