Tham khảo tài liệu 'principles of engineering mechanics (2nd edition) episode 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 196 Dynamicsofa body in three-dimensional motion giving the three scalar equations Mx Ixxdíìx dt ứylzzíìz ứzIyyíìy My lyydĩìy dt azlxxĩìx ứXIZZĩlZ Mz Izzdĩìz dt ứXIyyíìy- ứyIXA It must be emphasised that the above equations apply equally for moments about either a fixed point on the body or the centre of mass. The moments of inertia must of course be evaluated for the point considered. Rotation about a fixed axis This case is of practical interest in connection with the forces appearing at the bearings of an imperfect rigid motor. By imperfect we mean that the centre of mass of the rotor is not on the axis of rotation and or the axis of rotation does not coincide with a principal axis. Figure For the rotor shown in Fig. assume that the moment of inertia is known relative to the X- y- z-axes which are attached to the body. From equation with O x Uy 0 .t -lxz ứz Ly lyz ứz Lz Izz ứz The angular velocity of the body and of the axes ta I zk and is assumed to be constant. Thus i j k M A ta X L 0 0 ứz -Ixz z ỉyz z Izz z ilyz l 2 -jlxzb z2 In practice the products of inertia are small and may be considered to be due to the addition or subtraction of point masses to an otherwise perfect rotor in which case Ixz and lyz are easily calculated. A variation of the same problem occurs when a perfect rotor is misaligned as shown in Fig. . The angular velocity of rotor and axes referred to the rotating axes which are coincident with the principal axes is ta i z cos ak 1 Z sin aj so Lg lzz O cos ak Iyy 1 Z sin aj Hence assuming 1 Z to be constant Mg tứ X i j k 0 ozsina wzcosa 0 lyy i zsma ỉzz ứzcosa i ơz2 Iyy - lzz i sin 2a See example and refer to problem for the case of non-constant angular velocity. Euler s angles A convenient set of co-ordinates for describing the position of a rigid body are the Euler angles. Referring to Fig. the X- y- z-axes are attached to the body and the X- Y- Z-axes are fixed in space.
