Tham khảo tài liệu 'dynamics of mechanical systems 2009 part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 132 Dynamics of Mechanical Systems FIGURE Geometrical parameters and unit vectors of the piston connecting rod and crank arm system. where QP and are the angular velocity and angular acceleration respectively of the connecting rod QP. Because QP has planar motion we see from Figure that QP and a QP may be expressed as I . -ộnz and aQP -ộnz where nz nX X Uy is perpendicular to the plane of motion. Also the position vector QP may be expressed as QP l cos ộ nX -l sin ộ ny By carrying out the indicated operations of Eqs. and and by using Eqs. and vP and aP become vP -rD sin 0-1 ộ sin ộ nX rQ cos 0-1 ộ cos ny and . X. Observe from Figure that P moves in translation in the n direction. Therefore the velocity and acceleration of P may be expressed simply as vP X nx and aP X nx where X is the distance OP. By comparing Eqs. and with we have X -rD sin 0-1 ộ sin ộ Planar Motion of Rigid Bodies Methods of Analysis 133 0 rQcos 0-1 ộcos ộ x -rÓ2 cos 0 -l ộ sin ộ -l ộ2 cos ộ and 0 -rQ2 sin 0 -l ộ cos ộ lộ2 sin ộ Observe further from Figure that x may be expressed as x rcos 0 1 cos ộ and that from the law of sines we have l r 7 -r sin 0 sin ộ By differentiating in Eqs. and we immediately obtain Eqs. and . Finally observe that by differentiating in Eqs. and we obtain Eqs. and . Instant Center Points of Zero Velocity If a point O of a body B with planar motion has zero velocity then O is called a center of zero velocity. If O has zero velocity throughout the motion of B it is called a permanent center of zero velocity. If O has zero velocity during only a part of the motion of B or even for only an instant during the motion of B then O is called an instant center of zero velocity. For example if a body B undergoes pure rotation then points on the axis of rotation are .
