Tham khảo tài liệu 'dynamics of mechanical systems 2009 part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 82 Dynamics of Mechanical Systems FIGURE A rigid body B with a general orientation in a reference frame R. In matrix form these equations may be expressed as N Sn and n STN where N and n represent the columns of the Ni and ni and S is the matrix triple product S ABC where as before A B and C are 1 0 0 cp 0 sp ci sY 0 A 0 ca B 0 1 0 C sY cY 0 0 sa ca _ _ sP 0 cP_ _ 0 0 1_ Observe that by carrying out the product of Eq. with A B and C given by Eq. leads to Eq. see Problem . Finally observe that a body B may be brought into a general orientation in a reference frame R by successively rotating B an arbitrary sequence of vectors as illustrated in the following example. Example A 1-3-1 Euler Angle Rotation Sequence Consider rotating B about n1 then n3 and then n1 again through angles 01 02 and 03. In this case the configuration graph takes the form as shown in Figure . With the rotation angles being 01 02 and 03 the transformation matrices are 10 0 c2 s2 0 10 0 A 0 c1 s1 B s2 c2 0 C 0 c3 s3 0 s1 c1 _ _ 0 0 1_ 0 s3 c3 _ and the general transformation matrix becomes S ABC c2 c1s2 s2c3 c1c2c3 s1s3 s1c2c3 c1s3 s2s3 c1c2c3 s1c3 s1c2s3 c1c3 s1s2 Kinematics of a Rigid Body 83 i Ni Ni n. ni FIGURE Configuration graph defining the orientation of B in R see Figure . FIGURE Configuration graph for a 1-3-1 rotation. Hence the unit vectors are related by the expressions N1 c2n1 s2c3n2 - s2s3n3 N2 -C1s2n1 c1c2c3 - s1s3 n2 -c1c2c3 - s1c3 n3 N3 -s1s2n1 s1c2c3 c1s3 n2 - s1c2s3 c1c3 n3 and n1 c2N1 c1s2N2 - s1s2N3 n2 s2 Ni c1c2c3 - sis3 N2 3 cis3 N n3 -s2s3N1 -c1c2s3 - s1c3 N2 -s1c2s3 c1c3 N3 The angles 01 02 and 03 are Euler orientation angles and the rotation sequence is referred to as a 1-3-1 sequence. The angles a p and Y are called dextral orientation angles or Bryan orientation angles and the rotation sequence is a 1-2-3 sequence. Simple Angular Velocity and Simple Angular .