Page 42 - Introduction to Continuum Mechanics

Part B Transformation Matrix Between Two Rectangular Cartesian Coordinate Systems. 27

is called the transformation matrix between {e/} and {e/}. Using this matrix, we shall obtain,
in the following sections, the relationship between the two sets of components, with respect
to these two sets of base vectors, of either a vector or a tensor.

Fig.2B3

Example 2B 11.1

Let {e/ } be obtained by rotating the basis {e/} about the 63 axis through 30° as shown in
Fig. 2B.4. We note that in this figure, e 3 and e 3 coincide.
Solution, We can obtain the transformation matrix in two ways.
(i) Using Eq. (2B11.2), we have
//jr ..
( i» 2)
0 =--, (2i3 cos(e 1,e 3)=cos90 =G
j2 11=cos(e 1,ei)=cos30°=—, (2i2 =cos e e =cosl2 0 = 0
0
0
0
==
^2i=cos(e2,ei)=cos60 =-,j322 cos(e 2,ei)=cos30 =—,j223 ==c os(e2,e 3)=cos90 =0
=
(Q 3i=cos(e 3,ei)=cos90°=0, £> 32=cos(e 3,e 2)=cos90°=0, j233 cos(e 3,e 3)=cosO°= 1
(ii) It is easier to simply look at Fig. 2B.4 and decompose each of the e/ 's into its components
in the {e^e^} directions, i.e.,