Page 324 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 10.3 Transformation of Coordinates (Global Coordinates) 311

6 = 6. We can then include 6 in the transformation matrix as
u cos a sin a
V \ = -sin a cos a 1,2 (10.3-2)
0 0
Thus, the transformation matrix for any element making an angle a measured
counterclockwise from the horizontal is

c .s 0
i’l c 0 1 0 i’l

___ 0 0 1 1 < (10.3-3)
«2 C 5 0 ^2
i’2 0 - 5 c 0 ^2
1 0 0 1- ^ ^2 >
where c = cos a and .s = sin a. It is easily seen that the transformation matrix
developed for displacements also applies for the force vector.
In shorter notation, we can rewrite the transformation equations from local
to global coordinates as
r = Tr
(10.3-4)
F =TF
where T is the transformation matrix, and r, F and r, F are the displacement and
force in the local and global coordinates, respectively. We add to this the
relationship between r and T, which is the stiffness matrix:
hr (10.3-5)
and which we wish to write in the global system as F = kr. From Eq. (10.3-4), we
have
F = T~^F = T^F (10.3-6)
Here we have taken note that transformation matrices are orthogonal matrices and
j - \ ^ jT ^ Substituting for F from the stiffness equation and replacing r in terms
of r, we obtain
F = T^kr
= T^kTr = kr (10.3-7)
Thus, k for local coordinates is transformed to ~k for global coordinates by the
equation
k = T^kT (10.3-8)

^See Appendix C.