Tham khảo tài liệu 'introduction to continuum mechanics 3 episode 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 26 Tensors therefore Q Qfl QIIQr I Now ỊQỊ Qr and I 1 therefore I QI2 1 thus I QI 1 From the previous examples we can see that the value of 1 corresponds to rotation and -1 corresponds to reflection. 2B11 Transformation Matrix Between Two Rectangular Cartesian Coordinate Systems. Suppose e and e- are unit vectors corresponding to two rectangular Cartesian coordinate systems see Fig. . It is clear that e can be made to coincide with eý through either a rigid body rotation if both bases are same handed or a rotation followed by a reflection if different handed . That is e and e can be related by an orthogonal tensor Q through the equations ei Qei Qmfim . eí Ổllel Ổ21e2 Ỡ31e3 e2 Ỡ12el ổ22e2 ổ32e3 e3 Ỡ13el ổ23e2 ổ33e3 where QimQjm QmiQmj òịj or QQr QrQ l We note that Ổ11 ej-Qej ej-ei cosine of the angle between ej and ei Ỡ12 ei Qe2 er 2 cosine of the angle between ei and e2 etc. In general Qij cosine of the angle between e and eý which may be written Qi cos e eý Cj-eJ The matrix of these directional cosines . the matrix Q Qn Qn Ổ13 C 21 Q22 Q23 Ổ31 Ổ32 Ỡ33 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 Let e be obtained by rotating the basis e about the 63 axis through 30 as shown in Fig. . We note that in this figure e3 and e3 coincide. Solution. We can obtain the transformation matrix in two ways. i Using Eq. we have ổu cos e1 eỊ cos30 - C12 cos e1 e2 cosl20ơ 2 Ci3 cos ebe3 c s90o 0 Ỡ21 cos e2 ei cos60 Q22 cos e2 e2 cos30ơ -y- Ổ23 cos e2 e3 cos90 0 3i cos e3 ei cos90ơ 0 32 cos e3 e2 cos90o 0 33 cos e3 e3 cos0 l ii It is easier to simply look at Fig. and decompose each of the éị s into its components in the e1