Tham khảo tài liệu 'dynamics of mechanical systems 2009 part 14', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 632 Dynamics of Mechanical Systems Transformations between Absolute and Relative Coordinates As noted in the previous section it is desirable to use absolute orientation angles in computer analyses of multibody system dynamics and relative orientation angles for input and output data. Absolute angles provide for efficient numerical analysis but relative angles provide a simpler geometric description of the system. Thus it is desirable to have a means for readily converting from one to the other. Such conversions may be obtained using the transformation matrices. To develop this consider a typical multibody system S as in Figure . Let Bk be a typical body of the system and let Bj be its adjacent lower numbered body. As before let R be an inertial reference frame in which S moves. Let nsi njiz and nki i 1 2 3 be unit vector sets fixed in R Bj and Bk as shown in Figure . Let ak pk and Yk be dextral orientation angles defining the orientation of Bk in R and let ak pk and Yk be dextral orientation angles defining the orientation of Bk in Bj. Recall from Section that dextral orientation angles or Bryan angles may be defined as follows imagine the unit vectors nki of Bk to be mutually aligned with the unit vectors n0i of R. Then Bk may be brought into a general orientation dextral rotation of Bk about axes parallel to nk1 nk2 and nk3 through the angles ak pk and Yk. The relative orientation angles ak pk and Yk of Bk and Bj are defined similarly with the n0i replaced by the nji. If we know the absolute orientation angles ak pk and Yk we may use Eq. to obtain the transformation matrix SOK relating the n0i and the nki that is CPkCY k -CPkSY k SPk SOKmn Ca kSY k Sa kSBCY k CakCYk- SakSpkSYk -SakCPk _ SakSYk - CakSpkCYk SakCY k Ca kSpkSY k CakCPk _ where as before the SOKmn are defined as SOKmn n0m nkn Also as before S and C are abbreviations for sine and cosine. FIGURE A typical multibody system with typical adjacent bodies Bj
